MHD Waves in the Solar Atmosphere 9781108427661

Table of contents :
Cover......Page 1
Front Matter
......Page 3
MHD Waves in the Solar Atmosphere......Page 5
Copyright
......Page 6
Dedication
......Page 7
Contents
......Page 9
Preface
......Page 19
1 General Principles......Page 23
2 Waves in a Uniform Medium......Page 49
3 Magnetically Structured Atmospheres......Page 88
4 Surface Waves......Page 105
5 Magnetic Slabs......Page 124
6 Magnetic Flux Tubes......Page 161
7 The Twisted Magnetic Flux Tube......Page 204
8 Connection Formulas......Page 229
9 Gravitational Effects......Page 270
10 Thin Flux Tubes: The Sausage Mode......Page 303
11 Thin Flux Tubes: The Kink Mode
......Page 333
12 Damping......Page 382
13 Nonlinear Aspects......Page 419
14 Solar Applications of MHD Wave Theory
......Page 456
References
......Page 511
Index......Page 523

Citation preview

MHD Waves in the Solar Atmosphere This volume presents a full mathematical exposition of the growing field of coronal seismology which will prove invaluable for graduate students and researchers alike. Roberts’ detailed and original research draws upon the principles of fluid mechanics and electromagnetism, as well as observations from the TRACE and SDO spacecraft and key results in solar wave theory. The unique challenges posed by the extreme conditions of the Sun’s atmosphere, which often frustrate attempts to develop a comprehensive theory, are tackled with rigour and precision; complex models of sunspots, coronal loops and prominences are presented, based on a magnetohydrodynamic (MHD) view of the solar atmosphere, and making use of Faraday’s concept of magnetic flux tubes to analyse oscillatory phenomena. The rapid rate of progress in coronal seismology makes this essential reading for those hoping to gain a deeper understanding of the field. b e r na r d r o b e r t s is Emeritus Professor of Solar Magnetohydrodynamics at the University of St Andrews. His important contributions to the field over the past forty years have been recognized with an election to the Royal Society of Edinburgh in 1997, a Saltire Scottish Science Award in 1998, and, in 2010, the Royal Astronomical Society’s prestigious Chapman Medal, awarded for ‘investigations of outstanding merit in solar-terrestrial physics’.

MHD Waves in the Solar Atmosphere Bernard Roberts University of St Andrews, Scotland

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108427661 DOI: 10.1017/9781108613774 © Bernard Roberts 2019 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2019 Printed and bound in Great Britain by Clays Ltd, Elcograf S.p.A. A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Roberts, Bernard, 1946- author. Title: MHD waves in the solar atmosphere / Bernard Roberts (University of St Andrews, Scotland). Other titles: Magnetohydrodynamic waves in the solar atmosphere Description: Cambridge ; New York, NY : Cambridge University Press, 2019. | Includes bibliographical references and index. Identifiers: LCCN 2019002497 | ISBN 9781108427661 (hardback : alk. paper) Subjects: LCSH: Helioseismology. | Magnetohydrodynamic waves. | Fluid dynamics. | Solar atmosphere. | Electromagnetism. Classification: LCC QB539.I5 R63 2019 | DDC 523.7/6–dc23 LC record available at https://lccn.loc.gov/2019002497 ISBN 978-1-108-42766-1 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

To my wife Margaret and our sons Alastair, James, Michael and Richard.

Contents

Preface

page xvii

1

General Principles 1.1 Introduction 1.2 A Variety of Plasmas 1.2.1 The Sun 1.3 The Magnetohydrodynamic Equations 1.4 Some Properties of the MHD Equations 1.4.1 Fundamental Speeds 1.4.2 Magnetic Flux Tubes 1.4.3 The Induction Equation 1.4.4 Diffusion of Magnetic Field (Rm  1) 1.4.5 Advection of Magnetic Field (Rm  1) 1.4.6 The j × B Force 1.4.7 Energetics 1.5 Aspects of Wave Propagation 1.5.1 Linearization 1.5.2 Fourier Representation 1.5.3 Dispersion Relations, Phase Speed and Group Velocity

1 1 3 4 7 9 10 13 14 17 18 19 21 22 22 23 24

2

Waves in a Uniform Medium 2.1 Introduction 2.2 Wave Equations 2.3 Plane Waves 2.4 Sound Waves 2.5 Magnetohydrodynamic Waves 2.6 Alfv´en Waves 2.6.1 Phase Speed and Group Velocity 2.6.2 Perturbations in an Alfv´en Wave 2.7 Magnetoacoustic Waves 2.7.1 Dispersion Relation 2.7.2 Phase Speed Diagrams

27 27 29 33 35 37 38 38 40 42 42 45

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Contents

2.8 2.9

2.7.3 Group Velocity of Magnetoacoustic Waves 2.7.4 Perturbations in the Magnetoacoustic Waves Magnetoacoustic Waves: The Special Case cA = cs Two Physical Extremes 2.9.1 The Incompressible Fluid 2.9.2 The β = 0 Plasma

46 50 57 58 60 63

3

Magnetically Structured Atmospheres 3.1 Introduction 3.1.1 Wave Equations 3.1.2 Sound Waves 3.1.3 The Alfv´en Wave 3.2 Magnetohydrodynamic Waves: Cartesian System 3.2.1 Uniform Medium 3.2.2 The Case ky = 0 3.3 Magnetohydrodynamic Waves: Cylindrical Geometry System 3.4 Singularities 3.5 Phase Mixing of the Alfv´en Wave 3.6 Two Special Cases 3.6.1 The Incompressible Plasma 3.6.2 The β = 0 Plasma

66 66 67 70 71 71 73 74 75 77 78 79 80 80

4

Surface Waves 4.1 Introduction 4.2 Parallel Propagation 4.2.1 Boundary Conditions at an Interface 4.3 The Surface Wave Dispersion Relation 4.3.1 The Hydromagnetic Surface Wave of an Incompressible Medium 4.3.2 Surface Waves on a Magnetic–Non-Magnetic Interface 4.3.3 The β = 0 Medium 4.3.4 Properties of Surface Waves 4.4 Non-Parallel Propagation 4.4.1 The Incompressible Case 4.4.2 The β = 0 Plasma 4.5 Resonant Absorption at a Single Interface: The Incompressible Case

83 83 85 86 87

5

Magnetic Slabs 5.1 Introduction 5.2 The Incompressible Case: Hydromagnetic Surface Waves in a Slab 5.2.1 Sausage and Kink Modes 5.2.2 Dispersion Relations 5.2.3 Pressure and Longitudinal Motion 5.3 Compressible Effects 5.4 The Isolated Magnetic Slab

89 89 93 94 95 97 98 99 102 102 103 106 107 110 113 119

Contents

5.5

5.6

5.7 5.8

6

Trapped Waves: β = 0 Plasma 5.5.1 Trapped Sausage Modes 5.5.2 Sausage Mode Cutoffs 5.5.3 Trapped Kink Modes 5.5.4 Kink Mode Cutoffs 5.5.5 Dispersion Diagram 5.5.6 Sausage and Kink Mode Analogies Leaky Waves: β = 0 Plasma 5.6.1 Leaky Sausage Modes 5.6.2 Leaky Kink Modes 5.6.3 |kz a| < 1: Sausage and Kink Modes Impulsive Waves The β = 0 Slab: Epstein Profile 5.8.1 Epstein Kink Mode (n = 0) 5.8.2 Epstein Sausage Mode (n = 1)

Magnetic Flux Tubes 6.1 Introduction 6.2 Wave Equations 6.2.1 General Formalism 6.2.2 Torsional Alfv´en Waves 6.2.3 Sausage Waves: β = 0 Plasma 6.2.4 Fourier Representation 6.2.5 Ordinary Differential Equations 6.3 The Magnetic Flux Tube 6.4 Small kz a Expansions 6.4.1 General Aspects 6.4.2 Sausage (m = 0) Modes 6.4.3 The Kink (m = 1) Mode 6.4.4 Fluting (m ≥ 2) Modes 6.5 Large kz a Behaviour 6.5.1 Surface Waves 6.5.2 Body Waves 6.6 Leaky Modes 6.6.1 The General Dispersion Relation 6.6.2 The Thin Tube Limit: m ≥ 1 6.6.3 Leaky Kink (m = 1) Waves in a β = 0 Plasma 6.6.4 Leaky Sausage (m = 0) Waves: The Dense (ρ0  ρe ) Tube Limit in a β = 0 Plasma 6.7 Three Special Cases 6.7.1 The Incompressible Medium 6.7.2 The Isolated Flux Tube 6.7.3 The Embedded Flux Tube

ix

121 124 125 125 126 127 128 128 128 129 131 131 133 136 137 139 139 142 142 144 144 144 145 148 153 153 154 156 157 157 158 158 159 159 162 163 163 166 167 168 171

x

Contents

6.8

Perturbations 6.8.1 Kink and Fluting Modes 6.8.2 Sausage Mode 6.9 Comparison of Modes in a Slab and Tube: The β = 0 Plasma 6.10 Resonant Absorption in a Flux Tube: The β = 0 Plasma 6.10.1 Resonant Absorption: The Linear Profile 6.10.2 Resonant Absorption: The Sinusoidal Profile

175 176 178 178 179 181 181

7

The Twisted Magnetic Flux Tube 7.1 Introduction 7.2 Linear Perturbations 7.3 Fundamental Differential Equations 7.4 Boundary Conditions 7.5 The Twisted Flux Tube: Special Cases 7.5.1 The Straight Magnetic Tube (B0φ = 0) 7.5.2 The Incompressible Twisted Tube 7.6 The Compressible Twisted Tube: Small Twist 7.7 The β = 0 Tube: Effect of a Magnetically Twisted Annulus 7.7.1 Equations for β = 0 7.7.2 The Sausage (m = 0) Mode 7.7.3 Dispersion Relation for a Twisted Annulus 7.7.4 Dispersion Relation in the Thin Tube Limit, kz a  1

182 182 184 188 189 191 191 192 197 201 201 202 203 205

8

Connection Formulas 8.1 Introduction 8.2 The Twisted Magnetic Flux Tube 8.3 The Ideal (η = 0) Case 8.4 Alfv´en Singularity 8.5 Connection Formulas: Alfv´en Singularity 8.5.1 Resistive Effects 8.5.2 Differential Equations: Fourier Solution 8.5.3 The General Case  = 0 8.5.4 The Special Case  = 0: The Functions F and G 8.5.5 The Function F(τ ) 8.5.6 The Function G(τ ) 8.5.7 Physical Variables 8.6 Alfv´enic Connection Formulas: The Twisted Magnetic Tube 8.7 Alfv´enic Connection Formulas: The Magnetic Tube with No Twist 8.8 The Slow Mode Connection Formulas 8.9 The Thin Tube, Thin Boundary Approximation: General Formalism 8.10 The Thin Tube, Thin Boundary Approximation for a β = 0 Plasma 8.10.1 Equilibrium Conditions (β = 0) 8.10.2 Transition Layer: Linear Density Profile (β = 0) 8.10.3 Transition Layer: Sinusoidal Density Profile (β = 0) 8.10.4 Resonant Absorption Timescales

207 207 208 210 211 213 213 217 220 223 224 225 226 228 230 230 231 234 236 237 238 238

Contents

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8.11 Leakage Versus Resonance in Under-Dense Tubes 8.12 Thin Tube, Thin Boundary Dispersion Relation: The Photospheric Tube 8.13 Thin Tube, Thin Boundary Dispersion Relation: The Incompressible Tube 8.14 Cartesian Geometry: The Single Interface 8.14.1 Incompressible Medium 8.14.2 Transition Layer: Linear c2A (x)

241 244 245 247

Gravitational Effects 9.1 Introduction 9.2 Equilibrium 9.2.1 General Aspects 9.2.2 Isothermal Atmosphere 9.3 Acoustic-Gravity Waves: General Case 9.4 Acoustic-Gravity Waves: Differential Equations 9.5 Acoustic-Gravity Waves: The Isothermal Medium 9.6 Acoustic-Gravity Waves: General Features 9.7 Acoustic-Gravity Waves: Linear Temperature Profile 9.8 Acoustic-Gravity Waves: Vertical Propagation 9.9 The Klein–Gordon Equation 9.10 Significance of the Klein–Gordon Equation 9.10.1 Case  = 0: The Wave Equation 9.10.2 Case  = 0: The Solution of the Klein–Gordon Equation 9.11 Vertical Magnetic Field 9.11.1 Alfv´en Waves in a Stratified Medium 9.11.2 Magnetoacoustic Waves in a Stratified Medium 9.12 Horizontal Magnetic Field 9.12.1 Hydromagnetic Surface Waves 9.12.2 Magnetic Helioseismology

248 248 249 249 251 252 255 257 260 261 263 264 265 267 268 269 271 273 275 277 279

10 Thin Flux Tubes: The Sausage Mode 10.1 Introduction 10.2 Thin Flux Tube Theory: The Sausage Mode 10.3 Zeroth Order Equations: Uniform Tube 10.3.1 Equilibrium 10.3.2 Perturbations 10.3.3 Dispersion Relation 10.4 Zeroth Order Equations: Effects of Stratification 10.4.1 Stratified Equilibrium (g = 0) 10.4.2 Perturbations (g = 0) 10.5 The Rigid Tube 10.6 Magnetic Flux Tubes 10.7 The Isolated Photospheric Magnetic Flux Tube 10.7.1 Equilibrium of a Flux Tube

239 240

281 281 282 286 287 288 289 291 292 293 294 296 298 298

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10.7.2 Perturbations in a Photospheric Tube 10.7.3 Convective Collapse 10.8 The Straight Magnetic Flux Tube 10.9 Slow Modes in Coronal Loops 10.9.1 Standing Waves 10.9.2 Propagating Waves

299 302 304 306 308 309

11 Thin Flux Tubes: The Kink Mode 11.1 Introduction 11.2 The β = 0 Plasma 11.3 The β = 0 Plasma: Unidirectional Magnetic Field 11.3.1 General Formalism 11.3.2 Alfv´en Waves 11.4 The Uniform Tube in a Uniform Environment 11.4.1 Tubes of Arbitrary Radius 11.4.2 Thin Tubes: Dispersion Relation 11.4.3 Thin Tubes: Dispersive Effect 11.4.4 Thin Tubes: Perturbations 11.5 Thin Tubes: Multiple Scaling 11.6 Standing Kink Waves of an Unstructured Loop 11.7 Standing Kink Waves of a Structured Loop: Example 1 11.8 Standing Kink Waves of a Structured Loop: Example 2 11.8.1 Dispersion Relations for an Exponential Density Profile 11.8.2 Asymptotic Behaviour of Dispersion Relations 11.9 Non-Uniform Magnetic Field 11.9.1 Equilibrium 11.9.2 Perturbations 11.9.3 Transformed Coordinates 11.9.4 Special Case 11.9.5 Multiple Scalings 11.10 Radial Expansion Equations: Effect of Gravity 11.10.1 Equilibrium 11.10.2 Perturbations 11.10.3 Radial Expansion 11.10.4 The β = 0 Plasma in an Unstratified (g = 0) Medium 11.10.5 The Stratified (g = 0) Tube in a Field-Free Environment 11.10.6 Implications 11.11 Dispersive Correction for a Uniform Tube (g = 0)

311 311 312 314 314 315 316 316 318 319 321 323 327 329 334 334 335 338 338 339 342 343 344 348 348 349 350 351 353 355 357

12 Damping 12.1 Introduction 12.2 Damping Coefficients 12.3 The Incompressible Fluid 12.3.1 Formulation 12.3.2 Uniform State

360 360 361 363 363 365

Contents

12.4 12.5

12.6 12.7 12.8

12.9

12.10

12.11

12.12

12.13

12.3.3 Dispersion Relation in a Uniform Medium 12.3.4 Spatial Damping in a Uniform Medium 12.3.5 Temporal Damping in a Uniform Medium Phase Mixing of Alfv´en Waves: Basic Aspects Phase Mixing of Alfv´en Waves: Spatial Damping 12.5.1 Retention of Certain Terms 12.5.2 Approximations Phase Mixing of Alfv´en Waves: A Similarity Solution Phase Mixing of Alfv´en Waves: Temporal Damping 12.7.1 Phase Mixing in the Presence of Stratification Damping of the Slow Mode 12.8.1 Formulation: One-Dimensional Propagation 12.8.2 Wave-Like Equations 12.8.3 Fourier Form Damping of the Slow Mode by Viscosity Only (νth = 0, ν = 0) 12.9.1 Temporal Behaviour: Viscosity Only 12.9.2 Spatial Behaviour: Viscosity Only Damping of the Slow Mode by Thermal Conduction Only (ν = 0, κ = 0) 12.10.1 Temporal Behaviour: Thermal Conduction Only 12.10.2 Spatial Behaviour: Thermal Conduction Only Viscous and Thermal Damping of Slow Waves: Illustrations 12.11.1 Spatial Damping of Propagating Waves 12.11.2 Temporal Damping of Standing Waves 12.11.3 Viscosity Versus Thermal Conduction Thermal Conduction and Viscosity Combined 12.12.1 Temporal Behaviour 12.12.2 Spatial Behaviour Damping of the Slow Wave: Magnetic Effects

13 Nonlinear Aspects 13.1 Introduction 13.2 The Sausage Mode 13.3 Derivation of the Evolution Equation: The Interior 13.3.1 Linear Order 13.3.2 Nonlinear Order 13.3.3 The Environment 13.4 The Magnetic Slab: The Benjamin–Ono Equation 13.4.1 Derivation of the Evolution Equation 13.4.2 Recovery of Linear Theory 13.4.3 The Soliton Solution 13.5 The Magnetic Slab: Whitham’s Evolution Equation 13.6 The Magnetic Flux Tube 13.6.1 Whitham’s Evolution Equation for a Magnetic Tube 13.7 The Magnetic Tube: The Leibovich–Roberts Equation

xiii

365 367 367 368 370 373 374 375 378 381 381 381 382 383 384 385 385 387 388 389 390 390 391 392 393 393 394 395 397 397 399 400 402 402 404 405 405 407 407 409 410 410 413

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Contents

13.8 Effect of an External Magnetic Field (cAe = 0) and cse = cs 13.8.1 Magnetic Slab 13.8.2 Magnetic Tube 13.9 Dissipation in a Magnetic Slab: The Benjamin–Ono–Burgers Equation 13.10 Dissipative Effects: Sound Waves and Burgers’ Equation 13.10.1 Formulation 13.10.2 Nonlinear Perturbation Equations (g = 0) 13.10.3 Linear Case 13.10.4 Nonlinear Evolution 13.10.5 Effect of Gravity (g = 0) 13.11 Dissipative Effects: Slow Standing Waves and Burgers’ Equation 13.12 The Kink Mode 14 Solar Applications of MHD Wave Theory 14.1 Introduction 14.2 Radial Pulsations of a Coronal Tube: Rosenberg’s Approach 14.3 Radial Pulsations of a Coronal Tube: The Role of an Environment 14.4 Coronal Oscillations and Coronal Seismology 14.5 Standing Waves 14.5.1 Fast Kink Standing Modes 14.5.2 Slow Standing Modes 14.5.3 Fast Sausage Standing Modes 14.6 Fast Kink Modes and Resonant Absorption: The Seismology of Small Scales 14.7 Fast Kink Modes: Spatial and Temporal Damping 14.8 Fast Sausage Modes and Quasi-Periodic Pulsations 14.9 Propagating Waves 14.9.1 Fast Kink Mode: Propagating Waves 14.9.2 Fast Sausage Mode: Propagating Waves 14.9.3 Slow Modes: Propagating Waves 14.10 Prominence Oscillations 14.10.1 Equilibrium Force Balance 14.10.2 Transverse Oscillations: String (or Hybrid) Modes 14.10.3 Heavy Load on a String 14.11 Prominence Oscillations: Vibrations in Prominence Threads 14.11.1 Resonant Damping of Thread Oscillations 14.12 Prominence Oscillations: Longitudinal Vibrations 14.12.1 Model Formulation 14.12.2 The Klein–Gordon Equation 14.12.3 The Dispersion Relation 14.12.4 The Pendulum Mode and Prominence Seismology

414 415 416 417 419 419 420 422 423 425 426 430 434 434 435 437 439 442 443 446 448 450 452 453 455 455 455 456 456 458 459 461 462 466 466 467 468 469 473

Contents

xv

14.13 Period Ratios in Coronal Loops and Prominence Threads 14.13.1 Case κ < 1 14.13.2 Case κ > 1 14.14 Sunspots, Pores and Photospheric Flux Tubes

475 476 479 481

References Index

489 501

Preface

In this account we concentrate on providing a theoretical development that addresses the phenomenon of magnetic waves in the Sun’s atmosphere. There is no surprise that the solar atmosphere abounds with magnetic waves. After all, the presence of a magnetic field endows the medium with an elasticity that is proportional to the strength of the magnetic field, so any compression of the plasma results in a restoring response, and any twisting of the magnetic field leads to an attempt to return to an untwisted state: magnetic waves result. Now we are familiar with sound waves in the air around us. The added complexity in the solar atmosphere is the presence of magnetic field in a conducting medium, quite different from our everyday experience of sound in the open air or in a room where magnetism is relatively unimportant. Admittedly, in the Sun the magnetic field is relatively less important in the solar interior, where pressure forces tend to overwhelm whatever magnetic fields are there. But magnetism becomes more important in the visible layer of the solar atmosphere – the photosphere – where the field is marshalled into concentrations, becoming most evident in pores and sunspots. Moreover, in the higher atmosphere itself – in the solar corona – the magnetic field becomes all important, dominating much of the other forces (such as gravity) that may arise and effectively underwriting all that occurs there. In our account here we explore mathematical models of some of the physical effects that may arise in the solar atmosphere. Our treatment is based upon a magnetohydrodynamic view of the solar atmosphere in which detailed plasma effects are taken into account only in as far as they modify the large scale picture of phenomena. Consequently, our description is based upon a marriage of fluid mechanics and electromagnetism, and even here we discount the rapid effects of Maxwell’s electrodynamics being content to include the magnetic force as given by the Lorentz force, though Faraday’s law of induction interlinks magnetism and fluid so that phenomena are a result of magnetism and fluid dynamics acting together. Moreover, much of our description is based upon Faraday’s concept of a magnetic flux tube (or its cousin, the magnetic slab). In different layers of the solar atmosphere, the nature of a flux tube may be different, though wherever a flux tube occurs it gives rise to a communication channel, a one dimensional connection between one region and another in an otherwise three-dimensional medium. Accordingly, the guiding of magnetic waves plays a central role in our description. Waves are interesting in their own right as a physical phenomena, but they gain considerably in importance when we realise that waves carry information about the medium in xvii

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Preface

which they propagate. Understanding this affords us a seismology, a study of the medium from an exploration of the waves the medium supports. This notion, variously termed coronal seismology or MHD seismology, is of recent origin as new space observations of various solar objects allow us to deduce properties of that object, such as its magnetic field strength or the thickness of a layer of inhomogeneity or the size of the coefficient of damping of a wave, properties that may be difficult to ascertain by other means. The success of terrestrial seismology, studying the propagation of waves in the Earth’s interior, provides a general motivation, as does the success of helioseismology, the study of the Sun’s interior from sound vibrations detected at the solar surface. I can readily recall my own early involvement in coronal seismology. In the summer of 1982, I was visiting the Astronomy Department in ETH, Zurich, working principally with Professor J. O. Stenflo on photospheric magnetic fields. On one occasion, Dr. Arnold Benz (a well known solar radio astronomer) asked me to join him for lunch and, without preamble, posed to me the question: ‘how do you explain a one second oscillation in the solar corona?’. Without hesitation, I replied it must be ‘a magnetic wave within a coronal magnetic field’, a coronal loop oscillating therein. ‘But why such a short time as a second?’, challenged Benz. ‘That must be the transverse timescale of the oscillation, the period or timescale being determined by the transverse spatial scale and not the longitudinal one’, I responded. Benz then told me, casually, that this was a major problem in coronal physics, with many authors using the length of a loop to determine timescales. Anyway, I agreed to write out some of the details in my response to our lunchtime exchange, dotting the i’s and crossing the t’s and putting in the π ’s too. To guide my work and to explain my initial response to Benz’s question, I had in my head the wave diagrams computed by my then research student, Patricia Edwin, who was producing wave diagrams for a variety of solar circumstances. These diagrams were subsequently published in Edwin and Roberts (1982, 1983). When Pat started her research under my direction in St Andrews I had given her the task of adding a magnetic field to the outside of a magnetic structure (a slab), extending my own earlier work on this topic from a field-free environment to a magnetic environment. Also, I had asked her to carry out an investigation in cylindrical coordinates and not the Cartesian coordinates I had used earlier, this being appropriate for a magnetic flux tube. It is amusing to recall that at the time, in 1980, I had suggested this area of investigation partly because Pat was doing her PhD on a part-time basis, and I thought (wrongly, it transpired) that there was no urgency in this area and therefore the topic was entirely suitable for a part-time researcher! The theory developed in response to Benz’s question was subsequently published, first as a short article in the journal Nature (in Roberts, Edwin and Benz 1983) and then in a more detailed exposition in The Astrophysical Journal (in Roberts, Edwin and Benz 1984). I for one sat back expecting a minor revolution based upon these articles, which put forward the notion of coronal seismology rather fully. But no such development initially occurred, although the two articles were well cited. And there matters may have rested were it not for the launch in space of the TRACE instrument in 1998, which shortly thereafter saw directly oscillating loops and was able to measure their characteristics, which fortunately accorded reasonably well with the theories we had developed. Sadly, Patricia Edwin did not live to see the full development of the ideas expressed in our early papers, dying in 2007, though

Preface

xix

she must have noted with great interest the surge of activity in this area of our work that followed the launch of TRACE. Central to this development were the observations reported independently in 1999 by Dr. Markus Aschwanden (a former student of Benz’s) and colleagues in California and Professor Valery Nakariakov (then a postdoctoral researcher in St Andrews working with me) and co-researchers. The direct detection of waves in coronal magnetic structures provided an enormous boost to the subject. In our account here we have laid out in some mathematical detail the nature of magnetohydrodynamic waves in magnetic structures, beginning with the well known case of a uniform medium. Many books have described, in varying degrees of detail, the waves of a uniform medium, but it is important here in setting a benchmark by which the waves of a non-uniform medium may usefully be compared. Thereafter we turn to an examination of the various complexities introduced by magnetic structuring, covering the simplest case of a surface wave on a single interface before exploring the waves of a magnetic slab and the cylindrical magnetic flux tube. The waves in these two geometrical objects, the slab and the tube, are closely related in many respects. However, any treatment of a tube involves Bessel functions, a complication avoided in the slab. Thus, a knowledge of the results for a slab can often guide an exploration of a tube. Complications in addition to geometry, such as gravitational stratification, are then explored, before turning to the role of damping and nonlinearity. The level of mathematical detail we present corresponds to that which we perceive as necessary for anyone trying to understand what lies behind the main results. We view our treatment as hopefully providing a foundation for anyone wishing to work in the field of MHD waves in solar physics. Often the available review literature is content to omit such details making it difficult if not impossible to understand what is being presented. Finally, we end our treatment by presenting a number of solar illustrations which, in one way or another, help illuminate the general theory we have laid out earlier. Each illustration presented is taken from some aspect of solar physics, and represents to some extent a personal attempt to discuss available solar observations in the light of MHD wave theory. It is hoped – and indeed expected – that others will further extend the partial viewpoints that have occurred to me. It is a pleasure to record here the postgraduate students and post-doctoral researchers who have worked with me over the years. This is always a two way process, with each of us learning something from the other. The postgraduates include (in chronological order) Alec M. Milne, Andrew R. Webb, Patricia M. Edwin, W. Robert Campbell, David J. Evans, Alan J. Miles, Rekha Jain, Partha S. Joarder, Alan Johnston, Patrick Ferguson, Alastair MacDonald, Cheryl A. Mundie, Jason Smith, Mark Daniell, Keith Bennett, David Boddie, Claire Foullon, Lorna James, Gavin R. Donnelly, Michael P. McEwan, and Cicely K. Macnamara. Postdoctoral and other researchers include Iain C. Rae, Sami K. Solanki, Lorna M. Small, Kris Murawski, Valery Nakariakov, Ramon Oliver, Nagendra Kumar, Istvan Ballai, Robertus Erd´elyi, Jean Claude Thelen, Temury Zaqarashvili and Toni J. D´ıaz. I have learnt much from collaborations with colleagues, especially Joe Hollweg, Marty Lee and Terry Forbes (in Durham, New Hamphire), Michael Ruderman and Robertus Erd´elyi (in Sheffield), Valery Nakariakov and Erwin Verwichte (in Warwick), Marcel Goossens (in Leuven), and Jose Luis Ballester and Ramon Oliver (in Palma, Mallorca). Colleagues

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in St Andrews were always free for discussions, whatever their other commitments, and I especially benefited over many years from talks with Eric Priest, Alan Hood, Andrew Wright and Peter Cargill. A special mention is warranted for Professor Eugene Parker (Chicago), whose style of writing in solar physics has always been an inspiration for me and has helped show me how best to present physical ideas. I must confess to not always being the best of students, but his inspiration was always there. Finally, last but not least is the continuing inspiration afforded me by my hill walking companions Bob Grundy, Rod Cross and Frank Story, who have kept my mind stimulated through many an interesting discussion or debate, in science, religion, history, politics, economics, or whatever. And the walks were fun too. My sincerest thanks, and long may it continue! Bernard Roberts University of St Andrews

1 General Principles

1.1 Introduction Magnetohydrodynamics is concerned with the study of the interaction of a fluid with a magnetic field. It takes as its philosophy a continuum approach, describing its phenomena in macroscopic terms rather than in terms of particle motions. Thus, it is close in spirit to fluid mechanics, which studies the properties of fluids from such a continuum viewpoint. Magnetohydrodynamics – commonly abbreviated to MHD – may also be viewed from a particle approach, discussing the motions of charged particles (electrons or ions) in the presence of a magnetic field. This is the realm of plasma physics. In our account here we will consider the subject principally from the macroscopic viewpoint. The description of magnetic effects in magnetohydrodynamics is rooted in the celebrated equations of electromagnetism formulated by James Clerk Maxwell in 1864, though it is generally only the pre-Maxwellian form of the equations of electromagnetism that are used. The displacement current introduced by Maxwell is ignored on the basis that rapidly varying phenomena, such as electromagnetic waves, are best described from an electromagnetic viewpoint. Thus there are no electromagnetic waves or light in magnetohydrodynamics. Instead, the subject is concerned with relatively slow phenomena, such as sound waves or convective flows or field generation by dynamo action. It is to such phenomena that a fluid approach is particularly suited. By a fluid we mean a gas, plasma or liquid that may be treated from a continuum approach. In magnetohydrodynamics the fluid is a conductor of electricity, and motions within the fluid occur in the presence of an applied magnetic field. The current that flows throughout the volume of the fluid is determined by Ampere’s law which, when expressed in partial differential equation form, relates the current density to the ‘curl’ (a vector operator) of the magnetic field. Motions in the fluid are subject to a magnetic force, the Lorentz force (or j × B force), arising from the current density j and the magnetic (induction) field B; this force, together with any others that may act (such as pressure gradients or gravity), serves to define the motion of the fluid. Temporal changes in the magnetic field B are determined by Faraday’s law of induction, which links such changes in B to the ‘curl’ of the electric field E. The electric field is in turn related to the current density j through Ohm’s law, expressed in a form appropriate for a moving conductor (the fluid). Motions within the fluid are thus inextricably linked to the magnetic field embedded within it, so that movements of the fluid entail movements in the

1

2

General Principles

field, and vice versa. It is this intimate link of fluid and field that gives magnetohydrodynamics its distinctive nature. Magnetohydrodynamics, then, is the offspring from a marriage of fluid mechanics and electromagnetism. It was an offspring that took its time in developing. The basic physical laws and principles of its parents were well known by the end of the nineteenth century, but it was well into the twentieth century before the stirrings of magnetohydrodynamics took shape and then at first only in a somewhat sporadic fashion. By the 1940s, however, the subject was in full growth and has continued this way ever since. A brief account of the early years of development of magnetohydrodynamics is provided in Cowling (1962). The initially slow development of magnetohydrodynamics is in some ways surprising, given the pedigree of its parents. But early experiments with laboratory fluids such as mercury or sodium, aimed at investigating magnetohydrodynamic phenomena, were fraught with difficulties, principally connected with the liquids themselves and the maintenance of sufficiently strong magnetic fields. Strong ohmic attenuation of motions made comparison between theory and experiment somewhat qualitative, though reasonable agreement was obtained. However, it was through the application of magnetohydrodynamics to large-scale phenomena, such as exhibited in the magnetic fields of the Earth and its magnetosphere, the Sun and our Galaxy, that a spur to sustained development was provided. That spur has continued to the present day, increasing to ever greater effect as space and ground-based observations of, most notably, the Sun and the Earth’s magnetosphere give firm direction to magnetohydrodynamics. Moreover, the laboratory fluid has not been left behind, as detailed studies of fusion plasmas have revealed the utility of a magnetohydrodynamic description of certain phenomena as a valuable addition to a plasma approach. An early advance in magnetohydrodynamics, though paradoxically for some time it seemed more like a backward step, was made by T. G. Cowling who showed, in 1934, that a dynamo must have a non-symmetric component (Cowling 1934). That a magnetohydrodynamic fluid could support a wave motion, distinct from the familiar electromagnetic and sound waves of the parents, was not however realized until the early 1940s. In a brief half-page letter to the journal Nature, H. Alfv´en showed that in a perfectly conducting incompressible fluid a transverse wave may propagate along a homogeneous magnetic field; the speed of the wave was proportional to the strength of the applied magnetic field and inversely proportional to the square root of the mass density of the fluid in which the field was embedded (Alfv´en 1942a, b). Alfv´en termed this wave an ‘electromagnetic-hydrodynamic wave’, but it later became apparent that the Alfv´en wave was born! It seems that the term ‘Alfv´en wave’ entered into use with the work of V. C. A. Ferraro and J. W. Dungey (Ferraro 1954; Dungey 1954). For a recent general discussion of Alfv´en’s contribution to magnetohydrodynamic waves, see Russell (2018). Alfv´en was later, in 1950, awarded the Nobel prize for his contributions to magnetohydrodynamics. In his 1942 work Alfv´en made the suggestion that the observed latitudinal drift of sunspots on the Sun’s surface towards the equator may be a wave phenomenon controlled by wave motions (Alfv´en waves) deep below the solar surface (Alfv´en, 1942a). This direct linkage of sunspot drift with Alfv´en waves is not thought likely now but magnetohydrodynamic waves do arise in sunspots themselves.

1.2 A Variety of Plasmas

3

Alfv´en did not discuss what happens to wave motions in a fluid that is compressible. That extension, particularly important for astrophysical applications, was left to another brief letter to Nature, written by N. Herlofson (1950), and a more extensive treatment by H. C. van de Hulst (1951). Herlofson and van de Hulst made the remarkable theoretical discovery that in a magnetohydrodynamic medium there are in fact three modes of propagation open to the system: the Alfv´en wave (uninfluenced by the compressibility of the medium) and two compressible waves. There are three distinct wave speeds associated with these modes, and moreover these speeds depend upon the direction of propagation of the wave. In other words, magnetohydrodynamic waves are anisotropic. During the 1950s and early 60s, the theoretical properties of these waves were further explored; see, for example, Friedrichs and Kranzer (1958) and Lighthill (1960). More recently, developments in the subject of wave propagation have been motivated on a number of fronts: by the possibility of heating of laboratory plasmas by magnetohydrodynamic waves; by the observations of pulsations in the magnetosphere; by the direct observation of waves in the Sun’s corona and their use in coronal seismology; and by the realization that astrophysical plasmas generally, but most clearly the solar atmosphere, are likely to be strongly inhomogeneous. Perhaps above all has been the spur provided by the direct observations of magnetohydrodynamic waves in the solar corona, which has undoubtedly been a powerful stimulus in the further development of theoretical aspects. Underpinning much of these theoretical developments is the detailed study of magnetohydrodynamic wave motions in structured magnetic atmospheres, which are significantly different from those of a uniform medium, though an understanding of this simpler case is, of course, the basis for any study of a non-uniform medium. In any case, structured media provide wave guides for magnetohydrodynamic waves. We end this section with a brief comment about units. We are adopting the mks [metre kilogram second] system of units in our treatment, with electromagnetic quantities expressed in SI [System Internationale, rationalized mks] units. In magnetohydrodynamics it is convenient to regard the magnetic field B and the velocity u as the primary variables; other variables, such as the current density j and electric field E, are then of secondary interest, following from a knowledge of the primary variables if and when required. Generally, then, of the electromagnetic variables, in applications or illustrations of our equations we will only quote values of the magnetic field strength B (= |B|); in SI units, B is expressed in tesla (T). However, it frequently proves convenient to quote B in gauss (G), noting that 1 T = 104 G = 10 kG. 1.2 A Variety of Plasmas Magnetohydrodynamics has found application to a wide variety of plasmas, extending from the small scale of the laboratory plasma to the vast scale of the galactic medium. The fact that the Earth – and indeed many of the planets – has a magnetic field has prompted the development of magnetohydrodynamic theories aimed at describing its maintenance and temporal variation. This is the magnetohydrodynamic dynamo problem (see, for example, Moffatt 1978; Parker 1979a). In particular, extensive developments have taken place in connection with the Sun’s plasma: in its interior (where magnetic fields are both stored

4

General Principles

and manipulated) through to its surface (where fields are observed and measured in great detail); into the tenuous but enigmatic coronal outer atmosphere; and on into the solar wind that blows past the Earth and the other planets, interacting with their magnetic fields. The magnetospheres that envelop the magnetized planets are ever subject to variations in the solar wind, variations that commonly have their origin in events in the Sun’s lower atmosphere. The Earth’s magnetosphere, in particular, displays an array of phenomena that may be modelled using MHD and exhibits a variety of oscillations (Walker 2005; Wright and Mann 2006; Southwood, Cowley and Mitton 2015). Indeed, it is profitable to compare and contrast oscillatory phenomena in the Earth’s magnetosphere and in the Sun’s atmosphere (Nakariakov et al. 2016). However, our chief interest here is the solar atmosphere. Accordingly, we turn now to a brief overview of the Sun.

1.2.1 The Sun The most distinctive property of the Sun as a plasma is its size. With a radius of R◦ = 6.96 × 108 m, the Sun displays a wide variety of plasma conditions ranging from its hot and dense interior, out through its visible and relatively cool surface, and on into its hot but tenuous atmosphere. Gravitational stratification makes for a complicated plasma, doubly compounded by the fact that the Sun possesses a complex and often dynamic magnetic field. Magnetism is the cause of almost all the exotic phenomena displayed by the Sun; for without a magnetic field the Sun would be a very quiet and relatively uninteresting plasma indeed. The possession of a magnetic field is a property it shares with a wide range of stars, many of which must surely display yet more exotic phenomena than we see on the Sun, simply by virtue of their stronger magnetic fields; for to detect that a star has a magnetic field, that field must be about 102 times stronger than occurs in the Sun viewed as a star. The Sun’s size and the variety of phenomena it displays has led us to regard the plasma as made up of separate regions. This is a convenient view to take, though one should not overlook the fact that these different regions are connected to one another. The Sun’s interior, the region below the visible surface, is divided into three zones: an inner core, where nuclear reactions maintain the heat supply; a radiative zone, where the generated heat is distributed outwards by radiative transport; and, occupying the immediate layers below the visible surface, a convection zone, where heat transport is in the form of convective cells. Blending in with the top of the convection zone is the photosphere, the visible layer of the solar surface. The photosphere extends upwards for a height of about 500 km by which the Sun’s temperature has fallen to its lowest value, of about 4200 K. This is the temperature minimum. Higher still in the atmosphere, the temperature rises, at first slowly in the chromosphere but then rapidly as we enter the corona. The temperature of the chromosphere ranges from the temperature minimum value to some 5×104 K whereafter it rises steeply in a thin layer, known as the transition region, to the order of 106 K. This hot outer region is known as the corona. Gravitational stratification ensures that the plasma density falls off with height, so the chromosphere and more especially the corona are tenuous plasmas in comparison with the photosphere.

1.2 A Variety of Plasmas

5

Both the chromosphere and the corona are dominated by magnetism. That the Sun has a magnetic field was not known until 1908, when G. E. Hale used the then newly discovered Zeeman effect to measure the magnetic field of sunspots (Hale 1908), the dark blemishes frequently visible on the solar surface (the photosphere). Hale obtained field strengths of typically 3000 G. The field is sufficiently strong and covers a sufficiently large region – about 104 km across – that it locally modifies the convective transport of heat, resulting in a cool patch in the photosphere. Sunspots are magnetically complex structures. They are frequently observed in groups, where their magnetism tends to blend together to produce complex field patterns in the solar atmosphere that are somewhat similar to the patterns made by iron filings one sees around bar magnets in the school laboratory. But even in an isolated spot the field patterns detectable in the photosphere and chromosphere are complex. There are two distinct regions of a mature spot: its central cool umbra where the field is strongest and is predominantly vertical, and a surrounding penumbra where the field is weaker and has bent over towards the horizontal. The temperature in the umbra is typically 4000 K, compared with about 5000 K in the penumbra and 6000 K in the photosphere. The magnetism measured in sunspots at the solar surface is presumed to be generated and manipulated by flows deep within the interior. The solar interior, made up of about 90% hydrogen and 10% helium (there are also small amounts of heavier elements), is believed to consist of an inner core where nuclear reactions keep the plasma exceedingly hot, at some 1.6 × 107 K. The inner core occupies some 25% of the solar radius. The region outside the inner core and extending out to about 70% of the solar radius is the radiative zone, a region where energy transport is predominantly by radiation. The outer 30% of the solar interior is occupied by the convection zone, a region where convective cells carry the heat at the bottom of this zone out to the cooler solar surface some 2 × 108 m above. There are several scales of convection operating. The two most distinctive convective patterns of flow are the supergranules and the granules. Granules have horizontal sizes ranging between 200 km and 2000 km, with 1000 km providing a characteristic scale. The flows in granules are fairly vigorous, at some 1−3 km s−1 (about 2000−6000 miles per hour), to be compared with terrestrial wind speeds in hurricanes of perhaps 150 miles per hour and the Earth’s record wind speed of 230 miles per hour recorded on the top of Mt Washington in the USA. Supergranules, with a horizontal scale of about 3 × 104 km and so typically 30 times bigger than the granules, have flows of 0.1−0.4 km s−1 . Both these flow patterns are detectable in the Sun’s surface layers, with the granules enveloped by the supergranules. It is at the base of the convection zone and just below that magnetic field lines are believed to be manipulated by Coriolis forces and brought to the solar surface through buoyancy effects. Sunspots are the obvious locations of magnetism in the solar surface. But even away from sunspots there are smaller-scale concentrations of magnetic field. The smallest of these concentrations are the intense magnetic flux tubes, which typically occupy regions about 200 km across wherein magnetic fields of some 1−2 kG strength are confined by external gas pressure forces. The intense tubes are generally located in the regions between convective cells where downdraughts occur. Outside of sunspots, over 90% of the magnetic

6

General Principles

flux appearing in the Sun’s surface layers is in the form of concentrated flux tubes. Sunspots are known to support several types of wave motion. Above the photosphere the concentrations of magnetic field, be they in the small-scale intense flux tubes or the larger scale sunspots, rapidly spread out to fill the available space. This is simply a consequence of stratification. At the photospheric level the gas pressure is sufficient to confine the magnetic fields, once formed in a concentrated form. But the confining external pressure falls off exponentially fast from the photosphere to the chromosphere, decreasing by a factor of e (= 2.178 . . .), to some 37% of its value, over a distance of about 150 km, and this permits the confined magnetic fields to expand out in immediate response. By the mid-chromosphere the fields have filled the atmosphere and at coronal levels completely dominate the nature of the plasma. The coronal plasma is characterized by its low density – in terrestrial terms it would be regarded as almost a vacuum – and high temperature. The high temperature of the corona, in excess of 106 K, was discovered in the 1930s and it remains one of the great puzzles of solar physics: what effects conspire to reverse the strong decline in temperature from the interior of the Sun to its surface, producing an extremely hot outer atmosphere? The answer to this question is important not only for the Sun but for stellar physics in general, for a wide variety of stars are believed to possess a corona. Observations of the Sun’s corona from space have revealed that in X-ray and EUV wavelengths the corona appears not as an amorphous hot glow, as was commonly thought prior to the Skylab mission in the 1970s, but as a complex and structured atmosphere; for an extensive discussion see Aschwanden (2004) and Priest (2014). The basis for this structure is the ubiquitous presence of magnetism in the corona. Despite the overall complexity of the coronal plasma, it would appear that there are fundamentally two different coronae: regions in which the magnetic field lines are curved in the form of loops or arcades with their ends anchored in the dense photosphere, and regions where the field lines emanate from the photosphere but are then carried out into space. The regions with re-entrant magnetic fields – the magnetic loops – are the hottest and most dense parts of the corona; they glow the brightest in X-ray and EUV pictures of the Sun. These are the active regions. They are characterized by temperatures of 2−3 × 106 K, plasma densities of 1016 particles per m3 and have magnetic field strengths of about 102 G. The magnetic field, with its footpoints tied to the photosphere, confines the coronal plasma and heats it to its high temperature. By contrast, where the field is open the plasma blows out into space; these are the coronal holes, the source regions of the high-speed solar wind that blows from the Sun and flows on past the Earth. The plasma density in coronal holes, at 1014 particles per m3 , is two orders of magnitude less dense than that in the active regions. The plasma is also cooler, at 1.5−2 × 106 K, and the open magnetic field, with a strength of around 10 G, is a factor of 10 weaker than the field in active regions. These, then, are the two fundamentally different regions of the corona. Of particular interest here is the discovery by the space instrument TRACE (Transition Region And Coronal Explorer) that coronal loops support a variety of oscillations. Oscillations carry information about the medium in which they occur; such information may be used to obtain indirectly solar quantities that are otherwise difficult to measure. This is the new subject of coronal seismology.

1.3 The Magnetohydrodynamic Equations

7

There are, of course, many other structures in the corona in addition to the two fundamental forms. Of particular interest are quiescent prominences. In a sense, quiescent prominences are bits of the chromosphere that find themselves in a coronal environment. They are cool, dense structures, sometimes resembling a thin sheet of dense plasma, magically suspended in a tenuous corona. The source for their support is the magnetic field that threads through the prominence. They are generally passive structures, surviving for long periods (perhaps months) but then dramatically erupting, only to reform in much the same location shortly thereafter. In photographs of the chromosphere and corona they show up as thin (perhaps 6000 km across), filament-like, dark curves winding their way (for some 2 × 104 km) through the local magnetic structure; their height is about 5 × 104 km. Prominences have a typical density of 1017 m−3 , some two orders of magnitude larger than in their coronal surroundings, and a typical temperature of 7000 K. (There is evidence that the corona may be locally somewhat depleted in density in the neighbourhood of a prominence.) Quiescent prominences are observed to oscillate, a fact which may have important implications for coronal and prominence seismology.

1.3 The Magnetohydrodynamic Equations We have remarked above that magnetohydrodynamics is a combination of fluid mechanics and electromagnetism with Maxwell’s displacement current neglected. Here we describe the equations of this subject. We do not provide a derivation of these equations from basic principles; that route has been fully described elsewhere. Instead, we prefer to simply write down each of the relevant equations and to add some explanatory comments to illustrate various features of the equations. Derivations and discussions of the properties of the equations are given in, for example, Alfv´en (1950), Cowling (1957, 1976), Kendall and Plumpton (1964), Ferraro & Plumpton (1966), Jeffrey (1966), P. H. Roberts (1967), Boyd and Sanderson (1969, 2003), Parker (1979a, 2007) and Priest (1982, 2014). Solar applications are given special attention in Bray and Loughhead (1974), Parker (1979a), Priest (1982, 2014), Bray et al. (1991), Choudhuri (1998), Goossens (2003), Aschwanden (2004), Goedbloed and Poedts (2004), Goedbloed, Keppens and Poedts (2010), Narayanan (2013), Ryutova (2015) and Nakariakov et al. (2016). Consider a fluid with mass density ρ and motions u. Conservation of matter – the statement that matter is neither created nor destroyed within the system (so that there are no sources or sinks of matter) – is described by the equation ∂ρ + div ρu = 0. ∂t

(1.1)

Equation (1.1) is commonly referred to as the equation of continuity. The equation of momentum is the statement that changes in momentum are a result of forces acting in the fluid; it is Newton’s second law applied to a fluid. The momentum equation is   ∂u + (u · grad)u = −grad p + j × B + ρg + F. (1.2) ρ ∂t

8

General Principles

Here p denotes the fluid (or plasma) pressure and ρg is the force per unit volume on the fluid because of gravity of vectorial strength g (we will generally assume gravity to be uniform). B denotes the magnetic field that threads the fluid and j is the current density; these two terms produce the magnetic body force j × B, perpendicular to both B and j. Finally, there may also be other forces F acting, such as the viscous force. The magnetic field B is related to the current density j by Ampere’s law, namely μj = curl B,

(1.3)

where μ is the magnetic permeability of the fluid; generally it is assumed that μ = 4π × 10−7 henry m−1 , its value in free space. Temporal changes in the magnetic field B are related to spatial changes in the electric field E through Faraday’s law of induction: ∂B = −curl E. ∂t

(1.4)

There is a constraint on the magnetic field: it must be solenoidal, div B = 0.

(1.5)

This constraint is the statement that there are no magnetic monopoles: magnetic field lines have no ends, but either close upon themselves or are infinite in extent (which we may view as closing at infinity). There is thus a sharp contrast between magnetic field lines and electric field lines, for the latter originate in concentrations of charge and so may be viewed as emanating from a point. In view of the vector identity div curl ≡ 0, we see that equation (1.4) implies that ∂ (div B)/∂t = 0 and so, as a consequence of Faraday’s law of induction, div B is time independent (and thus is zero for all times if zero at any instant). The constraint (1.5) is stronger, though, insisting that the divergence of B is necessarily zero always. There is also an implied constraint on lines of current j, for the above vector identity taken with Ampere’s law (1.3) implies that div j = 0, and so lines of current density j (like lines of magnetic field) also have no ends. The electric field E is related to the current density j by Ohm’s law, as applied to a moving conductor – the fluid moving with an internal velocity u: j = σ (E + u × B),

(1.6)

where σ is the electrical conductivity of the fluid. Here E + u × B is the total electric field in the fluid, allowing for the induced electric field arising from the component of motion u across the field B. By combining equations (1.4) and (1.6) we may eliminate the electric field E:   1 ∂B = curl(u × B) − curl j . (1.7) ∂t σ

1.4 Some Properties of the MHD Equations

9

Using Ampere’s law (1.3), we may eliminate j to obtain ∂B = curl(u × B) − curl(η curl B), (1.8) ∂t where we have written η = 1/(μσ ); η is referred to as the magnetic diffusivity of the fluid, and has units m2 s−1 . Equation (1.8) is the general form of the magnetohydrodynamic induction equation. Changes in the fluid are generally considered to proceed according to an energy balance equation of the form   γ p ∂ρ ∂p + u · grad p = + u · grad ρ − (γ − 1)L, (1.9) ∂t ρ ∂t where L is the gain or loss function (energy per unit volume) and γ is the ratio of specific heats at constant pressure and constant volume. The term L includes contributions from thermal conduction and radiation. Mechanical heating from external sources as well as the Joule (or ohmic) heating may also be added to the right-hand side of (1.9). Joule heating, arising from the dissipation of current within the fluid, amounts to j2 /σ watts m−3 , for a current density of strength j (= |j|). Frequently all these heat losses or gains are considered to be negligible, and then isentropic (or adiabatic) conditions pertain:   γ p ∂ρ ∂p + u · grad p = + u · grad ρ . (1.10) ∂t ρ ∂t The ratio of specific heats, γ , is generally assumed to be constant. In numerical illustrations we take γ = 5/3, the value appropriate for a fully ionized gas. Heat losses are discussed in Chapter 12. The fluid we are considering will be treated as a perfect gas, for which the ideal gas law is kB (1.11) p = ρT, m ˆ where kB (= 1.38 × 10−23 J K−1 ) is the Boltzmann constant, T is the absolute temperature of the fluid in degrees kelvin (K), equal to the temperature in degrees celsius (◦ C) plus 273, and m ˆ is its mean particle mass. 1.4 Some Properties of the MHD Equations The above system of equations forms the basis for a description of waves in a magnetohydrodynamic fluid. However, as the physicist E. N. Parker says in the Preface of his 1979 monograph Cosmical Magnetic Fields, treating the physics of large-scale magnetic fields in fluids, ‘The fundamental equations of physics may contain all knowledge, but they are close-mouthed and do not volunteer that knowledge’ (Parker 1979a). Thus, in particular, the nature of wave propagation in a magnetic fluid, as described by the equations introduced earlier, is not transparent and indeed serves as the topic for this book. Certain basic features of the equations can, however, be immediately uncovered and these act as points of illumination in our general discourse. We set out these aspects here, as a preliminary to our more detailed discussion of the nature of wave propagation.

10

General Principles

1.4.1 Fundamental Speeds The magnetohydrodynamic equations have embedded within them the usual equations of acoustics, which follow by taking B ≡ 0. Consequently, the magnetohydrodynamic equations must contain the familiar sound speed cs , defined by   γ p0 1/2 cs = . (1.12) ρ0 Here ρ0 and p0 refer to the fluid density and pressure in the unperturbed state of the medium. The occurrence of such a speed as (1.12) is evident on general dimensional grounds. For, by balancing in the momentum equation (1.2) the acceleration term ρ(∂u/∂t) with the pressure force grad p we obtain the dimensional combination p Vτ −1 ∼ . Lρ Here V denotes a characteristic speed, L a characteristic length, τ a characteristic timescale, with p and ρ denoting a characteristic pressure and density; the use of ‘∼’ here denotes a dimensional balance. Writing V ∼ Lτ −1 then gives V 2 ∼ p/ρ, which leads to the combination (1.12) for a characteristic speed (though without the important factor of γ ) when we take the equilibrium pressure p0 and density ρ0 as representative values. A similar argument for a magnetic speed can be made by equating, in dimensional terms, the acceleration term ρ(∂u/∂t) with the magnetic force j × B. We obtain ρVτ −1 ∼ JB, where J and B are characteristic values of the current density and magnetic field. But from Ampere’s law (1.3) we have μJ ∼ BL−1 , which allows us to eliminate J. Setting V ∼ Lτ −1 then gives V 2 ∼ B2 /(μρ). We thus obtain a characteristic magnetic speed, a speed that arises in phenomena for which the magnetic force plays a role. We take this speed1 as  1/2 B20 cA = , (1.13) μρ0 choosing an equilibrium field strength B0 and a plasma density ρ0 as representative values of the magnetic field and fluid density. The speed cA defined by equation (1.13) is the Alfv´en speed, the speed obtained by Alfv´en in his short letter to Nature in 1942 (Alfv´en 1942a). It is central to all magnetohydrodynamic wave phenomena, just as the sound speed is central to all acoustic phenomena. The sound speed and the Alfv´en speed underpin all wave phenomena described by the magnetohydrodynamic equations. Other speeds also play an important role, but these are always constructed in terms of cs and cA . Accordingly, we consider the sound and Alfv´en speeds in a little more detail. Plasma Pressure and Density To begin with suppose our medium is a fully ionized hydrogen plasma, consisting of ne electrons and np protons (ions) in each unit volume of space. The total pressure is p = ne kB Te + np kB Tp 1 In the Gaussian cgs system of units, the Alfv´en speed is define as c = B /(4πρ )1/2 where the magnetic field strength B is A 0 0 0 in gauss (G) and the plasma density ρ0 is in grams per cubic centimetre (g cm−3 ).

1.4 Some Properties of the MHD Equations

11

where Te denotes the electron temperature and Tp the proton temperature. We will assume that the electron temperature and proton temperature are equal, so that Te = Tp = T, where T denotes the temperature of the medium (the plasma or fluid temperature). Then, p = (ne + np )kB T. Now charge neutrality requires that the number of electrons be the same as the number of protons, so that ne = np ; then total number of particles is n = ne + np = 2ne and the pressure is p = 2ne kB T = nkB T. The density ρ is determined by the number ne of electrons of mass me together with the number np of protons of mass mp in a unit volume, so that ρ = ne me + np mp . However, whilst the number of electrons equals the number of protons, the mass mp (= 1.673 × 10−27 kg) of a proton is much larger than the mass me (= 9.109 × 10−31 kg) of an electron, so we can neglect the electron mass and take ρ = ne mp =

1 nmp . 2

Accordingly, we can write p=

kB R ρT = ρT, μm ˆ p μˆ

ˆ ρ = μm ˆ p n = mn,

R=

kB mp

(1.14)

as a description of the plasma pressure p, density ρ and temperature T. Here m ˆ = μm ˆ p denotes the mean particle mass of the plasma; for a hydrogen plasma, μˆ = 1/2 and m ˆ = mp /2. If the plasma consists of a more complicated mixture of hydrogen and helium (and other elements too) then we can take relations (1.14) as still standing but now the factor μˆ is no longer 1/2; in the solar corona, to allow for the contributions from those other elements that make up the plasma, it is common to take μˆ ≈ 0.6 (see, for example, Aschwanden 2004). Sound Speed The ideal gas law (1.14) allows us to express the sound speed in terms of the square root of the temperature T0 of the medium:       γ p0 1/2 γ kB T0 1/2 γ kB T0 1/2 = = . (1.15) cs = ρ0 m ˆ μm ˆ p The mean particle mass m ˆ depends upon the nature of the plasma. With Boltzmann constant kB = 1.38 × 10−23 J K−1 , the proton mass mp = 1.673 × 10−27 kg (so that R = kB /mp = 8.25 × 103 m2 s−2 K−1 ) and an adiabatic index γ = 5/3, equation (1.15) gives a sound speed ˆ 1/2 m s−1 . cs = 1.17 × 102 (T0 /μ)

(1.16)

12

General Principles

In the solar corona, the plasma is mainly made up of hydrogen and helium, which produces a mean atomic weight of μˆ ≈ 0.6. Thus, in the corona the sound speed is cs = 151 T0 1/2 m s−1 .

(1.17)

A coronal temperature of, say, T0 = 106 K then yields a sound speed of cs = 151 km s−1 . The high temperature of the coronal plasma thus leads to a correspondingly high sound speed, far in access of the 340 m s−1 sound speed in the Earth’s atmosphere or the 1400 m s−1 sound speed in water. Lower in the solar atmosphere, a photospheric density of ρ0 = 3 × 10−4 kg m−3 ( = 3 × 10−7 g cm−3 ) and pressure of p0 = 2 × 104 N m−2 ( = 2 × 105 dynes cm−2 ) (see, for example, Parker 1979a, p. 212) produces (for γ = 5/3) a sound speed of cs = 10.5 km s−1 . Alfv´en Speed Turning to the Alfv´en speed, we have  1/2 B20 B0 cA = = 2.18 × 1016 × m s−1 . μρ0 (μn) ˆ 1/2

(1.18)

ˆ p n, for total We have taken μ = 4π × 10−7 henry m−1 and the plasma density as ρ0 = μm number density n (in particles per cubic metre), and the magnetic field strength B0 is in tesla (T). Thus, with μˆ ≈ 0.6 appropriate for the corona we obtain cA = 2.816 × 1016 ×

B0 m s−1 . n1/2

(1.19)

Thus, with a number density of say n = 1015 particles per m3 typical of the corona we obtain ρ0 = 2 × 10−12 kg m−3 (or 2 × 10−15 g cm−3 ) and an Alfv´en speed of cA = 0.890 × 109 B0 m s−1 . It is common to quote the magnetic field strength in gauss (G), noting that 1 T = 104 G. Then the expression for the Alfv´en speed reads cA = 2.816 × 1012 ×

B0 (G) m s−1 , n1/2

(1.20)

with B0 in gauss. For example, with a coronal field strength of B0 = 10−3 T (= 10 G), we obtain an Alfv´en speed of some cA = 890 km s−1 . In an active region the field is stronger; for example, with B0 = 10−2 T (= 102 G) in a medium with an electron number density ne = 1016 particles per m3 (1010 particles per cm3 ) and a total number density n = 2×1016 particles per m3 , we obtain cA ≈ 2000 km s−1 . Thus, despite the corona’s high temperature and correspondingly high sound speed, the tenuous nature of the coronal plasma acts to produce a yet higher Alfv´en speed. In the photosphere, it is usual to quote values of fluid density ρ0 directly, basing these values on model computations of the convection zone and atmosphere above. A density of ρ0 = 3 × 10−4 kg m−3 (= 3 × 10−7 g cm−3 ) is typical of the surface layers of the Sun (see, for example, Parker 1979a, p. 149). In a magnetic field of B0 = 0.15 T (= 1500 G) the corresponding Alfv´en speed is cA = 0.077 × 105 m s−1 . Thus in regions of strong

1.4 Some Properties of the MHD Equations

13

photospheric magnetic field the Alfv´en speed is some 7.7 km s−1 and sound and Alfv´en speeds are typically comparable.

1.4.2 Magnetic Flux Tubes It is convenient to introduce the notion of a magnetic flux tube. The concept of a magnetic flux tube goes back to the writings of Faraday and Maxwell. Faraday, with his non-mathematical approach to electromagnetic phenomena, pictured the movement and distortion of magnetic field lines. Maxwell later added the mathematical detail to give such an intuitive approach a more rigorous basis. Consider a curve drawn arbitrarily in a magnetic field. All the field lines passing through this curve are considered to be related, forming a single entity called a magnetic flux tube. Since the choice of the curve that relates the various field lines is entirely arbitrary, the flux tube thus formed is also an arbitrary collection of magnetic field lines. However, in Nature it is often found that a certain collection of field lines is of particular interest and form, giving special definition to those field lines. In the solar photosphere, for example, we have seen that isolated magnetic flux tubes occur, corresponding to concentrations of magnetic field surrounded by a field-free environment. In such objects the magnetic flux tube is given definition by the field itself. In the corona, magnetic loops are flux tubes given definition not so much by their field strength – the field may indeed be essentially uniform – but by the fact that certain field lines are loaded with more plasma or are at a higher temperature than other field lines; the flux tube is thus here given definition by an enhancement in the plasma density, or by temperature differences between one region or another. Certain types of prominence structures are also examples of this kind. The magnetospheres of planets also provide examples of flux tubes which are given definition by their magnetic field structure, being commonly twisted. One property of a magnetic flux tube follows immediately from the solenoidal constraint on a magnetic field. For with B satisfying equation (1.5), application of the divergence theorem in an arbitrary volume V yields   div B dV = B · dS = 0, (1.21) V

SV

where the volume V is enclosed by the surface SV and dS is the surface element pointing (by convention) out of the volume V. For a magnetic flux tube we may choose that surface to be the curved surface of the flux tube together with a ‘top’ surface St and a ‘bottom’ surface Sb to make a closed volume. Then, since no magnetic flux leaves the curved surface of the tube (on which B · dS = 0), equation (1.21) implies that   B · dS = B · dS, (1.22) St

Sb

where the cross-sectional surface element dS points in the same sense as B. In other words, the magnetic flux across any cross-section S of the tube is the same at all locations along the tube; it is thus an invariant of the motion. Since this property follows directly from the solenoidal constraint it is independent of whether we are considering dissipative effects

14

General Principles

(such as viscosity or diffusivity) or not. Stated loosely, the product of the field strength B of a tube and its normal cross-sectional area S is a constant. Thus, where a tube narrows its field strength is large, whereas in the expanded regions of the tube the field strength is correspondingly reduced.

1.4.3 The Induction Equation The induction equation (1.8) makes clear that there is a direct link in magnetohydrodynamics between fluid motions and the magnetic field, and that this link is independent of other properties of the fluid, such as its density and pressure, unless they enter indirectly through the magnetic diffusivity. In fact, in a plasma the electrical conductivity σ and magnetic diffusivity η are principally determined by the temperature of the medium. For a fully ionized hydrogen plasma at a temperature of T K, we have an electrical conductivity of σ ≈ 8 × 10−4 T 3/2 mho m−1 , leading to a magnetic diffusivity of (Parker 1979a, sect. 4.6 and 7.6; Priest 2014, sect. 2.1.5) η ≈ 109 T −3/2 m2 s−1 .

(1.23)

Thus the diffusivity varies quite strongly with the temperature of the plasma, being low in high temperature plasmas. Complications in the description of diffusivity arise when the level of ionization in the medium is very low or when the presence of the applied magnetic field is properly allowed for. Low ionization leads to a significant reduction in the value of η. The presence of a magnetic field renders the diffusivity as a tensor, with a different value along the field from that across the field. However, such complications will not be discussed here. To illustrate the link between the flow and the magnetic field embedded in the fluid, as described by the induction equation, it is convenient to discuss the nature of the induction equation for the case of constant magnetic diffusivity, ignoring the temperature dependence given in equation (1.23), except in so far as it provides an appropriate numerical value for η. In fact the complications introduced by a variable diffusivity η do not introduce anything of general significance as regards the nature of the link between the flow and the magnetic field. Accordingly, we consider the induction equation under the assumption that η is a constant. We may then simplify equation (1.8) by use of the vector identity curl curl ≡ grad div − ∇ 2 . Coupled with the solenoidal constraint (1.5) on the magnetic field, the above vector identity allows us to write equation (1.8) in the form ∂B = curl(u × B) + η∇ 2 B. ∂t

(1.24)

This is the induction equation for a medium with uniform magnetic diffusivity. The magnetic field B grows or diminishes in time according to the combined influences of a velocitydependent advective term (the first term on the right-hand side of equation (1.24)) and a velocity-independent diffusive term (the term involving η). These terms have quite distinct effects and, as we shall see, are generally of quite different magnitudes.

1.4 Some Properties of the MHD Equations

15

There is an interesting analogy between the evolution of magnetic field, as described by the induction equation (1.24), and the evolution of vorticity ω ≡ curl u in a viscous non-magnetic liquid with uniform density. Consider the momentum equation (1.2) in the absence of magnetism (B = 0) and gravity (g = 0), for a liquid with constant density ρ. The viscous force F in such a liquid is given by F = ρν∇ 2 u,

(1.25)

where ν is the kinematic viscosity (assumed constant) of the liquid. The momentum equation is thus   ∂u (1.26) + (u · grad)u = −grad p + ρν∇ 2 u. ρ ∂t Taking the ‘curl’ of the above equation, and noting the vector identity curl grad ≡ 0, we obtain ∂ω + curl[(u · grad)u] = curl (ν∇ 2 u). ∂t Then, making use of the above vector identities together with 1 grad(u · u) ≡ u × curl u + (u · grad)u, 2 we obtain ∂ω = curl (u × ω) + ν∇ 2 ω. ∂t

(1.27)

Thus we see that the vorticity ω in a uniform non-magnetic liquid evolves in time in much the same way as the magnetic field evolves in a magnetized fluid. There is a difference, however, in that whereas the vorticity is directly related to the motion (ω = curl u), the magnetic field B and motion u are not so related. This means that whereas the vorticity equation is nonlinear, involving a product of u with curl u, the induction equation is linear in B for a given motion u. Moreover, the kinematic diffusivity ν in a liquid is generally much smaller than the magnetic diffusivity η of a fluid. The kinematic viscosity of water, for example, is ν ≈ 10−6 m2 s−1 , some two orders of magnitude smaller than that of olive oil; the electrically conducting fluid mercury has a kinematic viscosity of ν ≈ 10−7 m2 s−1 , about a tenth that of water. These kinematic viscosities ν are several orders of magnitude smaller than the corresponding magnetic diffusivities η; in the case of liquid mercury, there is a difference of seven orders of magnitude. There are thus distinctive differences between the two systems. Nonetheless, the analogy can prove useful. The magnitudes of the two terms that make up the right-hand side of the induction equation (1.24) are easily estimated using dimensional considerations. Their ratio forms a number Rm , known as the magnetic Reynolds number (by analogy with the Reynolds number of a viscous fluid). We have Rm ∼ |curl (u × B)|/|η∇ 2 B|.

16

General Principles

This leads to Rm ∼ L−1 VB/(ηL−2 B), on noting that ‘curl’ involves a single division by a spatial scale L whereas ‘∇ 2 ’ involves division by L2 ; as before, V and B denote characteristic values of the motion and magnetic field strength, each assumed to vary on the same scale L. Thus Rm ∼

LV . η

(1.28)

This may be compared with the Reynolds number R of a viscous flow: R = LV/ν. The dimensionless number Rm determines which term on the right-hand side of the induction equation is dominant. If the magnetic Reynolds number is large (Rm  1) then the right-hand side of equation (1.24) is dominated by the advection term and diffusive effects are essentially negligible. Only if one follows the advected field for a long time would one detect the small diffusion of field that takes place when Rm  1. Precisely the opposite conclusion is reached when the magnetic Reynolds number is small. For with Rm  1 diffusive effects are dominant and the advection term is essentially negligible. Only if one follows the evolution of the magnetic field B for a long time would one see the influence of advection on the overall diffusion of the field. In the intermediate case, when Rm is of order unity, diffusion and advection play comparable roles in the evolution of the magnetic field. Now the size of Rm is determined not so much by whether a fluid is electrically a good or a poor conductor, but by the size L of the fluid in which the magnetic field is entrained and the flow V that is present. For these quantities can change by orders of magnitude, from circumstance to circumstance, over-shadowing the possible variations in η in all but the most severe cases (when large temperature differences can bring about correspondingly large changes in Rm , through changes in η). Generally, then, we find that for a liquid (such as mercury or molten iron) in laboratory circumstances Rm is small or of order unity, simply because L and V are relatively small. In astrophysical circumstances, however, Rm is large, simply because L (and perhaps V) are large. Thus laboratory systems tend to be dominated by diffusive effects, whereas astrophysical plasmas are largely free from the influences of diffusion. An exception in astrophysical systems occurs in regions where B may undergo a sudden reversal in direction or rapid variation; then the scale of variation of B is no longer necessarily the same as that of the flow and a local diffusion of magnetic field or magnetic reconnection may occur (see, for example, Priest and Forbes 2000; Priest 2014). To illustrate specifically the magnitude of Rm , consider a laboratory fluid with spatial extent L = 1 m and a flow of order V = 0.1 m s−1 ; for molten iron (with η = 0.06 m2 s−1 ), this produces a magnetic Reynolds number of order unity. For the Earth’s liquid core, with a scale of L = 3.5 × 106 m and a diffusivity of η = 3 m2 s−1 , fluid motions of say 0.1 mm s−1 (= 10−4 m s−1 ) produce a magnetic Reynolds number of order 102 . Turning to the Sun, there are flows of order 1 km s−1 (= 103 m s−1 ) on a scale of 103 km observed at its surface (in granules); with η = 105 m2 s−1 , we obtain an Rm of 104 . We turn now to a brief examination of the behaviour of the induction equation in the two extremes of small and large magnetic Reynolds number, corresponding to diffusive effects being either important or negligible.

1.4 Some Properties of the MHD Equations

17

1.4.4 Diffusion of Magnetic Field (Rm  1) Consider the extreme of the induction equation that arises for Rm  1, when the diffusion term η∇ 2 B dominates over advection. Then the induction equation reduces to ∂B (1.29) = η∇ 2 B. ∂t This reduction of the induction equation is exact when no flows are present (u = 0). Generally, though, we may regard equation (1.29) as giving the dominant short-term behaviour of the system when Rm  1. We recognize equation (1.29) as the vector form of the diffusion equation (or heat conduction equation). In a Cartesian coordinate system each component of the magnetic field satisfies the familiar (scalar) diffusion equation. The diffusion equation acts so as to smooth out irregularities or steep gradients in B. We note that the decay or leakage of magnetic field through the fluid is not an entirely passive process, for it is accompanied by ohmic heating j2 /σ (= μηj2 ), which is of order (η/μ)B2 L−2 in a field of characteristic strength B and spatial scale L. The process of field decay operates on a timescale τ , readily estimated from dimensional considerations. In dimensional terms, from equation (1.29) we have Bτ −1 ∼ ηBL−2 . Solving for the decay time τ = τ decay we obtain τ decay ∼ L2 /η.

(1.30)

Hence the magnetic field in a motionless conductor decays away on a timescale that is proportional to the electrical conductivity σ of the medium and proportional to the square of the spatial scale L: τ decay ∼ μσ L2 . The time τ decay gives an estimate of how long it takes a concentration of magnetic field to leak away by a factor of 1/e (with e = 2.718 . . . being Euler’s constant), reducing to some 37% of its initial value. Of course, the estimate provided by equation (1.30) is rather rough, as it ignores such factors as the geometry of the object and the precise choice for L. Only by solving the diffusion equation in a specific case can one determine τ decay more precisely. In the case of a sphere of radius a it turns out that τ decay = a2 /(ηπ 2 ), so that for the sphere we have an effective L of a/π. Such factors of π and the like are in fact significant if, as occurs here, they are squared or raised to a higher power. For the sphere the factor of π reduces the simple estimate of a2 /η for the decay time by an order of magnitude. The presence in τ decay of the square of L – an immediate consequence of the Laplacian operator ∇ 2 in the diffusion equation – makes for very short timescales in laboratory situations but very long ones in astrophysical circumstances. We may readily illustrate this wide range in decay (diffusion) times. For example, in a copper sphere (with η = 0.01 m2 s−1 ) of radius a = 1 m, the leakage time of the magnetic field through the sphere is of order τ decay = 10 s, and so rather short. By contrast, for a sphere the size of the Sun as a whole, with a radius of 6.96 × 108 m and a diffusivity of η = 1 m2 s−1 , the value given by equation (1.23) for a temperature of 106 K (roughly representative of the wide range in temperature in the solar interior from some 107 K in the core to 6000 K at the surface), we obtain a decay time of around 109 years (Cowling 1946, 1976), comparable with the

18

General Principles

determined age of the Sun (as 5 × 109 years). Thus, a primordial magnetic field entrapped within the Sun at its formation would still largely be entrapped today. Of course, we observe magnetic fields on the Sun changing on relatively short timescales (ranging from hours to years), so magnetic fields must be manipulated by forces that are more effective than the simple passive diffusion of the field. The decay time for the field in an individual object within the Sun is, of course, much shorter than the time for the Sun as a whole. For example, a sunspot with radius L = 107 m and diffusivity η = 103 m2 s−1 (corresponding to a temperature of 104 K, representative of the layers below the surface) we obtain a decay time of over 300 years, as originally estimated by Cowling. In fact, sunspots change their magnetism on much shorter times than this and so flows must play a significant part in determining their magnetic history. 1.4.5 Advection of Magnetic Field (Rm  1) Consider now the extreme of the induction equation arising when we neglect diffusivity. A fluid in which η = 0 is said to be a perfect (or ideal) conductor, for which the induction equation is simply ∂B = curl (u × B). (1.31) ∂t This equation is exact if η = 0 but applies more generally to the case of high magnetic Reynolds number, describing the short-term behaviour when Rm  1. The analogy between the induction equation and the vorticity equation for a liquid may be exploited here. We may immediately invoke Kelvin’s circulation theorem, which tells us that the vorticity ω in a flow is advected with the fluid. Accordingly, we may conclude that the magnetic field in a conducting fluid is advected with the flow. Alfv´en (1943) expressed this picturesquely, saying that the magnetic field lines are frozen into the flow: motions along the field lines leave them unchanged, whereas motions perpendicular to the field lines transport the lines with the flow. That the magnetic field is frozen into the flow in ideal magnetohydrodynamics has an immediate consequence for a magnetic flux tube. Since each magnetic field line moves with the fluid, it follows that a magnetic flux tube moves with the motion: the fluid entrained within a tube at a given time remains entrained within that tube during the subsequent motion of the fluid, though the tube itself may be distorted by the flow (though, as noted earlier, its magnetic flux remains invariant). Consider, then, an elemental flux tube with field strength B and normal cross-section dS. Then the flux BdS is a constant of the motion. If we consider a section of the flux tube of length ds measured between two cross-sections of the tube, then in the subsequent motion the mass ρdsdS contained between the two cross-sections is conserved during the motion. Thus, BdS = constant,

ρdsdS = constant.

(1.32)

Eliminating dS between these two invariants shows that B/ρ is proportional to the distance ds between neighbouring cross-sections. In other words, B/ρ increases during the motion

1.4 Some Properties of the MHD Equations

19

if that motion stretches the field line, whereas it decreases if the motion reduces the length of the field line between the two cross-sections. We can view the above discussion directly from the induction equation under ideal conditions and the equation of continuity. First rewrite the induction equation (1.31) by using the vector identity curl (u × B) ≡ u(div B) − B(div u) + (B · grad)u − (u · grad)B and setting div B = 0. Then, when combined with the equation of continuity (1.1), we obtain      B B ∂ + u · grad = · grad u. (1.33) ∂t ρ ρ Thus, if the right-hand side of the above equation is ignored we see that the quantity B/ρ is conserved during the motion. The effect then of a non-zero right-hand side is to cause an increase (decrease) in B/ρ if the motion stretches (compresses) the field lines, corresponding to B ∂u ρ ∂s being positive (negative). 1.4.6 The j × B Force Magnetic effects in the momentum equation (1.2) are determined by the j × B force, which acts perpendicular to the magnetic field B and the current j. Now, using Ampere’s law (1.3) and the vector identity 1 grad (B · B) ≡ B × curl B + (B · grad)B, 2 we obtain

 j × B = −grad

B2 2μ

 +

1 (B · grad)B, μ

(1.34)

where now B (= |B|) denotes the magnitude of the field B. Comparing this form of the magnetic force with the right-hand side of the momentum equation, we see immediately that we may regard the contribution − grad (B2 /2μ)

(1.35)

from the j × B force as acting as a pressure term, much the same as the fluid pressure force −grad p. Accordingly, we may introduce the magnetic pressure pm , pm =

1 B2 B·B= . 2μ 2μ

(1.36)

The magnetic pressure pm acts isotropically throughout the fluid, just as the plasma pressure p does.

20

General Principles

A magnetic field, then, has a magnetic pressure pm associated with it, augmenting the plasma pressure p; the magnetic pressure force increases in proportion to the square of the magnetic field strength. In illustrations we will commonly quote magnetic field strength in gauss; a field of B gauss produces a magnetic pressure of B2 /(80π ) Pa.

(1.37)

This is the magnetic pressure in pascals (Pa), the SI unit of pressure (1 Pa = 1 N m−2 ). (In cgs units with B in gauss, the magnetic pressure is B2 /8π dynes cm−2 .) For example, the Earth’s magnetic field of order 1/2 G produces a magnetic pressure of some 10−3 Pa (= 10−3 N m−2 ), eight orders of magnitude smaller than a typical atmospheric pressure of 1 bar (= 105 Pa). By contrast, a field of 3000 G, typical of a sunspot, produces a magnetic pressure of 3.6 × 104 Pa, which (at about 1/3 bar) is roughly comparable with the plasma pressure of 1.6 × 104 Pa at the solar surface. Equilibrium in the spot is achieved because the plasma within the magnetic field sinks to a level below the photosphere where the confining external plasma pressure is higher. The identification of magnetic pressure, acting in addition to any dynamical pressure, raises the question of their relative importance. This is decided upon by their ratio, commonly referred to as the plasma beta and defined by p p . (1.38) = 2 β≡ pm (B /(2μ) If we take p and B as represented by the equilibrium values p0 and B0 , we see that the plasma β is directly related to the ratio of the sound and Alfv´en speeds: β=

2 c2s . γ c2A

(1.39)

Thus a medium with high Alfv´en speed (cA  cs ) is a low-β plasma, and one with a low Alfv´en speed (cA  cs ) is a high-β plasma. A fluid with low plasma β has a correspondingly strong magnetic field, and generally speaking it is mechanically dominated by the magnetic forces. This is the circumstance in much of the Earth’s magnetosphere and in the upper atmosphere of the Sun. By contrast, in a high-β plasma magnetic forces are weak compared with the dynamical pressure force and magnetic effects are correspondingly less important; this is the circumstance pertaining in the Earth’s interior and below the Sun’s photosphere. Of course, in some situations β is of order unity, indicating that both magnetic and dynamical effects are of comparable importance. This situation typically pertains in magnetic concentrations in the surface layers of the Sun. The magnetic pressure term comprises only one part of the j × B force; there remains the term 1 (B · grad)B. (1.40) μ This term may be interpreted as a magnetic tension. The magnetic field behaves much as an elastic band, the tension in the band being B2 /μ per unit area, acting along the magnetic field. Accordingly, a distortion or bend introduced into a magnetic field line sets up a

1.4 Some Properties of the MHD Equations

21

tension force – just as with an elastic band – that acts so as to try to straighten out the field line. To see this in detail, consider a magnetic field line with arc distance s measured along it from a fixed point. Denote by sˆ a unit vector pointing along the magnetic field B. Then B = B(s)ˆs, where B(s) is the field strength at location s. The magnetic tension force is accordingly   1 ∂ ∂ B2 B2 ∂ sˆ 1 (B · grad)B = B(s) [B(s)ˆs] = sˆ + . (1.41) μ μ ∂s ∂s 2μ μ ∂s Now it is shown in discussions of vector calculus that ∂ sˆ 1 ˆ = n, ∂s Rc where Rc (s) is the radius of curvature of the field line at the location s and nˆ is the principal unit vector perpendicular to the field line at that location and pointing towards the centre of curvature. Hence   1 ∂ B2 B2 ˆ (B · grad)B = sˆ + n, (1.42) μ ∂s 2μ μRc and so

 j × B = −grad

B2 2μ



∂ + sˆ ∂s



B2 2μ

 +

B2 ˆ n. μRc

(1.43)

The contribution from the term acting in the direction of sˆ may be grouped with the contribution from the pressure term, these respective contributions cancelling out in the direction of the magnetic field (as they must do, since the j × B force is perpendicular to B). The term perpendicular to the magnetic field gives a tension force of magnitude B2 /μRc , which acts so as to straighten out any bends in the field. The sharper the bend in the field, the smaller is the radius of curvature Rc and so the larger is the tension force acting to straighten out the field.

1.4.7 Energetics As well as providing a magnetic pressure, the expression B2 /2μ gives the magnetic energy density (per unit volume) in the plasma. The total magnetic energy in a volume V is therefore  B2 W= dV. (1.44) V 2μ It is of interest to determine how W varies in time for a fixed volume V. Noting that B2 = B · B and j2 = j · j, we may invoke the induction equation (1.24) and Ampere’s relation (1.3) to obtain       dW 1 1 2 1 =− j + u · j × B dV + (u × B) × B − (j × B) · dS (1.45) dt μ SV σ V σ

22

General Principles

for surface SV (with vector area element dS) enclosing the volume V. Consider the case when V is the whole of space, with SV = S∞ being the sphere at infinity. Then the contribution from the surface integral is negligible if B declines sufficiently fast as r → ∞. For example, if B declines to zero faster than r−3/2 , a rate which ensures that W is finite (it diverges logarithmically if B ∼ r−3/2 ), then the surface integral of the j × B force is negligible. The result is   dW 1 2 [u · j × B]dV, (1.46) =− j dV − dt S∞ σ S∞ where the integrations are now over the whole of space. Thus, the fate of the magnetic energy in the system – whether it decays or grows – depends upon the two contributions on the right-hand side of equation (1.46). The contribution from Joule heating (the first term on the right-hand side of (1.46)) leads to an inexorable decline, to be balanced against the effect of the second term on the right-hand side of (1.46), the contribution from the fluid motions doing work against the opposing magnetic forces. When those motions are sufficiently vigorous and complicated so as to reverse the sign of the second term in equation (1.46), then W may grow in time: dynamo action is said to have occurred. To judge from the ubiquitous occurrence of magnetic fields in astrophysical objects, ranging from the planets to the stars and galaxies, Nature seems particularly adept at bringing about such an arrangement.

1.5 Aspects of Wave Propagation 1.5.1 Linearization Waves generated in a system are often of very small amplitude, a state of affairs that permits one to examine linear equations describing the temporal and spatial behaviour of the perturbations (disturbances) about an equilibrium state. The process of obtaining such equations for the perturbations is referred to as linearization. We may illustrate the process of linearization by considering the equation of continuity (1.1). Denote by ρ0 the value of the density in the basic state; suppose that there is no flow (u = 0) in the basic state. Then in the disturbed state we write density = ρ0 + ρ,

motion = u,

(1.47)

where ρ and u now denote the values of the perturbations in density and motion. Thus equation (1.1) becomes ∂ρ + div (ρ0 + ρ) u = 0. ∂t So far no approximation has been made, given that ρ0 is independent of time. But if we now suppose that the perturbation ρ is small, so that |ρ|  |ρ0 |, then the equation of continuity reduces to ∂ρ + div ρ0 u = 0. ∂t

(1.48)

1.5 Aspects of Wave Propagation

23

This represents a considerable simplification since it has resulted in a linear equation, and linear equations are relatively easy to solve whereas nonlinear equations (such as the original form of the continuity equation) are difficult. A further simplification occurs if the equilibrium state is a uniform one. For if the density ρ0 is a constant, then the linearized equation of continuity becomes ∂ρ + ρ0 div u = 0. ∂t

(1.49)

The equations describing the other perturbations may be treated in a similar fashion. For example, the linearized equation of motion (with F = 0) becomes ρ0

∂u = −grad p + j × B0 + j0 × B + ρg, ∂t

(1.50)

where p, B and j now denote the perturbations in fluid pressure, magnetic field and current density about an equilibrium with magnetic field B0 and current density j0 (with μj0 = curl B0 ). If the equilibrium magnetic field is a uniform one, with B0 a constant vector, then the current density j0 in the equilibrium is zero.

1.5.2 Fourier Representation In general, when the equilibrium is a uniform one, with constant density, pressure and magnetic field, then the linear equations describing the perturbations will have coefficients that are constants. This permits us to construct solutions with a sinusoidal or exponential behaviour. This is most conveniently done using a representation in terms of the complex exponential function. For an unbounded and uniform equilibrium it frequently proves convenient to consider a plane wave representation for the perturbations. By a plane wave representation we mean that at time t each perturbation may be expressed in the form f (ωt − k · r), where r denotes the position vector of the point (x, y, z) in a Cartesian coordinate system O, x, y, z. Here ω denotes the angular frequency of the perturbation, k = (kx , ky , kz ) is the wave vector, and the function f describes the shape or profile of the disturbance. The function P ≡ ωt − k · r

(1.51)

is known as the phase, and the equation P = constant describes a plane with normal vector k. The phase plane moves with a speed c, where c=

ω k

(1.52)

1

with k = |k| = (kx2 + ky2 + kz2 ) 2 denoting the magnitude of the wave vector. The disturbance moves in the direction of the vector k. Accordingly, the plane wave moves with a velocity c=

ω ek = c ek , k

(1.53)

where ek denotes a unit vector in the direction of propagation, k. The vector c is referred to as the phase velocity of the disturbance, with c being the phase speed.

24

General Principles

It is convenient to use a complex variable representation of a perturbation, such as the fluid motion u and the perturbed plasma density ρ, by writing u(x, y, z, t) = u1 exp i(ωt − k · r),

ρ(x, y, z, t) = ρ1 exp i(ωt − k · r),

(1.54)

where i2 = −1. Here u1 is a complex constant vector (i.e., a vector with components that are complex constants) and ρ1 is a complex constant. The modulus of these quantities, |u1 | and |ρ1 |, gives information about the amplitude of the motion u and the associated density variations ρ. Other perturbations (such as the pressure or magnetic field) may be expressed in a similar complex exponential form. The actual physical perturbations may be obtained by taking the real parts of the above complex representations, once the relationships between the various complex constants and vectors, such as ρ1 and u1 , are determined. The value of using a complex exponential representation is that it converts the various differential operators arising in the linear equations for the perturbations into simple algebraic scalar and vector forms. Thus, the relations ∂ ≡ iω, ∂t div ≡ −ik·,

∂ ≡ −ikz , ∂z grad ≡ −ik,

∇ 2 ≡ −k2 , curl ≡ −ik×

(1.55) (1.56)

provide algebraic scalar and vector representations (involving the scalar product (·) for div and the vector product (×) for curl) of the partial differential operators, converting such operators into relatively simple forms. Thus, for example, ∂ρ/∂t = iωρ and div u = −ik · u and so the linearized equation of continuity (1.49) becomes ωρ = ρ0 k · u = ρ0 (kx ux + ky uy + kz uz ),

(1.57)

where u = (ux , uy , uz ) and we have cancelled a common factor i. Thus, the density perturbation is related to the scalar product of k with u. Similar algebraic relationships arise from the other partial differential equations.

1.5.3 Dispersion Relations, Phase Speed and Group Velocity The system of linear algebraic equations that arise from a Fourier representation of the perturbations leads, in general, to a dispersion relation. This is a relationship between the frequency ω and the wave vector k. When the phase speed c = ω/k is independent of k, we say that the system is non-dispersive: all waves travel with the same speed c whatever their wavelength, 2π/k. However, in magnetohydrodynamics it turns out (see Chapter 2) that while the waves are non-dispersive the phase speed c nonetheless varies with direction: a wave propagating in the direction of the equilibrium field B0 , say, has a different speed from one propagating at an angle to the field. We say the medium is anisotropic. This anisotropy arises simply because the presence of an applied magnetic field in the equilibrium state introduces a preferred direction in the magnetohydrodynamic system, and this directionality is reflected in the behaviour of the phase speed c and phase velocity c.

1.5 Aspects of Wave Propagation

25

In addition to the phase velocity c, there is particular interest in the group velocity cg defined as (see, for example, Whitham 1974; Lighthill 1978)   ∂ω ∂ω ∂ω ∂ω cg ≡ , , ≡ . (1.58) ∂k ∂kx ∂ky ∂kz Thus the group velocity is the gradient in k-space of ω = ω(k) and may be determined once the dispersion relation ω = ω(k) is known. In general, cg is different from c, in both magnitude and direction, though in some systems – most notably for acoustic waves – the two speeds are equal. Since ω = kc, we may write ∂ ∂k ∂c ∂c (kc) = c+k =c+k , (1.59) ∂k ∂k ∂k ∂k on noting that ∂k/∂kx = kx /k, etc., and so ∂k/∂k = k/k. Now, as noted earlier, magnetohydrodynamic waves are not dispersive but are anisotropic, with a phase speed that is independent of k but a function of the angle of propagation of the wave relative to the direction of the applied magnetic field. Introduce the angle θ that the direction of propagation of a plane wave makes with a fixed direction, taken to be that of the applied magnetic field, and with it a unit vector eθ that is perpendicular to the wave vector k and points in the direction of increasing θ (see Figure 1.1). Magnetohydrodynamic waves are such that c = c(θ ). It is convenient to align our Cartesian coordinate system so that the plane wave vector k lies entirely in the xz-plane. Then we may write k = (k sin θ , 0, k cos θ ) and eθ = (cos θ , 0, − sin θ ), and so 1 dc ∂c (1.60) = eθ . ∂k k dθ Hence dc (1.61) eθ . cg = c + dθ Thus the group velocity is the vector sum of the phase velocity and a component perpendicular to the direction of propagation. The component of the group velocity in the direction of propagation is simply the phase speed: cg · ek = c. The component perpendicular to the direction of propagation may be positive, zero or negative; all three cases arise in magnetohydrodynamics and are discussed in Chapter 2. The difference between phase and group velocities is perhaps most readily illustrated for the special case of one-dimensional propagation of a dispersive wave, where k = (0, 0, k) with kz = k. Consider a travelling wave of the form A0 cos(ωt − kz). Then the sum of two such travelling waves with the same amplitude A0 but different frequencies ω1 , ω2 and wavenumbers k1 , k2 is

A0 cos(ω1 t − k1 z) + A0 cos(ω2 t − k2 z) = 2A(z, t) cos 12 (ω2 + ω1 )t − 12 (k2 + k1 )z , (1.62) where

(1.63) A(z, t) = A0 cos 12 (ω2 − ω1 )t − 12 (k2 − k1 )z . cg =

This simple addition underlies the well-known acoustic phenomenon of beats.

26

General Principles

c

k

cg

eq

Figure 1.1 The geometry of the phase and group velocities of a wave propagating at an angle θ to a fixed direction. The phase velocity c here makes an angle θ measured anti-clockwise from the horizontal axis (representing the direction of the applied magnetic field); θ = 0 corresponds to the horizontal axis (and alignment with the applied magnetic field). The group velocity cg in general makes a different angle, φ, with the applied magnetic field. The unit vector eθ , which acts perpendicular to c, is also indicated.

Now suppose that k1 and k2 are almost equal: k1 ≈ k2 ≈ k. Then we see that the addition of two cosine waves of equal amplitude gives a cosine wave travel with almost the same speed, ω ω2 + ω1 ≈ , k2 + k1 k

(1.64)

as the original pair of waves. But the effective amplitude A(z, t) of the resulting disturbance is a slowly varying function of both time and space, and in particular is of much greater wavelength than the original pair. The slowly varying, long wavelength, amplitude moves with the speed ω2 − ω1 . k2 − k1

(1.65)

In the limit k1 → k, k2 → k, ω1 → ω, ω2 → ω, this becomes dω/dk. Thus the wave packet A(z, t) moves with the group speed cg = dω/dk.

2 Waves in a Uniform Medium

2.1 Introduction In this chapter we discuss the nature of magnetohydrodynamic waves in a uniform medium. As pointed out in Chapter 1, most astrophysical plasmas are far from uniform and so one might argue that a discussion of waves in a uniform medium has little or no relevance to such plasmas. In fact, an understanding of wave phenomena in a uniform medium is a necessary preliminary to any understanding of their behaviour in other, more complicated, media. As noted in Chapter 1, there are two fundamental speeds in magnetohydrodynamics, namely the Alfv´en speed cA and the sound speed cs . Consider a plasma of constant density ρ0 and constant pressure p0 within which is embedded a uniform magnetic field B0 of strength B0 . The two fundamental speeds are then  cA =

B20 μρ0

1/2

 ,

cs =

γ p0 ρ0

1/2 ,

(2.1)

for adiabatic index γ and magnetic permeability μ. The directionality of the applied magnetic field B0 immediately signals an important difference between the propagation of sound in a field-free medium and its propagation in the presence of a magnetic field. For in the absence of a magnetic field there are no preferred directions in a uniform plasma, each direction being much as any other direction. Any directionality is introduced purely in the perturbations about the equilibrium and is not a property of the equilibrium state itself. The medium is isotropic. Thus, sound waves may propagate equally in all directions and with the same speed. An imposed equilibrium magnetic field, however, defines a preferred direction and so the direction of the applied magnetic field is picked out as distinct from other directions; the medium is anisotropic. Accordingly, waves in a magnetic medium are directional, with the direction of the equilibrium magnetic field given special preference. We consider an equilibrium magnetic field that is orientated along the z-axis of a Cartesian coordinate system x, y, z. Let ez denote a unit vector aligned with the z-axis of our coordinate system. Then the equilibrium magnetic field is B0 = B0 ez ,

(2.2) 27

28

Waves in a Uniform Medium

for constant field strength B0 . What wave motions may we expect? Any disturbance that compresses the plasma will tend to propagate like a sound wave, to a greater or lesser extent depending upon how much the magnetic field is compressed. But it is possible to disturb the equilibrium state without compressing the plasma. Such a disturbance may bend magnetic field lines but not compress them. Bending magnetic field lines generates currents which act to oppose the bending through the Lorentz force, which then try to restore the equilibrium state. The result is an oscillation, much as bending an elastic string under tension generates tension forces which act to restore equilibrium and thus give rise to a wave. The Lorentz force will, in general, generate both tension forces and pressure forces. For, as noted in Chapter 1, the Lorentz force associated with a magnetic field B and current density j is j × B. For small disturbances about a uniform field B0 this results in a body force j × B0 , for generated current j; we are here neglecting squares of perturbations. This force may be decomposed into (see Chapter 1)   B0 B0 ∂B j × B0 = −grad Bz + , (2.3) μ μ ∂z where B now denotes the perturbation in the magnetic field and Bz is its component in the direction of the equilibrium magnetic field. The Lorentz force acts in a direction perpendicular to the applied field B0 . The first term on the right-hand side of equation (2.3) represents a magnetic pressure force (arising as it does from the magnetic pressure B · B/2μ in a field B, when perturbed about the uniform equilibrium). The second term on the right-hand side of equation (2.3) corresponds to the tension force. Evidently, if no compressive forces arise in the disturbance, then Bz = 0.

(2.4)

(Strictly Bz = constant, which we may impose to be zero at some location, such as infinity, and so zero everywhere.) The disturbances we are envisaging, which bend but do not compress field lines, propagate much as a wave on an elastic string. This is the Alfv´en wave and it propagates with the familiar speed of a wave on an elastic string, namely  speed =

applied tension force mass density per unit length

1/2 .

(2.5)

Here the applied tension force is B20 /μ per unit length of field in a medium of volume density ρ0 , and so the wave propagates with the Alfv´en speed cA . Disturbances that compress the plasma will propagate with different speeds in different directions, reflecting the anisotropy of the medium. Since there are now two pressure contributions, the usual plasma pressure and the magnetic pressure, there arises the possibility of these two forces acting either in unison or in opposition. When acting together, a rapidly moving disturbance arises; this is the so-called fast magnetoacoustic wave. When the two pressure forces act in opposition, there results a slow moving disturbance; this is the so-called slow magnetoacoustic wave.

2.2 Wave Equations

29

The Alfv´en speed also describes the magnetic contribution to compressive waves, so that such modes are characterized by the Alfv´en speed cA and the sound speed cs . The fast wave is then associated with the speed cf defined through c2f = c2s + c2A .

(2.6)

By contrast, the slow wave is associated with the speed ct defined by 1 1 1 = 2+ 2; 2 cs ct cA

that is, c2t =

c2s c2A c2s + c2A

.

(2.7)

Thus, while the fast wave speed cf is both super-sonic and super-Alfv´enic, the slow wave speed ct is both sub-sonic and sub-Alfv´enic. The suffix f in the definition of cf is used to give reference to the fast wave; the suffix t, reserved for the slow wave, is used because of the common occurrence of the speed ct in the description of waves in magnetic flux tubes, a topic of particular importance in solar plasmas (see Chapter 6). Of course, as expected from the directionality imposed on the atmosphere by the applied magnetic field, wave propagation is only approximately characterized by these two speeds and we must turn to a more detailed analysis of the waves (given in Section 2.2) to see more precisely how such effects influence our modes. Nonetheless, the speeds defined through equations (2.6) and (2.7) serve as useful representatives for the fast and slow wave speeds. The additive combination of speeds in definition (2.6) is natural enough, but that introduced through equation (2.7) appears somewhat unusual. In fact, it is a familiar combination in studies of waves in elastic tubes (such as a blood vessel or a garden hose) where the compressibility of a mode arises through the combination of two effects, one stemming from the natural compressibility of the medium (fluid) and the other from the elasticity of the confining tube (see Lighthill 1978). In the case of a blood vessel or garden hose the speed ct is made up of the sound speed in the fluid and the elastic speed of the tube’s membrane. In the case of a magnetic flux tube the elastic speed is the Alfv´en speed. We return to this topic in greater detail in Chapter 6. Thus, the propagation of sound – that is, the propagation of a compression in the medium – is described by two speeds, cf and ct , for two distinct modes. Altogether, we now have four speeds, cs , cA , ct and cf , that characterize the propagation of disturbances in a magnetic atmosphere, and these speeds are ordered thus: ct ≤ min(cs , cA ) ≤ max(cs , cA ) ≤ cf .

(2.8)

2.2 Wave Equations We turn now to a more detailed description of wave motion in the presence of a uniform equilibrium magnetic field, again taken to be B0 = B0 ez aligned with the z-axis of a Cartesian coordinate system x, y, z. The field strength B0 , plasma pressure p0 , density ρ0 and temperature T0 of the equilibrium state are all taken to be constants. Conditions are assumed to be ideal (i.e., no energy gain or loss mechanisms, such as viscosity or thermal conduction), and gravitational effects are ignored. (Gravitational aspects are explored in Chapter 9; non-ideal effects in Chapter 12.)

30

Waves in a Uniform Medium

To describe wave motions about this equilibrium we consider the linearized form of the equations of ideal magnetohydrodynamics described in Chapter 1. Thus, the equation of mass continuity (1.1) yields the linearized equation ∂ρ (2.9) + div ρ0 u = 0, ∂t where ρ and u now denote the perturbations in density and velocity. Perturbations in density ρ may be related to those in pressure p through the energy equation for isentropic disturbances (equation (1.10)), yielding ∂ρ ∂p = c2s ∂t ∂t

(2.10)

for sound speed cs . Turning to the momentum equation (1.2), its linearized form is ∂u (2.11) = −grad p + j × B0 , ∂t where the perturbed current density j is related to the perturbation B in the magnetic field through Ampere’s law (equation (1.3)): ρ0

μj = curl B.

(2.12)

It is convenient to introduce the perturbation in total pressure (plasma plus magnetic). We write 1 pT ≡ p + B0 · B, (2.13) μ the sum of the plasma pressure perturbation p and the perturbation in magnetic pressure. Then the linearized momentum equation (2.11) combined with Ampere’s law (2.12) produces ∂u 1 (2.14) = −grad pT + (B0 ·grad) B. ρ0 ∂t μ This form of the momentum equation reveals clearly that wave motions are driven by a combination of two forces, a pressure gradient (the first term on the right-hand side of equation (2.14)) and a tension force (the second term on the right-hand side of equation (2.14)). In turn, the pressure contribution pT results from the sum of the thermal pressure and the magnetic pressure. The perturbation B in magnetic field is linked with the motion u through the induction equation, which in ideal magnetohydrodynamics is the linearized equation ∂B = curl (u × B0 ). ∂t This equation is supplemented by the solenoidal condition div B = 0.

(2.15)

(2.16)

Equation (2.16) may be viewed as an initial condition, since if it is satisfied at any given instant then the induction equation ensures that it is satisfied at all subsequent times (Chapter 1).

2.2 Wave Equations

31

Now the system of linear equations (2.9)–(2.16) may be analysed in a variety of ways. Our preference is to retain the partial differential equation form as much as possible; that way, a degree of generality is preserved. There arises the question of which variables to work in terms of and which variables to regard as secondary, deduced once the primary variables are determined. Our choice is to use a set of variables first introduced by Lighthill (1960), involving the compressibility , the shear  in the direction of the applied magnetic field, and the component ωz of the fluid’s vorticity in the direction of the applied magnetic field. The total pressure perturbation pT is also a convenient variable to introduce. To see how these variables arise, we begin by noting that we may expand the right-hand side of the induction equation (2.15) by using the standard vector identity: curl (u × B0 ) ≡ u div B0 − B0 div u + (B0 ·grad) u − (u·grad)B0 ,

(2.17)

which holds for any vectors u and B0 . For a uniform field B0 , we have curl (u × B0 ) = −B0 div u + (B0 ·grad) u. Thus the induction equation (2.15) yields ∂B = B0 ∂t



 ∂u − ez  , ∂z

(2.18)

(2.19)

where we have introduced the compressibility , the divergence of the velocity field u = (ux , uy , uz ):  ≡ div u =

∂uy ∂uz ∂ux + + . ∂x ∂y ∂z

(2.20)

Equation (2.19) shows that the component B⊥ ≡ (Bx , By , 0) of magnetic field perpendicular to the equilibrium field is related to the perpendicular component u⊥ ≡ (ux , uy , 0) of the motion by ∂u⊥ ∂B⊥ = B0 , ∂t ∂z

(2.21)

whereas the component Bz of the perturbed magnetic field in the direction of B0 satisfies   ∂uy ∂Bz ∂ux = −B0 div u⊥ = −B0 + . (2.22) ∂t ∂x ∂y Returning to the momentum equation (2.14), we may differentiate it once with respect to time and substitute the expression (2.19) into the tension term. The result is     2 ∂ 2u ∂ ∂pT 2 ∂ u ρ0 2 = −grad − ez + ρ0 cA , (2.23) ∂t ∂z ∂t ∂z2 where we have introduced the Alfv´en speed cA (defined in (2.1)). The total pressure perturbation pT arising in equation (2.23) may be related to the velocity u. For, from the definition (2.13) of pT , we have ∂p B0 ∂B ∂pT = + · , ∂t ∂t μ ∂t

(2.24)

32

Waves in a Uniform Medium

which, when combined with the induction equation (2.19), yields   ∂pT ∂u ∂p B0 = + B0 · − ez  . ∂t ∂t μ ∂z

(2.25)

But from the linearized isentropic equation (2.10) and the mass conservation equation (2.9) we have 1 ∂pT = c2A  − c2f , ρ0 ∂t

(2.26)

where we have written the shear in the direction of the applied magnetic field as ≡

∂uz . ∂z

(2.27)

Thus, equation (2.23) becomes   2 ∂ 2u ∂ 2 2 ∂ u = c grad  + c − e − grad  . z A f ∂z ∂t2 ∂z2

(2.28)

Equation (2.28), which involves only u and its components, is the basic equation for the motion. The use of the variables ,  and u goes back to Lighthill (1960); they have been used also by Cowling (1976) and Roberts (1981b, 1991b). We may extract information from equation (2.28) in a number of ways. For example, the divergence of equation (2.28) yields ∂ 2 = c2f ∇ 2  − c2A ∇ 2 , ∂t2

(2.29)

where ∇ 2 (≡ div grad) denotes the Laplacian operator in three dimensions. Consider the components of equation (2.28). The z-component (corresponding to the component in the direction of the applied magnetic field) is ∂ ∂ 2 uz = c2s 2 ∂z ∂t

(2.30)

2 ∂ 2 2∂  = c . s ∂t2 ∂z2

(2.31)

and so

The component of equation (2.28) perpendicular to the applied field gives 2 ∂ 2 u⊥ 2 ∂ u⊥ − c = grad⊥ (c2f  − c2A ) A ∂t2 ∂z2   1 ∂pT , = −grad⊥ ρ0 ∂t

where grad⊥ denotes the gradient operator in the plane perpendicular to B0 :   ∂ ∂ , ,0 . grad⊥ ≡ ∂x ∂y

(2.32)

2.3 Plane Waves

33

Thus we see that the component ωz of vorticity ω ≡ curl u in the direction of the applied magnetic field is propagated with the Alfv´en speed: ∂ 2 ωz ∂ 2 ωz = c2A 2 , 2 ∂t ∂z

(2.33)

where ωz ≡ ez · curl u =

∂uy ∂ux − . ∂y ∂x

The other two components of vorticity are in general related to the compressibility , as is evident from the curl of equation (2.23). The three quantities ωz ,  and  are in fact sufficient to determine the motion u, so our system of equations reduces to the solution of equations (2.29), (2.31) and (2.33). Moreover, these three equations separate into two classes, one class involving  and  and the other involving ωz . It follows immediately from equation (2.33) that any disturbance that possesses vorticity in the direction of the applied magnetic field will propagate that component of vorticity one-dimensionally along the magnetic field with the speed cA . The quantities  and  both satisfy a fourth-order wave equation. Eliminating  between equations (2.29) and (2.31) yields 2 2 ∂ 4 2 2∂  2 2 2∂  − c ∇ + c c ∇ = 0. s A f ∂t4 ∂t2 ∂z2

(2.34)

The quantity  satisfies this wave equation also. It is of interest to examine more closely the velocity components. It may be seen in equation (2.32) that ux and uy , the components of the motion in the plane perpendicular to the applied field, satisfy a wave-like equation involving the Alfv´en speed. By contrast, the component uz of motion in the direction of the magnetic field satisfies a wave-like equation involving the slow speed ct . For, the z-component of equation (2.23) combined with equation (2.26) yields   2 ∂ 2 uz c2s ∂ 1 ∂pT 2 ∂ uz − c = − . (2.35) t ∂t2 ∂z2 c2f ∂z ρ0 ∂t Thus, the components of the motion perpendicular to the applied field satisfy an equation that is quite different from that satisfied by the component of the motion in the direction of the applied magnetic field. This is simply another manifestation of the anisotropy imposed on our system by the magnetic field.

2.3 Plane Waves The array of partial differential equations we have presented here allows us to determine, quite generally, the behaviour of waves in a magnetic system. However, it frequently proves convenient to examine those equations for disturbances that are represented as plane waves. Such a representation is possible because the coefficients arising in the partial differential equations presented above are in each case constants, and so the system of equations admits of a Fourier analysis. By a plane wave representation we mean that each perturbation may

34

Waves in a Uniform Medium

be expressed in the form f (ωt − k · r), for angular frequency ω, wave vector k = (kx , ky , kz ), position vector r = (x, y, z), and profile (wave shape) f . This corresponds to a disturbance 1 moving with a speed ω/k in the direction of k. Here k = |k| = (kx2 + ky2 + kz2 ) 2 . It is convenient to use a complex variable representation writing u(x, y, z, t) = u1 exp i(ωt − k · r), p(x, y, z, t) = p1 exp i(ωt − k · r),

B(x, y, z, t) = B1 exp i(ωt − k · r),

ρ(x, y, z, t) = ρ1 exp i(ωt − k · r),

(2.36)

where u1 and B1 are complex constant vectors, p1 , ρ1 are complex constants, and i2 = −1. The actual physical perturbations may be obtained by taking the real parts of the above complex representations, once the relationships between the various complex constants and vectors p1 , ρ1 , B1 and u1 are determined. It should be noted that we may equally well use a Fourier representation in which exp i(ωt + k · r) is everywhere used in place of exp i(ωt − k · r) in (2.36); this changes the sign of a number of intermediate expressions, though where operators such as ∂ 2 /∂t2 or ∇ 2 arise the final results are independent of the choice. Either variable form may be chosen, though of course it is important to be consistent. Our preference here is for the form (2.36). With the representations (2.36), the various differential operators arising in the perturbation equations take on simple algebraic scalar and vector forms. Thus, ∂2 ≡ −ω2 , ∂t2 div ≡ −ik·,

∂2 ≡ −kz2 , ∂z2 grad ≡ −ik,

∇ 2 ≡ −k2 , curl ≡ −ik × .

(2.37)

These algebraic scalar and vector representations, involving the scalar product (·) for div and the vector product (×) for curl, convert the partial differential operators into relatively simple forms. For example, the equation of continuity (2.9) becomes ωρ = ρ0 k · u = ρ0 (kx ux + ky uy + kz uz ),

(2.38)

where we have cancelled a common factor i. Thus, the density perturbation is related to the scalar product of k with u. Notice that the above equation gives a relationship between the complex amplitudes ρ1 and u1 if we simply cancel the common exponential factor contained in the perturbations ρ and u. Similar algebraic relationships arise from the other partial differential equations that arise. Plane wave representations afford us a means of giving simple physical interpretations of the equations we are developing. For example, the induction equation expresses the concept of frozen-in flux (Chapter 1), whereby the magnetic field lines are tied to the motion. Accordingly, a component of motion perpendicular to the applied field carries the field lines with it, whereas a motion along the field leaves the field unchanged. To illustrate the former, consider the consequence of a simple displacement ξ of fluid which is everywhere aligned with the x-axis. The motion u is related to the displacement through u = ∂ξ/∂t. To be specific, consider a displacement ξ = ξx ex , where ξx (z, t) = ξ1 sin(ωt)sin(kz z),

(2.39)

2.4 Sound Waves

35

for real constant ξ1 . We are envisaging a sinusoidal motion perpendicular to the applied field, with a wavelength 2π/kz . Such a motion displaces the fluid sinusoidally in a direction perpendicular to the z-axis and so will distort the field lines in a similar sinusoidal pattern. A point such as z = 0, which suffers no displacement, nonetheless sees a change in the field at that location. Indeed, according to the induction equation, expressed through equation (2.19), the displacement ξ gives rise to a perturbation Bx given by Bx = B0

∂ [ξ1 sin(ωt)sin(kz z)] = B0 kz ξ1 sin(ωt)cos(kz z). ∂z

(2.40)

Thus, a point such as z = 0, which is a stationary point for the motion, sees the development of a field Bx in response to the shearing motion either side of the stationary point. A point such as kz z = π/2, however, sees a maximum displacement but no change in the magnetic field. Altogether, then, the field lines are distorted in a sinusoidal pattern determined by the motion. We return now to our analysis of the perturbation equations. In order to motivate and guide us in this analysis, we examine first the simpler problem of motions in the absence of a magnetic field. In a field-free atmosphere the equations of magnetohydrodynamics simply describe sound waves, to which we now turn.

2.4 Sound Waves We consider first the special case of a field-free atmosphere. In that way we are able to set out the properties of sound waves unencumbered by the complications of magnetism, thus providing some guidance as to the expected properties of such waves when a magnetic field is additionally present. In the absence of magnetic forces, any compression in the atmosphere is propagated as a sound wave and since there is now no preferred direction such propagation will be isotropic. In the absence of a magnetic field, our system of linear equations describing smallamplitude perturbations about a uniform equilibrium is simply ∂u ∂ρ ∂p ∂ρ ρ0 + ρ0 div u = 0, = −grad p, = c2s , (2.41) ∂t ∂t ∂t ∂t describing mass conservation, momentum balance, and isentropic energy exchange. Differentiating the isentropic equation with respect to time t and eliminating ρ through use of the continuity and momentum equations, yields ∂ 2p = c2s ∇ 2 p. (2.42) ∂t2 This is the three-dimensional wave equation for sound. The pressure perturbations p (and therefore the density perturbations ρ) are propagated isotropically with the sound speed cs . Similarly, the divergence  of the velocity field is propagated isotropically with the sound speed. However, the vorticity curl u is not propagated: in fact, it follows immediately from the momentum equation (on noting that curl grad ≡ 0) that ∂ (curl u) = 0. ∂t

(2.43)

36

Waves in a Uniform Medium

Vorticity, then, remains unchanged during the passage of a sound wave: any vorticity introduced into the system through an initial perturbation remains unchanged in time; and if the vorticity is initially zero, then it remains zero for all time. It is convenient at this stage to introduce the plane wave representation (2.36). Then it follows from equation (2.42) that any non-zero pressure perturbation has a frequency ω and wavenumber k (= |k|) that are related thus: ω2 = k2 c2s = (kx2 + ky2 + kz2 )c2s .

(2.44)

This is the dispersion relation of a sound wave. It describes how the frequency ω is related to the wavenumber k of the disturbance. The two solutions of equation (2.44), namely ω = −kcs and ω = kcs , correspond to propagation in the directions −k and k, respectively. The disturbance moves with speed cs , in either direction. Since there is no preferred direction in a uniform atmosphere devoid of magnetic field, the wave moves with the speed cs whatever its direction of propagation. Also, since the uniform field-free medium has no spatial lengthscale of its own, other than that introduced by the perturbation, all waves travel with the same speed, whatever their wavelength (= 2π/k): propagation is non-dispersive. The phase speed c ≡ ω/k, the speed with which an individual wave moves (see Chapter 1), is thus the same in all directions and is independent of the magnitude of the wave vector (and so of wavelength): c = cs . Consider a sound wave travelling in the k direction. Its phase velocity c is thus c = cs ek , where ek (= k/|k|) denotes a unit vector in the direction of propagation, k. The group velocity, cg , is defined in general by (see Chapter 1)   ∂ω ∂ω ∂ω ∂ω cg ≡ , , ≡ . (2.45) ∂k ∂kx ∂ky ∂kz Hence, for sound waves with ω = kcs and k = |k| = (kx2 + ky2 + kz2 )1/2 we have   kx ky kz cg = cs , , = cs ek , k k k

(2.46)

and so the group velocity is in the direction of propagation and is of magnitude cs . Thus, sound waves have cg = c, with cg = c = cs . What, then, are the characteristics of a sound wave in a uniform medium? We have seen that it propagates isotropically with the sound speed and without dispersion. What of its perturbations? From the isentropic relation in equation (2.41) we see that the pressure and density perturbations in the wave are related by p = c2s ρ,

(2.47)

ρ0 ωu = pk.

(2.48)

whereas the momentum equation gives

Hence u is parallel to k; that is, motions are longitudinal, being parallel to the direction of propagation. In terms of the individual components of the motion, we have ux =

kx p , ω ρ0

uy =

ky p , ω ρ0

uz =

kz p , ω ρ0

(2.49)

2.5 Magnetohydrodynamic Waves

37

with ω2 = k2 c2s = (kx2 + ky2 + kz2 )c2s . For a wave travelling in the direction of k, so that ω = kcs , we have   ρ p 1 p uz = = (2.50) = . 2 ρ0 γ p0 cs ρ0 cs For the case of an ideal plasma, the gas law (see equation (1.11) in Chapter 1) shows that p ρ T = + , p0 ρ0 T0

(2.51)

where T0 denotes the temperature in the equilibrium state and T the temperature perturbation. Thus, the relative change in temperature is related to the relative change in density by ρ T = (γ − 1) . T0 ρ0

(2.52)

The above relationships characterize the behaviour of sound waves.

2.5 Magnetohydrodynamic Waves The character of waves in the presence of a uniform magnetic field follows from the equations developed in Section 2.2. Consider a plane wave representation of eqs (2.33) and (2.34). A plane wave perturbation defines a plane through the vectors B0 and k. In our choice of a Cartesian coordinate system we are free to choose the orientation of the axes. We have chosen to align the z-axis with the applied magnetic field B0 , which leaves unspecified the orientation of the x, y-axes in the plane perpendicular to the applied field. It is convenient to choose their orientation so that the vectors k and B0 lie in the xz-plane, and then ky = 0 and k = (kx , 0, kz ),

k2 = kx 2 + kz2 .

There is no loss of generality in this choice, given that we are examining plane wave disturbances. The direction lying in the plane of k and B0 but perpendicular to k frequently arises. Denote by θ the angle that the propagation vector k makes with the applied magnetic field B0 . Then kx = k sin θ and kz = k cos θ . Write 1 (2.53) (kz , 0, −kx ) = (cos θ , 0, − sin θ ); k eθ is a unit vector perpendicular to k, pointing in the direction of increasing θ . In terms of the plane wave representation (2.36), equations (2.33) and (2.34) read  2 ω − kz2 c2A ωz = 0 (2.54) eθ =

and  4 ω − k2 c2f ω2 + k2 kz2 c2s c2A  = 0.

(2.55)

There are two possibilities to consider. Suppose that ωz = 0; then, to satisfy equation (2.54), we require that ω2 = kz2 c2A and equation (2.55) then implies that  = 0. Thus, we

38

Waves in a Uniform Medium

have a disturbance that propagates one dimensionally with the Alfv´en speed and involves no compression of the plasma (since  = 0). This is the Alfv´en wave. On the other hand, if ωz = 0 then (with  = 0) the expression in square brackets, [ ], in equation (2.55) must vanish. This latter possibility gives the magnetoacoustic wave branch. To describe the properties of the various modes that stem from these two branches, of non-compressive ( = 0) and compressive ( = 0) disturbances, we consider each in turn. We examine first the properties of the dispersion relation, determining the phase speed and group velocity, and then go on to examine the perturbations that characterize the various modes.

2.6 Alfv´en Waves 2.6.1 Phase Speed and Group Velocity Alfv´en waves are characterized by the dispersion relation ω2 = kz2 c2A ,

(2.56)

which holds whenever the vorticity ωz is non-zero and the compression  is zero. In terms of the angle θ that the propagation vector k makes with the applied magnetic field B0 , the Alfv´en wave dispersion relation is ω2 = k2 c2A cos2 θ,

(2.57)

where kz = k cos θ , k = |k|. Comparing with the dispersion relation (2.44) for sound waves, we see a sharp distinction: the isotropic nature of the sound wave, reflected in its dispersion relation by the presence of k2 , contrasts with the anisotropic dispersion relation of the Alfv´en wave, reflected in the singular presence of kz , the component of the wave vector in the direction of the applied magnetic field. We may further exhibit this anisotropy by noting the expression for the phase speed c (≡ ω/k) of an Alfv´en wave: c2 = c2A cos2 θ.

(2.58)

Evidently, then, the phase speed of an Alfv´en wave cannot exceed the Alfv´en speed cA . Moreover, the Alfv´en wave is unable to propagate across the magnetic field, for c2 = 0 and ω2 = 0 when θ = π/2. However, whatever the angle of propagation (θ = π/2), the group velocity cg of an Alfv´en wave is directed along (or anti-parallel to) the applied magnetic field and is of magnitude cA : ∂ω (2.59) = ± (0, 0, cA ) = ± cA ez . ∂k The ‘+’ sign applies for a wave moving in the direction of the applied field (for which − 12 π < θ < 12 π) and the ‘−’ sign for a wave going anti-parallel to the applied field (for which 12 π < θ < 32 π). The relations (2.58) and (2.59) can be exhibited in a geometrical fashion. For the phase speed we may plot c(θ ) as a function of θ in a polar representation. This is hardly necessary for the Alfv´en wave relation, which is of a fairly simple form. However, it proves cg ≡

2.6 Alfv´en Waves

39

particularly useful in more complicated situations – such as with magnetoacoustic waves – and serves to provide a convenient means of making visual comparisons between different wave modes. We consider, then, the phase speed c as a function of the angle θ that the direction of propagation k makes with a fixed line (conveniently chosen to be the direction of the applied magnetic field, the z-axis). This provides a polar representation of c = c(θ ), with c playing the role of the coordinate r in the standard polar coordinates (r, θ ). For each angle θ, we plot the point (r, θ ) = (c(θ ), θ ) by measuring radially a distance c(θ ) from the origin r = 0. With c being a function of θ the curve that we construct will generally be non-symmetric. However, since there is symmetry about the axis defined by the applied magnetic field – here the horizontal axis of the polar plot – a surface can be constructed simply by rotation of our curve about the axis of the applied field. Diagrams of this form were first discussed by Friedrichs and Kranzer (1958). The phase speed c of an Alfv´en wave propagating at an angle θ to the applied magnetic field is given by (see eq (2.58)) c(θ ) = cA |cos θ|.

(2.60)

For the angle range − 12 π ≤ θ ≤ 12 π, over which cos θ is positive or zero, this provides the polar representation of a circle of radius 12 cA with centre at (r = 12 cA , θ = 0). For the range of angle wherein cos θ is negative, namely 12 π < θ < 32 π , equation (2.60) produces another circle of radius 12 cA , save its centre is at c = 12 cA , θ = π . Thus, altogether we obtain a pair of circles touching at the origin. Rotation about the axis of symmetry, the zaxis, then produces a pair of touching spheres. This is the phase speed diagram of the Alfv´en wave; it is shown in Figure 2.1. Polar distance in Figure 2.1 is plotted in units of the Alfv´en speed. For example, with θ = 0 we obtain a polar distance of r = 1, corresponding to c/cA = 1 (i.e., c = cA ). Propagation in the opposite direction produces c = −cA . For comparison, Figure 2.2 shows the corresponding diagram for a sound wave. For a sound wave we have (from equation (2.44)) c2 = c2s ; the isotropy means there is no dependence on θ , the phase speed of a sound wave being the same at all angles θ to a fixed direction. Consequently the polar diagram for a sound wave is simply a circle of radius cs and centre the origin (c = 0); rotation about the axis of symmetry produces a sphere of radius cs and centre the origin. Thus spheres arise in the description of both sound waves and Alfv´en waves, but the displaced position of the spheres in the case of the Alfv´en wave points out that the two cases are in fact very different. This difference is even more marked in the diagram for the group velocity. The group velocity of the Alfv´en wave is given by equation (2.59). For propagation in the direction of the magnetic field, we have cg = cA ez . A diagram representing this information is not particularly useful, as the relation is explicit in any case. However, if we do represent this in diagrammatic form, plotting the polar distance cg (= |cg |) in the direction of cg , for each angle θ of the propagation direction, we simply obtain a point with coordinates r = cA , θ = 0, for cg is always directed along the magnetic field. Propagation in a direction opposite to B0 , corresponding to an angle range of 12 π < θ < 32 π , for which cg = −cA ez , produces another point, at r = cA , θ = π . Thus, altogether, for the Alfv´en wave we have the pair of points r = cA , θ = 0 and π. This is in sharp comparison to a sound wave, which

40

Waves in a Uniform Medium

Figure 2.1 The phase speed polar diagram for the Alfv´en wave, consisting of a pair of touching circles aligned with the applied magnetic field. The direction of the applied magnetic field is taken to correspond with the horizontal axis (θ = 0) in the polar diagram, with the angle θ in polar coordinates measured anti-clockwise from θ = 0; the vertical axis corresponds to θ = π/2. The plot shows in polar coordinates the wave phase speed c(θ ) as a function of θ. Note that c = 0 when θ = π/2, i.e., the wave is unable to propagate across the applied magnetic field. The phase speed c is measured in units of the Alfv´en speed cA (so that the point (1, θ = 0) corresponds to c = cA , θ = 0). Rotation of the circles about the axis of symmetry (the vertical axis) produces a pair of touching spheres.

produces a circle (sphere when rotated about the axis of symmetry) of radius cs , centred at the origin, just as arose for the phase speed.

2.6.2 Perturbations in an Alfv´en Wave Turning now to the perturbations, we note first that since  = 0 in an Alfv´en wave, then there are no pressure or density changes either. This follows from the mass continuity and isentropic equations, which give ∂p/∂t = ∂ρ/∂t = 0 when  = 0. Thus, pressure and density variations are independent of time and so if they are zero initially then they remain zero for all time. Accordingly, we take p = ρ = 0.

(2.61)

The Alfv´en wave is thus a silent disturbance. Also, equation (2.30) with  = 0 shows that uz = 0.

(2.62)

Thus, the Alfv´en wave involves no motions along the imposed magnetic field. It follows, then, from the induction equation (2.19), that the component of the perturbed field in the

2.6 Alfv´en Waves

41

Figure 2.2 The phase speed diagram for a sound wave, consisting of a single circle of radius cs . The plot is a polar representation of c(θ ) as a function of the polar angle θ = 0. The phase speed c is measured in units of the sound speed cs . Rotation of the circle about the vertical axis produces a sphere.

direction of the imposed field is zero and, from equation (2.13), that the total pressure perturbation is also zero: Bz = 0,

pT = 0.

(2.63)

The Alfv´en wave, then, must be driven by magnetic tension forces: any bends in the magnetic field lines are straightened by the tension in the field; any imposed torsion/twisting of the magnetic field generates a restoring tension force. The Alfv´en wave, then, consists of an incompressible motion u = u⊥ = uAlfv´en and a perturbed magnetic field B = B⊥ that lie solely in the plane perpendicular to the applied magnetic field. From equations (2.14) and (2.21) we see that both u⊥ and B⊥ satisfy the one-dimensional wave equation: 2 ∂ 2 u⊥ 2 ∂ u⊥ = c , A ∂t2 ∂z2

2 ∂ 2 B⊥ 2 ∂ B⊥ = c . A ∂t2 ∂z2

(2.64)

With plane wave vector k = (kx , 0, kz ), the constraint that div u = 0 leads to kx ux = 0, and so ux = 0 (for kx = 0). In other words, the motion in an Alfv´en wave is in the y-direction and (as equation (2.21) shows) the perturbed magnetic field is also in the ydirection. Thus, for kx = 0 we have uAlfv´en = (0, uy , 0),

B⊥ = −

kz B0 uAlfv´en , ω

(2.65)

and so the motion uAlfv´en associated with an Alfv´en wave is in the direction k × B0 and the perturbed field is in the opposite direction. Hence the Alfv´en wave is both transverse to the direction of propagation k and transverse to the applied magnetic field: uAlfv´en is perpendicular to both k and B0 . The actual magnitude uy of the motion is, of course, arbitrary since we are dealing with a linear system; we may specify at will the magnitude

42

Waves in a Uniform Medium

of any perturbation and then determine from the linear equations the relative magnitudes of all other perturbed quantities. An interesting observation follows from equation (2.65). If we calculate the magnetic energy density, B2 /2μ, of the perturbation magnetic field B, then we find that it equals the kinetic energy density, 12 ρ0 |u|2 , of the motion: B2 /2μ = 12 ρ0 |u|2 . The Alfv´en wave gives rise to an equipartition of energy between the energy of the perturbed magnetic field and the energy of the motion. Motions in the Alfv´en wave disturb the magnetic field, bending the field lines in accordance with the prescription of frozen-in flux (Chapter 1); the result is that a current j is generated which, by producing a Lorentz force j × B0 , acts so as to oppose the motion that bends the field lines. The magnetic perturbations B⊥ generated in an Alfv´en wave are given by equation (2.65) and, through Ampere’s law (2.12), are associated with a current j given by

u  kz y (2.66) eθ , μj = −ik × B⊥ = i uy B0 (−kz , 0, kx ) = −ikz B0 ω c where eθ (defined in equation (2.53)) is perpendicular to k. Thus, in general, the Alfv´en wave carries a current with non-zero component in the direction of the applied field; in other words, the Alfv´en wave supports a field-aligned current. This turns out to be a distinctive property of the Alfv´en wave. The presence of the factor of ‘i’ in the expression for the current j indicates a phase change in the temporal behaviour of the perturbation when compared with the motion u in the Alfv´en wave. For example, a motion uy = u0 cos(ωt − k · r) of amplitude u0 , corresponding to the real part of the complex representation (2.36), leads to a current j with time dependence sin(ωt − k · r) = cos(ωt − k · r − π/2). Finally, we may readily determine the electric field E and the motion’s vorticity ω. The electric field E = −u×B0 = −uy B0 ex , and so is in the negative x-direction (for uy B0 > 0). The vorticity ω = iuy (kz , 0, −kx ) is anti-parallel with j.

2.7 Magnetoacoustic Waves 2.7.1 Dispersion Relation Consider now the compressive disturbances. With  = 0, equation (2.55) leads to a quadratic equation for ω2 : ω4 − k2 c2f ω2 + k2 kz2 c2s c2A = 0.

(2.67)

This is the dispersion relation for magnetoacoustic waves. Like the relation for Alfv´en waves, it too exhibits anisotropy with kz2 being singled out in one of the terms, though like the relation for sound waves it also has the isotropic term k2 . This mixture of isotropy and anisotropy is reflected in the modes given by equation (2.67). Evidently, the two roots (in ω2 ) of equation (2.67) are positive (as we may expect on physical grounds, since no source of instability – corresponding to negative ω2 – here exists) for their product is k2 kz2 c2s c2A , and so positive, whereas their sum is k2 c2f . Thus, magnetoacoustic modes have phase speeds that cannot exceed cf = (c2A + c2s )1/2 , the fast magnetoacoustic speed.

2.7 Magnetoacoustic Waves

43

The primary quantity we want from equation (2.67) is ω as a function of k. However, there is interest too in the component of phase speed in the direction of the applied field, since this is singled out by the anisotropy in the system. The component of phase speed in the direction of the applied field is ω/kz , which follows from equation (2.67) if rearranged 2 ≡ k2 − k2 . Specifically, we have to determine k⊥ z 2 = k⊥

(ω2 − kz2 c2s )(ω2 − kz2 c2A ) c2f (ω2 − kz2 c2t )

.

(2.68)

2 > 0, it follows immediately that the square of the slow mode’s longitudinal Since k⊥ phase speed, ω2 /kz2 , lies between c2t and the minimum of c2s and c2A , whereas the fast mode has a longitudinal phase speed squared that is in excess of the maximum of c2s and c2A . In terms of the phase speed c (≡ ω/k), the dispersion relation (2.67) may be rewritten in the form

c4 − c2f c2 + c2t c2f cos2 θ = 0.

(2.69)

It is important to notice that the magnetoacoustic dispersion relation is symmetric in cs and cA , and so its properties are also symmetric in these speeds. The roots c2 = c2+ and c2 = c2− of the dispersion relation (2.69) are evidently such that c2+ + c2− = c2f ,

c2+ c2− = c2t c2f cos2 θ,

(2.70)

and so satisfy 0 ≤ c2 ≤ c2f for each of the roots. The corresponding frequencies squared 2 = k2 c2 and ω2 = ω2 = k2 c2 . The minus (−) root gives the slow are ω2 = ω+ + − − magnetoacoustic wave, the plus (+) root the fast magnetoacoustic wave, so 0 ≤ c2− ≤ c2+ ≤ c2f .

(2.71)

For θ = 0 the product of the roots is zero, and so c2− = 0 whilst c2+ reaches its maximum value of c2f . Thus, the slow wave is unable to propagate orthogonally to the applied magnetic field, a property it shares with the Alfv´en wave. Notice that equation (2.69) may be rewritten in the form (c2 − c2s )(c2 − c2A ) = c2s c2A sin2 θ,

(2.72)

revealing that c2− ≤ min (c2s , c2A ),

c2+ ≥ max (c2s , c2A ).

(2.73)

In other words, the phase speed c of the slow wave is less than or equal to the smaller of cs and cA , while the phase speed of the fast wave is greater than or equal to the larger of cs and cA . The phase speed relation (2.69) may, of course, be solved explicitly, giving the roots ⎡ 1/2 ⎤  2 1 c ⎦. c2 = c2± = c2f ⎣1 ± 1 − 4 t2 cos2 θ (2.74) 2 cf

44

Waves in a Uniform Medium

It is interesting to note in passing that we can write these two solutions in a compact form if we introduce a (real) angle 2α by writing (see, for example, Kendall and Plumpton 1964, p. 128) sin 2α = 2

cs cA ct cos θ = 2 cos θ . 2 cf cs + c2A

(2.75)

Then we can write the phase speed squared as c2 = c2f cos2 α,

c2 = c2f sin2 α

(2.76)

for the fast and slow magnetoacoustic waves. Now it is evident from the form of eq (2.74) that the fast wave, coming from the sum of the two factors in the dispersion relation, is only mildly influenced by the anisotropy imposed by the applied magnetic field, manifest in the occurrence of θ in the magnetoacoustic dispersion relation. The slow wave, however, steming from the difference in the two factors in eq (2.74), is strongly dependent upon the angle θ . The contrasting behaviour in the fast and slow waves is readily apparent when   c2 (2.77) 4 t2 cos2 θ  1. cf For under this circumstance the two roots of eq (2.74) are approximately c2 ≈ c2t cos2 θ

(2.78)

c2 ≈ c2f − c2t cos2 θ

(2.79)

for the slow mode, and

for the fast mode. Thus, the slow wave’s speed is less than ct and falls to zero at θ = π/2, whereas the fast wave propagates almost isotropically with a speed that is slightly below cf . The circumstances envisaged in the inequality (2.77) are rather wide; for they pertain whenever θ approaches π/2, thus reducing cos2 θ , and they pertain whenever 4c2t  c2f . This latter possibility arises whenever c2s and c2A are widely separated in magnitude, for c2t cannot exceed the minimum of c2s and c2A , and c2f is always greater than either of these speeds. Thus, whenever either c2s  c2A or c2s  c2A pertains, then the approximate expressions (2.78) and (2.79) for √ the slow and fast modes hold to a reasonably = 10cs , broadly typical of the solar corona, then high accuracy. For example, with c A √ √ giving 4c2t /c2f = 40/121 (≈ 0.331); and with cA = 10cs , ct = 10/11cs and cf = 11cs ,√ √ then ct = 100/101cs and cf = 101cs , giving 4c2t /c2f = 400/10201 (≈ 0.039). Finally, consider the two special cases of propagation perpendicular and parallel to the applied field. For perpendicular propagation, θ = π/2 and the slow wave is unable to propagate (ω2 = 0, c2 = 0) whereas the fast wave propagates at its maximum speed, cf . For parallel propagation, θ = 0 and we find that the relation (2.69) factorizes to give c2 = c2s ,

c2A ,

yielding a slow wave with c2 = min (c2s , c2A ) and a fast wave with c2 = max (c2s , c2A ).

2.7 Magnetoacoustic Waves

45

2.7.2 Phase Speed Diagrams The phase speeds of the magnetoacoustic waves follow from Equation (2.74), which gives c = c(θ ) for the two modes. Diagrams of this form for the magnetoacoustic waves were first discussed by Friedrichs and Kranzer (1958). There are three cases to consider, corresponding to whether cs < cA , or cs = cA , or cs > cA . In fact, since the dispersion relation (2.74) is symmetric in cs and cA , the two cases cs < cA and cs > cA may be discussed as one – say for cs < cA – with the other case following immediately by interchanging the speeds cs and cA . The third case, cs = cA , may be obtained as the limit of cA → cs from either of the cases cA > cs or cA < cs . But it is also of interest to examine cA = cs ab initio, for then its special features become more transparent; the special case cA = cs is treated in Section 2.8. Figures 2.3 and 2.4 show the behaviour of the phase speeds for typical cases, Figure 2.3 representative of the case cA > cs and Figure 2.4 representative of cA < cs . In fact, as noted above, because of the symmetry in the magnetoacoustic dispersion relation (2.69), the diagrams for cs < cA and for cs > cA are in fact the same except that the labels involving cs

Figure 2.3 The phase speed diagram for the fast and slow magnetoacoustic waves when cA > cs . √ The polar diagram shows c(θ ), drawn for cA = 5cs /4 (for which cf = cs 41/4 ≈ 1.601 cs ), representative of the case cA > cs . The phase speed c is plotted in units of the sound speed cs (so unity on the horizontal axis means c = cs ). The angle θ is measured anti-clockwise from the direction of the applied magnetic field, corresponding here to the horizontal axis, so that θ = 0 corresponds to alignment with the applied magnetic field and θ = π/2 is perpendicular to the applied field. Symmetry in the magnetoacoustic dispersion relation (2.69) means that the case cA < cs may be deduced from the case cA > cs simply by an appropriate interchange of labels cs and cA . The outer curve corresponds to the fast wave, and has c = max (cs , cA ) when θ = 0, and c = cf when θ = π/2 (corresponding to propagation of the fast wave perpendicular to the applied magnetic field); the slow wave is unable to propagate when θ = π/2. Comparing with Figure 2.1, we note the similarity of the fast wave with a pure sound wave and the slow wave with an Alfv´en wave.

46

Waves in a Uniform Medium

Figure 2.4 The phase speed diagram for the fast and slow √ magnetoacoustic waves when cA < cs . The diagram is drawn for cA = 4cs /5, for which cf = cs 41/5 ≈ 1.281 cs . The phase speed c(θ ) is plotted in units of the sound speed cs .

and cA are interchanged. The fast wave gives a spheroidal shape, centred on the origin and elongated in a direction perpendicular to the applied magnetic field (so perpendicular to the horizontal axis here). The slow wave gives rise to a pair of touching spheroids, somewhat resembling the Alfv´en wave. Notice that if either cA  cs or cA  cs then the simplifications (2.78) and (2.79) arise, and so c ≈ cf and c ≈ ct cos θ. Hence, in the extremes of either a strong magnetic field or a weak magnetic field, both the fast wave and the slow wave have phase speed diagrams that are made up of spheres. For the fast wave, its sphere is centred on the origin with a radius cf and propagation is essentially isotropic and similar to a sound wave. (If we allow for the small correction introduced by the term c2t cos2 θ in equation (2.79), then we may see that the sphere is in fact slightly elongated in a direction perpendicular to the applied field.) The phase speed diagram for the slow wave has touching spheres (circles in the plane) of radii 1 1 1 2 ct with centres at (c, θ ) = ( 2 ct , 0) and ( 2 ct , π ); this is similar to the phase speed spheres of an Alfv´en wave, save that ct (and not cA ) arises here. Accordingly, in either a strong (cA  cs ) or a weak (cA  cs ) field, the slow wave propagates in a strongly anisotropic manner, similar to an Alfv´en wave.

2.7.3 Group Velocity of Magnetoacoustic Waves General Features The group velocity cg = ∂ω/∂k may be calculated directly from the dispersion relations we have developed. However, it is of interest to argue from the general formalism presented

2.7 Magnetoacoustic Waves

47

in Chapter 1. The group velocity cg may then be written in terms of the phase velocity c (≡ c(θ )ek ) and the rate of change of the phase speed c(θ ) with respect to the angle θ :   dc cg = c + (2.80) eθ , dθ where the unit vector eθ = (cos θ, 0, − sin θ ) is perpendicular to k and points in the direction of increasing θ . Thus we see that the group velocity cg is the vector addition of the phase velocity c and the component dc/dθ perpendicular to ek . It is convenient to introduce two angles to describe the properties of cg . Denote by φ the angle that cg makes with the applied magnetic field and by ψ the angle that cg makes with the direction of propagation, ek ; then ψ = φ − θ. See Figure 1.1 in Chapter 1. From equation (2.80), we have 1 dc , c dθ and then the group speed cg (= |cg |) is given by tan ψ =

(2.81)

c2g = c2 sec2 ψ.

(2.82)

Evidently the group speed is greater than or equal to the phase speed: cg ≥ c. We may evaluate the above expressions in terms of the dispersion relations for the various modes. But before considering the magnetoacoustic modes it is of interest to apply equations (2.81) and (2.82) to the sound and Alfv´en waves discussed earlier. The dispersion relation for the sound wave gives c as independent of angle θ and so cg = c and therefore c2g = c2s (as obtained before). For an Alfv´en wave propagating in the direction of the magnetic field we have c = cA cos θ, with −π/2 ≤ θ ≤ π/2. Hence, dc/dθ = −cA sin θ and tan ψ = − tan θ; thus ψ = −θ. Hence, an Alfv´en wave propagating in the direction of the applied field (the ez direction) has group velocity cg = cA ez .

(2.83)

This is the result obtained earlier, which is in fact obvious from the form = of the dispersion relation. Turning to the magnetoacoustic waves, we note that the dispersion relation (2.69) may be differentiated to obtain dc/dθ: ω2

c2s c2A dc cos θ sin θ . = dθ c(2c2 − c2f )

kz2 c2A

(2.84)

Hence tan ψ =

c2s c2A 2c2 (2c2 − c2f )

sin 2θ .

(2.85)

Application of the dispersion relation (2.69) allows us to rewrite equation (2.85) in alternative forms: tan ψ =

c2s c2A (c4

− c2s c2A cos2 θ )

cos θ sin θ =

(c2

c2t cos θ sin θ . − 2c2t cos2 θ )

(2.86)

48

Waves in a Uniform Medium

The first form of ψ given in equation (2.86) allows a compact representation of the angle φ that cg makes with the applied magnetic field: with φ = ψ + θ , we obtain   c4 tan φ = tan θ . (2.87) c4 − c2s c2A The chief interest in formulas (2.85)–(2.87) is that they tell us by how much the group velocity cg departs from the directions k and B0 . We have noted earlier that the fast wave is somewhat akin to a sound wave; since for a sound wave its group velocity cg is parallel to k, we expect that for a fast wave its group velocity will be broadly aligned with k. This is in fact the case. Similarly, we have noted earlier that the slow wave is somewhat akin to an Alfv´en wave, and that the group velocity of an Alfv´en wave is precisely aligned with B0 ; thus we expect that the group velocity of a slow wave is broadly aligned with B0 . This too proves to be the case. In summary, then, for the fast magnetoacoustic wave the group velocity cg is closely aligned with the propagation vector k, whereas in the slow magnetoacoustic wave cg is closely aligned with the applied magnetic field B0 . To see all this in greater detail, consider first the fast magnetoacoustic wave. For the fast wave, c2 ≥ max (c2s , c2A ) and so c4 > c2s c2A cos2 θ. Thus, from equation (2.86) we find that ψ > 0 for 0 ≤ θ ≤ π/2. In other words, in Figure 1.1 in Chapter 1 the group velocity cg lies to the left of the phase velocity c. Now as θ → 0, c2 → max (c2s , c2A ), c2g → c2 and (from equation (2.86)) ψ → 0. Also, as θ → π/2 we see that c2 → c2f and ψ → 0. Thus the angle ψ is zero at both θ = 0 and π/2 and evidently attains a maximum at some intermediate angle. In summary: for the fast wave,  cf ek as θ → π/2, cg → (2.88) max (cs , cA )ek as θ → 0. Turning now to the slow magnetoacoustic wave, noting that c2 ≤ min (c2s , c2A ) we see from equation (2.86) the angle ψ is negative: Figure 1.1 applied to the slow wave gives a group velocity cg that lies to the left of the phase velocity c. In fact, the angle φ that cg makes with the applied field is also negative (for 0 < θ < π/2) since, with c4 < c2s c2A , equation (2.87) shows that φ has the opposite sign to θ . Thus, Figure 1.1 applied to the slow wave gives a group velocity lying to the left of B0 , even though the phase velocity points to the right. As θ → 0, c2 → min (c2s , c2A ) and so (from equations (2.85) and (2.87)) ψ → 0 and φ → 0 through negative values; also (from equation (2.82)), c2g → c2 . As θ → π/2, we have (from equation (2.78)) c2 → c2t cos2 θ (→ 0) and (from equation (2.86)) ψ → −π/2; so (from equations (2.82) and (2.87)) c2g → c2t , and φ → 0 through negative values. Hence φ is zero at both θ = 0 and π/2, and so has an extremum at some intermediate angle. Thus, the group velocity of the slow wave has a maximum departure from the direction of the applied field at an intermediate angle of propagation, being zero at both θ = 0 and π/2. In summary: for the slow wave,  ct ez as θ → π/2, (2.89) cg → min (cs , cA )ez as θ → 0.

2.7 Magnetoacoustic Waves

49

This feature of the fast and slow waves, for the group velocity of the fast wave to be closely aligned with the direction of propagation and for that of the slow wave to be closely aligned with the applied field, is readily illustrated in the extremes c2s  c2A and c2s  c2A . For in either case the phase speeds squared are simply c2 ≈ c2f for the fast wave and c2 ≈ c2t cos2 θ for the slow wave. Equation (2.85) applied to the fast wave then shows that tan ψ ≈

c2t sin 2θ 2c2f

(fast wave);

this attains a maximum at θ = π/4 when sin 2θ reaches its peak of unity. Then ψ ≈ c2t /(2c2f ) (on noting tan ψ ≈ ψ when ψ is small). Hence, for the fast wave  for cA  cs , 1 (c2A /c2s )−1 ψ≈ (2.90) 2 2 2 (cA /cs ) for cA  cs , and so ψ ≈ 0 whenever the sound and Alfv´en speeds are very different. In short, the fast wave’s group velocity is closely aligned with the direction of wave propagation k. In fact, even when the sound speed and Alfv´en speed are broadly comparable the angle ψ remains small; in the special case cA = cs (which is an extreme for the angle ψ), the largest value of ψ is tan−1 (1/2) radians (or 26.6 ◦ ), a result mentioned in Osterbrock (1961, p. 354); see also Section 2.8. Moreover, turning to the magnitude of cg we see by equation (2.82) that cg ≈ cf ; thus, overall the group velocity is closely aligned with the direction of propagation and has a magnitude of cf . If the magnetic field is weak (c2A  c2s ), then the fast wave’s group velocity is again close to the direction of propagation k but |cg | is now close to the sound speed. If the field is strong (c2A  c2s ), then the group velocity is still close to the k-direction but its magnitude is close to the Alfv´en speed. Turning now to the slow wave and considering the angle φ that cg makes with the applied magnetic field, we note that, in the extremes of weak or strong magnetic field, equation (2.87) with c2 ≈ c2t cos2 θ gives tan φ ≈ −

c2t sin 2θ cos2 θ , 2c2f

(slow wave).

√ Now the function sin 2θ cos2 θ has a peak value of 3 3/8, reached at θ = π/6. Hence, tan φ and φ are small and √  2 2 −1 for cA  cs , (cA /cs ) 3 3 (2.91) · |φ| ≈ 2 2 16 (cA /cs ) for cA  cs . A relation of this form is alluded to in Lighthill’s discussion (see Lighthill 1960, p. 417). It follows that cg is closely aligned with B0 (making a small negative angle φ with B0 ). The magnitude of cg is close to ct . Thus, in a weak field, |cg | is close to the Alfv´en speed; in a strong field, |cg | is close to the sound speed. Moreover, even in the special case cA = cs , when |φ| reaches a peak value (not a local maximum but the largest value) the angle φ is small, having a peak value of only |φ| = tan−1 (1/2) radians (corresponding to 26.6 ◦ ); see also Section 2.8.

50

Waves in a Uniform Medium

cA cs

cA

cs

cA

cs

Figure 2.5 The angle φ that the group velocity cg for the slow wave makes with the applied magnetic √ field, determined as a function of θ by equation (2.87) for the cases cA = cs , cA = 2cs and cA = 2cs ; note that φ ≤ 0. The curve for cA = cs starts at φ = −tan−1 (1/2) when θ = 0 and ends at φ = 0 when θ = π/2; it provides a bound for |φ|, in that for all cases with cA = cs we have |φ| ≤ tan−1 (1/2) ≈ 0.464 radians (equivalent to 26.6 ◦ ). For cA = cs , the maximum value of |φ| falls with increasing cA /cs .

We end this aspect of our discussion with an illustration of the behaviour of the angle φ that cg for the slow wave makes with the applied field. Figure 2.5 displays |φ| as a function of θ for the slow wave, showing also the upper bound that arises from the special case cA = cs and the rapidly falling angle φ for values of cA /cs larger than unity. Using the approximation (2.91), we note that when cA = 4cs , a condition representative of the corona, then |φ| ≈ 0.02 radians (corresponding to 1.2 ◦ ); the slow wave transports energy very close to the magnetic field lines. Similarly, the angle ψ that cg in the fast wave makes with the direction of propagation k is plotted in Figure 2.6, showing that for cA = cs the fast group velocity lies very close to the direction of propagation: ψ ∼ θ. Only the case cA = cs displays a somewhat different behaviour, where we find ψ → tan−1 (1/2) as θ → 0. Otherwise, the group velocity of the fast wave lies close to the direction of propagation k.

2.7.4 Perturbations in the Magnetoacoustic Waves We turn now to an examination of the perturbations associated with the magnetoacoustic waves, considering in some detail such variables as the velocity field, the magnetic field, and the pressure fields. In so doing we are able to bring out the various characteristics

2.7 Magnetoacoustic Waves

cA cs

cA

cs

51

cA

cs

Figure 2.6 The angle ψ that the group velocity cg of the fast wave makes with the direction k of wave propagation as a function of θ , as determined by equation (2.85) for the cases cA = cs , cA = √ 2cs and cA = 2cs . The curve for cA = cs provides an upper bound for all cases with cA = cs : |ψ| ≤ tan−1 (1/2) ≈ 0.464 radians. For cA = cs , the maximum value of |ψ| falls with increasing cA /cs and ψ ∼ θ .

of fast and slow magnetoacoustic waves, and how they compare with those of sound and Alfv´en waves. There are two fundamental directions in our system, namely that defined by the equilibrium state and that defined by the perturbation which we impose upon the equilibrium. Thus, our fundamental directions are defined by B0 and k, from which we may construct a third direction k × B0 , which is orthogonal to k and B0 ; these three directions, then, span the space of perturbations. As we have already seen, the Alfv´en wave is associated with the direction k×B0 . As we shall see in what follows, the magnetoacoustic waves are associated with the plane formed by k and B0 . The vector eθ lying in this plane and perpendicular to k is also useful. Velocity Field What is the velocity field in a magnetoacoustic wave? To answer this, consider equations (2.32) and (2.35) for ux , uy and uz . In terms of the plane wave representation with ky = 0 these equations yield (ω2 − kz2 c2A )ux = (ωpT /ρ0 )kx ,

(2.92)

(ω − kz2 c2A )uy (ω2 − kz2 c2t )uz

= 0,

(2.93)

= (c2s /c2f )kz (ωpT /ρ0 ).

(2.94)

2

52

Waves in a Uniform Medium

The possibility ω2 = kz2 c2A , with uy = 0, corresponds to Alfv´en waves and is discussed above. Suppose, then, that uy = 0, so that there are no motions perpendicular to the plane of k and B0 . Then, for non-zero kx and ωpT , we may solve eqs (2.92) and (2.94) to obtain ux u = (kx , 0, λkz ), (2.95) kx where the scale factor λ is given by λ=



c2s c2f



ω2 − kz2 c2A ω2 − kz2 c2t

 .

(2.96)

Equation (2.95) gives the velocity field in the magnetoacoustic waves. We see that u lies in the plane of k and B0 , since it may be written as the sum of a vector in the k direction and a vector in the direction of B0 . In expression (2.95) we have assumed that ux is non-zero. Only if the motion is purely in the z-direction does equation (2.95) not apply. Also, we may note that the possibility ω2 = kz2 c2t in equation (2.96) does not arise, for ω2 = kz2 c2t does not satisfy the dispersion relation (2.67); however, it turns out that modes with ω2 ≈ kz2 c2t do occur in magnetic flux tubes (see Chapter 6). There are a number of alternative expressions for λ that we should note. Using the dispersion relation (2.67), we may rewrite expression (2.96) in the various forms λ=

(ω2

2 c2 k⊥ c2 sin2 θ s = 2 s 2 . 2 2 − kz cs ) (c − cs cos2 θ )

(2.97)

In fact, we may eliminate entirely any explicit mention of the angle θ in the form of λ. Use of the dispersion relation (2.69) allows us to eliminated cos2 θ, giving λ = λ(θ ) = 1 −

c2A , c2

with the dependence upon θ arising through c(θ ). Evidently, λ < 1. With expression (2.98) for λ, we may write the velocity field in the form     c2A ux u= k− kz ez . kx c2

(2.98)

(2.99)

The motion u = uslow arising in the slow magnetoacoustic wave follows from equation (2.99) on taking c2 = c2− ; the motion u = ufast in the fast wave is given by taking c2 = c2+ . Observe that if cA = 0 then u is in the direction of k; in other words, the wave is longitudinal. We have recovered our earlier result that sound waves in the absence of a magnetic field are longitudinal (Section 2.4). The motions uslow and ufast arising in the two magnetoacoustic waves lie in the plane formed by k and B0 , but how are they related to one another? Denote by λ+ the value of λ in the fast wave, and by λ− its value in the slow wave, these values being determined by the roots c+ and c− of the magnetoacoustic dispersion relation. The form (2.68) of the dispersion relation, expressing k⊥ in terms of ω and kz , shows immediately that λ = λ− is negative for the slow wave, because the longitudinal phase speed is less than the minimum

2.7 Magnetoacoustic Waves

53

of cs and cA ; whereas for the fast wave λ = λ+ is positive (though less than unity), since here the longitudinal phase speed is greater than the maximum of cs and cA . This means that for the fast wave the parallel and perpendicular components of the motion, u and u⊥ , are in phase, whereas for the slow wave they are out of phase (by 180 ◦ ). Consider the relative directions of the two magnetoacoustic velocity fields, uslow and fast u . Taking the scalar product, we have uslow · ufast =

u2x (k⊥ 2 + kz2 λ+ λ− ). kx2

But, from the definitions of λ+ and λ− , we have    c2A c2A λ + λ− = 1 − 2 1− 2 . c+ c−

(2.100)

(2.101)

Utilizing expression (2.70) for the sum and product of c2+ and c2− , we are led to the result λ+ λ− = − tan2 θ = −

2 k⊥ . kz2

(2.102)

Thus, uslow · ufast = 0.

(2.103)

Hence, uslow is perpendicular to ufast . In fact, not only are the magnetoacoustic velocity fields perpendicular to each other they are also perpendicular to the motion, uAlfv´en , in the Alfv´en wave, for uAlfv´en lies in the ey -direction; so uslow · uAlfv´en = ufast · uAlfv´en = 0.

(2.104)

Thus, the Alfv´en wave, fast wave and slow wave velocity fields form an orthogonal triad of vectors. We may examine the motions uslow and ufast in greater detail. Denote by  the angle that the motion u of a magnetoacoustic wave makes with the applied magnetic field B0 . Then, from equations (2.92)–(2.94) describing the velocity components of the motion, it follows that 1 1 k⊥ = tan θ, (2.105) tan  = λ kz λ with λ = λ+ for the fast wave and λ = λ− for the slow wave. Using equations (2.96) and (2.97), we have   c2 c2 (c2 − c2s ) = tan  = tan θ . c2s c2A sin θcos θ c2 − c2A

(2.106)

These alternative forms that determine the angle  prove useful. To interpret equation (2.106), observe first that it is sufficient to examine just one of the magnetoacoustic modes, deducing the orientation of the other mode by invoking the fact of orthogonality. We consider the fast wave, for which λ = λ+ , with 0 ≤ λ+ ≤ 1. From equation (2.106)

54

Waves in a Uniform Medium

we see that, for propagation angles in the range 0 ≤ θ ≤ π/2, the angle  that the motion ufast makes with the applied magnetic field lies in the range 0 ≤  ≤ π/2. Consider how  varies as θ approaches zero. We have seen earlier that the phase speed c(θ ) approaches cs or cA as θ tends to zero. Consider the case cA > cs ; for the fast wave, c(θ ) tends to cA as θ approaches zero, and so by the first formula in equation (2.106) we see that tan  tends to ∞. Hence,  tends to π/2 as θ tends to zero. For the case in which the Alfv´en speed exceeds the sound speed, the fast wave’s motion, for propagation in the direction of the applied magnetic field, lies at right angles to the applied field. (Consequently, for propagation in the direction of the field, the slow mode’s motion is aligned with the field.) On the other hand, in the case cA < cs , the fast wave gives c(θ ) tends to cs as θ tends to zero. So, by the second formula in equation (2.106),  tends to zero as θ tends to zero. Turning to the limit of θ approaching π/2, corresponding to propagation directly across the applied magnetic field, we have seen earlier that for the fast wave c(θ ) tends to cf ; thus, equation (2.106) shows that  tends to π/2, independently of the relative magnitudes of cs and cA . Taken altogether, these results indicate that for the case cA < cs the fast wave is principally longitudinal (i.e., ufast is approximately parallel to k). The motion is aligned with k at θ = 0 and π/2, departing slightly from this alignment at other angles of propagation. The weaker the magnetic field, the closer is the alignment with k. When cA > cs , the motion in the fast wave is essentially perpendicular to the applied magnetic field B0 , whatever the direction of propagation. The tendency for the motions in the fast wave to be perpendicular to B0 is most marked in a strong magnetic field (cA  cs ). Since motions in the slow wave are always orthogonal to those in the fast wave, the orientation of the slow mode vector uslow is precisely opposite to that in the fast wave: in a strong magnetic field (cA  cs ), slow mode motions are principally aligned with the field; in a weak magnetic field (cA  cs ), those motions are transverse to the direction of propagation. We end by displaying the behaviour of the angle  as a function of θ for 0 ≤ θ ≤ π/2. Figure 2.7 shows the behaviour of a fast wave for the case cA ≥ cs . Magnetic Field The magnetic field induced by the motions in magnetoacoustic waves is frozen into the motion, and so takes on a closely related form: motions perpendicular to the field carry the field with them whereas motions along the field do not disturb the field. Specifically, we may determine the components of the perturbed magnetic field B by reference to the linearized induction equation (2.19): Bx = −B0

kz ux , ω

By = 0,

Bz = B0

2 ux k⊥ . kx ω

(2.107)

Thus, taken altogether we have B = B0

u  x

ω

(−kz , 0, kx ) = −B0

u  x

c

eθ ,

(2.108)

2.7 Magnetoacoustic Waves

cA cs

cA

cs

55

cA

cs

Figure 2.7 The orientation of the motion ufast in the fast wave, as measured by the angle  that the motions u make with the applied magnetic field;  is determined by equation (2.106). We have taken 0 ≤ θ ≤ π/2. In a strong field (cA  cs ), the motion ufast is perpendicular to the applied field ( ≈ π/2), whatever the angle θ of propagation. In the extreme cA  cs , corresponding to a weak magnetic field, ufast is aligned with the direction of propagation ( ≈ θ) and so the motion is longitudinal.

and so, as expected, B lies in the plane of k and B0 . Note that the direction (as opposed to the magnitude) of the field perturbation B is independent of which mode we are considering. Thus, for both the fast wave and the slow wave, B is perpendicular to k. Current, Electric Field and Vorticity The current generated in association with the perturbed magnetic field follows from Ampere’s law, equation (2.12):

u  x (2.109) ey . μj = curl B = −ik × B = ikB0 c Thus, the current density j in the magnetoacoustic modes is in the direction k × B0 (and so is aligned with the motions uAlfv´en in the Alfv´en wave). It follows, then, that any fieldaligned current, jz , must be carried solely by the Alfv´en wave. The magnetoacoustic waves carry current that is perpendicular to the applied field but are unable to propagate current that is in the direction of B0 . Consider the electric field generated by magnetoacoustic waves. We have E = −u × B0 = ux B0 ey , and so E is perpendicular to the plane of k and B0 .

(2.110)

56

Waves in a Uniform Medium

Turning now to the vorticity within the fluid, we have noted earlier, following equation (2.33), that the component ωz of vorticity in the direction of the applied magnetic field is carried by the Alfv´en wave. Any component of vorticity or current in the direction of the applied magnetic field that is induced in the system by (say) a disturbance is thus transported as an Alfv´en wave. The magnetoacoustic waves are responsible for the transport of components of vorticity and current that are perpendicular to B0 . Specifically, for the magnetoacoustic waves we have   c2 (2.111) curl u = −ik × u = −iux kz A2 ey , c where we have used expression (2.95) for the velocity field and equation (2.97) for λ. Thus, the vorticity is in the same direction as j (unless kz = 0, when ω = 0), though anti-parallel (i.e., perturbations opposite in sign) for acute (0 < θ < π/2) angles of propagation, and the magnetoacoustic waves carry no vorticity in the direction of the applied magnetic field. Pressure Fields Finally, we consider the pressure fields generated in the motion. We distinguish between the pressure p associated with the plasma and the pressure pm (= μ1 B0 · B) associated with the magnetic field. Consider first the z-component of the momentum equation (2.11): we have     c2A ux ω uz = ρ0 ω 1 − 2 , (2.112) p = ρ0 kz kx c where we have used equations (2.94) and (2.98) to eliminate uz . The magnetic pressure follows from equation (2.107) for Bz :   1 ux 2 k 2 c sin θ . pm = B0 Bz = ρ0 μ kx A c

(2.113)

Using the form (2.69) of the dispersion relation we may eliminate sin2 θ in favour of c(θ ) to give     2 ux c − c2s ω λ. (2.114) pm = ρ0 kx c2s The total pressure perturbation pT is the sum of p and pm :   ux k 2 (c − c2A cos2 θ ). pT = p + pm = ρ0 kx c

(2.115)

We may relate the pressure perturbations to the density perturbation ρ. From the equation of continuity (2.9) and expression (2.95) for the velocity field, we have   ux k 2 ρ = ρ0 (c − c2A cos2 θ ). (2.116) kx c3 Thus, we find that p = c2s ρ,

pm = (c2 − c2s )ρ,

pT = c2 ρ,

(2.117)

2.8 Magnetoacoustic Waves: The Special Case cA = cs

57

with c2 being determined by the dispersion relation (2.69). This may be compared with the simpler acoustic case (cA = 0) for which c2 = c2s , giving pm = 0 and p (= pT ) = c2s ρ. Thus, for both acoustic waves and magnetoacoustic waves, pT = c2 ρ. Consider the pressure ratios implied by the above expressions. We have   c2A pm 1 (2.118) = sin2 θ = 2 (c2 − c2s ). p cs c2 − c2A Thus we see that the perturbation pm in magnetic pressure is in phase with the perturbation p in plasma pressure when c2 > c2s , and 180 ◦ out of phase when c2 < c2s . Hence, the magnetic and plasma pressure perturbations, p and pm , are in phase for a fast wave and out of phase for a slow wave: p pm > 0 in a fast wave, p pm < 0 in a slow wave. 2.8 Magnetoacoustic Waves: The Special Case cA = cs The above general treatment of the magnetoacoustic waves considers the two cases cA > cs and cA < cs . From a physical point of view perhaps little more need be said, because we can always suppose that one or other of these situations pertains. However, it is of mathematical interest to examine briefly the special case when the sound and Alfv´en speeds are equal. With cA = cs the dispersion relation (2.69) factorizes to give c2 = c2s (1 ± sin θ ).

(2.119)

1 c = (1 − | sin θ |) 2 , cs

(2.120)

1 c = (1 + | sin θ |) 2 . cs

(2.121)

Thus, the slow wave gives

whilst the fast wave gives

The phase speed diagram for the fast and slow waves is given in Figure 2.8. Rotation of the polar curves c = c(θ ) about the axis of symmetry (θ = 0) produces a surface. For the fast wave the phase speed surface evidently resembles an apple, with the core of the apple corresponding to the axis of the applied magnetic field. The indentations leading to the apple core are at (c, θ ) = (cs , 0) and (cs , π ). The apple-like surface bulges √ out in√the direction orthogonal to the field (where θ = ±π/2), to give c = cf = cs 2 (= cA 2). By contrast, the slow wave gives rise to two lemon-like figures that are extended along the direction of the applied magnetic field. These lemon-like figures are enclosed by the fast wave, with the fast and slow surfaces touching at c = cs (= cA ), θ = 0 and π . Evidently, the fast wave is somewhat similar to a sound wave while the slow wave is closer in appearance to an Alfv´en wave. The angle φ that the group velocity cg makes with the applied magnetic field is given by (2.87). With c2 specified by equation (2.120), the slow wave leads to tan φ = −

(1 − sin θ )2 . (2 − sin θ ) cos θ

(2.122)

58

Waves in a Uniform Medium

Figure 2.8 The phase speed diagram for the fast and slow magnetoacoustic waves when cA = cs . Comparing with Figure 2.1, we may note the similarity of the fast wave with a pure sound wave and the slow wave with an Alfv´en wave.

In writing this form, we have for convenience assumed that 0 ≤ θ ≤ π , so that we may dispense with the modulus sign | | in the expression for c2 . Now the expression for | tan φ| arising here has its greatest value of 1/2, occurring as θ → 0; as θ → π/2 we obtain φ → 0. Accordingly, |φ| ≤ tan−1 (1/2) radians, corresponding to an upper limit of 26.6 ◦ . Figure 2.5, given earlier, displays the angle φ that the slow wave makes with the applied magnetic field (note that φ ≤ 0) for this case (along with cases when cA = cs ), and Figure 2.6 displays the behaviour of the angle ψ in the fast wave when cA = cs (as well as when cA > cs ). We turn now to the angle  that the motion u in the magnetoacoustic wave makes with the applied magnetic field B0 . We use equation (2.106) and again assume for convenience that 0 ≤ θ ≤ π. Then for the fast wave the angle , when cs = cA , is given by tan  = sec θ + tan θ   θ π + . = tan 4 2

(2.123)

Thus,  = π/4 + θ/2 for the fast wave. Also, since the motions in the slow wave are at right angles to those in the fast wave, when cs = cA the motion in the slow mode makes an angle  = 3π/4 + θ/2 with B0 .

2.9 Two Physical Extremes We consider here two special cases of the general magnetohydrodynamic system, namely that of an incompressible fluid and that of a β = 0 plasma (also sometimes referred to as a cold plasma). These cases arise sufficiently frequently that they justify a separate treatment. The case of an incompressible fluid is mathematically more tractable than the general compressible case and the special case has application to liquids (such as in the laboratory

2.9 Two Physical Extremes

59

or the Earth’s core). The second special case is that of a β = 0 plasma, which also has a number of features that allow a simpler treatment than the general compressible case. In this special case, the medium is compressible but now magnetic pressure dominates over fluid pressure. The β = 0 case finds direct application both in laboratory plasmas and in the Sun, especially in the coronal plasma. We begin our discussion by deriving the properties in these special cases from the general formalism of the previous sections, turning afterwards to obtaining those properties ab initio from the reduced forms of the magnetohydrodynamic equations that govern the two extreme cases. An incompressible fluid is one that suffers no changes in density as one moves with the fluid: the convective derivative of the density is zero. Thus ∂ρ + (u·grad)ρ = 0. ∂t

(2.124)

Accordingly, in view of the continuity equation (2.9), an incompressible fluid is characterized by the condition div u = 0.

(2.125)

Treating an incompressible fluid as a limiting case, from the viewpoint of a compressible plasma, we may see from the isentropic equation that we may obtain a description of the incompressible case if we choose the parameter γ (and with it the sound speed) to be arbitrarily large. For then the isentropic equation (2.10), combined with the equation of continuity (2.9), reduces simply to div u = 0. In a real plasma 1 ≤ γ ≤ 2, so letting γ → ∞ is simply a mathematical device to recover equation (2.125) from the isentropic equation. Specifically, then, we may let cs → ∞ in the general compressible plasma equations to obtain the corresponding incompressible results. The incompressible case is perhaps of most relevance to laboratory circumstances. Nonetheless, it remains of general interest, partly because it is more readily amenable to study than the compressible problem, though one must bear in mind its limitations in applications to astrophysical plasmas. The second case of particular interest is the cold (or low β) plasma. This corresponds to the special case of an arbitrarily small sound speed, obtained formally from the general case by letting cs → 0. It is of particular relevance to certain astrophysical plasmas, such as the solar corona and parts of the magnetosphere, where magnetism dominates (cs  cA ), and also finds application to the hot plasmas in laboratory fusion devices. Consider, then, these two limiting extremes of the general compressible case. The Alfv´en wave has a dispersion relation (equation (2.56)) that is independent of the sound speed and so this remains intact in either of the limits cs → 0 or ∞. Also, its properties (e.g., motions transverse to both k and B0 , and no density or pressure variations) are independent of any sound speed. Thus, the Alfv´en wave exists in both the incompressible fluid and the cold plasma. Turning now to the magnetoacoustic waves, consider first the incompressible (cs → ∞) case. From equation (2.67) (or (2.79)) we see that the phase speed of the fast wave goes to infinity in the incompressible extreme, and so the fast wave is lost from the system; the slow wave dispersion relation becomes ω2 = kz2 c2A , which coincides with that of

60

Waves in a Uniform Medium

the Alfv´en wave. However, a distinction remains, for whereas the Alfv´en wave has zero pressure variations (p = pm = pT = 0), the slow wave in the incompressible limit has only zero total pressure variation (pT = 0), with the individual pressures p and pm being in general non-zero. To see this, examine equation (2.117) in the limit cs → ∞, with c2 finite: we obtain ρ = 0 and pT = 0. But from equation (2.112) we see that pm is non-zero. So pm = −p, with both being non-zero; the plasma perturbation and magnetic pressure perturbation are exactly out of phase. The character of the magnetic field perturbation, the associated current and the vorticity are unchanged from the general case, as the directions of these vectors are independent of the sound speed (see equations (2.107)–(2.111)). Consider the β = 0 plasma case. With cs = 0, the dispersion relation for magnetoacoustic waves gives c2 = 0 for the slow wave and c2 = c2A (i.e., ω2 = k2 c2A ) for the fast wave. Thus now it is the slow wave that is lost from the system; the fast wave propagates isotropically at the Alfv´en speed. This mode, while having the Alfv´en speed as its characteristic speed, is nonetheless quite distinct from an Alfv´en wave, as the isotropic nature of its dispersion relation demonstrates. The fast wave in the cold plasma limit is commonly referred to as the ‘compressional Alfv´en wave’, but this terminology is misleading, implying a connection with the Alfv´en wave that does not exist. We prefer to label the mode as a fast wave, albeit in the extreme of zero sound speed, thus maintaining the natural link with the magnetoacoustic branch of disturbances. The compressional nature of the mode arises from magnetic pressure variations. For, from equation (2.116) we see that density disturbances arise unless the wave propagation is parallel to the applied magnetic field, i.e., ρ = 0 save when θ = 0. However, plasma pressure perturbations are zero (p = 0), as equation (2.117) makes clear. Magnetic pressure variations satisfy pm = c2A ρ, reminding one of pure sound waves but with magnetic pressure replacing plasma pressure (and cA replacing cs ). However, in contrast to the longitudinal motions in a sound wave, motions are here purely in the xy-plane and so transverse to the applied field B0 . This follows from equation (2.98) which gives λ = 0, showing that the component of motion in the z-direction is zero. Finally, as with the incompressible case, the characters of the magnetic field perturbation, the current and the vorticity, being directionally independent of the sound speed (see equations (2.108)–(2.111)), are the same as in the general magnetoacoustic case. We turn now to an ab initio treatment of the above special cases.

2.9.1 The Incompressible Fluid As noted above, an incompressible fluid is characterized by the condition div u = 0. The linearized equations of an incompressible system are accordingly ρ0

∂u 1 = −grad pT + (B0 ·grad) B, ∂t μ ∂u ∂B = B0 , ∂t ∂z

div B = 0,

div u = 0,

(2.126)

(2.127)

representing the equation of motion of an incompressible fluid with total pressure perturbation pT , and the induction equation of a (solenoidal) magnetic field. The equation of motion

2.9 Two Physical Extremes

61

is the same as in the general case. The induction equation is simplified from the general case by the solenoidal constraint on u. The above four equations describe our system of small amplitude disturbances about a uniform field B0 embedded in an incompressible fluid. The fluid density remains equal to its undisturbed value ρ0 , given that there is no motion in the undisturbed state. Since both u and B are divergence-free it follows from the divergence of the momentum equation that ∇ 2 pT = 0.

(2.128)

Thus, the total pressure perturbation pT satisfies Laplace’s equation. Consider an unbounded medium with no spatial changes in fluid (or magnetic) properties. In such a uniform medium of infinite extent the only solution of Laplace’s equation that has no singularities anywhere is a constant. Accordingly, for an unbounded medium, pT must be a constant. If we require it to be zero at infinity, it follows that pT = 0 everywhere. So, in an incompressible fluid, disturbances involve no change in the total pressure perturbation. Changes in fluid pressure are exactly balanced by magnetic pressure variations: p = −pm . If the fluid is not unbounded, then non-trivial solutions of Laplace’s equation arise for pT . Such a situation arises in a structured medium, including the important case of a magnetic flux tube, and will be discussed in detail in Chapters 4 and 6. Returning to the uniform unbounded medium, we note that with pT = 0 the equation of motion (2.126) and the induction equation (2.127) read ρ0

∂u 1 ∂B = B0 , ∂t μ ∂z

∂u ∂B = B0 . ∂t ∂z

(2.129)

2 ∂ 2B 2 ∂ B = c . A ∂t2 ∂z2

(2.130)

Thus 2 ∂ 2u 2 ∂ u = c , A ∂t2 ∂z2

The velocity field u and the magnetic field B (and so their respective Cartesian components) satisfy the one-dimensional wave equation, showing that disturbances are propagated along the applied magnetic field with the Alfv´en speed. Also, the vorticity curl u satisfies the same one-dimensional wave equation, as follows from the curl of the momentum equation and the curl of induction equation. The general solution of equation (2.130) is u = u0 (f , g, h),

(2.131)

where f , g and h are functions of x, y and (z ± cA t). These functions are arbitrary, save for requirements of differentiability and that the velocity u satisfy the solenoidal constraint. This solution includes both the possibility of a solution with pm = 0 (corresponding to an Alfv´en wave) and one with non-zero pm (corresponding to a slow mode in the incompressible limit). The Alfv´en wave solution may be illustrated by taking a disturbance of the form u = u0 f (z − cA t),

(2.132)

62

Waves in a Uniform Medium

which describes a wave propagating along the applied magnetic field with speed cA , in the positive z direction. The profile of the disturbance is described by the function f of variable (z − cA t), and may be arbitrary (though twice differentiable); the vector u0 , representing the direction of the motion, depends in general upon x and y. The solenoidal constraint (div u = 0) requires that uz = 0 for arbitrary f , so u is perpendicular to the applied magnetic field. Similarly, the perturbation B in the field is perpendicular to B0 (since div B = 0). A disturbance moving in the negative z-direction is of similar form to that given in equation (2.132), save (z − cA t) is replaced by (z + cA t). The sum of two such solutions provides the general D’Alembert solution, just as for the usual one-dimensional wave equation. With u given by equation (2.132), the induction equation (2.127) gives u0 B = −B0 f (z − cA t). (2.133) cA Thus 1 1 B = − u. B0 cA

(2.134)

Similarly, for a wave travelling in the negative z-direction 1 1 B= u. B0 cA

(2.135)

For either of these solutions, the magnetic pressure perturbation pm is zero (since uz = Bz = 0) and so also is the plasma perturbation (since pT = 0). This therefore describes an Alfv´en wave. For the Alfv´en wave solution, it may be seen that the magnetic energy density B2 /2μ per unit volume of the perturbed field B (B = |B|) is equal to the kinetic energy density 1 2 2 ρ0 u per unit volume of the motion u of magnitude u (= |u|): B2 1 = ρ0 u2 . 2μ 2

(2.136)

Thus, in an Alfv´en wave there is an equipartition of magnetic and kinetic energies. It is convenient to represent disturbances in terms of a plane wave, as in equation (2.36) with wave vector k = (kx , 0, kz ) = k(sin θ , 0, cos θ ). Alfv´en waves are characterized by being transverse to the applied magnetic field and the propagation vector. Thus we consider a solution of eqs (2.126) and (2.127) with ω2 = kz2 c2A and u = (0, uy , 0).

(2.137)

This motion satisfies the requirement that div u = 0, since ky = 0. For a wave moving in the k direction (so ω/kz > 0), the magnetic field is given by equation (2.134), namely B = −B0 (uy /cA )ey , and so is anti-parallel to u. Since the motions u are perpendicular to B0 , there is no electric field: E = −u × B0 = 0.

(2.138)

Moreover, there is no component of perturbed magnetic field in the direction of B0 , and so the magnetic pressure perturbation is zero (pm = B0 Bz /μ = 0). Consequently, p = 0 (since pT = 0). The current density j is given by

2.9 Two Physical Extremes

 μj = −i k × B = −iB0

uy cA



63

 (kz , 0, −kx ) = −i eθ kB0

uy cA

 .

(2.139)

The current is perpendicular to both k and B. Allowing for a phase change, as indicated by the factor i in equation (2.139), we conclude that j is in the direction (kz , 0, −kx ) and so makes an angle of π2 + θ with the z-axis. The vorticity is ikuy eθ , and so is anti-parallel to j. Thus we recover the expected properties of Alfv´en waves. Consider now a solution of eqs (2.126) and (2.127) with a non-zero motion along the magnetic field. Set uy = 0, so that the motions are entirely in the xz-plane. Then incompressibility requires that k · u = 0, i.e., kx ux + kz uz = 0. Hence uz u = (−kz , 0, kx ) = −u eθ , (2.140) kx and so the motion u lies in the xz-plane and is perpendicular to k. The field B is antiparallel to the motion. Since Bz is non-zero, the magnetic pressure perturbation is nonzero; the plasma pressure perturbation is consequently also non-zero (since pT = 0). Thus we recover the expected behaviour of the slow magnetoacoustic wave, albeit in the incompressible case. 2.9.2 The β = 0 Plasma The case of a zero β plasma is characterized by the neglect of the pressure term in the momentum equation, so the momentum equation is simply ρ0

∂u 1 ∂B = j × B0 = −grad pm + B0 , ∂t μ ∂z

(2.141)

where the total pressure perturbation is now pm (= B0 Bz /μ). Since the driving force in the momentum equation is perpendicular to the applied field B0 , there is no acceleration in this direction; accordingly, we take uz = 0 and u = (ux , uy , 0). The induction equation is much as before, namely   ∂B ∂u (2.142) = B0 − ez  , ∂t ∂z except now  ≡ div u =

∂uy ∂ux + . ∂x ∂y

(2.143)

The field B is of course solenoidal though, in contrast to the incompressible case, the motion u need not be. Density perturbations follow from mass continuity (2.9). From the z-component of the induction equation (2.142) we obtain 1 ∂pm = −c2A , ρ0 ∂t

(2.144)

and so the time derivative of the momentum equation leads to (cf. equation (2.28))   2 ∂ 2u ∂ 2 ∂ u 2 = c + c . (2.145) grad  − e z A A ∂z ∂t2 ∂z2

64

Waves in a Uniform Medium

In terms of components,   2 ∂ ∂2 2 ∂ − cA 2 ux = c2A , 2 ∂x ∂t ∂z



2 ∂2 2 ∂ − c A 2 ∂t2 ∂z

 uy = c2A

∂ . ∂y

(2.146)

The divergence of equation (2.145) leads to the wave equation ∂ 2 = c2A ∇ 2 , ∂t2

(2.147)

showing that any non-zero compression in the system (measured by a non-zero ) is propagated isotropically with the Alfv´en speed. This is the fast wave, with dispersion relation in the cold plasma case of ω2 = k2 c2A . The magnetic pressure is similarly propagated isotropically at the Alfv´en speed, with pm = c2A ρ. Accordingly, both the phase speed and the group velocity diagrams for the cold fast wave are simply spheres of radii cA , centred at the origin (cf. Figure 2.3). The various perturbation vectors for the fast wave are readily determined from the above and, as noted earlier, are the same as in the general case. For a plane wave with k = (kx , 0, kz ), we obtain uy = 0 and so the motion is u = (ux , 0, 0)

(2.148)

curl u = −ikz ux ey .

(2.149)

with associated vorticity

Thus By = 0 and so the perturbed magnetic field is B = (Bx , 0, Bz ) = B0 ux (−kz , 0, kx )/ω = −B0 (ux /c)eθ ,

(2.150)

with an electric field and current density   j = i B0 k2 /(μω) ux ey .

E = ux B0 ey ,

(2.151)

So B lies in the plane formed by k and B0 , and is perpendicular to k; the current j, electric field E and vorticity curl u are all perpendicular to k and B0 . The one-dimensional mode – the Alfv´en wave – remains, of course. This is the solution  = pm = 0. From equation (2.145), these incompressible (div u = 0) transverse (u ⊥ B0 ) motions satisfy the wave equation 2 ∂ 2u 2 ∂ u = c . A ∂t2 ∂z2

(2.152)

In particular, the z-component of the vorticity ωz is propagated with the Alfv´en speed cA : 2 ∂ 2 ωz 2 ∂ ωz = c , A ∂t2 ∂z2

ωz =

∂uy ∂ux − . ∂y ∂x

(2.153)

Thus we recover the dispersion relation ω2 = kz2 c2A describing the one-dimensional propagation of an Alfv´en wave.

2.9 Two Physical Extremes

65

The properties of the perturbations in the Alfv´en wave are readily determined. For a plane wave with k = (kx , 0, kz ), we have a motion u (with div u = 0), vorticity curl u and induced magnetic field B given by u = uy ey ,

curl u = ikuy eθ ,

B = −B0 (kz /ω)uy ey .

(2.154)

The electric field E and current j generated are E = −u × B0 = −uy B0 ex ,

j = iB0 (kz /ωμ)uy (−kz , 0, kx ).

(2.155)

Hence, as in the general case, the electric field E, current j and vorticity curl u in an Alfv´en wave all lie in the plane formed by k and B0 . The current and vorticity are in opposite directions (for ω/kz > 0), and perpendicular to k, as are the magnetic field B and motion u.

3 Magnetically Structured Atmospheres

3.1 Introduction Many astrophysical plasmas show evidence of being non-uniform, structured by the presence of a magnetic field. As discussed in Chapter 1, such structuring is particularly evident in the Sun’s atmosphere ranging from the photosphere to the corona. A strongly structured atmosphere presents propagation features that are not shared by the uniform medium discussed in Chapter 2. Here we examine such features for the simplest case of a structured medium, namely one in which the applied magnetic field is unidirectional with its field strength varying in a direction perpendicular to the field lines. The effects of gravity and other external forces are here ignored. Consider, then, an equilibrium magnetic field B0 = B0 ez .

(3.1)

The field strength B0 is assumed to be independent of the z-coordinate, though it may vary in a direction perpendicular to the z-axis. The solenoidal requirement (div B0 = 0) on the equilibrium field B0 is met since B0 is independent of z. The equilibrium plasma pressure p0 , density ρ0 and temperature T0 are each assumed to be independent of z but to vary in a direction perpendicular to the applied field. The requirement of magnetostatic balance (which follows from the momentum equation (1.2) with u ≡ 0; see Chapter 1) is simply that   B20 = 0, (3.2) grad p0 + 2μ and so any structure in the plasma pressure p0 is intimately linked with that in the magnetic field B0 . Provided the constraint (3.2) is satisfied (and p0 is everywhere positive), the profiles of equilibrium pressure p0 and field strength B0 are arbitrary functions of the coordinates perpendicular to ez , the unit vector in the direction of the applied field. Equation (3.2) simply ensures that the total pressure is uniform; where the magnetic pressure B20 /2μ is greatest the plasma pressure is correspondingly smallest. Profiles of equilibrium temperature and density are also arbitrary, save that the ideal gas law (p ∝ ρT; see Chapter 1) demands that where temperature is higher, the density is correspondingly lower, thus maintaining the same pressure. 66

3.1 Introduction

67

3.1.1 Wave Equations The equations of ideal magnetohydrodynamics are readily linearized about the equilibrium described by (3.1) and (3.2). The linear equation of continuity is ∂ρ + div ρ0 u = 0. ∂t

(3.3)

The linearized momentum equation is ρ0

∂u 1 1 = −grad pT + (B0 ·grad)B + (B·grad)B0 , ∂t μ μ

(3.4)

where we have introduced the perturbation in total pressure (the sum of the plasma pressure perturbation p and the perturbation in magnetic pressure), pT ≡ p +

1 B0 · B; μ

(3.5)

the magnetic field perturbation is B. The linearized induction equation is ∂B = curl (u × B0 ), ∂t

(3.6)

∂B = −B0 div u − (u·grad)B0 + (B0 ·grad)u, ∂t

(3.7)

which we can expand to give

where in the expansion of the vector identity for curl(u × B0 ) we have used the solenoidal condition div B0 = 0.

(3.8)

The perturbation magnetic field B also satisfies the solenoidal condition: div B = 0. Finally, the linear form of the isentropic equation is   γ p0 ∂ρ ∂p + u · grad p0 = + u · grad ρ0 , ∂t ρ0 ∂t

(3.9)

(3.10)

and the ideal gas law (T ∝ p/ρ) may be used to determine temperature variations T, if required: p ρ T = + . p0 ρ0 T0

(3.11)

The above system of equations determines the behaviour of the perturbations. While in many respects they are little different from the ones explored in Chapter 2 for the case of a uniform equilibrium state – indeed, that state is a special case of the above – the presence of non-uniformity in B0 and ρ0 considerably complicates any description of the behaviour of the waves.

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Magnetically Structured Atmospheres

The forms of the momentum equation (3.4) and induction equation (3.7) may be simplified a little by noting that for the equilibrium B0 = B0 ez , in which the field strength B0 is assumed to be independent of the z-coordinate, the operators involving B0 simplify. Thus, ∂ B, ∂z ∂ (B0 ·grad)u = B0 u, ∂z

(B0 ·grad)B = B0

(B·grad)B0 = ez (B · grad B0 ), (u·grad)B0 = ez (u · grad B0 ).

Then the momentum equation becomes

   ∂u B0 ∂B B0 = −grad pT + + ez B · grad , ρ0 ∂t μ ∂z μ

(3.12)

and the induction equation is  ∂B ∂u = B0 − ez B0 div u + u · grad B0 . ∂t ∂z

(3.13)

To progress further in our analysis we derive a general wave equation for the nonuniform case, generalizing equation (2.34) of Chapter 2 that pertains in a uniform medium. For this purpose it is convenient to consider the time derivative of the momentum equation (3.12). This involves ∂pT /∂t, the rate of change of the total pressure perturbation. So consider first the rate of change of the plasma pressure perturbation. From the isentropic equation (3.10) combined with the equation of continuity (3.3), we obtain ∂p = −γ p0 div u − u · grad p0 , ∂t

(3.14)

showing that plasma pressure changes are a consequence not only of volumetric changes (represented through the term involving div u) but also as a result of displacing the fluid up or down the equilibrium pressure field (represented by the term involving grad p0 ). Consider now how the magnetic pressure perturbation, pm = B0 · B/μ (= pT − p), evolves. Taking the scalar product of the induction equation (3.13) with B0 , we obtain   ∂uz ∂B 1 2 B0 · = B20 − B20 div u − u · grad B0 , ∂t ∂z 2 where uz is the component of the motion u along the direction of the equilibrium magnetic field B0 (so uz = ez · u). Thus   B20 ∂pm 2 ∂uz 2 = ρ0 cA − ρ0 cA div u − u · grad , (3.15) ∂t ∂z 2μ showing that magnetic pressure variations arise not only from volumetric changes (which compress or rarefy the plasma and in consequence the magnetic field) and fluid displacements along the gradient in the equilibrium magnetic field but also through motions along the field. Finally, combining results (3.14) and (3.15) we obtain ∂uz ∂pT = ρ0 c2A − ρ0 c2f div u, ∂t ∂z

(3.16)

3.1 Introduction

69

on using the fact that total pressure equilibrium (3.2) pertains. This result tells us how the total pressure perturbation evolves. The Alfv´en speed cA and the fast speed cf that arise in the above equations are defined as in Chapter 2; it is also convenient to introduce the sound speed cs and the slow speed ct :   2 1/2  B0 γ p0 1/2 −2 −2 2 2 2 −2 cf = cs + cA , ct = cs + cA , cs = , cA = . ρ0 μρ0 In keeping with the non-uniformity of the equilibrium state, these speeds will in general vary in a direction perpendicular to the applied magnetic field. We are now in a position to obtain the wave equation for u. Differentiating the momentum equation (3.12) with respect to time t and making use of the induction equation (3.13) we obtain, after a little algebra, the result   2 ∂ ∂pT 1 ∂ 2u 2 ∂ u 2 = c − c e grad div u − . (3.17) z A A ∂z ρ0 ∂t ∂t2 ∂z2 Equations (3.16) and (3.17) govern the total pressure perturbation pT and the motions u in an arbitrarily structured atmosphere, within which is embedded a unidirectional magnetic field B0 = B0 ez satisfying pressure balance (3.2). Various forms of the momentum equation may be considered. For example, we can eliminate the term involving div u, in preference to pT , by combining the velocity equation (3.17) with the pressure equation (3.16). The result is     2  2 2 cA 1 ∂ 2 pT ∂pT 1 ∂ 2u 2 ∂ u 2 ∂ uz − cA 2 = −ez 2 cA 2 − − grad . (3.18) ρ0 ∂z∂t ρ0 ∂t ∂t2 ∂z ∂z cf At this stage it is convenient to decompose explicitly the motion u into the component uz ez along the applied field and a component u⊥ perpendicular to that field, writing u = u⊥ + uz ez . Then, splitting the velocity equation (3.18) into components perpendicular and parallel to the applied magnetic field, we obtain   2  2  1 ∂ ∂pT 2 ∂ − c = − grad (3.19) u ⊥ ⊥ A 2 ρ0 ∂t ∂t2 ∂z for the component u⊥ perpendicular to the field, and  2 2 cs 1 ∂ 2 pT ∂ 2 uz 2 ∂ uz − c = − t 2 2 ∂t ∂z c2f ρ0 ∂z∂t

(3.20)

for the component uz along the applied field. In the above we have introduced the operator grad⊥ , the component of the gradient operator acting perpendicular to the direction of the applied magnetic field (i.e., perpendicular to the z-axis). In equation (3.20) for motions along the applied field, we see the natural occurrence of the slow speed ct . Equations (3.19) and (3.20) are coupled together through temporal and spatial variations in the total pressure perturbation pT , given by (3.16) and repeated here for ease of comparison, ∂uz ∂pT = ρ0 c2A − ρ0 c2f div u. ∂t ∂z

(3.21)

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Magnetically Structured Atmospheres

Alternatively, if instead we eliminate pT in preference to div u, using equation (3.17), then motions u⊥ perpendicular to the applied magnetic field satisfy  2   2  1 ∂ 2 ∂ 2 ∂uz 2 − c = grad c c div u + ρ u ρ (3.22) ⊥ 0 s 0 f ⊥ , ⊥ A 2 ρ0 ∂z ∂t2 ∂z while motions uz along the applied field are such that ∂ ∂ 2 uz = c2s (div u). 2 ∂z ∂t

(3.23)

Thus we see that equation (3.19) for the motion perpendicular to the applied field is the same in the non-uniform case as for the uniform case (see equation (2.32) of Chapter 2) and equation (3.20) for the parallel motion uz is the same as for the uniform case (see equation (2.35) of Chapter 2).

3.1.2 Sound Waves Before developing our treatment further it is worth noting at this stage the reduction of our equations in the special case of no magnetic field. In the absence of a magnetic field, pressure balance (3.2) requires that the equilibrium gas pressure p0 is a constant, since there is no longer a magnetic pressure to structure the atmosphere. However, the density distribution ρ0 may be non-uniform. With cA = 0, we obtain ct = 0, and so cf = cs and pT = p. Since in the absence of a magnetic field there is no anisotropy, the anisotropy evident in equations (3.16) and (3.18) must disappear. Thus, sound waves in a non-magnetic atmosphere that is structured in density satisfy   1 ∂p ∂ 2u = grad , (3.24) ρ0 ∂t ∂t2 with ∂p = −ρ0 c2s div u. ∂t

(3.25)

  1 ∂ 2p 2 = ρ0 cs div grad p , ρ0 ∂t2

(3.26)

Hence, pressure variations satisfy

and the motion u is given by ∂ 2u = c2s grad div u. (3.27) ∂t2 In the special case for which the equilibrium density is uniform, equation (3.26) simplifies to give the standard three-dimensional wave equation for the pressure perturbation: ∂ 2p = c2s ∇ 2 p, ∂t2 where ∇ 2 ≡ div grad is the three-dimensional Laplacian operator.

(3.28)

3.2 Magnetohydrodynamic Waves: Cartesian System

71

3.1.3 The Alfv´en Wave We return now to the magnetohydrodynamic problem, governed by equations (3.16) and (3.18), or the reduced forms in (3.19)–(3.23). Despite the obvious complexity of these equations we may note one simple solution, namely, an incompressible motion (div u = 0) for which there is no motion along the field (so uz = 0) and no total pressure variations (so pT = 0). Such a solution arises in a uniform medium and corresponds, of course, to the Alfv´en wave. Equation (3.16) is satisfied identically and equation (3.17) requires that the motion u⊥ perpendicular to the applied magnetic field satisfies the one-dimensional wave equation 2 ∂ 2 u⊥ 2 ∂ u⊥ = c . A ∂t2 ∂z2

(3.29)

Thus, motions in the plane perpendicular to the applied field are solutions of the wave equation, with general D’Alembert solution of the form u⊥ = f⊥ (z − cA t) + g⊥ (z + cA t)

(3.30)

for arbitrary functions f⊥ and g⊥ of coordinates (z − cA t) and (z + cA t), respectively. The vector functions f⊥ and g⊥ have an arbitrary dependence (subject to suitable differentiability) on the coordinates perpendicular to the applied magnetic field. This is the Alfv´en wave, the transverse vibration of each field line, in a magnetically structured medium. The presence of the Alfv´en speed cA , depending upon the coordinates perpendicular to the z-axis, within the functions f⊥ and g⊥ raises a special feature in the behaviour of the cross-field derivatives of the motion. We return to this feature shortly. To progress further in our discussion of the general magnetohydrodynamic case, it is useful to consider our equations in specific coordinate systems. There are two forms worth writing out in detail, namely that of a Cartesian coordinate system and that of a cylindrical coordinate system. We take each separately.

3.2 Magnetohydrodynamic Waves: Cartesian System In the Cartesian coordinate system x, y, z, we take the equilibrium magnetic field to be B0 = B0 (x)ez , and the equilibrium plasma pressure, density and temperature to be p0 (x), ρ0 (x) and T0 (x), respectively. Thus the equilibrium state is structured purely in terms of the x-coordinate, and the basic speeds cs (x), cA (x), ct (x) and cf (x) are, in general, all functions of x. The operator grad⊥ becomes  grad⊥ =

 ∂ ∂ , ,0 . ∂x ∂y

For a motion u = (ux , uy , uz ), equations (3.19) and (3.20) give

72

Magnetically Structured Atmospheres

∂ 2 ux ∂ 2 ux 1 ∂ 2 pT − c2A (x) 2 = − , 2 ρ0 (x) ∂x∂t ∂t ∂z ∂ 2 uy ∂ 2 uy 1 ∂ 2 pT 2 − c (x) = − , A ρ0 (x) ∂y∂t ∂t2 ∂z2  2  ∂ 2 uz cs (x) 1 ∂ 2 pT ∂ 2 uz 2 − c (x) = − , t ∂t2 ∂z2 c2f (x) ρ0 ∂z∂t

(3.31) (3.32) (3.33)

while equation (3.16) becomes

  ∂uy ∂ux ∂uz ∂pT 2 = −ρ0 (x)cf (x) + − ρ0 (x)c2s (x) . ∂t ∂x ∂y ∂z

(3.34)

Thus, we have four coupled equations for the four unknowns uy , uy , uz and pT . All other perturbation quantities (e.g., the magnetic pressure perturbation pm or the density perturbation ρ) may be deduced from these equations by using the relations presented earlier in Section 3.1. To progress with our analysis of these coupled equations, it is convenient to introduce Fourier components; we write ux (x, y, z, t) = ux (x) exp i(ωt − ky y − kz z), pT (x, y, z, t) = pT (x) exp i(ωt − ky y − kz z), with similar expressions for variables uy and uz . Here ω is the angular frequency of the perturbation and ky and kz are wavenumbers. The wave vector k = (0, ky , kz ) is of magnitude 1/2

k = (ky2 + kz2 ) and is confined to the yz-plane. Structuring in x prevents a similar Fourier analysis in this coordinate. Instead, we seek ordinary differential equations to determine the behaviour of the amplitudes ux (x) and pT (x). In terms of the Fourier forms, equation (3.31) yields the differential equation iω

dpT = −ρ0 (x)(kz2 c2A (x) − ω2 )ux , dx

(3.35)

while equations (3.32) and (3.33) yield the algebraic relations ωky pT = −ρ0 (x)(kz2 c2A (x) − ω2 )uy ,   c2s (x) 2 2 2 ρ0 (x)(kz ct (x) − ω )uz = −ωkz 2 pT . cf (x)

(3.36) (3.37)

A fourth relation between the four unknowns ux , uy , uz and pT is provided by equation (3.34) which gives the differential equation   1 c2s (x) dux uz = −iω pT ; (3.38) − iky uy − ikz 2 dx cf (x) ρ0 (x)c2f (x) alternatively, from equation (3.23), we may write   dux ω2 − iky uy − ikz 1 − 2 2 uz = 0. dx kz cs (x)

(3.39)

3.2 Magnetohydrodynamic Waves: Cartesian System

73

The above system of algebraic relations and first order ordinary differential equations completely describes the velocity and pressure perturbations. It may be reduced to a pair of coupled ordinary differential equations in the two unknowns ux and pT . Equations (3.36) and (3.37) give uy and uz in terms of pT . If now we eliminate uy and uz by use of equations (3.36), (3.37) and (3.39) we obtain a first order ordinary differential equation for ux : dux (kz2 c2A (x) − ω2 )(kz2 c2t (x) − ω2 ) dx

−1 4 2 2 2 2 2 2 2 ω − ω k c (x) + k k c (x)c (x) iωpT , = z s A f ρ0 (x)c2f (x)

(3.40)

to be considered together with equation (3.35). Equations (3.35) and (3.40) provide us with a pair of first order ordinary differential equations for the unknowns ux (x) and pT (x). From this pair of equations we may eliminate one or other variable to determine a single second order ordinary differential equation for one unknown. Eliminating pT in preference to ux gives (Goedbloed 1971a–c; Chen and Hasegawa 1974; Southwood 1974; Roberts 1981b)   d ρ0 (x)(kz2 c2A (x) − ω2 ) dux (3.41) = ρ0 (x)(kz2 c2A (x) − ω2 )ux , dx dx m2 (x) + ky2 where (following Roberts 1981b) we have introduced the expression m2 (x) =

(kz2 c2s (x) − ω2 )(kz2 c2A (x) − ω2 ) (c2s (x) + c2A (x))(kz2 c2t (x) − ω2 )

.

(3.42)

The sign of m2 (x) depends upon the magnitude of ω2 ; despite the notation, it may be positive or negative. Equations of the above forms also arise in a cylindrical geometry (see Appert, Gruber and Vaclavik 1974), and are more fully discussed in Section 3.3 and in Chapter 6. Alternatively, we may eliminate ux in terms of pT ; the result is the second order differential equation     dpT d 1 2 2 ρ0 (x)(kz2 c2A (x) − ω2 ) (x) + k (3.43) = m y pT . dx ρ0 (x)(kz2 c2A (x) − ω2 ) dx

3.2.1 Uniform Medium At this stage of our analysis it is of interest to recover the case of a uniform medium, for the above system of equations applies quite generally and in particular describes that special case too. With ρ0 , cs , cA and cf all constants, equation (3.41) becomes  2  2 2 2 d ux 2 2 (kz cA − ω ) − (m + ky )ux = 0, (3.44) dx2 where now m2 is a constant. Thus, either ω2 = kz2 c2A or

(3.45)

74

Magnetically Structured Atmospheres

d2 ux − (m2 + ky2 )ux = 0. (3.46) dx2 The first possibility is evidently the Alfv´en wave. The second possibility gives rise to an ordinary differential equation with constant coefficients, and this has bounded solutions of the form ux (x) = u0 exp(± ikx x),

(3.47)

for arbitrary constant u0 , provided the wavenumber kx satisfies kx2 + ky2 + m2 = 0;

(3.48)

ω4 − (c2s + c2A )(kx2 + ky2 + kz2 )ω2 + c2s c2A kz2 (kx2 + ky2 + kz2 ) = 0.

(3.49)

that is, provided

Thus we recover the standard dispersion relation for magnetoacoustic waves in a uniform medium. The expression −(ky2 +m2 ) is seen to be effectively the squared perpendicular wavenumber for the general inhomogeneous case. In the case of a uniform medium of unbounded extent m2 is seen to be negative. With m2 < 0, we see that the magnetoacoustic modes of an unbounded medium (for which kx2 , ky2 > 0) are such that ω2 /kz2 lies either between c2t and the smaller of c2s and c2A (corresponding to the slow magnetoacoustic wave) or is greater than the larger of c2s and c2A (corresponding to the fast magnetoacoustic wave). A full discussion of these modes was given in Chapter 2. 3.2.2 The Case ky = 0 Returning to the case of an inhomogeneous medium, it is worth noting the form of our equations for the special case of ky = 0, when disturbances are independent of the y-coordinate (so ∂/∂y ≡ 0) and k = (0, 0, kz ), k2 = kz2 . With ky = 0, equations (3.41) and (3.43) simplify by virtue of the cancelling of a factor (kz2 c2A (x) − ω2 ). Thus, the expression  2 2 2 kz2 c2A (x) − ω2 2 kz ct (x) − ω becomes cf 2 2 , m2 (x) + ky2 kz cs (x) − ω2 and so equation (3.41) reduces to (see, for example, Roberts 1981b)       kz2 c2t (x) − ω2 dux d ρ0 (x) c2s (x) + c2A (x) = ρ0 (x)(kz2 c2A (x) − ω2 )ux . dx kz2 c2s (x) − ω2 dx Moreover, with ky = 0 and k2 = kz2 , equation (3.40) becomes  2 2  k c (x) − ω2 dux , iωpT = −ρ0 (x)c2f (x) z2 t2 kz cs (x) − ω2 dx

(3.50)

(3.51)

where we have cancelled a common factor (kz2 c2A (x) − ω2 ). The total pressure perturbation pT is determined by the reduced form of equation (3.43) when ky = 0:   dpT d 1 = (kz2 c2s (x) − ω2 )pT . (3.52) ρ0 (x)c2f (x)(kz2 c2t (x) − ω2 ) dx ρ0 (x)(kz2 c2A (x) − ω2 ) dx

3.3 Magnetohydrodynamic Waves: Cylindrical Geometry System

75

Finally, from equation (3.36) we have (kz2 c2A (x) − ω2 )uy = 0,

(3.53)

showing that uy satisfies the wave equation. The motions ux and uz in the xz-plane occur independently of those in the y-plane. 3.3 Magnetohydrodynamic Waves: Cylindrical Geometry System In the cylindrical coordinate system r, φ, z, we take the equilibrium magnetic field to be B0 = B0 (r)ez ,

(3.54)

and the equilibrium plasma pressure, density and temperature to be p0 (r), ρ0 (r) and T0 (r), respectively. Thus the equilibrium state is structured purely in terms of the radial r-coordinate, and the basic speeds cs (r), cA (r), ct (r) and cf (r) are, in general, all functions of r. The operator grad⊥ becomes   ∂ 1 ∂ grad⊥ = , ,0 . ∂r r ∂φ For a motion u = (ur , uφ , uz ), equations (3.19) and (3.20) give ∂ 2 ur ∂ 2 ur 1 ∂ 2 pT 2 − c (r) = − , A ρ0 (r) ∂r∂t ∂t2 ∂z2 ∂ 2 uφ ∂ 2 uφ 1 1 ∂ 2 pT − c2A (r) 2 = − , 2 ρ0 (r) r ∂φ∂t ∂t ∂z  2  ∂ 2 uz cs (r) ∂ 2 uz 1 ∂ 2 pT 2 − c (r) = − , t 2 2 2 ∂t ∂z cf (r) ρ0 (r) ∂z∂t to be considered coupled with equation (3.16), which yields   ∂pT 1 ∂ 1 ∂uφ ∂uz = −ρ0 (r)c2f (r) (rur ) + −ρ0 (r)c2s (r) . ∂t r ∂r r ∂φ ∂z

(3.55) (3.56) (3.57)

(3.58)

Consider now the Fourier form of these equations. Write ur (r, φ, z, t) = ur (r) exp i(ωt − mφ − kz z)

(3.59)

for mode numbers m = 0, ±1, ±2, . . .. Symmetry in the basic equilibrium means that we may suppose that m ≥ 0 without loss of generality. With this Fourier representation, equation (3.55) yields dpT = −ρ0 (r)(kz2 c2A (r) − ω2 )ur . dr Equation (3.56) gives the algebraic relation iω

1 − ωm pT = ρ0 (r)(kz2 c2A (r) − ω2 )uφ , r

(3.60)

(3.61)

and equation (3.57) gives ρ0 (r)(kz2 c2t (r) − ω2 )uz = −ωkz

c2s (r) pT , c2f (r)

(3.62)

76

Magnetically Structured Atmospheres

while equation (3.58) is m 1 dur + ur − i uφ − ikz dr r r



 1 c2s (r) pT . uz = −iω 2 cf (r) ρ0 (r)c2f (r)

(3.63)

From these equations we may eliminate uφ and uz in favour of ur and pT , with the result that 1 d (rur ) (kz2 c2A (r) − ω2 )(kz2 c2t (r) − ω2 ) rdr      2 2 m m 1 =− ω4 − ω2 kz2 + 2 c2f (r) + kz2 kz2 + 2 c2s (r)c2A (r) iωpT . r r ρ0 (r)c2f (r) (3.64) Thus, our system may be reduced to the solution of a pair of first order ordinary differential equations relating pT and ur , namely equations (3.60) and (3.64). As expected, there is a close similarity between the equations that apply in a Cartesian coordinate system and those that apply in a cylindrical coordinate system. Comparing the various equations, we see that the role of the wavenumber ky in a Cartesian system is played in a cylindrical system by the effective ‘wavenumber’ m/r, so that the wave vector k = (0, ky , kz ) of a planar system becomes the effective ‘wave vector’

m  k = 0, , kz r in a cylindrical geometry; the magnitude of this wave vector is (kz2 + m2 /r2 )1/2 . We may eliminate pT between equations (3.60) and (3.64) to obtain   d ρ0 (r)(kz2 c2A (r) − ω2 ) 1 d (rur ) = ρ0 (r)(kz2 c2A (r) − ω2 )ur ,  2 dr r dr m2 (r) + mr2

(3.65)

with m2 given by m2 (r) =

(kz2 c2s (r) − ω2 )(kz2 c2A (r) − ω2 ) (c2s (r) + c2A (r))(kz2 c2t (r) − ω2 )

.

(3.66)

Again, we note the close similarity between the Cartesian and cylindrical systems: compare equations (3.41) and (3.65). As with the Cartesian system, the quantity m2 (r) may be positive or negative. Equation (3.65) is sometimes referred to as the Hain–L¨ust equation; an equation of related form first appeared in a paper by K. Hain and R. L¨ust dealing with considerations of stability of a cylindrical tube. In fact, Hain and L¨ust (1958) included a twisted field in their equilibrium state; the case of a twisted equilibrium field is considered in Chapter 7. Alternatively, we may eliminate ur in favour of the pressure perturbation pT , with the result:     dpT m2 1 2 2 2 1 d 2 ρ0 (r)(kz cA (r) − ω ) r = m (r) + 2 pT . (3.67) r dr ρ0 (r)(kz2 c2A (r) − ω2 ) dr r

3.4 Singularities

77

It is worth noting the reduction of the pressure equation (3.67) in the case when the medium is uniform. With ρ0 , cA and m all constants, equation (3.67) becomes r2

d2 pT dpT +r − (m2 r2 + m2 )pT = 0, 2 dr dr

(3.68)

on cancelling the factor (kz2 c2A − ω2 ). This is Bessel’s differential equation, with solutions Im (mr) and Km (mr) if m2 > 0 or Jm (|m|r) and Ym (|m|r) if m2 < 0. We discuss these solutions in detail in Chapter 6.

3.4 Singularities In the previous section we have derived the basic equations governing the propagation of magnetohydrodynamic waves in an atmosphere that is structured by a non-uniform, but unidirectional, magnetic field. These equations are evidently complex with properties that are far from clear. In this section and subsequently we explore the various properties of the waves, as deduced either from the governing partial differential equations (3.16), (3.19) and (3.20) or from the ordinary differential equations that apply in the Cartesian or cylindrical coordinate systems. Inspection of the first order ordinary differential equation that relates the gradient of the motion across the inhomogeneity to the total pressure perturbation, equation (3.40) in a Cartesian geometry or equation (3.64) in a cylindrical geometry, reveals that this equation is singular. Equations (3.40) and (3.64) are singular at locations x or r at which either of the factors (kz2 c2A − ω2 ) or (kz2 c2t − ω2 ) vanishes. Thus, provided cA and ct are distinct from one another, there are two singularities of the equations. Only when the medium is incompressible (cs infinite) do these singularities merge into one. The two singularities, corresponding to locations where either ω2 = kz2 c2A or ω2 = kz2 c2t , are referred to as the Alfv´en singularity and the slow (or cusp) singularity, respectively. They are distinctive features of the magnetohydrodynamic system, not possessed by the purely acoustic case. The occurrence of singularities associated with both the Alfv´en wave and the slow magnetoacoustic wave, but not the fast wave, is related to the fact that in a uniform medium these modes are unable to propagate across the applied magnetic field, that is, ω2 tends to zero if kz = 0 (see Chapter 2). The occurrence of these singularities was first established by K. Appert, R. Gruber and J. Vaclavik (1974); see also Goedbloed (1983) and Adam (1982, 1986). Consider the Alfv´en singularity in the Cartesian case. The mathematical nature of this singularity becomes clear if we suppose that near a singularity point x0 , where ω2 = kz2 c2A (x0 ), we may approximate the factor (kz2 c2A (x) − ω2 ) by the first term in its Taylor series, so that kz2 c2A (x) − ω2 = constant × (x − x0 ). Then the differential equation (3.40) becomes locally of the form (x − x0 )

dux = constant × pT , dx

78

Magnetically Structured Atmospheres

provided ct and cA are distinct. Similarly, the differential equation (3.35) is locally of the form dpT = constant × (x − x0 ) × ux . dx Thus, the system of differential equations has a simple pole at x = x0 . Hence, in the classification of the points of an ordinary differential equation (e.g., Bender and Orszag 1978), we see that the point x = x0 is a regular singularity of the system of differential equations. Much the same analysis applies to the slow singularity and also in a cylindrical geometry. Thus, both the Alfv´en and slow singularities are regular singularities.

3.5 Phase Mixing of the Alfv´en Wave We have seen in the Cartesian system that when ∂/∂y = 0 the motion uy is decoupled from the system and satisfies the wave equation (see equation (3.53)) ∂ 2 uy ∂ 2 uy 2 = c (x) . (3.69) A ∂t2 ∂z2 Oscillations in uy take place independently of those in ux or uz . This is the Alfv´en wave. However, the fact that the Alfv´en speed varies with x has interesting consequences. For consider the general solution of the wave equation (3.69), according to D’Alembert: uy (x, z, t) = f+ (x, z + cA (x)t) + f− (x, z − cA (x)t),

(3.70)

where f+ and f− are arbitrary functions of coordinates x and (z + cA (x)t) or (z − cA (x)t), subject only to suitable requirements of differentiability. With cA a function of x we see that the x-derivative of uy grows linearly in time t. This is the feature of phase mixing, discussed in detail by Heyvaerts and Priest (1983). We can illustrate phase mixing as follows. Consider the propagating wave uy = v0 sin kz [z − cA (x)t],

(3.71)

where v0 is the amplitude; the amplitude may be an arbitrary function of x, but for convenience we will take it to be a constant. Initially, the motion uy is in the form of a sine wave of wavelength 2π/kz and amplitude v0 ; subsequently, the initial sine wave profile is propagated with the local Alfv´en speed cA (x). This has consequences for the x-gradients of the motion, for with a variable cA (x) we see that ∂uy = −v0 cA  (x)kz t cos kz [z − cA (x)t], ∂x

(3.72)

where cA  (x) denotes the gradient of the Alfv´en speed across the field. It follows immediately from (3.72) that the x-gradient of uy grows linearly in time t (secularly), provided the medium is non-uniform (cA  (x) = 0). The timescale for this secular growth is τ mixing ≡

1 . cA  (x)

(3.73)

Regions in space where the Alfv´en speed is non-uniform lead to the build-up of x-gradients in the motion, on a timescale τ mixing .

3.6 Two Special Cases

79

A specific illustration of the secular growth timescale associated with phase mixing may be of interest. Consider the edge of a sunspot, where the Alfv´en speed changes from say 10 km s−1 in the spot to zero outside it, over a distance of order 102 km. Then cA  (x) is of order 10−1 s−1 , and phase mixing may be expected to occur on a timescale of about 10 s, which is rapid. But in the corona, where the Alfv´en speed is very large, changing from say 1000 km s−1 within an active region loop to 2000 km s−1 outside it, over a distance of order 103 km, the timescale is reduced to 1 second, and phase mixing is very rapid indeed. Phase mixing occurs also for standing waves. Consider the motion uy = u0 sin(kz z) cos[kz cA (x)t].

(3.74)

This represents a standing wave of amplitude u0 (which may be an arbitrary function of x), and is a solution of equation (3.69); if we choose kz = nπ/L for integer n, then the motion is zero at z = 0 and L. The motion is bounded by |u0 |, but x-gradients of the motion grow secularly on the phase mixing timescale. The characteristic spatial scale that develops across the field lines is  mixing  1 τ 1 mixing ≡ . (3.75) = L  kz cA (x)t t kz Thus, initially (t = 0) the motion is everywhere in the form of a sine profile (fixed at z = 0, L), but if the Alfv´en speed varies from location to location then neighbouring field lines begin to oscillate out of phase. Comparing the motion uy at two separate x-locations, we see that while the motion is initially in phase, thereafter the motion becomes out of phase, and may later be in phase again but later still out of phase again. Phase mixing is occurring. A similar phenomenon occurs in cylindrical geometry. Consider the wave equations (3.55)–(3.58) and set ∂/∂φ = 0, corresponding to symmetric disturbances. Then equation (3.56) reduces to ∂ 2 uφ ∂ 2 uφ = c2A (r) 2 , 2 ∂t ∂z

(3.76)

and the motion uφ is decoupled from the system. Equation (3.76) describes a torsional oscillation uφ . This is the torsional Alfv´en wave; it evidently exhibits phase mixing, much as in the Cartesian system, with radial gradients ∂uφ /∂r growing secularly on a timescale of (dcA /dr)−1 . Finally, we note that the build-up of gradients in an oscillation, due to phase mixing, may have important consequences for the damping of waves by dissipative processes. Such processes have been ignored here but are considered in Chapter 12.

3.6 Two Special Cases There are two special cases worth examining in detail, namely the incompressible plasma and the cold plasma. These cases may be deduced from the general case by examining the limits of cs → ∞ (incompressible) or cs → 0 (cold) case. This is mathematically equivalent to examining a plasma for which the adiabatic exponent γ is permitted to be

80

Magnetically Structured Atmospheres

arbitrarily large or arbitrarily small, rather than confined to the physically accepted range of between 1 and 2. In the cold case the equilibrium state satisfying (3.2) must be such that the magnetic field B0 is uniform, since setting cs = 0 implies p0 = 0 too. There is no such restriction in the incompressible case (save that (3.2) holds). In either case, the density ρ0 may have an arbitrary distribution in the coordinate perpendicular to the equilibrium magnetic field.

3.6.1 The Incompressible Plasma The case of an incompressible medium follows from the general case by taking the limit cs → ∞, for which ct → cA and cf → ∞. Thus the fast wave is lost from the system (or at least sent off to infinity), leaving the Alfv´en wave and the slow wave. The motion u satisfies div u = 0 and   2 ∂ 2u 1 ∂pT 2 ∂ u − cA 2 = − grad . (3.77) ρ0 ∂t ∂t2 ∂z In the Cartesian system the governing differential equations are dpT = −ρ0 (x)(kz2 c2A (x) − ω2 )ux , dx dux ρ0 (x)(kz2 c2A (x) − ω2 ) = −iωk2 pT , dx iω

(3.78) (3.79)

with k2 = ky2 + kz2 . Again, we note the regular singularity at points x = x0 where ω2 = kz2 c2A (x); this is the Alfv´en singularity. Elimination of pT yields the second order differential equation   dux d (3.80) ρ0 (x)(kz2 c2A (x) − ω2 ) = ρ0 (x)k2 (kz2 c2A (x) − ω2 )ux , dx dx whereas elimination of ux gives ρ0 (x)(kz2 c2A (x) − ω2 )

  dpT d 1 = k2 pT . dx ρ0 (x)(kz2 c2A (x) − ω2 ) dx

(3.81)

Equations (3.80) and (3.81) may be compared with (3.41) and (3.43), where it is seen that the incompressible form of the general equations simply replaces the effective squared wavenumber m2 (see equation (3.42)) by kz2 . Equations of the form (3.80) and (3.81) were first discussed by Barston (1964), Sedlacek (1971), Uberoi (1972) and Tataronis and Grossmann (1973); see also Roberts (1981b) and Hasegawa and Uberoi (1982). 3.6.2 The β = 0 Plasma The case of a zero β plasma (the cold case) is recovered from the general treatment by setting cs = 0 and requiring that B0 is a constant, though ρ0 may vary perpendicular to the direction of the applied magnetic field. Thus, as a consequence, although both density ρ0 and Alfv´en speed squared c2A may vary across the applied field, their product ρ0 c2A is a

3.6 Two Special Cases

81

constant. With cs = 0 we have ct = 0 and cf = cA . Thus the slow wave is lost from the system (or at least squeezed down to ω2 = 0), leaving the Alfv´en wave and the fast wave. There is no motion along the applied field (uz = 0), simply because the forces that drive wave motions in the β = 0 case are purely magnetic, arising from the j × B0 force, and so are perpendicular to the applied field. With these reductions, the governing wave equation for motions u⊥ perpendicular to the applied magnetic field is (see (3.19))   2 1 ∂ 2 u⊥ ∂pT 2 ∂ u⊥ − cA = − grad⊥ , (3.82) ρ0 ∂t ∂t2 ∂z2 as in the incompressible plasma, but now instead of the requirement that div u⊥ = 0 we have (see equation (3.16)) ∂pT = −ρ0 c2A div u⊥ . ∂t

(3.83)

It is convenient to introduce Cartesian coordinates, writing u⊥ = (ux , uy , 0). Then we may eliminate the pressure perturbation pT between equations (3.82) and (3.83) to give   ∂uy ∂ 2 ux ∂ ∂ux ∂ 2 ux 2 2 − c (x) = c (x) + , (3.84) A A ∂x ∂x ∂y ∂t2 ∂z2   ∂ 2 uy ∂ 2 uy ∂uy ∂ ∂ux 2 2 − c (x) = c (x) + . (3.85) A A ∂y ∂x ∂y ∂t2 ∂z2 Thus we obtain a pair of coupled wave-like equations for the velocity components ux and uy . Alternatively, we may retain pT and write equations (3.82) and (3.83) in differential equation form, yielding dpT = −ρ0 (x)(kz2 c2A (x) − ω2 )ux , dx 1 dux iωpT , − iky uy = − dx ρ0 (x)c2A (x) iω

(3.86) (3.87)

with uy and pT being related by ωky pT = −ρ0 (x)(kz2 c2A (x) − ω2 )uy .

(3.88)

Eliminating uy from this system leads to the pair of first order differential equations  2 2  k cA (x) − ω2 dux 1 (3.89) =− iωpT , dx ρ0 c2A kz2 c2A (x) − ω2 dpT (3.90) = −ρ0 (x)(kz2 c2A (x) − ω2 )ux , iω dx where k2 = ky2 + kz2 . Finally, elimination of pT gives  2 2   kz cA (x) − ω2 dux (kz2 c2A (x) − ω2 ) d ux , = dx k2 c2A (x) − ω2 dx c2A (x)

(3.91)

82

Magnetically Structured Atmospheres

whereas elimination of ux yields    2 2  c2A (x) k cA (x) − ω2 dpT d = pT . dx kz2 c2A (x) − ω2 dx kz2 c2A (x) − ω2

(3.92)

Of course, these equations are consistent with the general ones (3.41) and (3.43), from which they follow on setting cs = ct = 0 and noting that ρ0 (x)c2A (x) is a constant.

4 Surface Waves

4.1 Introduction In the previous chapter we have discussed the general nature of magnetohydrodynamic waves in a structured atmosphere. Here we apply that theory specifically to a structured medium which consists of two uniform media, with differing properties, that are joined at a single straight interface. This represents the simplest example of a structured medium and serves conveniently to illustrate the main new feature of such a medium, namely its ability to support surface waves which propagate along the interface. The distinctive property of a surface wave is that the wave is essentially confined to the vicinity of the interface along which it propagates, disturbing only slightly the medium far from the interface. We consider, then, the circumstances of a medium that is structured in the coordinate perpendicular to an applied magnetic field B0 . The equilibrium magnetic field is taken to be aligned with the z-axis of a Cartesian coordinate system x, y, z: B0 = B0 (x)ez .

(4.1)

Thus the medium is structured in x with variations in equilibrium pressure p0 (x), density ρ0 (x) and temperature T0 (x). The requirement of pressure balance in equilibrium (see equation (1.1) of Chapter 3) means that   B20 (x) d p0 (x) + = 0, dx 2μ

(4.2)

and so any spatial variation in magnetic pressure B20 /2μ is compensated for by spatial variations in the plasma pressure p0 . We are interested here in the special case when the equilibrium magnetic field changes rapidly from one location to another. In reality such a change must take place over a finite distance, but in many situations that distance is much shorter than the scale of the wave phenomena that we are interested in, so that in a sense the wave ‘sees’ a discontinuity in the equilibrium magnetic field strength. Accordingly, we consider the special case of a discontinuous profile. Suppose that the strength B0 (x) of the equilibrium magnetic field is given by 83

84

Surface Waves

Figure 4.1 The equilibrium configuration of a magnetic interface. The interface is the z-axis (x = 0) separating plasma in x < 0 from plasma in x > 0. In general, the magnetic field strength B0 , plasma density ρ0 , pressure p0 and temperature T0 of the plasma in x < 0 differs from those values Be , ρe , pe and Te , respectively, on the other side of the interface, in x > 0. The equilibrium must satisfy pressure balance (4.5). (From Jain and Roberts 1991.)

B0 (x) =

 Be ,

x > 0,

B0 ,

x < 0,

(4.3)

for constants B0 and Be . This represents an atmosphere in which conditions change discontinuously across the interface x = 0. The equilibrium plasma pressure p0 (x), density ρ0 (x) and temperature T0 (x) are taken to be structured in a similar way:  x > 0, p e , ρ e , Te , (4.4) p0 (x), ρ0 (x), T0 (x) = p0 , ρ0 , T0 , x < 0, for constants p0 , pe , ρ0 , ρe , T0 and Te . Quantities with a suffix ‘0’ refer to the region x < 0, and those with a suffix ‘e’ to the region x > 0. Figure 4.1 illustrates the equilibrium configuration. The various constants describing the equilibrium conditions either side of the interface cannot be chosen arbitrarily, for total pressure balance (4.2) must pertain: p0 +

B20 B2 = pe + e . 2μ 2μ

(4.5)

Combined with the ideal gas law, p0 (x) = kB ρ0 (x)T0 (x)/m, ˆ pressure balance implies a connection between the densities, sound speeds and Alfv´en speeds either side of the interface:

4.2 Parallel Propagation

85

c2 + 1 γ c2 ρe = s 12 2A , ρ0 c2se + 2 γ cAe

(4.6)

where cs and cA are the sound and Alfv´en speeds in x < 0 and cse , cAe are the corresponding speeds in x > 0. Notice that the interface x = 0 is a current sheet if B0 = Be , for then the equilibrium profile for field strength is a step function, the derivative of which gives rise to a delta function. In our view of the interface as a mathematical idealization of a region of rapidly changing magnetic field strength, the equilibrium current density j0 (x) (which follows from Ampere’s law) is essentially zero everywhere away from the interface x = 0 but is non-zero in the vicinity of the interface. Thus a surface concentration of current arises.

4.2 Parallel Propagation The starting point for our discussion is to consider the general differential equations governing motions in an arbitrarily structured medium, subject only to the requirement (4.2) of pressure balance. Those equations have been presented in detail in Chapter 3. To make principles clear we will begin with the special case of motions u = (ux , 0, uz ) that lie purely in the xz-plane, with a wave vector k (= (0, 0, kz )) that is aligned with the direction of the applied magnetic field. This is the case of parallel propagation. The more general case of non-parallel propagation is treated in Section 4.4. With ky = 0 and vy = 0, the component of motion ux perpendicular to the equilibrium magnetic field satisfies the second order differential equation (see equation (3.41) of Chapter 3)  2 2    k c (x) − ω2 dux d ρ0 (x)c2f (x) z2 t2 = ρ0 (x)(kz2 c2A (x) − ω2 )ux , dx kz cs (x) − ω2 dx

(4.7)

with the perturbation pT in total pressure given by  iωpT (x) =

−ρ0 (x)c2f (x)

 kz2 c2t (x) − ω2 dux . kz2 c2s (x) − ω2 dx

(4.8)

These equations apply throughout the structured medium. Now in each of the regions x > 0 and x < 0 the medium is uniform. So on either side of the interface equations (4.7) and (4.8) apply and their coefficients are constants. Thus, d2 ux − m2 ux = 0, dx2

(4.9)

where m2 =

(kz2 c2s − ω2 )(kz2 c2A − ω2 ) (c2s + c2A )(kz2 c2t − ω2 )

,

m2 takes on the values m20 in x < 0 and m2e in x > 0.

c2t =

c2s c2A c2s + c2A

;

86

Surface Waves

Notice that the sign of m2 is not specified; for kz2 > 0 and ω2 real, it may be positive or negative, depending upon the magnitude of ω2 /kz2 . To allow for the case m2 < 0, we write n2 ≡ −m2 . Then equation (4.9) has solutions  exp(± inx), for m2 < 0, ux ∝ exp(± mx), for m2 > 0. In a uniform unbounded medium, extending across −∞ < x < ∞, the solutions exp(± mx) are rejected as unbounded functions. So, in an infinite medium, solutions are of the form exp(± inx) with n2 > 0. Requiring disturbances to have an x-dependence of the form exp(−ikx x), for wavenumber kx in the x-direction, yields kx2 = n2 , which may be rewritten in the form ω4 − (kx2 + kz2 )(c2s + c2A )ω2 + (kx2 + kz2 )kz2 c2s c2A = 0.

(4.10)

Thus we recover the standard form of the dispersion relation describing the usual fast and slow magnetoacoustic waves of a uniform unbounded medium (Chapter 2), here confined to the xz-plane. In a medium consisting of two joined uniform media the argument that only solutions with m2 < 0, n2 > 0 arise no longer applies, for now the solution exp(−mx) is bounded in x > 0, provided we choose m > 0, and the solution exp(mx) is bounded in x < 0. Thus, if we impose the conditions that ux (x) tends to zero as x → +∞ and as x → −∞, then the appropriate solution of equation (4.9) is  ve exp(−me x), x > 0, (4.11) ux (x) = v0 exp(m0 x), x < 0, where v0 and ve are arbitrary constants, and m0 and me are chosen to be positive. Explicitly, m20 and m2e are m20 =

(kz2 c2s − ω2 )(kz2 c2A − ω2 ) (c2s + c2A )(kz2 c2t − ω2 )

m2e =

,

(kz2 c2se − ω2 )(kz2 c2Ae − ω2 ) (c2se + c2Ae )(kz2 c2te − ω2 )

,

(4.12)

with c2t =

c2s c2A c2s + c2A

,

c2te =

c2se c2Ae c2se + c2Ae

.

4.2.1 Boundary Conditions at an Interface The solutions (4.11) must be connected by imposing conditions of continuity across the moving interface displaced from the equilibrium state x = 0. We require that ux (x) and pT (x) are continuous across the undisturbed interface x = 0. We can obtain these conditions from either a mathematical or a physical viewpoint (see, for example, Roberts 1981b). Consider first the mathematical viewpoint. Equation (4.7) applies for arbitrary equilibrium profiles, subject only to pressure balance (4.2). Consider a discontinuous profile such as

4.3 The Surface Wave Dispersion Relation

87

(4.3) as arising from a smoothly, though rapidly, varying profile in some suitable limit. For example, the smoothly varying profile   1 1 x B0 (x) = (Be + B0 ) + (Be − B0 )tanh 2 2 l ranges from B0 as x → −∞ to Be as x → +∞, with a spatial scale l (> 0); the rapidity with which the profile changes at intermediate values of x depends upon the size of l. For very small l the profile, though continuous, closely resembles the discontinuous profile (4.3); in the limit l → 0, B0 (x) becomes the discontinuous profile. Now consider the governing differential equation (4.7) as applying to such rapidly varying profiles, which may become discontinuous in an appropriate limit. Firstly, it is evident that ux (x) must remain continuous in such a process. For suppose it developed a discontinuity at some location. Then its derivative dux /dx would have a stronger singularity, namely a delta function, at that location; but this would then be in contradiction to equation (4.7), giving rise to the derivative of a delta function with no other terms in equation (4.7) with which to achieve balance. Thus, we require that ux is continuous across an interface. Moreover, while dux /dx may be discontinuous across the interface, the expression within the curly brackets { } on the left-hand side of equation (4.7), namely,  2 2  kz ct (x) − ω2 dux 2 , ρ0 (x)cf (x) 2 2 kz cs (x) − ω2 dx must be continuous or otherwise its derivative would contain a delta function, in contradiction to the right-hand side of (4.7). Equation (4.8) shows that this latter requirement is equivalent to the continuity of the total pressure perturbation pT (x). Thus, even if ρ0 (x) and cA (x) are discontinuous across x = 0, we require that ux (x) and pT (x) be continuous. We may prefer to view our boundary conditions from another approach. Let the displaced interface be described by the equation x = ξ(z, t). Then x − ξ(z, t) = 0, and when inserted in the derivative following the motion we obtain:   d ∂ξ ∂ξ ∂ 0 = (x − ξ ) = + u · grad (x − ξ ) = − + ux − uz , dt ∂t ∂t ∂z where ux , uz may be determined either side of the interface. In the linearized calculation here ux , uz and ξ are taken to be small and so the product uz ∂ξ/∂z is negligible; thus ux (ξ , z, t) = ∂ξ/∂t, whichever side of the interface we evaluate ux . Hence ux must be continuous across the interface. Furthermore, consistent with our linear treatment we evaluate ux at the undisturbed location (x = 0) of the interface, corresponding to the first term in a Taylor series expansion of ux (ξ , z, t) about ξ = 0. Finally, we may argue that the pressure perturbation pT must be continuous for otherwise an unbalanced force arises on the disturbed interface. 4.3 The Surface Wave Dispersion Relation Armed with the matching conditions of continuity of ux (x) and pT (x) across the undisturbed interface x = 0, we turn now to a determination of the dispersion relation that governs motions. The continuity of the velocity ux (x) requires that ve = v0 , and so

88

Surface Waves

 ux (x) =

v0 exp(−me x),

x > 0,

v0 exp(m0 x),

x < 0,

(4.13)

with m0 and me chosen to be positive. There remains the condition of continuity of pT (x) which implies that  2 2  2 2   k c − ω2 kz cte − ω2 2 2 ρ0 (c2s + c2A ) z2 t2 = ρ (c + c ) m me . 0 e se Ae kz cs − ω2 kz2 c2se − ω2 This may be rewritten in the more compact form 2 2 2 −1 ρ0 (kz2 c2A − ω2 )m−1 0 + ρe (kz cAe − ω )me = 0.

(4.14)

Equation (4.14) is the dispersion relation describing the parallel propagation of magnetoacoustic surface waves at a single magnetic interface; it applies with m0 , me > 0. The dispersion relation is transcendental. One property of surface waves follows almost immediately from the dispersion relation (4.14). Denote by c the speed ω/kz of a wave propagating along the interface. Then (4.14) may be rewritten in the forms (cf. Roberts 1981b)   R 1 2 2 (4.15) (c2Ae − c2A ) = c2Ae − (c2 − c2A ), c = cA + 1+R 1 + R Ae where R=

ρe m0 ρ0 me

(4.16)

and so is positive. Suppose that cAe > cA ; then, since R > 0 it follows from the first equation in (4.15) that c2 > c2A , whereas from the second form it follows that c2 < c2Ae . Hence c2A < c2 < c2Ae . A similar argument applies if instead cA > cAe , with then c2Ae < c2 < c2A . Hence, in general, the speed c (≡ ω/kz ) of a surface wave must lie between the two Alfv´en speeds, cA and cAe , of the medium. Note that a surface wave propagates without dispersion, i.e., the speed c is independent of the wavenumber kz . This is as expected, given that the medium either side of the interface is uniform and unbounded and so has no natural lengthscale by which the scale of a wave (surface or otherwise) may be measured. Consequently all waves propagate with speeds that are independent of wavelength, and so are dispersionless. Thus the speed c is both the phase speed and the group speed of a surface wave propagating parallel to the interface. The forms in (4.15) suggest a single mode, but this is misleading. It must be remembered that R depends on c and so (4.15) is transcendental, and the number of its roots (and therefore the number of surface modes in the system) has yet to be determined. However, given that equation (4.14) (or (4.15)) describes magnetoacoustic waves, of which there are two (the fast and slow modes), we may expect that the surface wave dispersion relation (4.14) may possess up to two roots, which we may refer to as slow and fast surface waves (Roberts 1981b). Whether one, two or even no surface modes are actually permitted by the dispersion relation depends upon the detailed nature of the media either side of the interface.

4.3 The Surface Wave Dispersion Relation

89

Notice also that surface waves are compressive (div u = 0, pT = 0), coming from the magnetoacoustic branch of our equations (equation (4.7)). Therefore, the term ‘Alfv´enic surface wave’, sometimes seen in the literature, is in fact a misnomer, implying a nonexistent connection with the Alfv´en wave; it will not be used here. We consider now some special cases of the dispersion relation (4.14).

4.3.1 The Hydromagnetic Surface Wave of an Incompressible Medium The simplest case to consider is that of an incompressible fluid, when the surface wave dispersion relation applies to wave motion on the interface between two liquids. We can obtain the appropriate reduction of (4.14), or its equivalent forms in (4.15), as follows. When the sound speeds cs and cse tend to infinity, the effective wavenumbers m0 and me reduce to kz , taken to be positive. So R = ρe /ρ0 and equation (4.15) reduces to c2 =

ρ0 c2A + ρe c2Ae . ρ0 + ρe

(4.17)

The densities ρ0 and ρe of the two incompressible fluids are arbitrary. Thus, in an incompressible fluid there is only one surface wave and it propagates parallel , given by to the interface at a mean Alfv´en speed, cmean A   ρ0 c2A + ρe c2Ae 1/2 ≡ ; (4.18) cmean A ρ0 + ρe this speed is intermediate between the two Alfv´en speeds of the medium. It turns out that the speed defined by (4.18) plays an important role in magnetic flux tube waves, where it describes the kink mode (see Chapter 6). Now in the incompressible limit (cs  cA ) of magnetohydrodynamics the fast magnetoacoustic wave goes off to infinity and the slow wave remains, with the slow speed ct becoming the Alfv´en speed cA (see Chapter 2). So equation (4.17) describes the propagation of the slow surface wave in the incompressible limit. We refer to this as the hydromagnetic surface wave.

4.3.2 Surface Waves on a Magnetic–Non-Magnetic Interface A special case of wide application arises when one side of the interface is field-free, giving a magnetic–non-magnetic interface. For example, this circumstance has application to a sunspot penumbral magnetic field overlying an essentially field-free convection zone. Suppose that the region x > 0 is a field-free plasma (Be = 0), with the pressure balance condition (cf. equation (4.5)) now describing a balance of the total plasma plus magnetic pressure in the magnetic atmosphere with the plasma pressure in the field-free plasma: p0 + The density ratio ρe /ρ0 is now given by

B20 = pe . 2μ

(4.19)

90

Surface Waves

c2 + 1 γ c2 ρe = s 22 A . ρ0 cse

(4.20)

With cAe = 0, the magnetoacoustic dispersion relation (4.14), written in terms of the speed c (= ω/kz ), becomes 2 −1 ρ0 (c2A − c2 )m−1 0 = ρe c me ,

with m20 = kz2

(c2s − c2 )(c2A − c2 ) (c2s + c2A )(c2t − c2 )

,

(4.21)

  c2 m2e = kz2 1 − 2 cse

(4.22)

and m0 and me positive. Now we have seen that in general the speed of a surface wave lies between the two Alfv´en speeds of the media, and so here the speed c of a surface wave must lie below the Alfv´en speed cA of the magnetic atmosphere: 0 < c < cA . Moreover, since we require me > 0, we have c < cse . Thus c < min (cse , cA ). We also require that m0 > 0; an examination of the above expression for m20 shows that m20 > 0 for c lying between 0 and ct , or between min (cs , cA ) and max (cs , cA ). It is natural to identify these two regions with slow and fast surface waves, respectively. To unravel the structure of the dispersion relation (4.21) further, it is convenient to take a graphical approach. Rewrite (4.21) in the form f (c) = g(c), where f (c) = (c2A − c2 )1/2



c2t − c2 c2s − c2

1/2 ,

g(c) =

c2 , 2 (cse − c2 )1/2

(4.23)  =

ρe ρ0



c2se c2s + c2A

1/2 .

(4.24) The function f depends only upon the structure in the magnetic medium x < 0, together with c2 , and g depends only upon that in the field-free region (together with c2 ); the role of the parameter  is simply to give weight to these functions, but it does not change their basic structure. Note that f is zero at c2 = c2t and c2A , and goes off to ∞ as c2 → c2s ; g vanishes at c2 = 0 and goes off to ∞ as c2 → c2se . Now we have seen that c < min (cse , cA ); additionally, we see from the form of f that c cannot lie between ct and cs . Therefore, either c < ct or c > cs . Hence, if c < ct then a slow surface wave can arise, whereas if c > cs , then a fast surface wave can arise; these modes must also satisfy c < min (cse , cA ) and, then, whether they in fact occur will depend upon (4.23) being satisfied. Figure 4.2 gives a sketch of the behaviour of functions f and g. The occurrence of a slow wave, with c2 < min (c2se , c2t ), is clear for c2se < c2s , and when c2se > c2s it is evident that a fast wave arises. From such a graphical approach we may conclude that for a magnetic–non-magnetic interface there is always a slow surface wave, save only that the sound speed within the magnetic atmosphere is non-zero. Additionally, there is also a fast wave provided that the field-free medium is warmer than the magnetic medium and the Alfv´en speed exceeds the sound speed within the field, i.e., provided cse > cs and cA > cs .

4.3 The Surface Wave Dispersion Relation

f

91

g

Figure 4.2 The behaviour of f (c) and g(c) as functions of c, for c2A > c2s ; the illustration is for c2t < c2se < c2s < c2A . Specifically, with γ = 5/3, we have set c2A = 2c2s and c2se = 5c2s /6, giving √ c2t = c2A /3, ρe /ρ0 = 16/5 and  = 8 10/15. Modes are given by the intersection of f and g, with  simply scaling g. In the case here, for which cse < cs , there is one mode, a slow surface wave. (When cse > cs , there are two modes, a slow surface wave and a fast surface wave.)

We may give a more quantitative discussion, complementary to a graphical approach, by noting that the square roots embedded in m0 and me may be removed from the dispersion relation (4.21) by squaring. The result is a cubic in c2 : (c2t − c2 )(c2A − c2 )(c2se − c2 ) = 2 (c2s − c2 )c4 .

(4.25)

However, the process of squaring that leads to (4.25) may introduce spurious roots which satisfy the cubic but not the original dispersion relation (4.21) and its constraints that me > 0 and m0 > 0. We can illustrate (4.25) most simply by looking at cases that give rise to quadratic equations. This occurs: when (i) cs = cse ; (ii) cs = cA ; (iii) cs = 0; or when (iv)  = 1. Case (i) corresponds to the temperatures either side of the magnetic interface being equal, for which ρe > ρ0 . Case (ii) is when the sound and Alfv´en speeds in the magnetic field are equal, for which ρe /ρ0 > c2s /c2se . Case (iii) is when the magnetic pressure within the field dominates the plasma pressure there, so that we may set the plasma pressure p0 to zero and pressure balance is then simply pe = B20 /2μ. We may treat cases (i)–(iii) together. Case (iv) requires a separate discussion. For each of the four cases we may remove a factor from (4.25); such factors give spurious roots, leading either to me = 0 or m0 = 0 or c = 0, and are rejected. Then, for cases (i)–(iii), there remains an equation of the form (1 − 2 )c4 − (a2 + b2 )c2 + a2 b2 = 0, with  defined in equation (4.24). The three cases are covered by the choices: case (i) a = cA and b = ct ; case (ii) a = max (cse , ct ) and b = min (cse , ct ); case (iii) a = max (cse , cA )

92

Surface Waves

and b = min (cse , cA ). Thus, in each case, a ≥ b. The roots of the above quadratic are given by 2(1 − 2 )c2 = (a2 + b2 ) ± [(a2 − b2 )2 + 42 a2 b2 ]1/2 . A consideration of the plus (+) root shows that in each case it is spurious, because it violates the inequality c < min (cse , cA ). Thus, only the root 2(1 − 2 )c2 = (a2 + b2 ) − [(a2 − b2 )2 + 42 a2 b2 ]1/2

(4.26)

remains. However, observe that both sides of this equation vanish if 2 = 1, so the (1−2 ) may be factored out of the expression. This is done by multiplying top and bottom of the right-hand side appropriately so as to remove the square root expression from the numerator (opposite to the usual algebraic practice of clearing radicals from a denominator). The result is c2 =

2a2 b2 . a2 + b2 + [(a2 − b2 )2 + 42 a2 b2 ]1/2

(4.27)

Some specific illustrations are useful. To illustrate case (i) cs = cse , suppose additionally equal and consequently the density that cA = cse , so now all three speeds cse , cA and cs are√ √ ratio is ρe /ρ0 = 1+γ /2 and the factor  = (1+γ /2)/ 2. Then a = cA , b = ct = cA / 2 and equation (4.27) gives 2 c2 = ; 2 2 3 + (γ + 4γ + 5)1/2 cA with γ = 5/3, we obtain c ≈ 0.54 cA . This is below the slow speed ct and so the mode is a slow surface wave. Thus, in this illustration the slow surface wave propagates with a speed that is some 54% of the Alfv´en speed within the magnetic field. To illustrate case (ii) cs = cA , suppose additionally that the densities either side of the interface are equal (ρe = ρ0 ), so that c2se = (1 + γ /2)c2s together with cs = cA ; the field√ free medium is warmer than the magnetic region (cse > cs ). Also a = cse , b = ct = cA / 2 and 2 = (1 + γ /2)/2. Equation (4.25) then yields a wave speed of c ≈ 0.63 cA . This also is the slow surface wave. Finally, to illustrate case (iii) cs = 0, we set cse = cA , so that the sound speed in the field-free medium equals the Alfv´en speed in the field. Then ρe = (γ /2)ρ0 ,  = γ /2 and a = b = cA , yielding c ≈ 0.74 cA . In this case there can be no slow surface wave, since ct = 0. So this is the fast surface wave, propagating (under the circumstances illustrated) at about 74% of the Alfv´en speed. The above three cases demonstrate the occurrence of both fast and slow surface waves, but the equality of various speeds chosen for analytical tractibility prevented the simultaneous occurrence of both fast and slow surface waves. Such a possibility is apparent from our graphical treatment. In fact, the case (iv)  = 1 is sufficiently rich as to demonstrate all three possibilities, namely: the occurrence of a slow surface wave only; the occurrence of a fast surface wave only; and, finally, the simultaneous occurrence of both fast and slow surface waves.

4.3 The Surface Wave Dispersion Relation

93

Consider, then, the case  = 1. Besides the spurious root c2 = 0, equation (4.25) yields (c2t + c2A + c2se − c2s )c4 − (c2t c2A + c2t c2se + c2A c2se )c2 + c2t c2A c2se = 0;

(4.28)

with  = 1, the sound speed cse in the field-free medium satisfies c2se =

(c2s + 12 γ c2A )2 (c2s + c2A )

.

Observe that for 1 ≤ γ ≤ 2, c2s ≤ c2se ≤ c2s + c2A . The variety of cases embedded in equation (4.28) provides a convenient illustration of the various surface waves that can arise. For example, setting cs = 0 (giving some overlap with case (iii) above) equation (4.28) yields c2 = 0 together with c2 =

c2A c2se c2A + c2se

,

giving a speed that is smaller than either of cA or cse . Since ct = 0, this is the fast surface wave, propagating with a speed γ cA /(4 + γ 2 )1/2 ≈ 0.64 cA , for γ = 5/3. For our second example we set cA = cs (giving some overlap with case (ii) above), to obtain c2 =

c2t c2se c2t + c2se

(together with the spurious c2 = c2A ). This is the slow surface wave, propagating slower √ than either ct (= cA / 2) or cse ; with γ = 5/3 we obtain c ≈ 0.62 cA (compared with ct ≈ 0.71 cA and cse ≈ 1.29 cA ). Finally, to obtain both a slow and a √ fast surface wave √ we need to choose cA > cs , so set c2A = 2c2s . Then cse = (γ + 1)cA / 6 and ct = cA / 3. With γ = 5/3, solving the quadratic (4.28) gives c ≈ 0.55 cA for the slow surface wave and c ≈ 0.80 cA for the fast surface wave, to be compared with ct ≈ 0.58 cA and cse ≈ 1.09 cA . 4.3.3 The β = 0 Medium In the above sub-section we have seen that both fast and slow surface waves may exist on the interface between magnetic and nonmagnetic plasmas. A surface wave exists, even in the extreme when the plasma β (the ratio of the equilibrium plasma pressure to the magnetic pressure) is set to zero. From this, it might be thought that generally there is always a surface wave, fast or slow, on an interface between two plasmas, whatever the ordering of the various speeds that describe the equilibrium. In fact, this is not the case: in some circumstances no surface wave is permitted. We may demonstrate such a case by considering the circumstances when both sides of the interface are strongly magnetic. This situation is in fact of wide interest, for it arises in, for example, the corona and in the magnetosphere, both being plasmas where magnetic pressure dominates over plasma pressure. This is the case of a low β plasma. To consider the circumstances when both sides of an interface are magnetically dominated we set cs = cse = 0 (corresponding to p0 = pe = 0), giving the extreme of

94

Surface Waves

β = 0. In this extreme the equilibrium magnetic field must now be uniform (B0 = Be ), to satisfy equation (4.5), but across the interface density differences (ρ0 = ρe ) arise. With cs = cse = 0, equation (4.12) gives     c2 c2 2 2 2 2 m0 = kz 1 − 2 , me = kz 1 − 2 . (4.29) cA cAe Now for a surface wave we require both m20 > 0 and m2e > 0, and so c2 must be less than both c2A and c2Ae . But this contradicts the general requirement that the speed of a surface wave lies between the two Alfv´en speeds. Hence, surface waves are unable to propagate parallel to the magnetic interface between two β = 0 plasmas. So, considering parallel propagation, no surface waves exist on the magnetic interface between two β = 0 plasmas. However, as we shall see in Section 4.4, a surface wave may in fact propagate at an angle θ = 0 to the magnetic interface between two β = 0 plasmas.

4.3.4 Properties of Surface Waves There are two fundamental and distinctive properties of a surface wave. Firstly, its speed of propagation parallel to the interface is intermediate between the two Alfv´en speeds of the medium. The interface may support zero, one, or two surface waves propagating parallel to the interface, depending upon the ordering of the various speeds that describe the equilibrium. In the case of a magnetic–non-magnetic interface, then, the speed of a surface wave is less than the Alfv´en speed within the field. Such an interface always supports a slow surface wave, provided the sound speed cs within the field is non-zero; the slow surface wave propagates with a speed that is less than the smaller of the sound speed cse in the field-free medium and the slow speed ct within the magnetic region. When the fieldfree plasma is warmer than the magnetized plasma and the Alfv´en speed exceeds the sound speed within the field (so that both cse > cs and cA > cs ), then a fast surface wave arises and propagates with a speed that lies between cs and the smaller of cse and cA . The second distinctive property of a surface wave is that it is confined to within the neighbourhood of the interface on which it propagates. This makes a surface wave an attractive means of communication between different locations on or near the interface, the surface wave being guided by the interface. In general a surface wave penetrates a distance m−1 either side of the interface, with the effective wavenumber m taking values appropriate to the media either side of the interface and as determined by the dispersion relation (4.14). Motions ux , given by equation (4.13), are seen to decline by the exponential e in the penetration distance m−1 . Other quantities, such as the total pressure pT , decline on the same spatial scale. In the case of the hydromagnetic surface wave of an incompressible fluid the penetration distance is kz−1 either side of the interface. Denote by λ (≡ 2π/kz ) the wavelength of the surface wave. Then the hydromagnetic surface wave is confined to a region λ/2π either side of the interface; this corresponds to a penetration distance either side of the interface of about 16% of the wavelength. A specific illustration may be helpful. Consider the penumbral boundary of a sunspot, modelled by setting Be = 0 to represent the penumbral field overlying a field-free medium.

4.4 Non-Parallel Propagation

95

In sunspots compressibility and gravity effects are in fact important, but for illustrative purposes we will ignore this here. Suppose that the plasma density within the field is roughly comparable to that in the field-free medium, so that ρ0 ≈ ρe . Then, the hydro√ magnetic surface wave has a wave speed of cA / 2, where cA is the Alfv´en speed within the penumbral field, and so is roughly 70% of the Alfv´en speed within the field. For an assumed Alfv´en speed of 40 km s−1 (corresponding to a field strength of 1500 G and a chromospheric density of about 10−5 kg m−3 ), we obtain a surface wave speed of 28 km s−1 ; the surface wave penetrates into the field and field-free media either side of the penumbral interface a distance of about 160 km for a wavelength of λ = 103 km and period 25 s, rising to 1600 km for a wavelength of 104 km and period 250 s. For the case of a magnetic–non-magnetic interface in a compressible plasma the magnetoacoustic surface waves are effectively confined to a surface layer (skin depth) −1 of scale m−1 e on the field-free side of the interface and m0 on the magnetized side. From 2 2 2 the form of me (equation (4.22), we see that me < kz ; so a surface wave penetrates a greater distance into the field-free plasma than would be the case for the hydromagnetic surface wave of an incompressible fluid, the precise distance depending upon the speed c. A fast surface wave has propagation speed that is closer to the sound speed in the field-free medium than a slow surface wave, and so its penetration distance into the field-free region is correspondingly larger. To consider the penetration distance into the magnetic region, note first that m20 may be rewritten in the form m20 = kz2 +

kz2 c4 (c2s + c2A )(c2t − c2 )

.

Hence m0 > kz if c < ct , and m0 < kz if c > ct . Thus the penetration distance into the magnetic medium is greater than kz−1 for a fast surface wave, and is less than kz−1 for a slow surface wave. The slow surface wave is more closely confined to the interface than the fast surface wave. The fast magnetoacoustic surface wave penetrates further either side of the interface than would the hydromagnetic surface wave of an incompressible medium. The slow magnetoacoustic surface wave penetrates further into the field-free medium, but less into the magnetic region, than would the hydromagnetic surface wave. Finally, we can illustrate these general remarks by looking at the case  = 1 with 2 cA = 2c2s , for which c ≈ 0.55 cA in the slow surface wave and c ≈ 0.80 cA in the fast surface wave. Then, from equation (4.22), we obtain m−1 0 ≈ 0.48 λ in the fast wave and ≈ 0.09 λ in the slow wave, giving a penetration distance into the magnetic region of m−1 0 some 48% of a wavelength λ for the fast wave, falling to 9% of a wavelength for the slow −1 wave. In the field-free region, we have m−1 e ≈ 0.23 λ for the fast wave and me ≈ 0.18 λ for the slow wave, giving penetration distances of some 23% and 18% of a wavelength for the fast and slow waves, respectively. 4.4 Non-Parallel Propagation The case of non-parallel propagation, when the surface wave propagates at an angle to the applied magnetic field, may be treated along much the same lines as for parallel propagation. The governing differential equation is (cf. equation (3.41) of Chapter 3)

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 d ρ0 (x)(kz2 c2A (x) − ω2 ) dux = ρ0 (x)(kz2 c2A (x) − ω2 )ux , dx dx m2 (x) + ky2

(4.30)

where now the motion u has all three components, ux , uy and uz . The total pressure perturbation pT is given by

1 ω4 − ω2 k2 c2f (x) + kz2 k2 c2s (x)c2A (x) iωpT 2 ρ0 (x)cf (x) dux = −(kz2 c2A (x) − ω2 )(kz2 c2t (x) − ω2 ) . (4.31) dx The wave vector k = (0, ky , kz ) lies in the yz-plane, the plane of the magnetic interface, and 1/2

has magnitude k = (ky2 + kz2 ) . The wave vector k makes an angle θ to the direction of the applied field B0 (x), where kz = k cos θ , ky = k sin θ . Parallel propagation corresponds to θ = 0, perpendicular propagation to θ = π/2. Either side of the interface, where the medium is uniform, equation (4.30) applies in the reduced form d2 ux − (m2 + ky2 )ux = 0, dx2

(4.32)

where again m2 takes on the values m20 in x < 0 and m2e in x > 0. As with parallel propagation, we require that across an interface ux (x) and pT (x) are continuous. The solution of equation (4.32), subjected to ux (x) → 0 as x → ±∞, is  1 v0 exp[−(m2e + ky2 ) 2 x], x > 0, (4.33) ux (x) = 1 2 2 2 v0 exp[(m0 + ky ) x], x < 0, where v0 is an arbitrary constant and we have imposed the continuity requirement on ux (x). To satisfy the condition at ±∞ we require that 1

(m2e + ky2 ) 2 > 0,

1

(m20 + ky2 ) 2 > 0.

There remains the requirement of continuity of pT (x), which is equivalent to the continuity of ρ0 (x)(kz2 c2A (x) − ω2 ) dux dx m2 (x) + ky2 (the expression in { } in the differential equation (4.30)). Application of this condition leads to the dispersion relation ρ0 (kz2 c2A − ω2 )(m20 + ky2 )−1/2 + ρe (kz2 c2Ae − ω2 )(m2e + ky2 )−1/2 = 0.

(4.34)

This is the general dispersion relation describing surface waves at a magnetic interface. A relation of this form was given in Wentzel (1979) and Roberts (1981b). The general dispersion relation (4.34) reduces to (4.14) when ky = 0. Introduce the expression (Roberts 1981b)  2 1 ρe m0 + ky2 2 R≡ , (4.35) ρ0 m2e + ky2

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97

and note that R is positive. This extends our earlier definition of R to the general case; when ky = 0, R = R. Then we may rearrange the dispersion relation (4.34) into the forms (Roberts 1981b) ω2 R = c2A + (c2 − c2A ) 2 1 + R Ae kz 1 = c2Ae − (c2 − c2A ), 1 + R Ae

(4.36)

which may be compared with equation (4.15). It follows immediately that the longitudinal speed, ω/kz , of a surface wave lies between the two Alfv´en speeds of the medium. This property, then, is a general one, independent of the direction θ of propagation of a surface wave. It follows immediately that a magnetoacoustic surface wave cannot propagate perpendicular to the applied magnetic field, i.e., a surface wave cannot propagate at the angle θ = π/2. Introduce the speed c of a surface wave in the direction k in which it propagates by writing c ≡ ω/k. In the special case of parallel propagation (Section 4.2), when kz = k, the speed c of the surface wave is also the longitudinal speed ω/kz with which a surface wave propagates parallel to the interface; in general, though, the speeds c and ω/kz are not the same. In terms of c, equation (4.36) shows that surface waves satisfy min (c2A , c2Ae ) cos2 θ < c2 < max (c2A , c2Ae ) cos2 θ.

(4.37)

4.4.1 The Incompressible Case The simplest case to consider is that of an incompressible medium. When the sound speeds cs and cse are taken to be arbitrarily large, m0 and me become kz (> 0) and so R = ρe /ρ0 , the ratio of the densities of the two fluids. Thus dispersion relation (4.36) reduces to ρ0 c2A + ρe c2Ae ω2 = , ρ0 + ρe kz2

(4.38)

and so c2 =

ρ0 c2A + ρe c2Ae cos2 θ. ρ0 + ρe

(4.39)

It is worth noting two reductions of the incompressible, hydromagnetic wave. When the densities of the fluids are the same either side of the interface, so that ρe = ρ0 , then the square of the hydromagnetic wave speed ω/kz is given by B20 + B2e 1 ω2 = = (c2A + c2Ae ), 2 2μρ0 2 kz

(4.40)

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and so ω2 /kz2 is the mean of the squared Alfv´en speeds. Alternatively, when the fields either side of the interface are equal, so that Be = B0 , then 2B20 ω2 = , μ(ρ0 + ρe ) kz2

(4.41)

and so ω2 /kz2 is the square of the Alfv´en speed based upon the mean density, (ρ0 + ρe )/2, of the two fluids. 4.4.2 The β = 0 Plasma A case of particular interest is when the plasma-β is everywhere very small, as for the solar corona (Jain and Roberts 1991). In particular, if we set the sound speeds cs and cse to zero, producing a plasma-β that is zero, then we can simplify the dispersion relation (4.34). Consistent with a zero plasma-β, we set the equilibrium plasma pressures p0 and pe to zero, and then equilibrium pressure balance (4.5) means that B0 = Be . So the magnetic field is everywhere uniform, but density differences can still arise; such differences must satisfy ρ0 c2A = ρe c2Ae . Notice first that with cs = cse = 0,   ω2 c2 2 2 2 2 2 m = kz − 2 , m + ky = k 1 − 2 , (4.42) cA cA with appropriate values being taken in the regions x < 0 and x > 0. Thus, in the limit of zero plasma-β, the dispersion relation (4.34) reduces to 1 2 1 2 (c cos2 θ − c2 )(c2A − c2 )−1/2 + (c cos2 θ − c2 )(c2Ae − c2 )−1/2 = 0. cA A cAe Ae

(4.43)

Since we require that m2 + ky2 > 0 on both sides of the interface, equation (4.43) is subject to the constraint c2 < min (c2A , c2Ae ).

(4.44)

Altogether, then, the dispersion relation (4.43) has constraints (4.37) and (4.44). We may remove the square roots in equation (4.43) by squaring. The result is c4 − (c2Ae + c2A )c2 + c2Ae c2A (1 + sin2 θ ) cos2 θ = 0. Only one of the roots of this quadratic equation in c2 is acceptable:

1/2 1 1 2 c2 = (c2Ae + c2A ) − . (4.45) (cAe − c2A )2 + 4c2Ae c2A sin4 θ 2 2 Equation (4.45) is the dispersion relation for the fast surface wave on an interface between two zero-β plasmas. It was first obtained by Jain and Roberts (1991). Now we have seen earlier (Section 4.3.3) that in the case of a β = 0 plasma no surface wave can propagate parallel to the magnetic field. Also, it is a general property of all magnetoacoustic surface waves that they are unable to propagate perpendicular to the applied magnetic field. In other words, in the β = 0 plasma no surface exists when θ = 0 or π/2. Consider, then, the behaviour of c2 as a function of θ, as given by (4.45). When θ = 0,

4.5 Resonant Absorption at a Single Interface: The Incompressible Case

99

c2 = min (c2Ae , c2A ); this root must be rejected as not complying with the constraint (4.44). When θ = π/2 we obtain c2 = 0, again showing no propagation. But at any intermediate angle a surface wave is permitted. For example, consider the case c2Ae = 2c2A . When θ is close to zero, we obtain c2 /c2A ≈ 1 − 2 sin4 θ , showing that the speed c of the surface √ wave is very slightly below the Alfv´en speed cA . By θ = 45◦ , c2 has fallen to (3 − 3)c2A /2, giving c ≈ 0.80 cA ; and by θ = 51.5◦ , corresponding to sin4 θ = 3/8, c2 /c2A has fallen to 1/2, giving c ≈ 0.71 cA . By an angle of θ = 85◦ , c has fallen to one-tenth of the the constraints (4.37) and (4.44) are satisfied, namely, c < cA and Alfv´en speed cA . Also, √ cA cos θ < c < 2cA cos θ. Thus, a fast surface wave may propagate on the interface between two zero-β plasmas for all angles θ with 0 < θ < π/2, except strictly parallel (θ = 0) or strictly perpendicular (θ = π/2) to that field. When one of the Alfv´en speeds is much smaller or larger than the other or when the angle of propagation θ is close to π/2, then the condition  2 2  cA cAe cos2 θ  1 2 cA + c2Ae is met, and equation (4.45) reduces to (Jain and Roberts 1991)  2 2  cA cAe (1 + sin2 θ ) cos2 θ. c2 ≈ 2 cA + c2Ae

(4.46)

Within its range of validity, this solution satisfies the constraints (4.37) and (4.44). 4.5 Resonant Absorption at a Single Interface: The Incompressible Case We turn now to a consideration of the initial value problem for certain situations that arise in our discussion of MHD waves. We confine our discussion to the incompressible plasma. It turns out that such a treatment gives rise to surface waves that undergo a temporal decay or damping, even in an ideal medium in which there are no loss mechanisms for energy. Indeed, the damping is not related to any physical loss of energy but rather to the transfer of energy from one mode of oscillation to another. The topic has gained importance because of its possible indirect relationship with coronal heating and, perhaps more importantly, the direct detection of damping in coronal oscillations (see Chapters 6 and 14). This topic has been considered in varying degrees of detail by Tataronis and Grossmann (1973), Chen and Hasegawa (1974), Ionson (1978), Rae and Roberts (1981), Goedbloed (1983) and Lee and Roberts (1986) amongst others. All these analyses, directly or indirectly, are essentially based upon the pioneering work of Sedlacek (1971), who considered a similar topic in the context of plasma waves. Sedlacek’s approach applies with only minor modifications to MHD waves in an incompressible medium. Nonetheless, the importance of the fundamental problem gives full justification to subsequent work. A similar problem – of a magnetic flux tube (modelled in cylindrical coordinates) embedded in a β = 0 plasma – is of direct relevance to coronal observations (see Chapters 6 and 8). Consider a layer of inhomogeneity of width l that joins together two uniform regions of plasma. The governing differential equation for two-dimensional linear motions in an incompressible medium, with ky = 0 (so k = kz ), is

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     2 2 d 2 dux ρ0 (x) kz cA (x) − ω = ρ0 (x)kz2 kz2 c2A (x) − ω2 ux . dx dx

(4.47)

We are interested in the initial value problem for such a motion ux . In the main, the problem treated requires ρ0 (x)(kz2 c2A (x) − ω2 ) to be a constant or linear in x; this is the case for an equilibrium in which either (a) the density ρ0 is uniform but the magnetic pressure or Alfv´en speed squared (specifically B20 (x)) is linear in x, or (b) the magnetic field B0 is uniform but the plasma density ρ0 (x) is linear in x. In particular, we take an equilibrium for which ⎧ ⎪ x > l/2, ⎪ ⎨ρe , trans (4.48) ρ0 (x) = ρ (x), −l/2 < x < l/2, ⎪ ⎪ ⎩ρ , x < −l/2, 0

representing a plasma density that is ρ0 on the far left side of the medium changing to ρe on the far right side, with a non-uniform transition layer of width l connecting the two uniform regions. In the transition layer the equilibrium is assumed to have a monotonic behaviour in x. To be specific, within the transition layer the density is taken to be linear in x: ρ trans (x) =

1 1 x (ρ0 + ρe ) − (ρ0 − ρe ) 2 2 l

(4.49)

corresponding to a density that is ρ0 at x = −l/2 and ρe at x = +l/2, with the average value (ρ0 + ρe )/2 at x = 0. In the limit l → 0 the non-uniform layer is absent and we simply have a jump in plasma density from ρ0 to ρe . If the medium is uniform in density, then the equilibrium is taken to have a magnetic field B0 (x) that varies in the transition layer such that the Alfv´en speed squared has the form ⎧ 2 ⎪ x > l/2, ⎪ ⎨cAe , 2 1 (4.50) cA (x) = 2 (c2A + c2Ae ) − 12 (c2A − c2Ae )(x/l), −l/2 < x < l/2, ⎪ ⎪ ⎩c2 , x < −l/2. A

Solution of the linear problem shows that a disturbance has a frequency ω = ωR + iωI , corresponding to ux ∝ eiωt = e−ωI t · eiωR t , where  2 2  2 2 2 2 ρ0 cA + ρe cAe ωR ≈ kz ck = kz . ρ0 + ρe Not surprisingly, the kink speed ck arises to describe the surface wave in an incompressible medium. Additionally, the disturbance suffers a decay (|ux | ∝ e−ωI t ) at a rate ωI (> 0) given by (see, for example, Rae and Roberts 1981, eqn. (51)) ωI =

ρ0 ρe (c2Ae − c2A ) π kz ck . lkz 4 (ρ0 + ρe )2 c2k

(4.51)

This decay is not related to heating – the medium is ideal and so total energy is conserved – but to the transfer of energy from the global oscillation in ux into Alfv´enic disturbances (Lee 1980).

4.5 Resonant Absorption at a Single Interface: The Incompressible Case

101

In a medium with a uniform density, ρe = ρ0 , but an Alfv´en speed squared of the form (4.50), the rate of decay ωI is still given by (4.51), which now reduces to ωI =

(c2 − c2 ) π lkz Ae 2 A kz ck . 16 ck

(4.52)

This decay rate for a medium with uniform density was also obtained by Lee and Roberts (1986), who assumed a uniform density at the outset. In a medium with uniform magnetic field but non-uniform plasma density the result (4.51) becomes   π ρ0 − ρe (4.53) kz ck . ωI = lkz 8 ρ0 + ρe The timescale τ associated with the decay rate ωI is τ = 1/ωI , which for a uniform magnetic field is   ρ0 + ρe 1 8 . (4.54) τ= πlkz ρ0 − ρe kz ck Decay rates of this form were first deduced by Sedlacek (1970) in an analysis devoted to plasma waves but readily adapted to MHD waves (Chen and Hasegawa 1974; Ionson 1978; Rae and Roberts 1981; Lee and Roberts 1986; Parker 1991). A more physical approach is also possible; see, for example, Hollweg (1987). We end with a brief illustration using typical coronal parameters. For a medium with a uniform magnetic field, we consider a standing wave with wavenumber kz = π/L in a loop of length L = 130 Mm, with a transition layer of width l = 103 km. We take an Alfv´en = 103 km s−1 and a density contrast ρ0 = 10 ρe . Then the external Alfv´en speed speed cA √ is cAe = 10 cA = 3162 km s−1 and the kink speed is ck = 1348 km s−1 . The period of the standing wave is 2L/ck = 192.8 s. Finally, from equation (4.54) the associated decay time is τ = 3952 s (or 20.5 periods). Resonance of surface waves is discussed further in Chapter 6, Section 6.10 and Chapter 8, Sections 8.10.4 and 8.14.

5 Magnetic Slabs

5.1 Introduction In Chapter 4 we have established the form of surface waves on a single magnetic interface. We turn now to a consideration of the waves that arise in a magnetic slab. By a magnetic slab we mean a region of magnetic field that is delineated from its surroundings either by virtue of the magnetic field strength or the plasma density or the temperature within the region being different from that in the surroundings. Such structures model many of the magnetic features found in astrophysical applications. The magnetic slab is the Cartesian equivalent of a magnetic flux tube. The slab is mathematically easier to treat than a cylindrical structure, thus allowing us to present the basic features of wave propagation in a slab unburdened by the complexity that cylindrical geometry introduces in tubes. The case of a magnetic flux tube is treated separately in Chapter 6. We consider, then, a unidirectional equilibrium magnetic field B0 = B0 (x)ez ,

(5.1)

which is aligned with the z-axis of a Cartesian coordinate system x, y, z. The medium is structured in x with spatial variations in equilibrium plasma pressure p0 (x) and plasma density ρ0 (x). Pressure balance (see equation (3.2) of Chapter 3) means that   B20 (x) d p0 (x) + = 0, (5.2) dx 2μ showing that spatial variations in magnetic pressure, B20 (x)/2μ, are balanced by corresponding variations in plasma pressure, p0 (x). Suppose that the strength B0 (x) of the equilibrium magnetic field is given by  Be , |x| > a, (5.3) B0 (x) = B0 , −a < x < a, representing a magnetic field of strength B0 confined to a slab of width 2a, surrounded by a magnetic environment of field strength Be . The equilibrium plasma pressure p0 (x) and density ρ0 (x) are taken to be structured in a similar way:  p e , ρe , |x| > a, (5.4) p0 (x), ρ0 (x) = p0 , ρ0 , −a < x < a, 102

5.2 The Incompressible Case: Hydromagnetic Surface Waves in a Slab

103

Figure 5.1 The equilibrium configuration of a magnetic slab. A magnetic field B0 ez , confined to a slab of width 2a (≡ 2x0 ), is embedded in a magnetic environment with field Be ez . Within the slab (−a < x < a), the plasma density is ρ0 and the pressure is p0 ; in the environment (|x| > a), the density is ρe and the pressure is pe . Pressure balance (5.5) is maintained across the interfaces x = ±a. (After Edwin and Roberts 1982.)

for constants p0 , pe , ρ0 and ρe . Quantities with a suffix ‘0’ refer to the region within the slab (−a < x < a), and those with a suffix ‘e’ to the environment (|x| > 2a). Across the interfaces x = −a and x = a conditions change discontinuously; these interfaces are current sheets. As for the case of a single interface discussed in Chapter 4, total pressure balance must pertain: p0 +

B20 B2 = pe + e . 2μ 2μ

(5.5)

Figure 5.1 illustrates the equilibrium configuration. The fact that a magnetic slab possesses a lengthscale – the width of the slab – introduces an immediate and important distinction between waves in slabs (and also tubes) and waves in an unbounded uniform medium or on a single interface. In ideal MHD, waves in a uniform medium or surface waves on the interface between two unbounded media have no lengthscale imposed upon them by the equilibrium; any lengthscale that arises is introduced by the disturbance itself. Accordingly, the speed of a wave is independent of the wavelength of the wave (since all wavelengths are in a sense equivalent): wave propagation is non-dispersive. By contrast, a magnetic slab imposes a lengthscale against which the wavelength of a wave may be measured. Accordingly, in a magnetic slab we expect that waves of different wavelength travel with different speeds: wave propagation is dispersive.

5.2 The Incompressible Case: Hydromagnetic Surface Waves in a Slab The simplest case to discuss is that of an incompressible fluid. The general equation governing the motion of a fluid structured according to the presssure balance constraint (5.2)

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has been derived in Chapter 3. In an incompressible fluid (for which div u = 0), the motion u satisfies the wave equation (see (3.17) of Chapter 3)   ∂ 2u ∂ 2u ∂pT , (5.6) ρ0 2 − ρ0 c2A 2 = −grad ∂t ∂t ∂z where pT is the perturbation in the total pressure (plasma plus magnetic). This equation is simply the time derivative of the momentum equation, with the contribution from the magnetic tension force expressed in terms of u through use of the induction equation for an ideal fluid. Set u = (ux , uy , uz ). Then, in component form equation (5.6) gives ρ0 (x)

∂ 2 ux ∂ 2 ux ∂ 2 pT − ρ0 (x)c2A (x) 2 = − , 2 ∂x∂t ∂t ∂z

(5.7)

ρ0 (x)

∂ 2 uy ∂ 2 uy ∂ 2 pT 2 − ρ (x)c (x) = − , 0 A ∂y∂t ∂t2 ∂z2

(5.8)

∂ 2 uz ∂ 2 uz ∂ 2 pT 2 − ρ (x)c (x) = − . (5.9) 0 A ∂z∂t ∂t2 ∂z2 For our purposes here it is convenient to assume there are no variations along the ignorable coordinate of the slab (∂/∂y ≡ 0). Then equation (5.8) shows that the component of motion uy is decoupled from the motions ux and uz in the plane of the slab, and uy satisfies the Alfv´en wave equation. In a non-uniform medium this equation exhibits phase mixing, a topic discussed in Chapter 3. In order to bring out the basic properties of waves propagating in a magnetic slab, we remove the Alfv´en wave from our discussion by choosing uy = 0. Accordingly, we suppose motions to lie purely in the xz-plane and to be of the form   u = ux (x, z, t), 0, uz (x, z, t) , ρ0 (x)

with a similar form for the magnetic field perturbation. The governing equations are then (5.7) and (5.9), coupled through the incompressibility constraint ∂ux ∂uz + = 0. (5.10) ∂x ∂z We may eliminate the total pressure variations pT by taking the curl of equation (5.6) and noting the identity curl grad ≡ 0. The result is       2u 2u 2u ∂ 2 uz ∂ ∂ ∂ ∂ ∂ z x x − c2A (x) 2 − c2A (x) 2 ρ0 (x) ρ0 (x) = . (5.11) ∂x ∂z ∂t2 ∂z ∂t2 ∂z A further differentiation with respect to z allows us to eliminate uz through use of the incompressibility constraint (5.10), the result being        ∂2 ∂ 2 ∂ux ∂ 2 ux ∂ 2 ux ∂2 ∂ 2 2 − cA (x) 2 − cA (x) 2 ρ0 (x) + 2 ρ0 (x) = 0. (5.12) ∂x ∂x ∂t2 ∂z ∂z ∂t2 ∂z This is the basic wave equation we require. It describes the behaviour of slow magnetoacoustic waves in the incompressible limit.

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105

To progress further we Fourier analyse our equations in t and z, supposing that perturbations are of frequency ω and longitudinal wavenumber kz ; the x-component of motion is taken to be of the form ux (x, z, t) = ux (x) exp i(ωt − kz z),

(5.13)

with similar forms for all other perturbations. Then, equation (5.12) shows that the amplitude ux (x) of such transverse motions satisfies the ordinary differential equation   d 2 2 2 dux (5.14) ρ0 (x)(kz cA (x) − ω ) = ρ0 (x)kz2 (kz2 c2A (x) − ω2 )ux . dx dx This is the fundamental equation governing incompressible motions in a magnetically structured medium. Other perturbation quantities, such as the total pressure perturbation pT (x) and the motion component uz (x), follow from solution of this equation. Specifically, equations (5.9) and (5.10) show that pT (x) is given by ωkz2 pT (x) = iρ0 (x)(kz2 c2A (x) − ω2 )

dux , dx

(5.15)

with the component of motion uz directed along the magnetic slab determined by the incompressibility constraint (5.10): uz (x) = −

i dux . kz dx

(5.16)

Now in a uniform medium the differential equation (5.14) gives either that ω2 = kz2 c2A

(5.17)

d2 ux = kz2 ux . dx2

(5.18)

with ux (x) arbitrary, or

The first possibility corresponds to slow waves in the incompressible extreme; each field line is able to oscillate with the local Alfv´en speed, without disturbing its neighbours. Equation (5.15) shows that in this mode there are no variations in the total pressure: pT = 0. The second possibility, given by equation (5.18), corresponds to the surface modes described in Chapter 4. Since the media both inside and outside the magnetic slab are uniform, the above considerations apply directly to such a slab. Consider the surface modes given by equation (5.18). The solution of equation (5.18) appropriate to a magnetic slab is ⎧ ⎪ x > a, ⎪ ⎨ae exp(−|kz |x), (5.19) ux (x) = a0 cosh(|kz |x) + b0 sinh(|kz |x), −a < x < a, ⎪ ⎪ ⎩b exp(|k |x), x < −a, e

z

where ae , be , a0 and b0 are arbitrary constants. In writing this solution we have imposed the condition that disturbances are bounded at infinity (ux (x) → 0 as x → ±∞). Equation (5.19) describes the component of motion ux ; these motions are evidently confined to the

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neighbourhood of the magnetic slab, penetrating into the environment (|x| > a) a distance of order |kz |−1 either side of the slab. In writing the solution (5.19) we have retained the possibility that the wavenumber kz along the slab may be negative; if it is positive, then we may dispense with the modulus signs on kz . The four constants, ae , be , a0 and b0 , arising in the description of the transverse motion ux may in fact be related to one another through the application of the boundary conditions that pertain across the interfaces x = −a and x = a. Our equations being linear, one constant remains arbitrary, leaving unspecified the actual magnitude of the vibration ux . A direct application of the matching conditions across the slab boundaries would in fact result in a system of four linear equations for the constants ae , be , a0 and b0 . But considerations of symmetry (the equilibrium structure of the slab is mirror symmetric about the z-axis) suggest that such a system could be factorized into two separate systems, one being symmetric about the z-axis and the other anti-symmetric. Rather than derive these systems in this way, we instead take advantage of the symmetry in the equilibrium structure and seek particular solutions of Equation (5.19) that possess the appropriate symmetry. In this way the algebraic complexity in the calculations is reduced. There is no loss of generality in this approach. 5.2.1 Sausage and Kink Modes The symmetry of the equilibrium configuration means that we may take the transverse component of motion ux to be either zero or non-zero on the axis of the slab. The choice ux = 0 when x = 0 corresponds to taking a0 = 0, giving ux (x) ∝ sinh(|kz |x) within the slab (|x| < a). This gives an odd function of x: ux (−x) = −ux (+x), |x| < a. In keeping with this, we expect that be = −ae , so that ux is an odd function for all x. In the ensuing motions the central axis of the slab (the z-axis) is undisturbed; the geometrical configuration is symmetrically perturbed about the central axis. The transverse motions ux (x) are equal in magnitude but opposite in direction at points symmetrically placed either side of the slab’s central axis (the z-axis); the motion is either simultaneously towards or simultaneously away from the central axis of the slab. We refer to this wave as a sausage mode, in view of the fact that the vibrating slab (and more so a vibrating cylindrical tube) has some resemblance to an array of sausages suspended in a butcher’s shop. This term is used also in laboratory studies to describe instabilities of such geometrical objects as slabs or tubes. The second choice of solution involves setting b0 = 0, with a0 = 0. In this case, ux (x) ∝ cosh(kz x) in the slab. Thus, transverse motions are in unison either side of the slab centre and ux is an even function of x: ux (−x) = ux (+x). Whereas in the sausage mode the slab centre was undisturbed in the motion, here the central axis of the slab partakes in the motion (since ux = 0 at the slab centre). Thus the cross-section of the slab is displaced as a whole, the motion of the slab resembling that of a moving snake. We refer to this wave as the kink mode. Note that the definitions of sausage and kink modes are geometrical ones; they simply describe the shape of an oscillating structure. Accordingly, more than one type of mode

5.2 The Incompressible Case: Hydromagnetic Surface Waves in a Slab

107

Figure 5.2 The geometry of sausage and kink modes in a magnetic slab. The sausage mode represents a geometrically symmetric pulsation of the slab, with transverse motions being an odd function of x; the central axis of the slab is undisturbed. The kink mode represents a geometrically asymmetric vibration of the slab, with transverse motions ux being an even function of x; the central axis of the slab is disturbed in a serpentine fashion and the slab vibrates as a whole. A consideration of the motions in the hydromagnetic sausage and kink modes shows that in the sausage mode motions along the slab dominate over transverse ones: the motions are principally longitudinal (i.e., |uz | > |ux |). In the kink mode the situation is reversed and motions are principally transverse (i.e., |ux | > |uz |).

may be referred to as a sausage wave or a kink wave, though in the incompressible case discussed here only one sausage wave and one kink wave arise; we see later (Section 5.3) that when compressibility is admitted more than one sausage or kink wave may arise. The geometrical forms of the sausage and kink waves are sketched in Figure 5.2.

5.2.2 Dispersion Relations In the case of the sausage mode, transverse motions are taken to be of the form ⎧ ⎪ x > a, ⎪ ⎨ae exp(−|kz |x), ux (x) = b0 sinh(|kz |x), −a < x < a, ⎪ ⎪ ⎩−a exp(|k |x), x < −a. e

z

For the kink mode, transverse motions are of the form ⎧ ⎪ x > a, ⎪ ⎨de exp(−|kz |x), ux (x) = a0 cosh(kz x), −a < x < a, ⎪ ⎪ ⎩d exp(|k |x), x < −a. e

(5.20)

(5.21)

z

The use of the modulus signs on kz is unnecessary in the expression cosh(kz x) because it is an even function of kz x.

108

Magnetic Slabs

For each mode, the two constants arising in the description of the motions are related to one another through the boundary conditions that pertain across the two interfaces of the slab. We have seen in Chapter 4 that across such an interface the motion ux and the total perturbed pressure pT must be continuous. Thus, ux

and

ρ0 (x)(kz2 c2A (x) − ω2 )

dux dx

(5.22)

must be continuous across x = −a and across x = a. Symmetry allows us to discuss the sausage and kink modes together, applying the boundary conditions at x = a (the conditions at x = −a then being automatically satisfied). Write  sausage mode, b0 sinh(|kz |x), (5.23) u(x) = a0 cosh(kz x), kink mode, so the transverse motion ux (x) within the slab is denoted by u(x). Continuity of ux leads to ae = b0 exp(|kz |a) sinh(|kz |a)

(5.24)

de = a0 exp(|kz |a) cosh(kz a)

(5.25)

in the sausage mode, and

in the kink mode; the constants a0 and b0 remain arbitrary. It remains to satisfy the pressure balance condition. This yields ρ0 (kz2 c2A − ω2 )

u (a) + ρe (kz2 c2Ae − ω2 ) = 0, |kz |u(a)

(5.26)

where now cA (= B0 /(μρ0 )1/2 ) denotes the Alfv´en speed within the slab (there being no confusion with the general Alfv´en speed cA (x) used above) and cAe (= Be /(μρe )1/2 ) denotes the Alfv´en speed in the environment. Here u (a) denotes the derivative of the function u(x) evaluated at x = a. Through choice of the function u(x) in (5.23), the dispersion relation (5.26) applies to both the sausage and the kink hydromagnetic modes in a magnetic slab. Introducing the speed c ≡ ω/|kz | of the wave along the magnetic slab, we may rewrite (5.26) in the forms   R 1 2 2 c = cA + (5.27) (c2Ae − c2A ) = c2Ae − (c2 − c2A ), 1+R 1 + R Ae where R ≡ (ρe /ρ0 )|kz |

u(a) . u (a)

Explicitly, the form of R for the sausage and kink modes is given by  sausage mode, (ρe /ρ0 )tanh(|kz |a), R= (ρe /ρ0 )coth(|kz |a), kink mode. Note that R is positive.

(5.28)

(5.29)

5.2 The Incompressible Case: Hydromagnetic Surface Waves in a Slab

109

Now the forms of the dispersion relation given in (5.27) are precisely those forms that arises for surface waves on a single magnetic interface (cf. equation (4.15) of Chapter 4), save here the expression for R is modified (and is now a function of |kz |a). Since R is positive, it follows immediately that c2 lies between c2A and c2Ae . Hence, the speed c of a hydromagnetic surface wave, sausage or kink, in a magnetic slab lies between the Alfv´en speeds, cA and cAe , of the slab and its environment. The explicit forms of the dispersion relations for the two modes follow from (5.27) and (5.29). The sausage wave has dispersion relation c2 =

ρe c2Ae + ρ0 c2A coth(|kz |a) , ρe + ρ0 coth(|kz |a)

(5.30)

c2 =

ρe c2Ae + ρ0 c2A tanh(|kz |a) . ρe + ρ0 tanh(|kz |a)

(5.31)

and the kink wave satisfies

It may be noted that for either mode the speed c of the wave along the magnetic slab depends upon the magnitude of the wavenumber kz measured relative to the slab half-width a. Accordingly, wave propagation is dispersive: waves of different wavelengths travel with different speeds. It is of interest to examine two extremes of the relations (5.30) and (5.31). We consider modes that have either short or long wavelengths. Short wavelength modes have |kz |a  1, corresponding to wavelengths λ (= 2π/|kz |) very much shorter than 2π a; this is likely to be the case in wide slabs. By contrast, long wavelength modes are such that |kz |a  1, corresponding to wavelengths λ that are very much less than 2π a; this case is of particular interest in thin (or slender) slabs or tubes, which have widths that are much smaller than the wavelength of a given wave. When |kz |a  1 we may approximate tanh(|kz |a) by unity, and so R ≈ ρe /ρ0 for both sausage and kink modes. Hence, both sausage and kink waves propagate with a speed c given approximately by c2 =

ρ0 c2A + ρe c2Ae . ρ0 + ρe

(5.32)

We recognize in (5.32) the mean Alfv´en speed, cmean , of a magnetic interface (cf. (4.17) A in Chapter 4). This is to be expected in that the limit of large |kz |a may be viewed as corresponding to the case of two widely separated magnetic interfaces, each of which is able to support surface waves. Because tanh(|kz |a) rapidly approximates to one for quite moderate |kz |a (for example, 0.9 < tanh(|kz |a) < 1 for |kz |a > 1.5), hydromagnetic sausage and kink modes propagate with a speed that is close to the mean Alfv´en speed of the slab and its environment for all but the longest waves in thin slabs. Turning to long waves (satisfying |kz |a  1, λ  2π a), for which we may note that tanh(|kz |a) ≈ |kz |a, we find that the square of the speed c is given approximately by  c2 − (ρe /ρ0 )(c2A − c2Ae )|kz |a, sausage mode, 2 (5.33) c ≈ A c2Ae − (ρe /ρ0 )−1 (c2Ae − c2A )|kz |a, kink mode.

110

Magnetic Slabs

Note the asymmetry in the ratio of densities occurring in equation (5.33). For the sausage mode (5.33) gives a speed c that is slightly slower than the slab Alfv´en speed cA if cA > cAe , but slightly faster than cA if cA < cAe ; for the kink mode, the speed c is faster than the Alfv´en speed cAe in the environment if cA > cAe , but slower than cAe if cA < cAe . Of particular interest is the case of a magnetic slab embedded in a field-free (Be = 0) environment. Since here the Alfv´en speed in the environment is zero, both sausage and kink modes propagate with a speed that is less than the Alfv´en speed cA in the slab. With cAe = 0, the dispersion relation for the sausage mode is simply c2 =

ρ0 coth(|kz |a) ρ0 c2 = c2 , ρe + ρ0 coth(|kz |a) A ρ0 + ρe tanh(|kz |a) A

(5.34)

and the dispersion relation for the kink mode is c2 =

ρ0 tanh(|kz |a) ρ0 c2A = c2 . ρe + ρ0 tanh(|kz |a) ρ0 + ρe coth(|kz |a) A

(5.35)

In the short wavelength limit of a wide slab both modes propagate with the mean Alfv´en speed, which for a slab in a field-free environment is  1/2 ρ0 mean cA = cA . (5.36) ρ0 + ρe In the case of equal√densities, ρe = ρ0 , the mean Alfv´en speed for a slab in a field-free environment is cA / 2, some 71% of the slab’s Alfv´en speed. In the opposite extreme, of a long wave in a thin slab, equation (5.34) shows that the sausage mode propagates with a speed just below the Alfv´en speed cA in the slab and the kink mode propagates very slowly, with a speed that is proportional to the square root of |kz |a. The form of the relations (5.34) and (5.35) is displayed in Figures 5.3 and 5.4.

5.2.3 Pressure and Longitudinal Motion It is of interest to examine the total pressure perturbation pT (x) and longitudinal motions uz (x) arising in the propagation of hydromagnetic waves. We consider the sausage and kink modes separately. Sausage Mode Consider first the total pressure perturbation. Since pT (x) is related, through equation (5.15), to the derivative of ux the total pressure perturbation is an even function of x: pT (−x) = pT (x). Inside the slab, we have pT (x) = ib0 ρ0 (c2A − c2 )

|kz | cosh(kz x), ω

|x| < a.

(5.37)

The presence of the factor i (= eiπ/2 ) indicates that the total pressure variations pT are π/2 out of phase with the transverse motions ux . Evidently, the amplitude |pT | of pressure variations reaches a peak on the boundaries of the slab, declining to |b0 ρ0 (c2A − c2 )/c| at the centre of the slab. In the short wavelength limit of a wide slab, |kz |a  1, this decline

5.2 The Incompressible Case: Hydromagnetic Surface Waves in a Slab

111

Figure 5.3 The wave speed c = ω/|kz | of the hydromagnetic sausage (equation (5.34)) and kink cA /10). (equation (5.35)) waves in a magnetic slab of width 2a, for cA > cAe (specifically, cAe = √ = cA / 2. The We have taken the slab and environment fluid densities equal (ρ0 = ρe ), giving cmean A solid curve represents the sausage mode, the dashed curve the kink mode. [Notational changes from the present text: VA ≡ cA , VAe ≡ cAe , k ≡ kz and x0 ≡ a.] (After Edwin and Roberts 1982.)

Figure 5.4 The speed c of the hydromagnetic sausage and kink waves in a√magnetic slab with = cA / 2. The solid curve cA < cAe (specifically, cAe = 3cA /2). Here ρ0 = ρe , giving cmean A represents the sausage mode, the dashed curve the kink mode. [Notational changes: VA ≡ cA , VAe ≡ cAe , k ≡ kz and x0 ≡ a.] (After Edwin and Roberts 1982.)

112

Magnetic Slabs

is very marked, pressure variations being considerably larger on the slab boundaries than in the slab centre. But in the opposite extreme, of long waves in a thin slab, |kz |a  1, the variation between slab centre and the boundaries is slight, the factor cosh(kz a) being close to unity (e.g., 1 ≤ cosh(kz a) < 1.1 for 0 ≤ kz a < 0.44). Consider the longitudinal component uz of the incompressible (div u = 0) motions of the slab, given by equation (5.16). Inside the slab, uz (x) = −ib0 sgn(kz ) cosh(kz x),

|x| < a,

(5.38)

where sgn(kz ) denotes the sign of the wavenumber kz :  1, kz > 0, sgn(kz ) = −1, kz < 0.

(5.39)

Again, the presence of the factor i in equation (5.38) indicates a phase change of π/2 when uz is compared with ux . We see that in a thin slab the longitudinal motions vary only weakly across the slab. Comparing the magnitudes of the two components of motion inside the slab, we have |ux | = | tanh(kz x)|, |uz |

|x| < a,

(5.40)

showing that in the hydromagnetic sausage mode, motions are predominantly longitudinal, satisfying |uz | > |ux | for all wavelengths. In a thin slab (|kz |a  1), the longitudinal motions are strongly dominant (|uz |  |ux |). For example, with kz a = 1/10 we obtain |uz | > 100|ux |, and so transverse motions are at most 1% of longitudinal ones. We may draw together our findings as follows. We give perturbations in real form. Denote by u0 the amplitude of the longitudinal motions at the centre of the slab. Then, within the slab longitudinal motions of the form uz (x, z, t) = u0 cosh(kz x) cos(ωt − kz z),

|x| < a,

(5.41)

|x| < a,

(5.42)

may arise, and are accompanied by transverse motions ux (x, z, t) = −u0 sinh(kz x) sin(ωt − kz z),

with ω and kz related through the dispersion relation (5.30). The variations in the total pressure are given by pT (x, z, t) = −ρ0

(kz2 c2A − ω2 ) u0 cosh(kz x) cos(ωt − kz z), ωkz

|x| < a.

(5.43)

Kink Mode For the kink mode, denote by u0 the magnitude of ux (x = 0) at the centre of the slab. Then the transverse component of motion inside the slab is ux (x) = u0 cosh(kz x). The longitudinal component, uz (x), follows from equation (5.16); inside the slab, we obtain uz (x) = −iu0 (kz ) sinh(kz x),

|x| < a.

(5.44)

The total pressure variations pT (x) in the kink mode follow from equation (5.15): pT (x) = iu0 ρ0 (c2A − c2 )

kz sinh(kz x), ω

|x| < a.

(5.45)

5.3 Compressible Effects

113

Thus, both the longitudinal motions and the total pressure perturbation are odd functions of x: uz (−x) = −uz (x), pT (−x) = −pT (x). Comparing the two components of motion inside the slab, we have |ux | = | coth(kz x)|, |uz |

|x| < a,

(5.46)

showing that in the hydromagetic kink mode motions are predominantly transversal, satisfying |ux | > |uz | at all wavelengths. In a thin slab (|kz |a  1), the longitudinal motions are very weak and transverse motions dominate (|ux |  |uz |). Finally, we may illustrate the form of the motion and the associated variations in the total pressure as follows. In real form, the kink mode may exhibit motions u = (ux (x, z, t), 0, uz (x, z, t)) of the form ux = u0 cosh(kz x) cos(ωt − kz z), with pressure variations pT (x, z, t) = −ρ0 u0



kz ω

uz = u0 sinh(kz x) sin(ωt − kz z),

|x| < a, (5.47)

 (c2A − c2 ) sinh(kz x) sin(ωt − kz z),

|x| < a,

(5.48)

with ω (= |kz |c) and kz related by the dispersion relation (5.31). 5.3 Compressible Effects We turn now to a consideration of the effects of compressibility on the modes of a magnetic slab. We know from the investigation of surface waves in Chapter 4 that zero, one or two surface waves are able to propagate on a magnetic interface, depending upon the particular orderings of the propagation speeds either side of the interface. We may expect, then, that a similar situation will arise in a magnetic slab. Additionally, we may expect that modes other than surface waves will arise, waves that occupy the interior of the slab even if the slab is wide. For clearly, in the limit of an arbitrarily wide magnetic slab, we may expect to obtain the equivalent of the usual fast and slow waves of an unbounded medium. The equilibrium state for a compressible medium is simply that of pressure balance, equations (5.2) and (5.5), just as for the incompressible case. Additionally, the equilibrium temperature T0 (x) (and with it the sound speed) may be structured much as the plasma pressure and density:  Te , |x| > a, (5.49) T0 (x) = T0 , −a < x < a. Combined with the ideal plasma law, p0 (x) = kB ρ0 (x)T0 (x)/m, ˆ pressure balance implies a connection between the densities, sound speeds and Alfv´en speeds inside and outside of the magnetic slab: c2 + 1 γ c2 ρe = s 12 2A , ρ0 c2se + 2 γ cAe

(5.50)

where cs (= (γ p0 /ρ0 )1/2 ) is the sound speed within the slab (−a < x < a) and cse (= (γ pe /ρe )1/2 ) the sound speed in the environment ( |x| > a).

114

Magnetic Slabs

As for the incompressible case, we may obtain the governing wave equation by examining the time derivative of the momentum equation, namely,   2 ∂pT ∂ 2u 1 2 ∂ u 2 ∂ − cA 2 = −ez cA div u − grad . (5.51) ∂z ρ0 ∂t ∂t2 ∂z Comparing with equation (5.6), we see that the compression div u enters into the motion, indicating the presence of sound waves. With the Alfv´en wave again removed by setting uy = 0 and requiring that ∂/∂y ≡ 0, we find that the motion u = (ux (x, z, t), 0, uz (x, z, t)) satisfies the coupled wave equations (Roberts 1981b; see also Chapter 3) ∂ 2 ux ∂ 2 ux 1 ∂ 2 pT 2 − c (x) = − A ρ0 (x) ∂x∂t ∂t2 ∂z2   2 2 2 (x) u ∂ c ∂ uz 1 ∂ 2 pT z s 2 − c (x) = − , t ∂t2 ∂z2 c2f (x) ρ0 (x) ∂z∂t

(5.52) (5.53)

with ∂pT ∂ux ∂uz = −ρ0 (x)c2f (x) − ρ0 (x)c2s (x) . ∂t ∂x ∂z

(5.54)

Here cs (x) (= (γ p0 (x)/ρ0 (x))1/2 ) denotes the sound speed in the medium, cf (x) is the fast magnetoacoustic speed and ct (x) is the slow cusp speed, with c2f (x) = c2s (x) + c2A (x),

c2t (x) =

c2s (x)c2A (x) c2s (x) + c2A (x)

.

(5.55)

For perturbations of the form u = (ux (x), 0, uz (x)) exp i(ωt − kz z)

(5.56)

the wave equations (5.52) to (5.54) lead to an ordinary differential equation for the transverse component ux (x) of motion (Roberts 1981b, c; see also equation (3.50) of Chapter 3):     kz2 c2t (x) − ω2 dux d 2 2 (5.57) ρ0 (x)(cs (x) + cA (x)) 2 2 = ρ0 (x)(kz2 c2A (x) − ω2 )ux . dx kz cs (x) − ω2 dx The total pressure perturbation pT (x) exp i(ωt − kz z) is given by   kz2 c2t (x) − ω2 dux 2 iωpT (x) = −ρ0 (x)cf (x) 2 2 , kz cs (x) − ω2 dx

(5.58)

and the longitudinal component uz (x) follows from equations (5.53) and (5.58): uz (x) = −

ikz c2s (x) 2 (kz c2s (x) − ω2 )

dux . dx

(5.59)

In the incompressible limit of cs → ∞ these equations reduce to equations (5.14), (5.15) and (5.16) discussed earlier.

5.3 Compressible Effects

115

As in the incompressible hydromagnetic case, the media inside and outside the magnetic slab are uniform, and in a uniform medium equation (5.57) reduces to d2 ux − m2 ux = 0, dx2

(5.60)

where m2 =

(kz2 c2s − ω2 )(kz2 c2A − ω2 ) kz2 c2s c2A − ω2 (c2s + c2A )

;

(5.61)

m2 takes on values appropriate for the regions inside and outside the slab. Inside the slab, where m2 = m20 , the general solution of equation (5.60) is ux (x) = a0 cosh(m0 x) + b0 sinh(m0 x),

|x| < a,

(5.62)

for arbitrary constants a0 and b0 , with the effective wavenumber m0 given by m20 =

(kz2 c2s − ω2 )(kz2 c2A − ω2 ) kz2 c2s c2A − ω2 (c2s + c2A )

.

(5.63)

The geometrical considerations of sausage and kink waves given in the incompressible case apply here also, so we may discuss the cases of ux (x) being an odd function (ux (−x) = −ux (x)) or an even function (ux (−x) = ux (x)). Symmetric pulsations of the slab involve transverse motions that on opposite sides of the slab are either towards or away from the slab centre (the z-axis), which itself remains undisturbed. This is the sausage mode, for which we select a0 = 0 and ux (x) = b0 sinh(m0 x),

|x| < a.

(5.64)

For the kink mode we choose b0 = 0, with a0 = 0, so that ux (x) = a0 cosh(m0 x),

|x| < a.

The two modes may be treated together by writing  b0 sinh(m0 x), sausage mode, u(x) = a0 cosh(m0 x), kink mode.

(5.65)

(5.66)

Notice that there is no restriction on m20 ; it may be positive or negative (or indeed complex) and still ux will satisfy the differential equation (5.60) with m = m0 . Similarly, there is no restriction on m0 , which may be chosen positive if m20 > 0 or purely imaginary if m20 < 0. For m20 > 0 equation (5.64) gives motions that are of exponential form, declining from the slab boundary. These waves are clearly the equivalent of the surface waves of a single magnetic interface (discussed in Chapter 4). Accordingly, following Roberts (1981b, c), we refer to those modes with m20 > 0 as the surface waves of a magnetic slab; on the other hand, when m20 < 0, then the motions within the slab are oscillatory in nature (corresponding to motions that persist across the whole of the slab interior) and we refer to those modes with m20 < 0 as the body waves of the magnetic slab. In contrast to the surface waves that have amplitudes that decline as we move away from the slab boundaries,

116

Magnetic Slabs

Figure 5.5 A sketch of the form of ux (x) in a slab for surface waves and body waves. Surface modes have maximum amplitude |ux (x)| on the boundaries of the slab, declining to smaller values away from the slab boundaries. In a wide slab the decline is very marked so that surface waves hardly disturb the central region of the slab. Body waves decline in amplitude outside the slab but are oscillatory within the slab.

body waves persist throughout the slab interior. The distinction between surface and body waves is illustrated in Figure 5.5. Consider now the motion in the environment of the slab, where m2 = m2e . Introduce 2 ne = −m2e . Then the general solution of equation (5.60) may be written in the forms  ae exp(−me x) + be exp(me x), m2e > 0, ux (x) = (5.67) ae exp(−ine x) + be exp(ine x), m2e < 0, where m2e =

(kz2 c2se − ω2 )(kz2 c2Ae − ω2 ) kz2 c2se c2Ae − ω2 (c2se + c2Ae )

.

(5.68)

The constants ae and be are arbitrary. Suppose that m2e > 0. Then, in the region x > a, the appropriate solution of equation (5.60) that declines to zero as x → ∞ is ux (x) = ae exp(−|me |x),

x > a,

(5.69)

giving a motion that is confined to the neighbourhood of the slab. When the condition on m2e is not satisfied, leaky waves may arise; see Section 5.6. Altogether, then, with the constraint that m2e > 0, transverse motions are of the form ⎧ ⎪ x > a, ⎪ ⎨ae exp(−|me |x), (5.70) ux (x) = u(x), −a < x < a, ⎪ ⎪ ⎩±a exp(|m |x), x < −a e

e

for arbitrary constant ae . In (5.70), for the environment region x < −a, the ‘+’ sign applies for the kink mode, and the ‘−’ sign applies for the sausage mode. This solution has the property that ux (x) → 0 as x → ±∞.

5.3 Compressible Effects

117

We may now apply the boundary conditions that pertain across an interface, namely that the transverse motion ux and the total pressure perturbation pT are continuous across x = ±a. Continuity of ux gives ae = u(a) exp(|me |a).

(5.71)

Continuity of pT leads to the dispersion relation connecting ω2 with kz2 : ρ0 (kz2 c2A − ω2 )

1 u (a) + ρe (k2 c2 − ω2 ) = 0. |me | z Ae m20 u(a)

(5.72)

With the choice of mode in (5.66), equation (5.72) provides the dispersion relation for both sausage and kink magnetoacoustic waves in a magnetic slab. The dispersion relation (5.72) is subject to the constraint m2e > 0. Writing c2 = ω2 /kz2 , we may cast (5.72) into the familiar forms  c = 2

c2A

+

 R 1 (c2Ae − c2A ) = c2Ae − (c2 − c2A ), 1+R 1 + R Ae

(5.73)

where now  R≡

ρe ρ0



m20 u(a) . |me | u (a)

(5.74)

Equation (5.73) is formally the same as the surface wave dispersion relation (4.15) of Chapter 4 and the hydromagnetic dispersion relation (5.27) for incompressible waves in a magnetic slab. However, there is a significant difference here in that the expression (5.74) for R depends upon the speed c, whereas for the surface wave case R is simply the ratio of plasma densities and for the hydromagnetic slab case R is a function of the wavenumber kz . The presence of c2 in R renders the dispersion relation (5.73) transcendental. The explicit forms for R are  (ρe /ρ0 )(m0 /|me |)tanh(m0 a), sausage mode, R= (5.75) (ρe /ρ0 )(m0 /|me |)coth(m0 a), kink mode. Note that the nature of R depends upon the nature of m20 . If m20 > 0, then R > 0. If m20 < 0, then m0 is imaginary and R may be positive or negative, depending upon the value of |m0 |. For R > 0 it follows immediately from the form of (5.73) that c2 lies between c2A and 2 cAe . Now surface waves in a magnetic slab correspond to the case m20 > 0, for which R > 0. Thus, we may conclude that the magnetoacoustic surface waves of a magnetic slab have propagation speeds c that lie between the Alfv´en speed of the slab and that of the environment. This statement generalizes the conclusion reached earlier for hydromagnetic surface waves in an incompressible slab. The explicit forms of the dispersion relations for sausage and kink modes follow from (5.73) and (5.75). The dispersion relations are subject to the constraint m2e > 0. From now on it is convenient to choose the positive root me of m2e , dispensing with the modulus signs

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Magnetic Slabs

on me ; our dispersion relations, then, are subject to the constraint me > 0. For sausage waves, we have c2 =

2 −1 ρe c2Ae m−1 e + ρ0 cA m0 coth(m0 a) −1 ρe m−1 e + ρ0 m0 coth(m0 a)

;

(5.76)

.

(5.77)

for kink waves, the governing dispersion relation is c2 =

2 −1 ρe c2Ae m−1 e + ρ0 cA m0 tanh(m0 a) −1 ρe m−1 e + ρ0 m0 tanh(m0 a)

Dispersion relations (5.76) and (5.77) are subject to the constraint that me > 0, but the sign of m20 remains open. The forms (5.76) and (5.77) are suitable for considering surface modes, for which m20 > 0. They are also applicable to the case m20 < 0, for which m0 is purely imaginary, but for this case – corresponding to body waves – it is better to replace the hyperbolic functions by trigonometric functions, allowing explicitly for m20 < 0. To do this write n20 = −m20 , so that n20 > 0 when m20 < 0, and choose m0 = in0 . Note the general relations (Abramowitz and Stegun 1965, sect. 4.5) sin(ix) = i sinh x,

cos(ix) = cosh x,

tan(ix) = i tanh x,

(5.78)

which connect the hyperbolic functions with the circular (trigonometric) functions. Then −1 −1 −1 m−1 0 coth(m0 a) = −n0 cot(n0 a) and m0 tanh(m0 a) = n0 tan(n0 a); note the sign difference between these two formulas. Thus, the dispersion relation (5.76) describing sausage modes may be rewritten in the form c2 =

2 −1 ρe c2Ae m−1 e − ρ0 cA n0 cot(n0 a) −1 ρe m−1 e − ρ0 n0 cot(n0 a)

,

(5.79)

.

(5.80)

while kink modes are given by c2 =

2 −1 ρe c2Ae m−1 e + ρ0 cA n0 tan(n0 a) −1 ρe m−1 e + ρ0 n0 tan(n0 a)

The forms (5.79) and (5.80) are appropriate for body (n20 > 0, m20 < 0) waves; they are subject to the constraint me > 0. Note that in the incompressible case m20 becomes kz2 , which is positive, and so no hydromagnetic body waves arise. Consider the case of short waves in a wide magnetic slab, corresponding to |kz |a  1. The form (5.63) for m20 suggests that m20 → ∞ as kz2 → ∞, provided c2 does not approach either c2s or c2A . Suppose that m0 > 0. Then, for a wide slab we may assume that m0 a is large and positive, so both tanh(m0 a) and coth(m0 a) approach unity. Thus both (5.76) and (5.77) reduce to 2 2 −1 2 2 ρ0 m−1 0 (cA − c ) + ρe me (cAe − c ) = 0.

(5.81)

We recognize here the dispersion relation governing surface waves at a single magnetic interface (cf. equation (4.14) in Chapter 4). As discussed in Chapter 4, this equation admits

5.4 The Isolated Magnetic Slab

119

of zero, one or two solutions, depending upon the orderings of the various speeds, with (when a mode exists) c lying between the two Alfv´en speeds, cA and cAe . The roots given by equation (5.81) provide the asymptotes to which the propagation speeds c of the sausage and kink surface modes in a magnetic slab approach in the limit of large |kz |a.

5.4 The Isolated Magnetic Slab We turn again to an examination of the special case of a magnetic slab embedded in a fieldfree environment, corresponding to Be = 0 and cAe = 0. We refer to such a slab as being isolated, in that the magnetic field in the slab is confined purely by an external plasma pressure: p0 +

B20 = pe . 2μ

(5.82)

The densities inside and outside the isolated slab satisfy c2 1 c2 ρe = 2s + γ 2A . ρ0 2 cse cse

(5.83)

The isolated slab is of particular interest in that it corresponds closely to the circumstances that prevail in the solar photosphere, where concentrations of magnetic field occur in the form of isolated tubes and perhaps also in more elongated slab-like structures (see Chapter 1). When the sound speeds inside and outside the slab are equal, pressure balance shows that ρ0 < ρe , i.e., the presence of the magnetic field within the slab leads to a partial evacuation of the plasma within the field. We are concerned here with the construction of the dispersion curves c = c(kz ) that describe the speed of propagation of waves along an isolated magnetic slab. Setting cAe = 0 in equation (5.76), the sausage surface mode is given by c2A ρe m0 =1+ tanh(m0 a), ρ0 me c2

(5.84)

with the sausage body modes given by c2A ρe n0 =1− tan(n0 a). ρ0 me c2

(5.85)

The kink surface and body waves follow from the above on replacing tanh by coth in equation (5.84) and tan by − cot in equation (5.85). Relations (5.84) and (5.85) are subject to the constraint that me > 0, where now   c2 2 2 me = kz 1 − 2 . (5.86) cse Thus the requirement that me > 0 governing the various dispersion relations presented above becomes simply c2 < c2se ; the speed of propagation of a wave along an isolated magnetic slab is less than the sound speed in the environment.

120

Magnetic Slabs

The transcendental nature of the sausage mode dispersion relations (5.84) and (5.85), or their kink mode equivalents, means that ultimately we must examine these relations either graphically or numerically. Nonetheless, we may obtain considerable insight into the expected form of the dispersion curves by first examining certain limiting circumstances. Consider the limit of a wide slab, corresponding to |kz |a  1. Supposing that m0 a is large and positive for |kz |a large, so that tanh(m0 a) ≈ 1, then the surface mode dispersion relation (5.84) reduces to (cf. equation (5.81)) 2 2 −1 2 ρ0 m−1 0 (cA − c ) = ρe me c ,

(5.87)

with the kink mode satisfying the same relation. This is the dispersion relation governing the propagation of surface waves on a magnetic–non-magnetic interface (see Section 4.3 of Chapter 4). It is applicable provided m20 a2 is large and positive (and me > 0). Now m20 is positive for wave speed c (taken positive) lying below the sub-sonic, sub-Alfv´enic cusp speed ct in the slab, and also for c lying between cA and cs . The first possibility corresponds to slow surface waves, the second to fast surface waves. So, provided the requirement me > 0 is met, there arises the possibility of both fast and slow surface waves occurring on an isolated magnetic slab. We have seen in Chapter 4 that both fast and slow surface waves occur on a magnetic–non-magnetic interface provided the magnetic region is cooler than the field-free region (i.e., cs < cse ) and the Alfv´en speed exceeds the sound speed inside the field (i.e., cA > cs ). These conditions are likely to be met in the solar photosphere. With the ordering of speeds cs < cse < cAe , appropriate for photospheric conditions, an isolated magnetic slab supports the propagation of a slow surface wave in the band c < ct and a fast surface wave in the band cs < c < cse . Consider now the opposite extreme of long waves in a slab, |kz |a  1. We assume that tanh(m0 a) ≈ m0 a. Then, dispersion relation (5.84) for the sausage mode becomes   ρe 2 2 (5.88) (c2A − c2 )me = c m0 a. ρ0 Since the approximation of the tanh function used here is applicable whether m20 is positive or negative, equation (5.88) applies to both surface and body waves. A detailed examination of (5.88) reveals that the only possible solutions are of the forms c2 → c2t or c2 → c2se as |kz |a → 0. Specifically, we find     ρe cse (c2s − c2t ) |k |a , |kz |a  1, (5.89) c2 ≈ c2t 1 − z ρ0 (c2s + c2A )(c2se − c2t )1/2 provided ct < cse . This gives the slow surface wave (since c2 < c2t and m20 > 0). The terms in the dispersion relation (5.89) simplify somewhat in the special case cse = cs , corresponding to the same temperature inside and outside the slab. With cse = cs , the slow surface wave gives c2 ≈ c2t − 2αct |kz |, where α=

1 ρe 2 ρ0



ct cA

|kz |a  1, 3 act .

(5.90)

5.5 Trapped Waves: β = 0 Plasma

121

In a similar way, we may explore the solution of (5.88) for c2 close to c2se . The result is  2   2  c4se ρe c2se − c2s 2 2 2 2 k a (5.91) c ≈ cse 1 − , |kz |a  1, z ρ0 (c2s + c2A )2 c2se − c2t provided cs < cse or cse < ct . The kink mode may be treated similarly and leads to   ρe |kz |a  1. |kz |a, c2 ≈ c2A ρ0

(5.92)

This describes the propagation of the slow surface kink mode. All of the above approximate results are based upon the assumption that m0 a tends to zero as kz a tends to zero (so that the approximation tanh(m0 a) ≈ m0 a is valid). Inspection of expression (5.63) for m20 shows that this assumption is valid provided c2 does not tend to c2t . In fact, such a possibility arises for body waves. Consider the sausage mode. Equation (5.85) shows that n0 tan(n0 a) must remain finite as n0 → ∞; this requires that n0 a tend to the roots of tan(n0 a). Thus, n0 a → jπ for integers j (= 1, 2, 3, . . .). This condition determines the behaviour of c2 : the sausage body mode has   4 k2 a2 c (5.93) |kz |a  1. c2 ≈ c2t 1 + t 2 2z 2 , c2s cA j π The kink body mode satisfies a relationship of the same form, except the integer j is replaced by (j − 12 ), so that the kink body modes propagate faster than the sausage body modes. All this behaviour is displayed in the dispersion curves of Figure 5.6. 5.5 Trapped Waves: β = 0 Plasma The second special case we consider is that of a β = 0 plasma. This case is of particular interest for the corona, where magnetism dominates much of the behaviour. We can examine this case from the general compressible one simply by setting sound speeds to zero. However, in view of the importance of the β = 0 plasma and its role in displaying clearly many features of wave propagation, it is of interest to rework this case from the start rather than just deduce its properties from the general dispersion relations. To begin then we note that in the case of a β = 0 plasma the equilibrium consists of a uniform magnetic field with a plasma density inhomogeneity. With a uniform equilibrium magnetic field, we have generally that ρ0 (x)c2A (x) = a constant. A convenient starting point for our analysis is thus to write down the partial differential equations that apply to the case of a β = 0 plasma with a uniform equilibrium magnetic field. Moreover, we consider two-dimensional motions in the xz-plane, with uy = 0 and ky = 0. Accordingly, from Chapter 3, we have ∂ 2 ux 1 ∂ 2 pT ∂ 2 ux − c2A (x) 2 = − , 2 ρ0 (x) ∂x∂t ∂t ∂z

(5.94)

122

Magnetic Slabs

Figure 5.6 The dispersion curves of a magnetic slab of width 2a under photospheric conditions, representative of an isolated magnetic slab. The diagram gives the wave speed ω/kz as a function of dimensionless wavenumber kz a, showing solutions of the dispersion relations (5.79) and (5.80) for a slab under photospheric conditions, namely cAe < cs < cA . Specifically, cAe = cs /2, cse = 3cs /2 and cA = 2cs ; these values give ρ0 ≈ 1.8ρe and ct ≈ 0.89cs . Full curves correspond to sausage waves and dashed curves to kink waves. [Note the notational changes from the present text: VA ≡ cA , VAe ≡ cAe , C0 ≡ cs , CT ≡ ct , CTe ≡ cte , Ce ≡ cse , k ≡ kz and x0 ≡ a.] (After Edwin and Roberts 1982.)

with ∂ux ∂pT = −ρ0 (x)c2A (x) . ∂t ∂x Eliminating pT we obtain the two-dimensional wave equation   2u 2u ∂ ∂ ∂ 2 ux x x = c2A (x) + . ∂t2 ∂x2 ∂z2

(5.95)

(5.96)

5.5 Trapped Waves: β = 0 Plasma

123

In terms of Fourier components, this yields the second order differential equation   d2 ux ω2 2 − kz ux = 0. + 2 dx2 cA (x)

(5.97)

Consider a magnetic slab of width 2a consisting of a uniform medium of plasma density ρ0 inside the slab (−a < x < a) surrounded by a uniform medium with plasma density ρe outside the slab (|x| > a); the equilibrium magnetic field is everywhere uniform (B0 = B0 ez ). Accordingly, ρ0 c2A = ρe c2Ae .

(5.98)

Now within the slab the differential equation reduces to d2 ux + n20 ux = 0, dx2

|x| < a,

(5.99)

where n20 =

ω2 − kz2 . 2 cA

(5.100)

Inside the slab the general solution of the differential equation for ux is ux = A0 cos(n0 x) + C0 sin(n0 x),

−a < x < a,

(5.101)

where A0 and C0 are arbitrary constants. The associated total pressure perturbation pT inside the slab is given by pT = −

 ρ0 c2A dux ρ0 c2A  =− −n0 A0 sin(n0 x) + n0 C0 cos(n0 x) . iω dx iω

(5.102)

Similarly, the region outside the slab is also uniform and so we may write the general solution of the differential equation applying outside the slab as ux = Ae exp(−me x) + Ce exp(me x),

|x| > a.

(5.103)

|x| > a.

(5.104)

Alternatively, we may write the solution in the form ux = De exp(ine x) + Ee exp(−ine x), Here we have written n2e =

ω2 − kz2 , c2Ae

m2e = kz2 −

ω2 . c2Ae

(5.105)

Ae , Ce , De and Ee are arbitrary constants (possibly complex). Equations (5.101), (5.103) and (5.104) provide general solutions of the differential equation irrespective of the nature of n20 and n2e , which may be positive, negative or indeed complex. It is only when we apply boundary conditions that we select different forms of the solution.

124

Magnetic Slabs

5.5.1 Trapped Sausage Modes Trapped waves correspond to choosing the exponential decay solution in the region outside the slab, taking Ce = 0 and requiring that me > 0. Inside the slab, we can choose to discuss separately those solutions that are sausage modes and do not displace the centre of the slab (requiring ux = 0 at x = 0) and kink modes, which do disturb the slab centre (ux = 0 at x = 0). Thus, for trapped sausage modes  Ae exp(−me x), x > a, ux = (5.106) C0 sin(n0 x), 0 ≤ x < a, and

 iωpT =

ρe c2Ae me Ae exp(−me x),

x > a,

−ρ0 c2A n0 C0 sin(n0 x),

0 ≤ x < a.

(5.107)

With the equilibrium condition ρ0 c2A = ρe c2Ae and the requirement of continuity of ux and pT across x = a, we obtain the dispersion relation for trapped sausage modes, namely (Edwin and Roberts 1982; Roberts, Edwin and Benz 1983) 1 −1 tan(n0 a) = . n0 a me a

(5.108)

Equation (5.108) is subject to the constraint me > 0, which corresponds in a β = 0 plasma to the requirement that ω2 < kz2 c2Ae . Substituting for n0 and me , equation (5.108) becomes 1/2 1/2  2    2 c − c2A c − c2A cAe tan |kz |a = − . (5.109) cA c2A c2Ae − c2 This is the dispersion relation for the sausage mode in a β = 0 plasma slab. The periodicity of the tan function (with a period of π ) arising in the dispersion relation means that if one solution arises then an infinity of related harmonic solutions arise also; we expect to see a multiplicity of sausage modes. These are all body waves. In fact, there are no surface waves. To see this, note that for a surface wave we have n20 < 0. Then we may write n20 = −m20 and therefore n0 = im0 . Noting that tan(ix) = i tanh(x), the dispersion relation (5.108) becomes 1 −1 tanh(m0 a) = , m0 a me a leading to a contradiction since the left side is positive (for positive or negative m0 ) whereas the right side is negative. Hence, there are no surface sausage waves in a coronal slab (with β = 0).This argument applies specifically to sausage modes but an almost identical argument applies to kink modes. So, only body waves arise in a β = 0 plasma slab. Now the constraint me > 0, corresponding to waves being trapped within the slab, implies that c < cAe . Also, the fact that only body waves are permitted, so n20 > 0, means that c > cA . Thus, the body modes of a zero-β magnetic slab have propagation speed c lying in the range cA < c < cAe .

5.5 Trapped Waves: β = 0 Plasma

125

It follows then that only a magnetic slab with an Alfv´en speed cA that is less than the Alfv´en speed cAe in the environment, so cA < cAe , is able to support the propagation of trapped magnetoacoustic waves. Equivalently, only those magnetic slabs that have a plasma density that is higher than that in the environment, so ρ0 > ρe , are able to support trapped waves (Roberts 1985b).

5.5.2 Sausage Mode Cutoffs It is interesting to note that the sausage modes have a cutoff: for the principal sausage mode to arise as a trapped mode it is necessary that kz exceeds a critical value. To see this, examine the dispersion relation (5.109) in the limit as the speed c approaches the Alfv´en speed cAe in the slab’s environment. As c2 → c2Ae , the right-hand side of (5.109) tends to −∞ and accordingly the tan function on the left-hand side of (5.109) tends also to −∞, requiring that 1/2  c2 − c2A π |kz |a → 2 c2A from above. Thus, propagation of trapped sausage waves occurs only for those wavenumbers kz that exceed a critical cutoff value given by  1/2 c2A π cutoff kz = . (5.110) 2a c2Ae − c2A Associated with the cutoff wavenumber is the cutoff frequency ωcutoff and associated period Pcutoff = 2π/ωcutoff , determined by setting ωcutoff = kzcutoff cAe : 1/2  πcAe 4a ρ0 1 cutoff ωcutoff = , P = − 1 . (5.111) 2a ( ρρ0e − 1)1/2 cAe ρe These are the cutoffs associated with the sausage mode. In the extreme ρ0  ρe the sausage mode cutoffs become π cA π cA 4a kzcutoff ∼ , ωcutoff ∼ . (5.112) , Pcutoff ∼ 2a cAe 2a cA 5.5.3 Trapped Kink Modes Trapped kink waves are treated in much the same way as trapped sausage modes. Kink modes move the slab centre (x = 0), which corresponds to  Ae exp(−me x), x > a, ux = (5.113) A0 cos(n0 x), 0 ≤ x < a, with

⎧ 2 ⎨ ρe cAe m A exp(−m x), e e e pT = ρ iωc2 ⎩ 0 A n A sin(n x), iω

0 0

0

x > a, 0 ≤ x < a.

(5.114)

126

Magnetic Slabs

With the equilibrium condition ρ0 c2A = ρe c2Ae and the requirement of continuity of ux and pT across x = a, the dispersion relation for trapped kink modes is (Edwin and Roberts 1982; Roberts, Edwin and Benz 1983) n0 a tan(n0 a) = me a.

(5.115)

Equation (5.115) is subject to the constraint me > 0, corresponding to ω2 < kz2 c2Ae . Written out more fully, we have ⎤ ⎡ 1/2 1/2   2 c2 − c2A c − c2A cAe ⎦ ⎣ |kz |a = . (5.116) cot cA c2A c2Ae − c2 This is the dispersion relation for the trapped kink mode in a β = 0 plasma slab. Just as with the sausage modes, the periodicity of the cot function (with a period of π ) means that if one solution arises then an infinity of related harmonic solutions arise also; we expect to see a multiplicity of kink modes. We can examine the form of the dispersion relations (5.115) or (5.116) in the thin tube limit. For |kz a|  1 we expect n0 a to be small, and so c2 → c2Ae . Set c2 = c2Ae (1 − 0 ) where 0 is a small quantity to be determined. Then the kink mode dispersion relation is simply (n0 a)2 = me a, from which it follows that   c2Ae − c2A 1/2 , 0 = |kz a| c2A Hence,

 0 = (kz a)

2

(5.117)

c2Ae − c2A c2A

2    ρ0 c2 = c2Ae 1 − (kz a)2 −1 , ρe

2

 = (kz a)2

|kz |a  1.

ρ0 −1 ρe

2 .

(5.118)

This relation gives the square of the speed of propagation c (= ω/kz ) of the principal kink mode in a step function slab.

5.5.4 Kink Mode Cutoffs Much the same as for the sausage waves, the kink modes have cutoffs too. The kink mode cutoff corresponding to me = 0 follows from the zeros of the tan function:  1/2 c2A π π 1 cutoff = = (5.119) kz 1/2 . 2 2 a cAe − cA a ρ0 − 1 ρe The associated cutoff frequency and period of the kink mode are 1/2  πcAe 2a ρ0 1 cutoff cutoff ω = P = −1 . 1/2 ,

a cAe ρe ρ0 − 1 ρe

(5.120)

5.5 Trapped Waves: β = 0 Plasma

In the extreme ρ0  ρe the kink mode cutoffs become   π cA π ρe 1/2 π cA cutoff kz ∼ = , ωcutoff ∼ , a cAe a ρ0 a

127

Pcutoff ∼

2a . cA

(5.121)

5.5.5 Dispersion Diagram The general dispersion relations (5.79) and (5.80), or their β = 0 forms (5.109) and (5.116), require a graphical or numerical solution. This was done originally by Edwin and Roberts (1982), with some corrections to the curves given in Edwin and Roberts (1983). Figure 5.7 gives the wave speed diagram, displaying the wave speed ω/kz as a function of the dimensionless wavenumber kz a for a low β plasma. The case of a β = 0 plasma is simply the extreme of a low β plasma exhibited in Figure 5.7; in that extreme the slow modes are absent and only the fast body waves (including the principal kink mode) remain. The slow waves are mildly dispersive (their speed ω/kz only varies weakly

Figure 5.7 The dispersion curves of a magnetic slab of width 2a under coronal conditions. The diagram gives the wave speed ω/kz as a function of dimensionless wavenumber kz a, showing solutions of the dispersion relations (5.79) and (5.80) for a magnetic slab under coronal conditions, namely cs < cA < cAe . Specifically, cAe = 5cs , cse = cs /2 and cA = 2cs ; these values give ρ0 ≈ 4.9ρe and ct ≈ 0.89cs . The fast waves are strongly dispersive, the slow waves only weakly dispersive. Full curves correspond to sausage waves and dashed curves to kink waves. For a β = 0 plasma slab, the slow modes are absent and only the fast modes remain. [Notational changes from the present text: VA ≡ cA , VAe ≡ cAe , C0 ≡ cs , CT ≡ ct , CTe ≡ cte , Ce ≡ cse , k ≡ kz and x0 ≡ a.] (After Edwin and Roberts 1982, 1983.)

128

Magnetic Slabs

with longitudinal wavenumber kz ), but the fast waves are strongly dispersive, and possess cutoffs. While the fast sausage and kink modes behave similarly in many ways, there is one important distinction between them in that, whereas all fast sausage modes have a non-zero wavenumber cutoff, the principal kink mode propagates for all kz ; only the higher radial kink harmonics exhibit a non-zero wavenumber cutoff.

5.5.6 Sausage and Kink Mode Analogies Following Edwin and Roberts (1982, 1983), it is interesting to note that dispersion relations (5.109) and (5.116) have analogies in other studies. The dispersion relation (5.109) for the sausage wave is analogous to the dispersion relation governing acoustic waves in an ocean layer. This case was studied extensively by C. L. Pekeris in the 1940s (Pekeris 1948; Ewing, Jardetzky and Press 1957). The kink wave finds an analogy with seismic waves in the Earth’s crust, first studied by A. E. H. Love in 1908 (see Love 1911). In view of these important links, the fast magnetoacoustic waves in a zero-β magnetic slab are sometimes termed magnetic Pekeris and magnetic Love waves (Roberts, Edwin and Benz 1983, 1984).

5.6 Leaky Waves: β = 0 Plasma 5.6.1 Leaky Sausage Modes The treatment of leaky waves is similar to trapped waves. For sausage modes, we write the form of ux as  Ae exp(−ine x), x > a, (5.122) ux = C0 sin(n0 x), 0 ≤ x < a, with an associated pressure perturbation pT given by  ρe c2Ae ine Ae exp(−me x), iωpT = −ρ0 c2A n0 C0 cos(n0 x),

x > a, 0 ≤ x < a.

(5.123)

The dispersion relation follows from the continuity conditions of ux and pT : 1 i tan(n0 a) = , n0 a ne a

(5.124)

with (as earlier) n20 =

ω2 − kz2 , c2A

n2e =

ω2 − kz2 . c2Ae

(5.125)

Equation (5.124) is the dispersion relation describing leaky sausage waves in a β = 0 magnetic slab; it is subject to the constraint that real (ne ) > 0 and real (ω) > 0 (given a time dependence of eiωt ) to ensure that the wave is outwardly propagating laterally. It may be noted that the leaky wave dispersion relation (5.124) corresponds to the trapped wave dispersion relation (5.108) with me replaced by ine .

5.6 Leaky Waves: β = 0 Plasma

129

It is of interest to ask what happens as kz → 0. We assume that as kz → 0, the frequency ω tends to a finite constant; the alternative is that |ω| → ∞. Then, for a finite ω the leaky mode dispersion relation (5.124) gives   ωa cAe tan . (5.126) =i cA cA Write ω = ωR + iωI , for real part ωR and imaginary part ωI . Given that the tan function has a periodicity of π , we have from (5.126) that ωR a = nπ (5.127) cA where n = 0, 1, 2, 3, . . ., and the imaginary part satisfies   ωI a cAe tan i . =i cA cA Since (Abramowitz and Stegun 1965) tan(iz) = i tanh(z), we have



ωI a tanh cA

 =

cAe . cA

(5.128)

Since the real function tanh x cannot exceed unity (see Abramowitz and Stegun 1965), the equation for ωI has a solution provided cAe < cA , and then (see Terradas, Oliver and Ballester 2005)     1 + cAe /cA 1 cAe ωI a = tanh−1 (5.129) = ln , cAe < cA . cA cA 2 1 − cAe /cA In the extreme cAe  cA we have ωI ∼

cAe , a

with a corresponding leakage time τ leakage (= 1/ωI ) of a τ leakage = . cAe

(5.130)

(5.131)

If cAe > cA , then equation (5.126) has no solution; the assumption that ω → a finite constant as kz → 0 must be false.

5.6.2 Leaky Kink Modes The treatment of leaky kink waves is similar to that of leaky sausage waves. For kink modes, we write the form of ux as  Ae exp(−ine x), x > a, (5.132) ux = A0 cos(n0 x), 0 ≤ x < a,

130

Magnetic Slabs

with an associated pressure perturbation given by  ρe c2Ae ine Ae exp(−ine x), iωpT = ρ0 c2A n0 A0 sin(n0 x),

x > a, 0 ≤ x < a.

(5.133)

The dispersion relation follows from the continuity conditions of ux and pT : n0 a tan(n0 a) = ine a.

(5.134)

This is the dispersion relation describing leaky kink waves in a β = 0 magnetic slab; it is subject to the constraints real (ne ) > 0, real (ω) > 0 (given a time dependence of eiωt ). The constraint is to ensure that the wave is outwardly propagating in the lateral direction. As with the sausage mode, it may be noted that the leaky wave dispersion relation (5.134) corresponds to the trapped dispersion relation (5.115) with me replaced by ine . We note too that the dispersion relation (5.134) can be rewritten in the form 1 i cot(n0 a) = − . n0 a ne a For small kz we have



ωa cot cA

 = −i

cAe . cA

The function cot is periodic with period π, so ωR a = nπ . cA

(5.135)

(5.136)

(5.137)

Also, from the imaginary part we obtain   ωI a cAe cot i . = −i cA cA Hence

 coth

which we rewrite in the form



ωI a cA

ωI a tanh cA

 =

cAe , cA

=

cA . cAe



For this equation to have a solution we require cA < cAe (and so ρ0 > ρe ). Altogether, then, we have cA ωR = nπ , a     cA 1 + (cA /cAe ) cA cA −1 tanh ln , cA < cAe . = ωI = a cAe 2a 1 − (cA /cAe ) In the extreme cAe  cA , we may expand the log term to give   c2 cA ρe 1/2 ωI ∼ = A , ρ0  ρe , a ρ0 acAe

(5.138)

(5.139)

(5.140)

5.7 Impulsive Waves

with a corresponding leakage time τ leakage (= 1/ωI ) of acAe ρ0  ρe . τ leakage = 2 , cA

131

(5.141)

5.6.3 |kz a| < 1: Sausage and Kink Modes Consider the typical coronal loop situation for which ρ0 > ρe ,

cA < cAe .

For the sausage mode, no finite ωR exists as kz a → 0 and instead ωR → ∞. This suggests that there is no preferred frequency for small kz a and instead the cutoff frequency for the sausage mode will play an important role: ωcutoff =

πcAe 1 1/2 .

2a ρ0 − 1 ρe

(5.142)

Turning to the kink mode, the principal kink mode exists for all kz (since ρ0 > ρe ). For kz a → 0, the kink mode dispersion relation yields (see equation (5.118)) 2    2 c −1 c2 ∼ c2Ae 1 − (kz a)2 Ae c2A 2    2 2 ρ0 = cAe 1 − (kz a) −1 . (5.143) ρe However, lateral overtones of the kink mode suffer a cutoff and for small kz a they are leaky. The first leaky kink mode possesses a frequency ωR and an associated damping timescale ωI−1 where     cA cA cA 1 + (ρe /ρ0 )1/2 1 + (cA /cAe ) ωI = ln = ln ωR = π , , ρ0 > ρe . a 2a 1 − (cA /cAe ) 2a 1 − (ρe /ρ0 )1/2 (5.144) This damping is fairly weak.

5.7 Impulsive Waves The presence of cutoffs in the dispersion curves plays an important role when the wave is excited impulsively. This was first demonstrated by Pekeris (1948) for sound waves in an ocean layer, and it was realized by Roberts, Edwin and Benz (1983, 1984) that much the same feature must occur when fast magnetoacoustic waves are wave-guided in magnetic structures. Figure 5.8 illustrates the resulting signature of an impulsively excited sausage mode in a magnetic slab. Impulsively excited fast waves are thought to be involved in an interpretation of rapid pulsations detected in an eclipse (Williams et al. 2001, 2002). Such waves are usually associated with flares. When impulsively excited, the dispersion in a wave is important for it determines the detailed nature of the resulting disturbance. The fast magnetoacoustic

132

Magnetic Slabs

Figure 5.8 The impulsively generated signal of a fast sausage wave in a magnetic slab. The sketch shows the signature of the wave at an observational location z = h from an impulsive source located at z = 0 which initiated the wave motion at time t = 0. The signal has three distinct phases. First is the periodic phase, wherein the wave amplitude gradually grows in time and exhibits an almost periodic disturbance; this phase begins after a time t = h/cAe . Then at time t = h/cA the periodic phase gives way to the quasi-periodic phase, which is stronger in amplitude and more irregular in behaviour. Finally, at a time depending upon the waves group velocity cg , the quasi-periodic phase gives way to a decay phase, as the wave and its wake pass by the observation point z = h. [Notational changes from the present text: VA ≡ cA , VAe ≡ cAe and cg ≡ Cg .] (After Roberts, Edwin and Benz 1983, 1984, drawing on Pekeris 1948.)

waves in a magnetic slab are strongly dispersive, and this geometrical dispersion results in a distinctive wave–packet behaviour for the fast sausage mode (and similarly for the harmonics of the fast kink modes). This was first realised by Roberts, Edwin and Benz (1983, 1984), who drew on the analogy between the waves in a magnetic slab and the ducted sound waves in an ocean layer (Edwin and Roberts 1982, 1983). In an extensive analysis, Pekeris (1948) demonstrated that an impulsive signal in such a sound wave duct will carry a complex signature of the form sketched in Figure 5.8. The same diagram must apply to the fast waves in a magnetic slab. Indeed, numerical simulations have shown directly such features for both slabs (see Murawski and Roberts 1993a, b, 1994; Nakariakov et al. 2004; Nakariakov, Pascoe and Arber 2005; Nakariakov, Hornsey and Melnikov 2012) and tubes (see Oliver, Ruderman and Terradas 2014, 2015; Shestov, Nakariakov and Kuzin 2015). When impulsively excited, the fast sausage mode produces a wave packet, the nature of which is defined by the internal and external Alfv´en speeds of the slab (or tube), together with the detailed form of the group velocity for the mode (which depends specifically upon the plasma density profile that is modelled; see Nakariakov and Oraevsky 1995). Wavelet analysis has shown that a wave train may resemble a ‘crazy tadpole’ (Nakariakov, Pascoe and Arber 2005), consisting of a narrow spectrum tail preceding a broadband head. Pascoe, Nakariakov and Kupriyanova (2013) give a good summary of more recent developments. All the features of an impulsively generated wave can be determined theoretically: in particular, the wave packet exhibits rapid oscillations. The timescale τ pulse for such oscillations in a slab is of the order of the Alfv´en travel time across the slab, specifically, τ pulse =

4a cAe



ρ0 −1 ρe

1/2 .

(5.145)

5.8 The β = 0 Slab: Epstein Profile

133

For example, in a slab of width 2a = 103 km and an external Alfv´en speed of cAe = 1.5 × 103 km s−1 with the plasma density inside the tube being ten times that in the environment (so ρ0 = 10ρe ), yielding an internal Alfv´en speed of cA = 474 km s−1 then the timescale in the guided wave is τ pulse = 4 s. Similar results hold for a magnetic flux tube (see Chapter 6). 5.8 The β = 0 Slab: Epstein Profile The special case of a β = 0 plasma discussed in Section 5.5 lends itself to various extensions. Instead of considering an equilibrium density profile that consists of a discontinuous step function we may consider a smoothly varying profile ρ0 (x). Thus the equilibrium is one of a uniform magnetic field with a density profile ρ0 (x) that varies arbitrarily across the field; the Alfv´en speed cA (x) is then such that ρ0 (x)c2A (x) is a constant. Accordingly, the governing ordinary differential equation (5.97) is rewritten here for convenience:   ω2 d2 ux 2 − kz ux = 0, + 2 (5.146) dx2 cA (x) while changes in the pressure perturbation pT are given by (see equation (5.95)) dux . (5.147) dx We are interested in solutions of equation (5.146) that tend to zero far from the centre of the structure: ux → 0 as x → ±∞. Such solutions correspond to trapped waves. As for the case of a discrete step function slab, there are an infinity of modes determined by their various lateral harmonic structures. Consider the specific density profile

x . (5.148) ρ0 (x) = ρe + (ρ0 − ρe )sech2 a This gives a density profile that is symmetric about x = 0, takes the value ρ0 at x = 0 and the value ρe as x → ±∞; the profile has a characteristic spatial scale of a. A profile of this form was discussed by Epstein (1930) in the context of radio wave propagation and it is of interest in quantum mechanics (Landau and Lifshitz 1958) and fibre optics (Adams 1981). Discussions in solar physics are given in Nakariakov and Oraevsky (1995) and Cooper, Nakariakov and Williams (2003). The profile is of special interest since it turns out to permit an analytical treatment of a medium with a continuous density and moreover is of value in numerical simulations (Pascoe, Nakariakov and Arber 2007) of wave propagation in magnetic atmospheres. Other continuous profiles are, of course, of interest too; see Edwin and Roberts (1988), Terradas, Oliver and Ballester (2005), Li, Habbal and Chen (2013), Lopin and Nagorny (2015) and Yu et al. (2015). In the Epstein density profile (5.148), the square of the Alfv´en speed, c2A (x) (= B20 /μρ0 (x)), associated with the profile is iωpT = −ρ0 c2A (x)

c2A (x) =

c2Ae [(c2Ae /c2A ) − 1) + cosh2 ( ax )]

cosh2

x a

,

(5.149)

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Magnetic Slabs

giving cA (x) varying from cA ≡ cA (x = 0) (= (B20 /μρ0 )1/2 ) at x = 0 to cAe (= (B20 /μρe )1/2 ) as x → ±∞. The density profile (5.148) allows an exact analytical solution of equation (5.146). To see this, we begin by making a change of spatial variable; set s = tanh (x/a).

(5.150)

Note that this change of variable compresses x, which varies from −∞ to +∞, to a dimensionless variable s, which varies in a one–one (monotonic) fashion from −1 to +1. A little algebra shows that the differential operators d/dx and d2 /dx2 transform to d2 1 2 d d2 = 2 (1 − s2 )2 2 − 2 s(1 − s2 ) . 2 ds dx a ds a

d d 1 = (1 − s2 ) , dx a ds Hence equation (5.146) becomes 2u x ds2

2 2d

(1 − s )

dux − 2s(1 − s ) + ds 2



    ω2 a2 ω2 a2 ω2 a2 2 2 2 − k a − ) ux = 0. + (1 − s z c2Ae c2A c2Ae

That is, transverse motions ux are governed by the second order differential equation     d λ2 2 dux (5.151) (1 − s ) + ν(1 + ν) − ux = 0, ds ds (1 − s2 ) where



λ =a 2

2

kz2

ω2 − 2 cAe



 =

kz2 a2

c2 1− 2 cAe



 ,

ν(1 + ν) = ω a

2 2

1 1 − 2 2 cA cAe

 (5.152)

and c = ω/kz is the wave speed. Equation (5.151) is Legendre’s equation, with solutions the associated Legendre functions. However, it proves convenient to recast this into the hypergeometric equation. Set ux = (1 − s2 )λ/2 V(s). Then

(5.153)

  dux λs 2 λ/2 dV = (1 − s ) − V ds ds (1 − s2 )

and

    dux d2 V dV λ 2 s2 d (1 − s2 ) = (1 − s2 )λ/2 (1 − s2 ) 2 − 2(1 + λ)s − λV + V . ds ds ds ds (1 − s2 ) A further scaling of s proves useful: write

1 1 − s, 2 2 which transforms the variable s extending from s = −1 to s = +1 to a variable ξ extending from ξ = 1 to ξ = 0. Then our governing equation becomes ξ=

ξ(1 − ξ )

dV d2 V + [C − ξ(1 + A + B)] − ABV = 0, dξ dξ 2

(5.154)

5.8 The β = 0 Slab: Epstein Profile

135

where the parameters A, B and C are such that AB = λ(1 + λ) − ν(1 + ν) = (λ − ν)(1 + λ + ν),

1 + A + B = 2(1 + λ),

C = 1 + λ.

Equation (5.154) is the canonical form of the hypergeometric equation (Abramowitz and Stegun 1965), representing all second order linear ordinary differential equations with three regular singularities (here at ξ = 0, 1 and ∞). The parameters A, B and C may be taken as A = λ − ν,

B = 1 + λ + ν,

C = 1 + λ,

(5.155)

and so (through relations (5.152)) depend upon wavenumber kz and frequency ω. Now the hypergeometric equation (5.154) has solutions (Abramowitz and Stegun 1965) F(A, B; C; ξ )

and

ξ (1−C) F(1 + A − C, 1 + B − C; 2 − C; ξ )

valid in the region of ξ = 0. Here the function F(A, B; C; ξ ) has series representation F(A, B; C; ξ ) = 1 +

A(A + 1)B(B + 1) ξ 2 AB ξ+ + · · ·, C C(C + 1) 2!

convergent in the region |ξ | < 1. In terms of the variable s, these solutions are   1 (1 − s2 )λ/2 F λ − ν, 1 + λ + ν; 1 + λ; (1 − s) 2 and

  1 (1 − s2 )λ/2 (1 − s)−λ F 1 + A − C, 1 + B − C; 2 − C; (1 − s) 2

valid near s = 1. Since F(A, B; C; 0) = 1, it follows that the first solution satisfies the boundary condition ux → 0 as s → +1 (ξ → 0) provided λ > 0, but the second solution is singular and is rejected. Accordingly, we take as our solution   1 (5.156) ux = u0 (1 − s2 )λ/2 F λ − ν, 1 + λ + ν; 1 + λ; (1 − s) 2 for arbitrary constant u0 . With λ > 0, this solution satisfies the boundary condition ux → 0 as s → +1; the condition that λ > 0 implies that c2 < c2Ae . There remains the condition that ux → 0 as s → −1 (ξ → +1). To apply this condition we need the value of the hypergeometric function on the unit circle; we have (Abramowitz and Stegun 1965) F(A, B; C; 1) =

(C)(C − A − B) , (C − A)(C − B)

which holds (for real parameters A, B and C) provided C − A − B > 0 and C = 0, −1, −2, . . .. But applied to the above, we see that here C − A − B = −λ and so is negative; accordingly we cannot apply this form to determine the behaviour as s → −1. However, the linear transformation (Abramowitz and Stegun 1965) F(A, B; C; ξ ) = (1 − ξ )C−A−B F(C − A, C − B; C; ξ )

136

Magnetic Slabs

gives parameters that satisfy the requirement for evaluation at ξ = 1, for now C −(C −A)− (C − B) = A + B − C = λ > 0. But then the term (1 − ξ )C−A−B = ((1 + s)/2)λ results in a singular behaviour as s → −1; specifically, as s → −1     1 1 − s λ/2 F 1 + λ, −ν; 1 + λ; (1 − s) ux = u0 1+s 2  λ/2 2 (1 + λ)(λ) ∼ . 1+s (λ − ν)(1 + λ + ν) Thus, there is a singularity at s = −1; it would seem that no solution is possible! However, the above result also shows that there is an implicit assumption that λ − ν is not a negative integer or zero, for otherwise the gamma function (λ−ν) is itself singular and 1/(λ − ν) is zero. Thus, for a solution to exist that satisfies ux → 0 as s → −1 we must take λ − ν = −n

(5.157)

where n = 0, 1, 2, 3, . . .. This constraint on the parameters λ and ν is in fact the dispersion relation for modes in the Epstein profile. The condition (5.157) implies     1 1 F λ − ν, 1 + λ + ν; 1 + λ; (1 − s) = F −n, 1 + λ + ν; 1 + λ; (1 − s) 2 2 is a polynomial in 12 (1 − s) of degree n. The simplest case is n = 0 when we may take the polynomial to be unity (F = 1); the case n = 1 leads to a polynomial of degree 1 which gives F = s, etc. Altogether, then, the first two solutions are   1 − s λ/2 n=0: ux = u0 with λ = ν, 1+s and



n=1:

1−s ux = u0 s 1+s

λ/2 with

λ = ν − 1.

These are respectively the principal kink and sausage modes of the Epstein profile. It is of interest to write our solutions in terms of the original coordinate system; for clarity, we present each case separately. 5.8.1 Epstein Kink Mode (n = 0) The principal kink mode is ux = u0 sechλ

x a

,

(5.158)

with the power λ (> 0) given by 

c2 λ = |kz a| 1 − 2 cAe

1/2 (5.159)

5.8 The β = 0 Slab: Epstein Profile

137

and so dependent upon kz and c. Also, since λ = ν we have   1 1 2 2 − 2 ν(1 + ν) = ω a . c2A cAe Since λ(= ν) > 0 we require that c2 < c2Ae and c2A < c2Ae . The solution (5.158) has the property that ux = 0 for all finite x. In particular, the centre of the slab is disturbed (ux (x = 0) = u0 = 0); hence the terminology of referring to this as the principal kink mode. The principal kink mode has dispersion relation λ = ν, which may be rearranged into the form   1/2  c2 c2 = |kz a| 2 − 1 . (5.160) 1− 2 cAe cA Written in terms of the expressions n0 and me that arise in the step function slab, this relation becomes (on multiplication by a) me a = (n0 a)2 ,

(5.161)

which is exactly the same as the thin tube form of the kink mode dispersion relation in a step function slab. Thus, c2A < c2 < c2Ae ; this corresponds to a density enhancement (ρ0 > ρe ). The dispersion relation (5.160) is the equivalent relation to equation (5.116) obtained for the discontinuous density profile. The dispersion relation (5.160) is straightforward to investigate, either by squaring it to obtain a quadratic in c2 /c2Ae or by regarding it as an equation for |kz a| as a function of c2 between c2A and c2Ae . It is immediately evident that c → cAe for small kz a and that c → cA for large kz a. It follows immediately from the dispersion relation (5.160) or (5.161) that (see equation (5.118)) 2    ρ0 −1 c2 = c2Ae 1 − (kz a)2 , ρe

|kz |a  1.

(5.162)

5.8.2 Epstein Sausage Mode (n = 1) The principal sausage mode is (see also Cooper, Nakariakov and Williams 2003)

x

x ux = u0 sechλ tanh (5.163) a a with dispersion relation ν = 1 + λ, that is 

  1/2 c2 c2 2 2 − 1 kz a − 3kz a 1 − 2 − 2 = 0. c2A cAe

The dispersion relation (5.164) is a quadratic equation for kz a as a function of c2 .

(5.164)

138

Magnetic Slabs

The dispersion relation has a cutoff: when c2 = c2Ae , the cutoff wavenumber kzcutoff being given by 1/2 √  c2A 2 cutoff kz = . (5.165) a c2Ae − c2A The wavenumber kz must exceed the cutoff value for the sausage mode to be trapped. This occurrence of a cutoff is to be expected from the behaviour of the sausage mode in the case of a discrete profile. Comparing the cutoffs that arise in the two cases, we see that the critical wavenumber differs only by a numerical factor: in the discrete √ slab the numerical factor is π/2 (see equation (5.110)), whereas in the Epstein profile it is 2.

6 Magnetic Flux Tubes

6.1 Introduction In this chapter we turn to a consideration of the waves in a cylindrical magnetic flux tube. This is similar to the case of a magnetic slab discussed in Chapter 5, the magnetic slab being slightly easier to consider because it involves trigonometric and hyperbolic functions, whereas we may anticipate Bessel functions in any treatment of a cylindrical tube. However, the magnetic flux tube is important in its own right, simply because many concentrations of magnetic field occurring in laboratory or astrophysical plasmas are more appropriately modelled as tubes rather than slabs. Accordingly, we give here a discussion devoted to cylindrical geometry. The similarities between waves in a magnetic tube and a magnetic slab will be pointed out, as appropriate. The magnetic flux tube may be viewed as a communication channel, linking one region in a plasma with another. This introduces a one-dimensionality, a directness of communication. Flux tubes may carry waves, flows and energy from one site to another. Accordingly, it is important to know what are the basic modes of oscillation of a magnetic flux tube. Viewed geometrically, there are sausage modes, kink modes, fluting modes and torsional modes, each of which disturbs the tube in a distinctive geometrical manner. Figure 6.1 displays some of the various geometrical patterns that the boundary of the flux tube may make when it is in oscillation. If the vibrating tube maintains a circular cross-sectional form centred about the axis of symmetry of the undisturbed tube, so that the central axis of the tube remains motionless, then this is the sausage mode; in the sausage oscillation, a crosssection of the tube simply expands or contracts, but always maintaining a circular form. Looked at from the side, the vibrating tube resembles a string of sausages suspended from a point, hence the term ‘sausage’ mode (see Chapter 5). By contrast, in the kink mode the vibrating tube suffers a bulk displacement, with the central axis of the tube swaying from side to side, as in the motion of a snake. In the kink mode the tube moves as a whole, with very little compression (because the tube is not squeezed); this is a global oscillation of the tube with the central axis of the tube being displaced in the vibration. The kink and sausage modes are the two most significant modes of vibration of the tube. However, there are also various higher order distortions of the tube boundary which leave the central axis of the tube undisturbed. These distortions of the boundary may range in shape from elliptical to highly crinkled; these are the fluting modes. Finally, we may imagine the tube remaining circular in shape but suffering a twisting or torsional motion,

139

140

Magnetic Flux Tubes B

B

Figure 6.1 The modes of oscillation of a magnetic flux tube, illustrating the various distortions that an oscillating tube can perform. The short horizontal arrows indicate the local transverse motions. A symmetric squeezing and rarefying of the tube, which leaves the central axis of the tube undisturbed, is termed the sausage mode; see the left side of the sketch. By contrast, when a bulk or global movement of the tube occurs, displacing the central axis of the tube, then this is the kink mode; see the right side of the sketch. There is also the possibility that the central axis of the tube remains undisturbed but the tube boundary is distorted from its circular shape, producing a crinkly boundary; these are the fluting modes (not shown above). Finally, the tube may also be rotated back and forth about its central axis; this results in torsional oscillations (also not shown in the sketch). (From Morton et al. 2012.)

so the central axis of the tube is undisturbed but the boundary of the tube rotates; this is the torsional mode. We return to the geometrical description of the modes once we have introduced a Fourier representation; see Section 6.2.4 below. The modes of a tube may be classified further according to their spatial nature in the radial coordinate r: disturbances that inside the tube are oscillatory in r are referred to as body waves, and those that are exponential in form are the surface waves. Basically, in a wide tube the surface waves are confined to near the boundary of the tube and do not penetrate far into the centre of the tube, whereas body modes disturb the centre of a wide tube; however, in a thin tube, the centre is disturbed for both surface and body modes. In addition to the modes that are trapped within a tube, disturbing the tube’s environment only slightly, there arise leaky waves. Leaky waves are generated by motions within the tube that result in an outflow of wave energy: the tube is a generator for waves in its environment. These classifications may be applied to both fast and slow magnetoacoustic waves, so the description becomes complicated. Finally, in addition to the magnetoacoustic waves, a tube may support incompressible disturbances; these are the torsional Alfv´en waves. The characteristic speeds that govern wave propagation in a magnetic flux tube are readily established. Consider a plasma of density ρ0 and pressure p0 within which is embedded a magnetic field B0 of strength B0 . The sound speed cs and Alfv´en speed cA are defined by  1/2   γ p0 1/2 B0 2 cs = , cA = , (6.1) ρ0 μ0 ρ0

6.1 Introduction

141

where μ0 is the magnetic permeability and γ the adiabatic index of the plasma. From these two speeds it is natural to construct combinations that we may associate with the fast or slow magnetoacoustic waves. Accordingly, we introduce a fast magnetoacoustic speed, cf , through the combination c2f = c2s + c2A ,

(6.2)

and a slow speed, ct , through the slightly less natural construction −2 −2 c−2 t = cs + cA .

This expression may be rearranged to give  1/2 c2s c2A cs cA = . ct = 2 2 cf cs + cA

(6.3)

(6.4)

Thus the fast speed is super-sonic and super-Alfv´enic (cf ≥ cs , cA ), and the slow speed is sub-sonic and sub-Alfv´enic (ct ≤ cs , cA ). The slow speed proves to be particularly significant for waves in a magnetic flux tube. Indeed, a speed equivalent to ct is common to a variety of elastic tubes, with the role of the Alfv´en speed being played by the appropriate elastic speed of the physical situation. For example, in the case of a blood vessel the equivalent of the speed cA is simply the elastic speed in the membrane of the blood vessel. In this case, the speed of sound cs in blood is much larger than the elastic speed, so effectively cs  cA giving ct ≈ cA ; wave propagation in a blood vessel proceeds with a speed that is close to the elastic speed of the membrane walls. For water contained in a pipe, the relative magnitudes of the two basic speeds depends upon the material of the pipe. In a metal pipe, the elastic speed of the metal membrane is much larger than the speed of sound in water, and so the effective propagation speed is close to the sound speed in water (about 1.4 km s−1 ). In a plastic pipe, the orderings in the two speeds are reversed and the effective propagation speed is close to the elastic speed of the plastic (about 10 m s−1 ), lying far below the speed of sound in water. A detailed description of such waves in elastic tubes is given in Lighthill (1978). The bulk motion of a tube in the kink mode of oscillation results in the tube disturbing its environment in almost equal measure to its cross-sectional size. The result is that the characteristic speed for the kink wave involves both the density ρ0 of the plasma inside the tube and the density ρe of the plasma in the environment. At least for a thin tube, we may view the kink mode as somewhat akin to an Alfv´en wave, except that the kink mode disturbs the environment of the tube: both waves are driven principally by the magnetic tension force. In the general case of a tube of field strength B0 embedded in a magnetic environment of field strength Be , the tension force in the magnetic field in the environment plays a role too, producing the kink speed ck :  1/2  1/2 B20 /μ0 + B2e /μ0 ρ0 c2A + ρe c2Ae ck = = . (6.5) ρ0 + ρe ρ0 + ρe The speed ck is intermediate between the two Alfv´en speeds of the media, lying between the Alfv´en speed cA of the tube and the Alfv´en speed cAe of the environment.

142

Magnetic Flux Tubes

Two extremes of expression (6.5) are of particular interest. The case where the field is everywhere uniform (B0 = Be ), so that the tube is defined solely by virtue of a plasma density (or temperature) contrast (ρ0 = ρe ), produces a kink speed  1/2 2ρ0 ck = cA . (6.6) ρ0 + ρe √ So, for the extreme of a high density tube, ρ0  ρe , we have ck = 2cA ; thus, the kink speed in a dense coronal flux tube is some 41% higher than the Alfv´en speed within the tube. The second case of interest is when a flux tube is embedded in a field-free (Be = 0) environment, typical of a photospheric tube; in this case, the kink speed is given by  1/2  1/2 B20 /μ0 ρ0 ck = = cA . (6.7) ρ0 + ρe ρ0 + ρe Accordingly, in an isolated tube, the speed ck is sub-Alfv´enic. Some specific numerical illustrations may be helpful. In a coronal tube, for example, we may take a tube Alfv´en speed of cA = 103 km s−1 and a sound speed of cs = 200 km s−1 , which produce a kink speed of ck = 1414 km s−1 (in a high density tube), a slow speed of ct = 196 km s−1 and a fast speed of cf = 1020 km s−1 . On the other hand, in an isolated magnetic flux tube typical of photospheric conditions, we may consider a tube of field strength B0 = 2 kG and plasma density ρ0 = 2.2 × 10−4 kg m−3 , embedded in a field-free environment. Then the Alfv´en speed is cA = 12 km s−1 . For a plasma pressure of p0 = 0.84 × 104 Pa and γ = 5/3, we find a photospheric sound speed of cs = 8 km s−1 . Thus, the slow speed in the tube is ct = 6.7 km s−1 and the fast speed is cf = 14.4 km s−1 . Finally, supposing that ρ0 = ρe /2, so that the interior of the photospheric tube is 50% evacuated by the magnetic field, we obtain ck = 6.9 km s−1 , or roughly 60% of the Alfv´en speed. 6.2 Wave Equations 6.2.1 General Formalism The above discussion makes plausible the speeds that are likely to arise in a description of the modes of oscillation of a magnetic flux tube, but it is too imprecise to stand on its own. Accordingly, we turn now to a detailed investigation of the linear modes of oscillation of a tube. The starting point for our discussion is the equilibrium arrangement of a straight magnetic field that is structured in the radial direction. We consider an equilibrium magnetic field B0 = B0 (r)ez aligned with the z-axis of a cylindrical polar coordinate system r, φ, z. The equilibrium plasma pressure p0 (r) and density ρ0 (r) are also structured in radius r, with total pressure balance (see equation (3.2) of Chapter 3) requiring that   B20 (r) d p0 (r) + = 0; (6.8) dr 2μ

6.2 Wave Equations

143

again we see that spatial variations in magnetic pressure, B20 (r)/2μ, are balanced by corresponding variations in plasma pressure, p0 (r). Small amplitude motions u about the equilibrium (6.8) are readily shown to satisfy the wave equation   2 ∂ 2u ∂pT 1 2 ∂ u 2 ∂ − c = −e c grad (div u) − , (6.9) z A A ∂z ρ0 ∂t ∂t2 ∂z2 where cA (r) (= B0 (r)/(μρ0 (r))1/2 ) denotes the Alfv´en speed within the field. Equation (6.9) follows from the time derivative of the momentum equation combined with the induction equation (see Chapter 3). Note the anisotropic nature of (6.9), evident in the explicit presence of a term in the direction of ez , the direction of the applied magnetic field, and the explicit mention of the z-coordinate. Additional to equation (6.9) is the equation describing the evolution of the perturbation in the total pressure, pT (r, φ, z), the sum of the plasma pressure perturbation and the perturbation in the magnetic pressure: ∂uz ∂pT = ρ0 c2A − ρ0 (c2s + c2A )div u, ∂t ∂z

(6.10)

where cs (r) (= (γ p0 (r)/ρ0 (r))1/2 ) is the sound speed within the plasma. Equation (6.10) is simply a combination of the isentropic equation and the induction equation. Again we note the anisotropic nature of the equation, explicit in the separate presence of the flow component uz in the direction of the applied magnetic field. For a motion u(r, φ, z) = (ur , uφ , uz ), the components of equation (6.9) give ∂ 2 ur 1 ∂ 2 pT ∂ 2 ur 2 − c (r) = − , A ρ0 (r) ∂r∂t ∂t2 ∂z2 ∂ 2 uφ ∂ 2 uφ 1 1 ∂ 2 pT − c2A (r) 2 = − , 2 ρ0 (r) r ∂φ∂t ∂t ∂z   ∂ 2 uz c2s (r) ∂ 2 uz 1 ∂ 2 pT 2 − c (r) = − . t ∂t2 ∂z2 c2s (r) + c2A (r) ρ0 (r) ∂z∂t

(6.11) (6.12) (6.13)

Note that the slow speed ct is a function of the radial coordinate r: c2t (r) =

c2s (r)c2A (r) c2s (r) + c2A (r)

.

Finally, from (6.10) the evolution of pT (r, φ, z) is given by   ∂pT 1 ∂ 1 ∂uφ ∂uz 2 2 = −ρ0 (r)(cs (r) + cA (r)) (rur ) + − ρ0 (r)c2s (r) . ∂t r ∂r r ∂φ ∂z

(6.14)

The wave equations (6.11)–(6.13) coupled with the evolution equation (6.14) are the governing equations describing the linear motions of a cylindrically symmetric inhomogeneous magnetic field, such as found in a magnetic flux tube. The form of equations (6.11)– (6.13) is reminiscent of coupled oscillator equations, but it must be remembered that the unknown motion u and pressure perturbation pT are also related through (6.14).

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6.2.2 Torsional Alfv´en Waves There is one solution of the general equations we may readily note. Observe that when the perturbations are independent of the azimuthal angle φ, so that ∂/∂φ = 0, then azimuthal motions uφ are decoupled from (6.11)–(6.14) and satisfy ∂ 2 uφ ∂ 2 uφ 2 = c (r) . A ∂t2 ∂z2

(6.15)

Thus, azimuthal motions satisfy the one-dimensional wave equation. Axisymmetric motions uφ (r, z, t) may take place independently of motions in the r- and z-directions. When purely azimuthal motions take place without generating any pressure changes, so that u = uφ (r, z, t)eφ and pT = 0, then these motions are torsional Alfv´en waves. The r-dependence in the motions uφ of a torsional Alfv´en wave, satisfying the wave equation (6.15), are determined simply by the means through which the wave is generated. It is evident from equation (6.15) that torsional Alfv´en waves are subject to phase mixing (see Chapters 3 and 12). 6.2.3 Sausage Waves: β = 0 Plasma There is another reduction of the coupled wave equations (6.11)–(6.14) that we should note. In the case of axisymmetric oscillations, so that ∂/∂φ = 0, and for a β = 0 plasma (so that we can set cs = 0), the total pressure variations pT may be eliminated from our coupled wave equations to give a single partial differential equation for the radial motions ur ,   ∂ 2 ur ∂ 2 ur ∂ 1 ∂ 2 2 ρ0 (r)cA (r) (rur ) . ρ0 (r) 2 − ρ0 (r)cA (r) 2 = ∂r r ∂r ∂t ∂z Further, noting that consistent with cs = 0 we also have a uniform equilibrium magnetic field (and so ρ0 (r)c2A (r) is a constant), the above equation reduces to   ∂ 2 ur ∂ 2 ur 1 ∂ur ∂ 2 ur 1 2 = cA (r) + − 2 ur + . (6.16) r ∂r ∂t2 ∂r2 r ∂z2 Thus, for this special case we obtain a single wave-like equation for the radial motion ur . Equation (6.16), which describes linear sausage mode oscillations in a radially structured medium with β = 0, was first given by Nakariakov, Roberts and Murawski (1997), and further investigated in Nakariakov, Hornsey and Melnikov (2012) and Chen et al. (2015a, b).

6.2.4 Fourier Representation Returning to the general equations (6.11)–(6.14), to progress further we introduce the Fourier form of these equations. Write ur (r, φ, z, t) = ur (r) exp i(ωt − mφ − kz z),

(6.17)

with similar forms for all other perturbation quantities. The mode number m describes the geometrical form, in the φ-coordinate, of the perturbations. In order for the perturbations

6.2 Wave Equations

145

to be uniquely described after a rotation through an angle of φ = 2π , m must be an integer (m = 0, ±1, ±2, . . .). The symmetry of the equilibrium state, of a straight magnetic field with equilibrium quantities being functions of r alone, allows us to consider positive or zero integers only: m ≥ 0. The case m = 0 corresponds to symmetric motions of the tube, disturbances being independent of φ. We have noted above that torsional Alfv´en waves may arise when m = 0 and such oscillations involve no pressure changes (pT = 0) or motions in the radial (ur = 0) or longitudinal (uz = 0) directions. But, in addition to torsional Alfv´en waves, there may arise compressional motions (pT = 0) which perturb the field axisymmetrically; these are the sausage modes (m = 0). Modes with m = 1 are the kink modes, giving a global disturbance of the tube. Finally, waves with m ≥ 2 are the fluting modes of the tube. We can view the modes in greater detail by considering the cross-sectional shape of the magnetic tube as it is distorted by the motion (6.17). Suppose that the undisturbed magnetic tube has the circular cross-section r = a, a tube of radius a. Then the boundary of the moving magnetic tube may be described by the real part of the equation r = a + ξ0 exp i(ωt − mφ − kz z),

(6.18)

where ξ0 is a complex constant. In terms of real variables, the vibrating magnetic flux tube may be taken to have a cross-section of the form r = a + ξ(z, t) cos mφ,

(6.19)

for suitable ξ(z, t) describing the variation with z and t. The assumption of linearity means that |ξ |  a, |ξ0 |  a. How is the tube disturbed by the motion? Figure 6.2 illustrates the various geometrical shapes. The undisturbed configuration is, of course, a circle of radius a, centre r = 0: r = a. In the m = 0 mode, the boundary of this circle is displaced to r = a + ξ(z, t), which again is a circle (for given z and t): the tube r = a may be viewed as simply expanding and contracting in time (or z), but always preserving its circular shape, centred on the origin r = 0. The tube simply ‘breathes’ in and out. In the m = 1 mode, the disturbed boundary becomes r = a + ξ(z, t) cos φ, for |ξ |  a, which also gives a circle, though now the circle is laterally displaced from the undisturbed position. In contrast with the sausage (m = 0) mode, in the kink (m = 1) mode the centre of the tube moves in response to the perturbation. Finally, consider the fluting modes of the tube, for which m ≥ 2. For m = 2 the boundary of the tube is elliptical; for m = 3 it is a smoothed triangular shape, like the three petals of a flower. For higher m, the tube boundary becomes increasingly more crinkled, almost castellated, much like a column in certain architectural styles. In these fluting modes the centre of the tube remains undisturbed, just as with the sausage (m = 0) mode. Consequently, only the kink (m = 1) mode displaces the centre of the tube. 6.2.5 Ordinary Differential Equations We return now to our investigation of the properties of the wave equations (6.11)–(6.14). We adopt the Fourier form (6.17) for ur , with similar forms for uφ , uz and pT . Then equation (6.11) yields the first order ordinary differential equation

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Magnetic Flux Tubes

(a) m = 0

(b) m = 1

(c) m = 2

(d) m = 10

Figure 6.2 The geometry of the various modes of oscillation of a tube, indicating the location of a cross-section during the motion for modes m = 0, 1, 2 and 10. The sketches are polar plots of the equation r/a = 1 + ξ cos mφ, for small ξ ; specifically, for ease of visibility we have set ξ = 0.1. (a) The sausage mode (m = 0) leaves the centre of the circle r = a undisturbed during the motion, but the circumference ‘breathes’ in and out. (b) In the kink mode (m = 1) the circle r = a is bodily (globally) displaced, its centre moving from r = 0; the motion of the central axis resembles that of a snake. In the fluting modes (m ≥ 2), the centre (r = 0) of the circle r = a remains undisturbed but its boundary is distorted, in simple shapes for moderate m but becoming almost serrated or castellated, though smoothly, for larger m; the cases (c) m = 2 and (d) m = 10 are illustrated.

iω

dpT = −ρ0 (r)(kz2 c2A (r) − ω2 )ur , dr

(6.20)

relating ur and pT . Equations (6.12) and (6.13) give algebraic relations that relate uφ and uz to pT : ρ0 (r)(kz2 c2A (r) − ω2 )uφ = −ω

m pT r

ρ0 (r)(kz2 c2t (r) − ω2 )uz = −ωkz



c2s (r) pT . c2s (r) + c2A (r)

(6.21) (6.22)

6.2 Wave Equations

147

Finally, equation (6.14) relates ur , uφ , uz and pT :   dur m 1 1 c2s (r) pT . + ur − i uφ − ikz uz = −iω 2 2 2 dr r r cs (r) + cA (r) ρ0 (r)(cs (r) + c2A (r))

(6.23)

From these equations we may eliminate uφ and uz in favour of ur and pT , giving

  1 d − ρ0 (r) c2s (r) + c2A (r) kz2 c2A (r) − ω2 kz2 c2t (r) − ω2 (rur ) r dr       m2 2 m2 2 4 2 2 2 2 2 2 cs (r) + cA (r) + kz kz + 2 cs (r)cA (r) iωpT . = ω − ω kz + 2 r r

(6.24)

Thus, we have obtained a pair of first order ordinary differential equations, (6.20) and (6.24), that relate ur and pT . In fact, equation (6.24) may be simplified a little and our pair of coupled differential equations conveniently gathered together as: dpT = −ρ0 (r)(kz2 c2A (r) − ω2 )ur , dr   m2 2 2 2 1 d 2 (rur ) = − m (r) + 2 iωpT , ρ0 (kz cA (r) − ω ) r dr r iω

(6.25) (6.26)

where m2 (r) is defined by m2 (r) =

(kz2 c2s (r) − ω2 )(kz2 c2A (r) − ω2 ) (c2s (r) + c2A (r))(kz2 c2t (r) − ω2 )

.

(6.27)

The nature of m2 (r) proves important in determining the physical nature of the modes that the system supports. The differential equations (6.25) and (6.26) are the principal equations governing the behaviour of waves in our system; the behaviour of any of the other quantities, such as the flow component uz or the density perturbation ρ, may be determined once these equations are solved. We may eliminate one or other variable between the pair of first order differential equations (6.25) and (6.26) to produce a second order ordinary differential equation. Eliminating pT between these equations yields   d ρ0 (r)(kz2 c2A (r) − ω2 ) 1 d (6.28) (rur ) = ρ0 (r)(kz2 c2A (r) − ω2 )ur ,  2 dr r dr m2 (r) + m2 r

whereas eliminating ur in favour of the pressure perturbation pT yields     dpT m2 1 2 2 2 1 d 2 r = m (r) + 2 pT . ρ0 (r)(kz cA (r) − ω ) r dr ρ0 (r)(kz2 c2A (r) − ω2 ) dr r

(6.29)

We should note the close similarity between the cylindrical system and the Cartesian one. For example, we may compare directly equation (6.25) with equation (3.36) of Chapter 3, and equation (6.22) with (3.38). Comparing equations (6.21) and (3.36), we note that the role of m/r in a cylindrical system is played by ky in a Cartesian system.

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There are slight differences between equations (6.23) and (3.40), reflecting the geometrical differences of the two systems; for example, in a cylindrical geometry combinations involving rur tend to arise. Notice also that the important expression (6.27) for m2 (r) is precisely of the form that arises for a slab (cf. equation (5.59) in Chapter 5). Equation (6.28) is sometimes referred to as the Hain–L¨ust equation (see Chapter 3).

6.3 The Magnetic Flux Tube We are interested in solutions of equations (6.28) and (6.29) that are appropriate for a magnetic flux tube. Consider then a uniform magnetic field B0 ez confined to a tube of radius r = a, the tube being embedded in a magnetic environment with field strength Be (assumed uniform):  Be , r > a, (6.30) B0 (r) = B0 , r < a. The equilibrium plasma pressure p0 (r), density ρ0 (r) and temperature T0 (r) are structured similarly:  pe , ρe , Te , r > a, p0 (r), ρ0 (r), T0 (r) = (6.31) p0 , ρ0 , T0 , r < a, for constants p0 , pe , ρ0 , ρe , T0 and Te . Figure 6.3 illustrates the equilibrium.

Figure 6.3 The equilibrium configuration of a magnetic tube embedded in a magnetic environment. Within the tube (r < a), the field strength is B0 , the plasma density is ρ0 and the plasma pressure is p0 ; in the environment (r > a), the field strength is Be , the plasma density is ρe and the pressure is pe . Pressure balance (equation (6.32)) is maintained across the boundary r = a. (From Edwin and Roberts 1983.)

6.3 The Magnetic Flux Tube

149

Quantities with a suffix ‘0’ refer to the region within the tube (r < a), and those with a suffix ‘e’ to the environment (r > a). However, it is convenient to denote the sound speed within the tube by cs (= (γ p0 /ρ0 )1/2 ), the Alfv´en speed within the tube by cA (= (B20 /μρ0 )1/2 ), and the slow speed by ct ; an index is retained where it is needed to distinguish quantities in the environment from those within the tube, or where it proves clearer to maintain a suffix for the interior. Now the interface r = a is a current sheet across which conditions change discontinuously (see Chapter 4), while preserving total pressure balance: p0 +

B20 B2 = pe + e . 2μ 2μ

(6.32)

Combined with the ideal gas law, p0 (r) = kB ρ0 (r)T0 (r)/m, ˆ pressure balance implies a connection between the plasma densities, sound speeds and Alfv´en speeds inside and outside the magnetic flux tube: c2 + 1 γ c2 ρe = s 12 2A . ρ0 c2se + 2 γ cAe

(6.33)

Here cse (= (γ pe /ρe )1/2 ) and cAe (= (B2e /μρe )1/2 ) denote the sound and Alfv´en speeds in the environment ( r > a) of the tube. The fact that the media inside and outside the flux tube are uniform affords us considerable simplification in the governing differential equations (6.28) and (6.29). The equation for the pressure perturbation is particularly convenient to consider. For the uniform medium inside the tube (where the density, sound speed and Alfv´en speed are all constants, and m2 becomes the constant m20 ), equation (6.29) yields r2

d2 pT dpT +r − (m20 r2 + m2 )pT = 0, 2 dr dr

(6.34)

where m20 =

(kz2 c2s − ω2 )(kz2 c2A − ω2 ) kz2 c2s c2A − ω2 (c2s + c2A )

=

(kz2 c2s − ω2 )(kz2 c2A − ω2 ) (c2s + c2A )(kz2 c2t − ω2 )

.

(6.35)

We have cancelled a factor (kz2 c2A − ω2 ) corresponding to the possibility of Alfv´en waves, which can be considered separately from the magnetoacoustic modes. Equation (6.34) is a form of Bessel’s differential equation, with solutions the modified Bessel functions Im (m0 r) and Km (m0 r). (This is easy to see if we first write R = m0 r, so that the operator d/dr becomes m0 d/dR and the operator d2 /dr2 becomes m20 d2 /dR2 .) The spatial nature of this solution depends upon the nature of m0 , whether m0 > 0 or otherwise. However, at this stage we may leave open the nature of m0 ; equation (6.34) applies whatever the nature of m0 , be it positive (if m20 > 0) or purely imaginary (if m20 < 0), or indeed complex. However, the solution involving the Bessel function Km should be rejected since it is singular at r = 0 and we require solutions that are bounded at the centre of the tube. Accordingly, we take pT = A0 Im (m0 r), where A0 is an arbitrary constant.

r < a,

(6.36)

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Magnetic Flux Tubes

In the environment of the flux tube (where the sound speed is cse , the Alfv´en speed is cAe , the slow speed is cte and the medium is uniform) equation (6.34) again applies, but now with m20 replaced by m2e , the value of m2 (r) in the environment. Thus, solutions of the form pT ∝ Im (me r) and Km (me r) arise. For me > 0 the solution Im (me r) grows exponentially fast in r, whereas Km (me r) decays exponentially. Thus Im (me r) produces pressure variations that are unbounded at infinity and is to be rejected. Accordingly, requiring that me > 0, we take pT (r) = Ae Km (me r),

r > a,

(6.37)

where m2e =

(kz2 c2se − ω2 )(kz2 c2Ae − ω2 ) kz2 c2se c2Ae − ω2 (c2se + c2Ae )

=

(kz2 c2se − ω2 )(kz2 c2Ae − ω2 ) (c2se + c2Ae )(kz2 c2te − ω2 )

(6.38)

and Ae is an arbitrary constant. The requirement me > 0 of a declining pressure field outside the tube (pT → 0 as r → ∞) places constraints on the range of wave speed ω/kz : from expressions (6.38) it is apparent that either ω2 /kz2 < c2te or ω2 /kz2 lies between c2se and c2Ae . It remains to match the two solutions (6.36) and (6.37) across the boundary of the tube. As discussed in earlier chapters, the appropriate boundary conditions are that the radial velocity component ur and the pressure perturbation pT must both be continuous across r = a. We may also view these conditions from equation (6.29) determining pT in an arbitrarily structured plasma. For it is evident from that equation that pT must be continuous across any interface separating differing values of ρ0 (r), cA (r) or m2 (r); for otherwise the second derivative of pT could not be balanced by the other terms that occur in (6.29). In a similar fashion we may argue from equation (6.28) that ur must be continuous. Taken together with equation (6.25), which relates ur to the derivative of pT , we thus conclude that pT

and

iω ρ0 (r)(kz2 c2A (r) − ω2 )

dpT dr

(6.39)

must be continuous across r = a. Continuity of pT (r) immediately yields A0 Im (m0 a) = Ae Km (me a).

(6.40)

Matching the derivative of pT according to the second condition in (6.39), and multiplying by a factor of a, results in 1 1 A0 m0 aIm  (m0 a) = Ae me aKm  (me a), ρ0 (kz2 c2A − ω2 ) ρe (kz2 c2Ae − ω2 ) where the dash ( ) denotes the derivative of the Bessel functions concerned: e.g., Im  (x) ≡

dIm (x) , dx

Im  (m0 a) ≡

dIm (x) dx

evaluated at x = m0 a.

(6.41)

6.3 The Magnetic Flux Tube

151

Finally, we may eliminate the constants A0 and Ae (assumed non-zero) between equations (6.40) and (6.41). The result is 1 ρ0 (kz2 c2A − ω2 )

m0 a

Im  (m0 a) Km  (me a) 1 m a = . e Im (m0 a) Km (me a) ρe (kz2 c2Ae − ω2 )

(6.42)

Equation (6.42) is the general dispersion relation governing magnetoacoustic waves in a uniform magnetic flux tube embedded in a uniform magnetic environment. Several authors have examined special cases or aspects of this dispersion relation (e.g. McKenzie 1970; Roberts and Webb 1978, 1979; Wilson 1980), with Edwin and Roberts (1983) giving the most detailed discussion. The leaky form of this dispersion relation, when Hankel functions replace modified Bessel functions in the tube environment, was first presented by Zaitsev and Stepanov (1975); see also Meerson, Sasorov and Stepanov (1978), Spruit (1982) and Cally (1986). Leaky waves are discussed separately in Section 6.6. The dispersion relation (6.42) is subject to the requirement that me > 0, ensuring that disturbances decay far from the tube (as r → ∞). However, the dispersion relation is valid whatever the nature of m0 . The restriction me > 0 imposed on the flux tube dispersion relation (6.42) means that the amplitude of a wave declines with radius r (> a), so that far from the tube there is no appreciable disturbance: waves are essentially confined to the tube. But inside the tube (for r < a) the disturbance may be oscillatory or nonoscillatory (exponential). Modes that are oscillatory inside the tube are the body waves (m20 < 0); modes that are non-oscillatory (exponential) inside the tube are surface waves (m20 > 0). Since me > 0, both body and surface waves are confined to the tube, disturbing the surroundings by an exponential tail which extends out into the environment a distance of order m−1 e . The dispersion relation (6.42) applies for both surface and body modes. The case of body waves (for which m0 is purely imaginary, since m20 < 0) may be dealt with from (6.42) by noting that the Bessel functions Im and Im  now have imaginary arguments. In fact, we may relate the modified Bessel functions Im to the Bessel functions Jm through the relation Im (iz) = (−i)m Jm (z), and then the dispersion relation (6.42) transforms into a relation involving Jm and Km and their derivatives. But it is perhaps clearer to consider body waves afresh. We may do this by noting that when m20 < 0 we may write n20 = −m20 , and then the solution (6.36) within the tube may be written in the form pT = A0 Jm (n0 r),

r < a,

(6.43)

for appropriate arbitrary constant A0 . Then the dispersion relation governing body waves in a magnetic flux tube is 1 ρ0 (kz2 c2A − ω2 )

n0 a

Jm  (n0 a) Km  (me a) 1 m a = , e Jm (n0 a) Km (me a) ρe (kz2 c2Ae − ω2 )

(6.44)

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with n20 =

(ω2 − kz2 c2A )(ω2 − kz2 c2s ) (c2s + c2A )(ω2 − kz2 c2t )

.

(6.45)

Equation (6.44) is subject to the constraint that me > 0. The dispersion relation (6.44), or its alternative form (6.42), is transcendental, determining in a complicated way the relationship of ω to kz (other parameters being fixed). The nature of the dispersion relation is not transparent and it requires a combination of analytical and numerical investigations to draw out the behaviour of ω as a function of wavenumber kz . In this it is convenient to determine the wave (or phase) speed c = ω/kz of the various waves; ω then follows as a function of kz from the relation ω = kz c once the speed c(kz ) is known as a function of kz . There are a number of general conclusions to be drawn from the dispersion relations (6.42) or (6.44). The most important general point is that the waves of a flux tube are dispersive, the speed c of the waves along the tube varying with the dimensionless wavenumber kz a. Short waves (for which kz a  1, producing a wavelength λ (= 2π/kz ) that is much shorter than 2π a) propagate at speeds that differ from the speeds of long waves (for which kz a  1, producing a wavelength λ that is much longer than 2π a). We can draw another general conclusion by rewriting our dispersion relation (6.42) in a form first encountered in the description of surface waves on a single magnetic interface (see Chapter 4). Writing   ρe m0 a Im  (m0 a) Km (me a) Rm = − , (6.46) ρ0 me a Im (m0 a) Km  (me a) then we may rearrange the dispersion relation (6.42) into the forms c2 = c2A +

Rm 1 (c2 − c2A ) = c2Ae − (c2 − c2A ). 1 + Rm Ae 1 + Rm Ae

(6.47)

This rearrangement of the basic dispersion relation disguises its transcendental nature, now buried in the definition of Rm . Nonetheless, treated with appropriate care, this rewrite can prove useful. Another form of the basic dispersion relation is also of interest. We can introduce a form that displays the kink speed ck , buried in the above expressions. By an appropriate rearrangement of either of the forms (6.47), we obtain  ρe  − R m ρ c2 = c2k + (c2k − c2Ae ) 0 . (6.48) 1 + Rm We use these various forms of the dispersion relation to draw out information about the speeds of wave propagation in a magnetic flux tube. In fact, the similarity of the form (6.47) and the dispersion relation for waves on a single magnetic surface (Chapter 4) suggests we can examine more generally when surface waves may arise. To consider the possibility of surface waves occurring, suppose that m0 is positive (so that both m0 and me are positive). Equation (6.47) is precisely the form of the dispersion relation describing surface waves on a single magnetic interface. Now for surface waves on a single interface, the dispersion

6.4 Small kz a Expansions

153

relations (6.47) apply with Rm > 0, and such waves have the property that c2 lies between c2A and c2Ae . It follows immediately that provided Rm > 0, with Rm defined by (6.46), then the surface waves of a magnetic tube have longitudinal phase speed c that lies between the Alfv´en speed inside the tube and the Alfv´en speed of the tube’s environment. Inspection of expression (6.46) shows that the requirement Rm > 0 is met for modes with m0 > 0 and me > 0, since then the modified Bessel functions are such that Im (z), Im  (z) and Km (z) are each positive whenever z > 0, whereas Km  (z) < 0 (since Km (z) is an exponentially decreasing function of z, whereas Im (z) is an exponentially increasing function of z). Thus Rm > 0 whenever m0 , me > 0, producing surface waves on the boundary of the tube. Hence, any surface waves of a magnetic flux tube propagate with a phase speed c that lies between cA and cAe . However, it may be that there are no modes with m0 > 0 and me > 0, in which case this argument fails (for then Rm is no longer of one sign but is oscillatory). In this case, body waves may arise but there are no surface waves. We now turn to an examination of the dispersion relation in the extremes of small or large wavenumber kz , measured accordingly as to whether kz a  1 or kz a  1. 6.4 Small kz a Expansions 6.4.1 General Aspects Consider the dispersion relation (6.42) (or its equivalent forms (6.44), (6.47) and (6.48)) in the limit of a thin tube, corresponding to waves being much longer than the radius of the tube; specifically, we consider the case of small kz a. Interest then centres on the behaviour of terms such as m0 a or me a. Consider the expression m20 a2 = (kz a)2

(c2s − c2 )(c2A − c2 ) (c2s + c2A )(c2t − c2 )

.

(6.49)

As kz a → 0 we may expect that m0 a → 0; this must happen provided c2 remains finite and does not tend to c2t . The speed of the wave is limited by the requirement that me > 0, so we can rule out the possibility c2 → ∞. However, the possibility c2 → c2t as kz a → 0 is in fact realized, and corresponds to slow waves. In this case, it may still happen that m0 a → 0 but now we have to examine carefully the manner in which c2 → c2t ; similar comments may be made about me a. We treat the case of slow waves separately below, considering first the more straightforward case. Suppose then that m0 a → 0,

me a → 0

as

kz a → 0.

To see the implications of this we need to know the behaviour of the Bessel functions that arise in the dispersion relation (6.42) (or (6.44)). From mathematical handbooks (see, for example, Abramowitz and Stegun 1965) we note the relations for the derivatives of the Bessel functions Jm , Im and Km : m m Im  (z) = Im+1 (z) + Im (z), Jm  (z) = −Jm+1 (z) + Jm (z), z z m  (6.50) Km (z) = −Km+1 (z) + Km (z). z

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Magnetic Flux Tubes

We also need the behaviour of the Bessel functions as z → 0: Jm (z) ∼

( 12 z)m , m!

Im (z) ∼

( 12 z)m , m!

m ≥ 0;

Km (z) ∼

(m − 1)! 2( 12 z)m

,

m ≥ 1.

(6.51)

The above relations may all be used for m = 1, 2, 3, . . ., but for the case m = 0 we need to proceed differently. We have J0 (z) ∼ 1 and I0 (z) ∼ 1 as z → 0, and for the function K0 (z) we may use its asymptotic behaviour as z → 0, namely   1 (6.52) z − γE , K0 (z) ∼ − ln 2 where ln denotes the natural logarithm and γE (≈ 0.5772) is Euler’s constant. However, in what follows it is convenient to retain K0 (z) explicitly, replacing it by its asymptotic behaviour only if this proves useful. The above relations may be combined to determine the behaviour of the terms zIm  (z)/Im (z) and zKm  (z)/Km (z), forms of which arise in the dispersion relation with the z being chosen appropriately. Specifically, as z → 0, z

Jm  (z) z2 ∼m− , Jm (z) 2(m + 1)

z

Im  (z) z2 ∼m+ , Im (z) 2(m + 1)

m ≥ 0.

(6.53)

For expressions involving the Bessel functions Km it is useful to discuss the cases m = 0, m = 1 and m ≥ 2 separately. As z → 0, we have ⎧ z2 ⎪ −m − 2(m−1) , m ≥ 2, ⎪ ⎨  Km (z) (6.54) z ∼ −1 − z2 K0 (z), m = 1, ⎪ Km (z) ⎪ 1 ⎩− , m = 0. K0 (z)

The difference in the behaviour of relations involving Km (z) and those involving Jm (z) or Im (z) arises because the modified Bessel function K0 (z) behaves differently from the modified Bessel functions K1 (z), K2 (z), etc., in that as z → 0 these functions all tend to infinity through inverse powers of z, but K0 (z) tends to infinity logarithmically. We can now use these relations to determine the behaviour of the magnetic tube dispersion relation when kz a is small. It is convenient to discuss separately the cases m = 0 and m ≥ 1, corresponding to sausage modes or kink and fluting modes.

6.4.2 Sausage (m = 0) Modes Consider the case m = 0 which corresponds to sausage modes. Noting the small me a behaviour indicated in relation (6.54), the dispersion relation (6.44) gives    2 cA − c2 J1 (n0 a) ρ0 1 n0 a ∼ . (6.55) 2 2 J0 (n0 a) ρe K (m cAe − c 0 e a) Notice the right-hand side of this relation tends to zero as me a → 0.

6.4 Small kz a Expansions

155

There are two possibilities to consider here. Suppose first that as kz a → 0, we have n0 a → 0. Then our dispersion relation gives   ρ0 c2A − c2 1 2 2 1 n a = , 2 0 ρe c2Ae − c2 K0 (me a) valid for |kz a|  1. Thus we have (c2 − c2 )(c2A − c2 ) 1 ∼ (kz a)2 s 2 (c2s + c2A )(c2 − c2t )



ρ0 ρe



c2A − c2 c2Ae



− c2

1 . K0 (me a)

Hence, c2 ∼ c2t (since kz2 a2 K0 (me a) → 0 as kz a → 0) and we have c ∼ 2

c2t

1 + 2



ρe ρ0



 (c2Ae

− c2t )

c2s c2s + c2A

2 (kz a)2 K0 (λt |kz |a),

kz2 a2  1,

(6.56)

where λ2t =

(c2se − c2t )(c2Ae − c2t ) (c2se + c2Ae )(c2te − c2t )

.

When cAe > ct (as in the corona), we have c2 > c2t and the mode described by relation (6.56) is a slow body wave. However, if cAe < ct , typical of the photosphere when the environment may be field-free (cAe = 0), then it is a slow surface wave. In the above we have written |kz | to allow for the possibility that kz may be negative, and it is assumed that λt > 0. A relation of the form (6.56) was given by Roberts and Webb (1978) for the case cAe = 0. We should note that at first sight the result that c2 → c2t as kz a → 0 raises a concern, since it implies that n20 /kz2 becomes unbounded, whereas we have assumed that n0 a → 0 as kz a → 0. In fact, there is no contradiction because substituting (6.56) into the expression for n20 a2 shows that it behaves like 1/(K0 (λt |kz |a)) as kz a → 0 and so tends to zero (since K0 (λt |kz |a) tends to infinity). There is another way that relation (6.55) may be satisfied. It may be that as kz a → 0, we have J1 (n0 a) → 0, and so n0 a → j1,s ,

(6.57)

where j1,s denotes the zeros of the Bessel function J1 . (Note that j1,1 = 3.8317, j1,2 = 7.0156, etc.) Rewritten as an expression for c2 , we have     1 c2t 2 2 2 (kz a) . (6.58) c = ct 1 + 2 j1,s c2s + c2A There are an infinite number of modes (s = 1, 2, 3, . . .); these are slow sausage waves, and c2 ≈ c2t to good approximation. For cA  cs we see that the correction to the basic speed is very small. A relation of the form (6.58) was noted in Roberts and Webb (1979), for cAe = 0, and in Edwin (1984).

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Magnetic Flux Tubes

6.4.3 The Kink (m = 1) Mode Consider the case m = 1, which corresponds to the kink mode. For kink (or fluting) modes, relation (6.48) is perhaps the most convenient form of the dispersion relation to work with, since it gives precedence to the speed ck which, it transpires, is central to any description of kink (or fluting) modes. To leading order, we see that as z → 0 (for m = 1), zIm  (z)/Im (z) ∼ m and

zKm  (z)/Km (z) ∼ −m.

(6.59)

Hence, applied to Rm we see that as kz a → 0, Rm ∼ ρe /ρ0 ,

m = 1.

(6.60)

Substitution of Rm = ρe /ρ0 into the dispersion relation (6.48) then gives c2 → c2k

as

kz a → 0.

(6.61)

Thus, provided me > 0, long wavelength (kz a  1) kink (m = 1) modes propagate with a speed close to ck . In fact, each of the statements (6.59)–(6.61) holds generally for m ≥ 1. Thus, not only the kink mode but also the fluting modes have the speed ck in the limit of small kz a. We can improve on the above result if we retain higher order terms in our expansions of the Bessel functions. In fact, using the expansions (6.53) and (6.54) noted earlier for zIm  (z)/Im (z) and zKm  (z)/Km (z), we obtain (for m = 1)     ρe ρe 1 ρe − R ∼ (6.62) (me a)2 K0 (me a). , Rm ∼ m ρ0 1 + (me a)2 K0 (me a) ρ0 ρ0 Thus, from equation (6.48) we obtain that the square of the speed c of the kink (m = 1) mode is given by   ρe c2 ∼ c2k − (6.63) (c2Ae − c2k )(me a)2 K0 (me a). ρ0 + ρe In all of these relations, the term me is calculated with c = ck , and so m2e a2 = (kz a)2

(c2se − c2k )(c2Ae − c2k ) (c2se + c2Ae )(c2te − c2k )

Also, as noted earlier, we require that me > 0. Finally, noting that   ρe c2A − c2k = − (c2Ae − c2A ), ρ0 + ρe

 c2Ae − c2k =

.

ρ0 ρ0 + ρe

 (c2Ae − c2A ),

we can rewrite the above result for c2 as follows: for the kink (m = 1) mode,     c2Ae − c2A ρ0 ρe 2 2 2 2 c ∼ ck 1 − λk (kz a) K0 (λk |kz |a) , (ρ0 + ρe )2 c2k where λ2k =

(c2se − c2k )(c2Ae − c2k ) (c2se + c2Ae )(c2te − c2k )

(6.64)

6.5 Large kz a Behaviour

157

and it is assumed that λk > 0. Relation (6.64) holds quite generally for the trapped (me > 0) kink mode in a thin tube, whether the medium is compressible or incompressible. Special cases simply select specific forms of the expression for λk . A relation of the form (6.64) was first obtained by Edwin and Roberts (1983). 6.4.4 Fluting (m ≥ 2) Modes Consider the fluting modes, given by setting m = 2, 3, 4, . . .. We have ⎡ ⎤ m20 a2   ρe ⎣ m + 2(m+1) ⎦ , m ≥ 2, Rm ∼ m2e a2 ρ0 m + 2(m−1) and so expanding binomially we obtain     m20 a2 ρe m2e a2 ρe 1 − Rm ∼ − − , ρ0 ρ0 2m m + 1 m − 1

m ≥ 2.

Thus, the dispersion relation (6.48) reads     m20 a2 ρe m2e a2 1 2 2 2 2 − , c ∼ ck − (ck − cAe ) ρ0 + ρe 2m m + 1 m − 1

(6.65)

(6.66)

(6.67)

for the fluting (m = 2, 3, 4, . . .) modes. In these expressions, the squared effective wavenumbers m20 and m2e are calculated with c2 = c2k . We can rewrite (6.67) in the form      m20 a2 c2Ae − c2A m2e a2 ρ0 ρe 1 2 2 c ∼ ck 1 + − , m ≥ 2. (6.68) 2m m + 1 m − 1 (ρ0 + ρe )2 c2k The expression (6.68) for c2 shows that the fluting modes propagate with the speed ck and have a dispersive correction that is of order (kz a)2 . The approximate dispersion relation (6.68) holds quite generally for fluting modes m ≥ 2 in a thin tube, whether the medium is compressible or incompressible. Its form simplifies a little in special cases simply through reductions in the expressions for m20 and m2e . Some special cases are treated in Section 6.7. 6.5 Large kz a Behaviour Complementary to the behaviour of the dispersion relation in the limit of small kz a is the behaviour for large |kz |a, corresponding to wavelengths much shorter than the tube cross-section. We need the behaviour of the Bessel functions at infinity. As z → +∞ (Abramowitz and Stegun 1965),  1/2  1/2 π π e−z , Km  (z) ∼ − e−z . (6.69) Km (z) ∼ 2z 2z Thus, as |kz |a → ∞,

 Rm ∼

ρe ρ0



m0 a Im  (m0 a) . me a Im (m0 a)

(6.70)

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Magnetic Flux Tubes

This requires that me a → +∞ as |kz |a → +∞, which must occur except possibly if c2 tends towards c2te . 6.5.1 Surface Waves The behaviour of Rm now depends upon whether m20 > 0 or m20 < 0. If m20 > 0, so we are discussing surface waves, then we may invoke the asymptotic behaviour that, as z → +∞, Im (z) ∼

ez , (2πz)1/2

Im  (z) ∼

ez . (2π z)1/2

(6.71)

Thus, excluding the possibility that c2 tends to c2t , we conclude that, as |kz |a → +∞,   ρe m0 a , (6.72) Rm ∼ ρ0 me a irrespective of the value of m. But relation (6.72) is precisely the expression that arises in the description of surface waves at a single magnetic interface (Chapter 4). Thus, provided m20 > 0, the asymptotic value that the wave speed c tends to when kz a is large is determined by the transcendental surface wave dispersion relation (4.15) for a single magnetic interface (see Chapter 4), and is independent of the value of m. Consequently, following Chapter 4, we can conclude that the asymptotic values of the surface wave speeds in a magnetic flux tube must lie between the Alfv´en speeds cA and cAe , be they sausage modes, kink modes or fluting modes. 6.5.2 Body Waves Body waves behave differently. Returning to equation (6.70), consider the possibility that m20 < 0. In this case it is more convenient to work in terms of n0 rather than m0 and to note that the form of the dispersion relation for body waves is precisely that for surface waves save that m0 is replaced by n0 and the modified Bessel functions Im (m0 a) involving m0 become Bessel functions Jm (n0 a) involving n0 . Thus,   ρe n0 a Jm  (n0 a) Rm ∼ . (6.73) ρ0 me a Jm (n0 a) Then, noting that, as |kz |a → +∞,  1/2   2 1 π Jm (z) ∼ cos z − mπ − πz 2 4

(6.74)

and similarly that

    Jm  (n0 a) m 1 π 1 π ∼ − tan z − mπ − ∼ − tan z − mπ − , Jm (n0 a) z 2 4 2 4

the dispersion relation reads       1 π ρ0 me c2 − c2A tan n0 a − mπ − ∼ 2 4 ρe n0 c2 − c2Ae with me > 0 and n0 > 0.

(6.75)

(6.76)

6.6 Leaky Modes

159

The form of (6.76), with the occurrence of the periodic tangent function, suggests immediately that we are dealing with an infinity of body modes. Also, (6.76) may be compared with the relations that arise for a magnetic slab (Chapter 5), where analogies with Pekeris (1948) and Love (1911) waves were pointed out; we see here then the cylindrical equivalents of magnetic Pekeris and magnetic Love waves.

6.6 Leaky Modes 6.6.1 The General Dispersion Relation There remains the possibility that the condition me > 0 may not be met, either because m2e < 0 or me is complex. In such a situation the vibrations within the flux tube act so as to generate waves outside the tube which propagate out to r → ∞. This topic arises in particular with the kink mode in an under-dense plasma in the special case of a β = 0 plasma. Waves of this form are referred to as leaky waves; the waves propagate within the flux tube but leak or radiate some of their energy to the outside medium. The study of leaky waves from magnetic flux tubes was in the main begun by Zaitsev and Stepanov (1975) and Spruit (1982), who were concerned with leaky waves both for coronal tubes (for which the plasma β is small or zero) and photospheric tubes (for which the environment of the tube is field-free). Further aspects were explored by Cally (1986). Early related studies of the leakage dispersion relation for a photospheric tube include Roberts and Webb (1979). More recently, Goossens and Hollweg (1993) and Goossens et al. (2009) have assessed the importance of wave leakage as compared with resonant absorption brought about by a spatially varying (radially) Alfv´en profile, concluding that resonant absorption is usually the dominant effect. This is discussed in Section 8.11 of Chapter 8. Although the treatment of leaky waves is in fact very similar to the treatment of trapped waves, it is convenient to give a self-contained discussion. We consider a magnetic flux tube in the absence of twist. We consider a plasma density of ρ0 inside a tube of radius a, surrounded by a uniform medium with plasma density ρe . As we have seen earlier, in a uniform medium and working in cylindrical coordinates the total pressure perturbation pT satisfies the differential equation (see equation (6.34); see also Chapter 3, Section 3.3) r2

d2 pT dpT +r + (n20 r2 − m2 )pT = 0, dr dr2

(6.77)

where n20 = −m20

(6.78)

with m20 being defined in equation (6.35) and n20 in equation (6.45). This is Bessel’s differential equation, with Bessel function solutions Jm (n0 r) and Ym (n0 r). Accordingly, inside the tube the total pressure perturbation is pT (r) = A0 Jm (n0 r) + A1 Ym (n0 r),

r < a.

(6.79)

160

Magnetic Flux Tubes

Since the tube has no twist, for convenience we may assume that m ≥ 0; there is no loss of generality in this assumption. The sausage mode corresponds to m = 0, the kink mode to m = 1, and fluting modes are for integer m ≥ 2. The constants A0 and A1 are arbitrary. Now the solution (6.79) is required to be bounded at the centre of the tube (r = 0), which allows us to select the Bessel function Jm (n0 r), applying in r < a, rejecting the solution Ym (n0 r) as being unbounded at r = 0.1 This is the case whatever the nature of the constant n0 (assumed non-zero), whether it is positive, negative or indeed complex. Accordingly, with n0 = 0 we take A1 = 0. Now in the environment of the tube, in r > a, we also have Bessel’s equation with Bessel function solutions, and now we need to select our solution depending upon which physical situation we wish to model. If we are interested in waves that are largely confined to within the tube, then we select the exponentially decaying Bessel function Km (me r), for an appropriate real and positive me . This was the choice made earlier. But if are interested in waves that may leak from the tube, propagating radially away from the tube, then we select the appropriate Hankel function solution of Bessel’s equation that represents a radially outgoing wave. Accordingly, the differential equation (6.77) for pT has a solution of the form  r > a, Ae Hm (ne r), pT (r) = (6.80) A0 Jm (n0 r), r < a, where n2e = −m2e

(6.81)

with m2e being defined in equation (6.38). Here Hm denotes a Hankel function (see Abramowitz and Stegun 1965). Hankel functions play the role of a complex exponential (just as the ordinary Bessel functions are akin to cosine and sine functions, and modified Bessel functions are akin to exponential (growth or decay) functions). There are two (1) (2) Hankel functions, Hm = Hm (ne r) and Hm = Hm (ne r), both solutions of (6.77); their asymptotic behaviour as |ne r| → ∞ is  1/2 1 π 2 (1) ei(ne r− 2 mπ − 4 ) , Hm (ne r) ∼ πne r  1/2 1 π 2 (2) Hm (ne r) ∼ e−i(ne r− 2 mπ − 4 ) . (6.82) πne r Now, for an assumed time dependence of the form2 eiωt = eiωR t · e−ωI t with ω = ωR + iωI for real and imaginary parts ωR and ωI , we obtain a wave that is (2) outwardly propagating (in r) if we choose the function Hm (ne r) together with ωR > 0 and real (ne ) > 0. 1 We could of course also have chosen the modified Bessel function I (m r); either choice is permissible. m 0 2 The alternative choice e−iωt is made in Chapter 8 and in Goossens and Hollweg (1993), for example. For this choice we (1) would need to choose the Hankel function Hm for an outwardly propagating wave.

6.6 Leaky Modes

161

Now pT is continuous across r = a, so (2) A0 Jm (n0 a) = Ae Hm (ne a).

(6.83)

Additionally, we require that ur is continuous across r = a. Now, from equation (6.20) we have ⎧  iω ⎨ n A H (2) (ne r), r > a, ρe (ω2 −kz2 c2Ae ) e e m ur (r) = (6.84) iω  ⎩ r < a, 2 2 2 n0 A0 Jm (n0 r), ρ0 (ω −kz cA )

and so continuity of ur across r = a gives 1 1 (2)  n0 A0 Jm  (n0 a) = ne Ae Hm (ne a). ρ0 (ω2 − kz2 c2A ) ρe (ω2 − kz2 c2Ae )

(6.85)

Finally, elimination of the constants A0 and Ae yields the relation (2) 

1 Jm  (n0 a) Hm (ne a) 1 n ne a (2) a = . 0 2 2 2 2 2 2 Jm (n0 a) ρ0 (ω − kz cA ) ρe (ω − kz cAe ) Hm (ne a)

(6.86)

Equation (6.86) is the dispersion relation3 for leaky waves in a magnetic flux tube. Versions of this equation were first given by Zaitsev and Stepanov (1975) and Meerson, Sasorov and Stepanov (1978). An equation of this form, though with cAe = 0, was presented by Roberts and Webb (1979), but they did not consider leaky waves much further. A detailed discussion was first given by Spruit (1982) and Cally (1986), who considered both coronal and photospheric tubes. Recent applications, especially to the axisymmetric (m = 0) modes, have been given by, for example, Kopylova et al. (2007), Nakariakov, Hornsey and Melnikov (2012) and Pascoe, Nakariakov and Kupriyanova (2013). It is of interest to note that the leaky dispersion relation (6.86) becomes the trapped wave dispersion relation (6.44) if we simply replace me by ine a (see, for example, Kopylova et al. 2007). To see this, note first the relations (Abramowitz and Stegun 1965, chap. 9) m m (2) (2) (2)  Hm (x) = Hm (x) − H1+m (x), Km  (x) = Km (x) − K1+m (x), x x 1 1 1 (2) (xe− 2 πi ). (6.87) Km (x) = − πi e− 2 mπ i Hm 2 Then, setting me = ine (so m2e = −n2e ) it readily follows that (2) 

Km  (me a) Hm (ne a) = ne a (2) . me a Km (me a) Hm (ne a)

(6.88)

Hence the close correspondence between the leaky wave dispersion relation and the trapped wave dispersion relation.

3 The choice of a time dependence of the form e−iωt (as opposed to the form e+iωt adopted here) leads to a dispersion relation (2) (2)  (1) of precisely the same form, except that the Hankel function Hm and its derivative Hm is replaced by Hm and its derivative (1)  Hm .

162

Magnetic Flux Tubes

6.6.2 The Thin Tube Limit: m ≥ 1 As usual, we can examine the case of the dispersion relation (6.86) when the tube is thin (corresponding to assuming kz a  1). Then we can simplify the ratios of Bessel functions. As kz a → 0, the behaviour of the Jm Bessel functions is given in (6.51). For the Hankel function we have (2) 

Hm (x) (2)

Hm (x)

(2)

=

m Hm+1 (x) − (2) . x Hm (x)

(6.89)

(2)

Noting that Hm (x) may be expressed in terms of the Bessel functions Jm (x) and Ym x) (2) as (see Abramowitz and Stegun 1965) Hm (x) = Jm (x) − iYm (x), we can expand the Bessel functions Jm (x) and Ym (x) in series in x. To leading order in x considered small in magnitude (so |x|  1), we obtain     (2)  (x) 1 1 2 m m Hm 1 + 2π i x , m ≥ 1, (6.90) ∼− (2) x 4 m! (m − 1)! Hm (x) with terms neglected being of order x ln |x| or x2 . We retain the leading terms in the expansion of the real and imaginary parts of the expansion. Hence, in a thin tube the dispersion relation for leaky waves with m ≥ 1 becomes 1 1 + ρ0 (ω2 − kz2 c2A ) ρe (ω2 − kz2 c2Ae ) =

1 2 2 1 1 1 n0 a − 2πi 2 2 2 2 m(m + 1) ρ0 (ω − kz cA ) 2 ρe (ω − kz2 c2Ae )



1 2 2 n a 4 e

m

1 . m! (m − 1)!

In fact, we can go a little further and now neglect the term involving n20 a2 , which simply provides a correction to the real part of the frequency ω. Then   1 1 1 1 1 2 2 m + = −2πi ne a . 2 2 2 2 2 2 2 2 2 m! (m − 1)! ρ0 (ω − kz cA ) ρe (ω − kz cAe ) ρe (ω − kz cAe ) 4 (6.91) This is the dispersion relation for the leaky kink (m = 1) and fluting (m = 2, 3, 4, . . .) modes in a thin flux tube. Writing ω = ωR + iωI and assuming that ωI is small compared with ωR , we obtain to leading order in |kz a|  1 that    c2Ae − c2A 1 ρ0 ρe 1 2 2 m 2 2 2 ω = kz ck , ωI = −π a n ωk , (6.92) e 2 2 4 m! (m − 1)! (ρ0 + ρe ) ck where ωk (= kz ck ) is the kink mode frequency. The wavenumber n2e is calculated at ω2 = kz2 c2k . An equivalent expression for the leakage rate ωI in (6.92) was given by Spruit (1982; see his eqn. (46)), Goossens and Hollweg (1993; see their eqn. (72)) and Goossens et al. (2009; see their eqn. (65)).4 This relation shows that ωI /ωk is of order (kz a)2 for the kink 4 Goossens and Hollweg (1993) and Goossens et al. (2009) assume a time dependence of e−iωt , which leads to choosing the (1) (2) Hankel function Hm in place of Hm ; this also means that in place of ωI they have −ωI . Spruit (1982) assumes a time dependence of the form e+iωt , the same as here.

6.6 Leaky Modes

163

(m = 1) mode, and of higher order still (namely, (kz a)2m ) for the fluting modes. Thus, leakage is generally a small effect.

6.6.3 Leaky Kink (m = 1) Waves in a β = 0 Plasma So far we have not specified in detail the form of n2e (= −m2e ), save that it is defined through (6.38), so we can take the reduction a little further if we consider a specific case. For example, for the kink (m = 1) mode in a β = 0 plasma, the equilibrium requires that ρ0 c2A = ρe c2Ae , and then (6.92) simplifies to give π ωI = 8



ρ0 − ρe ρ0 + ρe

2 (kz a)2 ωk ,

ρ0 < ρe .

(6.93)

In a β = 0 plasma, wave leakage of the kink mode (and also the fluting modes) arises whenever ρ0 < ρe ; otherwise, trapped waves occur. This result for ωI , indicating a decay as energy in the kink mode leaks radially out to infinity when ρ0 < ρe , shows that leakage is small (an order (kz a)2 effect) (Spruit 1982; Cally 1986; Goossens et al. 2009). In fact, the effect is readily dominated by resonant absorption, whereby the decay of kink motions into transverse Alfv´enic motions is brought about through mode coupling in a thin transition layer within the flux tube boundary (Goossens and Hollweg 1993; Goossens et al. 2009). This topic is discussed further in Chapter 8.

6.6.4 Leaky Sausage (m = 0) Waves: The Dense (ρ0  ρe ) Tube Limit in a β = 0 Plasma The analysis presented earlier for the kink and fluting modes is based upon (6.90) and so does not apply when m = 0. Consider instead the general dispersion relation (6.86) for (2) (2) the case m = 0. Noting that J0  = −J1 and (H0 ) = −H1 , the general leaky mode dispersion relation (6.86), for m = 0, may be written in the form (2)

ρ0 (ω2 − kz2 c2A )

1 J0 (n0 a) 1 H0 (ne a) = ρe (ω2 − kz2 c2Ae ) . n0 a J1 (n0 a) ne a H (2) (ne a)

(6.94)

1

This is the dispersion relation for leaky sausage modes. For complex ω, in general its solutions must be investigated numerically. However, for one case it is amenable to an analytical investigation. The case that can be treated analytically corresponds to a dense (ρ0  ρe ) flux tube. This case was first considered in the context of a photospheric tube (see Zaitsev and Stepanov 1975), but in fact this has limited application since generally in the photosphere we can expect that ρ0 ≈ ρe . However, it is of relevance to the corona. Notice first that as ρe /ρ0 → 0 we expect from (6.94) that J0 (n0 a) → 0, suggesting that n0 a → j0,s as ρe /ρ0 → 0. Here j0,s denotes the zeros of the Bessel function J0 (so J0 (j0,s ) = 0 for integer s = 1, 2, 3, . . .);

164

Magnetic Flux Tubes

in particular, the first zero (see Abramowitz and Stegun 1965) is j0,1 = 2.4048, the second zero is j0,2 = 5.5201, etc. Accordingly, we look for a solution of the form n0 a = j0,s + z1 ,

ρe /ρ0  1,

(6.95)

for complex z1 where |z1 |  j0,s . Then, expanding by Taylor series we have J0 (n0 a) ≈ J0 (j0,s ) + (n0 a − j0,s )J0  (j0,s ) ≈ −z1 J1 (j0,s ), since J0 (j0,s ) = 0 and J0  = −J1 . Substituting the above expansion into the dispersion relation (6.94) we obtain    2 (2) ω − kz2 c2Ae ρe 1 H0 (ne a) z1 = −j0,s , ρ0 ω2 − kz2 c2A ne a H1(2) (ne a)

(6.96)

where the right-hand side has to be calculated when n0 a = j0,s . Suppose that |ne a|  1, an assumption to be checked a posteriori. Then we can replace the Hankel functions by their leading behaviour for small |ne a|, taking   (2) H0 (x) 1 + π i , |x| → 0, (6.97) ∼ −x log(x/2) + γ E (2) 2 H (x) 1

with γE denoting Euler’s constant. This gives the leading behaviour for complex x of a particular ratio of Hankel functions when |x| → 0. Accordingly, with x = ne a we obtain        2 ω − kz2 c2Ae 1 ρe 1 n0 a = j0,s 1 + ne a + γE + π i , (6.98) log ρ0 2 2 ω2 − kz2 c2A with terms involving ω and ne here to be calculated to the lowest order in ρe /ρ0 (when n0 a = j0,s ). Now the main case of interest here is the corona, so consider a β = 0 plasma with equilibrium ρ0 c2A = ρe c2Ae . Then n20 =

ω2 − kz2 c2A

Thus, ω2 = c2A (n20 + kz2 ) and n2e a2

=

c2A

n20 a2 c2Ae

 +

,

c2A

c2A c2Ae

n2e =

 −1

kz2 a2

=

ω2 − kz2 c2Ae c2Ae 

c2A

j20,s c2Ae

+

.

c2A c2Ae

(6.99)

 − 1 kz2 a2 .

Hence, for |kz a|  1 (including kz = 0), we may take n0 a = ωa/cA , and then

ne a =

cA j0,s , cAe

   1/2  1 1 ρe 1 log ne a = log j0,s ≈ ln(ρe /ρ0 ), 2 2 ρ0 2 

6.6 Leaky Modes

165

to leading order in ρe /ρ0 . Thus, to leading order in the small parameter ρe /ρ0 ,      cA π ρe 1 z1 = j0,s j0,1 ln + γE + i ρ0 2 cAe 2 and so (for β = 0, |kz a|  1)        ωa 1 ρe ρe π ρe = j0,s 1 + ln +i , cA 2 ρ0 ρ0 2 ρ0

ρe  ρ0 .

(6.100)

Considering specifically the first mode (for which s = 1 and j0,s = j0,1 ) and taking ω = ωR + iωI , for the real part ωR and imaginary part ωI , we have        π ρe cA 1 ρe ρe cA , ωI = j0,1 , ρe  ρ0 , ln ωR = j0,1 1 + 2 ρ0 ρ0 a 2 ρ0 a (6.101) and then   ωI π ρe ∼ (6.102) , ρe  ρ0 . ωR 2 ρ0 Results of the form (6.101) and (6.102) were first given in a brief analysis by Zaitsev and Stepanov (1975), though there appears to be a minor error in their analysis.5 Meerson, Sasorov and Stepanov (1978) give the correct result for a low β plasma, appropriate for the corona, in agreement with equations (6.101) and (6.102); see their eqn. (3.5) with the parallel wavenumber set to zero (kz = 0 in our notation).6 See also Kopylova et al. (2007), who give a numerical treatment of the leaky wave dispersion relation, and Zaitsev and Stepanov (2008) and Lopin and Nagorny (2014) for more recent discussions. The associated sausage mode period Psaus and decay time τsaus (= 1/ωI ) are given for large ρ0 /ρe by     2π a a 2 ρ0 a ρ0 a ≈ 2.6 , τsaus = ≈ 0.26 , (6.103) Psaus = j0,1 cA cA πj0,1 ρe cA ρe cA and so τsaus

1 = 2 π



ρ0 ρe

 Psaus .

(6.104)

The results (6.101) and (6.102) indicate that in a dense coronal flux tube the sausage mode with kz a  1 is leaky: any initiated symmetric wave propagates radially outwards in concentric rings, with the wave decaying in time as the wave radiates out to infinity. Nonetheless, an oscillation period Psaus is established that depends upon the travel time across the radius a. It should be noted that the convenient split of the Hankel function expression into real and imaginary parts that initiated our analysis of the sausage mode dispersion relation 5 There is an error in Zaitsev and Stepanov (1975) where the equilibrium relation between the density ratio ρ /ρ and the 0 e

propagation speeds following from pressure balance is wrongly given (see the equation following their eqn. (17)) and modifies their final result (eqn. (18)); moreover, the case investigated (cA  cs , cAe  cse ) has limited applicability. Nonetheless, the case cA  cs , cAe  cse discussed here leads to results that are similar to those obtained originally by Zaitsev and Stepanov (1975). 6 Note the different time dependence used by Zaitsev and Stepanov (1975) and also Meerson, Sasorov and Stepanov (1978), who assume an e−iωt dependence compared with the eiωt used here.

166

Magnetic Flux Tubes

required that |ne a|  1. Thus we require that ωR a/cAe  1, and so need j0,1 cA  cAe , that is, ρ0  j20,1 ρe (or ρ0  5.78 ρe ). We also require that ωI  ωR , which is met when ρ0  ρe . For coronal tubes that are not so extreme in their densities when compared with the surroundings we can expect some departures from the simple estimates provided here, and some other approximation or a numerical treatment (see, for example, Guo et al. 2016) may be required. One implication of the result (6.104) is that only in very dense flux tubes are we likely to see the global sausage mode established, because otherwise τsaus is so short as to quench any oscillation. This topic is further discussed in Chapter 14. Finally, we may note that the analysis given above for a β = 0 plasma applies more generally to a β = 0 medium, with only minor algebraic changes to the expressions for the wavenumbers n0 and ne . In particular, for a fully compressible medium the expressions for these wavenumbers when kz = 0 are simply n20 =

ω2 , c2s + c2A

n2e =

ω2 . c2se + c2Ae

(6.105)

Then ⎧ ⎛ ⎫  1/2 ⎞  ⎨ ⎬ c2s + c2A a ρe 1 ⎠ + γE , ⎝ j0,1 ω = j + j ln R 0,1 0,1 ⎭ ρ0 ⎩ 2 (c2s + c2A )1/2 c2se + c2Ae   π ρe a ωI = j0,1 (6.106) , ρe  ρ0 . 2 ρ0 (c2s + c2A )1/2 These general results for the leaky sausage mode reduce to equation (6.101) when cs = 0 and cse = 0. The ratio of the damping frequency to the oscillation frequency remains the same as for a β = 0 medium, namely ωI π ∼ ωR 2



ρe ρ0

 ,

ρe  ρ0 .

(6.107)

6.7 Three Special Cases Returning to a consideration of trapped waves, we note that there are three particular cases of the general theory that deserve special attention as they apply either to a number of important physical situations or they are mathematically easier to consider and so instructive. The first case is that of an incompressible medium. This case has perhaps limited direct application, though it is of interest in laboratory plasmas, but it is informative as it presents some mathematical simplifications that make the discussion of its dispersion relation more transparent. The second case is that of an isolated magnetic flux tube, in which the tube resides in a field-free environment; this is typified by magnetic flux tubes in the solar photosphere. The third case is that of a flux tube embedded in a strong magnetic field; this case is typified by coronal loops. We consider each case separately.

6.7 Three Special Cases

167

6.7.1 The Incompressible Medium As we have remarked earlier, the dispersion relation (6.42) is transcendental, so the number of its modes and their nature is not immediately apparent. However, the transcendental aspect is removed in the case of an incompressible fluid. An incompressible fluid is one for which the sound speeds exceed any other speed in the system; this corresponds to taking the limit cs → ∞ and cse → ∞ in the general theory; we can also regard this as a fictitious plasma with the property that the adiabatic index γ is allowed to tend to infinity. Alternatively, it may be discussed from first principles by replacing the isentropic equation by the condition that motions u satisfy div u = 0, the incompressible condition. Here it is convenient to deduce the incompressible case directly from the general dispersion relation for a compressible medium, letting cs and cse approach infinity. In the incompressible limit of infinite sound speeds, the tube speed ct becomes cA and the external tube speed cte becomes cAe . The squares of the transverse wavenumbers m0 and me reduce to m20 = kz2 ,

m2e = kz2 .

Thus m0 = me = |kz |, the modulus sign allowing for the possibility of negative kz ; otherwise we can restrict attention to positive kz . The restriction me > 0 that arises in the general compressible case is now seen to be automatically satisfied for an incompressible medium. Hence we see that in an incompressible medium    ρe Im (|kz |a) Km (|kz |a) Rm = − . (6.108) ρ0 Im (|kz |a) Km  (|kz |a) Thus the dispersion relation (6.47) (or equivalently (6.42)) now gives the wave speed c explicitly in terms of kz a (and the equilibrium parameters ρ0 , ρe , cA and cAe ): c2 =

c2A + Rm c2Ae . 1 + Rm

(6.109)

This resolution of the transcendental nature of the compressible dispersion relation that arises in the incompressible limit makes for a more straightforward discussion of the incompressible case. Written out in full, the dispersion relation (6.109) for a magnetic flux tube in an incompressible medium is Im  (|kz |a) Km  (|kz |a) 1 1 = . ρ0 (ω2 − kz2 c2A ) Im (|kz |a) ρe (ω2 − kz2 c2Ae ) Km (|kz |a)

(6.110)

A dispersion relation of this form was derived and discussed in Dungey and Loughhead (1954), Uberoi and Somasundaram (1980) and Edwin and Roberts (1983). The behaviour of c2 for small kz a may be deduced directly from the incompressible form of the dispersion relation with Rm given by (6.108), or alternatively we may examine (6.56), (6.64) and (6.68) in the incompressible limit. When cs → ∞ and cse → ∞, λt and λk both tend to unity and ct → cA . Thus, for the sausage (m = 0) mode, in a thin tube (|kz a|  1), equation (6.56) reduces to   1 ρe (6.111) (c2Ae − c2A )(kz a)2 K0 (|kz |a). c2 ∼ c2A + 2 ρ0

168

Magnetic Flux Tubes

Considering the kink (m = 1) mode, equation (6.64) reduces to     c2Ae − c2A ρ0 ρe 2 2 2 (kz a) K0 (|kz |a) , c ∼ ck 1 − (ρ0 + ρe )2 c2k while for the fluting (m ≥ 2) modes equation (6.68) gives     2 − c2 c ρ ρ 1 0 e Ae A c2 ∼ c2k 1 − (kz a)2 . (ρ0 + ρe )2 m(m2 − 1) c2k

(6.112)

(6.113)

We may also examine the behaviour of the dispersion relation in the limit of large kz a, corresponding to wavelengths much shorter than the tube cross-section. In the general case of a compressible medium we needed to consider both surface and body waves, but here only the surface wave case occurs. Accordingly, we deduce (either directly from the dispersion relation or from (6.72)) that as |kz |a → ∞ ρe Rm ∼ , ρ0 irrespective of the mode number m. Hence, in an incompressible medium c2 ∼ c2k as |kz |a → ∞ for all mode numbers m. It is thus evident that the kink speed ck plays an important role for both large and small wavelengths.

6.7.2 The Isolated Flux Tube The special case of an isolated magnetic flux tube follows from the general discussion on setting the Alfv´en speed in the environment to zero: cAe = 0. The equilibrium configuration is here simply one in which the magnetic field of the tube is confined by a higher external pressure pe : p0 +

B20 = pe , 2μ

(6.114)

and the densities inside and outside the isolated tube are related by ρe c2 1 c2 = 2s + γ 2A . ρ0 2 cse cse

(6.115)

For sound speeds cs and cse that are roughly comparable, pressure balance (6.114) implies that the magnetic field has partially evacuated the interior of the tube: ρ0 < ρe . The speed ck is  1/2 ρ0 cA (6.116) ck = ρ0 + ρe and so is sub-Alfv´enic. The profile of Alfv´en speed across the structure is of interest; inside the tube, the Alfv´en speed is non-zero, whereas outside the tube the Alfv´en speed has fallen to zero. This resembles a top hat, with the outer rim of the hat modelling the region of zero Alfv´en speed and the high part of the hat modelling the interior of the tube where the Alfv´en speed is non-zero.

6.7 Three Special Cases

169

Figure 6.4 The dispersion diagram of Edwin and Roberts (1983), showing solutions of the dispersion relation (6.42) for a magnetic flux tube under photospheric conditions, namely cAe < cs < cse < cA . Specifically, cAe = cs /2, cse = 3cs /2 and cA = 2cs ; with γ = 5/3, these values give ρ0 ≈ 0.57ρe , ct ≈ 0.89 cs and ck ≈ 0.63 cA ≈ 1.27 cs . The diagram displays the phase speed c (= ω/kz ) as a function of the dimensionless wavenumber kz a for fast and slow magnetoacoustic waves in a magnetic flux tube in an environment with low Alfv´en speed. [Notational changes from the present text: VA ≡ cA , VAe ≡ cAe , C0 ≡ cs , CT ≡ ct , CTe ≡ cte , Ce ≡ cse , and k ≡ kz .] (After Edwin and Roberts 1983.)

With the external Alfv´en speed zero, cAe = 0, the wavenumber me is given by  m2e

=

kz2

c2 1− 2 cse

 (6.117)

and so the requirement that me > 0 means that c < cse ; taken with the observation that ck lies below the tube’s Alfv´en speed, this means that the propagation speed of the kink mode lies below both the Alfv´en speed in the tube and the sound speed in the environment. With cAe = 0, the dispersion relation (6.47) reduces to c2 =

c2A , 1 + Rm

(6.118)

170

Magnetic Flux Tubes

Figure 6.5 The dispersion diagram showing solutions of the dispersion relation (6.42) for a magnetic flux tube under photospheric conditions (namely cAe < cs < cse < cA ), with the same parameter values as used in Figure 6.4. Specifically, cAe = cs /2, cse = 3cs /2, cA = 2cs and γ = 5/3, giving ρ0 ≈ 0.57 ρe , ct ≈ 0.89 cs and ck ≈ 0.63 cA ≈ 1.27 cs . The diagram displays the dimensionless phase speed c/cs (where c = ω/kz ) as a function of the dimensionless wavenumber kz a for fast and slow magnetoacoustic waves in a magnetic flux tube embedded in an environment with low Alfv´en speed. The hatching shows a region where the waves are leaky. Torsional Alfv´en waves are indicated by a horizontal line at c = cA . (Courtesy E. Verwichte.)

with Rm defined generally in (6.46). Written out in full, equation (6.118) gives c2A ρe m0 a Im  (m0 a) Km (me a) = 1 − . ρ0 me a Im (m0 a) Km  (me a) c2

(6.119)

The requirement that me > 0 implies c2 < c2se , and so the speed of propagation of any magnetoacoustic wave along an isolated magnetic flux tube is less than the speed of sound in the environment. The earlier discussion of the kink and fluting modes holds generally, so these modes arise in the isolated tube and propagate with a speed close to ck when kz a is small. The corrections to the speed of the kink and fluting modes obtained earlier, in equations (6.64) and (6.68), also apply generally. In particular, for kz a  1, the kink (m = 1) mode

6.7 Three Special Cases

171

propagates with the speed

    ρe 1 c ∼ ck 1 + (λk kz a)2 K0 (λk |kz |a) , 2 ρ0 + ρe

(6.120)

where λ2k = 1 −

c2k , c2se

c2k < c2se .

(6.121)

Numerically obtained solutions of the general dispersion relation are displayed in Figure 6.4, following the work of Edwin and Roberts (1983). Specifically, Edwin and Roberts considered the case of a low Alfv´en speed in the environment of the tube as generally representative of the isolated tube. A recently computed version of such a figure, which also displays the first two fluting modes (m = 2 and 3), is shown in Figure 6.5, provided courtesy of Erwin Verwichte.

6.7.3 The Embedded Flux Tube The special case of a flux tube embedded in a strong magnetic field is of particular importance as it includes the case of a coronal loop. We are interested in the case where the Alfv´en speed in the tube and the Alfv´en speed in the environment are larger than the corresponding sound speeds. The dispersion diagram for such a case is given in Figure 6.6, obtained originally in Edwin and Roberts (1983). The diagram shows the occurrence of two sets of modes, namely fast and slow body (n20 > 0) waves. There are no surface waves. The fast waves are strongly dispersive, and arise only if cAe > cA ; the flux tube is a waveguide for fast waves (as well as for slow waves). If cAe < cA , the fast waves are leaky and propagate energy away from the region of high Alfv´en speed (see Section 6.6). Thus fast body waves are trapped in regions of low Alfv´en speed, typically corresponding to regions of high plasma density. The slow waves are also dispersive, but their dispersion is much weaker than arises in the fast waves. It is evident from Figure 6.6 that the kink mode is rather special: it exists for all kz a, whereas other modes have cutoffs that require the wave to possess a sufficiently large kz a (longitudinal wavelength sufficiently small) in order to propagate as a trapped wave. Moreover, the kink mode has very little compression (Spruit 1982), reminding one of an Alfv´en wave. Ruderman and Roberts (2002) referred to the mode as a global wave, and Goossens et al. (2009) described the Alfv´enic nature of the wave and used the term Alfv´enic kink wave, but concluding that the wave is distinctive and not readily described in simple generic terms. Figure 6.7, provided courtesy of Erwin Verwichte, is a more recently computed plot of the dispersion curves, showing the fluting modes (m ≥ 2) as well as the kink and sausage modes. Embedded Tube: β = 0 Plasma The behaviour of the fast waves, including the global kink mode, in a strong magnetic field can perhaps best be understood by looking at the extreme of a β = 0 plasma. This

172

Magnetic Flux Tubes

Figure 6.6 The dispersion diagram of Edwin and Roberts (1983), showing solutions of the dispersion relation (6.42) for a magnetic flux tube under coronal conditions, namely cs < cA < cAe . Specifically, cAe = 5 cs , cse = cs /2 and cA = 2 cs , giving ρ0 ≈ 4.9 ρe , ct ≈ 0.89 cs and ck ≈ 1.38 cA ≈ 0.55 cAe . The diagram gives the phase speed c (= ω/kz ) as a function of the dimensionless wavenumber kz a for fast and slow magnetoacoustic body modes in a coronal magnetic flux tube. Solid curves correspond to sausage waves, dashed curves to kink waves. [Notational changes from the present text: VA ≡ cA , VAe ≡ cAe , C0 ≡ cs , CT ≡ ct , CTe ≡ cte , Ce ≡ cse , and k ≡ kz .] (After Edwin and Roberts 1983.)

corresponds to setting the sound speeds to zero. With cs = 0 and cse = 0, the corresponding tube speeds are also zero, so we lose the slow modes of the tube. What remains is the fast waves, and for these modes the extreme of β = 0 provides a good description of the fast waves in an embedded tube. Notice first that when we set the sound speeds to zero, this constrains the equilibrium state since we are in effect setting pressures to zero; consequently pressure balance (6.32) requires that the pressure of the magnetic field is the same inside and outside the tube. Unless the field reverses its direction, this corresponds to requiring that the field is everywhere uniform, so B0 = Be . The flux tube, then, is distinguished purely by density differences: the tube may be denser (or less dense) than its surroundings. The densities and the Alfv´en speeds are related through c2 ρe = 2A . ρ0 cAe

(6.122)

Considering a dense tube, ρ0 > ρe , we see that the β = 0 (coronal) embedded tube is distinguished by a profile in Alfv´en speed that resembles an inverted top hat: the Alfv´en

6.7 Three Special Cases

173

Figure 6.7 The dispersion diagram for solutions of the dispersion relation (6.42) for a magnetic flux tube under coronal conditions, namely cs < cA < cAe . The figure shows the dimensionless speed c/cs (the phase speed ω/kz in units of the sound speed cs inside the tube) of a wave for the four azimuthal mode numbers m = 0, 1, 2 and 3. The curves are for the same parameter values as used in Figure 6.6, namely cAe = 5 cs , cse = cs /2 and cA = 2 cs . The slow modes emanate from c = ct , and display only weak dispersion as their speed c varies from ct in the thin tube limit (kz a  1) to cs at large kz a. (shown in the enlargement in the bottom figure). The fast waves consist of the global kink (m = 1) mode and the fluting (m ≥ 2) modes, all emanating from c = ck . Also shown are the transverse harmonics of fast waves, which have finite wavenumber cutoffs. The hatching indicates the region where waves are leaky. The torsional Alfv´en waves are shown as solid horizontal lines at ω/kz = cA , cAe . (Courtesy E. Verwichte.)

speed is lower inside the tube than it is outside, simply because the plasma density is higher inside the tube. This is in contrast to the isolated tube which produced a profile in Alfv´en speed that resembled a top hat (in the usual configuration for placing a hat upon ones head). With cs = cse = 0, the effective wavenumbers m0 and me reduce to  m20

=

kz2

c2 1− 2 cA



 ,

m2e

=

kz2

c2 1− 2 cAe

 .

(6.123)

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Magnetic Flux Tubes

Thus, to satisfy the condition me > 0 we require c2 < c2Ae ; the fast waves propagate with a speed c that is below the external Alfv´en speed cAe . Embedded Tube: Trapped Kink (m = 1) Modes Considering the kink mode, we see that the general result (6.64), applied to an embedded tube with cs = cse = 0, gives     2  2 c − c ρ e Ae k c2 ∼ c2k 1 − (6.124) (λk kz a)2 K0 (λk |kz |a) , ρ0 + ρe c2k where now λ2k = 1 −

c2k c2Ae

,

(6.125)

with c2k < c2Ae . Noting that ρ0 c2A = ρe c2Ae , we can rewrite the above into the alternative form     c2Ae − c2A ρ0 ρe 2 2 2 c ∼ ck 1 − (6.126) (λk kz a) K0 (λk |kz |a) . (ρ0 + ρe )2 c2k A relation of this form was first given by Edwin and Roberts (1983). We can simplify (6.126) a little further noting that for a β = 0 medium we have   2ρ0 ρ0 − ρe λ2k = . c2A , c2k = ρ0 + ρe ρ0 + ρe Accordingly, the thin tube dispersion relation for the kink mode when β = 0 is   ρ0 ρe (c2Ae − c2A ) 2 2 2 2 ω = kz ck 1 − (λk kz a) K0 (λk |kz a|) , |kz a|  1, (6.127) (ρ0 + ρe )2 c2k and the wave speed c (= ω/kz ) of the kink mode is given by    2 − ρ ρ 1 0 e c2 = c2k 1 − (kz a)2 K0 (λk |kz a|) , 2 ρ0 + ρe

|kz a|  1.

(6.128)

Since we require λ2k > 0 for trapped waves, it follows that in a β = 0 medium we require ρ0 > ρe , that is, the magnetic flux tube must be over-dense. If the tube is under-dense, so that ρ0 < ρe , then the wave is no longer trapped and leakage occurs (see Section 6.6). Embedded Tube: Trapped Fluting (m ≥ 2) Modes The fluting modes are analysed in much the same way as the kink mode. Again we suppose that β = 0, so that the equilibrium magnetic field is uniform and we have ρ0 c2A = ρe c2Ae . In the long wave limit of a thin tube (|kz a|  1), the dispersion relation for m ≥ 2 simplifies to ρ0

(ω2

1 1 + 2 2 2 − kz cA ) ρe (ω − kz2 c2Ae ) =

n20 a2 m2e a2 1 1 − . ρ0 (ω2 − kz2 c2A ) 2m(m + 1) ρe (ω2 − kz2 c2Ae ) 2m(m − 1)

(6.129)

6.8 Perturbations

175

Setting ω2 = kz2 c2k (1 + m ), where m is considered small compared with unity, and substituting into relation (6.129) we obtain m , finding that fluting (m ≥ 2) modes are described approximately by     1 1 ρ0 − ρe 2 2 2 2 2 ω = kz ck 1 − (6.130) (kz a) , |kz a|  1, 2 ρ0 + ρe (m2 − 1) with the wave speed c of the fluting modes given by   1 1 ρ0 − ρe 2 2 ck 2 (kz a)2 , c2 = c2k − 2 ρ0 + ρe (m − 1)

|kz a|  1.

(6.131)

This result agrees with equation (6.68) when that expression is simplified by setting cs = 0, cse = 0.

6.8 Perturbations It is of interest to examine the nature of the perturbations (eigenfunctions) that arise in the modes of oscillation of a magnetic flux tube. We confine our attention to trapped waves, and concentrate on the case of a thin tube (|kz a|  1). As seen earlier, within a flux tube the total pressure perturbation pT is given simply as pT = A0 Im (m0 r).

(6.132)

The constant A0 may be chosen arbitrarily. The motion u = (ur , uφ , uz ) connected with the variations in the pressure pT is given by equations (6.20)–(6.22), yielding ur =

ρ0

(ω2

A0 iω ωm A0 m0 Im  (m0 r), uφ = Im (m0 r), 2 2 2 2 2 − kz cA ) ρ0 (ω − kz cA ) r   c2s 1 uz = ωkz A0 Im (m0 r). (6.133) c2s + c2A ρ0 (ω2 − kz2 c2t )

The associated displacement ξ = (ξr , ξφ , ξz ), satisfying u=

∂ξ = iωξ , ∂t

(6.134)

has components ξr =

A0 1 1 A0 m0 Im  (m0 r), ξφ = −im Im (m0 r), ρ0 (ω2 − kz2 c2A ) ρ0 (ω2 − kz2 c2A ) r   c2s 1 ξz = −ikz A0 Im (m0 r). (6.135) 2 2 2 cs + cA ρ0 (ω − kz2 c2t )

One quantity of particular interest is the compression div u, which provides a measure of the variations in plasma density associated with the wave motion: div u =

1 ∂uφ 1 ∂uz ∂ur + ur + + . ∂r r r ∂φ ∂z

Using (6.20)–(6.22) together with (6.14), after some algebra we obtain

176

div u =

Magnetic Flux Tubes

iω3 ρ0 (c2A

+ c2A )(kz2 c2t

− ω2 )

pT ,

div ξ =

ω2 ρ0 (c2A

+ c2A )(kz2 c2t

− ω2 )

pT . (6.136)

Relations such as these, together with the dispersion relation (6.42), determine in detail the actual perturbation. Of specific interest here is the case of a thin magnetic flux tube, when we can take |kz a|  1 and so expand the various relations in power series in the parameter kz a. To do this we need the standard expansion of the modified Bessel function and its derivative (Abramowitz and Stegun 1965). Specifically, for m ≥ 0 we have   m &  m  ∞ 1 2 ( 14 z2 )k 1 1 1 4z z = z + · · · , (6.137) 1+ Im (z) = 2 k! (1 + m + k) m! 2 1+m k=0

where k! denotes the factorial of the summation integer k and (m) is the Gamma function of m (note that (1 + m) = m!). Coupled with the expansion of Im (z), we have the expansion of its derivative (which we can obtain by using the relations that relate the derivative to the Bessel functions themselves):     1 1 1 m−1 (m + 2) 2  m ≥ 1. (6.138) z 1+ z +··· , Im (z) = 2 2 (m − 1)! 4m(m + 1) These relations cover the kink (m = 1) and fluting (m ≥ 2) modes. For the sausage (m = 0) mode, we have I0  (z) = I1 (z). 6.8.1 Kink and Fluting Modes Now the leading order behaviour of the pressure perturbation pT follows from the expansion of Im , giving m  1 1 m0 r , m ≥ 0. (6.139) pT ∼ A0 2 m! Thus, the total pressure perturbation is zero at the centre of the tube (r = 0) for all modes save the sausage wave. It is of interest to compare the pressure variations with another quantity determined by linear theory. For the kink and fluting modes, the radial component of motion or displacement provides a convenient reference speed or measure of displacement. Consider the radial component of the displacement; for small kz a, m−1  1 1 1 1 m r m , m ≥ 1. (6.140) ξr = A0 0 0 2 ρ0 (ω2 − kz2 c2A ) 2 m! Write 1 1 aCm = A0 m0 2 ρ0 (ω2 − kz2 c2A )



1 m0 r 2

m−1

1 , m!

m ≥ 1.

Here ω2 needs to be determined by the dispersion relation that applies for small kz a, with m ≥ 1; thus we can set ω2 = kz2 c2k in the above relationship between A0 and Cm . Then ξr = aCm rm−1 .

6.8 Perturbations

177

In particular, for the kink (m = 1) mode we have ξr = aC1 ,

aC1 =

1 1 m0 . A0 2 2 ρ0 (ω − kz2 c2A )

(6.141)

Thus, the radial displacement in the kink mode is a constant; the constant C1 is the fractional measure of ξr when compared with the radius a of the tube (so, for example, a displacement of the tube that is 50% of the radius a has C1 = 1/2, and so on). The azimuthal displacement in the kink mode in a thin tube is equal in magnitude to the radial component, but out of phase by π/2: ξφ ∼ −iξr . By contrast, the longitudinal component ξz is much smaller than the radial component:   ξz 1 c2 − c2A r  uz c2s = = i(kz a) . ξr ur a c2s + c2A m c2t − c2 Thus, in the kink or fluting modes (for which c2 = c2k ) the longitudinal motion is of order kz a smaller than the transverse motion (Spruit 1982): |uz | ∼ |(kz a)ur |. The associated pressure variations in the kink mode are

r . (6.142) pT ∼ C1 ρ0 (c2k − c2A )(kz a)2 a Thus the total pressure variations in the kink mode are of order (kz a)2 , and so all but negligible. The total pressure variations in the fluting mode are smaller still, of order (kz a)m+1 :  pT ∼ C1 ρ0 (c2k − c2A )

m0 kz

m−1

m 1 m+1 r a) , (k z a 2m−1 m! 1

m ≥ 1.

(6.143)

In the special case of a β = 0 plasma, with ρ0 c2A = ρe c2Ae , the pressure variation in the kink mode is  

r ρ0 − ρe 2 , (kz a)2 pT ∼ ρ0 cA C1 ρ0 + ρe a and the associated displacement compression and plasma density ρ are given as  

r ρ ρ0 − ρe = −div ξ . , (kz a)2 div ξ ∼ −C1 ρ0 + ρe a ρ0 Thus, in the kink (and also in the fluting) modes there is an almost negligible compression and associated change in plasma density. Longitudinal motions are small (of order kz a), falling to zero in a β = 0 plasma. The above results are in agreement with those obtained by Goossens et al. (2009).

178

Magnetic Flux Tubes

6.8.2 Sausage Mode Finally, we turn to the sausage (m = 0) mode. With m = 0, the azimuthal motion is absent (uφ = 0) and the wave consists of a longitudinal motion combined with a radial disturbance. In fact, the radial motion is smaller than the longitudinal flow:   ur 1 c2s − c2 r  = i(kz a) . (6.144) uz 2 a c2s Thus ur is of order kz a of uz : the sausage mode involves mainly longitudinal motions (Roberts and Webb 1978: Spruit 1982; Spruit and Roberts 1983). With c2 = c2t ,  

r ur 1 c2s = i(kz a) . (6.145) uz 2 a c2s + c2A In the β = 0 case, the longitudinal motion is zero, but in this case the sausage mode has a finite kz cutoff which prevents the mode from existing for arbitrarily small kz a, save as a leaky wave. 6.9 Comparison of Modes in a Slab and Tube: The β = 0 Plasma It is of interest to bring together our results for the various modes, to see differences and similarities. We do this for the case of a β = 0 plasma, pertinent to the corona, and we suppose that ρ0 > ρe . We compare the results for a slab geometry (step function and Epstein density profiles) and a cylindrical tube. Consider first the principal kink mode. In both slab and cylinder it propagates for all wavenumbers kz . In a slab, whether step function or Epstein density profiles, for small kz a we have (see Chapter 5)  2   ρ 0 c2 = c2Ae 1 − (kz a)2 −1 (6.146) , |kz a|  1. ρe In a thin flux tube, the principal kink mode has speed c given by (see equation (6.128))    2 − ρ ρ 1 0 e c2 = c2k 1 − (kz a)2 K0 (λk |kz a|) , |kz a|  1, (6.147) 2 ρ0 + ρe where

 c2k

=

2ρ0 ρ0 + ρe

 c2A ,

λ2k =

ρ0 − ρe . ρ0 + ρe

In the extreme ρ0  ρe , for |kz a|  1 these relations reduce to  '  2 ( c2Ae 1 − (kz a)2 ρρ0e , slab, 2 c = 2' ( 1 2 tube. ck 1 − 2 (kz a) K0 (|kz a|) ,

(6.148)

By contrast, the sausage mode has a finite wavenumber cutoff in both slab and tube geometries. For the principal sausage mode we have  1/2  1/2 c2A 1 1 ρe cutoff =C =C , (6.149) kz a c2Ae − c2A a ρ0 − ρe

6.10 Resonant Absorption in a Flux Tube: The β = 0 Plasma

where the constant C is

⎧ ⎪ ⎪π/2, ⎨ √ C= 2, ⎪ ⎪ ⎩j , 0,1

discrete slab, Epstein slab,

j0,1

(6.150)

cylindrical tube.

The associated cutoff period Pcutoff is     2π a a ρe 1/2 ρe 1/2 =D Pcutoff = 1− 1− C cA ρ0 cA ρ0 where the constant D is given by ⎧ ⎪ ⎪ ⎨4,√ D = π 2, ⎪ ⎪ ⎩ 2π ,

179

(6.151)

discrete slab, Epstein slab,

(6.152)

cylindrical tube.

In the extreme of a high density structure, ρ0  ρe , we have a Pcutoff = D . cA

(6.153)

6.10 Resonant Absorption in a Flux Tube: The β = 0 Plasma The kink mode resonance of a surface wave in an incompressible medium with a transition region joining one side of the interface to the other was discussed from the viewpoint of an initial value problem in Chapter 4, Section 4.5. Here we turn to a closely related topic, the kink oscillations of a magnetic flux tube in which there is a narrow transition region joining the interior of a flux tube to its magnetic environment. This topic was discussed in Sakurai, Goossens and Hollweg (1991) from the point of view of so called connection formulas, which we treat in Chapter 8. Another approach, from an initial value problem, was carried out in Ruderman and Roberts (2002), who considered a flux tube of radius a with a thin layer of thickness l centred on the boundary r = a, the tube being embedded in β = 0 plasma. An involved initial value treatment – in which the effect of weak viscosity is also included – leads to the determination of the frequency ωR of a kink oscillation generated in the medium from initial conditions; the oscillation exhibits a decay at the rate ωI (for |ur | ∝ e−ωI t , ωI > 0). Ruderman and Roberts (2002; see their eqn. (56)) obtained a result equivalent to ωR = kz ck ,

ωI =

π ρ0 (rA ) 2 2 (ρ0 − ρe )2 ωk , k c 2a |A | z A (ρ0 + ρe )3

(6.154)

where ωk (= kz ck ) denotes the kink mode frequency. The resonance point is at r = rA , occurring inside the thin transition layer where the density changes from ρ0 inside the tube to ρe outside the tube. The resonance point r = rA is where ωk2 = kz2 c2A (rA ), the location where the kink mode frequency matches the local Alfv´en frequency. The density ρ0 (rA ) is the plasma density ρ0 (r) calculated at the resonance point r = rA . Noting that for a β = 0 plasma we have

180

Magnetic Flux Tubes

c2k =

2B20 , μ(ρ0 + ρe )

c2A (r) =

ρ0 (rA ) =

1 (ρ0 + ρe ). 2

B20 , μρ0 (r)

(6.155)

it follows immediately that (6.156)

Finally, A is a measure of the slope at r = rA of the Alfv´en frequency squared: ) B2 ρ0  (rA ) A = −(kz2 c2A (r)) )r=r = kz2 0 2 . A μ ρ0 (rA )

(6.157)

Thus, eliminating A in preference to the slope in the density ρ0 (r) we obtain ωI =

π (ρ0 − ρe )2 ωk . 8a (ρ0 + ρe )|ρ0  (rA )|

(6.158)

This formula gives the decay rate due to resonant absorption on the thin boundary (l  a) of a magnetic flux tube in a β = 0 plasma. The decay rate depends inversely upon the slope ρ0  (r = rA ) of the equilibrium plasma density at the resonance point r = rA within the transition layer. Although formulas of the form (6.154) and (6.158) were newly obtained in the initial value treatment given by Ruderman and Roberts (2002) they were not new to solar physics. In fact, such formulas had arisen in a quite distinct treatment of the problem, leading to the topic of connection formulas derived by Sakurai, Goossens and Hollweg (1991). Moreover, a similar formula for the damping rate had arisen in a Cartesian coordinate system explored by Hollweg and Yang (1988), who correctly suggested on the basis of their Cartesian results the expected equivalent form in a cylindrical geometry. Nonetheless, despite these important connections, the application of the decay formula in a cylindrical geometry was perhaps only fully brought to the fore when related to detailed observations of loop oscillations. This was done specifically in Ruderman and Roberts (2002) and Goossens, Andries and Aschwanden (2002). Ruderman and Roberts (2002) applied their initial value treatment to the particular observation of a decaying coronal loop oscillation observed and analysed by Nakariakov et al. (1999). Goossens, Andries and Aschwanden (2002) derived their results from connection formulas theory, and applied their work to observations of 11 oscillating loops. All these works gave encouragement to the interpretation of the observed decay as being due to resonant absorption. The topic of connection formulas is discussed at length in Chapter 8. To simplify further the expression (6.158) for the decay rate ωI , we must specify the density profile in the transition region on the boundary of the flux tube. Two profiles have received particular interest: the case of a linear profile (which has ρ0  (r) constant in the layer), and the sinusoidal profile, introduced by Ruderman and Roberts (2002), for which the density and its slope are smooth functions.

6.10 Resonant Absorption in a Flux Tube: The β = 0 Plasma

181

6.10.1 Resonant Absorption: The Linear Profile The linear profile for the density has a transition region on the edge of the tube: 1 (a − r) l l (ρ0 + ρe ) + (ρ0 − ρe ) , a− a, (0, 0, B0e ), (7.50) B0 = (0, Ar, B0z ), r < a, corresponding to a magnetic flux tube with uniform twist (A = 0) within a radius r = a but no twist outside the tube (in r > a), where the external longitudinal field is taken to be everywhere a constant, B0e . The densities ρ0 inside and ρe outside the flux tube are assumed to be constants. Then ωA2 =

(mA + kz B0z )2 , μρ0

2 ωAe = kz2

B20e μρe

are constants. The case of no twist in the tube corresponds to A = 0. 6 There is an abuse of notation here in that the coefficients have had common factors cancelled out; provided this is recognized,

there is no risk of confusion.

194

The Twisted Magnetic Flux Tube

The coefficients D, C1 , C2 and C3 for an incompressible medium with uniform twist, uniform longitudinal magnetic field and uniform density follow from the expressions in (7.49): D = ρ0 (ω2 − ωA2 ),   m2 2 + kz , C2 = − r2

C1 = −

2mAfB , μr

(7.51)

C3 = ρ02 (ω2 − ωA2 )2 − 4ρ0 ωA2

A2 . μ

Thus, D, rC1 and C3 are all constants. Accordingly, the differential equation (7.36) for pT reduces to   1 dpT m2 d2 pT + + κ02 pT = 0, − r dr dr2 r2 where the constant κ02 is defined by



κ02 = kz2 1 −

2 4A2 ωA 2 )2 μρ0 (ω2 − ωA

(7.52)

(7.53)

 .

(7.54)

The Incompressible Case: Derivation of the Dispersion Relation Equation (7.53) is a form of the modified Bessel’s equation (see Abramowitz and Stegun 1965) with solutions pT ∝ Im (κ0 r), Km (κ0 r) for modified Bessel functions Im and Km In order to allow for the possibility of negative m, we take the solutions in the form pT ∝ I|m| (κ0 r), K|m| (κ0 r). This is permitted because the Bessel equation involves m2 and this is also |m|2 . However, m remains within the expression for κ02 through the definition of ωA2 , and this creates an asymmetry between modes with positive m and modes with negative m. Only in the absence of twist, when A = 0, does this asymmetry disappear. Goossens, Hollweg and Sakurai (1992) take a similar approach in their discussion of the compressible tube. For a solution that is finite at the tube centre r = 0, we need to choose the function I|m| ; then inside the tube pT = A0 I|m| (κ0 r), for arbitrary constant A0 . Similarly, outside the twisted tube we consider the medium to be incompressible with a uniform longitudinal magnetic field of strength Be0 and uniform density ρe , but no twist. Thus, B0φ is discontinuous at r = a and a surface current arises on the tube boundary. The differential equation (7.53) again applies, but now with A = 0, and the solution outside the tube that is bounded as r → ∞ is pT = Ae K|m| (|kz |r) for arbitrary constant Ae . Thus, altogether we have  r > a, Ae K|m| (|kz |r), pT (r) = A0 I|m| (κ0 r), r < a. We now need to apply the boundary conditions (7.41) which allow us to relate the inside of the tube with its environment. To do this we use the second equation in (7.32) to determine ξr . Inside the tube, where the twist is uniform, we have ξr = Thus,

D dpT C1 pT . + C3 dr C3

(7.55)

7.5 The Twisted Flux Tube: Special Cases

 ξr (r) = A0

195



C1 D κ0 I|m|  (κ0 r) + I|m| (κ0 r) , C3 C3

r < a,

(7.56)

a prime  denoting the derivative of the function (so I|m|  (x) denotes the derivative with respect to x of the function I|m| (x) and I|m|  (κ0 r) denotes the value of the derivative function I|m|  (x) at x = κ0 r). Outside the tube, where the twist is assumed to be zero, we have C1 = 0 and so 1 |k |K  (|kz |r), r > a. 2 ) z |m| ρe (ω2 − ωAe   Continuity of ξr and pT − B20φ /(μr) ξr across r = a is achieved provided   1 C1 D A0 |k |K  (|kz |a) κ0 I|m|  (κ0 a) + I|m| (κ0 a) = Ae 2 ) z |m| 2 C3 C3 ρe (ω − ωAe ξr = Ae

(7.57)

(7.58)



  A2 a D I|m|  (κ0 a) C1 A0 I|m| (κ0 a) 1 − κ0 + = Ae K|m| (|kz |a). μ C3 I|m| (κ0 a) C3

(7.59)

These continuity relations allow us to determine Ae /A0 through two expressions, which must therefore be compatible. Consequently, we must have    K|m|  (|kz |a) A2 a D I|m|  (κ0 a) C1 2 2 κ0 + + = 1. (7.60) ρe (ω − ωAe ) C3 I|m| (κ0 a) C3 |kz |K|m| (|kz |a) μ Equation (7.60) is the dispersion relation for a uniformly twisted magnetic flux tube embedded in an environment that consists of a uniform straight external magnetic field. The plasma densities ρ0 and ρe inside and outside the tube are uniform. The expression (7.60) may be rewritten in the form I

 (κ

a)

2 )κ a |m| 0 − 2mω √ A (ω2 − ωA 0 I|m| (κ0 a) A μρ0 2

2 )2 − 4ω2 A (ω2 − ωA A μρ0

K

=

 (|k

a|)

z |kz a| K|m| |m| (|kz a|)

ρe 2 ρ0 (ω

2 )+ − ωAe

K|m|  (|kz a|) A2 μρ0 |kz a| K|m| (|kz a|)

.

(7.61)

A relation of this form, though written for the case of positive m, was first obtained by Dungey and Loughhead (1954)7 and later by Bennett, Roberts and Narain (1999). The dispersion relation (7.61) applies to an incompressible magnetic flux tube with uniform twist. It has been discussed in some detail by Bennett, Roberts and Narain (1999) and Erd´elyi and Fedun (2006a). Erd´elyi and Fedun (2006a) also allow for twist in the environment of the tube. The effect of a twisted annular region between the tube and its environment has been explored in Erd´elyi and Carter (2006) and Carter and Erd´elyi (2007, 2008). In the absence of twist, A = 0 and then κ02 = kz2 . Moreover, without loss of generality, we can confine attention to m ≥ 0; then relation (7.61) reduces to 2 ) ρ0 (ω2 − ωA

 Km  (kz a) 2 Im (kz a) ) = ρe (ω2 − ωAe . Km (kz a) Im (kz a)

(7.62)

7 There is a sign error in one of the terms given in Dungey and Loughhead (1954); see Bennett, Roberts and Narain (1999) for

clarification.

196

The Twisted Magnetic Flux Tube

We have recovered the dispersion relation (6.110) discussed in Chapter 6 and obtained originally in Dungey and Loughhead (1954); see also Uberoi and Somasundaram (1980) and Edwin and Roberts (1983). The Incompressible Case: The Thin Tube The general dispersion relation (7.61) has been investigated numerically by Bennett, Roberts and Narain (1999) and Erd´elyi and Fedun (2006a). The dispersion relation allows for the occurrence of unstable modes, corresponding to ω2 < 0, a discussion of which would take us too far afield. The sausage (m = 0) mode has been investigated by Bennett, Roberts and Narain extensively. The kink (m = 1) mode has been treated numerically but not analytically. Accordingly, we end our discussion here by examining the case of an incompressible twisted thin tube, corresponding to kz a  1. Notice first that for small x we have (see Chapter 6) x

I|m|  (x) ∼ |m|, I|m| (x)

x

K|m|  (x) ∼ −|m|, K|m| (x)

|m| ≥ 1.

So, for the thin tube limit of kz a  1, the dispersion relation (7.61) for kink and fluting modes with |m| ≥ 1 reads 2 )|m| − 2mω √ A (ω2 − ωA A μρ0 2 )2 (ω2 − ωA

2 A2 − 4ωA μρ0

∼

−|m| ρe 2 ρ0 (ω

2 )− − ωAe

A2 μρ0 |m|

.

(7.63)

We can seek a solution of the approximate relation (7.63) that is valid for small twist by expanding in powers of A, writing ω2 = ω02 + Aω1 + A2 ω2 + · · · 2 is also a function of A. Then, for coefficients ω0 , ω1 , etc. to be determined. Recall that ωA for the lowest order term (equivalent to A → 0) we obtain     2 2 B B 0z ρ0 ω02 − kz2 + ρe ω02 − kz2 0e = 0, μρ0 μρe

yielding ω02 = kz2 c2k ,

c2k =

B20z + B20e μ(ρ0 + ρe )

.

(7.64)

Proceeding to terms of order A and A2 , we find (after considerable algebra) that     m m − sgn (m) B0z , ω2 = . (7.65) ω1 = 2kz m − sgn (m) μ(ρ0 + ρe ) μ(ρ0 + ρe ) Here sgn denotes the sign of the term: sgn (m) = Note that m = |m| sgn (m).



+1,

m > 0,

−1,

m < 0.

7.6 The Compressible Twisted Tube: Small Twist

Thus, for small twist (|Aa|  |B0z |) we obtain     2B0z kz A mA2 2 2 2 ω = kz ck + m − sgn (m) + , μ(ρ0 + ρe ) μ(ρ0 + ρe )

197

|m| ≥ 1.

(7.66)

Curiously, the approximate dispersion relation (7.66) does not appear to have been derived elsewhere, for an incompressible medium. However, it is interesting to note that a relation of precisely the same form, for small twist, has been given; see Goossens, Hollweg and Sakurai (1992; their eqn. (93)), though their analysis is in fact for a β = 0 plasma and not the incompressible case treated here. Furthermore, Ruderman (2007) has developed a thin tube theory for a twisted tube in a β = 0 plasma (see Chapter 11), which gives a relation of the form (7.66); cf. Ruderman (2007, eqn. (40)). Thus, for small twist, the wave speed in a thin tube agrees in the two extremes of an incompressible or β = 0 plasma. Finally, we note that it is apparent from the relation (7.66) that in a weakly twisted tube the dispersive correction to the simple relation ω2 = kz2 c2k vanishes for the case of a kink mode with either m = 1 or m = −1; to a good approximation for small twist, the kink wave propagates with the kink speed, ck , of an untwisted tube. By contrast, the fluting modes (|m| ≥ 2) all suffer dispersive effects. This distinction in the behaviour between modes with |m| = 1 and modes with |m| = 1 appears to be a consequence of the particular assumed equilibrium profile of the twist (Terradas and Goossens 2012), emphasizing the importance of exploring a variety of twist profiles. In this context it is interesting to note that Ruderman (2015) has extended his thin tube theory (Ruderman 2007; see also Chapter 11), for a β = 0 plasma, that allows for a profile of twist that is radially everywhere continuous across the tube boundary (and so may be non-uniform or indeed zero or non-zero anywhere within the tube or its environment).

7.6 The Compressible Twisted Tube: Small Twist We now return to a discussion of the fully compressible case of a twisted flux tube. It transpires that under the assumption of small twist, corresponding to B20φ  B20z , the compressible equations permit a solution in terms of confluent hypergeometric functions. This was first pointed out by Erd´elyi and Fedun (2007, 2010), who also gave a detailed analysis accompanied by numerical results. Consider then the system of equations describing the twisted compressible flux tube. The equilibrium state satisfies equation (7.2). We have in mind the possibility of a medium in which the plasma pressure p0 may be negligible, the β = 0 plasma. Accordingly, we suppose that the equilibrium pressure p0 is a constant and then equilibrium is achieved by the magnetic pressure balancing the magnetic tension:  2  2 B20φ 1 d B0φ + B0z =− . . (7.67) dr 2μ μ r We suppose that inside the tube the twist is such that B0φ = Ar

(7.68)

198

The Twisted Magnetic Flux Tube

for constant A. Then pressure balance means that the longitudinal magnetic component B0z varies with r according to 

B0z

2A2 2 r = B0z (0) 1 − 2 B0z (0)

1/2 .

(7.69)

Accordingly, fB , ωA and cA are not constants but vary with r. However, for 2A2 a2  B20z (0) the variation with r is slight and we can suppose that ωA , fB and cA can be treated as constants. We assume that the density ρ0 is a constant, so the sound speed cs is also a constant. The coefficients C1 , C2 , C3 and D that arise for this equilibrium are given by   2A 4 2 D2 m2 4 2 C2 = ω − C4 + kz , (1 − α02 A2 ), (ω Ar − mfB C4 ), C3 = C1 = 2 μr C4 r (7.70) where 2 )C4 , D = ρ0 (ω2 − ωA

α0 =

(μρ0

2ωA 1/2 ) (ω2

2) − ωA

and (following Erd´elyi and Fedun 2010) we have written 2 C4 = (c2s + c2A )(ω2 − ωc2 ) = (c2s + c2A )ω2 − c2s ωA .

In writing the coefficient C3 in (7.70) we have neglected a term in A4 , assumed negligible for small B20φ /B20z . With ρ0 and p0 assumed to be constants, we see that α0 , D and C3 are constants, and then equation (7.36) for pT reduces to d2 pT 1 dpT + + XpT = 0, r dr dr2

(7.71)

where   C2 C3 − C12 C3 rC1  + rD C3 D2    4 2 2 4ω4 A2 ω m 4A2  4 2 2 2 2 ω = − + k A ) − Ar − mf C + . (1 − α B 4 z 0 C4 μD r2 μ2 r2 D2

X=

(7.72) It may be noted that in the limit of an incompressible (cs → ∞) fluid C4 → ∞ and D → ∞, and the expression for X reduces to the form given by equation (7.53). Now we may write the expression for X in the form X=

1 {X0 + r2 X1 + r4 X2 }, r2

where the constants X0 , X1 and X2 are given as

7.6 The Compressible Twisted Tube: Small Twist

 X0 = −m2 ,

X1 =

199



 ω4 4ω4 A2  − kz2 (1 − α02 A2 ) + 1 + mAα0 , C4 μD

X2 = −

4ω8 A4 . μ2 D2

Set x = λr2

(7.73)

with the constant λ to be chosen at our convenience. Effectively, we are changing from the coordinate r to r2 . Then the first and second order differential operators associated with r change according to d d d = 2λr = 2λ1/2 x1/2 , dr dx dx and equation (7.71) becomes 4λx

d2 d d2 = 4λx + 2λ , 2 2 dx dr dx

d2 pT dpT + 4λ + XpT = 0. dx dx2

(7.74)

(7.75)

Finally, put pT = x−1/2 Q so that dpT = x−1/2 dx



 1 dQ − Q , dx 2x

d2 pT = x−1/2 dx2

(7.76) 

 3 d2 Q 1 dQ − + 2Q . x dx dx2 4x

(7.77)

This choice of variable eliminates the first order term in the second order differential equation for pT . Then, on division by 4λx, equation (7.75) yields   d2 Q X2 X1 1 + X0 + + + Q = 0. 4λx dx2 4λ2 4x2 It remains to choose the constant λ. If we set λ2 = −X2 , so that 4 2 2 4 λ= A ω , λ2 = 2 2 A4 ω8 , μ|D| μ D

(7.78)

then this produces −1/4 for the constant term in the coefficient of Q, with the result that we obtain the standard form of Whittaker’s equation (see, for example, Whittaker and Watson 1969, chap. XVI; Abramowitz and Stegun 1965, chap. 13):

⎫ ⎧ 1 m2 ⎬ 4 − 4 d2 Q ⎨ 1 κ + − + + Q = 0, (7.79) ⎩ 4 ⎭ x dx2 x2 with 1 X1 = κ= 4λ 4λ



 1 ω4 2 − kz (1 − α02 A2 ) + (1 + mAα0 ) sgn (D). C4 2

(7.80)

An equation of the form (7.79) was first obtained and investigated for a twisted flux tube by Erd´elyi and Fedun (2010); the special case m = 0, corresponding to sausage modes, was explored in Erd´elyi and Fedun (2007). Equation (7.79) has solutions

200

The Twisted Magnetic Flux Tube − 2x

e

x

1 m 2+ 2

 M

 1 m + − κ, 1 + m, x , 2 2

− 2x

e

x

1 m 2+ 2

 U

 1 m + − κ, 1 + m, x , 2 2

where M and U are the confluent hypergeometric functions. Requiring pT to be finite at the tube centre (r = 0), we choose the solution   x m 1 m pT = A0 e− 2 x 2 M + − κ, 1 + m, x , r < a, 2 2

(7.81)

where A0 is an arbitrary constant. This applies for m = 0, 1, 2, . . .. The case of negative m requires a separate treatment, not covered here. We can now determine ξr by use of the second equation in (7.32), with the result     1 m C1 D − 2x m 1/2 1/2  ξ r = A0 e x 2 M , r < a. (7.82) 2λ x M − M + M + C3 2 2x C3 Here M stands for M( 12 + m2 − κ, 1 + m, x) and M  is the derivative of the function M( 12 + m 2 2 − κ, 1 + m, x) with respect to x (= λr ). Outside the tube we take a uniform untwisted magnetic field with a uniform plasma density ρe and plasma pressure pe (and corresponding sound and Alfv´en speeds, cse and cAe ). Then the total pressure perturbation pT and radial displacement ξr are (see Chapter 6) pT = Ae Km (me r),

ξr = Ae

De Km  (me r), C3e

r > a,

(7.83)

with m2e =

(kz2 c2se − ω2 )(kz2 c2Ae − ω2 ) kz2 c2se c2Ae − ω2 (c2se + c2Ae )

.

(7.84)

We require me > 0 (so pT and ξr tend to zero as r → ∞). Ae is an arbitrary constant, and De and C3e denote the values of the terms D and C3 in the untwisted environment (r > a) of the twisted tube. Then, matching these solutions by application of the continuity condition (7.41) at r = a produces the result    C3e Km (me a) A2 a M m + 2λaD − λaD + D + C1 = 1. (7.85) μ C3 C3 De me Km  (me a) M a The functions M and M  are evaluated at x = λa2 . The terms D, De , C1 , C3 and C3e are evaluated as r → a (either for r < a or r > a, as appropriate). Equation (7.85) is the dispersion relation describing the modes of a compressible magnetic flux tube (with small uniform twist) embedded in an untwisted uniform magnetic field. It applies for m = 0, 1, 2, . . ., and requires me > 0. The dispersion relation (7.85) may be rewritten in the form   (1 − α02 A2 ) De Km (me a) A2 D + = . (7.86)  C4e me aKm  (me a) μ C4 m(1 − α0 A) + λa2 (sgn (D) − 1) + 2λa2 M M   Here M stands for the confluent hypergeometric function M 12 + m2 − κ, 1 + m, x evaluated at x = λa2 , and M  is its derivative with respect to x, also evaluated at x = λa2 . The form

7.7 The β = 0 Tube: Effect of a Magnetically Twisted Annulus

201

(7.86) of the dispersion relation was first obtained by Erd´elyi and Fedun (2010),8 who also give an extensive numerical investigation of the various modes and discuss how the dispersion relation may be reduced to the special cases of a straight compressible tube or a twisted incompressible tube. 7.7 The β = 0 Tube: Effect of a Magnetically Twisted Annulus The complexity of the fully twisted field equations in a compressible medium has meant that there are only a few solutions available in the literature that guide us as to the expected mode behaviour. Hence our particular interest in previous sections in the incompressible flux tube. Here we examine the compressible medium in the low β limit for a twisted flux tube with a rather specific form of twist, which nonetheless offers some insight into the behaviour of wave modes. In particular, the sausage mode is amenable to treatment. 7.7.1 Equations for β = 0 We begin by writing down the governing system of differential equations and we examine the form of the coefficients in the limit of a β = 0 plasma. Recall that the equilibrium is one in which the equilibrium magnetic field is such that pressure balance is achieved through balancing the magnetic pressure against the tension force in a twisted tube (see equation (7.67)):  2  2 B20φ 1 d B0φ + B0z =− . . (7.87) dr 2μ μ r Now the general system of differential equations describing the linear perturbations about this equilibrium are as discussed earlier: d d D pT = C3 ξr − C1 pT . (7.88) (rξr ) = C1 rξr − C2 rpT , dr dr In the special case of a β = 0 medium, the coefficients of the differential equations follow from the general forms (7.33)–(7.35) on setting the sound speed to zero: ' ( D = ω2 c2A · ρ0 (ω2 − ωA2 ) ,      2 2 ω2 2mfB ω2 m2 2 2 2 2 2 C1 = ω cA · − B0φ , − + kz B C2 = ω cA · , μr 0φ c2A μr2 r2 c2A D

 C3 =

ω2 c2A

·

ρ02 (ω2

2 2 − ωA )

+ 2ρ0 (ω

2

2 B0φ − ωA )

μ

+



B0φ r



4B20φ 

B20φ (ω2 μ2 r2 c2A



2 2 2 − ωA ) − ωA B0z



,

with 8 There are notation differences in the coefficients D and D in Erd´elyi and Fedun (2010) and those coefficients used here. Also, e there is a typographical error in their term involving A2 /μ, which is wrongly given as A2 /μ2 , and it would appear that they

have implicitly assumed D > 0.

202

The Twisted Magnetic Flux Tube 2 ωA =

fB2 , μρ0

c2A =

B20φ + B20z μρ0

fB =

,

m B0φ + kz B0z . r

Now the coefficients C1 , C2 , C3 and D each contain a common factor, ω2 c2A , which we may divide out; then the system of differential equations (7.88) applies with the new coefficients,9 for a β = 0 plasma and for any mode number m,   2B20φ ω2 2mfB ω2 m2 2 2 2 D = ρ0 (ω − ωA ), C1 = − B0φ , C2 = 2 − + kz , μr c2A μr2 r2 cA   4B20φ  2 B0φ  2 2 2 2 2 2 B0φ 2 2 B0φ (ω2 − ωA . + ) − B20z ωA C3 = ρ0 (ω − ωA ) + 2ρ0 (ω − ωA ) 2 2 2 μ r μ r cA (7.89) 7.7.2 The Sausage (m = 0) Mode The above system is still quite complicated, despite the simplification of cs = 0. Only a few solutions are known, such as the case of an untwisted magnetic field (Bφ = 0) treated in Chapter 6. Accordingly, it is of interest to examine a special solution pointed out by Mikhalyaev (2005) and further explored in Mikhalyaev and Khongorova (2012) and Khongorova, Mikhalyaev and Ruderman (2012); see also Mikhalyaev and Solov’ev (2005).These authors were interested in sausage waves in a twisted magnetic flux tube. Accordingly, consider sausage modes (m = 0) for which the coefficients (7.89) reduce to 2B20φ ω2 ω2 , C = − kz2 , 2 μr c2A c2A   4B20φ  2 2  B0φ  2 2 2 2 2 2 B0φ C3 = ρ0 (ω − ωA ) + 2ρ0 (ω − ωA ) + B0φ ω − B20z kz2 c2A , (7.90) 2 2 2 μ r μ r cA D = ρ0 (ω2 − ωA2 ),

C1 =

where now 2 ωA = kz2

B20z μρ0

.

(7.91)

Expressions (7.90) and (7.91) give the coefficients for the case of the sausage (m = 0) mode in a β = 0 plasma. The governing equation for the total pressure perturbation pT is still equation (7.36). There is a further reduction of the coefficients in the extreme when the field is purely azimuthal, for then (with B0z = 0) we obtain D = ρ0 (r)ω2 ,

1 C1 = 2ρ0 (r)ω2 , r

C2 =

ω2 − kz2 . c2A (r)

(7.92)

The expression for C3 remains complicated. However, Mikhalyaev (2005) has pointed out that for the special case of an purely azimuthal magnetic field B0φ in a medium with plasma density ρ0 (r) with the property that 9 There is an abuse of notation here, calling different expressions the same name, but the result should not be confusing.

7.7 The β = 0 Tube: Effect of a Magnetically Twisted Annulus

203

1 1 (7.93) , ρ0 ∝ 2 , r r there is a reduction in the form of the coefficient C3 , which then yields (see also Mikhalyaev and Khongorova 2012; Khongorova, Mikhalyaev and Ruderman 2012) B0φ ∝

C3 = ρ02 (r)ω4 .

(7.94)

There is a related reduction in the coefficients that arise in the differential equation (7.36) for pT . In the special case represented by the equilibrium (7.93) the governing equation for pT reduces to (Mikhalyaev and Khongorova 2012; Khongorova, Mikhalyaev and Ruderman 2012)10 d2 pT 3 dpT + − m2e pT = 0, 2 r dr dr

(7.95)

where m2e = kz2 −

ω2 , v2A0

v2A0 =

B20φ (r) μρ0ann (r)

.

Here vA0 denotes the Alfv´en speed within the twisted field region where the plasma density is ρ0ann (r); through the choice of the equilibrium, vA0 is a constant. Following Khongorova, Mikhalyaev and Ruderman (2012), we have chosen vA0 = cAe (so that the Alfv´en speed vA0 is equal to the Alfv´en speed cAe in the far environment). Thus, we obtain the Bessel differential equation for rpT , with solution 1 (7.96) {α0 I1 (me r) + α1 K1 (me r)} r for constants α0 and α1 ; I1 (me r) and K1 (me r) are modified Bessel functions. This is the form of pT within the twisted region where the equilibrium form (7.93) holds. pT =

7.7.3 Dispersion Relation for a Twisted Annulus It is interesting that Bessel functions arise in the description of both an untwisted magnetic field (Chapter 6) and a purely azimuthal field (albeit of the form (7.93)). Mikhalyaev and Khongorova (2012) and Khongorova, Mikhalyaev and Ruderman (2012) apply such solutions to a flux tube consisting of a longitudinal magnetic field making an inner core together with a purely azimuthal annulus about the inner core, and a longitudinal external magnetic field providing a surrounding environment. The annulus occupies the region R1 < r < R2 ; the limit R1 → R2 removes the annulus and the equilibrium becomes the case of a uniform field contained within a coronal tube of radius R2 . This is the case discussed at length in Chapter 6. The equilibrium state, then, consists of a tube with an azimuthal annulus of field:

10 There is a typographical error in Khongorova, Mikhalyaev and Ruderman (2012, eqn. (20)) where the term involving their λ2 is wrongly given as +λ2 pT instead of −λ2 pT ; it is given correctly in Mikhalyaev and Khongorova (2012, eqn. (11)).

204

The Twisted Magnetic Flux Tube

B0 (r) =

⎧ ⎪ ⎪ ⎨Be ez ,

r > R2 ,

B0φ (r)eφ , ⎪ ⎪ ⎩B e ,

R1 < r < R2 ,

(7.97)

0 ≤ r < R1 ,

0 z

with a plasma density of the form

⎧ ⎪ ⎪ ⎨ρe , ρ0 (r) = ρ0ann (r), ⎪ ⎪ ⎩ρ , 0

r > R2 , R1 < r < R2 ,

(7.98)

0 ≤ r < R1 .

In the annular region between r = R1 and r = R2 the magnetic field B0φ (r) and plasma density ρ0ann (r) vary in r according to (7.93). With this choice of equilibrium we may formulate the dispersion relation for sausage (m = 0) modes; that relation consists of Bessel functions of zero order arising in both the inner core region r < R1 and the outer region r > R2 , where the field is uniform, together with Bessel functions of order unity arising in the annular region R1 < r < R2 , where the field is twisted. In the absence of an annular region, corresponding to the limit R2 → R1 , we recover the simple embedded flux tube discussed in detail in Chapter 6. The dispersion relation may be obtained by imposing  continuity of ξr (or ur ) and pT − B20φ /(μr) ξr at each of the boundaries r = R1 and r = R2 (see condition (7.41)). The result is a dispersion relation of the form (Mikhalyaev and Khongorova 2012; Khongorova, Mikhalyaev and Ruderman 2012; Bahari 2017) XY = WZ, where (in the form given by Khongorova, Mikhalyaev and Ruderman 2012)   J1 (n0 R1 ) R1 ω2 X = J0 (n0 R1 )K0 (me R1 ) − K1 (me R1 ) , K0 (me R1 ) + n0 R1 me v2A0   K1 (me R2 ) R2 ω2 Y = I0 (me R2 )K0 (me R2 ) + I1 (me R2 ) , I0 (me R2 ) − me R2 me v2A0   K1 (me R2 ) R2 ω2 2 Z = K0 (me R2 ) + K1 (me R2 ) , K0 (me R2 ) + me R2 me v2A0   J1 (n0 R1 ) R1 ω2 W = J0 (n0 R1 )I0 (me R1 ) − I1 (me R1 ) , I0 (me R1 ) − n0 R1 me v2A0

(7.99)

(7.100)

with n20 =

ω2 − kz2 . c2A

(7.101)

Equation (7.99) is the sausage mode dispersion relation for a tube with an annular region embedded in a β = 0 medium. Relation (7.99) is too complicated to reveal much information transparently, and generally a numerical investigation is necessary (see Mikhalyaev and Khongorova 2012; Khongorova, Mikhalyaev and Ruderman 2012; Bembitov, Mikhalyaev and Ruderman 2014). However, when we look at long waves in a thin tube (kz R2  1)

7.7 The β = 0 Tube: Effect of a Magnetically Twisted Annulus

205

then the principal sausage mode (with pT having no nodes in r) turns out to satisfy a simple dispersion relation. 7.7.4 Dispersion Relation in the Thin Tube Limit, kz a  1 To see how this comes about, note that (Abramowitz and Stegun 1965, chap. 9) 1 1 1 x, I1 (x) ∼ x, K1 (x) ∼ as x → 0. 2 2 x We employ these relations in (7.100) to see the form of those relations in the limit of small me R2 , me R1 and n0 a, supposing that kz a is small. For the moment, we will retain the Bessel function K0 in the expressions. We have     R22 ω2 1 ω2 1 2 2 2 X∼ K0 (me R1 ) − , Y∼ 1 + me R2 K0 (me R2 ) − 2 , 2 m2e v2A0 m2e R22 2vA0 J0 (x) ∼ 1,

I0 (x) ∼ 1,

1 Z∼ 2 me R22



J1 (x) ∼

ω2 K0 (me R2 ) + m2e v2A0

 W∼

,

1 R21 ω2 + 2 . 2 4vA0

Now the term R22 ω2 /(2v2A0 ) arising in Y is negligible when compared to unity; to see this note that we may introduce the wave speed c = ω/kz and write R22 ω2 2v2A0

=

c2 1 (kz R2 )2 2 . 2 vA0

Thus, assuming that c is finite as kz a → 0, it is evident that R22 ω2 /(2v2A0 ) is small compared to unity, for kz R2  1. Similar comments apply to R21 ω2 /(2v2A0 ). Moreover, we may note that the term involving m2e R22 K02 (me R2 ) arising in Y is negligible when compared with unity. Hence we have   1 ω2 1 X∼ , K0 (me R1 ) − , Y∼ 2 2 2 2 me vA0 me R22 1 Z∼ m2e R22



ω2 K0 (me R2 ) + m2e v2A0

 ,

W∼

1 . 2

In these expressions we have retained the dominant large term behaviour (such as K0 (me R1 ) in X) as well as the order unity terms such as ω2 /(m2e v2A0 ). Thus a term such as ω2 /(m2e v2A0 ) is retained as of order unity, whereas a term such as R22 ω2 /(2v2A0 ) is neglected. Substituting these approximate expressions into the dispersion relation (7.99), following a cancellation of a common factor 1/m2e a2 we obtain ω2 =

 1 2 2 v m K0 (me R1 ) − K0 (me R2 ) , 2 A0 e

which from the definition of m2e yields   K0 (me R1 ) − K0 (me R2 ) . ω2 = kz2 v2A0 2 + K0 (me R1 ) − K0 (me R2 )

(7.102)

206

The Twisted Magnetic Flux Tube 3.5 3

Ai

ω /V k

2.5

4

3

2

5

1

2 0

1.5 1 0.5 0 0

1

2

3

4

5

kb

Figure 7.2 The dimensionless speed ω/(kz cA ) as a function of dimensionless wavenumber kz R1 (≡ kz b) of a sausage wave in a β = 0 tube with an annulus R1 < r < R2 consisting of an equilibrium magnetic field that is entirely twisted (no longitudinal component). The Alfv´en speed within the annulus is taken to be three times the Alfv´en speed within the inner core of the tube (vA0 = cAe = 3cA ) and R2 = 2R1 (so the annulus extends from r = R1 to r = 2R1 ). The numbers by the curves denote the radial harmonic, corresponding to the number of nodes in the total pressure perturbation pT (r). The principal harmonic, denoted by a 0, has no nodes in r; this mode is described by the approximate dispersion relation (7.103) when kz R2  1. [Notational changes from the present text: b ≡ R1 , VAi ≡ cA , k ≡ kz .] (From Khongorova, Mikhalyaev and Ruderman 2012.)

Finally, since for small x we have K0 (x) ∼ − ln x, we have   ln(R2 /R1 ) , kz2 R22  1. ω2 ≈ kz2 v2A0 2 + ln(R2 /R1 )

(7.103)

Relation (7.103) was first obtained by Khongorova, Mikhalyaev and Ruderman (2012), and describes the principal sausage mode in a β = 0 plasma with an annulus of twisted magnetic field in R1 < r < R2 . The relation is of particular interest because it shows explicitly that the principal sausage mode in such a twisted magnetic flux tube has no small wavenumber cutoff. Apart from the principal mode given by (7.103) when kz R2  1, it is necessary to carry out a numerical investigation of the dispersion relation (7.99). This is done in Mikhalyaev and Khongorova (2012) and Khongorova, Mikhalyaev and Ruderman (2012); see also Bembitov, Mikhalyaev and Ruderman (2014). The result of such an investigation is displayed in Figure 7.2, and may be compared with Figure 6.6 in Chapter 6 for an untwisted tube under coronal conditions. The sharp contrast in the behaviour of the principal mode, depending upon whether there is a twisted annulus or not, is of considerable interest. However, it is not at present clear whether this is a feature of all twisted annular tubes or whether it requires specific profiles of twist and equilibrium density structure to exhibit the feature. Recent further studies of this model of loop oscillations have been made in Bahari (2017) and Lim, Nakariakov and Moon (2018).

8 Connection Formulas

8.1 Introduction In this chapter we continue our discussion of twisted magnetic flux tubes begun in Chapter 7 by examining how resistive effects may be incorporated into a description of the way magnetohydrodynamic waves behave in a flux tube. In general terms resistive effects are not important in the behaviour of such waves, simply because under astrophysical conditions viscous and magnetic Reynolds numbers are so large that such effects become negligible. However, there is an important exception, namely when we examine so called resonant effects, associated either with the Alfv´en speed or with the magnetohydrodynamic slow (or cusp) speed. In ideal magnetohydrodynamics there are two singularities associated with these speeds (see Chapter 4), and it is in seeking to describe the behaviour of our waves when these singular points are important that viscous or resistive effects play a role. We show here how resistive effects operate in the Alfv´en and cusp resonant layers. The analysis we present is algebraically intensive but it has two rewards. Firstly, it reveals the local behaviour of the perturbations in the resonant layers, which turn out to be expressible in terms of explicit analytical functions such as F and G (which are themselves conveniently defined in terms of integral representations). Secondly, it permits a means of avoiding the details of the resonant layers and replacing them with jump conditions, leading to the so called connection formulas. Such formulas may be used to derive approximate dispersion relations. Key papers in this area include Sakurai, Goossens and Hollweg (1991), Goossens, Hollweg and Sakurai (1992) and Goossens, Ruderman and Hollweg (1995), which established the basic properties. Useful reviews of this extensive topic are provided by Goossens and Ruderman (1995), Goossens (2008), and Goossens, Erd´elyi and Ruderman (2011). In the main, our own treatment follows theirs though occasionally we have preferred a somewhat different, though related, approach. The theory is developed for a twisted magnetic flux tube in a compressible medium. This is the case described in Chapter 7. The equilibrium magnetic field is   B0 = 0, B0φ (r), B0z (r) ,

(8.1)

consisting of a longitudinal component B0z (r) and an azimuthal component B0φ (r), each of which may vary radially with r; there is no radial component of magnetic field (B0r = 0). The important case of a straight untwisted (B0φ = 0) magnetic flux tube follows as a special 207

208

Connection Formulas

case of the twisted tube. Extremes of a compressible medium, such as the incompressible case or the β = 0 plasma, are readily recovered from the general case. We do not examine the role of steady flows. The treatment when such flows are included is in fact similar to the equilibrium (no flow) case discussed here (see, for example, the original treatment by Goossens, Hollweg and Sakurai (1992) and Erd´elyi (1997) and the review by Goossens, Erd´elyi and Ruderman (2011)), but a discussion of flows is outside the coverage here.

8.2 The Twisted Magnetic Flux Tube Consider then the equilibrium state of a twisted magnetic flux tube in which the magnetic field B0 and plasma pressure p0 are in magnetostatic pressure balance. Expressed in cylindrical coordinates r, φ, z with the z-axis aligned with the central axis of the flux tube, the equilibrium field B0 = 0, B0φ (r), B0z (r) and equilibrium pressure p0 (r) are related by   B20φ 1 B2 d p0 + 0 = − (8.2) dr 2μ μ r where B0 = (B20φ + B20z )1/2 denotes the strength of the equilibrium magnetic field. The equilibrium magnetic field satisfies the solenoidal condition, div B0 = 0.

(8.3)

Equation (8.2) expresses a balance between the total magnetic pressure force and the tension force in the twisted magnetic field. The special case B0φ = 0 corresponds to a magnetic tube without twist. The governing system of linear equations describing perturbations about the equilibrium (8.2) is taken to be (see, for example, Goossens 2003) ρ + div (ρ0 ξ ) = 0, ρ0

p + ξ · grad p0 = c2s (ρ + ξ · grad ρ0 ),

∂ 2ξ 1 1 = −grad pT + (B0 · grad) B + (B · grad) B0 , μ μ ∂t2   ∂B ∂ξ div B = 0. = curl × B0 + η∇ 2 B, ∂t ∂t

(8.4)

We have assumed non-ideal magnetohydrodynamics with the induction equation taken to include diffusivity η. This is the same system of equations discussed in Chapter 7, save there conditions were taken to be ideal (η = 0). The sound speed cs that arises in equations (8.4) is related to the equilibrium pressure p0 and plasma density ρ0 through γ p0 c2s = . ρ0 We consider linear perturbations about the equilibrium state (8.2) of a twisted flux tube, and assume a time and spatial dependence of the form1 f (r)e−iωt+imφ+ikz z

(8.5)

1 For ease of comparison with the published literature on this topic, we again assume here a time dependence of the form e−iωt .

8.2 The Twisted Magnetic Flux Tube

209

for amplitude function f (r). Here ω denotes the frequency of the disturbance (assumed in general as complex), kz is the longitudinal wavenumber, and m is the azimuthal wavenumber (taken to be an integer). As for the ideal case treated in Chapters 6 and 7, the linear form of the equation of continuity provides the plasma density perturbation ρ: ρ = −ρ0  ξr − ρ0 (div ξ ),

(8.6)

and the equation of isentropic energy exchange gives the pressure perturbation p: p = −p0  ξr − ρ0 c2s (div ξ ).

(8.7)

A prime  denotes the radial derivative of an equilibrium quantity (e.g., ρ0  denotes the radial derivative, dρ0 /dr, of the equilibrium density ρ0 (r)). From the momentum equation, the components of the displacement ξ = (ξr , ξφ , ξz ) are seen to satisfy the system of equations   d 1 2 ρ0 ω2 ξr = (pT ) − ifB Br − B0φ Bφ , (8.8) dr μ r   m 1 1 d ifB Bφ + (rB0φ )Br , (8.9) ρ0 ω2 ξφ = i pT − r μ r dr 1 (8.10) ρ0 ω2 ξz = ikz pT − [ifB Bz + B0z  Br ], μ where again the total pressure perturbation pT is pT = p +

1 1 B0 · B = p + (B0φ Bφ + B0z Bz ). μ μ

(8.11)

The factor fB that arises in the momentum equation is m (8.12) fB = B0φ + kz B0z . r Notice that fB depends upon the azimuthal order m and the longitudinal wavenumber kz , as well as the equilibrium magnetic field. The perturbation magnetic field B must satisfy the induction equation, taken here in the resistive form. Generally, resistive terms are considered to be negligible under astrophysical conditions. However, it turns out that resistive effects may play a role in certain thin layers. Accordingly, we retain the diffusive term in the induction equation. However, it is convenient to make an important simplification, namely we treat the equilibrium terms as constants when they are multiplied by the diffusivity η; in terms which do not involve η, we retain the non-uniformity of the equilibrium state. Thus, in the ideal limit of η = 0, the system of equations pertains generally to the equilibrium state described by the force balance (8.2). Furthermore, it is convenient to treat the induction diffusive term η∇ 2 B as providing a term η(∂ 2 B/∂r2 ) only; these simplifications, applying only in thin layers where η is assumed to be nonzero, aid the description of the role of dissipative effects without the attendant increase in

210

Connection Formulas

algebraic complexity that a fuller discussion would require. Both these simplifications were made by Goossens, Ruderman and Hollweg (1995). With these simplifications, the three components of the induction equation yield   η d2 (8.13) Br = ifB ξr , 1−i ω dr2     B0φ  η d2 1−i ξr , (8.14) Bφ = ifB ξφ − B0φ (div ξ ) − r ω dr2 r   η d2 (8.15) Bz = ifB ξz − B0z (div ξ ) − B0z  ξr . 1−i ω dr2 These expressions for the components of the perturbation field B allow determination of the forces in the momentum equation in terms of the displacement ξ . 8.3 The Ideal (η = 0) Case As shown in Chapter 7, the ideal system of equations for a twisted magnetic flux tube leads to a pair of ordinary differential equations for the radial displacement ξr and total pressure perturbation pT (Appert, Gruber and Vaclavik 1974; Sakurai, Goossens and Hollweg 1991): d d D pT = C3 ξr − C1 pT . (rξr ) = C1 rξr − C2 rpT , dr dr The coefficients of the differential equations are D

(8.16)

2 D = ρ0 (c2s + c2A )(ω2 − ωA )(ω2 − ωc2 ), (8.17) 2 2 4 2mfB B0φ ω − (c2s + c2A )(ω2 − ωc2 ) 2 B0φ , C1 = μr μr   2 m 4 2 2 2 2 2 C2 = ω − (cs + cA )(ω − ωc ) + kz , r2    ω4 B40φ B2 B0φ B0φ  2 2 0φ C3 = D ρ0 (ω2 − ωA )+2 . + 4 2 2 − 4ρ0 (c2s + c2A )(ω2 − ωc2 )ωA μ r μ r μr2 (8.18)

Again, a prime  denotes the derivative of an equilibrium quantity with respect to the radial distance r. The frequencies ωA and ωc that arise in the above are referred to as the Alfv´en and cusp (or slow mode) frequencies, and are defined through 2 = ωA

2 f2 1 m B0φ + kz B0z = B , μρ0 r μρ0

ωc2 =

c2s 2 ωA , c2s + c2A

(8.19)

with the Alfv´en speed cA defined in terms of the field strength B0 by c2A =

B20φ B2 B20 = + 0z . μρ0 μρ0 μρ0

(8.20)

8.4 Alfv´en Singularity

211

According to the theory of ordinary differential equations (see, for example, Bender and Orszag 1978), equations (8.16) are singular at locations where the coefficients of the leading order terms vanish; here this corresponds to D = 0. The expression for D shows that it vanishes at those locations r = r0 where either 2 (r0 ), ω2 = ωA

(8.21)

ω2 = ωc2 (r0 ).

(8.22)

or where

These locations are referred to as the Alfv´en singularity and the slow or (cusp) singularity, respectively. In our discussion here (and in the literature describing this problem) we 2 (r) and ω2 (r) are such that near a singular point r assume that the spatial profiles of ωA 0 c the factor D vanishes at r = r0 as a simple zero, proportional locally to the linear factor (r − r0 ) (as opposed, say, to a quadratic point vanishing as (r − r0 )2 ). In the theory of ordinary differential equations, such singularities are termed regular singularities, and as such the method of Frobenius applies to such singular points, giving rise to a solution of the Frobenius form ∞ &

an (r − r0 )n+α ,

n=0

where the power α (which may be non-integral) and the relationship between the coefficients an has to be determined by the detailed behaviour of the differential equations. Additionally, the method may lead to the determination of solutions that contain logarithm terms. A recent discussion using the Frobenius method on equations (8.16) has been given by Soler et al. (2013). We consider the Alfv´en singularity and the cusp singularity separately, beginning with the Alfv´en singularity (see Section 8.8 for the cusp singularity).

8.4 Alfv´en Singularity To begin with we will suppose ω to be real. Consider the behaviour of the differential equations (8.16) in the vicinity of the Alfv´en singularity r = r0 = rA , for which 2 (rA ). ω2 = ωA

(8.23)

2 (r) may be expanded near the singular point r = r in a Suppose that the spatial profile ωA A Taylor series of the form 2 (r) )) dωA 2 2 ) ωA (r) = ωA (rA ) + (r − rA ) · + · · ·. dr )r=rA

Then ω

2

2 − ωA (r)

2 (r) )) dωA ) = −(r − rA ) · +··· dr )r=rA

= sA + · · ·,

212

Connection Formulas

2 (r) at r = r , where s = r − rA and A denotes the negative of the slope of ωA A ) dω2 ) , A = − A )) dr r=rA

which we assume is non-zero (A = 0). Then, to leading order in an expansion in powers of s we have that near s = 0 (where r = rA ) 2 A · s, D = ρ0 c2A ωA 2 arising in D is calculated at and so locally D is linear in s. Each of the terms ρ0 , c2A and ωA 2 2 the point r = rA (where ωA (rA ) = ω ): so now ρ0 = ρ0 (r = rA ) and c2A = c2A (r = rA ). Similarly, we may examine the behaviour of the coefficients C1 , C2 and C3 near the point r = rA . Only the leading behaviour of these coefficients is needed, and this corresponds to the values of C1 , C2 and C3 at r = rA :

C1 = −

2 2B0φ B0z ωA fB gB , μ2 ρ0 rA

C2 = −

2 ωA g2 , μρ0 B

4 C3 = −4ωA

B20φ B20z 2 μ2 rA

,

(8.24)

where fB =

m B0φ + kz B0z , r

gB =

m B0z − kz B0φ . r

Again, in these expressions, terms are evaluated at r = rA . Notice that C2 C3 = C12 . With these expansions, it follows that near the point r = rA the pair of differential equations (8.16) become d gB X(s), ξr = ds ρ0 B20 d 2 B0φ B0z sA pT = fB X(s), ds μρ0 rA B20 sA

(8.25) (8.26)

where X(s) = gB pT (s) −

2 B0φ B0z fB ξr (s). μrA

(8.27)

We have noted that with s = r − rA , the operator d/dr becomes d/ds. It follows immediately from the local equations (8.25) and (8.26) that s

dX =0 ds

(8.28)

and so X = constant = X0 . Thus, gB pT (s) −

2 B0φ B0z fB ξr (s) = X0 μrA

for constant X0 , a result first noted by Sakurai, Goossens and Hollweg (1991).

(8.29)

8.5 Connection Formulas: Alfv´en Singularity

213

Returning to equations (8.25) and (8.26), we see that ξr is determined locally by sA

d gB X0 , ξr = ds ρ0 B20

(8.30)

a first order differential equation in which the terms on the right-hand side are all constants. Suppose first that s > 0 (corresponding to r > rA ). Then equation (8.30) has integral ξr =

gB X0 ln s + E+ , ρ0 B20 A

s > 0.

Here E+ is an arbitrary constant of integration. Now suppose that s < 0. Then we can set s1 = −s and d/ds = −d/ds1 , and then equation (8.30) yields ξr =

gB X0 ln s1 + E− , ρ0 B20 A

s1 > 0,

where E− is also an arbitrary constant of integration. Finally, we can dispense with the variable s1 and write the single result  s > 0, E+ , gB ξr (s) = (8.31) X0 ln |s| + ρ0 B20 A E− , s < 0. This expression is logarithmically singular at s = 0. The behaviour of pT is much the same as for ξr , with the result  s > 0, F+ , 2 B0φ B0z pT (s) = fB X0 ln |s| + 2 μρ0 rA B0 A F− , s < 0,

(8.32)

where F− and F+ are arbitrary constants of integration. These expressions for the local behaviour of ξr and pT near the Alfv´en singularity have been obtained in ideal magnetohydrodynamics. We now turn to examine how the singular behaviour may be resolved by dissipative effects.

8.5 Connection Formulas: Alfv´en Singularity 8.5.1 Resistive Effects The singular behaviour of ξr and pT in the vicinity of r = rA (that is, at s = 0) may be resolved by including some non-ideal effect, such as viscosity or resistivity. Sakurai, Goossens and Hollweg (1991) and Goossens, Ruderman and Hollweg (1995) have explored how such effects modify the governing equations. The general point to note is that under astrophysical conditions away from a singular point non-ideal effects can be considered to be negligible, but in the vicinity of a singular point these effects become important because in such regions large gradients develop. The governing system of resistive linear equations follows from the components (8.8)– (8.10) of the momentum equation combined with the components (8.13)–(8.15) of the

214

Connection Formulas

induction equation, determining the magnetic forces. The three components of the momentum equation may then be written as (see Goossens, Ruderman and Hollweg 1995)       B20φ B0φ  dpT 2 B0φ 2 2 ρ0 ω L{ξr } − ρ0 ωA ξr = L ξr + i fB ξ φ − div u − B0φ dr μ r r r (8.33)

  B0φ im 2 2 ρ0 ω2 L{ξφ } − ρ0 ωA ξφ = L(pT ) + ifB div u − ξr r μ r B 0z 2 ξz = ikz L(pT ) + ifB ρ0 ω2 L{ξz } − ρ0 ωA div u, μ

(8.34) (8.35)

where L is the operator η d2 . ω dr2 The system of equations (8.33)–(8.35) may be manipulated to yield the pair of ordinary differential equations   d D (8.36) (rξr ) = C1 rξr − C2 rpT , dr   d (8.37) (pT ) = C3 ξr − C1 pT , D dr L=1−i

where C1 , C2 and C3 are as defined in equation (8.18): 2 2 4 2mfB B0φ ω − (c2s + c2A )(ω2 − ωc2 ) 2 B0φ , μr μr   2 m 4 2 2 2 2 2 C2 = ω − (cs + cA )(ω − ωc ) + kz , r2    ω4 B40φ B2 B0φ B0φ  2 2 0φ C3 = D ρ0 (ω2 − ωA )+2 . + 4 2 2 − 4ρ0 (c2s + c2A )(ω2 − ωc2 )ωA μ r μ r μr2 (8.38)

C1 =

The operator D arising in equations (8.36) and (8.37) is given as   2 D = ρ0 (c2s + c2A )(ω2 − ωc2 ) ω2 L − ωA , that is,

 D=

ρ0 (c2s

+ c2A )(ω2

− ωc2 )

ω

2

2 − ωA

d2 − iωη 2 dr

 .

(8.39)

In the absence of diffusivity, when η = 0, D reduces to the multiplicative factor D defined in equation (8.17), and the system (8.36) and (8.37) reduces to the first order differential equations (8.16) discussed earlier. For non-zero diffusivity (η = 0), the system (8.36) and (8.37) is a pair of third order differential equations. Our interest here is in the way non-zero diffusivity (η = 0) modifies the behaviour of the system in a small region round the point r = rA , defined to be the location where

8.5 Connection Formulas: Alfv´en Singularity

215

an oscillation is in resonance with the Alfv´en frequency ωA . In the treatment of Sakurai, Goossens and Hollweg (1991) and Goossens, Ruderman and Hollweg (1995) the frequency ω is assumed to be real (and positive), but here we follow Tirry and Goossens (1996) and allow the frequency ω to be complex. We write ω = ωR + iωI ,

(8.40)

where ωR and ωI are the real and imaginary parts of the frequency ω. The real part ωR is taken to be in resonance with the Alfv´en frequency ωA at a point r = rA located within the transition layer, so that ωR = ωA (rA ).

(8.41)

Notice first that the local behaviour of the Alfv´en frequency near r = rA is given by its Taylor expansion about r = rA : 2 2 (r) = ωA (rA ) − sA + · · ·, ωA

(8.42)

where, as before, s = r − rA and A = −

2 )) dωA ) . dr )r=rA

(8.43)

2 is locally of the form Accordingly, the operator ω2 L − ωA 2 ω 2 L − ωA = sA + 2iωA (rA )ωI − iηωA (rA )

d2 , ds2

where we have set ωR = ωA (rA ) and neglected terms involving ηωI and ωI2 . This requires that |ωI |  |ωR |, an assumption that needs to be checked a posteriori. We carry out such a check in Section 8.10. We may now expand the operator D, obtaining   2 d 2 sA + 2iωA (rA )ωI − iωA η 2 , (8.44) D = ρ0 c2A ωA ds with A defined in equation (8.43) and the coefficients in expression (8.44) determined at r = rA . Similarly, we may expand the coefficients C1 , C2 and C3 about the point r = rA . Only the lowest order terms are required, evaluating C1 , C2 and C3 at r = rA : 2 2B0φ B0z ωA fB gB , C1 = − μ2 ρ0 rA

ω2 C2 = − A g2B , μρ0

C3 =

4 −4ωA

B20φ B20z 2 μ2 rA

,

(8.45)

with fB and gB defined as earlier (and to be evaluated at r = rA ): fB =

m B0φ + kz B0z , r

gB =

m B0z − kz B0φ . r

(8.46)

216

Connection Formulas

Consequently, the local forms of the differential equations (8.36) and (8.37) are    d2 gB dξr sA + 2iωA ωI − iωA η 2 X(s), (8.47) = ds ds ρ0 B20    d2 2 B0φ B0z dpT fB X(s). (8.48) = sA + 2iωA ωI − iωA η 2 ds μρ0 rA B20 ds In place of (8.28), we now find that X(s) satisfies    d2 dX sA + 2iωA ωI − iωA η 2 = 0. ds ds

(8.49)

For non-zero diffusivity (η = 0), equations (8.47)–(8.49) are no longer singular at s = 0. Equations equivalent to (8.47)–(8.49) but for the case ωI = 0 were first given by Sakurai, Goossens and Hollweg (1991) and Goossens, Ruderman and Hollweg (1995). The case ωI = 0 was given by Tirry and Goossens (1996). Now we neglect the effects of non-zero η everywhere except where it produces terms that are comparable to the contribution sA . In the above equations this means we can set η = 0 everywhere except in a layer near s = 0 where ) ) ) d2 )) ) |sA | is comparable to )ωA η 2 ) . ) ds ) This suggests a spatial scale δA such that

) ) )ω η) ) A ) |δA A | ∼ ) 2 ) ; ) δA )

accordingly, we may define a diffusive spatial scale δA through   ωA η 1/3 δA = |A |

(8.50)

where we assume for convenience that ωA > 0. Within this scale (that is, for |s| < δA ) diffusive effects are important, whereas outside this region (in |s| > δA ) dissipative effects can be taken as negligible and conditions are effectively ideal (that is, we may set η = 0). Under astrophysical conditions we can expect the diffusive spatial scale δA to be small, and consequently diffusive effects are confined to a small region |s| < δA near rA . It is convenient to scale distance s against the diffusive spatial scale δA , setting s τ= ; δA τ provides a dimensionless measure of distance on a scale of δA , with τ = ±1 corresponding to s = ±δA and |τ | → +∞ corresponding to |s|  δA . Then d 1 d = , ds δA dτ

1 d2 d2 = . 2 dτ 2 ds2 δA

Thus, in the region |s| < δA the system of equations (8.47) and (8.48) may be written in the form (Goossens, Ruderman and Hollweg 1995; Goossens and Ruderman 1995; Tirry and Goossens 1996)

8.5 Connection Formulas: Alfv´en Singularity

217





dξr 1 d2 gB X(τ ), + i sgn (A ) τ +  =i dτ dτ 2 ρ0 B20 |A |   2 B0φ B0z dpT d fB X(τ ), + i sgn ( ) τ +  = 2i A 2 dτ dτ ρ0 B20 μrA |A |

(8.51) (8.52)

where the sgn function is simply

⎧ ⎪ ⎪ ⎨1, sgn (A ) = 0, ⎪ ⎪ ⎩−1,

A > 0, A = 0,

(8.53)

A < 0.

The parameter , non-zero when ω is not real, is defined by (see Tirry and Goossens 1996)2 =−

2ωA ωI . δA |A |

(8.54)

Each of the coefficients arising on the right-hand side of (8.51) and (8.52) is calculated at r = rA , that is at s = 0, and so they are constants. Recalling that X(s) = gB pT (s) −

2 B0φ B0z fB ξr (s), μrA

a linear combination of equations (8.51) and (8.52) produces  2   d dX + i sgn (A ) τ +  = 0. 2 dτ dτ

(8.55)

(8.56)

8.5.2 Differential Equations: Fourier Solution We are interested in the solution of equations (8.51), (8.52) and (8.56) under the requirement that dX dξr dpT , and dτ dτ dτ are bounded in the limits τ → −∞ and τ → +∞. It is perhaps a surprising result that the behaviour of our system within the resistive layer is governed by homogeneous or inhomogeneous ordinary differential equations of very specific forms. This is an attractive and pleasing feature of our analysis: the behaviour within the resistive layer is governed entirely by equations of the form  2  d + i sgn ( ) τ +  (τ ) = α. (8.57) A dτ 2 The form of the constant α on the right-hand side of equation (8.57) varies with the dependent variable  in the physical system that we choose to examine, be it the τ -derivative 2 In fact Tirry and Goossens (1996) take their  to be of opposite sign to the form we have used here; since it turns out that

ωI ≤ 0 we have chosen a definition of  that gives rise to a positive parameter. Goossens, Erd´elyi and Ruderman (2011) have also made such a notational change.

218

Connection Formulas

of the radial displacement ξr , or the total pressure perturbation pT , or the quantity X. The constant α may be zero or non-zero (in fact, imaginary). Notice first that the term sgn (A ) is not a significant complication. Indeed, if we introduce the new variable τˆ = sgn (A ) τ then we have d d = sgn (A ) , dτ dτˆ

2 d2 d2 2 d = [sgn ( )] = . A dτ 2 dτˆ 2 dτˆ 2

Hence equation (8.57) becomes  d2   + i τˆ +   = α. 2 dτˆ

(8.58)

This is formally the same equation as (8.57) for the case A > 0. Hence, provided the boundary conditions are not such as to modify the problem, the solution of (8.58) follows from the solution of (8.57) for A > 0, save only that τˆ replaces τ . This observation affords us some simplification in our treatment of the problem. Equations of the form (8.57) but with  = 0 were first solved by Sakurai, Goossens and Hollweg (1991) in terms of Airy functions, though the behaviour of the solution is not transparent. Goossens, Ruderman and Hollweg (1995) considered in effect the case  = 0 and obtained solutions by seeking contour integral representations of possible solutions; this led to a relatively simple and convenient form of the solution. Finally, Tirry and Goossens (1996) considered the case  = 0 and employed Fourier transforms to seek solutions, obtaining the same integral representation as Goossens, Ruderman and Hollweg (1995). Tirry and Goossens’s solution by a Fourier approach was only presented briefly, in an appendix, in the original publication, but it is of wider interest warranting our exploration here. Accordingly, consider equation (8.57) for the case A > 0: the governing differential equation is  d2 (τ )  + i τ +  (τ ) = α. 2 dτ Introduce Fourier representations  ∞ ˆ (τ )e−ikτ dτ , (k) = F () = −∞

1 (τ ) = 2π

(8.59)



∞

−∞

ikτ ˆ dk. (k)e

(8.60)

*∞ Multiply equation (8.59) by the factor e−ikτ and form the integral −∞ over the variable τ . We need the Fourier transform of the second derivative term as well as the transform of a term τ :     d2  d d ˆ ˆ = ik(k), F . F = (ik)2 (k), F(τ ) = i dτ dk dτ 2

8.5 Connection Formulas: Alfv´en Singularity

219

We also require the Fourier representation of the Dirac delta function (see, for example, Lighthill 1958):  ∞  ∞ 1 1 eikτ dk = e−ikτ dk. δ(τ ) = 2π −∞ 2π −∞ Accordingly, equation (8.59) yields ˆ   d(k) ˆ + k2 −  (k) = −2π αδ(k). dk

(8.61)

ˆ This is a first order linear ordinary differential equation for (k), with an integrating factor      k k3 − k . k2 −  dk = exp exp 3 0 Now, from (8.61) we obtain      d k3 k3 ˆ (k) exp − k = −2π αδ(k) exp − k . dk 3 3 We integrate this equation from k = −∞ to k with the result      k u3 k3 ˆ δ(u) exp − k = A − 2π α − u du, (k) exp 3 3 −∞

(8.62)

where A is an arbitrary constant of integration. The analysis of (8.62) now depends upon the sign of k. If k < 0, then the integral on the right-hand side of the above is zero, because the delta function is everywhere zero within the domain of integration −∞ to k. Thus,   3 k ˆ k < 0. (k) = A exp − + k , 3 This function is exponentially unbounded as k → −∞ unless we choose the constant ˆ A = 0. Thus, (k) = 0 when k < 0. On the other hand, if k > 0 then the integral on the right-hand side of (8.62) is now unity, since the delta function δ(k) now contributes to the integral. Hence  3 k > 0, −2π α exp(− k3 + k), ˆ (k) = 0, k < 0. Finally, we have 1 2π



∞

ikτ ˆ dk (k)e    ∞ k3 = −α exp − + k eikτ dk. 3 0

(τ ) =

−∞

(8.63)

Equation (8.63) provides the solution of equation (8.57) in the case A > 0, subject to the boundary conditions that  is bounded as τ → ±∞. However, as remarked earlier, we can now immediately extend our solution to allow for A being positive or negative,

220

Connection Formulas

simply by replacing τ by τ sgn (A ). Accordingly, replacing the dummy variable k in the integral by a dummy variable u, we have    ∞ u3 exp − + u eiuτ sgn (A ) du (τ ) = −α 3 0  ∞ 1 3 = −α eiuτ sgn (A )− 3 u +u du. (8.64) 0

An integral of this form, though for  = 0, was first obtained by Goossens, Ruderman and Hollweg (1995), who employed a contour integral approach to obtain their solution. The case with  = 0 was first obtained by Tirry and Goossens (1996), using a Fourier approach.3 We may write the solution for (τ ) as (τ ) = −αF (τ ) where the function F (τ ) is defined by  ∞ 3 e[iuτ sgn (A )+u−u /3] du. F (τ ) =

(8.65)

(8.66)

0

Equations (8.65) and (8.66) provide a solution of the differential equation (8.59) that is bounded as |τ | → ∞. It follows immediately from the solution (8.65) that if the constant α on the right-hand side of our differential equation is zero, then the only bounded solution is the trivial one (τ ) = 0. 8.5.3 The General Case  = 0 We begin by examining the case  = 0 and in fact show that this case can be related to the situation that arises when  = 0. Consider the nature of equations (8.51), (8.52) and (8.56) with  = 0. Their behaviour is encapsulated in the functions  and F, determined by the forms   2 d + i sgn ( ) τ (τ ) = 0, (8.67) A dτ 2  2  d + i sgn ( ) τ F(τ ) = −1. (8.68) A dτ 2 Consider the differential equation (8.67). This is a form of Airy’s differential equation, which may be written in the general form (Abramowitz and Stegun 1965) d2 (x) − x(x) = 0, (8.69) dx2 with x treated as a complex variable. The general solution of Airy’s equation (8.69) is a linear combination of the Airy functions Ai(x) and Bi(x). 3 There is an obvious typographical error in the form of the solution (their (A4)) given in Tirry and Goossens (1996), where sin

has been written instead of sgn .

8.5 Connection Formulas: Alfv´en Singularity

221

To see how this applies here, we change the variable τ arising in equation (8.67) so as to obtain Airy’s equation. Set x = i sgn (A ) τ . Then d2 d2 d2 = [i sgn (A )]2 2 = − 2 , 2 dτ dx dx

d d = i sgn (A ) , dτ dx

and equation (8.67) becomes (8.69). Accordingly, in terms of the variable τ we may write the solution of (8.67) as (τ ) = A1 Ai (i sgn (A ) τ ) + B1 Bi (i sgn (A ) τ ), where A1 and B1 are arbitrary constants. Now we are interested in solutions of (8.67) that are bounded for τ → −∞ and τ → +∞, or at least do not grow exponentially large with τ (see Goossens and Ruderman 1995). As we have seen earlier, it turns out that there are no such solutions other than the trivial one, for which A1 = B1 = 0 and (τ ) ≡ 0. Consider the complex function F (τ ):  ∞ 3 F (τ ) = e[iuτ sgn (A )+u−u /3] du. (8.70) 0

It is perhaps of interest to verify the solution we have obtained by use of Fourier analysis. To do this, we form the derivatives of F (assuming differentiation under the integral sign is permissible), obtaining  ∞ dF 3 eiuτ sgn (A ) · ue(u−u /3) du, = i sgn (A ) dτ 0  ∞ d2 F 3 =− eiuτ sgn (A ) · u2 e(u−u /3) du. 2 dτ 0 It then follows that d2 F + i sgn (A )τ F + F = Z, dτ 2 where



∞

Z=

eiuτ sgn (A ) · e(u−u

3 /3)

[iτ sgn (A ) +  − u2 ]du

0 ∞

d [iuτ sgn (A )+u−u3 /3] }du {e du )∞ ) 3 = {eiuτ sgn (A ) · e(u−u /3) })) = −1. =

0

0

Hence, F satisfies the inhomogeneous ordinary differential equation  2  d + i sgn ( ) τ +  F (τ ) = −1. A dτ 2

(8.71)

222

Connection Formulas

This is equation (8.57) with  = F and with the constant on the right-hand side being α = −1. Using elementary properties of the even function cos and the odd function sin we can rewrite the expression (8.70) for the function F (τ ) as F (τ ) = FC (τ ) + i sgn (A ) FS (τ ) where



FC (τ ) =

∞

cos(uτ ) · e(u−u

3 /3)

 du,

FS (τ ) =

0

∞

sin(uτ ) · e(u−u

(8.72)

3 /3)

du. (8.73)

0

The integrals FC (τ ) and FS (τ ) are real; the integral FC (τ ) involving the cosine function is an even function of τ , and the integral FS (τ ) involving the sine function is an odd function of τ . Now the Riemann–Lebesgue lemma (see, for example, Bender and Orszag 1978) applies to integrals of this form, and states that  b  b f (x) cos(λx)dx or f (x) sin(λx)dx → 0 as λ → ±∞, a

a

*b

provided only that the integral a |f (x)|dx exists. It follows immediately that FC (τ ) → 0 and FS (τ ) → 0 as τ → ±∞. Thus, the function F (τ ) is zero at infinity: F (τ ) → 0 as

τ → ±∞.

(8.74)

Thus, the function F (τ ) defined in (8.70) provides a solution of equation (8.71) that is zero as |τ | → +∞. Not only is the function F (τ ) zero at infinity (τ → ±∞), but it is bounded for all τ . In fact, it follows immediately from the defining integral of F that  ∞ 3 e(u−u /3) du. (8.75) |F (τ )| ≤ F (τ = 0) = 0

Thus, F (0) provides an upper bound (which depends upon the value of ). A numerical evaluation of the integral F (0) gives F (0) = 1.2879 for  = 0, F (0) = 1.9150 for  = 1/2 and F (0) = 3.0543 for  = 1. Associated with the function F (τ ) is the function G (τ ), defined by   ∞  [iuτ sgn (A )+u] −1 e 3 G (τ ) = (8.76) · e−u /3 du, u 0 with the two functions F (τ ) and G (τ ) being related by the result dG (τ ) = i sgn (A )F (τ ). dτ It follows from this result that



τ

i sgn (A ) 0

F (u)du = G (τ ) − G (0).

(8.77)

8.5 Connection Formulas: Alfv´en Singularity

223

8.5.4 The Special Case  = 0: The Functions F and G The case  = 0 corresponds to ω being real (ωI = 0). This corresponds to the problem first investigated by Sakurai, Goossens and Hollweg (1991) and Goossens, Ruderman and Hollweg (1995), who specified that they were interested in the driven problem in which a specific frequency is maintained by a driver. It is of interest to ask what happens in this situation. Denote by F(τ ) the value of the function F (τ ) when  = 0 and by G(τ ) the value of the function G (τ ) when  = 0. Thus, the functions F(τ ) and G(τ ) are defined by  ∞ 3 F(τ ) = eiuτ sgn (A ) · e−u /3 du (8.78) 0

and 

∞

G(τ ) =



0

 ei uτ sgn (A ) − 1 −u3 /3 du, e u

(8.79)

with the two functions being related through dG = i sgn (A )F(τ ). dτ

(8.80)

Functions of the form of F and G seem first to have been discussed in the thesis by Boris (1968), but their application to solar physics – specifically to resonant layers – began with Ruderman, Tirry and Goossens (1995) for an incompressible medium and Goossens, Ruderman and Hollweg (1995) for a compressible medium. Consider the difference between G (τ ) and G(τ ):   ∞  u e −1 3 · e−u /3 cos(uτ )du G (τ ) − G(τ ) = u 0   ∞  u e −1 3 + i sgn (A ) · e−u /3 sin(uτ )du. u 0 The Riemann–Lebesque lemma applies to these integrals, since the integral   ∞  u e −1 3 · e−u /3 du u 0 exists (the integrand is well behaved both at u = 0 and at infinity) and so it follows immediately that lim {G (τ ) − G(τ )} = 0,

τ →−∞

lim {G (τ ) − G(τ )} = 0.

τ →+∞

Thus, lim G (τ ) = lim G(τ ).

τ →±∞

τ →±∞

(8.81)

Since much of our interest here focusses on the behaviour of G (τ ) as τ → ±∞, we can transfer that interest to an examination of the simpler function G(τ ) in place of G (τ ), and similarly F(τ ) instead of F (τ ). Accordingly, we concentrate on the case  = 0.

224

Connection Formulas

8.5.5 The Function F(τ ) Consider the complex function F(τ ), defined in equation (8.78). We have already seen that F (τ ) satisfies the differential equation (8.71). Exactly the same argument applies to F(τ ), save only that we set  = 0. Thus, F satisfies the inhomogeneous ordinary differential equation (8.68). Let us take a closer look at the nature of the function F. Application of Euler’s relation (eix = cos x + i sin x) for the exponential function, and exploiting the fact that cos is an even function whereas sin is an odd function, allows us to rewrite the defining integral expression for F in a form that reveals its real and imaginary parts, namely  ∞  ∞ 3 3 e−u /3 cos(τ u)du + i sgn (A ) e−u /3 sin(τ u)du F(τ ) = 0

0

= FC + i sgn (A )FS ,

(8.82)

where the real integrals FC and FS are given by  ∞  3 e−u /3 cos(τ u)du, FS (τ ) = FC (τ ) = 0

∞

e−u

3 /3

sin(τ u)du.

(8.83)

0

Thus, Real (F(τ )) = FC (τ ) and Imag (F(τ )) = sgn (A )FS (τ ). Application of the Riemann–Lebesgue lemma shows that FC → 0 and FS → 0 as τ → ±∞. Thus, the function F(τ ) is zero at infinity: F(τ ) → 0 as

τ → ±∞.

Thus, the function F(τ ) defined in (8.78) provides a solution of the differential equation (8.68) that is zero as |τ | → +∞. Furthermore, it may be noted that  ∞ 3 |ei uτ sgn (A ) · e−u /3 |du |F(τ )| ≤ 0 ∞ 3 e−u /3 du = F(0). = 0

Now the exponential integral is readily evaluated to give    ∞ 1 1 3 e−u /3 du = 2/3  = 1.2879, |F(τ )| ≤ 3 3 0 where (x) is the usual Gamma function (Abramowitz and Stegun 1965). Thus, the function F, which vanishes as |τ | → +∞, is everywhere bounded by its value F(0) at τ = 0. Note that F(0) = 1.2879. It is of interest to examine in a little more detail the behaviour of FC , FS (and so F) as τ → ±∞. If we carry out an integration by parts on the integral FC we obtain )  )∞ 1 ∞ 2 −u3 /3 1 3 FC (τ ) = e−u /3 sin(τ u) )) + u e sin(τ u)du τ τ 0 0  1 ∞ 2 −u3 /3 = u e sin(τ u)du. τ 0

8.5 Connection Formulas: Alfv´en Singularity

Similarly,

225

)  )∞ 1 ∞ 2 −u3 /3 1  −u3 /3 cos(τ u) )) − u e cos(τ u)du FS (τ ) = − e τ τ 0 0  1 1 ∞ 2 −u3 /3 = − u e cos(τ u)du. τ τ 0

Thus, since by the Riemann–Lebesgue lemma the integrals that remain on the right-hand sides of these expressions give zero in the limit τ → ±∞, it follows that τ FC (τ ) → 0 and

FS (τ ) ∼

1 τ

as

τ → ±∞.

(8.84)

8.5.6 The Function G(τ ) Consider now the complex function G(τ ), defined in equation (8.79). From the defining expression (8.79), noting the even and odd properties of the cosine and sine functions, we may write G(τ ) = GC (τ ) + i sgn (A ) GS (τ ) where

∞  1 − cos(uτ ) 



GC (τ ) = − 0

u

−u3 /3

e

 du,

GS (τ ) = 0

(8.85) ∞  sin(uτ ) 

u

e−u

3 /3

du.

(8.86) Notice that GC (τ ) is an even function of τ and that GC (τ ) ≤ 0; GS (τ ) is an odd function of τ . We cannot directly apply the Riemann–Lebesgue lemma to the function GS (τ ) because *∞ 3 the integral 0 u1 e−u /3 du diverges (the integrand being unbounded at u = 0). However, by elementary rearrangement we have   ∞  ∞ 3 1 1 − e−u /3 sin(uτ )du − sin(uτ )du. GS (τ ) = u u 0 0 Now the first integral is connected with the Si function (see Abramowitz and Stegun 1965, p. 232) and can be evaluated directly,  ∞ 1 π sin(uτ )du = sgn (τ ) . u 2 0 The second integral is in a form suitable for application of the Riemann–Lebesgue lemma. Moreover, we can carry out an integration by parts giving     3 π 1 ∞ 1 − e−u /3 1 ∞ −u3 /3 ue cos(uτ )du + GS (τ ) = sgn (τ ) − cos(uτ )du. 2 τ 0 τ 0 u2 Now these integrals are subject to the Riemann–Lebesgue lemma and so tend to zero as τ → ±∞. Hence,   π 1 as τ → ±∞ GS (τ ) = sgn (τ ) + o 2 τ

226

Connection Formulas

Figure 8.1 The real and imaginary parts of the functions F(τ ) and G(τ ), plotted for the case A > 0 (when sgn (A ) = 1). Note that generally FC = Real (F), FS = sgn (A ) Imag (F), GC = Real (G) and GS = sgn (A ) Imag (G). For A > 0 this gives the functions FC , FS , GC and GS . (Based on Goossens, Ruderman and Hollweg 1995; see also Goossens, Erd´elyi and Ruderman 2011.)

and so GS (τ ) is close to ± π2 for τ → ±∞. Here o( τ1 ) indicates an undetermined function which has the property that it tends to zero faster than 1/τ as τ → ±∞. Hence, we have π π lim GS (τ ) = . (8.87) lim GS (τ ) = − , τ →−∞ τ →+∞ 2 2 Finally, we can examine the leading asymptotic behaviour of the function GC (τ ). Goossens, Ruderman and Hollweg (1995) have shown that 2 1 as τ → ±∞, (8.88) GC (τ ) ∼ − ln |τ | − γE − ln 3 3 3 where γE (= 0.5772) is Euler’s constant. Thus, the function GC (τ ) is unbounded at infinity, but only weakly. We end this discussion of the functions F and G by displaying their real and imaginary parts in Figure 8.1, sketches conveniently given in Goossens, Ruderman and Hollweg (1995); see also Goossens, Erd´elyi and Ruderman (2011). It is evident that the figure displays all the features we have stressed in the above analysis.

8.5.7 Physical Variables We have seen above that the solution of the differential equation (8.67) for the variable (τ ) is the trivial one,  = 0. Applied to equation (8.56) with  = 0, it follows that dX = 0, ds where X0 is a constant of integration.

X = X0 ,

(8.89)

8.5 Connection Formulas: Alfv´en Singularity

227

Returning to F as a solution of equation (8.68), applied to equation (8.51) with  = 0 for dξ/dr, it follows immediately that i gB dξr X0 F(τ ). =− dτ |A | ρ0 B20 Thus, i gB X0 ξr (τ ) = C − |A | ρ0 B20



τ

F(u)du, 0

where C is a constant of integration. As before, the terms ρ0 , B0 and gB are calculated at the point r = rA . Now the functions G(τ ) and F(τ ) are related through equation (8.80). Hence, ξr (τ ) = C −

gB X0 G(τ ). ρ0 B20 A

(8.90)

The solution of equation (8.52) for pT follows immediately (since the defining differential equations for dξr /dτ and dpT /dτ differ only by a multiplicative constant): pT (τ ) = P0 −

2B0φ B0z X0 fB G(τ ) μρ0 rA B20 A

(8.91)

where P0 is a constant of integration. The three constants of integration, X0 , C and P0 , are related through the requirement that X is a constant; thus, X0 = gB P0 −

2B0φ B0z CfB . μrA

Of particular interest is the component of displacement given by ξ⊥ =

B0z ξφ − B0φ ξz . B0

(8.92)

This arises in the expression for ξ × B0 : ξ × B0 = B0z ξφ er − B0φ ξz er − B0z ξr eφ + B0φ ξr ez = B0 ξ⊥ er − B0z ξr eφ + B0φ ξr ez . Returning to the perturbation equations (8.34) and (8.35) with diffusivity, we can subtract the equation for ξz from the equation for ξφ and expand terms about the resonance point r = rA . The result is the local equation (Sakurai, Goossens and Hollweg 1991; Goossens, Ruderman and Hollweg 1995)     B0φ B0z d2 i ρ0 sA + 2iωA ωI − iωA η 2 ξ⊥ = ξr . gB pT − 2fB (8.93) B0 μrA ds Consequently, 

 d2 1 + i sgn ( ) τ +  ξ⊥ = − X(τ ). A 2 ρ0 B0 δA |A | dτ

(8.94)

228

Connection Formulas

Now, as we have seen earlier X(τ ) = constant = X0 , and so the right-hand side of (8.94) is a constant; accordingly, comparing with equation (8.71) we deduce that the solution of (8.94) is X0 F (τ ). ρ0 B0 δA |A |

ξ⊥ =

(8.95)

Thus, apart from a multiplicative constant the variable ξ⊥ is the function F (τ ),  = 0, or F(τ ) for  = 0. For the case  = 0 equation (8.95) is simply ξ⊥ =

X0 F(τ ). ρ0 B0 δA |A |

(8.96)

8.6 Alfv´enic Connection Formulas: The Twisted Magnetic Tube The solutions (8.90) and (8.91) give the behaviour of ξr and pT in the region of the Alfv´en singularity r = rA where ωR = ωA . In the scaled variable τ , this behaviour is relatively simple, but since the spatial scale δA is small under astrophysical conditions, this behaviour is confined to a small region round the point r = rA . Of interest now is to ask what are the changes in the variables ξr and pT as we pass from one region where conditions are closely ideal to another region where they are also ideal. To this end, we determine the change (if any) that occurs in ξr and pT as we pass through the point r = rA . For any function f (τ ), define [[f (τ )]] = lim f (τ ) − lim f (τ ). τ →+∞

τ →−∞

(8.97)

The quantity [[f (τ )]] is referred to as the jump in the function f as we traverse the Alfv´en point rA . If [[f (τ )]] = 0, then the function does not change in value as we traverse from one side of the region around τ = 0 to the other; on the other hand, if [[f (τ )]] = 0, then the function f suffers a jump in value. Applied to the function F, we have seen that it is zero at both τ → −∞ and τ → +∞. Accordingly, [[F(τ )]] = 0.

(8.98)

Thus, there is no jump in the function F(τ ) as we move from τ → −∞ to τ → +∞. However, the function G exhibits a different behaviour. We have [[G(τ )]] = [[GC (τ ) + i sgn (A )GS (τ )]] = [[GC (τ )]] + i sgn (A )[[GS (τ )]]. Now GC (τ ) is an even function of τ so its jump is zero: [[GC (τ )]] = 0. However, as follows from (8.87), there is a jump in GS (τ ) of π : [[GS (τ )]] = π .

8.6 Alfv´enic Connection Formulas: The Twisted Magnetic Tube

229

Hence the jump in the function G(τ ) is (Goossens, Ruderman and Hollweg 1995)4 [[G(τ )]] = iπ sgn (A ).

(8.99)

In terms of the physical variables, it then follows immediately from equations (8.90) and (8.91) that the jumps in ξr and pT are [[ξr ]] = −iπ [[[pT ]] = −iπ

1 gB X0 , |A | ρ0 B20

(8.100)

1 2B0φ B0z fB X0 , |A | μρ0 rA B20

(8.101)

where we recall that A = −

2 )) dωA ) . dr )r=rA

Each term on the right-hand side of equations (8.100) and (8.101) is calculated at the location r = rA . Equations (8.100) and (8.101) are the jump relations for a twisted tube. They have been derived here under the assumption that ωA > 0, made in Section 8.5 in connection with the definition of the transition layer scale δA (which we require positive). These relations are commonly referred to as connection formulas, appropriate since they enable us to link ideal solutions in one location with ideal solutions in another location, The transition region between the two ideal regions may be treated in various ways, the application of the functions F and G being particularly attractive. The jump relations for a twisted tube were first obtained by Sakurai, Goossens and Hollweg (1991) and further explored by Goossens, Hollweg and Sakurai (1992). Goossens, Ruderman and Hollweg (1995) exploited the functions F and G and their properties. The initial investigations of jump conditions were made explicitly assuming that ω is real, corresponding to the so-called driven problem (in which a driver of specified frequency maintains the ensuing motion). A derivation of the jump conditions that allows for complex ω was presented in Tirry and Goossens (1996), and so applies not only to a driven problem but also to an eigenvalue problem, when a complex frequency defines an oscillation and a temporal evolution (a decay); such a motion is sometimes referred to as a quasi-mode (see, for example, Goossens 1991; 2008). Broad reviews of the subject of connection formulas, including the role of basic state flows (not treated in our discussion), are given in Goossens and Ruderman (1995), Goossens (2008) and Goossens, Erd´elyi and Ruderman (2011). An overview is provided by Goossens et al. (2008), emphasizing dispersion relations. The strength of the jumps given by (8.100) and (8.101) is independent of the diffusivity η. In particular, then, the jump relations apply in the limit η → 0 and so to an ideal medium. Thus, (8.100) and (8.101) give the jumps in the ideal system too. Looking back at expressions (8.31) and (8.32) that we obtained for the ideal case, we see that [[ξr ]] = lim ξr (s) − lim ξr (s) = E+ − E− , s→0+

s→0−

[[pT ]] = F+ − F− ,

the logarithm term in ξr (s) and pT (s) cancelling because it involves a modulus term |s|. 4 There is a typographical error in the reviews by Goossens (2008) and Goossens, Erd´elyi and Ruderman (2011), which wrongly

give the jump in G as πi. This is correct only if A > 0.

230

Connection Formulas

It is easy to see from the solutions (8.90) and (8.91) that forming the expression (8.55) for X we see the cancellation of the term involving G(τ ), demonstrating explicitly that X is a constant (the value X0 of X at r = rA ).

8.7 Alfv´enic Connection Formulas: The Magnetic Tube with No Twist The special case of a straight magnetic tube in which twist is absent (B0φ = 0) and the magnetic field is purely longitudinal is important in its own right. We can view it as following from the case of a twisted tube, on setting the twist to zero, or we can consider the situation afresh starting from a straight (untwisted) equilibrium field. Viewed as following from the case of a twisted flux tube, we set the twist to zero by taking B0φ = 0 and then B20 = B20z ,

gB =

m B0z , rA

fB = kz B0z .

(8.102)

The connection formulas (8.100) and (8.101) then reduce to [[ξr ]] = −iπ

1 mB0z 1 X0 , |A | rA ρ0 B20

[[pT ]] = 0,

(8.103)

with the terms ρ0 , B0z and B20 calculated at r = rA . We require ωA > 0. Thus, there is no jump in the total pressure perturbation: pT = constant = pT0 , say. Moreover, m X= B0z pT0 = X0 , (8.104) rA a constant, and the jump in ξr is given by [[ξr ]] = −iπ

1 m2 pT0 , 2 rA ρ0 (rA )|A |

(8.105)

with ωA > 0. This is the Alfv´enic jump relation in a flux tube with no twist. It is interesting to note that the jump relations at an Alfv´en singularity are independent of the sound speed, and so apply whether the sound speed cs is large or small. It follows, then, that the same relations hold whether the medium is a β = 0 plasma (corresponding to cs → 0) or is an incompressible fluid (corresponding to cs → ∞). Only the form of the equilibrium state may differ in these two extremes. It is also interesting to note that the jump [[ξr ]] in ξr depends upon the square of the azimuthal wavenumber m, but depends upon the longitudinal wavenumber kz only through the expression A and the point rA .

8.8 The Slow Mode Connection Formulas The slow mode also has a singular point which may be treated in much the same way as the Alfv´enic singularity. The slow mode singularity occurs at a point r = rc , where ωR2 = ωc2 (rc ) =

c2s 2 ωA . c2s + c2A

(8.106)

8.9 The Thin Tube, Thin Boundary Approximation: General Formalism

231

For a compressible flux tube the jump conditions that emerge are (Goossens, Hollweg and Sakurai 1992; see also Ruderman and Goossens 1996; Goossens et al. 2009; Yu, Van Doorssellaere and Goossens 2017)   1 c2s 1 [[ξr ]] = −iπ C, (8.107) 2 s |c | c2s + c2A ρ0 c2A ωA [[pT ]] = −iπ where

 Cs =

c2s c2s + c2A

2ω2 B20φ 1 1 Cs , 2 |c | ρ0 c2A ωA μr

 2 pT − 2ω2 ωA

B20φ μr

ξr ,

c = −

(8.108)

dωc2 . dr

(8.109)

All the terms on the right-hand side of (8.107) and (8.108) are evaluated at the singular point rc . The jumps (8.107) and (8.108) depend upon the azimuthal wavenumber m only 2 and  . through the expressions ωA c In general, both singular points (one associated with the slow mode, the other with the Alfv´enic mode) may contribute to the jump in a variable, depending upon the equilibrium configuration and whether one or both singular points are involved, and therefore which continua (continuous spectra) are involved. In the case of a β = 0 plasma the sound speed is effectively set to zero and so the cusp spectrum is absent and only the Alfv´enic jump arises. In the case of an incompressible fluid, which we may view as the limit β → ∞, the Alfv´en and cusp spectra merge into one and both jumps are additively combined.The incompressible case is discussed in further detail below (see Section 8.13). In the case of a tube without twist, so that B0φ = 0, the jump conditions (8.107) and (8.108) for the slow mode reduce to  2 1 c2s kz2 pT0 , [[pT ]] = 0, (8.110) [[ξr ]] = −iπ ρ0 (rc )|c | c2s + c2A where pT0 denotes the total pressure at the resonance point r = rc ; since [[pT ]] = 0, pT is uniform within the transition layer. The jump [[ξr ]] given in (8.110) is independent of m but depends upon kz explicitly through the term kz2 and implicitly through c (which is independent of m when B0φ = 0).

8.9 The Thin Tube, Thin Boundary Approximation: General Formalism In this section we examine how jump relations such as (8.103) and (8.110) may be used to obtain a dispersion relation. We consider a magnetic flux tube that is free from twist: the interior of the tube consists of a uniform medium (constant Alfv´en speed, sound speed and plasma density) and the external medium is also uniform. These two uniform regions are connected across the tube boundary by a transition layer, where physical quantities change rapidly but smoothly over a layer of thickness l. Since the tube has no twist, we exploit the jump conditions (8.103) and (8.110) as appropriate.

232

Connection Formulas

Consider an equilibrium density profile ρ0 (r) structured radially in r. To be specific we take ⎧ ⎪ r ≥ a + 12 l, ⎪ ⎨ρe , (8.111) ρ0 (r) = ρtrans (r), a − 12 l < r < a + 12 l, ⎪ ⎪ 1 ⎩ρ , r ≤ a − l. 0

2

This represents a flux tube with uniform internal density ρ0 and uniform external density ρe , the two regions being linked by a transition layer of width l where the density ρ0 (r) = ρtrans (r) changes monotonically from ρ0 at r = a − 12 l to ρe at r = a + 12 l. A similar profile in the equilibrium magnetic field B0z (r) or in the magnetic pressure (effectively B20z (r)) or the Alfv´en speed cA (r) may be prescribed. We are particularly interested in a thin layer, l  a, when the transition layer is much thinner than the tube radius a. For l → 0 the transition is abrupt; this is the case (l = 0) discussed in detail in Chapter 6. Now outside of the transition layer the medium is uniform and the total pressure perturbation pT (r) is given by (see Chapter 6)  Ae Km (me r), r > a + 12 l, (8.112) pT (r) = A0 Jm (n0 r), r < a − 12 l, where n20 = −

(kz2 c2s − ω2 )(kz2 c2A − ω2 ) (c2s + c2A )(kz2 c2t − ω2 )

,

m2e =

(kz2 c2se − ω2 )(kz2 c2Ae − ω2 ) (c2se + c2Ae )(kz2 c2te − ω2 )

.

In the above, Jm and Km are Bessel functions and modified Bessel functions of order m, with derivatives Jm  and Km  . Since the tube has no twist, for convenience we may assume that m ≥ 0; there is no loss of generality in this assumption. The sausage mode corresponds to m = 0, the kink mode to m = 1, and fluting modes are for integer m ≥ 2. The constants A0 and Ae are arbitrary. The medium is fully compressible, as the expressions for n0 and me make clear. Special cases such as a β = 0 plasma or an incompressible plasma result in simplifications in the expressions for n0 and me , and are treated separately below. Now the pressure condition [[pT ]] = 0 means that     1 1 pT r = a − l = pT r = a + l (8.113) 2 2 and so we have

) ) A0 Jm (n0 r))r=a− 1 l = Ae Km (me r))r=a+ 1 l . 2

(8.114)

2

There remains the jump condition on the radial displacement ξr (r). To exploit this we need to examine the connection between ξr and the pressure perturbation pT . Now the radial displacement ξr (r) is related to the total pressure perturbation pT (r) by (see equation (6.25), Chapter 6) ξr (r) =

ρ0

(r)(ω2

dpT 1 . 2 2 − kz cA (r)) dr

(8.115)

8.9 The Thin Tube, Thin Boundary Approximation: General Formalism

233

Thus, the jump in ξr (r) across the transition layer is     1 1 [[ξr ]] = ξr r = a + l − ξr r = a − l 2 2 1 1 me Km  (me a) − A0 n0 Jm  (n0 a), = Ae 2 2 2 2 ρe (ω − kz cAe ) ρ0 (ω − kz2 c2A )

(8.116)

where in writing these expressions we have taken account of the fact that the transition layer is thin (so l  a) and therefore a ± l/2 ≈ a. Expression (8.116) for the jump in ξr may be further simplified if we assume that the tube is also thin, corresponding to assuming kz a  1 (which in turn corresponds to waves that are much longer than the radius of the tube). Then we can simplify the expressions involving Bessel functions (see Chapter 6). For m = 0, as kz a → 0 we have Jm  (n0 a) m ∼ , Jm (n0 a) n0 a

Km  (me a) m ∼− . Km (me a) me a

These relations follow directly from standard formulas for the derivative of a Bessel function, combined with the leading behaviour of the Bessel functions for small argument (see, for example, Abramowitz and Stegun 1965, chap. 9). It then follows (since [[pT ]] = 0), for m = 0, that ) ) )  ξr )) 1  )) ξr )) 1 ξr r=a+ 1 l − ξr )r=a− 1 l = [[ξr ]] = − 2 2 pT pT pT )r=a+ 1 l pT )r=a− 1 l 2 2 m m + . = ρ0 (ω2 − kz2 c2A ) ρe (ω2 − kz2 c2Ae )

(8.117)

We now need to specify the jump relations. The two jump conditions we have for a tube without twist (for which [[pT ]] = 0) are the Alfv´enic and slow mode (cusp) jumps, given by equations (8.105) and (8.110):

[[ξr ]]Alfven

1 m2 = −iπ 2 pT0 , ρ (r rA 0 A )|A |

[[ξr ]]slow

1 = −iπ kz2 ρ0 (rc )|c |



c2s c2s + c2A

2 pT0 .

(8.118) It is interesting to note from these relations that in a thin tube with m = 0 the slow mode jump is greatly outweighed by the Alfv´enic mode jump: [[ξr ]]slow ∼ [[ξr ]]Alfven



c2s c2s + c2A

2

kz2 a2 . m2

(8.119)

This follows upon assuming that pT is comparable for the two singular points, namely rA ∼ a and rc ∼ a, and that |c | ∼ |A |. We turn now to an examination of the jumps in several different circumstances, considering the situations separately.

234

Connection Formulas

8.10 The Thin Tube, Thin Boundary Approximation for a β = 0 Plasma Consider a zero β medium. With the sound speed effectively zero the cusp singularity is entirely absent and so the jump condition is simply the Alfv´enic one: [[ξr ]] = −iπ

1 m2 pT0 . 2 ρ (r rA 0 A )|A |

(8.120)

Thus, applying the jump condition (8.120) leads to me Km  (me a) n0 Jm  (n0 a) 1 1 − ρe (ω2 − kz2 c2Ae ) Km (me a) ρ0 (ω2 − kz2 c2A ) Jm (n0 a) = −iπ

1 m2 . 2 ρ0 (rA )|A | rA

(8.121)

This is the dispersion relation for a magnetic flux tube of arbitrary radius a but with a thin transition layer of width l on the boundary of the tube. In the absence of a transition layer, the right-hand side is absent and we recover the dispersion relation discussed in Chapter 6. For the case m = 0, describing the kink and fluting modes, the right-hand side of (8.121) shows that damping of the modes is occurring, an illustration of the process of resonant absorption. Turning to expression (8.117) for a thin tube (and cancelling a common factor of m), the dispersion relation reduces to ρ0

(ω2

1 m a 1 1 + = iπ 2 . 2 2 2 2 2 − kz cA ) ρe (ω − kz cAe ) rA |A | ρ0 (rA )

(8.122)

This is the dispersion relation for m = 0 modes in a thin tube (kz a  1) with a thin transition layer (l  a) on the boundary of the tube. It is sometimes referred to as the thin tube, thin boundary approximation (Goossens et al. 2008). Dispersion relations of this form were first obtained by Goossens, Hollweg and Sakurai (1992). Relations such as (8.122) provide a convenient means of determining the speed of the wave and its damping scales, without the need to solve the complicated initial value problem (see Chapters 4 and 6). In principle, it is easy to add complications to the description of the equilibrium and then solve a dispersion relation for the basic wave speed and damping rate of a given mode. Notice that the dispersion relation (8.122) is independent of n0 and me . Thus, it follows that Equation (8.122) holds quite generally for a thin magnetic tube, whether the medium is compressible or incompressible; in particular, it also holds in a β = 0 plasma. A dispersion relation of the form (8.122) has been introduced by Soler et al. (2009) as a means of investigating oscillations in prominence threads (see the discussion in Chapter 14, Section 14.10). The dispersion relation (8.122) for the kink and fluting modes can be rearranged into the form ρ0 (ω2 − kz2 c2A ) + ρe (ω2 − kz2 c2Ae ) = iπ

1 m a ρ ρ (ω2 − kz2 c2A )(ω2 − kz2 c2Ae ). 2 ρ (r ) 0 e |A | rA 0 A (8.123)

8.10 The Thin Tube, Thin Boundary Approximation for a β = 0 Plasma

235

With ω = ωR + iωI , where ωR and ωI are the real and imaginary parts of ω, we have ω2 = ωR2 − ωI2 + 2iωR ωI ≈ ωR2 + 2iωR ωI on assuming that ωI2  ωR2 (an assumption that we must check a posteriori). Thus, relation (8.123) leads to ρ0 (ωR2 − kz2 c2A ) + ρe (ωR2 − kz2 c2Ae ) + 2i(ρ0 + ρe )ωR ωI 1 m a = iπ ρ ρ (ω2 − kz2 c2A + 2iωR ωI )(ωR2 − kz2 c2Ae + 2iωR ωI ). (8.124) 2 ρ (r ) 0 e R |A | rA 0 A To lowest order of approximation (effectively, taking l/a → 0 and so A → ∞) the dispersion relation (8.124) yields ρ0 (ωR2 − kz2 c2A ) + ρe (ωR2 − kz2 c2Ae ) = 0, and so ωR2 = ωk2 = kz2 c2k ,

c2k =

ρ0 c2A + ρe c2Ae . ρ0 + ρe

(8.125)

Here ωk2 is the square of the kink frequency. Thus, to leading order in l/a, whatever the value of the mode number m = 0, the wave propagates with the kink speed ck (so, to leading order, ω2 ≈ kz2 c2k ). Turning now to the imaginary part of equation (8.124), we find (for m = 0) 2(ρ0 + ρe )ωk ωI = mπ Noting that ωk2

− kz2 c2A

 =

ρe ρ0 + ρe

a 2 | | rA A

1 ρ0 ρe (ωk2 − kz2 c2A )(ωk2 − kz2 c2Ae ). ρ0 (rA ) 

 kz2 (c2Ae

− c2A ),

ωk2

− kz2 c2Ae

ρ0 =− ρ0 + ρe

 kz2 (c2Ae − c2A ),

we obtain ωI = −

ρ02 ρe2 (c2Ae − c2A )2 2 1 mπ a k ω . 2 | | ρ (r ) (ρ + ρ )2 (ρ c2 + ρ c2 ) z k 2 rA 0 e A 0 A 0 A e Ae

(8.126)

Expressions (8.122)–(8.126) hold quite generally for a thin magnetic tube with a thin boundary, whether the medium is compressible or incompressible. In particular, (8.126) gives the rate ωI in terms of the kink frequency ωk of the m = 0 modes. Now in the derivation of the Alfv´enic jump condition we assumed that ωk > 0. Thus, the result (8.126) shows that ωI < 0. Also, since disturbances are proportional to e−iωt = eωI t · e−iωR t , a negative ωI corresponds to a decay, and so ωI is a damping rate. The decay is the signature of the kink and fluting modes giving up their energy to Alfv´enic oscillations: the lateral motions in kink and fluting oscillations of the flux tube excite motions in the transverse eφ -direction. (The sausage mode, for which m = 0, is not covered by the above results derived for m = 0 and we would need to return to the fuller form (8.121) to assess the damping brought about by the cusp singularity.) Otherwise, the kink (m = 1) mode and all fluting (m ≥ 2) modes suffer a decay.

236

Connection Formulas

To reduce expression (8.126) further requires a more detailed consideration of the transition layer, and this is most conveniently carried out for specific assumptions about the equilibrium and the plasma. Accordingly, we turn to a discussion of such specific conditions. 8.10.1 Equilibrium Conditions (β = 0) The assumption of a β = 0 plasma requires an equilibrium with a uniform magnetic field, so Be = B0 and then ρ0 c2A = ρe c2Ae . The flux tube is defined simply by the plasma density profile. The kink frequency is given by   ρ0 k2 c2 , ωk2 = 2 ρ0 + ρe z A and so equation (8.126) becomes ωI = −

1 mπ a ρ0 (ρ0 − ρe )2 c2A kz2 ωk . 2 4 rA |A | ρ0 (rA ) (ρ0 + ρe )2

(8.127)

2 (r) at r = r and ω = ω , we have ρ (r ) = ρ c2 /c2 and so Also, with ωR2 = ωA A R k 0 A 0 A k

ρ0 (rA ) =

1 (ρ0 + ρe ). 2

(8.128)

Thus, ωI = −

mπ a ρ0 (ρ0 − ρe )2 2 2 c k ω . 2 | | (ρ + ρ )3 A z k 2 rA 0 e A

(8.129)

This is a convenient form of the expression for ωI in a thin tube with β = 0. It expresses ωI in terms of ωk , a measure of the oscillation frequency. It is evident from (8.129) that the larger the value of m, the larger is the value of |ωI | and the more rapid is the damping of the m = 0 modes: the fluting modes m = 2, 3, . . . are damped more rapidly than the kink (m = 1) mode. Forms equivalent to expression (8.129), for the kink mode (m = 1), have been given by Goossens, Hollweg and Sakurai (1992; see their eqn. (77)5 ) and Ruderman and Roberts (2002; see their eqn. (56)). The treatment by Goossens, Hollweg and Sakurai (1992) is based upon the connection formula approach and is in fact for an incompressible medium, whereas here we are concerned with a β = 0 plasma. Incompressible plasmas are discussed further in Section 8.13 below. The treatment by Ruderman and Roberts (2002) was based upon solution of the initial value problem for a β = 0 plasma; see Chapters 4 and 6. Finally, we can take the reduction of expression (8.129) for ωI a little further by looking more closely at A . We have ) B2 ρ0  (rA ) ρ0  (rA ) d  2 )) = ρ0 c2A kz2 2 , (8.130) A = − = 0 kz2 2 ωA (r) ) dr μ ρ0 (rA ) ρ0 (rA ) r=rA 5 As noted in Goossens et al. (2009), eqn. (77) in Goossens, Hollweg and Sakurai (1992), which is for an incompressible 2 − ω2 | in their eqn. (77) should be squared. medium, has a typographical error: the term |ωAe A

8.10 The Thin Tube, Thin Boundary Approximation for a β = 0 Plasma

237

where the prime  denotes the derivative with respect to r. Then, we obtain (see also Van Doorsselaere et al. 2004) ωI = −

1 mπ a (ρ0 − ρe )2 ωk . 2 |ρ  (r )| ρ + ρ 8 rA 0 A 0 e

(8.131)

To simplify equation (8.131) further we must look at a specific density profile that arises in the transition layer where ωR = ωA (rA ) is satisfied, evaluating |ρ0  (rA )|. We illustrate this for two profiles: a linear density profile and a sinusoidal density profile, each profile connecting ρ0 and ρe across a layer of width l. It turns out that both profiles give rise to the same resonance point, namely r = rA = a, and both profiles have the same density ρ0 (rA ) at r = rA . However, the density profiles differ in their slopes ρ0  (rA ) at r = rA . 8.10.2 Transition Layer: Linear Density Profile (β = 0) Consider a linear density profile in the transition layer; specifically, we take    1 1 1 1 a − l < r < a + l. ρ0 (r) = ρtrans (r) = ρ0 − (ρ0 − ρe ) r − a − l , l 2 2 2 (8.132) This gives a density profile in the transition layer that changes linearly from ρ0 on the inner edge (r = a− 12 l) to ρe on the outer edge (r = a+ 12 l), over a distance l. The density profile’s slope is a constant at all points within the layer. The point at which ωR2 = ωk2 = kz2 c2k occurs at r = rA = a, and then ρ0  (rA ) = − Thus, equation (8.131) gives ωI = −

(ρ0 − ρe ) , l

rA = a.

  mπ l ρ0 − ρe ωk . 8 a ρ0 + ρe

(8.133)

(8.134)

We have discarded the modulus sign on the difference in densities because, as we have seen in Chapter 6, for the kink mode to exist as a free mode of oscillation (in the absence of a transition layer, l = 0) we require that ρ0 > ρe . Note too the different sign to ωI to that given in Chapter 6, Section 6.10; this is simply due to the notation used here, of e−iωt , whereas in Chapter 6 we used e+iωt . Now the result (8.134) was obtained by a derivation (see Section 8.5) that assumes |ωI |  |ωR |, an assumption that needs to be checked at some stage. We can carry out such a check here. Indeed, it is evident that the condition is met provided that l  a. In fact, the multiplicative factors in (8.134) favour the assumption. For example, with l/a = 1/10 we have (π/8)(l/a) = 0.039, showing that |ωI |  |ωR | is indeed satisfied. Moreover, even for l/a = 1/2, which is not a small fraction, we have (π/8)(l/a) = 0.196, which is reasonably small compared with unity. Thus, the assumption that |ωI |  |ωR | is readily met provided l  a. If we introduce Pk = 2π/ωk as the period of a kink (m = 1) wave and τdamping = −1/ωI as the damping time of the kink mode, then for a linear transition layer of width

238

Connection Formulas

l the damping time is related to the period by (Goossens, Hollweg and Sakurai 1992; Roberts 2008; see also Hollweg and Yang 1988)   4 a ρ0 + ρe (8.135) τdamping = 2 Pk . π l ρ0 − ρe The damping time τdamping is inversely proportional to l/a, the layer thickness as a fraction of the tube radius a. 8.10.3 Transition Layer: Sinusoidal Density Profile (β = 0) Another profile of interest is the sinusoidal density profile, first discussed by Ruderman and Roberts (2002). The profile is defined in the transition layer by6      ρ0 ρe 1 1 ρe π ρ0 (r) = ρtrans (r) = 1+ a − l < r < a + l. − 1− sin (r −a) , 2 ρ0 ρ0 l 2 2 (8.136) Again, this gives a density profile in the transition layer that changes smoothly from ρ0 on the inner edge (r = a − 12 l) to ρe on the outer edge (r = a + 12 l), over a distance l. The point rA is again simply a, and the slope of the ρ0 (r) at r = rA is π ρ0  (rA ) = − (ρ0 − ρe ), rA = a. (8.137) 2l Then, with ρ0 > ρe we obtain   1 l ρ0 − ρe (8.138) ωI = − m ωk . 4 a ρ0 + ρe Accordingly, the damping time and period of the kink (m = 1) mode for a sinusoidal density profile are related by   2 a ρ0 + ρe (8.139) Pk . τdamping = π l ρ0 − ρe This result is equivalent to equation (73) in Ruderman and Roberts (2002).

8.10.4 Resonant Absorption Timescales Formulas such as (8.135) and (8.139) have been widely used to understand observed coronal oscillations (see, for example, the discussion in Roberts (2008), Goossens (2008), Goossens et al. (2009) and Soler et al. (2013)). It is interesting to note that Hollweg and Yang (1988) considered a Cartesian planar geometry for their investigation of coronal loop oscillations and deduced results equivalent to (8.135), and suggested that coronal loop oscillations might damp rapidly as a consequence of resonant absorption processes. The result (8.139) was first applied directly to an observed oscillation by Ruderman and Roberts (2002), who deduced that the observed damping and period of a kink oscillation 6 In the profile considered by Ruderman and Roberts (2002) the transition region is confined to a region of width l extending

over a − l < r < a and ending at r = a; here we have displaced the transition region a distance l/2, so that the inhomogeneous layer is still of width l but is now symmetrically centred around the tube boundary r = a.

8.11 Leakage Versus Resonance in Under-Dense Tubes

239

in a coronal loop was consistent with a loop flux tube having a thin density transition layer of width l = 0.23 a. Goossens, Andries and Aschwanden (2002) examined 11 oscillating loops, concluding that the observed damping was consistent with a layer of width l ranging from 0.16 a to 0.46 a. Roberts (2008) and Goossens (2008) give discussions. Taking the linear and sinusoidal profiles together, we may write   a ρ0 + ρe τdamping = C (8.140) Pk , l ρ0 − ρe where the factor C depends upon the precise form of the density structure within the thin transition layer of width l. For the two cases evaluated above we have  4/π 2 , linear profile, C= (8.141) 2/π , sinusoidal profile. Thus, the exact form of the thin density layer has a bearing on the resulting decay time τdamping , modified by a multiplicative factor of order C = (2/π)2 = 0.4053 (linear transition layer) or C = 2/π = 0.6366 (sinusoidal layer). Other profiles are also of interest (see, for example, Soler et al. 2013). In a transitional layer that is no longer thin, the thin boundary approximation (l  a) is no longer applicable and a numerical investigation becomes necessary, though the analytical results holding for a thin boundary provide a useful guide. Van Doorsselaere et al. (2004), Terradas, Oliver and Ballester (2006), Arregui et al. (2007) and Soler et al. (2013) provide a detailed appraisal of the situation. 8.11 Leakage Versus Resonance in Under-Dense Tubes The result (8.140) allows us to compare the damping timescale for resonant absorption with the damping time for wave leakage of the kink mode (see the calculation of leaky waves in a β = 0 situation in Chapter 6). This was first done by Goossens and Hollweg (1993); see also Goossens et al. (2009). In a dense flux tube with ρ0 > ρe there is no leakage of the kink mode but if the tube is under-dense, so that ρ0 < ρe , then leakage arises. The two timescales to compare are τdamping , given in (8.140), and τ leakage = 1/|ωI |, the timescale of wave leakage that follows from the decay rate |ωI | determined by the imaginary part of a complex ω (see the discussion of wave leakage in Chapter 6). We have     a ρ0 + ρe 4 ρ0 + ρe 2 1 τdamping = C τ leakage = 2 Pk . (8.142) Pk , l ρ0 − ρe ρ0 − ρe π (kz a)2 Thus, the ratio of these two timescales is    τ leakage 4 ρ0 + ρe l 1 = , 2 τdamping a ρ0 − ρe (kz a)2 Cπ

ρ0 < ρe .

(8.143)

If we set kz = π/L, where L is the length of a loop, and for convenience choose a resonance layer with a linear density profile (for which C = 4/π 2 ), and consider a low density flux tube with ρ0 = 13 ρe , say, then we obtain    2 τ leakage 2 L l = 2 . τdamping a a π

240

Connection Formulas

If this ratio of timescales is much greater than unity, then leakage is unimportant and the dominant process in an under-dense tube is resonant absorption (in an over-dense tube, ρ0 > ρe , and there is no leakage of the kink mode). If the ratio is small, then leakage dominates over resonant absorption. If the ratio is about one, then both processes make comparable contributions to the decay timescale of a kink oscillation in an under-dense flux tube. Thus, applied to coronal loops, it is evident that for very long loops, L  a, the ratio of timescales will be large and consequently the dominant process is resonant absorption unless the layer thickness l is extremely small. We can rewrite our condition for resonant absorption to be dominant over radial wave leakage in an under-dense flux tube as π2 1 l . > a 2 (L/a)2 For example, in a long loop with L = 102 a we obtain τ leakage > τdamping for l > 5 × 10−4 a, a demand readily met and showing that resonant absorption is the dominant process in all but the thinest resonant layers. This conclusion has been reached by Goossens and Hollweg (1993) and Goossens et al. (2009). However, matters are less clear-cut for shorter loops. For example, a loop with L = 50 a has a timescale ratio of about 500 (l/a), giving a ratio that is less than unity for a resonant layer l < a/500. Finally, in a short loop with L = 10 a, we require only that l < a/50 for the time ratio to be greater than unity and resonant absorption to be dominated by wave leakage.

8.12 Thin Tube, Thin Boundary Dispersion Relation: The Photospheric Tube The photospheric tube has not been considered much in the context of the decay of the kink mode but we can give a discussion similar to that presented for a β = 0 plasma. Somewhat similar discussions have been given by Soler et al. (2009) for prominence threads (see Chapter 14). We consider a magnetic flux tube embedded within a compressible field-free medium. The interior and exterior of the tube are related through a thin transition layer of width l, over which both the magnetic field and the plasma density undergo a rapid spatial variation. The derivation of the thin tube, thin boundary layer dispersion relation proceeds as in Section 8.9, save now the squared wavenumbers n20 and m2e are slightly different: n20 = −

(kz2 c2s − ω2 )(kz2 c2A − ω2 ) (c2s + c2A )(kz2 c2t − ω2 )

,

m2e =

kz2 c2se − ω2 . c2se

In the thin tube limit, the resulting dispersion relation is (see equation (8.122)) ρ0

(ω2

1 m a 1 1 + = iπ 2 . 2 2 2 − kz cA ) ρe ω rA |A | ρ0 (rA )

(8.144)

Again, we may solve the dispersion relation by writing ω = ωR + iωI , where ωR and ωI are the real and imaginary parts of ω with the assumption that ωI2  ωR2 . The result is that   ρ0 ωR2 = ωk2 = kz2 c2k , c2k = (8.145) c2 , ρ0 + ρe A

8.13 Thin Tube, Thin Boundary Dispersion Relation: The Incompressible Tube

241

and the imaginary part of the dispersion relation (8.144) yields ωI = −

ρ0 ρe2 mπ a k2 c2 ωk . 2 2 rA |A | ρ0 (rA )(ρ0 + ρe )2 z A

(8.146)

To simplify expression (8.146) much further we need to consider in detail the equilibrium profiles in the transition region on the boundary of the flux tube. Suppose that within the transition region the plasma density is given by    1 1 1 1 ρ0 (r) = ρtrans (r) = ρ0 − (ρ0 −ρe ) r − a − l , a − l < r < a + l (8.147) l 2 2 2 and the square of the magnetic field B20z is    1 2 2 21 B0z (r) = B0 − B0 r− a− l , l 2

1 1 a− l 12 l, (8.167) pT (x) = A0 e+|kz |x , x < − 12 l. c = −kz2

Expression (8.167) satisfies the condition that pT (x) → 0 as x → ±∞, and it gives the behaviour of pT outside of the transition region − 12 l < x < 12 l. Across the transition region, the jump condition (8.166) that [[pT ]] = 0 requires     1 1 pT x = l = pT x = − l . 2 2 Thus Ae = A0 . Now, whether the medium is compressible or incompressible, ξx and pT are linked by (see Chapter 3) ξx (x) =

ρ0

(x)(ω2

dpT 1 . 2 2 − kz cA (x)) dx

So, outside of the transition region we have ⎧ −|kz | ⎨ A e−|kz |x , ρe (ω2 −kz2 c2Ae ) e ξx (x) = |kz | +|kz |x , ⎩ 2 2 2 A0 e ρ0 (ω −kz cA )

(8.168)

x > 12 l, x < − 12 l.

(8.169)

Hence, the jump (8.166) in ξx is [[ξx ]] =

ρe



−|kz | |kz |  Ae e−|kz |l/2 −   A0 e+|kz |l/2 . 2 2 2 − kz cAe ρ0 ω − kz2 c2A

ω2

Accordingly, applying the jump condition (8.166) for an incompressible fluid we obtain −

ρe



ω2

|kz | |kz |  Ae e−|kz |l/2 −   A0 e|kz |l/2 2 2 2 − kz cAe ρ0 ω − kz2 c2A = −iπ kz2

1 pT0 . ρ0 (xc )|c |

(8.170)

Since e±|kz |l/2 ≈ 1 for |kz l|  1 and pT0 = pT (x = 0) = A0 , we finally obtain (Goossens 1991; see his eqn. (7.37)) ρ0

(ω2

1 1 1 + = iπ |kz | . 2 2 2 2 2 ρ0 (xc )|c | − kz cA ) ρe (ω − kz cAe )

(8.171)

This is the dispersion relation for an incompressible wave on a sharply changing thin interface connecting two uniform media. Comparing (8.171) with the thin tube, thin boundary dispersion relation (8.122) that applies in a β = 0 medium, we see that the two relations are the same provided we

8.14 Cartesian Geometry: The Single Interface

247

2 in the tube result by |k |. Accordingly, with ω = ω + iω for real and replace ma/rA z R I imaginary parts ωR and ωI , we may immediately deduce (from the earlier analysis of the tube in a β = 0 plasma, leading to equations (8.125) and (8.126)) the approximate solution (Goossens 1991)

ωR2 = kz2 c2k ,

ωI = −

ρ02 ρe2 (c2Ae − c2A )2 π 1 |kz |kz2 ωk . 2 ρ0 (xc )|c | (ρ0 + ρe )2 (ρ0 c2A + ρe c2Ae )

(8.172)

8.14.2 Transition Layer: Linear c2A (x) To progress further we must specify the equilibrium in greater detail. Suppose that the equilibrium is such that the square of the Alfv´en speed is constant save for within the transition region, where we take c2A (x) to be linear in x. Specifically, ⎧ 2 ⎪ x > 12 l, ⎪ ⎨cAe , (8.173) c2A (x) = 12 (c2A + c2Ae ) − (c2A − c2Ae )(x/l), − 12 l < x < 12 l, ⎪ ⎪ 1 ⎩c2 , x < − l, A

2

giving c2A (x) = c2A at x = −l/2 and c2A (x) = c2Ae at x = l/2. Also, we assume that the density is uniform (so that ρe = ρ0 ). Evidently c = −kz2 (c2Ae − c2A )/l,

(8.174)

and then taking ωk = |kz |ck (so ωk > 0) we obtain ωI = −

π 2 |c2Ae − c2A | , k l 16 z ck

(8.175)

with an associated timescale τdecay (= 1/|ωI |) given by τdecay =

ck 16 . 2 2 πlkz |cAe − c2A |

(8.176)

These results are in agreement with the analyses by Rae and Roberts (1981; see their eqn. (51) for uniform density), Goedbloed (1983) and Lee and Roberts (1986; see the expression for the decay rate γ in their section II (d)). See also Ruderman, Tirry and Goossens (1995). An illustration of such results on coronal oscillations was discussed in the reviews by Roberts (2000, 2008). These earlier results were generally obtained from an initial value analysis (see Chapters 4 and 6). Here we have obtained these results without the need for the more elaborate initial value approach, illustrating the usefulness of the connection formulas.

9 Gravitational Effects

9.1 Introduction So far in our discussion of MHD waves we have ignored the effect of gravity. In many situations this is a good approximation, but in other circumstances it is important to include the influence of a gravitational force. What in general terms are the expected effects of gravity? Gravity enters into discussion through the momentum equation, which we may write in the form (see equation (1.2), Chapter 1) ρ

∂u + ρ(u · grad)u = −grad p + j × B + ρg. ∂t

(9.1)

As in Chapter 1, p denotes the plasma pressure, ρ is the plasma density and u is the flow velocity of the plasma. The plasma moves under the combined action of the pressure force, −grad p, the magnetic force j × B arising from the magnetic field B and its associated current j (given by Ampere’s law), and the gravitational force ρg per unit volume. It is convenient to introduce a Cartesian coordinate system with the z-axis chosen to be aligned with gravity and pointing downwards; then we may write g = gez , where g is the strength of the gravitational acceleration, and the gravitational force acting on the medium is ρgez . We assume gravity to be uniform, so that g is a constant. In the solar plasma, g = 274 m s−2 . It is important to note that an alternative choice of coordinate system, commonly adopted in much of the literature, takes the z-axis pointing vertically upwards, opposite to the direction of gravity; in such a system, the gravitational force acting on the plasma is −ρgez . This choice of coordinates leads to differences in some of the important expressions and equations that arise, so care should be taken in comparing one system with another. There are several consequences of including gravity: firstly, we see that gravity introduces a directionality (here the z-axis) in the medium. This is in addition to any directionality introduced by the magnetic field. Indeed, in the absence of a magnetic field, the directionality introduced by gravity remains and sound waves lose their isotropic nature, just as in the presence of a magnetic field (but in the absence of gravity) we have seen how compressive disturbances – MHD waves – lose their isotropy because of magnetism and so become anisotropic in nature. Secondly, gravity introduces stratification, since an equilibrium under gravity tends to have its mass loaded preferentially lower in the atmosphere rather than in the upper regions, if it is to be in stable equilibrium; in short, the medium is stratified. Thus, just as the introduction of a magnetic field creates a preferred direction and may support structuring, so the inclusion of gravity creates a preferred direction and 248

9.2 Equilibrium

249

tends to layer the medium through stratification. Thirdly, the presence of gravity introduces a lengthscale in the medium, by which disturbances are measured; this causes waves to be dispersive, since it allows us to distinguish between long and short wavelengths by reference to the lengthscale imposed by gravity. In fact, the presence of gravity introduces several lengthscales, as we may consider how various equilibrium quantities, such as the pressure or density, vary with depth z. Moreover, the presence of a lengthscale in the equilibrium means that there is also a timescale (period or its reciprocal, frequency), determined by the travel time of a wave over a lengthscale. Thus, waves behave differently in the presence of gravity, depending upon whether their period or frequency is higher or lower than the imposed timescale or frequency determined by the strength of gravity. Finally, the presence of gravity implies a force that acts in addition to any other forces in the system, raising the possibility of a wave motion driven by this force. It turns out that this introduces a new mode of oscillation, the gravity wave, which propagates in a manner quite distinct from how sound or magnetohydrodynamic waves propagate. We turn in greater detail to these aspects now.

9.2 Equilibrium 9.2.1 General Aspects Denote by a suffix 0 an equilibrium quantity so that the equilibrium pressure and density are p0 and ρ0 . Then, according to equation (9.1), in a magnetic field B0 the forces are in equilibrium when grad p0 = j0 × B0 + ρ0 gez .

(9.2)

There is, of course, a vast array of possible equilibria described by equation (9.2), in which a mixture of the effects of magnetism and gravity may play a role. However, the basic aspects of stratification are best understood in the absence of magnetism or when the magnetic field is so strong that it can only be balanced by being force-free, so that j × B = 0. The solar corona is an example of a force-free medium. To be specific, we consider (9.2) in the absence of a magnetic field. Then, with B0 = 0, equilibrium pressure balance requires that grad p0 = ρ0 gez .

(9.3)

Suppose that equilibrium variables such as p0 and ρ0 are functions of z only. Then the equilibrium constraint (9.3) is just p0  (z) = gρ0 (z),

(9.4)

where we have found it convenient to introduce a prime ( ) to denote the derivative with respect to z of an equilibrium quantity. We retain the use of ∂/∂z for the derivative of a perturbation. Equation (9.4) is an ordinary differential equation relating p0 (z) and ρ0 (z). If we introduce also the ideal gas law, p=

kB ρT, m ˆ

(9.5)

250

Gravitational Effects

where kB (= 1.38 × 10−23 J K−1 ) is the Boltzmann constant, T is the absolute temperature of the plasma and m ˆ is its mean particle mass, we may express (9.4) in terms of the equilibrium temperature T0 of the medium; thus p0 (z) 0

(9.6)

kB T0 (z) p0 . = mg ˆ gρ0

(9.7)

p0  (z) = where 0 (z) =

The quantity 0 (z) has the dimension of length and provides a measure of how rapidly the equilibrium pressure p0 (z) changes with depth z (or height −z); it is the local pressure scale height in the medium. In the absence of gravity, the scale height is infinite (0 → ∞ as g → 0), indicating that the equilibrium pressure does not change in height over any finite distance; in short, the medium is uniform. Roughly speaking, a medium with weak stratification under gravity has a large pressure scale height, whereas a strongly stratified medium has a short scale height. Of course, what is meant by large and small 0 (z) has yet to be made clear. We shall see shortly that in the solar corona the pressure scale height is large, of order 105 km, whereas in the photosphere it is short, of order 102 km. As well as the pressure scale height 0 we may also introduce a scale height for any other equilibrium quantity; the density scale height H0 (z) is of particular interest. The two scale heights we use are thus 0 (z) =

p0 (z) , p0  (z)

H0 (z) =

ρ0 (z) . ρ0  (z)

(9.8)

The differential equation (9.6) may be integrated formally to give p0 (z) = p0 (0)eN0 (z) , where p0 (z = 0) denotes the pressure at z = 0, an arbitrary reference level, and  z ds . N0 (z) = 0 0 (s)

(9.9)

(9.10)

The plasma density ρ0 (z) may be deduced from this expression and use of the ideal gas law: ρ0 (z) = ρ0 (0)

0 (0) N0 (z) . e 0 (z)

(9.11)

Of course, these formal expressions for p0 and ρ0 cannot be determined further until we specify the scale height 0 (z). There are some interconnections between the equilibrium quantities that are of interest. For a given mean particle mass m ˆ and constant g, 0 is directly proportional to the equilibrium temperature T0 . Also, since the sound speed cs = (γ p0 /ρ0 )1/2 we have the relation c2s = γ g0 .

(9.12)

9.2 Equilibrium

251

It then follows that (c2s ) = γ g0  = γ g −

c2s , H0

0  = 1 −

0 , H0

(9.13)

with γ and g taken to be constants. The presence of gravity, then, introduces a preferred direction and it also creates an inhomogeneity in the medium: the medium is stratified. This stratification is in addition to any structuring that a magnetic field may impose. Moreover, the presence of a spatial scale 0 (or H0 ) means that all waves have a unit of length against which they may be measured, and this implies the occurrence of dispersion. We have seen in earlier chapters how magnetic structuring introduces dispersion in wave propagation, so gravity adds further to this effect. The presence of a lengthscale introduced by gravity introduces also a timescale, once a specific speed is added. In the acoustic case, this may be taken to be the time it takes a sound wave to propagate over the distance H0 and back again: 2H0 /cs ; in terms of frequency, this suggests that gravity imposes on the system a frequency of order a =

cs . 2H0

(9.14)

Thus we may anticipate that a frequency of this form may play an important role in wave propagation. For reasons that will become clear in further developments, we refer to a as the acoustic cutoff frequency. In dimensional terms there are other ways we may form a frequency using gravity. For example, both (g/H0 )1/2 and g/cs have the dimension of frequency (s−1 ), so we may expect that they too will be involved in any description of wave propagation in a stratified medium. In fact, it turns out that the combination 2g =

g − H0



g cs

2 (9.15)

plays an important role in the description of waves in a stratified atmosphere. The frequency g is referred to as the buoyancy frequency (or Brunt–V¨ais¨al¨a frequency); 2g may be positive or negative, which also has significant implications.

9.2.2 Isothermal Atmosphere In an isothermal medium (for which 0 and cs are constants), N0 (z) = z/0 and H0 = 0 , and the equilibrium pressure and density both grow exponentially fast on a scale 0 , growing by a factor of e in one scale height 0 and by a factor of 10 in a distance of 2.3 0 . Thus, we may view the equilibrium pressure or density as roughly uniform over a distance much less than the scale height 0 (i.e., for |z|  0 ), but structured over distances comparably to or larger than 0 . For waves that have wavelengths much less than a scale height, then, we can expect that the effects of gravity are negligible, but for waves of wavelength comparable to 0 then gravity plays an important part.

252

Gravitational Effects

In an isothermal atmosphere, the expressions (9.14) and (9.15) for the acoustic cutoff and buoyancy frequencies yield   γg g γ −1 2 2 a = , g = . (9.16) 40 0 γ With γ = 5/3, we obtain √  1/2  1/2 15 g g a = = 0.6455 , 6 0 0 and

+ g =

24 a = 25

√  1/2  1/2 10 g g = 0.6325 . 5 0 0

Thus, in an isothermal atmosphere the buoyancy frequency g lies 2% below the acoustic frequency a . It is common to express the acoustic and buoyancy frequencies in terms of cyclic frequency, taking νa ≡ a /2π and νg ≡ g /2π in hertz (Hz). For example, with γ = 5/3 and g = 274 m s−2 , a sound speed of cs = 7.5 km s−1 in an isothermal atmosphere with scale height 0 = H0 = 125 km, broadly appropriate for the solar photosphere, gives a = 0.03 s−1 , with a corresponding acoustic cyclic frequency of νa = 0.0048 Hz (or 4.8 mHz) and a buoyancy frequency of νg = 4.7 mHz. The periods associated with these frequencies are νa−1 = 208 s and νg−1 = 213 s. By contrast, under coronal conditions a sound speed of cs = 200 km s−1 gives 0 = 87 Mm and νa = 0.18 mHz, with a corresponding period of 91.7 minutes. Thus, in the corona νa is fairly small and the effect of stratification is consequently less significant. It is also of interest to compare the solar results with those for the Earth’s atmosphere. For air of temperature 20◦ C (giving T0 = 313 K) with an adiabatic index γ = 1.4, the sound speed is 340 m s−1 ; thus, with a terrestrial gravitational acceleration g = 9.81 m s−2 we obtain a = 0.020 s−1 and νa = 3.2 mHz (with an associated period of 312 s); the terrestrial values of the buoyancy frequency are g = 0.018 s−1 and νg = 2.86 mHz (with an associated of period of 349 s). So far our comments have been entirely of a general nature, based upon dimensional analysis, but in the next section we consider in more detail the linear motions of a medium stratified under gravity.

9.3 Acoustic-Gravity Waves: General Case The simplest case in which to explore the effects of stratification is to consider our system in the absence of a magnetic field. In effect, then, we are examining the influence of gravity on a sound wave. However, it turns out that the presence of the gravitational restoring force permits a mode of oscillation in its own right, distinct from that of the sound wave. This is the gravity wave. Its occurrence is not surprising, given that gravity introduces a restoring force that is separate from the compressibility that drives a sound wave. The combined

9.3 Acoustic-Gravity Waves: General Case

253

system of waves that arise under the influence of stratification are known as acousticgravity waves. The governing system of linear equations for a plasma (or a gas) are: continuity, ∂ρ + div ρ0 u = 0; ∂t

(9.17)

momentum, ρ0

∂u = −grad p + gρ ez ; ∂t

isentropic conditions, γ p0 ∂p + u · grad p0 = ∂t ρ0



 ∂ρ + u · grad ρ0 . ∂t

(9.18)

(9.19)

In these linear equations, p now denotes the perturbed pressure (so the full pressure is p0 + p, made up of the equilibrium pressure p0 plus the perturbation pressure p), ρ is the perturbed density, and u is the perturbation flow. The temperature perturbation T may be determine from the linearized ideal gas law once p and ρ are known: T p ρ = − . T0 p0 ρ0

(9.20)

In analysing these equations, we have to decide which variables to eliminate and which we wish to elect as primary variables to be solved for. A number of choices are available. One choice of variables is to eliminate perturbations in favour of the perturbed pressure p and the perturbed momentum flux density ρ0 u; this is favoured by Lighthill (1978). Another choice, favoured by Lamb (1932), is to eliminate variables in favour of uz and div u. There are advantages and disadvantages in either choice, and we consider both. One immediate deduction from the momentum equation follows straightforwardly. Dividing first by ρ0 and then taking the curl of the momentum equation (9.18), we obtain     ρ 1 ∂ grad p + g curl ez . curl u = −curl (9.21) ∂t ρ0 ρ0 Now the vector identity curl (f A) = f curl A + (grad f ) × A allows us to treat the first term on the right-hand side of (9.21), showing that it produces a term that is perpendicular to the z-axis. Similarly, the second term on the right-hand side of (9.21) also lies perpendicular to the z-axis. Consequently ωz , the z-component of the vorticity curl u, satisfies ∂ (9.22) ωz = 0. ∂t In other words, the component of vorticity in the direction of gravity is not propagated by the motions; if ωz is initially zero, then it remains zero for all subsequent times. This echoes back to the property of a sound wave in a uniform (g = 0) medium, which preserves entirely any initial vorticity (all three components of the vorticity remain unchanged in time); see Chapter 2.

254

Gravitational Effects

Consider the divergence of the momentum equation (9.18): ∂ρ ∂ div (ρ0 u) = −∇ 2 p + g ; ∂t ∂z combined with continuity, this yields ∂ 2ρ ∂ρ +g (9.23) = ∇ 2 p. 2 ∂z ∂t Also, from the isentropic equation we may relate density and pressure perturbations with ρ0 uz :    1 ∂p p0 ∂ρ  − ρ = 2 + uz 0 ∂t cs ∂t c2s 1 ∂p 1 2 = 2 (9.24) − g (ρ0 uz ), g cs ∂t where we have used the equilibrium condition p0  (z) = gρ0 (z) to eliminate p0  (z) and have introduced the buoyancy frequency g (defined in equation (9.15)). From the z-component of momentum we obtain ∂ ∂p (9.25) + (ρ0 uz ). gρ = ∂z ∂t We may use equation (9.25) to eliminate ρ from equation (9.24), obtaining   g ∂p ∂ 2p ∂2 2 + g (ρ0 uz ). (9.26) − = ∂z∂t c2s ∂t ∂t2 Also, consider equation (9.23) and eliminate the time derivative of ρ by use of (9.24): 1 ∂ 2p ∂ 1 ∂ρ − ∇ 2 p = 2g (ρ0 uz ) − g . 2 2 g ∂t ∂z cs ∂t Finally, we eliminate the term involving gρ by use of equation (9.25):   1 ∂ 2p 1 2 ∂ ∂ 2 − ∇⊥ p =  − (ρ0 uz ), g g ∂z ∂t c2s ∂t2

(9.27)

(9.28)

where 2 ∇⊥ =

∂2 ∂2 + ∂x2 ∂y2

denotes the (two-dimensional) Laplacian in the horizontal plane. Equations (9.26) and (9.28) provide a pair of coupled partial differential equations determining p and ρ0 uz . The anisotropy introduced by gravity is manifest in equations (9.26) and (9.28) in the preferential role played by the operator ∂/∂z. Instead of working with the pressure variable p we may follow Lamb (1932) and use the variable  ≡ div u, which provides a measure of the compressibility in the motion (since  = 0 corresponds to an incompressible medium). A connection between p and  follows from the adiabatic equation (9.19), which gives ∂p = −gρ0 uz − ρ0 c2s , ∂t

(9.29)

9.4 Acoustic-Gravity Waves: Differential Equations

255

where we have used the continuity equation (9.17) and the equilibrium condition (9.4) in writing (9.29). Then, eliminating p in favour of ρ0 uz and  from equations (9.26) and (9.28), we obtain     ∂ g ∂2 2 +  u ) = (9.30) − (gρ0 uz + ρ0 c2s ), (ρ 0 z g ∂z c2s ∂t2     ∂ 1 2 ∂2 1 ∂2 2 2 − ∇⊥ (gρ0 uz + ρ0 cs ) = (ρ0 uz ). (9.31) −  ∂z g g ∂t2 c2s ∂t2 Equations (9.30) and (9.31) govern our system.

9.4 Acoustic-Gravity Waves: Differential Equations The fact that our perturbation equations (9.30) and (9.31) (or (9.26) and (9.28)) involve only a z-dependence from the equilibrium suggests that we may Fourier analyse in terms of the other variables x, y and t. Accordingly, we write   u = ux (z), uy (z), uz (z) e i(ωt−kx x−ky y) for frequency ω and horizontal wavenumbers kx and ky , with similar expressions other perturbation quantities. We retain the z-dependence, given that our system general a non-uniform behaviour in this variable. Equations (9.30) and (9.31) lead to the coupled ordinary differential equations   d g d − ω2 ρ0 uz , (ρ0 c2s ) + (gρ0 uz ) = gρ0  + dz dz H0     2 2 2 gk⊥ k⊥ cs d 1 − (ρ0 uz ) + ρ0 uz = 1 − 2 ρ0 , dz H0 ω2 ω

for all has in

(9.32) (9.33)

where k⊥ = (kx2 + ky2 )1/2 is the magnitude of the horizontal wavevector k⊥ = (kx , ky , 0). We can combine these equations so as to leave a single derivative term in each equation. Equation (9.33) is already in the appropriate form, but (9.32) and (9.33) combine to yield     2 2 c2 g2 k⊥ gk⊥ d s 2 2 − ω ρ0 uz + (9.34) (ρ0 cs ) = ρ0 . dz ω2 ω2 The density ρ0 can be divided out of equations (9.33) and (9.34) in favour of uz and , yielding ω2 c2s

d 2 2 2 cs ) = (g2 k⊥ − ω4 )uz + g(γ ω2 − k⊥ dz   2 2 c2 k⊥ gk⊥ duz s + 2 uz = 1 − 2 . dz ω ω

(9.35) (9.36)

Equations (9.35) and (9.36) are the desired equations that relate uz and ; they were first given by Lamb (1932). They are both first order ordinary differential equations. It is interesting to note that the variation in the coefficients of the differential equations (9.35)

256

Gravitational Effects

and (9.36) depends entirely upon the distribution of c2s (z), the square of the sound speed, with no direct dependence upon ρ0 . Equations (9.35) and (9.36) determine how sound behaves in a stratified atmosphere and, as we shall see, they determine also the behaviour of a new mode – the gravity wave. Equations of this form underpin developments in the field of helioseismology, where they and their generalizations in spherical coordinates provide a means of probing the solar interior, when combined with suitable data. It is common in helioseismology to term the wave solutions of (9.35) and (9.36), or their equivalents in spherical coordinates, the p and g modes (p for pressure, g for gravity), a terminology that goes back to Cowling (1941). Other perturbation quantities can be related to uz and . For example, the equation of continuity gives iωρ = −ρ0  − ρ0  uz ,

(9.37)

and the horizontal components of the momentum equation give (on cancellation of a common factor i) ωρ0 ux = kx p,

ωρ0 uy = ky p,

(9.38)

with iωp = −gρ0 uz − ρ0 c2s .

(9.39)

Also, from the definition of  = div u we may eliminate the horizontal components ux and uy by use of (9.38), giving =

k2 ∂uy ∂ux ∂uz ∂uz + + = − i ⊥ p. ∂x ∂y ∂z ∂z ωρ0

We may eliminate uz from (9.35) and (9.36), with the result that (Lamb 1932)  2  2   2 c2 2  2 ω − k⊥ gk⊥ d2  cs γ g d g cs s + + − − (γ − 1) +  = 0. dz2 c2s c2s dz c2s ω2 c2s c2s

(9.40)

(9.41)

Thus, we obtain a second order ordinary differential equation for the unknown  (= div u). Alternatively, if we eliminate  in preference to uz then we obtain the differential equation       2  2 gk⊥ g2 ρ0 k⊥ d ρ0 c2s duz ρ0 c2s − + ρ0 + 2 uz = 0, 2 c2 2 c2 2 c2 )ω2 dz dz ω ω2 − k⊥ ω2 − k⊥ (ω2 − k⊥ s s s (9.42) 2 c2 . Equation (9.42) is more complicated in appearance than (9.41) so it provided ω2 = k⊥ s is often preferable to investigate (9.41) first and then to deduce the flow uz by use of (9.35). However, it sometimes proves useful to consider (9.42) directly. Consider, then, equation (9.41) for the compression . It proves helpful to eliminate the first derivative term in (9.41), which we can do as follows. Set (z) = Z(z)Q(z). Our aim then is to choose the function Z(z) so that the resulting equation for Q has no first derivative term. Now

dQ d =Z + Z  Q, dz dz

dQ d2  d2 Q = Z + 2Z  + Z  Q. 2 2 dz dz dz

9.5 Acoustic-Gravity Waves: The Isothermal Medium

257

So to eliminate the first derivative term in dQ/dz, which arises on the use of (9.41), we choose Z such that  2  γg cs 2Z  + 2 = 0; + Z c2s cs that is, we choose Z(z) so that d [ln(Z 2 ) + ln(c2s ) + ln(p0 )] = 0, dz

(9.43)

where here we have used (9.8) to rewrite the term involving γ g/c2s : 1 p0  d γg = = = ln p0 . 0 p0 dz c2s Integrating equation (9.43) shows that Z 2 p0 c2s is a constant, and so (since c2s = γ p0 /ρ0 ) the quantity ρ0 1/2 c2s Z is a constant. Accordingly, we may take   ρ0 (0) 1/2 c2s (0) (9.44) Z(z) = ρ0 (z) c2s (z) and 

ρ0 (0) (z) = ρ0 (z)

1/2

c2s (0) Q(z). c2s (z)

(9.45)

With this choice of Z it follows that Q satisfies the ordinary differential equation d2 Q + κ 2 (z)Q = 0, dz2

(9.46)

  2 g ω2 2a  2 κ (z) = 2 − 2 (1 + 2H0 ) + k⊥ −1 . cs cs ω2

(9.47)

where 2

We see here the occurrence of the two frequencies 2a and 2g ,  2a =

cs 2H0

2 ,

2g =

g − H0



g cs

2 .

9.5 Acoustic-Gravity Waves: The Isothermal Medium The simplest case to consider is that of an isothermal atmosphere, for which the temperature T0 , the sound speed cs and the scale heights H0 and 0 are all constants (given γ and g are constants). This permits a simple description of the oscillations, allowing us to see more clearly the distinction between sound waves and gravity waves. In an isothermal medium, H0 = 0 ,

H0  = 0,

a =

cs , 20

g =

2 (γ − 1)1/2 a , γ

258

Gravitational Effects

and accordingly κ 2 is a constant. Thus equation (9.46) takes on a simple form, and its solutions are simple functions: Q = Q0 e±ikz z , where Q0 is an arbitrary constant and kz2 = κ 2 , i.e., kz2

  2 g ω2 − 2a 2 = + k⊥ −1 . c2s ω2

(9.48)

2 and ω2 , k2 may take positive or negative values, dependDespite the notation, for real k⊥ z 2 2 ing upon ω and k⊥ . Accordingly, the vertical dependence of Q(z) is purely oscillatory if kz2 > 0, indicating a vertically propagating disturbance. However, if kz2 < 0, then Q(z) is exponential (growth or decay) in nature, indicating (when boundary conditions require a decay) a vertically evanescent disturbance Q. This is true even in the special case when 2 = 0, corresponding to vertical propagation (which we discuss in detail in Section 9.8); k⊥ in this case, sound is propagated vertically if ω2 > 2a , whereas if ω2 < 2a then the solution (9.48) has an exponential nature (growth or decay) in z but no vertical propagation. Accordingly, we refer to a as the acoustic cutoff frequency. 2 = 0 we need to examine when k2 > 0 and when k2 < 0; More generally, when k⊥ z z thus the zeros of kz2 become of interest (there being no infinities of kz2 to concern us). Accordingly, we display in Figure 9.1 the sign of kz2 in the k⊥ , ω-plane, taking k⊥ (in units of a /cs ) as the horizontal axis and ω (in units of a ) as the vertical axis. On the vertical axis, when k⊥ = 0, we have already noted that kz2 > 0 for ω > a and so the region ω > a gives propagation. Equation (9.48) may be rearranged into the form 2 2 cs = 0, ω4 − (2a + k2 c2s )ω2 + 2g k⊥

(9.49)

2 + k2 . We consider the case when there is vertical propagation, k2 > 0. The where k2 = k⊥ z z quadratic nature (in ω2 ) of this relation suggests the presence of two modes. We anticipate the presence of sound waves modified by gravity, and indeed this corresponds to the larger of the two roots of (9.49); the smaller of the two roots corresponds to the new mode, the gravity wave. In fact, we can solve (9.49) to yield

  2 c2 2 1/2  4k⊥ 1 2 s g 2 2 . ω = (a + k cs ) 1 ± 1 − 2 2 (a + k2 c2s )2 2

(9.50)

In the absence of gravity (when both a and g are zero), equation (9.50) gives the dispersion relation for sound waves in a uniform medium (ω2 = k2 c2s ) with the smaller root of the quadratic yielding ω2 = 0 when g = 0. Also, when k⊥ = 0 or g = 0 the roots of the quadratic equation become ω2 = 2a + k2 c2s and ω2 = 0, corresponding to the vertical propagation of the sound wave (see Section 9.8) and the non-propagation of the gravity wave either for k⊥ = 0 or when the buoyancy frequency is zero (g = 0). The gravity wave, then, is unable to propagate vertically or in a neutrally buoyant medium with g = 0.

9.5 Acoustic-Gravity Waves: The Isothermal Medium

259

Figure 9.1 The diagnostic diagram of acoustic-gravity waves propagating in an isothermal medium, with the x-axis corresponding to k⊥ cs / a and the y-axis to ω/a . The diagram indicates those regions of the wavenumber–frequency plane (the k⊥ , ω-plane) that have kz2 > 0 (corresponding to vertical propagation) and those regions where kz2 < 0 (corresponding to vertical evanescence or nonpropagation). The dividing curves are obtained by solving equation (9.48) with kz2 = 0 for ω (in units of a ) as a function of k⊥ (in units of a /cs ). The solid curve corresponds to the acoustic branch, giving ω = a for k⊥ = 0 and ω ∼ k⊥ cs for large k⊥ . The dotted curve corresponds to the gravity branch, giving ω = 0 when k⊥ = 0 with an asymptote ω ∼ g for large k⊥ . For γ = 5/3, g = 0.98 a .

Consider (9.50) when 2 c2 2 4k⊥ s g

(2a + k2 c2s )2

 1;

we may expand the square root term binomially to obtain ω2 ≈ 2a + k2 c2s

(9.51)

for the plus (+) root, and ω2 ≈

2 c2 2 k⊥ s g

(2a + k2 c2s )2

(9.52)

for the minus (−) root. The plus root evidently corresponds to the acoustic wave, reducing to ω2 = k2 c2s in the absence of gravity (a = 0). The minus root gives the gravity wave. 2 /k2 , For wavelengths such that k2 c2s  2a , the two roots give ω2 = k2 c2s and ω2 = 2g k⊥ respectively. The relations (9.51) and (9.52) may also be used to give approximate expressions for the curves that separate the ω, k⊥ space into regions of vertical propagation (kz2 > 0) and vertical evanescence (kz2 < 0); see Figure 9.1. The separating curves are given by 2 . Accordingly, the setting kz2 = 0, and so (9.51) and (9.52) apply with k2 replaced by k⊥ approximate relations (9.51) and (9.52) apply when 2 c2 2 4k⊥ s g 2 c2 )2 (2a + k⊥ s

 1.

260

Gravitational Effects

In fact, this inequality is readily met for small or large k⊥ cs , since in either extreme the expression on the left of the inequality rapidly approaches zero. In fact, this expression has a maximum value of 2g / 2a , achieved when k⊥ cs = a , and this maximum is itself less than unity (since a > g ). Thus, the two modes are generally well separated. The sound wave propagates with ω2 > 2a and the gravity wave has ω2 < 2g . Thus, we may view a and g as providing cutoffs for the sound and gravity waves, with sound waves having frequencies above a and gravity waves having frequencies below g .

9.6 Acoustic-Gravity Waves: General Features Equation (9.46) with κ(z) defined by (9.47) affords us a means of describing in general terms some of the properties of its solutions that we may expect to arise. We have already seen that in an isothermal atmosphere, for which the sound speed cs , the density scale height H0 , the acoustic frequency a and the buoyancy frequency g are all constants, the quantity κ is also a constant. Thus, in an isothermal atmosphere it is of interest to discuss regions of the parameter space where κ 2 changes sign, for a change of sign in κ 2 implies a change in the spatial character of the perturbation Q(z). For κ 2 > 0, equation (9.46) indicates an oscillatory z-dependence in Q(z), the equation being equivalent to that of simple harmonic motion (though here in terms of spatial variable z, rather than time). For κ 2 < 0, equation (9.46) indicates an exponential behaviour in z. This behaviour can also be described in terms of the concavity of the function Q(z), it being concave towards the z-axis for κ 2 > 0 and concave away from the z-axis if κ 2 < 0. In an isothermal medium these properties apply to the whole of space. Moreover, other variables, such as the velocity uz or the pressure p, have essentially the same character as Q, since they are related to Q through factors (such as the amplification factor Z(z)) that do not change the oscillatory nature of the solution. Thus, in an isothermal medium the concept of wave cutoff is a robust one. In a non-isothermal medium, we may still discuss the concavity of Q(z), but now our conclusions are restricted to certain spatial regions, and the concept of cutoff is no longer robust. Moreover, our conclusions are restricted to each perturbation variable because each has its own effective κ 2 . In short, in contrast to an isothermal medium, a non-isothermal medium produces behaviour that is local in z and therefore does not carry general implications. The stratified nature of the medium leads also to wave refraction, whereby a wave motion is turned away from the stratified direction so that the wave motions are unable to reach certain spatial levels. We may illustrate this most simply by considering the special case of zero gravity, for with g = 0 the frequencies a and g are zero and equation (9.47) reduces to κ 2 (z) =

ω2 2 . − k⊥ c2s (z)

(9.53)

Although we have set gravity to zero, we have retained the possibility of a variable sound speed cs (z), though the equilibrium pressure p0 must be uniform; the equilibrium density ρ0 (z) is non-uniform, though ρ0 c2s is a constant.

9.7 Acoustic-Gravity Waves: Linear Temperature Profile

261

If we ignore the spatial variation in c2s , or consider it to be so slight that we may regard the medium as uniform, then equation (9.46) suggests a local dispersion relation of the form kz2 =

ω2 2 − k⊥ . c2s

(9.54)

This, of course, agrees exactly with (9.48) if the medium is isothermal and g = 0. Consider now what happens as we move spatially from a region in which cs is low to one in which cs is high, just as occurs when we move from the solar photosphere to the solar interior. We fix ω and k⊥ and ask what happens to kz2 . Suppose that in the region where cs is small we obtain kz2 > 0, corresponding to vertical propagation. Then, at greater depths where cs has increased in magnitude we expect a smaller kz2 and ultimately we see that (9.54) predicts a depth for which kz2 = 0; as we pass through this depth, the local character of the motion changes from vertically propagating to vertically evanescent. The wave propagating in the higher reaches of the atmosphere where cs is relatively small is refracted away from the vertical and becomes horizontal, and this happens by a depth z = zc at which kz2 falls to zero. Setting kz2 = 0, we obtain c2s (z = zc ) =

ω2 , 2 k⊥

an equation that determines zc for a given ω, k⊥ and sound speed profile. Such refraction effects are central to the behaviour of p modes in the solar interior. The sound wave is confined to a cavity below which it is unable to penetrate. Of course, these considerations here are based upon a crude application of our equations, but much the same behaviour may be demonstrated more formally. We turn to this topic in the next section. 9.7 Acoustic-Gravity Waves: Linear Temperature Profile The differential equation (9.41) may in principle be solved in a number of special cases. The case of a linear temperature profile, giving a profile for c2s (z) that is linear in z, is of particular interest. Consider, then, the profile     z z c2s (z) = c20 1 + , 0 (z) = 0 (0) 1 + , (9.55) z0 z0 

giving a uniform gradient c2s = c20 /z0 in the square of the sound speed and an associated uniform gradient 0  (z) = 0 (0)/z0 in the pressure scale height; the two profiles being related by c2s = γ g0 . For z0 > 0, the sound speed squared c2s increases linearly with depth z, being c20 at the reference level z = 0, where the pressure scale height is 0 (0) = c20 /γ g. Since the equilibrium temperature T0 is proportional to the square of the sound speed, this case corresponds to a uniform temperature gradient. For a given sound speed c0 at z = 0, the spatial scale z0 determines the steepness of the slope in c2s . With c2s (z) or 0 (z) specified by equation (9.55) we may evaluate the integral (9.10), obtaining   z0 z N0 (z) = ln 1 + . (9.56) 0 (0) z0

262

Gravitational Effects

The equilibrium pressure and density profiles follow immediately:     ρ0 (z) z m+1 z m p0 (z) , . = 1+ = 1+ p0 (0) z0 ρ0 (0) z0

(9.57)

Here m+1=

γg c2s



(9.58)

defines the index m in terms of the gradient in c2s . These expressions lead to a density scale height that increases linearly with depth z and a squared buoyancy frequency that decreases inversely with depth:   1 1 g 2g = m − (1 + m) , H0 = (z + z0 ), m γ (z + z0 ) while the acoustic cutoff frequency is given by 2a =

m2 c20 c2s = . 4z0 (z + z0 ) 4H02

It proves convenient to write  = e−k⊥ (z+z0 ) u so that

  du d = e−k⊥ (z+z0 ) − k⊥ u , dz dz

(9.59)

  2 du d2  −k⊥ (z+z0 ) d u 2 =e − 2k⊥ + k⊥ u . dz dz2 dz2

Further, writing s = 2k⊥ (z + z0 ), with k⊥ = 0, so that d d = 2k⊥ , dz ds

2 d2 2 d = 4k , ⊥ dz2 ds2

we may transform equation (9.41) into du d2 u + (m + 2 − s) + αu = 0, ds ds2 where the parameter α is defined by   gk⊥ (m + 1) ω2 1 + 2 m − (1 + m) − (m + 2). 2α = γ gk⊥ γ ω s

(9.60)

(9.61)

Equation (9.60) is a form of Kummer’s equation. In fact, the canonical form of Kummer’s equation is (Abramowitz and Stegun 1965) du d2 u + (b − s) − au = 0, (9.62) 2 ds ds from which we identify the parameters as a = −α and b = m + 2. Kummer’s equation (9.62) has independent solutions M(a, b, s) and U(a, b, s), where the Kummer functions M and U are confluent hypergeometric functions (Abramowitz and Stegun 1965) and are available in computer algebra systems such as MAPLE. s

9.8 Acoustic-Gravity Waves: Vertical Propagation

263

Accordingly, the solution of equation (9.60) may be written in the form u = AM(−α, m + 2, s) + BU(−α, m + 2, s), where A and B are arbitrary constants. The solution for  then follows as   = e−k⊥ (z+z0 ) AM(−α, m + 2, 2k⊥ z0 + 2k⊥ z) + BU(−α, m + 2, 2k⊥ z0 + 2k⊥ z) . (9.63) These solutions are important in magnetic helioseismology (see Section 9.12.2). Finally, we note that in the case of an atmosphere stratified neutrally or adiabatically, so that the buoyancy frequency is identically zero (g = 0), then for the linear case discussed here we have m = 1/(γ − 1) and the parameter 2α = mω2 /(gk⊥ ) − (m + 2); for γ = 5/3, m = 3/2.

9.8 Acoustic-Gravity Waves: Vertical Propagation It is useful to consider the special case of vertical propagation, supposing that motions u are entirely vertical and that all perturbation variables depend upon z and t only. We have seen in the previous section that gravity waves are unable to propagate vertically. Accordingly, by examining the case of vertical propagation we are removing gravity waves from consideration and concentrating upon the effect of stratification on sound waves propagating vertically. We consider the general case of a non-isothermal atmosphere (arbitrarily stratified sound speed cs (z)). Consider, then, motions of the form u = uz ez , with uz = uz (z, t), and associated pressure fluctuations p = p(z, t) and density variations ρ = ρ(z, t). Then with ∂/∂x ≡ 0 and ∂/∂y ≡ 2 ≡ 0, we may consider the reduced form of the general equations (9.26) 0, giving ∇⊥ and (9.28). However, it is in fact more illuminating to return to the fundamental linearized equations afresh, applied specifically to the case of parallel propagation. Accordingly, consider the one-dimensional forms of the continuity equation (9.17) and the momentum equation (9.18), ∂ρ ∂ + (ρ0 (z)uz ) = 0, ∂t ∂z ρ0 (z)

∂uz ∂p =− + gρ(z), ∂t ∂z

(9.64)

(9.65)

together with the one-dimensional form of isentropic energy exchange (see (9.19)),   ∂ρ ∂p   2 + p0 (z)uz = cs (z) + ρ0 (z)uz . (9.66) ∂t ∂t We may combine the equations of continuity and isentropy to yield ∂uz ∂p = −p0  (z)uz − ρ0 c2s . ∂t ∂z Then, taking the time derivative of the momentum equation (9.65) we obtain ρ0

∂ 2 uz ∂ 2 uz ∂uz 2 = ρ c (z) + [p0  + (ρ0 c2s ) − gρ0 ] + [p0  − gρ0  ]uz . 0 s ∂z ∂t2 ∂z2

(9.67)

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Gravitational Effects

The coefficient of the uz term vanishes by virtue of the equilibrium condition p0  = gρ0 , and the coefficient of the term in ∂uz /∂z simplifies to (ρ0 c2s ) = γ gρ0 . Accordingly, vertical motions uz (z, t) satisfy the equation (Lamb 1909, 1932) ∂ 2 uz ∂ 2 uz ∂uz 2 = c (z) + γg . s 2 2 ∂z ∂t ∂z

(9.68)

9.9 The Klein–Gordon Equation It is convenient to remove the term involving ∂uz /∂z from equation (9.68). We may do this by following much the same procedure as outlined in the general case discussed in Section 9.4. Specifically, we set uz (z, t) = Z (z)Q (z, t) with the aim of choosing the function Z (z) so that the resulting equation for Q has no ∂Q /∂z term. Now ∂ 2 Q ∂ 2 uz = Z ,  ∂t2 ∂t2

∂Q ∂uz = Z + Z  Q , ∂z ∂z

∂ 2 Q ∂Q ∂ 2 uz = Z + 2Z  + Z  Q .  ∂z ∂z2 ∂z2

Substituting these results into (9.68) and dividing by Z gives     2   ∂Q ∂ 2 Q Z  2 ∂ Q 2 Z 2 Z = c + 2c + γ g + γ g + c Q . s s s Z ∂z Z Z ∂t2 ∂z2 We now choose Z (z) so that the coefficient of the ∂Q /∂z term is zero, requiring d [ln(Z 2 ) + ln(p0 )] = 0; dz

(9.69)

1/2

that is, Z 2 p0 is a constant and so Z ρ0 cs is also a constant. Accordingly, we take  1/2 ρ0 (0)c2s (0) (9.70) Z (z) = ρ0 (z)c2s (z) and then

 uz (z, t) =

ρ0 (0)c2s (0) ρ0 (z)c2s (z)

1/2 Q (z, t).

(9.71)

Thus, after some algebra, we see that Q (z, t) satisfies ∂ 2 Q ∂ 2 Q 2 − c (z) + 2s (z)Q = 0, s ∂t2 ∂z2

(9.72)

where 2s (z) =

c2s [1 − 20  (z)]. 420

(9.73)

Equation (9.72) is the Klein–Gordon equation, a generalization of the wave equation (see Morse and Feshbach 1953). Thus, we see that the vertical propagation of a sound wave in a stratified atmosphere leads to the Klein–Gordon equation. The quantity s (z) has the dimensions of frequency which, as discussed earlier in general terms, imposes a timescale

9.10 Significance of the Klein–Gordon Equation

265

on the system. In an isothermal atmosphere, 0  = 0 and s reduces to the acoustic cutoff frequency a . It should be noted that the function Z that arises here is distinct from the function Z in the general case discussed in Section 9.3. Here Z 2 p0 is a constant, whereas in Section 9.3 it is Z 2 p0 c2s that is shown to be a constant. The reason for the difference is that in a stratified medium every variable has its own z-dependence. In our treatment here we have considered the vertical velocity uz , whereas in the earlier general treatment we considered the compression ; for the case of vertical propagation,  becomes ∂uz /∂z and this behaves differently from uz .

9.10 Significance of the Klein–Gordon Equation The Klein–Gordon equation arises in many wave propagation problems, including the case of a vertically propagating sound wave in a stratified atmosphere and the case of waves in a thin magnetic flux tube (see Chapters 10 and 11). It is of interest, therefore, to better understand its properties. Here we investigate the solution of the initial value problem, making clear the importance of the imposed timescale that the Klein–Gordon equation automatically introduces into any solution. We write the Klein–Gordon equation in the form ∂ 2Q ∂ 2Q − c2 2 + 2 Q = 0, 2 ∂t ∂z

(9.74)

where the propagation speed c and the frequency  are constants. In the case of a vertically propagating sound wave in an isothermal atmosphere, the speed c is the sound speed cs and  is the acoustic frequency a : c = cs and  = a . The significance of the Klein–Gordon equation is that it introduces a timescale imposed by the equilibrium. The presence of  imposes a natural timescale, −1 , making the Klein– Gordon equation distinct from the one-dimensional wave equation, to which it reduces when  = 0. Indeed, equation (9.74) has a simple solution of the Fourier form Q(z, t) = Q0 e i(ωt−kz z) , where the frequency ω and wavenumber kz are related by the dispersion relation ω2 = kz2 c2 + 2 ,

(9.75)

and Q0 is an arbitrary constant. It is evident from the form of this dispersion relation that wave propagation (requiring kz2 > 0) occurs only if ω2 > 2 , i.e., only for frequencies ω above the cutoff value . For frequencies below the cutoff , kz2 < 0 and so there is no propagation; the motion is evanescent. A distinctive consequence of the presence of the frequency  is seen if we construct a general Fourier solution of the form (see, for example, Whitham 1974)  ∞  ∞ A+ (κ)e i(κz+ω(κ)t) dκ + A− (κ)e i(κz−ω(κ)t) dκ, (9.76) Q(z, t) = −∞

−∞

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Gravitational Effects

where the function ω(κ) is defined through the dispersion relation (9.75): ω(κ) = (κ 2 c2 + 2 )1/2 .

(9.77)

The functions A+ (κ) and A− (κ) are determined by initial conditions that are applied to the Klein–Gordon equation. We may conveniently illustrate (9.76) by considering the initial conditions: Q(z, t = 0) = 0,

∂Q z (z, t = 0) = A0 δ( ). ∂t 0

(9.78)

Here δ(s) denotes the Dirac delta function of variable s. These initial conditions correspond to Q being generated from rest (Q = 0 when t = 0) by an impulsive source concentrated at the point z = 0 (and so ∂Q/∂t is initially zero everywhere except at the origin z = 0, where it is infinite). The delta function has been written as a function of the dimensionless variable z/0 , the spatial scale 0 being chosen to non-dimensionalize z. The * ∞delta function δ(s) has the property that it is zero everywhere except at s = 0 and that −∞ δ(s)ds = 1. We have also included a weighting of the delta function source through the constant A0 , but since the Klein–Gordon equation is linear it is clear that this weighting will only provide a multiplicative constant to the solution. Other solutions can be constructed in much the same way or through use of the Laplace transform (see, for example, Sutmann, Musielak and Ulmschneider 1998). In the case of a sound wave, these conditions correspond to waves being generated initially from rest (no vertical motion at t = 0) by an impulsive source at z = 0. Applying the initial condition on Q requires that  ∞ [A+ (κ) + A− (κ)]eiκz dκ = 0. −∞

Thus we take A− (κ) = −A+ (κ), and then  ∞ Q(z, t) = 2iA+ (κ)eiκz sin[ω(κ)t]dκ. −∞

From this expression we calculate the time derivative of Q as  ∞ ∂Q(z, t) 2iA+ (κ)ω(κ)eiκz cos[ω(κ)t]dκ, = ∂t −∞ and then

)  ∞ ∂Q(z, t) )) = 2iA+ (κ)ω(κ)eiκz dκ. ∂t )t=0 −∞

Thus, the second initial condition requires that    ∞ z iκz 2iA+ (κ)ω(κ)e dκ = A0 δ . 0 −∞ Recalling the Fourier representation of the delta function (Lighthill 1958), namely  ∞  ∞ 1 δ(s) = e−2π iκs dκ = eiκs dκ 2π −∞ −∞

9.10 Significance of the Klein–Gordon Equation

267

(the second expression following from the first representation of δ(s) by a change of integration variable from κ to −2π κ), allows us to equate expressions to obtain 2iA+ (κ)ω(κ) = A0 0 · Thus, we obtain 1 Q(z, t) = A0 0 · 2π



∞ −∞

1 . 2π

1 iκz e sin[ω(κ)t]dκ. ω(κ)

Finally, on expanding the exponential term through use of Euler’s relation (eix = cos x+ i sin x) and noting that ω(κ) is an even function of κ, we obtain  1 ∞ 1 sin[ω(κ)t] cos[κz]dκ. (9.79) Q(z, t) = A0 0 · π 0 ω(κ) This integral expression provides a solution of the Klein–Gordon equation (9.74) subject to the initial conditions (9.78). An expression of this form was first obtained by Lamb (1909, 1932) in his investigation of the vertical propagation of sound in an atmosphere. A discussion of its use in the description of magnetohydrodynamic waves in magnetic flux tubes was first given by Roberts (1981a) and Rae and Roberts (1982); see also Spruit and Roberts (1983). The expression (9.79) is deceptively simple looking but it is far from clear what information it is giving. Fortunately, the integral that arises may be evaluated (see below) in terms of the Bessel function J0 . But before giving the general treatment, it may be helpful to consider the special case of  = 0, when the Klein–Gordon equation reduces to the one-dimensional wave equation. 9.10.1 Case = 0: The Wave Equation In the special case  = 0, the function ω(κ) reduces to cκ (for κ > 0) and then expression (9.79) reads  1 ∞ 1 sin[cκt] cos[κz]dκ. (9.80) Q(z, t) = A0 0 · π 0 cκ Recalling the familiar trigonometric formula (Abramowitz and Stegun 1965, p. 72) 2 sin A cos B = sin(A + B) + sin(A − B), we may rewrite Q in the form Q = A0 0 ·

1 2π

 0

∞

  1 sin[(z + ct)κ] − sin[(z − ct)κ] dκ. κc

(9.81)

The integrals arising here may be evaluated by using the result (Abramowitz and Stegun 1965, p. 78) ⎧ ⎪ β > 0, ⎪ 1/2,  ∞ ⎨ 1 1 sin(βκ)dκ = 0, β = 0, ⎪ π 0 κ ⎪ ⎩−1/2, β < 0.

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Gravitational Effects

Allowing for symmetry about z = 0, the result is  0, |z| > ct, Q(z, t) = A0 0 · 1 −ct < z < ct. 2c ,

(9.82)

Thus, we obtain two wavefronts moving symmetrically apart about z = 0 with the speed c; there is no oscillation present within this disturbance. The solution (9.82) can also be obtained directly from the wave equation by use of D’Alembert’s well-known general solution. The general solution of the one-dimensional wave equation is Q = f (z − ct) + g(z + ct), for arbitrary functions f and g. Allowing for the initial condition that Q = 0 at t = 0, we have Q = f (z − ct) − f (z + ct).

(9.83)

We then require that −2cf  (z) = A0 · δ



z 0



in order to satisfy the initial condition on ∂Q/∂t, resulting in    1 z z δ f (z) = −A0 · dz; 2c −∞ 0

(9.84)

this gives zero if z < 0 (because then the delta function is zero * ∞everywhere within the integration region) and −A0 0 /(2c) if z > 0 (for then the result −∞ δ(s)ds = 1 applies). The result is Q(z, t) = −A0 0 ·

1 [H(z − ct) − H(z + ct)], 2c

where H(z) denotes the Heaviside function,  1, H(z) = 0,

(9.85)

z > 0, z < 0.

Accordingly, we recover the result (9.82).

9.10.2 Case = 0: The Solution of the Klein–Gordon Equation Returning to the general case of  = 0, we consider the expression (9.79), namely  1 1 ∞ 1/2 Q(z, t) = A0 0 · sin[(κ 2 c2 + 2 ) t] cos[κz]dκ. (9.86) 1/2 π 0 (κ 2 c2 + 2 ) Now Lamb (1909; see also Erd´elyi 1954) has evaluated an integral of this form, giving the result (following a minor rewrite):  ,  1 J0 ( β 2 − α 2 ), β 2 > α 2 , 1 ∞ 1 2 1/2 2 sin[(1 + k ) β] cos[kα]dk = π 0 (1 + k2 )1/2 0, β 2 < α2.

9.11 Vertical Magnetic Field

269

Here J0 denotes the Bessel function of zero order. Applied to (9.86), we obtain  0, |z| > ct, Q(z, t) = A0 0 · 1  2 2 2 1/2 ], |z| < ct. 2c J0 [ c (c t − z )

(9.87)

This is the desired solution of the Klein–Gordon equation (9.74) subject to the initial conditions (9.78). Notice that this solution recovers the simpler result (9.82) for the wave equation if we proceed to the limit  → 0, for then the Bessel function J0 is evaluated at zero, producing unity; thus we recover (9.82). The importance of the result (9.87) is that it demonstrates the presence of two wavefronts that move with the speed c in opposite directions away from the generation source at z = 0. The disturbance ahead of a wavefront is at rest, just as with the simpler case of the wave equation. But behind the wavefront an oscillating wake is set up; the wake oscillates at the frequency , the natural frequency of the medium. Thus, the cutoff frequency  is significant in that we may expect any impulsively excited wave (satisfying the Klein– Gordon equation (9.74)) to exhibit a wake that oscillates at the frequency . For locations well behind the wavefront (so that z2  c2 t2 ) and for times much longer than the period of the wake (so that t  1), we may approximate the Bessel function by its asymptotic value (Abramowitz and Stegun 1965, p364)  1/2

2 π cos x − , x → +∞, J0 (x) ∼ πx 4 resulting in 1 Q(z, t) ∼ A0 0 · 2c



2 π t

1/2

π cos t − , 4

|z|  ct.

(9.88)

The amplitude of the oscillation Q behind the wavefront declines in time as the wave moves on. What happens to the various components of motion depends on how they are related to Q. In the case of a vertically propagating sound wave in an isothermal atmosphere, the vertical motions u are amplified in height. This follows from (9.71), which shows that |u| ∝ −1/2 p0 |Q|; so, for bounded |Q|, |u| grows like ez/(20 ) in an isothermal atmosphere, giving an e-folding distance of two pressure scale heights, 20 . For example, with 0 = 125 km this amounts to an increase by a factor of e in 250 km. Thus, a 1 km s−1 photospheric wave motion would be amplified to a value equal to a sound speed of 7.5 km s−1 in about 500 km, emphasizing the likelihood of shocks or other physical effects (e.g. radiative losses or viscous heating) occurring and acting to complicate this picture. On the other hand, in the corona a wave amplitude at the base of a coronal loop would be amplified by a factor of only 1.78 in reaching the apex of a loop of height 100 Mm.

9.11 Vertical Magnetic Field We turn now to a consideration of the combined effects of gravity and magnetism. We examine first the case of a vertical magnetic field and then in Section 9.12 we turn to a

270

Gravitational Effects

horizontal magnetic field. We consider a uniform magnetic field B0 ez . The equilibrium magnetic field is taken to be pointing downwards in the z-direction, aligned with gravity. The uniform magnetic field offers no support against gravity so the medium is stratified hydrostatically, with the equilibrium pressure p0 (z) and density ρ0 (z) being related by p0  = gρ0 (z).

(9.89)

Again, a prime  denotes differentiation of an equilibrium quantity with respect to depth z. The linear system of equations describing the perturbations about this equilibrium are readily determined and may be written in the form ∂ρ = −ρ0  − ρ0  uz , ∂t ∂B ∂u = B0 − B0  ez , ∂t ∂z

∂u B0 ∂B = −grad pT + + gρ ez , ∂t μ ∂z ∂p = −p0  uz − ρ0 c2s . ∂t

ρ0

(9.90)

Here the perturbing motion is u = (ux , uy , uz ) with compression  = div u, and B = (Bx , By , Bz ) is the perturbed magnetic field. It is convenient to assume that ∂/∂y ≡ 0 (so that the y-axis is an ignorable coordinate). It is also convenient to introduce the total pressure perturbation pT , given by pT = p +

B0 Bz . μ

(9.91)

The four linear equations (9.90), arising from the equations of continuity, momentum (with a magnetic field and gravity), ideal induction and isentropic conditions, define our system. Differentiating the momentum equation with respect to time t and using the other three equations to eliminate variables, we obtain     ∂ 2u ∂pT ∂ 2 ux ∂ 2 ux 2 ρ0 2 = −grad ex − + ρ0 cA ez − g(ρ0  + ρ0  uz ) ez . (9.92) ∂t ∂z∂x ∂t ∂z2 We need also an expression for the time derivative of pT , which we can obtain easily from the definition (9.91) and the expressions arising in equations (9.90): ∂uz ∂pT = −ρ0 (c2s + c2A ) + ρ0 c2A − p0  uz ∂t ∂z ∂ux = −ρ0 c2s  − ρ0 c2A (9.93) − gρ0 uz , ∂x where we have used the equilibrium condition (9.89). Thus, combining equations (9.92) and (9.93) we find that linear compressible motions (ux , uy , uz ) with ∂/∂y = 0 satisfy the partial differential equations   ∂ 2 ux ∂ 2 ux ∂ 2 ux ∂ ∂uz 2 2 2 2 − cA (z) 2 − (cs (z) + cA (z)) 2 = cs (z) + g (9.94) 2 ∂z ∂x ∂t ∂z ∂x ∂ 2 uy ∂ 2 uy − c2A (z) 2 = 0 2 ∂t ∂z   2u ∂ 2 uz ∂ ∂uz ∂ ∂ux z 2 2 − c (z) − γ g (z) = c + (γ − 1)g . s s 2 2 ∂z ∂z ∂x ∂t ∂z

(9.95) (9.96)

9.11 Vertical Magnetic Field

271

Equations of this form were first derived by Ferraro and Plumpton (1958; see also Ferraro 1954).1 They apply for sound speed cs (z) and Alfv´en speed cA (z), each functions of depth z. The coupled partial differential equations (9.94) and (9.96) describe fast and slow waves in the presence of stratification. Transverse motions uy are uncoupled from the magnetoacoustic waves and satisfy the wave equation with variable speed cA (Ferraro 1954); these are Alfv´en waves in a stratified atmosphere. It may be noted that in the absence of gravity, g = 0, equations (9.94) and (9.96) may be combined to yield the familiar fourth order wave equation for magnetoacoustic waves (see Chapter 2): 2 2 ∂ 4 2 2 ∂ 2 2 2 ∂ 2 − (c + c ) ∇  + c c s s A 2 ∇  = 0, A ∂t4 ∂t2 ∂z

(9.97)

where  = div u is the divergence of u and ∇ 2 here denotes the two-dimensional Laplacian: ∂ 2 ∂ 2 + 2. ∂x2 ∂z Equation (9.95) becomes the simple one-dimensional wave equation when g = 0. ∇ 2 =

9.11.1 Alfv´en Waves in a Stratified Medium It is of interest to examine the Alfv´en wave equation (9.95) in an isothermal medium, for which a solution is readily obtained. It is convenient to use the height s as our variable in place of depth z (note that s = −z). In an isothermal medium, the equilibrium density declines exponentially fast with height s so the Alfv´en speed cA increases with height (but at half the rate):  1/2     B20 s s ρ0 (s) = ρ0 (0) exp − = cA (0) exp , cA (s) = , (9.98) 0 (μρ0 (s)) 20 for constant scale height 0 . Accordingly, for a time dependence of the form eiωt , the Alfv´en wave equation yields   d2 uy s ω2 exp − + (9.99) uy = 0. 0 ds2 c2A (0) Introduce a new variable in place of s: set   s 2ω0 2ω0 exp − . ζ = = cA (0) 20 cA (s) Then d 1 d =− , ds 20 dζ

(9.100)

  2 d 1 d d2 = , ζ2 2 + ζ dζ ds2 dζ 420

1 Ferraro and Plumpton choose the z-axis to be pointing vertically upwards, whereas here we have taken the z-axis as pointing

vertically downwards. This produces sign changes in certain expressions, when comparing the two formulations.

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Gravitational Effects

and so equation (9.99) becomes ζ2

d2 uy duy +ζ + ζ 2 uy = 0. 2 dζ dζ

(9.101)

Equation (9.101) is Bessel’s equation for Bessel functions of zero order, with general solution uy = A0 J0 (ζ ) + A1 Y0 (ζ ) for arbitrary constants A0 , A1 . Hence, in terms of the original variable s we have       s s 2ω0 2ω0 uy = A0 J0 exp − exp − + A1 Y0 . cA (0) 20 cA (0) 20

(9.102)

A solution of this form was obtained by Ferraro and Plumpton (1958; see also Ferraro 1954). To illustrate the solution (9.102) we consider a coronal loop of length L which is symmetric about its apex. To treat this case we imagine the loop to be straightened out and to be symmetric about its apex, so that we may treat one half of the loop stratified vertically from its base at s = 0 to its apex at s = L/2. For a disturbance vanishing at the base we require uy = 0 at s = 0. Accordingly, we take    J0 (0 )  L − s  − s  0≤s≤ , Y0 0 e 20 , (9.103) uy = A0 J0 0 e 20 − Y0 (0 ) 2 where 0 =

2ω0 . cA (0)

(9.104)

We consider an Alfv´en mode of oscillation that disturbs the apex (s = L/2) of the loop, requiring that we satisfy the boundary condition ∂uy =0 ∂s

at s =

L . 2

This implies   − L  J1 0 e 40 J0 0 ) . =   − L  Y0 0 ) Y0 0 e 40

(9.105)

This is the dispersion relation determining the dimensionless frequency 0 of standing Alfv´en waves in a gravitationally stratified isothermal loop of length L. We should contrast relation (9.105) with what arises in an unstratified loop. With g = 0, we have d2 uy ω2 uy = 0. + ds2 c2A (0)

(9.106)

The solution to this differential equation is simply sin and cos functions of ωs/cA . Requiring that uy = 0 at the base s = 0 gives   ωs L , 0≤s≤ . (9.107) uy = A2 sin cA (0) 2

9.11 Vertical Magnetic Field

273

For the fundamental mode of the loop as a whole, we choose duy /ds = 0 at the loop apex, giving ω = π cA (0)/L,

(9.108)

with a corresponding period in a uniform loop of Punif = 2L/cA (0); cA (0) denotes the Alfv´en speed at the loop base (which is also the Alfv´en speed throughout the uniform loop). In the stratified loop, the corresponding period is Pstrat = 4π 0 /(0 cA (0)) with 0 determined by the transcendental equation (9.105). Now dispersion relation (9.105) does not appear to have been investigated in the literature. However, we may readily illustrate the information to be obtained from it. Suppose we have an isothermal coronal loop of length L = 100 Mm (so the loop apex is at z = 50 Mm) in a medium with scale height 0 = 90 Mm, and suppose that the Alfv´en speed at the loop base is cA (0) = 200 km s−1 . In a uniform unstratified medium (g = 0) these conditions admit of a period Punif = 2L/cA (0) = 1000 s. By contrast, the period of a standing wave in a stratified loop is Pstrat = 4π 0 /(0 cA (0)) = 827 s, with 0 ≈ 6.84 determined by a simple numerical investigation of relation (9.105). For a loop that is twice the length, at L = 200 Mm, we obtain Punif = 2000 s and Pstrat = 1377 s (for 0 ≈ 4.105). Finally, it is of interest to notice what happens if instead of solving the wave equation (9.95) we were content with an estimate of the period. The simplest estimate is that provided by the uniform medium derivation given above, but we can see from our illustration that at least for long loops stratification has a significant effect and the uniform medium provides at best a rather crude estimate. An improved estimate comes from determining the travel time along the loop. Denote by τ travel the travel time along the whole tube and back again. This is four times the travel time from the base (s = 0) of the loop to the apex (s = L/2):  L/2  1 80  τ travel = 4 ds = 1 − e−L/(40 ) , cA (s) cA (0) 0 the integration being carried out by elementary means. (In the limit 0 → ∞, or equivalently for L  0 , this result reduces to τ travel = 2L/cA (0), the fundamental period of oscillation in a uniform tube.) For the parameters chosen above, namely a loop of length L = 200 Mm in an atmosphere with scale height 0 = 90 Mm and an Alfv´en speed at the base of cA (0) = 200 km s−1 , we obtain a travel time of τ travel = 1534 s, or about 11% above the period determined from relation (9.105).

9.11.2 Magnetoacoustic Waves in a Stratified Medium Magnetoacoustic waves satisfy the coupled wave-like equations   ∂ 2 ux ∂ 2 ux ∂ 2 ux ∂ ∂uz 2 2 2 2 = cA (z) 2 + (cs (z) + cA (z)) 2 + cs (z) + g , 2 ∂z ∂x ∂t ∂z ∂x   ∂ 2 uz ∂ 2 uz ∂uz ∂ ∂ux 2 2 = c (z) + γ g (z) + c + (γ − 1)g . s s 2 2 ∂z ∂z ∂x ∂t ∂z

(9.109) (9.110)

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Gravitational Effects

This is a complicated set of coupled partial differential equations which hides its behaviour from us. It is, however, possible to obtain series solutions of equations (9.109) and (9.110), as first shown by Ferraro and Plumpton (1958), but the information so gleaned remains limited. Moreover, solutions in terms of special functions are also possible (Zhugzhda 1979; Hollweg 1979; Zhugzhda and Dzhalilov 1982; Leroy and Schwartz 1982; Schwartz and Leroy 1982; Moreno-Insertis and Spruit 1989; Hasan and ChristensenDalsgaard 1992; Cally 2001; Hague and Erd´elyi 2016; Mather and Erd´elyi 2016), but again the information obtained remains limited. Numerical schemes are often required, especially where specific solar objects such as sunspots are being considered (see, for example, Scheuer and Thomas 1981). Nonetheless, we may extract some information from equations (9.109) and (9.110) through exploring the equations under special conditions. Vertical Propagation Perhaps the most obvious special case requiring our attention is that of vertical propagation, when the partial differential equations decouple. Suppose that ∂/∂x = 0, so that there is no dependence upon the transverse coordinate x. Then equations (9.109) and (9.110) become ∂ 2 ux ∂ 2 uz ∂ 2 uz ∂uz ∂ 2 ux 2 2 = c (z) , = c (z) + γg . (9.111) s A 2 2 2 2 ∂z ∂t ∂z ∂t ∂z The equation for transverse motions ux coincides with equation (9.95) for Alfv´en waves, and so may be treated much as discussed in Section 9.11.1. The second equation is precisely equation (9.68) which arises in the description of acoustic-gravity waves in a non-magnetic atmosphere – see Section 9.8 – and, as noted earlier, this equation may itself be written in the form of the Klein–Gordon equation: ∂ 2Q ∂ 2Q 2 − c (z) + 2s (z)Q = 0, s ∂t2 ∂z2 where 1

uz = e− 2 N0 (z) Q,



z

N0 (z) = 0

ds 0 (s)

(9.112)

(9.113)

and s is the acoustic cutoff frequency of a non-magnetic atmosphere: 2s (z) =

c2s [1 − 20  (z)]. 420

(9.114)

Thus, we expect any phenomena described by equations (9.109) and (9.110) to display features in common with an Alfv´en wave in a stratified medium (described in Section 9.11.1) combined with the Klein–Gordon equation and its cutoff frequency. The Slow Mode The Klein–Gordon equation arises from system (9.109) and (9.110) in another way also. Roberts (2006) has pointed out a way of extracting information about the slow mode in these equations. Introduce a scaling of the form x = X,

ux = Ux ,

(9.115)

9.12 Horizontal Magnetic Field

275

leaving other variables (z, t and uz ) unscaled. Then ∂ 1 ∂ = . ∂x  ∂X Under this scaling the form of equation (9.110) is unchanged but equation (9.109) reads    2 2   2  2 ∂uz 2 ∂ Ux 2 ∂ 2 ∂ 2 ∂ cs + cA + cs − cA 2 Ux . (9.116) +g = ∂z ∂X ∂X 2 ∂t2 ∂z For  2  1 we can neglect the right-hand side of equation (9.116) and then integrate once with respect to X; in terms of the original variables, the result is  ∂ux  2 ∂uz (9.117) = −c2s − guz . cs + c2A ∂x ∂z Equations (9.110) and (9.117) are the governing equations giving an approximate description of slow waves in a stratified magnetic atmosphere (Roberts 2006). Eliminating ux between these two equations results in  4   2 ∂ 2 uz ct ∂uz g c2t c2t 2 ∂ uz 2 − c − γ g + (9.118) +  uz = 0, t g 0 c2s ∂t2 ∂z2 c4s ∂z c2A where g is the buoyancy frequency defined earlier (in equation (9.15)). We may cast our governing equation in terms of the Klein–Gordon equation. Set   ρ0 (z)c2t (z) 1/2 Q= uz (9.119) ρ0 (0)c2t (0) and then after some algebra we obtain (Roberts 20062 ) 2 ∂ 2Q 2∂ Q − c + 2str (z)Q = 0, t ∂t2 ∂z2

where

 2str (z)

=

c2t

1 420



ct cs

4

    1 1 g c2t c2t 2 + γ g 4 + 2 g + . 2 0 c2s cs cA

(9.120)

(9.121)

An alternative derivation of an equation equivalent to (9.120) was given by Webb (1980); see also Webb and Roberts (1978), Moreno-Insertis and Spruit (1989) and Hague and Erd´elyi (2016). It turns out that expression (9.121) for the cutoff frequency str in a uniform vertical magnetic field embedded in a stratified atmosphere arises also in a thin vertical magnetic flux tube; see Chapter 10, Section 10.8. Applications of this type of theory to sunspots and pores is discussed in Chapter 14, Section 14.14. 9.12 Horizontal Magnetic Field We turn now to a consideration of the combined effects of gravity and magnetism in the case when the equilibrium magnetic field is horizontal. Consider an equilibrium magnetic field 2 Roberts (2006) chooses the z-axis to be aligned vertically upwards, anti-parallel to the direction of gravity. This results in

some expressions changing sign.

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Gravitational Effects

B0 = (B0 (z), 0, 0) which is aligned with the x-axis. The field strength may vary with depth z, so there is an associated equilibrium current j0 = (B0  (z)/μ) ey . The tension force is zero but there is a pressure force − grad (B20 /2μ) acting in the z-direction, with equilibrium achieved by 

B2 p0 + 0 2μ



= gρ0 (z).

(9.122)

Equation (9.122) reduces to (9.4) in the absence of a magnetic field or in a uniform magnetic field. Perturbations about this equilibrium may be investigated much as for the vertical magnetic field case. Again, we take a perturbation u = (ux , uy , uz ) and suppose that ∂/∂y = 0. We may eliminate all variables in favour of u and (for algebraic convenience) the compression . The perturbation u satisfies the linear momentum equation ρ0

∂u = −grad p + j0 × B + j × B0 + gez ρ, ∂t

(9.123)

where j0 = B0 /μey is the current in the equilibrium state and 

∂By ∂Bx ∂Bz ∂By , − , μj = − ∂z ∂z ∂x ∂x

 (9.124)

gives the perturbation current density j. We may differentiate the momentum equation with respect to t and then use the components of the induction equation ∂Bx ∂ = − (B0 uz ), ∂t ∂z

∂uy ∂By = B0 , ∂t ∂x

∂uz ∂Bz = B0 , ∂t ∂x

(9.125)

to eliminate the perturbed magnetic field. After some algebra, we obtain    2 ∂ 2 ux ∂uz 2 ∂ ux 2 ∂ = cs 2 + g + cs , ∂z ∂x ∂t2 ∂x

(9.126)

∂ 2 uy ∂ 2 uy = c2A 2 , (9.127) 2 ∂t ∂x       B0 B0 2B0 B0 ∂uz ∂ 2 uz ∂ 2 uz ∂ 2 uz 2 2 ∂ = c + + c + (γ − 1)g − γ . + g + s A μρ0 ∂z ∂z μρ0 ∂t2 ∂x2 ∂z2 (9.128) The component in the ignorable direction, ey , satisfies a one-dimensional wave equation. The components ux and uz satisfy complicated coupled partial differential equations. If we take a Fourier dependence of the form ux (x, z, t) = ux (z) exp (iωt − ikx x),

9.12 Horizontal Magnetic Field

277

then the magnetoacoustic motions satisfy coupled ordinary differential equations. We may eliminate ux in favour of uz . Then uz satisfies the second order differential equation   d ρ0 (c2s + c2A )(ω2 − kx2 c2t ) duz dz dz (ω2 − kx2 c2s )     g2 ρ0 kx2 ρ0 c2s 2 2 2 2 + ρ0 (ω − kx cA ) + gkx − 2 (9.129) uz = 0, ω2 − kx2 c2s (ω − kx2 c2s ) where cA (z) = (B20 /μρ0 )1/2 is the Alfv´en speed in the medium and ct (z) is the slow speed (see Chapter 2 ) given by cs cA ct = . 2 (cs + c2A )1/2 Equation (9.129) was first discussed by Goedbloed (1971a) and Adam (1977); see also Roberts (1985b), Campbell and Roberts (1989) and Miles and Roberts (1992). It reduces to equation (9.42) in the absence of a magnetic field (cA = 0), and to equation (3.50) in Chapter 3 in the absence of gravity (g = 0). Notice that equation (9.129) is singular at ω2 = kx2 c2t , which is associated with the occurrence of the slow mode’s continuous spectrum (see Chapter 3).

9.12.1 Hydromagnetic Surface Waves One special case of our general system of equations that we should take note of is that of an incompressible medium. We can discuss this case directly by replacing the isentropic equation (9.19) with the requirement that div u = 0, but it is here easier to obtain the incompressible case as a limit of the general compressible one. We may do this by allowing the adiabatic index γ to tend to infinity; this causes the sound speed cs to tend to infinity (see Chapter 2). In the incompressible limit cs → ∞, the cusp speed ct becomes the Alfv´en speed cA and equation (9.129) reduces (on multiplication by −kx2 ) to     d  2 2 2 duz 2 2 2 2 (9.130) ρ0 (ω − kx cA ) − kx ρ0 (ω − kx cA ) − gρ0 uz = 0. dz dz Equation (9.130) governs the behaviour of two-dimensional incompressible motions (ux , 0, uz ) in the presence of a horizontal magnetic field (with equilibrium pressure balance determined by equation (9.122)). The simplest case to consider is that of a fluid with uniform density ρ+ embedded within a uniform horizontal magnetic field B0 = B+ ex (of strength B+ ) lying above a fluid of uniform density ρ− embedded within a uniform horizontal magnetic field B− ex (of strength B− ), so that   ρ+ , B+ , z > 0, z > 0, ρ0 (z) = B0 (z) = (9.131) ρ− , B− , z < 0, z < 0. In general there is a discontinuity in density ρ0 (z) and magnetic field strength B0 (z) at z = 0.

278

Gravitational Effects

Within each half of the plane, z > 0 or z < 0, the medium is uniform and so Equation (9.130) applies with ρ0  = 0. Thus, in both z > 0 and z < 0   2u d z (ω2 − kx2 c2A ) − kx2 uz = 0. (9.132) dz2 The factor (ω2 − kx2 c2A ) = 0 corresponds to the possibility of Alfv´en waves. Aside from this, there remains the possibility d2 uz − kx2 uz = 0 dz2 with solution uz (z) =

 A+ e −kx z , A−

e kx z ,

z > 0, z < 0,

(9.133)

(9.134)

for constants A− and A+ . We have imposed the condition that uz (z) is bounded as z → ±∞, choosing kx > 0. However, we may relate the solutions either side of the interface by imposing the condition that uz (z) is continuous across the interface at z = 0, and then A− = A+ . There is a second condition to be imposed on (9.134). To see this consider the differential equation (9.130) that holds for an arbitrary distribution of equilibrium density ρ0 (z) and magnetic field strength B0 (z). Of particular interest is the term involving ρ0  . Imagine a profile in ρ0 (z) that changes rapidly but continuously from ρ− in z < 0 to ρ+ in z > 0 over a narrow layer centred on z = 0. Such a profile produces a large ρ0  within the narrow layer. It is evident that for g = 0 the term   duz ρ0 (z) ω2 − kx2 c2A (z) dz arising in (9.130) cannot be continuous across z = 0, for then its derivative would not be able to balance the term involving gkx2 ρ0  (z)uz that would arise within the narrow layer. Consequently, ρ0 (ω2 − kx2 c2A )duz /dz must be discontinuous across z = 0. In fact, it must be that   duz (9.135) + gkx2 ρ0 (z)uz ρ0 (z) ω2 − kx2 c2A (z) dz is continuous across z = 0. We may see this more formally by integrating equation (9.130) across a thin layer around z = 0 that extends from z = − to z = +, where  is small. Then    +

+  2  duz + 2 2 2 = −gkx uz (0) ρ0  (z)dz = −gkx2 uz (0) ρ0 (z) . ρ0 (z) ω − kx cA (z) − dz − − Hence, letting  → 0, we obtain continuity of the quantity in (9.135). Applied to the solution (9.134), we obtain −kx A+ ρ+ (ω2 − kx2 c2A+ ) + gkx2 A+ ρ+ = kx A− ρ− (ω2 − kx2 c2A− ) + gkx2 A− ρ− ,

9.12 Horizontal Magnetic Field

279

where cA+ is the Alfv´en speed in the upper layer (z > 0) and cA− is the Alfv´en speed in the lower layer (z < 0). Finally, since A+ = A− , we obtain   ρ+ − ρ− ω2 = kx2 c2k + (9.136) gkx , ρ+ + ρ− where ck is the weighted mean Alfv´en speed of the layers: c2k =

ρ+ c2A+ + ρ+ c2A+ ρ+ + ρ−

.

(9.137)

Equation (9.136) is the well-known dispersion relation for hydromagnetic surface waves on an interface between two incompressible magnetized fluids (Chandrasekhar 1961; Priest 2014, chap. 7). In the absence of a magnetic field, we see that if ρ− > ρ+ , so that a denser fluid sits on top of a lighter fluid, then ω2 < 0 and the interface separating the two incompressible fluids is unstable; this is the Rayleigh–Taylor instability. In the stable case of ρ− < ρ+ , surface waves propagate according to the dispersion relation (9.136). In the extreme ρ−  ρ+ , we obtain ω2 = gkx , the dispersion relation for water waves on deep water (Lighthill 1978). The presence of a magnetic field provides a stabilizing influence, but, whatever the magnetic field strengths B− and B+ , instability may occur for sufficiently small wavenumber kx . Only if gravity is absent can we have a stable surface wave in all situations, and this then propagates with the mean speed ck . We note the occurrence of the mean Alfv´en or kink speed ck , familiar from our earlier discussion of waves on an interface or in magnetic flux tubes (see Chapters 4 and 6). Its occurrence in the discussion of surface waves is one of the earliest instances in which this important speed arises (Kruskal and Schwarzschild 1954). Its occurrence in flux tubes was first noted in Ryutov and Ryutova (1976), and subsequently developed by many authors especially Parker (1979a), Spruit (1982) and Edwin and Roberts (1983).

9.12.2 Magnetic Helioseismology Returning to the compressible case, we note that equations such as (9.129) have been used in an attempt to assess the influence of a chromospheric magnetic field on the behaviour of p and f modes, the fundamental modes of oscillation of the Sun as a global object (Campbell and Roberts 1989; Evans and Roberts 1990b, 1991, 1992; Jain and Roberts 1993, 1994, 1996). Similar studies have been made to assess the role of a magnetic field buried below the photosphere or convection zone (Roberts and Campbell 1986; Foullon and Roberts 2005). Reviews of these aspects of magnetic helioseismology are given in Roberts (1996) and Erd´elyi (2006a, b); see also the extensive recent discussion in Pinter and Erd´elyi (2018). To illustrate the sort of analysis that arises in these models we consider a field-free medium extending over z > 0 and representing the solar interior; the medium is taken to have a linear profile in c2s , modelling the run of sound speed in the solar interior quite well. Then the divergence  of velocity u is given by  = 0 ei(ωt−kx x) e−kx (z+z0 ) U(−α, m + 2, 2kx z0 + 2kx z),

z > 0,

(9.138)

280

Gravitational Effects

where 0 is an arbitrary constant and kx is the wavenumber aligned with the horizontal equilibrium magnetic field in the x-direction. This solution follows from equation (9.63) on imposing the boundary condition that the kinetic energy in the motion is bounded as z → +∞, which selects the confluent hypergeometric function U but rejects the function M. The constant z0 provides a measure of the slope in c2s , with z0 = c2s (z = 0)/(c2s (z)) . The upper region, z < 0, is taken to have a horizontal magnetic field, for which motions uz satisfy equation (9.129). Solutions in the magnetic atmosphere depend in detail upon what the magnetic field is there. Two cases have been investigated in some detail: the case of a constant Alfv´en speed, requiring a magnetic field strength that declines with height (Campbell and Roberts 1989); and the case of a uniform magnetic field (Evans and Roberts 1990b; Jain and Roberts 1993). Matching solutions in the magnetic atmosphere to the field-free zone below (where solution (9.63) applies) leads to a dispersion relation of the form   ω2 U  (−α, m + 2, 2kx z0 ) ω2 1 ω2 2 − 1+ (9.139) + (m + 1) = Fatm , gkx U(−α, m + 2, 2kx z0 ) kx z0 gkx gkx where U  denotes the derivative of the U function with respect to z: U  (−α, m + 2, z) =

d U(−α, m + 2, z). dz

The expression Fatm depends upon which case is being considered in the magnetic atmosphere: in the case of a constant Alfv´en speed it takes on a simple algebraic form, but for a uniform magnetic field it involves hypergeometric functions. Solutions of the dispersion relation (9.139) may be examined numerically or, in the case kx z0  1, analytically. Such studies lead to an understanding of how the magnetic chromosphere may influence the p and f modes of oscillation of the Sun. In the limit kx z0  1, and with the interior taken as neutral (so that its buoyancy frequency is zero), the dispersion relation (9.139) has the approximate solutions   n 2 (9.140) gkx , ω ≈ 1+ m for n = 1, 2, 3, . . . corresponding to the p modes, and n = 0 corresponding to the f mode. The corrections to these basic frequencies depend upon the magnetic atmosphere. Such calculations as these are used to argue how the modes may vary through the solar cycle, modelled as a change in the magnetic atmosphere (Evans and Roberts 1992; Jain and Roberts 1993, 1994, 1996).

10 Thin Flux Tubes: The Sausage Mode

10.1 Introduction This chapter is in the main concerned with describing how sausage modes may propagate in a thin magnetic flux tube. The equilibrium magnetic field on which a wave may propagate is assumed to be of the form B0 = (B0r , 0, B0z ),

(10.1)

representing (in cylindrical coordinates r, φ, z) a field that may have both a radial component B0r (r, z) and a longitudinal component B0z (r, z). This corresponds to a cylindrical tube the cross-section of which may vary along the tube. It includes the special but important case of a uniform tube for which B0r = 0 and B0z is a constant. There is however no twist: B0φ = 0. (Twist in a uniform magnetic flux tube is investigated in Chapter 7.) The complexity of a non-uniform tube is such that in the main progress is possible only through numerical means or through approximate analytical methods. It is such approximate analytical methods that we are concerned with here. The region of the solar atmosphere lying between the photosphere and the corona is extremely complicated. It contains an obvious energy source for wave motions in the granules and supergranules that reside in the convection zone, and it is a region of the Sun where stratification effects are most pronounced. Added to this is the complex architecture of the magnetic field, which in the photosphere tends to occur in concentrated structures (ranging from small-scale intense flux tubes through to pores and sunspots) but the field has spread out to fill the available plasma in the upper chromosphere and corona. This complex magnetic architecture, forming three-dimensional structures overlying regions that are largely field-free, is difficult to model and the description of MHD waves is especially complicated (see the reviews by Roberts 2004, 2008). One area where detailed theoretical studies are available is that of thin flux tubes. By assuming that magnetism is confined to a thin tube, which nonetheless may expand outwards with height, it is possible to give a detailed description of the various modes of oscillation of the tube. The applicability of such theories to regions high above the photosphere, where the field has expanded out, is difficult to assess fully, but it seems reasonable to suppose that many features predicted for thin tubes are useful markers as to what might actually occur in this complex zone. We have seen in Chapter 6 that certain modes of a magnetic flux tube can be described by approximate dispersion relations, derived from appropriate exact dispersion relations 281

282

Thin Flux Tubes: The Sausage Mode

that are however transcendental. The question arises: can we obtain such approximate dispersion relations not from first deriving an exact dispersion relation and then approximating its behaviour but from an ab initio treatment which looks at the outset for an approximate result? Such a treatment has the virtue of simplicity and also offers the potential for an extension to allow for physical effects perhaps not included in a first treatment. In particular, we have in mind the effect of gravity. To be specific, consider the dispersion relation for sausage modes in a uniform magnetic flux tube of radius a in the limit of long waves in a thin tube, which corresponds mathematically to longitudinal wavenumbers kz that satisfy the extreme kz a  1. The effects of gravity are ignored. We have seen in Chapter 6 that the sausage mode has a wave speed c that is determined approximately by

c2 ∼ c2t +

1 2



ρe ρ0

 (c2Ae − c2t )

(c2s − c2t ) (kz a)2 K0 (λt |kz |a), (c2s + c2A )

kz a  1,

(10.2)

where

λ2t =

(c2se − c2t )(c2Ae − c2t ) (c2se + c2Ae )(c2te − c2t )

and it is assumed that λt > 0. Equation (10.2) gives the square of the phase speed c (= ω/kz ) of the sausage mode as it propagates along the flux tube. The expression shows that the wave moves with approximately the cusp speed ct , and is accordingly a slow magnetoacoustic wave. The second term in (10.2), involving kz2 a2 and the Bessel function K0 , gives the dispersive correction to the square of the wave speed. Our interest here is in developing an approximate theory of the wave modes of a thin magnetic flux tube. Such a theory must be sufficiently accurate so as to recover the basic nature of the waves, as revealed in particular in relations such as (10.2). In this chapter we discuss the sausage mode, leaving to Chapter 11 a discussion of the kink mode.

10.2 Thin Flux Tube Theory: The Sausage Mode To develop a system of equations that give an approximate description of the waves we begin by considering the simple case of a uniform magnetic flux tube, the central axis of which is aligned with the z-axis of a cylindrical polar coordinate system (r, φ, z). To begin with we ignore the complications introduced by gravity, though we return to this issue at a later stage. Consider then the equations of ideal magnetohydrodynamics under isentropic conditions (see Chapter 1). We consider a flow u = (ur , 0, uz ) that has no twisting motion (uφ = 0)

10.2 Thin Flux Tube Theory: The Sausage Mode

283

and is symmetrical about the z-axis with ∂/∂φ = 0. In cylindrical polars, the equation of continuity may be written as ∂ρ 1 ∂ ∂ + (ρur ) + (ρur ) + (ρuz ) = 0, ∂t ∂r r ∂z

(10.3)

that is, ∂ρ ∂ρ ∂ρ + ρ + ur + uz = 0, ∂t ∂r ∂z where  denotes the compression:

(10.4)

∂uz 1 ∂ur + ur + . (10.5) ∂r r ∂z We expand all dependent variables in Taylor series about the z-axis (r = 0), assuming that the central axis of the tube is undisturbed so that there is no radial motion there (though there is a longitudinal motion); this assumption means that we are excluding kink modes but including both sausage and fluting modes. In this approximate theory there is no distinction between sausage and fluting modes, both being characterized by the same propagation speed in the thin tube limit of kz a  1. For convenience, we refer to these waves as sausage modes, as distinct from kink waves (which have a non-zero transverse motion ur at r = 0). With ur = 0 at r = 0, we have  = div u =

ur = rur (1) + r2 ur (2) + · · ·,

uz = uz (0) + ruz (1) + r2 uz (2) + · · ·,

ρ = ρ (0) + rρ (1) + r2 ρ (2) + · · ·,

 = (0) + r(1) + r2 (2) + · · ·,

(10.6)

where ur (1) , ur (2) , uz (0) , uz (1) , uz (2) , ρ (0) , ρ (1) , ρ (2) , (0) , (1) , (2) , etc. are all functions of z and t. This is the expansion procedure used originally by Roberts and Webb (1978), who focussed attention on the zeroth order effects represented by such terms as uz (0) , ρ (0) and (0) . A systematic treatment of higher order terms is also possible, and shows that the Taylor series in fact contain either odd or even terms in powers of r, as appropriate; for example, in turns out that ur (2) = 0, uz (1) = 0 and ρ (1) = 0 (Ferriz-Mas and Schussler 1989; Ferriz-Mas, Schussler and Anton 1990; Zhugzhda 1996, 2002). Moreover, the more terms that are carried in the expansion, the more accurate is the representation of the original unexpanded system (Stix 2004). In our treatment here only the leading order terms will be discussed. Further aspects of the thin tube theory are discussed in Zhugzhda (1996, 2002) and Zhugzhda and Goossens (2001). Substituting these expansions into the continuity equation (10.4) gives ∂ (0) (ρ + rρ (1) + · · ·) + (ρ (0) + rρ (1) + · · ·)((0) + r(1) + · · ·) ∂t + (rur (1) + r2 ur (2) + · · ·)(ρ (1) + 2rρ (2) + · · ·) ∂ + (uz (0) + ruz (1) + · · ·) (ρ (0) + rρ (1) + · · ·) = 0. (10.7) ∂z We may equate the various terms in this equation in powers of r; generally, only the lowest powers of r are of interest. In particular, the zeroth order equation (equivalent to taking r → 0) gives

284

Thin Flux Tubes: The Sausage Mode

∂ρ (0) ∂ρ (0) + ρ (0) (0) + uz (0) = 0. ∂t ∂z

(10.8)

This is the equation of continuity at zeroth order. Consider next the momentum equation. To begin with we ignore the complications introduced by gravity, including the pressure and magnetic forces only. Then   ∂u ρ + (u · grad)u = −grad p + j × B ∂t     B2 B = −grad p + + · grad B, (10.9) 2μ μ where B (= |B|) denotes the magnitude of the magnetic field B. For a flow u = (ur , 0, uz ) and magnetic field B = (Br , 0, Bz ), this becomes       ∂u ∂ ∂ B2 ρ + (u · grad)u = −grad p + + Br + Bz B, ∂t 2μ ∂r ∂z with B2 = B2r + B2z ; we have noted that



 ∂ ∂ (B · grad)B = Br + Bz B. ∂r ∂z

Consider the radial component of the momentum equation (10.9), expanding p and B in Taylor series in r: p = p(0) + rp(1) + r2 p(2) + · · ·, Br = rBr (1) + r2 Br (2) + · · ·,

(1) Bz = B(0) + r2 Bz (2) + · · ·. z + rBz

Symmetry about the central axis of the tube, the z-axis, requires that Br = 0 at r = 0; hence (0) the absence of a term Br . The radial component of the momentum equation yields p(1) +

1 (0) (1) B B = 0. μ z z

(10.10)

Consider the total pressure field within the tube, made up of the plasma pressure p and the magnetic pressure B2 /2μ: p+

2 B2 + B2z B2 B2 (B(0) z ) =p+ r = p + z + O(r2 ) = p(0) + + {terms in r2 } + · · ·. 2μ 2μ 2μ 2μ

Thus, the total pressure field is constant across the tube to within order r2 : (0)

p+

B2 (Bz )2 = p(0) + , 2μ 2μ

to good approximation. In other words, the total pressure is represented (to good approxi(0) mation) by the zeroth order pressure field, p(0) + (Bz )2 /2μ.

10.2 Thin Flux Tube Theory: The Sausage Mode

285

We assume that the total pressure within the tube is balanced on the tube boundary by the pressure just outside the tube, so that p+

B2 = πe , 2μ

(10.11)

where πe denotes the external pressure field on the boundary of the flux tube and includes the equilibrium external pressure and any variations in that pressure that may be brought about by disturbances. Thus, (0)

p(0) +

(Bz )2 = πe . 2μ

(10.12)

Turning to the momentum equation, we note that the longitudinal component yields ρ (0)

∂u(0) ∂u(0) ∂p(0) z z + ρ (0) u(0) =− . z ∂t ∂z ∂z

(10.13)

The magnetic field plays no role in the longitudinal force because j × B is perpendicular to the z-axis (to leading order in r). Equation (10.13) is the zeroth order form of the longitudinal momentum equation. Consider the induction equation. We have ∂B = u(div B) − B(div u) + (B · grad)u − (u · grad)B ∂t     ∂ ∂ ∂ ∂ u − ur + uz B = −B + Br + Bz ∂r ∂z ∂r ∂z on noting that div B = 0. Expansion in r leads, to zeroth order, to the result (0)

∂Bz ∂t

(0)

(0) + B(0) = −B(0) z  z

(0)

∂uz ∂Bz − u(0) . z ∂z ∂z

(10.14)

It should be noted that the solenoidal condition (div B = 0) for B is ∂Bz 1 ∂Br + Br + = 0, ∂r r ∂z which on expansion in r yields (0)

2B(1) r +

∂Bz ∂z

(1)

= 0.

(10.15)

(0)

This relation may be used to determine Br once Bz is known. Note that the solenoidal (0) condition does not impose a constraint on the z-dependence of Bz . Similarly, the expansion (10.6) applied to div u gives (0) = 2ur (1) +

∂uz (0) . ∂z

(10.16) (0)

This relation may be used to determine ur (1) in terms of uz and (0) .

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Thin Flux Tubes: The Sausage Mode

Finally, we have the isentropic condition, which we take in the form   ∂p γ p ∂ρ + u · grad p = + u · grad ρ . ∂t ρ ∂t

(10.17)

Expanding in powers of r produces     ∂p(0) ∂ ∂ ∂p(1) +r + · · · + u r + uz (p(0) + rp(1) + · · ·) ∂t ∂t ∂r ∂z      γ (p(0) + rp(1) + · · ·) ∂ρ (0) ∂ ∂ ∂ρ (1) (0) (1) = +r + · · · + ur + uz (ρ + rρ + · · ·) ∂t ∂t ∂r ∂z (ρ (0) + rρ (1) + · · ·) with ur and uz also to be expanded. To lowest order in r this yields   (0) (0) ∂ρ (0) (0) ∂p(0) ∂p ∂ρ γ p + u(0) = (0) + u(0) . z z ∂t ∂z ∂t ∂z ρ

(10.18)

This is the zeroth order form of the isentropic energy equation. It may be noted that we can eliminate the explicit occurrence of  using a combination (0) of the equations of continuity and induction by considering the quantity ρ (0) /Bz ; in particular, we have     ∂ ρ (0) ρ (0) ∂B(0) 1 ∂ρ (0) z − (0) = (0) ∂t B(0) ∂t ∂t Bz Bz z   (0) ∂ ρ (0) uz =− , (0) ∂z Bz on use of equations (10.8) and (10.14). Hence,     ∂ ρ (0) ∂ ρ (0) u(0) z + = 0. (0) ∂t B(0) ∂z Bz z

(10.19)

(0) If we write Bz A(z, t) = 0 for constant 0 , then we can cast equation (10.19) in the form

∂ ∂ (0) (ρ A) + (ρ (0) Au(0) z ) = 0. ∂t ∂z

(10.20)

A may be interpreted as the cross-sectional area of a thin tube, since flux conservation in a thin tube gives Bz A = constant.

10.3 Zeroth Order Equations: Uniform Tube It is convenient to gather together the zeroth order equations for the sausage mode. For ease of reading it is also convenient to make a notation change, now that the expansions in r have been performed. We write uz (z, t) = u(0) z ,

ρ(z, t) = ρ (0) ,

p(z, t) = p(0) ,

Bz (z, t) = B(0) z

10.3 Zeroth Order Equations: Uniform Tube

287

so that uz , ρ, p and Bz now denote the longitudinal velocity, the plasma density, the pressure and the magnetic field in the centre of the tube. In ideal MHD and ignoring gravity, the thin tube (or zeroth order) equations for the sausage mode are: ∂ ∂ (ρA) + (ρAuz ) = 0, ∂t ∂z 

∂uz ∂uz + uz ρ ∂t ∂z

p+

 =−

∂p , ∂z

B2z = πe , 2μ

(10.21)

(10.22)

(10.23)

Bz A = 0 ,

(10.24)

  ∂p ∂ρ γ p ∂ρ ∂p + uz = + uz . ∂t ∂z ρ ∂t ∂z

(10.25)

The external pressure is πe , and the magnetic flux of the tube is the constant 0 . Equations (10.21)–(10.25) form a nonlinear system of equations, similar in form to the one-dimensional equations of a non-magnetic gas. We recognize equation (10.21) as the continuity equation of a one-dimensional fluid in an elastic tube with cross-sectional area A (see Lighthill 1978); the tube’s cross-section may change in response to motions or it may be rigid (∂A/∂t = 0). Equation (10.22) is the one-dimensional momentum equation. Equation (10.23) expresses pressure balance in an elastic tube in which the total pressure within the tube is made up of the fluid pressure p and the magnetic pressure B2z /2μ, the sum of which is in balance with the surrounding pressure field πe ; pressure balance is related to area change through magnetic flux conservation (Bz A = constant), equation (10.24). Finally, motions take place under isentropic conditions as expressed by equation (10.25).

10.3.1 Equilibrium In the absence of any flow (uz = 0) or changes with time (∂/∂t = 0) the equilibrium form of equations (10.21)–(10.25) requires that the equilibrium pressure is uniform, p = p0 for constant p0 , so that ∂p/∂z = 0 to satisfy the momentum equation (10.22). Denote by B0 the strength of the uniform equilibrium magnetic field within the magnetic flux tube (Bz = B0 ) and pe the uniform pressure in the environment of the tube. Magnetic flux conservation (Bz A = 0 = constant) then requires that the equilibrium cross-sectional area A of the tube is also uniform (A = A0 = constant). We also assume that the density ρ0 within the tube is uniform; given a uniform equilibrium pressure p0 and density ρ0 , the ideal gas law gives a uniform temperature T0 within the tube. Total pressure balance requires that p0 +

B20 = pe . 2μ

(10.26)

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Thin Flux Tubes: The Sausage Mode

10.3.2 Perturbations Consider the linearized equations describing perturbations about the equilibrium (10.26). We write ρ = ρ0 + ρ1 ,

p = p0 + p1 ,

Bz = B0 + Bz1 ,

A = A 0 + A1 ,

uz = uz1 .

Then, the linear system of thin tube equations (10.21)–(10.25) becomes ∂ ∂uz1 (ρ0 A1 + A0 ρ1 ) + ρ0 A0 = 0, ∂t ∂z p1 +

1 B0 Bz1 = πe1 , μ

B0 A1 + A0 Bz1 = 0,

ρ0

∂uz1 ∂p1 =− , ∂t ∂z ∂ρ1 ∂p1 = c2s , ∂t ∂t

(10.27)

where πe = pe + πe1 is the pressure in the environment of the tube (made up of the equilibrium pressure pe and its perturbation πe1 ) and cs (= (γ p0 /ρ0 )1/2 ) denotes the sound speed within the tube. We may readily manipulate these equations so as to eliminate Bz1 and p1 , obtaining the result c2 1 ∂A1 ∂ρ1 1 ∂πe1 = 2 + ρ0 A2 , ∂t cs ∂t cs A0 ∂t where cA (= B0 /(μρ0 )1/2 ) denotes the Alfv´en speed in the tube. Combined with the continuity equation for a tube, this gives the result 1 ∂A1 1 ∂πe1 c2 ∂uz1 1 = − 2t − . A0 ∂t cA ∂z (c2s + c2A ) ρ0 ∂t We see here the occurrence of the sub-sonic, sub-Alfv´enic slow mode tube speed ct , defined by c2t =

c2s c2A c2s

+ c2A

,

1 1 1 = 2+ 2. 2 cs ct cA

(10.28)

Finally, we may differentiate the longitudinal momentum equation with respect to time t and use (10.27) to eliminate p1 , ρ1 and A1 in preference to uz1 and πe1 , with the result that disturbances uz (= uz1 ) satisfy the wave equation 2 ∂ 2 uz c2t 1 ∂ 2 πe1 2 ∂ uz − c = − . t ∂t2 ∂z2 c2A ρ0 ∂z∂t

(10.29)

An equation of the form (10.29) was first given in Roberts (1981c), derived for a slab geometry, following a related discussion for a cylindrical geometry given in Roberts and Webb (1979). If instead of eliminating A1 in preference to πe1 we retain A1 , then we may readily show that ∂ 2 uz 1 ∂ 2 A1 ∂ 2 uz − c2s 2 = c2s . 2 A0 ∂z∂t ∂t ∂z

(10.30)

10.3 Zeroth Order Equations: Uniform Tube

289

This equation shows clearly that in a rigid tube (∂A1 /∂t = 0) sound is simply propagated longitudinally along the tube at the sound speed cs . It is the elasticity of the tube that results in fluctuations in the cross-sectional area A1 and with it a propagation speed ct that is reduced below the sound speed. 10.3.3 Dispersion Relation Results such as equations (10.29) and (10.30) may be used to derive approximate dispersion relations such as (10.2). To see this, consider the thin tube equations for the sausage mode. It proves convenient to work in terms of the compression  = div u. The zeroth order continuity equation is ∂ρ (0) ∂ρ (0) + ρ (0) (0) + uz (0) = 0. ∂t ∂z When linearized about a uniform static equilibrium by writing ρ (0) = ρ0 + ρ1 ,

(0) = 1 ,

the zeroth order continuity equation yields ∂ρ1 + ρ0 1 = 0. ∂t It is convenient to introduce Fourier forms with z, t dependence of the form

(10.31)

exp(iωt − ikz z). Then the equation of continuity gives iωρ1 = −ρ0 1 ,

(10.32)

whilst the wave equation (10.29) yields c2t 1 ωkz πe1 . c2A ρ0

(10.33)

iωp1 = iωc2s ρ1 = −ρ0 c2s 1 .

(10.34)

(kz2 c2t − ω2 )uz = − Also, from equations (10.27) we have iωρ0 uz = ikz p1 , So

kz c2s ρ0 1 , πe1 = 3 (kz2 c2t − ω2 )(c2s + c2A )1 . (10.35) iω2 iω We need to calculate the radial component of velocity because it is this component of velocity that is continuous across the tube boundary (as discussed in Chapter 6). The radial component of velocity is not directly involved in the thin tube equations (10.27) but it is implicit in the form of the continuity equation involving 1 . From equation (10.31), we have 1 ∂ρ1 ρ1 1 = − = −iω , ρ0 ∂t ρ0 uz = −

whereas equation (10.16) gives

290

Thin Flux Tubes: The Sausage Mode

∂uz (0) = 2ur (1) − ikz uz (0) = 2ur (1) − ikz uz . ∂z Thus, to the lowest order in r to which we are working,   kz2 c2s 1 (1) ur = rur = r 1 − 2 1 2 ω   k2 c2 ω2 1 = r 1 − z 2s uz , r < a, 2 ω ikz c2s 1 = 2ur (1) +

(10.36)

where in simplifying the expression we have used relation (10.35). Expression (10.36) shows that for the sausage mode in a thin flux tube the radial motions ur are much smaller than the longitudinal ones uz . We can illustrate this by setting the wave speed equal to the tube speed, so that ω/kz = ct ; then, calculating ur on r = a we have ) ) 2 ) ur ) 1 ) ) ≈ |kz a| ct . )u ) 2 c2A z For example, in the photospheric case when cA = cs and kz a = 1/10 we obtain |ur /uz | ≈ 1/40. In the coronal case ur is even smaller: with cA = 5 cs and kz a = 1/10, we obtain |ur /uz | ≈ 1/500. In either case, the radial flow is all but negligible. Returning to the derivation of an approximate dispersion relation we note that we require that the radial motion is continuous across the tube boundary r = a. Outside the tube, in the tube’s environment, we have a uniform medium for which we have already seen that the radial motion ur is related to the gradient of the total pressure perturbation pT , so that (see equation (2.12) in Chapter 6) ur = −

dpT iω , ρe (kz2 c2Ae − ω2 ) dr

r > a.

(10.37)

Accordingly, requiring the two expressions (10.36) and (10.37) for ur in r < a and r > a to agree in the limit as r approaches a means that ) dpT )) 1 (ω2 − kz2 c2s ) ω  = . (10.38) a 1 2 ω2 iρe (kz2 c2Ae − ω2 ) dr )r=a Finally, eliminating 1 between equations (10.35) and (10.38) results in (ω2 − kz2 c2t )(c2s + c2A ) +

1 ρe 2 πe (ω − kz2 c2s )(ω2 − kz2 c2Ae ) dp ) T) 2 ρ0

= 0.

(10.39)

dr r=a

An expression of this form, though for the field-free case cAe = 0, was first given by Roberts and Webb (1979). Equation (10.39) may be referred to as the general dispersion relation for a sausage mode in a thin tube. The final step in our argument is to specify the form of the pressure perturbation in the environment of the tube. In Chapter 6 we have seen that we may write the total pressure perturbation, pT , in the form pT = Ae K0 (me r), with me (> 0) defined by

r>a

10.4 Zeroth Order Equations: Effects of Stratification

m2e =

(kz2 c2se − ω2 )(kz2 c2Ae − ω2 ) (c2se + c2Ae )(kz2 c2te − ω2 )

We then have πe1 = pT (r = a) = Ae K0 (me a)

and

291

.

) dpT )) = Ae me K0  (me a), dr )r=a

the prime  here denoting the derivative of the Bessel function (so K0  (me a) = dK0 (x)/dx evaluated at x = me a). Thus the dispersion relation (10.39) becomes 1 K0 (me a) 2 ρ0 (kz2 c2t − ω2 )(c2s + c2A ) = . a ρe (kz2 c2Ae − ω2 )(kz2 c2s − ω2 ) me K0 (me a)

(10.40)

This relation determines the speed of a wave in the flux tube. We can simplify further by noting that −1 K0 (x) ∼  K0 (x) xK0 (x) so we obtain the result 1 ω2 ∼ kz2 c2t + kz2 2



ρe ρ0

 (c2Ae − c2t )

as x → 0, (c2s − c2t ) (kz a)2 K0 (me a). (c2s + c2A )

(10.41)

This is equivalent to equation (10.2), equivalent to a result first obtained (for the fieldfree environment case, cAe = 0) by Roberts and Webb (1978). Thus we have recovered a complicated dispersion relation by using the thin tube equations, coupled with a description of the disturbance in the environment of the flux tube.

10.4 Zeroth Order Equations: Effects of Stratification The utility of the thin tube equations has been demonstrated in the above. The thin tube equations provide a tractable system that offers an approximate description of the behaviour of the sausage waves in the limit of long waves in a thin tube. We now consider the complications introduced by gravity. The introduction of gravity adds an additional force term, gρez , to the equation of momentum. This complicates our description of the waves mainly because it complicates the equilibrium state which becomes stratified. Thus, generally our equations are complicated by non-uniformity. The presence of gravity has no direct effect on the equations of continuity, of traverse pressure balance, of induction or of isentropic energy balance; only the longitudinal component of momentum has the addition of an extra term. Accordingly, in the presence of gravity we have the following set of nonlinear thin tube equations to describe the sausage mode: ∂ ∂ (ρA) + (ρAuz ) = 0, ∂t ∂z   ∂uz ∂uz ∂p ρ + uz =− + ρg, ∂t ∂z ∂z

(10.42)

(10.43)

292

Thin Flux Tubes: The Sausage Mode

p+

B2z = πe , 2μ

(10.44)

Bz A = 0 ,

(10.45)

  ∂p ∂p ∂ρ γ p ∂ρ + uz = + uz . ∂t ∂z ρ ∂t ∂z

(10.46)

In the above, g denotes the local gravitational acceleration (g = 274 m s−2 in the solar surface and atmosphere). Again, the external pressure field is πe and the constant 0 is the magnetic flux of the tube. In this description it is important to note that we have chosen the z-axis to point downwards, aligned with gravity. The system of equations (10.42)–(10.46) describes the sausage mode in a thin flux tube stratified by gravity. Equations of this form were first presented in Roberts and Webb (1978), derived by the expansion procedure about r = 0 This system of equations includes the special cases of an incompressible medium or an isothermal plasma, cases that have been considered in complementary discussions by Parker (1974) for the incompressible case and by Defouw (1976) for the isothermal case.

10.4.1 Stratified Equilibrium (g = 0) The equilibrium state of a thin magnetic flux tube standing vertically in a field-free atmosphere is one of hydrostatic pressure balance in the vertical direction, coupled with transverse pressure balance. At the centre of a thin flux tube standing vertically in a stratified atmosphere, the magnetic field provides no support for the plasma (since the j × B force is perpendicular to the magnetic field and so on the axis of the tube it acts purely transversely), which is accordingly stratified hydrostatically with the equilibrium pressure p0 (z) related to the equilibrium density ρ0 (z) by p0  = gρ0 .

(10.47)

We use a prime ( ) here to denote the derivative with respect to depth z of an equilibrium quantity; thus p0  ≡ dp0 (z)/dz. Equation (10.47) has formal solution (see Chapter 9) p0 (z) = p0 (0)eN0 (z) ,

(10.48)

where p0 (0) denotes the plasma pressure at z = 0, an arbitrary reference level, and N0 (z) is an integrated scale determined by the pressure scale height 0 (z):  z ds 1 kB T0 (z) p0 . (10.49) , N0  = , 0 (z) = = N0 (z) = 0 (z) mg ˆ gρ0 0 0 (s) The plasma density ρ0 (z) may be deduced from (10.48) combined with the ideal gas law ˆ where kB (= 1.38 × 10−23 J K−1 ) is the Boltzmann constant, T0 is the p0 = kB ρ0 T0 /m,

10.4 Zeroth Order Equations: Effects of Stratification

293

absolute temperature of the plasma and m ˆ is its mean particle mass. The plasma density ρ0 (z) is given by ρ0 (z) = ρ0 (0)

0 (0) N0 (z) . e 0 (z)

(10.50)

10.4.2 Perturbations (g = 0) Small perturbations of the form A = A0 + A1 ,

Bz = B0z + B1z ,

ρ = ρ0 + ρ1 ,

p = p0 + p1

about the stratified equilibrium (10.47) result in the linearized system ∂ ∂ ∂uz ∂p1 [ρ0 (z)A1 + A0 (z)ρ1 ] + [ρ0 (z)A0 (z)uz ] = 0, ρ0 (z) =− + gρ1 , ∂t ∂z ∂t ∂z 1 p1 + B0z (z)B1z = πe1 , B0z (z)A1 + A0 (z)B1z = 0, μ   ∂ρ1 ∂p1   2 + p0 (z)uz = cs (z) + ρ0 (z)uz . (10.51) ∂t ∂t Our aim is to analyse this system so as to determine the behaviour of, say, the longitudinal velocity uz , extending the result (10.29) to allow for the effects of stratification that occurs in the presence of gravity (g = 0). The equilibrium state of the flux tube is one of hydrostatic equilibrium, equation (10.47). It proves helpful to consider just three of the five equations in system (10.51), leaving aside the equations that link pressure perturbations with those in the magnetic field. We return to the two magnetic equations at a later stage. Consider, then, the three equations ∂ ∂ ∂uz ∂p1 [ρ0 (z)A1 + A0 (z)ρ1 ] + [ρ0 (z)A0 (z)uz ] = 0, ρ0 (z) =− + gρ1 , ∂t ∂z ∂t ∂z   ∂ρ1 ∂p1   2 + p0 (z)uz = cs (z) + ρ0 (z)uz (10.52) ∂t ∂t together with the equation of hydrostatic equilibrium, p0  = gρ0 .

(10.53)

These equations apply rather generally to longitudinal waves in any elastic tube stratified by gravity; the magnetic flux tube is simply one illustration of the equations. The unstratified tube is recovered by setting g = 0. We begin by rewriting the continuity equation to give ∂ρ1 /∂t, namely ∂uz ρ0 ∂A1 (ρ0 A0 ) ∂ρ1 uz − = −ρ0 − . ∂t ∂z A0 A0 ∂t

(10.54)

Thus, the combination of terms that arises in the isentropic equation may be expressed in the form ∂ρ1 ∂uz A0  ρ0 ∂A1 uz − + ρ0  uz = −ρ0 − ρ0 . ∂t ∂z A0 A0 ∂t

(10.55)

294

Thin Flux Tubes: The Sausage Mode

Accordingly, we can then use the isentropic equation to solve for pressure variations:   ∂p1 A0  ∂uz ρ0 c2s ∂A1 = − gρ0 + ρ0 c2s − , (10.56) uz − ρ0 c2s ∂t A0 ∂z A0 ∂t where we have used the equilibrium requirement that p0  = gρ0 . Turning to the momentum equation, differentiated with respect to t and dividing through by ρ0 , we have     2 ∂ 2 uz (ρ0 c2s ) ∂uz 2 ∂ uz 2 A0 − cs 2 − + cs 2 ρ0 A0 ∂z ∂t ∂z         2 ∂A 1 A A ) ∂ c (ρ 1 ρ g ∂A1 0 0 0 0 1 s − −g − . gρ0 + ρ0 c2s uz = ρ0 A0 ρ0 A0 ρ0 ∂z A0 ∂t A0 ∂t This equation may be simplified slightly by use of the equilibrium constraint (10.47):        2 A0  ∂ 2 uz ∂uz 2 ∂ uz 2 A0 2 A0 − cs 2 − γ g + cs + g(γ − 1) − cs uz A0 ∂z A0 A0 ∂t2 ∂z 1 ∂ = ρ0 ∂z



ρ0 c2s ∂A1 A0 ∂t

 −

g ∂A1 . A0 ∂t

(10.57)

Equation (10.57) applies quite generally to a fluid in an elastic tube (∂A1 /∂t = 0), including the extreme when the tube is rigid, so that there are no temporal variations of cross-sectional area (∂A1 /∂t = 0). If the tube is elastic then ∂A1 /∂t = 0 and we need to specify how the cross-sectional area of the tube varies in time. It proves convenient to consider some specific cases separately, namely the isolated magnetic flux tube of the photosphere, and the straight magnetic flux tube more typical of the corona. But we begin with the case of a rigid tube.

10.5 The Rigid Tube Consider the case of a rigid (∂A1 /∂t = 0) tube standing vertically in a gravitational medium and filled with a compressible fluid (a gas). For the case of a rigid tube the righthand side of equation (10.57) is zero and the governing equation for longitudinal motions uz reduces to        2 A0  ∂uz ∂ 2 uz 2 ∂ uz 2 A0 2 A0 − cs 2 − γ g + cs + g(γ − 1) − cs uz = 0. (10.58) A0 ∂z A0 A0 ∂t2 ∂z If the rigid tube is straight (A0  = 0), then equation (10.58) reduces further to 2 ∂ 2 uz ∂uz 2 ∂ uz − c − γg = 0. s 2 2 ∂z ∂t ∂z

(10.59)

Accordingly, we recover the governing equation for a vertically propagating sound wave in a stratified atmosphere, discussed in Chapter 9, Section 9.8.

10.5 The Rigid Tube

295

We wish to cast equation (10.58) into the form of the Klein–Gordon equation. Set uz = Z(z)Q(z, t). Our aim is to choose Z(z) so as to eliminate the ∂uz /∂z term. Note first that ∂ 2 uz ∂ 2Q = Z , ∂t2 ∂t2

∂uz ∂Q =Z + Z  Q, ∂z ∂z

∂Q ∂ 2 uz ∂ 2Q = Z + 2Z  + Z  Q. ∂z ∂z2 ∂z2

Substituting these expressions into equation (10.58) allows us to collect terms and divide throughout by Z; we obtain   2  ∂ 2Q 2∂ Q 2Z 2 A0 ∂Q − c − 2c + γ g + c s s s Z A0 ∂z ∂t2 ∂z2           Z A0 Z A0  A0 − c2s + (γ − 1)g + γ g + c2s + c2s Q = 0. Z A0 Z A0 A0 Accordingly, we choose Z(z) such that 2c2s

Z A0  = 0. + γ g + c2s Z A0

Thus, 

ρ0  c2 A0  γ g A0  Z =− − s2 − . 2 =− 2 − Z A0 ρ0 A0 cs cs Hence we may take

 Z(z) =

and then

 uz (z, t) =

ρ0 (0)c2s (0)A0 (0) ρ0 (z)c2s (z)A0 (z)

ρ0 (0)c2s (0)A0 (0) ρ0 (z)c2s (z)A0 (z)

1/2 ,

1/2 Q(z, t).

(10.60)

The result is that in a rigid tube Q satisfies the Klein–Gordon equation 2 ∂ 2Q 2∂ Q − c + 2rigid Q = 0, s ∂t2 ∂z2

where 2rigid

(10.61)

⎧   2 ⎫   1 2 ⎨ ρ0  c2s A0  1 ρ0  c2s A0  ⎬ = cs + 2 + − + 2 + 2 ⎩ ρ0 A0 2 ρ0 A0 ⎭ cs cs         c2s A0  A0  ρ0  1 2 A0 2 A0 + 2 + − (γ − 1)g . γ g + cs − cs + 2 A0 ρ0 A0 A0 A0 cs (10.62)

The expression for 2rigid can be simplified somewhat. First observe that from the equilibrium requirement that p0  = gρ0 we see that

296

Thin Flux Tubes: The Sausage Mode 

(ρ0 c2s ) ρ0  c2s 1 = + = , ρ0 0 ρ0 c2s c2s and so 2rigid



ρ0  c2 A0  1 A0  + s2 + = + ρ0 A0 0 A0 cs

      2   1 2 1 0  A0 1 A0  2 1 A0  = cs − 2 − + + −1 . 2 A0 2 A0 γ 0 A0 220 0

(10.63)

In a straight (A0 = 0) and rigid tube filled with an isothermal (0  = 0) plasma, expression (10.63) reduces to 2rigid = c2s /(420 ) and we recover the usual cutoff frequency, rigid = cs /(20 ), for an isothermal medium (see Chapter 9). In an isothermal tube with exponentially varying cross-section, such that A0 (z) = A0 (0)e−αz for constant α, expression (10.63) reduces to (Roberts 1981a)   c2s 4 2 2 (1 + α0 ) − α0 . rigid = γ 420 When α = 0 this recovers the usual result for an isothermal atmosphere. It turns out that the particular case of a rigid tube that is expanding exponentially at a rate for which α0 = 1/2 is of special interest, since this rate of expansion arises naturally for a magnetic flux tube in an isothermal atmosphere (discussed further below). With α0 = 1/2 and A0 (z) = A0 (0)e−z/(20 ) , then   cs 9 2 1/2 rigid = . (10.64) − 20 4 γ √ For γ = 5/3 this gives a cutoff frequency rigid that is 21/20 times the isothermal value of cs /(20 ), an increase of almost 2.5%.

10.6 Magnetic Flux Tubes We return now to a consideration of the elastic tube case. We have in mind two specific cases of a magnetic flux tube, standing vertically and stratified hydrostatically. In the first case the tube is isolated and expands with height as the confining pressure in the tube’s environment falls off with height; this is the case of a photospheric flux tube. The second case is where the tube is embedded in a magnetic atmosphere, such as with a coronal loop. The two cases differ in our treatment simply in the form of the Alfv´en speed: in the photospheric case, the Alfv´en speed varies with depth z in such a fashion that the ratio of Alfv´en speed to sound speed is a constant. In the coronal flux tube, this is no longer the case but we can instead assume that the equilibrium magnetic field is uniform (and then the Alfv´en speed varies with depth z purely because the density ρ0 does). Our starting point is the general equation (10.57), namely        2 A0  ∂uz ∂ 2 uz 2 ∂ uz 2 A0 2 A0 − cs 2 − γ g + cs + g(γ − 1) − cs uz A0 ∂z A0 A0 ∂t2 ∂z   1 ∂ ρ0 c2s ∂A1 g ∂A1 = − . (10.65) ρ0 ∂z A0 ∂t A0 ∂t

10.6 Magnetic Flux Tubes

297

The terms on the right-hand side of (10.65) depend upon the elasticity of the tube, which is itself determined by magnetic pressure balance and conservation of magnetic flux, so that equation (10.65) is to be combined with p1 +

1 B0z (z)B1z = πe1 , μ

B0z (z)A1 + A0 (z)B1z = 0.

(10.66)

We can use these relations in combination with equation (10.56) for the temporal change in the pressure perturbation to solve for ∂A1 /∂t. Specifically,       ∂ A1 −1 1 ∂πe1 2 ∂uz 2 A0 + cs + g + cs = 2 uz , (10.67) ∂t A0 ∂z A0 cf ρ0 ∂t where c2f (z) = c2s + c2A . We use this result to obtain an expression for the right-hand side of equation (10.65). First note that      c4 ∂ 2 uz ∂ A1 ∂ c2s ∂πe1 ∂ ρ0 (z)c2s (z) − ρ0 s2 2 =− 2 ∂z ∂t A0 ∂z cf ∂t cf ∂z  2  2       4 cs cs cs ∂uz 2 A0 2 A0 − 2 gρ0 + ρ0 cs uz . − 2 gρ0 + ρ0 cs + ρ0 2 A0 ∂z A0 cf cf cf It then follows that     1 ∂ c2s ∂πe1 g 1 ∂πe1 c4 ∂ 2 uz g ∂A1 1 ∂ ρ0 c2s ∂A1 − =− + 2 − s2 2 2 ρ0 ∂z A0 ∂t A0 ∂t ρ0 ∂z cf ∂t cf ρ0 ∂t cf ∂z  4            c4 c2 A0  A0  c A0  1 g 1 ∂uz − s2 + − 2 g + c2s − ρ0 s2 ρ0 s2 g + c2s uz . ρ0 ∂z ρ0 A0 A0 cf A0 cf cf cf Hence, substituting in equation (10.65) we obtain  4    2 c4s  cs A0  1 ∂ 2 uz 2 ∂ uz 2 A0 ∂uz − ct 2 + 2 + ρ0 2 − γ g − cs ρ0 A0 ∂z ∂t2 ∂z cf A0 cf            c2 A0  A0  g A0  1 A0 − 2 g + c2s − (γ − 1)g + ρ0 s2 g + c2s − c2s uz ρ0 A0 A0 A0 A0 cf cf   1 ∂ c2s ∂πe1 g 1 ∂πe1 =− + 2 . (10.68) 2 ρ0 ∂z cf ∂t cf ρ0 ∂t This is the general form of the equation satisfied by vertical motions uz in a thin magnetic flux tube, stratified vertically according to hydrostatic pressure balance (p0  (z) = gρ0 (z)). No other equilibrium condition has been imposed in the derivation of the general equation. Accordingly, it applies to a range of problems, for each of which the various coefficients arising in the partial differential equation simplify somewhat. We consider two special cases: the case of an isolated photospheric magnetic flux tube in which the equilibrium magnetic field varies with height as the tube expands out, and the case of a coronal flux tube in which the magnetic field is taken to be uniform with height.

298

Thin Flux Tubes: The Sausage Mode

10.7 The Isolated Photospheric Magnetic Flux Tube 10.7.1 Equilibrium of a Flux Tube The medium inside a flux tube is assumed to be stratified hydrostatically. It is usual also to assume that the plasma outside the thin flux tube is field-free and in static equilibrium. Of course, in the dynamic environment of the solar photosphere the reality is that flows and sound waves (p-modes) complicate matters, but such complications will be ignored here. We assume then that the environment of the magnetic flux tube is field-free (zero magnetic field) and hydrostatically stratified, so that the external pressure pe (z) satisfies pe  = gρe ,

(10.69)

where ρe (z) denotes the plasma density in the environment of the tube. Accordingly,  z  ds kB Te (z) pe pe (z) = pe (0) exp ; (10.70) , e (z) = =  (s) mg ˆ gρ e e 0 here pe (0) denotes the external plasma pressure at z = 0 and Te (z) denotes the temperature in the environment of the tube. The interior and exterior of the magnetic flux tube are connected by transverse pressure balance, which requires that the sum of the internal plasma pressure p0 (z) and the magnetic pressure B20z (z)/(2μ) equal the confining external equilibrium pressure pe (z): p0 (z) +

B20z (z)

= pe (z).

(10.71)

= g(ρe − ρ0 ).

(10.72)

2μ

Accordingly, we have 

B20z (z)



2μ

We confine attention to the case when the temperatures inside and outside the flux tube are equal and 0 = e . Then the magnetic pressure varies with depth z according to  2   2  B0z B0z 1 = , (10.73) 2μ 2μ 0 which is the same rate that the plasma pressure p0 (z) exhibits. Thus, the equilibrium state of the thin tube is one for which p0 (z) = p0 (0)eN0 (z) , B0z (z) = B0z (0)eN0 (z)/2 ,

ρ0 (z) = ρ0 (0)

0 (0) N0 (z) , e 0 (z)

A0 (z) = A0 (0)e−N0 (z)/2 ,

(10.74)

where N0 (z) is defined in equation (10.49). Here B0z (0) denotes the value of the equilibrium magnetic field B0z (z) at the reference level z = 0 and A0 (0) denotes the cross-sectional area of the thin flux tube at that level (z = 0). Thus the magnetic field strength B0z (z) increases with depth z at half the rate as the plasma pressure p0 (z), whilst the cross-sectional area correspondingly decreases to conserve magnetic flux (B0z (z)A0 (z) = constant).

10.7 The Isolated Photospheric Magnetic Flux Tube

299

In the special case of an isothermal atmosphere, for which 0 is a constant, N0 is simply z/0 (the variable z in units of the constant scale height 0 ) and the equilibrium is one of simple exponential stratification: p0 (z) = p0 (0)ez/0 , B0z (z) = B0z (0)ez/(20 ) ,

ρ0 (z) = ρ0 (0)ez/0 , A0 (z) = A0 (0)e−z/(20 ) .

(10.75)

It is important to note that the equilibrium state of our flux tube, whether isothermal or non-isothermal, is one for which the plasma β, the ratio of the plasma pressure p0 (z) to the magnetic pressure B20z (z)/2μ, is here a constant, even though the two pressures involved vary with z: β=

p0 (z) 2 c2s = = constant; γ c2A (B20z (z)/2μ)

here we have written β in terms of the sound speed cs (z) = (γ p0 /ρ0 )1/2 and the Alfv´en speed cA (z) = (B20z /(μρ0 ))1/2 within the tube. Furthermore, the ratio of the sound speed to the Alfv´en speed is a constant: cs (z)/cA (z) is a constant. The demands of pressure balance (10.71) mean that the flux tube is partially evacuated by the magnetic field. With temperatures inside and outside the flux tube being assumed equal, the plasma density ρ0 (z) inside the tube is related to the density ρe (z) in the field-free environment by ρ0 p0 β c2s = = . = ρe pe 1+β c2s + 12 γ c2A

(10.76)

For example, in a tube with sound and Alfv´en speeds equal, cs = cA , we have ρ0 /ρe = 2/(γ + 2), which for γ = 5/3 produces ρ0 /ρe = 6/11, and so ρ0 is some 54% of the environment’s density of ρe . In a stronger field producing an Alfv´en speed that is twice the sound speed (cA = 2cs ), we obtain β = 3/10 and a field which produces a magnetic pressure that is some 75% of the confining external pressure, evacuating the tube to such an extent that the tube density ρ0 falls to 23% of the external density ρe . 10.7.2 Perturbations in a Photospheric Tube Our starting point is the general equation (10.68) for uz . Given the hydrostatic equilibrium (10.74), for which p0  = gρ0 , we note first that   A0  A0  1 1 c2s = γ g0 , =− , g + c2s = 1 − γ g; A0 20 A0 2 moreover, cs (z)/cf (z) is a constant. We may then calculate the coefficients in the partial differential equation (10.68) as follows:   ∂uz c4  A0  1 c4 A0  coefficient of + = s2 ρ0 s2 − γ g − c2s ∂z ρ0 A0 cf A0 cf =−

c2t , 20

300

Thin Flux Tubes: The Sausage Mode



       c2 A0   g A0  1 A0  A0 coeff of uz = − 2 g + c2s − (γ − 1)g ρ0 s2 g + c2s − c2s ρ0 A0 A0 A0 A0 cf cf   2 2 c 1 c = 2t 1 + γ A2 2g . 2 cs cA Here we have introduced the buoyancy frequency g through the relation      ρ0 g g 1 2g = g − 2 = 1 − − 0  . ρ0 0 γ cs

(10.77)

Accordingly, the velocity equation for vertical motions uz in a photospheric magnetic flux tube in hydrostatic equilibrium (p0  (z) = gρ0 (z)) is   2 ∂ 2 uz c2t ∂uz c2t 1 ∂ 2 πe1 c2t 2 g 1 ∂πe1 1 c2A 2 ∂ uz − c −  = − + γ + 2 . 1 + u z t g 2 2 2 20 ∂z 2 ρ ∂z∂t ∂t2 ∂z2 c cA cA 0 cf ρ0 ∂t s (10.78) Equation (10.78) is the general equation governing longitudinal motions uz in an elastic and thin magnetic flux tube vertically stratified by gravity. It may be noted that in the limit of a strong magnetic field, cA /cs → ∞, (10.78) reduces to the rigid tube result, equation (10.58). Also, in the absence of gravity (g = 0) the general equation (10.78) reduces to ∂ 2 uz c2 1 ∂ 2 πe1 ∂ 2 uz − c2t 2 = − 2t , 2 ∂t ∂z cA ρ0 ∂z∂t and we have recovered equation (10.29) discussed earlier. The partial differential equation (10.78) may be transformed into the form of the Klein– Gordon equation, following much the same procedure as used for a rigid tube. In fact, the same scaling factor Z(z) works in both the rigid tube and the elastic flux tube cases because in a thin flux tube the propagation speed ct (z) is proportional to cs (z), given that c2A /c2s is a constant. Thus, either cs or ct may be used; we choose ct since this is the fundamental propagation speed for the sausage wave in a thin flux tube. We set uz = Z(z)Q(z, t) and choose Z(z) such that 2c2t

Z c2 + t = 0. Z 20

Thus, Z 1 1 = − N0  , =− Z 40 4 with N0 defined in equation (10.49). Hence   N0 , Z(z) ∝ exp − 4 which we can also write in the form Z(z) ∝ [ρ0 (z)c2t (z)A0 (z)]−1/2 . Hence we may take

10.7 The Isolated Photospheric Magnetic Flux Tube

 Z(z) = and

 uz (z, t) =

ρ0 (0)c2t (0)A0 (0) ρ0 (z)c2t (z)A0 (z)

ρ0 (0)c2t (0)A0 (0) ρ0 (z)c2t (z)A0 (z)

301

1/2 (10.79)

1/2 Q(z, t),

(10.80)

and then equation (10.78) is transformed into an equation of the Klein–Gordon form, namely   2 2 ∂ 2Q 1 c2t 1 g ∂πe ∂ πe 2∂ Q 2 − ct 2 + saus Q = + 2 , (10.81) − Z c2A ρ0 ∂t∂z ∂t2 ∂z cs ∂t where 2saus

    c2t 3γ 4 1 2 2 = + ct g 2 + 2 −3 + γ 4cs 1620 cA     c2t 9 1 c2s 1 3  = 2 − 0 + − + 2 − 0  1− . 4 16 2γ γ 0 γ cA

(10.82)

Klein–Gordon equations of this form were first derived in Roberts (1981a) and Rae and Roberts (1982), though using a somewhat different approach. The expression in (10.82) involving explicitly the derivative 0  arises in Roberts and Webb (1978).1 Notice that in the limit of cA  cs expression (10.82) reduces to the rigid tube result 1 (10.63) for a rigid tube with cross-sectional area A0 (z) = A0 (0)e− 2 N0 (z) . In the special case of an isothermal magnetic flux tube, for which 0  = 0 and the tube cross-sectional area is A0 (z) = A0 (0)e−z/(20 ) , the general expression (10.82) for the square of the cutoff frequency reduces to    c2t 9 1 c2s 1 2 saus = 2 − + 2 1− . (10.83) γ 0 16 2γ γ cA An alternative version of this expression which displays more clearly the tube’s geometrical and elasticity (measured by the plasma β) contributions is     9 2 3 2 2 β 2 (10.84) saus = − − − 2ac , 4 γ 2 γ β + γ2 where ac = cs /(20 ) is the acoustic cutoff frequency in an isothermal atmosphere. The plasma β of the flux tube is related to the sound speed cs and Alfv´en speed cA through β = 2c2s /(γ c2A ). An expression of the form (10.83) for the sausage mode cutoff frequency in an isothermal thin tube was first obtained by Defouw (1976) and further investigated in Roberts and Webb (1978) and Rae and Roberts (1982). Expression (10.84) shows that the cutoff frequency saus is determined mainly by the geometry of the thin tube, the rate at which it expands with height. Indeed, comparing 1 Note that the z-axis is taken to point upwards in Roberts and Webb (1978), Roberts (1981a) and Rae and Roberts (1982), so

this introduces a sign change in certain derivative terms when compared with the expressions given here.

302

Thin Flux Tubes: The Sausage Mode

(10.84) with (10.64), we see that the term independent of β gives precisely the cutoff frequency of a rigid tube with cross-sectional area A0 (z) = A0 (0)e−z/(20 ) . The elasticity of the tube enters through the term involving β, which acts so as to reduce the cutoff below its value in a rigid tube. However, for γ = 5/3 the effect is not large. We can give a numerical illustration. For a photospheric flux tube with an Alfv´en speed √ that is twice the tube’s sound speed, so cA = 2cs , then the slow mode’s speed is ct = 2cs / 5 and the slow wave in a thin tube propagates at some 90% of the sound speed; for cs = 7.5 km s−1 and cA = 15 km s−1 , this produces a sub-sonic, sub-Alfv´enic tube speed of ct = 6.7 km s−1 . With these values, the tube’s cutoff frequency amounts to saus = 1.016 ac , and so is very close to the acoustic cutoff frequency of an isothermal atmosphere. Even if we allow β to vary over its complete range, the effect amounts to at most 4.4% and for β restricted to be less than one, the reduction in cutoff frequency amounts to no more than 2%. Only if we allow γ to depart markedly from 5/3 do we find that saus may depart significantly from the acoustic cutoff value. Finally, we note that if we neglect the forcing terms involving ∂πe1 /∂t on the right-hand side of the partial differential equation (10.78) and we Fourier analyse by taking a time dependence of the form eiωt , writing uz (z, t) = uz (z)eiωt , then (10.78) leads to an ordinary differential equation of the form  2    d2 uz 1 duz 1 ω 1 c2A + (10.85) + 2 − 2 1 + γ 2 2g uz = 0, 20 dz 2 cs dz2 ct cA which can be recast as  2    ω − 2g 1 duz 1 1 d2 uz + + + γ 2g uz = 0. 1 − 20 dz 2 dz2 c2s c2t

(10.86)

An equation of the form (10.86) was first derived by Roberts and Webb (1978), using the zeroth order expansion equations.

10.7.3 Convective Collapse The equation of motion (10.85) or (10.86) may be rewritten in the form     2  γ c2 ρ0 c d ρ0 c2t duz + ω2 − 2g 2t + 2t uz = 0. dz B0z dz B0z 2cs cA

(10.87)

This is in the standard form of the Sturm–Liouville equation, namely,   duz d p(z) + q(z)uz = −λr(z)uz . dz dz Here the coefficients p(z) and r(z) of the general Sturm–Liouville equation are required to be positive in the domain of the equation, whereas the sign of the coefficient q(z) determines the nature of the eigenvalues λ. The boundary conditions that are added to the Sturm– Liouville equation are of the general form α1 uz + α2 (duz /dz) = 0, for constants α1 and α2 , the conditions being applied at the ends z = a and z = b of the domain a ≤ z ≤ b.

10.7 The Isolated Photospheric Magnetic Flux Tube

303

A number of theorems govern the nature of the solutions to the Sturm–Liouville equation and its eigenvalues λ for the given associated boundary conditions (see, for example, Ince 1944; Coddington and Levinson 1955). Of particular interest here is the result that λ ≥ 0 if q < 0. Comparing equation (10.87) with the general Sturm–Liouville equation we see that ρ0 c2t , p(z) = B0z

ρ0 r(z) = , B0z

ρ0 c2t q=− B0z c2A



1 c2 1 + γ A2 2 cs

 2g

and λ = ω2 . Accordingly, the conditions p > 0 and r > 0 are met, and the sign of q is determined by the sign of −2g . Thus, ω2 ≥ 0 if 2g > 0. If 2g < 0, then ω2 may or may not be negative. In other words, 2g < 0 is a necessary condition for instability (ω2 < 0). In fact, this is the well-known Schwarzschild condition for convection to occur in a non-magnetic atmosphere (see, for example, Priest 2014, sect. 1.3.3). It is of interest to exhibit the behaviour of our system in a simple way. Following Webb and Roberts (1978), we can do this by looking at the case when the flow uz within the tube vanishes at two specific levels, say at z = 0 and at z = h, so that motions are confined to occur between the level z = 0 representing the solar photosphere and the depth z = h (representing some location below the solar surface where the perturbed flow uz falls to zero). Then, the ordinary differential equation (10.87) together with the boundary conditions uz = 0 at z = 0 and z = h forms a classical Sturm–Liouville problem. It then immediately follows from the above that if 2g > 0 then so also is ω2 ; that is, longitudinal tube waves occur if 2g > 0. However, if 2g < 0, then there arises the possibility of modes for which ω2 < 0, corresponding to unstable motions. An instability may arise in the flux tube, and the condition for the occurrence of this instability is the standard Schwarzchild criterion (2g < 0) for convection in a compressible (nonmagnetic) atmosphere. The consequence of the instability in a flux tube is that it leads to the intensification of any vertical downdraught uz within the magnetic flux tube, as lighter material is transported down into the stratified tube, allowing less dense material higher within the tube to take its place. Transverse pressure balance then ensures that the tube is squeezed in by the surrounding medium, resulting in a concentration of the magnetic field. In contrast, any upflow generated within the unstable stratified flux tube leads to a weakening of the magnetic field, as denser material deep within the tube is transported to higher levels. This process by which a magnetic flux tube in the convection zone may collapse in on itself until the magnetic field is strong enough so as to resist the process is sometimes referred to as convective collapse (Spruit and Zweibel 1979; Spruit 1979) or the superadiabatic effect (Parker 1978, 1979a). Aspects of the instability have been worked out in Parker (1979a), Webb and Roberts (1978), Spruit and Zweibel (1979) and Spruit (1979). We can give a simple approximate treatment that illustrates the main features of this process very conveniently. Following Webb and Roberts (1978), we note that   z sin(kz z) u(z) = u0 exp − 40

(10.88)

304

Thin Flux Tubes: The Sausage Mode

may be substituted in equation (10.85), treating 0 as a constant (even though we know that it varies with z), with the result that (10.85) is satisfied provided that   ω2 1 1 γ 2 = kz + + 2 + 2 2g . 2cs c2t 1620 cA Now the boundary condition that uz = 0 at z = 0 is automatically satisfied by our choice of approximate solution. There remains the second boundary condition, that uz = 0 at z = h; this is satisfed by taking kz h = nπ for integers n. In particular, the fundamental mode n = 1 has the dispersion relation (Webb and Roberts 1978)     2 1 1 γ 2 2 π ω = ct + + 2 + 2 2g . (10.89) h2 2cs 1620 cA This relation shows immediately that if 2g < 0 then there is the possibility of a mode with ω2 < 0 arising, depending upon the depth h. For any given depth h, a sufficiently strong magnetic field will quench the instability. In particular, setting ω2 = 0 we see that the instability is quenched if the magnetic field in the tube is sufficiently strong so as to give a plasma β (= 2c2s /(γ c2A )) such that   1 1 2π 2 20 1 + β = − γ −1 + , 2g < 0. 2  8 h ( γ − 0 ) For depths h much in excess of π 0 the term involving h2 is negligible and we obtain the estimate 1 , 2g < 0. 1+β = 1  8(0 + γ − 1) Such relations as this provide a useful estimate of how strong the magnetic flux tube must be in order not to collapse further in consequence of a downflow uz occurring. For example, with γ = 5/3 and 0  = 1/2, we obtain a critical value of β = 1/4. Of course, more accurate treatments of this process are needed to complement the simple illustration here. Webb and Roberts (1978) give an exact solution of the eigenvalue problem as well as the approximate treatment sketched here, and Spruit and Zweibel (1979) evaluate the ω2 = 0 case for a realistic solar convection zone. Such treatments reveal that kilogauss magnetic fields are required for a photospheric flux tube to be stable to convective collapse (see the review in Spruit and Roberts (1983)). Weiss and Proctor (2014) offer a perspective in the general context of magnetoconvection.

10.8 The Straight Magnetic Flux Tube For our second illustration of the sausage mode in a stratified magnetic tube we return to the general equation (10.68) and examine its reduction when applied to a magnetic flux tube that is of uniform cross-section. We consider a uniform magnetic field B0 = B0 ez inside the magnetic tube, so that A0  = 0. This is the sort of situation that might occur within a sunspot or in a coronal loop. We are of course ignoring any effects due to curvature that in

10.8 The Straight Magnetic Flux Tube

305

reality may have a bearing, especially upon the case of a coronal loop. Nonetheless, it is of interest to consider a magnetic flux tube that is of uniform cross-section, though containing a plasma that is stratified in density and pressure according to hydrostatic equilibrium (p0  = gρ0 ). Particular interest then focusses on the vertical profile of the Alfv´en speed. This example is in contrast to the case of an isolated photospheric magnetic tube for which the cross-sectional geometry of the tube, expanding with height, is of central importance. Here only the stratified nature of the plasma plays a role. Hydrostatic equilibrium in a uniform magnetic field means that the Alfv´en speed varies with depth z because of the stratification in the plasma density ρ0 (z), with p0  (z) = gρ0 (z). In consequence, the Alfv´en speed squared, c2A , and the fast speed squared, c2f = c2s + c2A , vary according to 

(c2A ) = −c2A

c2 ρ0  = − A (1 − 0  ), ρ0 0

(c2f ) = c2f

c2 0  − A. 0 0

Consider, then, the general equation (10.68) of longitudinal motion. With B0 = constant and A0 = constant we may simplify the coefficients arising in the partial differential equation, giving   ∂uz c4  A0  1 c4 A0  coeff of + = s2 ρ0 s2 − γ g − c2s ∂z ρ0 A0 cf A0 cf   4 c 1 = ρ0 s2 − γ g ρ0 cf = −γ g

c4A c4f

,

         c2s 1 g A0  2 A0 2 A0 2 A0 coeff of uz = − 2 g + cs − (γ − 1)g ρ0 2 g + cs − cs ρ0 A0 A0 A0 A0 cf cf   2 2 c g g = ρ0 s2 − 2 ρ0 cf cf   2 c g c2t = 2t 2g + . 0 c2s cA Hence, longitudinal motions uz (z, t) in a vertical magnetic flux tube with uniform (A0  = 0) cross-section (and uniform field) satisfy   2 c4A ∂uz g c2t c2t ∂ 2 uz 2 ∂ uz 2 − ct 2 − γ g 4 + 2 g + uz 0 c2s ∂t2 ∂z cf ∂z cA   1 ∂ c2s ∂πe1 g 1 ∂πe1 =− + 2 . ρ0 ∂z c2f ∂t cf ρ0 ∂t

(10.90)

An equation of this form, though with no forcing terms on the right-hand side of (10.90), was given by Roberts (2006); however, the derivation in Roberts (2006) was quite different from that given here (and the z-axis there was pointing upwards). It may be noted that in the limit of a strong magnetic field, cA /cs → ∞, equation (10.90) reduces to the rigid tube

306

Thin Flux Tubes: The Sausage Mode

result, equation (10.59). Notice also that if g = 0 then equation (10.90) reduces to equation (10.29). As with earlier examples, we can cast equation (10.90) in the form of the Klein–Gordon equation. We neglect the terms on the right-hand side of (10.90), arising from terms involving ∂πe1 /∂t. Then, setting 1/2  ρ0 (z)c2t (z) uz (z, t) (10.91) Q(z, t) = ρ0 (0)c2t (0) we obtain the Klein–Gordon equation (Roberts 2006) ∂ 2Q ∂ 2Q 2 − c (z) + 2str (z)Q = 0, t ∂t2 ∂z2 where

 2str (z)

=

c2t

1 420



ct cs

4

(10.92)

    1 1 g c2t c2t 2 + γ g 4 + 2 g + . 2 0 c2s cs cA

(10.93)

The derivation of expression (10.93) given in Roberts (2006) is different from the one presented here.2 We can gain some insight into expression (10.93) by looking at its behaviour in the extremes that the speed ct (z) may take. In a very strong magnetic field with cs /cA  1, corresponding say to the upper atmosphere of a sunspot or in a coronal loop, we have ct = cs ,

2str (z) =

c2s (1 − 20  ), 420

cs  cA .

(10.94)

Thus we see that in a strong magnetic field the slow wave propagates one-dimensionally as a sound wave that is constrained to run along rigid magnetic field lines; the frequency str has become the acoustic cutoff frequency of a stratified atmosphere. In the opposite extreme of a weak field (cA  cs ), such as might arise in the sub-photospheric layers of a sunspot, we have   g 1 ct = cA , 2str (z) = 2g = (10.95) 1 − − 0  , cs  cA , 0 γ giving the Alfv´en speed as the speed of propagation and the buoyancy frequency g as the cutoff frequency.

10.9 Slow Modes in Coronal Loops Consider the specific application of our analysis to a coronal loop. In this we follow the treatment in Roberts (2006). To apply our discussion to a coronal loop, we need to employ boundary conditions that allow in part for the curved geometry of a loop; we can do this by treating only that section of a loop that extends from the apex to one of the loop’s footpoints, assuming symmetry about the apex. This circumvents some of the problems 2 In Roberts (2006), the z-axis was taken to point upwards; this results in the derivative term in 2 taking a different sign. str

10.9 Slow Modes in Coronal Loops

307

associated with loop geometry. We suppose that the medium is isothermal (0  = 0); with the exception of the footpoint regions, the temperature variation along a coronal loop is not pronounced, so the assumption of isothermality is reasonable. Now in an isothermal medium with uniform magnetic field the Alfv´en speed decreases with distance z from the apex (z = 0) exponentially fast, on a scale of twice the pressure scale height: 

z cA (z) = cA (0) exp − 20

 ,

0 ≤ z ≤ L/2.

The loop length is L and we have represented the Alfv´en speed over one-half of the loop, extending from its apex at z = 0 to its footpoint at z = L/2. So, for example, in a loop of length L = 200 Mm with pressure scale height 0 = 5 × 104 km the Alfv´en speed decreases by a factor of e (≈ 2.7) from the apex to the loop footpoint. The square of the cutoff frequency str in an isothermal loop with strong magnetic field (cA  cs ) is given by expansion of (10.93), which (for γ = 1) yields 2str ≈

          c2t 1 1 2 c2s c2s 1 2 c2s − 1 − ≈ 1 − 1 + 4 1 − , γ γ 20 4 c2A 420 c2A

c2A  c2s .

(10.96) Thus, magnetism decreases the ratio of 2str /c2t and it decreases the propagation speed ct ; the result is that the cutoff frequency str falls from its value in the absence of a magnetic field. In the presence of a strong magnetic field, with γ = 5/3 we obtain str ≈

   cs 41 c2s 1− , 20 50 c2A

c2A  c2s .

(10.97)

The effect is small; for a sound speed of cs = 150 km s−1 and an Alfv´en speed of cA = 103 km s−1 it amounts to a reduction in str of 1.8% below its acoustic value, ac = cs /(20 ). A similar reduction occurs in the propagation speed ct , which falls below the sound speed cs by about 1.1%. With cs = 150 km s−1 a gravitational constant of g = 274 m s−2 and γ = 5/3, we obtain a coronal pressure scale height of 0 = 5 × 104 km (or 50 Mm), leading to an acoustic cutoff frequency of ac = 1.5 × 10−3 s−1 (with corresponding period 2π/str = 4189 s or almost 70 minutes). Consider the Klein–Gordon equation (10.92) which we may Fourier analyse in time by writing uz (z, t) = uz (z)eiωt ,

Q(z, t) = Q(z)eiωt

for frequency ω. Then Q satisfies the ordinary differential equation d2 Q + dz2



 ω2 − 2str (z) Q=0 c2t (z)

with 2str given by (10.96) or (10.97) for isothermal conditions. It is convenient to discuss standing waves and propagating waves separately.

(10.98)

308

Thin Flux Tubes: The Sausage Mode

10.9.1 Standing Waves We can illustrate aspects of these calculations by considering standing slow waves in a loop. Such waves have been detected in coronal loops (see the reviews by Wang 2004, 2016). To apply the above analysis to this case we assume that cA  cs . Moreover, we use a local analysis, whereby we treat the coefficients of the ordinary differential equation (10.98) as effectively constants. A more rigorous treatment would solve explicitly the differential equation. However, we can expect the two treatments to give broadly similar results. Accordingly, we take the solution of equation (10.98) to be of the form (treating ct and str as approximately constant) 1/2  1/2   2  2 ω − 2str ω − 2str z + Q1 sin z , Q = Q0 cos c2t c2t with Q0 and Q1 being arbitrary constants. With z = 0 chosen to be the loop apex, we choose a solution Q of (10.98) to be either an even function about z = 0 or an odd function. Consider the case of an even function (the case of an odd function is treated similarly). We have 1/2   2 ω − 2str z . Q = Q0 cos c2t We also require that the vertical motions vanish at the base of the loop, thereby modelling the effect of the dense photosphere below the tenuous corona. Thus we require uz = 0 (and so Q = 0) at the loop base z = L/2. Accordingly, we require 1/2  L π ω2 − 2str = − + nπ (10.99) 2 2 2 ct for integer n (= ±1, ±2, . . .). The fundamental mode is given by setting n = 1; the mode has the period P = 2π/ω with P=

1 2L 1 2L · · 1/2 = 1/2 , 2 ct c t 1 + (Lstr /π ct ) 1 + (L/Lc )2 )

(10.100)

where Lc = π ct /str denotes the effective length introduced by stratification. In the absence of gravity (g = 0), equation (10.100) reduces to P = 2L/ct , the period being the time it takes a slow mode to propagate from one end of a loop of length L to the other end, and back again. Thus, in the presence of gravity the period of the standing slow wave is reduced by the effect of stratification; the square root factor in (10.100) indicates by how much the period P = 2L/ct obtained in the g = 0 case (Edwin and Roberts 1983; Roberts, Edwin and Benz 1984) is modified as a consequence of stratification. The importance (or otherwise) of stratification is thus determined by the magnitude of (L/Lc )2 ; if L  Lc , then stratification is unimportant (as regards the period of a slow wave) but if this term is of order unity or larger then stratification has a pronounced effect. Since ct ∼ cs and str ∼ ac , the acoustic values, we have Lc ∼ 2π 0 and the importance of gravity

10.9 Slow Modes in Coronal Loops

309

is determined by the magnitude of (L/2π 0 )2 . The effect of gravity on short (L  Lc ) loops is negligible, their period being determined principally by the travel time 2L/ct in an unstratified loop. Many of the hot loops observed by the SUMER (Solar Ultraviolet Measurements of Emitted Radiation) instrument are short in the above sense. A sound speed of cs = 370 km s−1 (corresponding to the high temperature of 6.3 MK) leads to a pressure scale height of 0 = c2s /(γ g) = 3 × 105 km; thus Lc = 1900 Mm, far larger than the observed typical loop length of L = 191 Mm. Hence, gravitational effects on the period P of slow modes in hot SUMER loops are negligible, and the period is determined essentially by the travel time 2L/ct . Accordingly, an observed period of P = 17.6 minutes implies a slow mode speed of ct = 362 km s−1 and this in turn implies an Alfv´en speed of cA = 1750 km s−1 . In contrast to SUMER loops, loops observed by the instrument TRACE are much cooler than the hot SUMER loops and have sound speeds of order cs = 150 km s−1 ; this produces Lc = 300 MK, which is comparable with the lengths of many TRACE loops.

10.9.2 Propagating Waves Consider the possibility of slow waves propagating in a loop. If insufficient time has elapsed for the wave to reach the far end of a loop, or if decay mechanisms act efficiently to prevent a slow wave surviving to the far ends of a loop, then the wave will propagate freely. The observations by De Moortel et al. (2002a, b) would seem to be in this category, since the waves appear to penetrate only about 10% of a loop from the footpoint before vanishing. We can envisage two aspects: that the waves are impulsively excited, or that the waves are driven by an oscillation in the lower atmosphere. Impulsively excited waves are likely to arise when an energetic event, such as a flare, occurs; driven waves are likely to arise from p-modes (5 minute period) or from chromospheric oscillations (3 minute period). An impulsively excited wave obeying the Klein–Gordon equation leads to a wave front travelling with the speed ct and trailing behind it a wake that oscillates at the cutoff frequency str . Thus, we can expect a disturbance to propagate slightly sub-sonically, at speed ct , and trailing a wake with an oscillation period of some 70 minutes. This is so long that it is difficult to observe. Consider a driven oscillation prescribed at the footpoint (z = 0) of a loop, Q = Q0 cos(ω0 t),

(10.101)

where Q0 is the amplitude and ω0 the frequency of the driver. The solution of the Klein– Gordon equation (subject to an outgoing wave condition) may be formulated (Rae and Roberts 1982); we are interested in the case ω0 > str here, since in the corona str is small. For large times t, we have a solution of the form   z Q ∼ Q0 cos ω0 t − , cg

(z  ct t,

ω0 > str ),

(10.102)

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Thin Flux Tubes: The Sausage Mode

where cg is the group velocity of the wave, determined by the frequency ω0 of the driver:  1/2 ω02 cg = ct . (10.103) ω02 − 2str This describes a wave moving vertically with the group speed cg (> ct ), the base (z = 0) oscillating according to (10.101). With ω0 = 0.035 s−1 (corresponding to a 3 minute period) and str = 1.5 × 10−3 s−1 (corresponding to a 70 minute period), we obtain a speed of propagation that is slightly above the tube speed ct . This effect may be difficult to detect in the corona.

11 Thin Flux Tubes: The Kink Mode

11.1 Introduction We turn now to a consideration of the kink mode in a thin magnetic flux tube. As in Chapter 10, we are in the main concerned with an equilibrium magnetic field of the form B0 = (B0r , 0, B0z )

(11.1)

representing in cylindrical coordinates r, φ, z a field that may have both a radial component B0r (r, z) and a longitudinal component B0z (r, z). Again, this corresponds to a cylindrical tube, the cross-section of which may vary along the tube, and it includes the special case of a uniform tube, though twist is ignored (B0φ = 0). In view of the wide discussion of kink modes in solar plasmas, following their detection with space instruments, our treatment here will be largely self-contained. The governing equations of ideal magnetohydrodynamics are taken in the form ∂ρ + div ρu = 0, ∂t

ρ

∂u + ρ(u · grad)u = −grad p + j × B + ρg, ∂t

∂B = curl (u × B), div B = 0, ∂t   ∂p γ p ∂ρ kB + u · grad p = + u · grad ρ , p = ρT. ∂t ρ ∂t m ˆ

(11.2)

This system of ideal magnetohydrodynamic equations comprises the equation of continuity, the momentum equation (with pressure and magnetic forces included), the ideal induction equation, the solenoidal condition on the magnetic field, the isentropic equation of energy exchange, and the ideal gas law applied to a plasma. To begin with the effects of gravity are ignored (setting g = 0) but we return to those effects later (see Section 11.10). We are interested in the behaviour of the kink mode as described by this system. We have seen in Chapter 6 how kink oscillations in the absence of gravity may be described in linear theory applied to the simple case of a uniform flux tube embedded in a uniform medium (field-free or magnetic). The characteristic speed associated with kink modes turns out to be the speed ck , defined as  1/2 ρ0 c2A + ρe c2Ae ; (11.3) ck = ρ0 + ρe 311

312

Thin Flux Tubes: The Kink Mode

it plays a dominant role. Here cA denotes the Alfv´en speed within a magnetic flux tube of plasma density ρ0 , and cAe is the Alfv´en speed in the environment of the tube where the plasma density is ρe . In this chapter we examine this mode using approximate descriptions. Such descriptions have the virtue that in principle they permit a generalization to allow a consideration of a wide range of effects, including structuring along the tube, that are not easy to include in a general linear theory. We begin our discussion by looking at the special case of a β = 0 plasma, a situation of particular interest and importance in the corona. 11.2 The β = 0 Plasma We are interested in the application of system (11.2) to a medium that is dominated by magnetic forces, the corona being the main example. We accordingly ignore the plasma pressure terms entirely and also the effect of gravity, leaving any description of such effects (including the slow mode) to a separate treatment. With the plasma pressure p = 0, the plasma temperature T = 0 and gravity ignored (g = 0), the system of equations (11.2) becomes   ∂u B2 1 ∂ρ + div ρu = 0, ρ + ρ(u · grad)u = −grad + (B · grad)B, ∂t ∂t 2μ μ ∂B = curl (u × B), ∂t

div B = 0,

(11.4)

where B (= |B|) denotes the magnitude of the magnetic field B. Equations (11.4) are the β = 0 plasma equations. They are also referred to as the cold (because temperature T = 0) or pressureless (because the plasma pressure p = 0) plasma equations. In the momentum equation we have used the familiar result   B2 1 j × B = −grad + (B · grad)B, 2μ μ expressing the magnetic force in terms of the magnetic pressure and tension forces. Equations (11.4) provide a useful magnetohydrodynamic description of a medium that is dominated by magnetic forces. The corona is a good illustration of such a medium, even though the corona is a hot plasma; the point is that its dynamics can often be described by system (11.4) because the corona is a medium with an Alfv´en speed cA that is much larger than the sound speed cs , even though the sound speed in the corona is large compared with the typical sound speeds in lower parts of the Sun, such as the chromosphere or photosphere. In the corona a typical sound speed cs is of order 102 km s−1 and a typical Alfv´en speed cA is of order 103 km s−1 . These order of magnitude values imply a plasma β of β=

6 2 c2s = ≈ 10−2 γ c2A 500

and so the assumption that β = 0 is reasonable for the wave dynamics.

11.2 The β = 0 Plasma

313

A static equilibrium field B = B0 and equilibrium current j = j0 described by the system of equations (11.4) satisfies j0 × B0 = 0,

div B0 = 0,

(11.5)

giving an equilibrium magnetic field B0 that is force-free. The equilibrium plasma density ρ0 is taken to be a function of the spatial coordinates. The system of equations (11.4) may be linearized about the force-free equilibrium (11.5) by writing ρ = ρ0 + ρ1 ,

u = u1 ,

B = B0 + B1

in an expansion of variables in which we discard squares of perturbation terms. The resulting system of linear equations is ∂ρ + div (ρ0 u) = 0, ∂t

ρ0

∂u 1 1 = −grad pm + (B0 · grad)B + (B · grad)B0 , ∂t μ μ

∂B = curl (u × B0 ), ∂t

div B = 0.

(11.6)

In writing the linear system (11.6) we have found it convenient to discarded the suffix on the perturbation quantities, distinguishing terms by noting that equilibrium quantities carry a suffix ‘0’ whereas now u refers to the perturbation motion, B is the perturbation magnetic field and ρ is the perturbed plasma density; pm denotes the perturbation magnetic pressure, pm =

1 B0 · B. μ

(11.7)

Notice that, if desired, we may expand curl (u × B0 ) and simplify the vector expansion by noting that div B0 = 0: the resulting linearized induction equation may then be written in the form ∂B = −B0 (div u) + (B0 · grad)u − (u · grad)B0 . ∂t

(11.8)

It is convenient to work in terms of the linear displacement ξ given in terms of the perturbation velocity u by writing (see, for example, Bernstein et al. 1958; Goossens 2003) u=

∂ξ . ∂t

(11.9)

Working in terms of ξ means that we can carry out an integration with respect to time t, both in the equation of continuity and in the induction equation; this assumes that initial conditions are such as to select the particular solution we describe. Following such an integration, system (11.6) then becomes ρ = −div (ρ0 ξ ),

ρ0

∂ 2ξ 1 1 = −grad pm + (B0 · grad)B + (B · grad)B0 , 2 μ μ ∂t

B = −B0 (div ξ ) + (B0 · grad)ξ − (ξ · grad)B0 ,

div B = 0.

(11.10)

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Thin Flux Tubes: The Kink Mode

11.3 The β = 0 Plasma: Unidirectional Magnetic Field 11.3.1 General Formalism An important special case of the system (11.10) is that of a unidirectional equilibrium magnetic field. We take the equilibrium magnetic field B0 to be aligned with the z-axis of a cylindrical polar coordinate system r, φ, z, so that B0 = B0 ez .

(11.11)

The constraint div B0 = 0 means that B0 is independent of z; we take B0 to be a constant. Thus we consider an equilibrium magnetic field that is everywhere uniform. The equilibrium current j0 is zero, and the equilibrium constraint j0 × B0 = 0 is trivially satisfied. However, the plasma embedded within the equilibrium magnetic field need not be uniform; we consider an equilibrium plasma density ρ0 that may be spatially structured. We have in mind an equilibrium density that is structured along the equilibrium magnetic field as well as across the field, such as a magnetic flux tube filled with plasma that is stratified along the tube. For a uniform equilibrium magnetic field B0 = B0 ez and for displacement ξ = (ξr , ξφ , ξz ) and perturbation magnetic field B = (Br , Bφ , Bz ) we have ∂B ∂ξ , (B0 · grad)ξ = B0 , (B · grad)B0 = 0, (ξ · grad)B0 = 0. ∂z ∂z Accordingly, the β = 0 system (11.10) gives us a linear momentum equation (B0 · grad)B = B0

ρ0

∂ 2ξ B0 ∂B = −grad pm + , μ ∂z ∂t2

(11.12)

with the magnetic pressure perturbation pm being pm =

B0 Bz . μ

(11.13)

The perturbation magnetic field B satisfies ∂ξ , div B = 0. (11.14) ∂z Once the displacement ξ is known, variations in density ρ may be determined by B = −ez B0 div ξ + B0

ρ = −div (ρ0 ξ ).

(11.15)

Notice first that the z-component of the momentum equation (11.12) gives ∂ 2 ξz = 0, (11.16) ∂t2 which we take as giving ξz = 0 and uz = 0, corresponding to no displacement (or motion) in the z-direction, appropriate since there are no forces acting in the direction of the applied magnetic field (plasma pressure and gravitational forces being here neglected). Thus, in the zero β plasma with uniform equilibrium magnetic field B0 ez we take a displacement ξ and velocity u of the form ξ = (ξr , ξφ , 0),

u = (ur , uφ , 0).

11.3 The β = 0 Plasma: Unidirectional Magnetic Field

315

Substitution of B from (11.14) into the momentum equation (11.12) results in    2 2  ∂ξ ∂ 2 ∂ 2 − c − e ρ c div , (11.17) ξ = −grad p ρ0 m z 0 A A 2 ∂z ∂t2 ∂z where cA (= B20 /(μρ0 )1/2 ) is the Alfv´en speed in the medium; cA varies spatially with density structuring. The radial and azimuthal components of equation (11.17) are  2  2 2  2  ∂ 1 ∂ ∂ ∂ 2 ∂ 2 ∂ − cA 2 ξr = − pm , ρ0 − cA 2 ξφ = − (11.18) pm . ρ0 ∂r r ∂φ ∂t2 ∂z ∂t2 ∂z Also, from equation (11.14) we obtain Bz = −B0 div ξ ,

(11.19)

and so the magnetic pressure perturbation pm (= B0 Bz /μ) is pm = −

B20 div ξ = −ρ0 c2A div ξ . μ

(11.20)

Consider the second time derivative of the magnetic pressure perturbation:  2  B20 ∂ 2 pm B0 ∂ 2 Bz ∂ ξ = = − div μ ∂t2 μ ∂t2 ∂t2    1 B0 ∂B 2 = −ρ0 cA div −grad pm + ρ0 μ ∂z on use of (11.12). Using the vector identity div (λw) = λ div w + w · grad λ, for vector w and scalar λ, after some rearrangement we finally obtain ∂ 2ξ ∂ 2 pm 2 2 2 − c ∇ p = c (grad ρ ) · . (11.21) m 0 A A ∂t2 ∂t2 Equation (11.21) shows that the total pressure (magnetic pressure) perturbation in a β = 0 plasma satisfies a three dimensional wave-like equation with a coupling term that is related to the gradient in the equilibrium plasma density. An equation of this form has been obtained in Ruderman and Roberts (2002; see their eqn. (5)). We note that in the special case when the equilibrium density ρ0 is purely a function of z or is a constant, then the second term on the right-hand side of equation (11.21) vanishes, because ξz = 0, and so the magnetic pressure perturbation is a solution of the threedimensional wave equation (see D´ıaz, Oliver and Ballester 2002; McEwan et al. 2006) ∂ 2 pm = c2A (z)∇ 2 pm . ∂t2

(11.22)

11.3.2 Alfv´en Waves There is one solution of (11.22) we should not overlook and this is the simple solution pm = 0. With pm = 0, equation (3.15) shows that div ξ = 0, that is, the displacement is

316

Thin Flux Tubes: The Kink Mode

incompressible. There remain the radial and transverse components of the displacement, which satisfy the one-dimensional wave equation ∂ 2 ξr ∂ 2 ξr = c2A 2 , 2 ∂t ∂z

∂ 2 ξφ ∂ 2 ξφ = c2A 2 . 2 ∂t ∂z

(11.23)

The two components are related by 1 ∂ξφ 1 ∂ξr + ξr + = 0. ∂r r r ∂φ

(11.24)

In the special case of perturbations being independent of the azimuthal angle φ, so that ∂/∂φ = 0, then we see that ξr and ξφ can propagate one-dimensionally with the Alfv´en speed. However, the incompressibility condition means that ξr = 0 if we are to avoid a singular behaviour at r = 0. These are torsional Alfv´en waves, propagating with the local Alfv´en speed (see also Goossens et al. 2009). We can also consider disturbances with a φ-dependence, such as the kink oscillations, but we do not pursue this issue. Instead, we turn back to the general case of disturbances with non-zero magnetic pressure perturbation, pm = 0.

11.4 The Uniform Tube in a Uniform Environment 11.4.1 Tubes of Arbitrary Radius In view of the importance of the special case of a uniform tube in a uniform environment we give here a self-contained development of the linear theory applicable for a β = 0 plasma containing a straight magnetic flux tube. The theory is of course a special case of the treatment in Chapter 6, where the general case of a compressible medium is discussed. But the wide applicability of the β = 0 case, and the simplifications the assumption β = 0 introduces, justifies a specific treatment. Our starting point is the equation for pressure variations in the case where the equilibrium density is a constant inside the tube (and also a constant outside the tube); then the magnetic pressure perturbation pm satisfies (11.22), the wave equation in three dimensions. Written in cylindrical coordinates (r, φ, z), we have  2  1 ∂ ∂2 1 ∂2 ∂ ∂ 2 pm 2 = cA + + 2 pm . (11.25) + r ∂r r2 ∂φ 2 ∂t2 ∂r2 ∂z Since the medium is uniform in t, φ and z we can Fourier analyse in those coordinates, writing pm (r, φ, z, t) = pm (r)ei(ωt−mφ−kz z) for frequency ω, azimuthal mode number m and longitudinal wavenumber kz . Here we are interested specifically in the kink mode, for which m = 1 (or m = −1). Then pm (r) satisfies the ordinary differential equation   1 dpm 1 ω2 d2 pm 2 + − 2 − kz pm = 0. (11.26) + r dr dr2 r c2A

11.4 The Uniform Tube in a Uniform Environment

317

Equation (11.26) is recognized to be a form of the Bessel equation (see Abramowitz and Stegun 1965). Write n20 =

ω2 − kz2 . c2A

Then equation (11.26) has solutions J1 (n0 r) and Y1 (n0 r), where J1 and Y1 are Bessel functions of order unity. Imposing the condition that pm is finite at r = 0 removes the solution Y1 , which is unbounded at the origin. Accordingly, we select the solution pm = A0 J1 (n0 r),

r < a,

with A0 being an arbitrary constant. Similarly, outside the tube differential equation (11.26) for pm applies, but now the Alfv´en speed cA is replaced by its value cAe in the environment (r > a); moreover, it proves convenient to write m2e = kz2 − Then pm satisfies

ω2 . c2Ae

  1 dpm 1 d2 pm 2 + + − m pm = 0, e r dr dr2 r2

r > a,

(11.27)

with solutions I1 (me r) and K1 (me r) for modified Bessel functions I1 and K1 . We impose the condition that pm declines to zero as me r → +∞, which rejects I1 (me r) as exponentially unbounded but selects the solution K1 (me r), which declines to zero exponentially fast as me r → +∞. Physically, we are selecting the solution that corresponds to the wave within the flux tube being confined to the tube and its immediate environment, not disturbing the medium far from the tube. Taken altogether our solution of the wave equation for pm is  A1 K1 (me r), r > a, pm (r) = (11.28) A0 J1 (n0 r), r < a, where n20 =

ω2 − kz2 , c2A

m2e = kz2 −

ω2 , c2Ae

(11.29)

with cA denoting the Alfv´en speed inside the tube and cAe denoting the Alfv´en speed in the tube’s environment. The selection of this solution means that we require me > 0, and so ω2 < kz2 c2Ae . We have written the solution in a form appropriate for n20 > 0, though in fact the solution applies even if n20 < 0. With n20 > 0 and me > 0 it follows that kz2 c2A < ω2 < kz2 c2Ae . The β = 0 plasma under discussion here has a uniform magnetic field, the tube being distinguished by density contrast only, and then equilibrium pressure balance means that ρ0 c2A = ρe c2Ae . We require that the total pressure perturbation is continuous across the tube boundary (see Chapter 6); here the pressure perturbation consists of magnetic pressure only, so that

318

Thin Flux Tubes: The Kink Mode

pm (r+ ) = pm (r− ); that is, the value of pm (r) is the same whether we approach r → a from values of r less than a or greater than a. Then A1 = A0

J1 (n0 a) K1 (me a)

and the magnetic pressure perturbation pm is  0 a) A0 KJ11 (n r > a, (me a) K1 (me r), pm (r) = A0 J1 (n0 r), r < a.

(11.30)

There remains the condition that the displacement ξr (r) is also continuous across the tube boundary (see Chapter 6). From the radial component of the momentum equation, we have dpm 1 ξr (r) = . (11.31) ρ0 (r)(ω2 − kz2 c2A (r)) dr Noting that inside the tube (in r < a) the Alfv´en speed is cA (r) = cA and the plasma density ρ0 (r) = ρ0 , whereas outside the tube (in r > a) the Alfv´en speed is cA (r) = cAe and the equilibrium plasma density is ρ0 (r) = ρe , continuity of ξr across r = a means that ) ) dpm )) dpm )) 1 1 = . (11.32) ρ (ω2 − k2 c2 ) dr ) ρ (ω2 − k2 c2 ) dr ) 0

z A

r=a−

e

z Ae

r=a+

Application of this constraint results in the dispersion relation 1 J1  (n0 a) 1 K1  (me a) + = 0, n0 J1 (n0 a) me K1 (me a)

(11.33)

where the prime here denotes the derivative function (evaluated, as appropriate, at n0 a or me a): ) ) dJ1 (x) )) dK1 (x) ))   , K1 (me a) = . J1 (n0 a) = dx )x=n0 a dx )x=me a Dispersion relation (11.33) describes the kink modes in a β = 0 plasma; it has to meet the condition me > 0, requiring ω2 < kz2 c2Ae . 11.4.2 Thin Tubes: Dispersion Relation Relations of the form (11.33) have been discussed in Chapter 6. Here we are particularly interested in the case of a thin tube. Suppose that kz a is small. Then inspection of the expressions for n0 a and me a shows that these quantities too are small. Now the Bessel functions Jν (x) and Kν (x) are such that (see Abramowitz and Stegun 1965) as x → 0,  −ν ( 12 x)ν 1 1 , K0 (x) ∼ − ln(x), , J0 (x) ∼ 1, Kν (x) ∼ (ν) x Jν (x) ∼ (1 + ν) 2 2 where here ν is a positive integer and (ν) denotes the gamma function of ν (with the property that for integer ν, (1 + ν) = ν!, the factorial of ν). Also 1 J1  (x) = J0 (x) − J1 (x), x

1 K1  (x) = −K0 (x) − K1 (x). x

11.4 The Uniform Tube in a Uniform Environment

319

Armed with these properties we may deduce that as kz a → 0, 1 J1  (n0 a) ∼ , J1 (n0 a) n0 a

K1  (me a) −1 ∼ , K1 (me a) me a

and then the dispersion relation (11.33) reduces to n20 = m2e .

(11.34)

This is the dispersion relation for the kink mode in a thin (kz a  1) tube in a β = 0 plasma. The dispersion relation (11.34) may be readily solved to give ω2 = 2kz2

c2A c2Ae c2A + c2Ae

.

(11.35)

Since the equilibrium requires that ρ0 c2A = ρe c2Ae , the dispersion relation (11.35) may also be written in the form ω2 = kz2 c2k ,

(11.36)

where ck is the kink mode speed introduced earlier, c2k =

ρ0 c2A + ρe c2Ae . ρ0 + ρe

With the kink mode satisfying ω2 = kz2 c2k , we find that the mode numbers n0 and me are given by   ρ0 − ρe n20 = m2e = kz2 . (11.37) ρ0 + ρe Since we require me > 0, we also require ρ0 > ρe ; that is, the plasma density ρ0 within the flux tube must exceed the density ρe in the environment.

11.4.3 Thin Tubes: Dispersive Effect Having seen how the basic dispersion relation ω2 = c2k kz2 for the kink mode may be obtained, we can in fact determine the leading dispersive corrections for kz a  1. To do this it is convenient to employ the thin tube result for the behaviour of the magnetic pressure perturbation pm within the tube, but leave open the form of pm outside the tube. Thus, we write the solution for the magnetic pressure perturbation in the form  pme (r), r > a, pm (r) = 1 (11.38) 2 A0 n0 r, r < a. Here pme denotes the magnetic pressure perturbation in the tube’s environment. Continuity of the magnetic pressure perturbation means that pme (a) =

1 A0 n0 a. 2

(11.39)

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Thin Flux Tubes: The Kink Mode

We also require continuity of the radial component of the displacement, so ) ) dpm )) dpme )) 1 1 = ; ρ0 (ω2 − kz2 c2A ) dr )r=a− ρe (ω2 − kz2 c2Ae ) dr )r=a+ thus,

) 1 dpme )) 1 · A0 n0 = . ρ0 (kz2 c2A − ω2 ) 2 ρe (kz2 c2Ae − ω2 ) dr )r=a+ 1

Introducing Re = −a

) 1 dpme )) pme (a) dr )r=a+

(11.40)

we can write the resulting dispersion relation (for wave speed c = ω/kz ) in the form c = 2

c2A ρρ0e Re + c2Ae ρ0 ρe Re

+1

=

ρ0 c2A + ρe c2Ae + ρ0 c2A (Re − 1) . ρ0 + ρe + ρ0 (Re − 1)

Assuming for the moment that (Re − 1) is small (we show below, a posteriori, that this is the case) we can expand binomially the expression for c2 to obtain   ρ0 (11.41) c2 ∼ c2k − (c2k − c2A )(Re − 1). ρ0 + ρe There remains the determination of Re . Now the pressure perturbation in the environment of the tube is determined by the three-dimensional wave equation (11.26). If we non-dimensionalize the radial variable r against the tube radius a, writing r¯ = r/a, ∂/∂r = a∂/∂ r¯ and ∂ 2 /∂r2 = a2 ∂ 2 /∂ r¯ 2 , we find that equation (11.26) becomes   1 dpme a2 ω2 1 d2 pme 2 + − (kz a) pme = 0. (11.42) − 2 pme + r¯ d¯r d¯r2 r¯ c2Ae Now the term in square brackets is small for kz a  1 because we can expect the characteristic time of a wave to propagate a distance L along the tube with an Alfv´enic speed cAe to be L/cAe and so its characteristic frequency ω is of order cAe /L and consequently the term a2 ω2 /c2Ae is of order (kz a)2 , for kz of order 1/L. (For a standing wave in a loop we take L to be the length of the loop; this case is discussed in Section 11.6.) Accordingly, the determination of pme is dominated by the other terms in the differential equation, namely the equation d2 pme 1 dpme 1 + − 2 pme = 0. r¯ d¯r d¯r2 r¯

(11.43)

This differential equation has solutions r¯ and r¯ −1 , and so for a solution that is bounded as r¯ → +∞ we take pme = Ae /¯r, that is pme = Ae a/r. It then follows immediately that Re = 1. Thus, c2 = c2k and ω2 = c2k kz2 . If we wish to determine the dispersive correction to the relation ω2 = c2k kz2 , determining the way the speed c departs from ck when kz a is small, then we need to calculate pme more precisely. It turns out that the dominant term in the interior may still be taken as pm = A0 n0 r/2; corrections arising from expanding the Bessel function J1 (n0 r) beyond

11.4 The Uniform Tube in a Uniform Environment

321

its first term turn out to be small compared with the contribution from the environment. Accordingly, we retain the leading thin tube result for pm in r < a but in the environment we consider the modified Bessel function solution, namely pme (r) = A1 K1 (me r),

r > a.

(11.44)

We can then calculate Re from (11.40), finding that Re = −me a

K1  (me a) , K1 (me a)

Re − 1 = me a

K0 (me a) . K1 (me a)

(11.45)

From the above expression it is evident that Re does indeed tend to unity for small kz a. Finally, if we replace K1 (me a) by the first term in its expansion for small kz a, namely 1/(me a), we obtain (see also Chapter 6)     2  cAe − c2k ρe 2 2 2 (11.46) (λk kz a) K0 (λk |kz |a) , c ∼ ck 1 − ρ0 + ρe c2k where λ2k = 1 −

c2k c2Ae

,

(11.47)

with c2k < c2Ae . Noting the equilibrium relation that ρ0 c2A = ρe c2Ae , we can rewrite (11.46) in the alternative form     c2Ae − c2A ρ0 ρe 2 2 2 c ∼ ck 1 − (11.48) (λk kz a) K0 (λk |kz |a) . (ρ0 + ρe )2 c2k These relations make clear the dispersive correction of the kink mode in a β = 0 plasma. A relation of the form (11.48) was first given by Edwin and Roberts (1983).

11.4.4 Thin Tubes: Perturbations It is of interest to examine more closely the perturbations in the thin tube case. Consider the perturbations inside the tube when kz a  1. Taking the first term in the Taylor series for the Bessel function J1 , we have J1 (n0 r) ∼ 12 n0 r, r < a. Thus, the magnetic pressure perturbation pm inside the tube is approximated by 1 r < a. A0 n0 r, 2 The radial displacement ξr inside the tube follows from equation (11.31), pm (r) ∼

ξr (r) =

1 dpm A0 , ∼ 2 2 dr ρ0 cA n0 2ρ0 c2A n0

r < a.

Thus, inside the tube the radial displacement is a constant (to the lowest order in kz a) and the magnetic pressure perturbation is directly proportional to the radius r. Set ξr = C0 a for constant C0 . Then inside the tube we have (Goossens et al. 2009)   r ρ0 − ρe r < a, (11.49) ξr ∼ C0 a, (kz a)2 , pm (r) ∼ C0 ρ0 c2A ρ0 + ρe a

322

Thin Flux Tubes: The Kink Mode

for arbitrary (dimensionless) constant C0 . Thus, with the radial displacement ξr taken to be of order unity, the associated magnetic pressure variation is of order (kz a)2 , for small kz a. Using the Taylor series for the Bessel function J1 , it is in fact easy to calculate the next correction to the expansion of these variables. For example, in the expansion for pm we obtain        1 r 2 2 ρ0 − ρe 2r 2 ρ0 − ρe pm (r) ∼ C0 ρ0 cA 1 − (kz a) (kz a) , r < a, ρ0 + ρe a 8 ρ0 + ρe a indicating a reduction in pm . However, the reduction is small; for example, with kz a = 1/2 (a not particularly small value of kz a) we obtain a reduction of at most 1/32 compared with unity; for kz a = 1/10, the reduction is at most 1/800 compared with unity. Returning to the leading order expansions given in (11.49) we note that various other perturbation quantities are readily calculated. For example, the azimuthal component of the displacement is given by equation (11.18) as ξφ (r) = −iaC0 ,

r < a,

and from equations (11.15) and (3.15) the density perturbation ρ is given as   r 1 ρ0 − ρe ρ(r) = −ρ0 div ξ , div ξ = − pm ∼ −C0 (kz a)2 , ρ0 + ρe a ρ0 c2A

r < a.

Thus, the kink mode has very little compression; the wave is almost incompressible (Spruit 1982; Goossens et al. 2009). Following Roberts (1985b) and Goossens et al. (2009), we can also examine the relative role of the forces in the momentum equation. From equation (11.12) we see that motions of the tube are determined by a combination of the magnetic tension T and the magnetic pressure force P. The tension force T acting on the tube is T =

B0 ∂B μ ∂z

= −ρ0 (r)c2A (r)kz2 ξ − ikz pm ez , where in the second expression we have used equations (11.14) and (3.15) and have taken a Fourier representation of a perturbation f (r, φ, z, t) in the form f (r, φ, z, t) = f (r)ei(ωt−mφ−kz z) with m = 1. The magnetic pressure force P is   ∂pm 1 ∂pm ∂pm P = −grad pm = − , , ∂r r ∂φ ∂z = ρ0 (r)(kz2 c2A (r) − ω2 )(ξr er + ξφ eφ ) + ikz pm ez , where we have made use of the components of the momentum equation (11.12). Comparing the expressions for the forces T and P, we note first that the components in the z-direction cancel out, expected since the j × B0 force acts perpendicular to the equilibrium magnetic

11.5 Thin Tubes: Multiple Scaling

323

field. The tension and magnetic pressure forces acting perpendicular to the tube, T ⊥ and P ⊥ , are given by T ⊥ = −ρ0 (r)c2A (r)kz2 ξ (r),

P ⊥ = ρ0 (r)(kz2 c2A (r) − ω2 )ξ (r).

(11.50)

Evidently, the tension force T is directed in opposition to the displacement ξ , as we would expect, whereas the magnetic pressure force P ⊥ is aligned with the tension but acts either in unison with the tension or in opposition, depending upon whether ω2 < kz2 c2A (r) or ω2 > kz2 c2A (r). For the kink mode, with ω2 = kz2 c2k and cA < ck < cAe , it follows that inside a tube the magnetic pressure force acts in unison with the tension (opposing the motion), whereas outside the tube the magnetic pressure force opposes the tension force. Specifically, the tension force is  −ρe c2Ae kz2 ξe , r > a, (11.51) T⊥ = 2 2 −ρ0 cA kz ξ0 , r < a, where ξ0 denotes the value of the displacement inside the tube and ξe is the value of the displacement outside the tube. Noting that     ρ0 − ρe 2 ρ0 − ρe 2 c2Ae − c2k = + cA , c , c2A − c2k = − ρ0 + ρe ρ0 + ρe Ae we obtain P⊥

⎧   ⎨+ρe c2 k2 ρ0 −ρe ξe , Ae z ρ0 +ρe = ⎩−ρ0 c2 k2  ρ0 −ρe ξ0 , A z ρ0 +ρe

Thus, P⊥

⎧   ⎨− ρ0 −ρe T ⊥ , ρ0 +ρe =  ⎩ ρ0 −ρe T ⊥ , ρ0 +ρe

r > a,

(11.52)

r < a.

r > a,

(11.53)

r < a.

Again we see that inside the tube the magnetic pressure force and the tension force are in unison, whereas outside the tube the forces act in opposition. Given ρ0 > ρe , the tension force is larger than the magnetic pressure force. As an illustration, with ρ0 = 2ρe the tension force inside the tube is three times the magnetic pressure force there; however, for ρ0  ρe the two forces are broadly comparable.

11.5 Thin Tubes: Multiple Scaling We now turn to the task of determining the properties of the kink mode by assuming at the outset that we are dealing with a thin tube. Our aim is to obtain approximate expressions that agree with the results obtained above for a uniform tube in a uniform medium, but without first considering the case of a finite tube radius. In principle, such an approximate theory has the virtue that it may be extended to include effects otherwise excluded in the case of a finite tube. In particular, we have in mind the role of stratification along the tube, though in principle many other effects can be explored in an appropriate thin tube theory. We follow Ruderman, Verth and Erd´elyi (2008; see also the review by Ruderman and Erd´elyi 2009)

324

Thin Flux Tubes: The Kink Mode

and consider the use of scaled coordinates, exploiting the applied mathematical concepts of multiple scales analysis (see, for example, Bender and Orszag 1978; Nayfeh 1973). Our system of equations is for the case of a unidirectional equilibrium magnetic field in a β = 0 plasma. These are the coupled equations for the displacement and the equation of the variation of magnetic pressure perturbation pm (see equations (11.18) and (11.21)). For easy of reference we gather them together here. The components of the momentum equation give  2  2 2  2  ∂ 1 ∂ ∂ ∂ 2 ∂ 2 ∂ − c = − , ρ − c (11.54) ρ0 p pm , ξ r m 0 A 2 A 2 ξφ = − ∂r r ∂φ ∂t2 ∂z ∂t2 ∂z and the magnetic pressure perturbation pm is given by pm =

B2 B0 Bz = − 0 div ξ = −ρ0 c2A div ξ , μ μ

(11.55)

with pm satisfying ∂ 2 pm ∂ 2ξ = c2A ∇ 2 pm + c2A (grad ρ0 ) · 2 . (11.56) 2 ∂t ∂t Density perturbations ρ may be determined from the linearized equation of continuity, if needed, but do not affect ξ and pm . For scaled coordinates we set Z = z,

τ = t.

(11.57)

We are interested in the case when the parameter  is small, but for the moment we can leave the value of  as arbitrary. The choice of scaled variables (11.57) is based upon the idea that for a thin tube the variation of the wave in the longitudinal direction is slower than the variation in the perpendicular direction. Consider then a thin magnetic flux tube of radius a and length L. When z varies from 0 to (say) the tube middle, a distance of L/2, then the scaled variable Z varies from 0 to L/2; thus, Z varies on a scale of a if we take L = 2a. Also, since we expect a wave to take a time of order L/cA to travel over a distance comparable with L, by taking τ = t we find that when t varies from 0 to L/2cA , the travel time of an Alfv´enic wave over a distance L/2, then the scaled variable τ varies from 0 to L/2cA ; if we choose L = 2a then the scaled variable τ varies from 0 to a/cA , the radial timescale. We note from (11.57) that 2 2 ∂ ∂2 ∂ ∂ ∂ ∂2 2 ∂ 2 ∂ =  , =  . = , = , ∂z ∂Z ∂t ∂τ ∂z2 ∂Z 2 ∂t2 ∂τ 2 Then, in transformed coordinates, equation (11.21) for the rate of change of the magnetic pressure becomes  2 2  ∂ 2 pm ∂ 2ξ ∂ 1 ∂ 1 ∂2 2 2 ∂ 2 = c (z) + +  + pm +  2 c2A 2 · grad ρ0 , (11.58) A 2 2 2 2 2 r ∂r r ∂φ ∂τ ∂r ∂Z ∂τ

whilst the transverse components (11.18) of the momentum equation yield  2  2 2  2  ∂ 1 ∂ ∂ ∂ 2 ∂ 2 2 ∂ − c = − ,  ρ − c  2 ρ0 p pm . (11.59) ξ ξφ = − r m 0 A A 2 2 2 2 ∂r r ∂φ ∂τ ∂Z ∂τ ∂Z

11.5 Thin Tubes: Multiple Scaling

325

Equations (11.58) and (11.59) provide the partial differential equations of our system. We require pm and ξr to be continuous across the tube boundary r = a. So far in our analysis, we have made no explicit use of the definition of , which may be arbitrary. Indeed, we could choose  = 1 and then we would recover the original system of linear equations that we started from. But we are interested in applying the scaled system of equations to the special case of a thin coronal flux tube. The tube is thin in the sense that its radius a is much smaller than the tube’s representative longitudinal scale L, so that the dimensionless parameter  = a/(2L) is very much less than unity. Regarding  as a small quantity means that we can order terms in the above equations and set up series expansions in powers of . In fact, inspection of our system reveals that the parameter  enters the equations specifically in the form  2 , so we may instead expand our variables in powers of  2 . (Equivalently, an expansion in powers of  would result in the coefficients of all odd powers of  being zero, only even powers giving non-zero contributions.) With  2  1, an expansion of the partial differential equation (11.58) gives, to lowest order in  2 , the result  2  ∂ 1 ∂ 1 ∂2 2 0 = cA (z) + (11.60) + pm . r ∂r r2 ∂φ 2 ∂r2 For the kink mode we may choose mode number m = 1, so that the perturbations have an e−imφ dependence with m = 1, i.e., an e−iφ dependence (corresponding to cos φ or sin φ dependences), and then (11.60) gives ∂ 2 pm 1 ∂pm 1 + − 2 pm = 0. 2 r ∂r ∂r r

(11.61)

Solutions with pm ∝ r or pm ∝ r−1 arise. Accordingly, requiring that pm is bounded both inside and outside the tube means that inside the tube we choose the solution that is proportional to r, whereas outside the tube we choose the solution that declines to zero at infinity (pm → 0 as r → ∞), meaning that pm varies inversely proportional to r. Altogether, then, we take  A0 (Z, τ )r, r < a, pm = 1 Ae (Z, τ ) r , r > a, for arbitrary functions A0 and Ae of the variables Z and τ . Requiring continuity of pm across the tube radius r = a, so that pm (r = a− ) = pm (r = a+ ), means that aA0 (Z, τ ) = Ae (Z, τ ) 1a , and then the solution of (11.61) may be written in the form  r < a, A0 (Z, τ )r, (11.62) pm = a2 A0 (Z, τ ) r , r > a with A0 an arbitrary function of Z and τ .

326

Thin Flux Tubes: The Kink Mode

These expressions for pm allow us to constrain ξr . From equation (11.59) applied separately to the regions r < a and r > a, we have  2 2  ∂ 2 ∂  2 ρ0 − c r < a, (11.63) ξr = −A0 (Z, τ ), A ∂τ 2 ∂Z 2 and

 2

 ρe

2  a2 ∂2 2 ∂ − c = A (Z, τ ) , ξ r 0 Ae ∂τ 2 ∂Z 2 r2

r > a,

(11.64)

where ρe is the plasma density and cAe (= (B20 /(μρe ))1/2 ) the Alfv´en speed in the environment (r > a) of the tube. Expressions (11.63) and (11.64) show that pm enters into the calculation as an order  2 term, treating ξr as an order unity term. Moreover, (11.63) shows that inside the tube ξr is independent of r , so that to the lowest order in  2 the radial displacement ξr does not vary across the tube (for r < a). Outside the tube, in r > a, the influence of the magnetic pressure on the radial displacement ξr (and with it the radial motions ur ) declines rapidly, inversely proportional to r2 . We apply equations (11.63) and (11.64) on the tube boundary r = a, with the result that  2  2 2  2  ∂ ∂ 2 ∂ 2 2 ∂ − c =  ρ − c (11.65) A0 (Z, τ ) = − 2 ρ0 ξ ξre , r0 e A Ae ∂τ 2 ∂Z 2 ∂τ 2 ∂Z 2 where ξr0 is the radial displacement on the boundary as r → a− (that is, ξr0 = ξr as r approaches r = a from within the tube (r < a)), and ξre is the radial displacement on the boundary as r → a+ (that is, ξre = ξr as r approaches r = a from outside the tube (r > a)). Now we require that the radial displacement ξr is continuous across r = a, so that ξr0 = ξre . Accordingly,  2  2 2  2  ∂ ∂ 2 ∂ 2 ∂ − c + ρ − c ρ0 ξ ξr0 = 0. r0 e A Ae ∂τ 2 ∂Z 2 ∂τ 2 ∂Z 2 Thus, transverse motions ξr (= ξr0 ) within a thin magnetic flux tube in a β = 0 plasma satisfy the wave equation 2 ∂ 2 ξr0 2 ∂ ξr0 = c , k ∂τ 2 ∂Z 2

where ck denotes the kink speed:



ck =

ρ0 c2A + ρe c2Ae ρ0 + ρe

(11.66)

1/2 .

(11.67)

The radial displacement ξr within the tube is independent of r and so ξr does not vary with r across the tube (though it does decline with r in the environment of the tube). Accordingly, in the kink mode a thin tube is displaced globally – the tube moving as a single body or string – and the displacement ξr (= ξr0 ) within the tube satisfies the wave equation (11.66). In the original coordinates, this gives ∂ 2 ξr ∂ 2 ξr 2 = c (z) . k ∂t2 ∂z2

(11.68)

11.6 Standing Kink Waves of an Unstructured Loop

327

Thus, we obtain the wave equation for the kink mode in a β = 0 plasma. An equation of this form was first derived using multiple scales by Ruderman, Verth and Erd´elyi (2008), following an alternative approach given earlier by Dymova and Ruderman (2005). The photospheric case, for which β = 0, also gives rise to the same wave equation, derived originally in a variety of ways (see Ryutov and Ryutova 1976; Parker 1979a; Spruit 1982; Edwin and Roberts 1983). Taking a time dependence of the form exp(iωt) we find that equation (11.68) leads to the ordinary differential equation ω2 d 2 ξr + 2 ξr = 0. 2 dz ck (z)

(11.69)

The above derivation applies in particular to a medium in which the equilibrium density ρ0 (z) inside the tube and the density ρe (z) in the environment are functions of z. This includes the special case of uniform densities, discussed in Chapter 6. It is interesting to observe that the argument above, given for the case of mode number m = 1, applies with only minor changes for mode numbers m = 2, 3, . . . and leads to the same wave equation (11.68); thus fluting modes also satisfy the kink mode wave equation. The only exception to this is the sausage mode (m = 0). Notice too that, from the fact that pm turns out to be of order  2 , it follows from equation (11.20) that div ξ is also of order  2 ; in other words, the kink wave does not compress the medium significantly (any compressions arising at order  2 or higher), though it is not strictly incompressible: div ξ is small but not zero. Thus, to lowest order in  2 we have ∂ξr 1 ∂ξφ 1 + ξr + = 0, ∂r r r ∂φ

(11.70)

and so with a φ-dependence of the form e−iφ we obtain iξφ =

∂ (rξr ). ∂r

(11.71)

Similarly, since pm = (B0 /μ)Bz we see that Bz is of order  2 and the magnetic perturbation in the kink mode is mainly perpendicular to the axis of the magnetic flux tube, the component along the tube being much smaller.

11.6 Standing Kink Waves of an Unstructured Loop Before examining the standing kink waves of a structured loop, it is as well to make clear the calculation of such oscillations in the simpler case of a uniform loop. Accordingly, consider the solution of equation (11.69) for the case when the propagation speed ck is a constant, say ck = ck (0), the value of kink speed at the point z = 0. We consider a loop of length L, with its apex at z = 0. We require that there is no transverse displacement at the loop footpoints, so that ξr = 0

at z = ±L/2.

(11.72)

328

Thin Flux Tubes: The Kink Mode

The equilibrium plasma density ρe in the environment of the coronal flux tube is taken to be some multiple of the internal plasma density ρ0 , so that ρe = χρ0 with χ a constant. Then, equation (11.69) becomes simply d 2 ξr ω2 (1 + χ )ρ0 ξr = 0, + dz2 (2B20 /μ)

(11.73)

an equation with constant coefficients. This equation possesses the general solution ξr = A0 cos α0 z + A1 sin α0 z,

(11.74)

where A0 and A1 are arbitrary constants (determined by initial conditions) and the constant α0 is given by α02 =

ω2 ω2 (1 + χ )ρ0 (0) = 2 2 (2B0 /μ) ck (0)

(11.75)

for a kink mode speed ck (which is a constant, and so equal to the kink mode speed ck (0) at the loop apex). Symmetry about the apex z = 0 means that we can examine separately solutions that are either even or odd about the apex. We refer to such solutions as the even modes or the odd modes, as appropriate. Consider the even modes, which correspond to the solution ξr = A0 cos α0 z,

(11.76)

giving ξr (−z) = ξr (z). For this solution to satisfy the boundary condition that ξr = 0 at z = L/2, we require 1 cos α0 L = 0 2

(11.77)

with solutions 1 π α0 L = (2n − 1) (11.78) 2 2 for integer n = 1, 2, 3, . . .. The case n = 1 gives a mode that has no nodes (ξr = 0) within the tube (−L/2 < z < L/2), the displacement vanishing only at the ends z = ±L/2 of the tube; this is the fundamental mode of the tube as a whole. Its frequency ω = ω1 is given by π ω1 = ck (0) (11.79) L and the corresponding period P1 (= 2π/ω1 ) is P1 =

2L ≡ Pkink . ck (0)

(11.80)

The period Pkink denotes the fundamental mode of the whole loop. In an unstructured loop, we see that the fundamental period is the time it takes the wave, propagating with a speed ck (0), to go from one end of a loop to the other end and back again. Consider the odd modes, which arise from the solution ξr = A1 sin α0 z,

(11.81)

11.7 Standing Kink Waves of a Structured Loop: Example 1

329

giving ξr (−z) = −ξr (z). For this solution to satisfy the boundary condition that ξr = 0 at z = L, we require 1 sin α0 L = 0 2

(11.82)

1 α0 L = nπ 2

(11.83)

with solutions

for integer n = 1, 2, 3, . . . . (The case n = 0 is of no interest since it gives ξr ≡ 0.) The case n = 1 gives a mode that has one node within the tube (−L/2 < z < L/2), namely at the apex z = 0: the displacement vanishes at the apex z = 0 of the loop (as well as at the ends z = ±L/2). This is the first overtone of the whole flux tube. Its frequency ω = ω2 is given by ω2 =

2π ck (0) L

(11.84)

and the corresponding period P2 (= 2π/ω2 ) is P2 =

L . ck (0)

(11.85)

This is the period of the first overtone of the loop. Finally, we note that the period ratio P1 /(2P2 ) in a uniform loop is unity, just as it would be for waves on a uniform string. However, as we shall see in the examples below, in a non-uniform loop the period ratio departs from unity; this provides a potentially useful diagnostic tool for the analysis of the extent of longitudinal structuring in coronal loops where multiple harmonics have been detected; see Verwichte et al. (2004), Van Doorsselaere, Nakariakov and Verwichte (2007) and Duckenfield et al. (2018) for some observational examples.

11.7 Standing Kink Waves of a Structured Loop: Example 1 We turn now to a consideration of a structured loop. The density structuring along a loop could arise for a number of reasons, not least because of gravity; however, we ignore the direct influence of gravity on the waves here, allowing only for stratification in the equilibrium state. We give two examples. In the first example, we treat a case where the differential equation (11.69) may be solved in terms of elementary mathematical functions, such as the natural logarithm. This case draws on the work of Dymova and Ruderman (2006). In the second example, discussed in Section 11.8 below, we discuss a case where Bessel functions arise. Consider, then, an equilibrium plasma density of the form ρ0 (z) =

ρ0 (0) 2

[1 − (1 − κ) 4z ]2 L2

,

r < a,

(11.86)

330

Thin Flux Tubes: The Kink Mode

where

 κ=

ρ0 (0) ρ0 (L/2)

1/2 .

(11.87)

Again, the magnetic flux tube is of length L, representing a coronal loop of length L which has been straightened out. The apex of the loop is taken to be at z = 0, and the loop’s footpoints are at z = ±L/2. The density profile is assumed to be symmetric about the apex z = 0. The parameter κ is a measure of how strongly the equilibrium density is longitudinally structured; an unstructured uniform loop has κ = 1, whereas a loop with plasma density that is much higher at the loop base than at the loop apex has κ  1. For κ ≈ 1, ρ0 (z) ≈ ρ0 (0) for −L/2 ≤ z ≤ L/2; however, for κ  1, ρ0 (z) ≈ ρ0 (0) over much of the coronal tube in −L/2 < z < L/2, except near the footpoints at z = ±L/2 where there is a region in which the density ρ0 (z) changes rapidly to the value ρ0 (0)/κ 2 at the base. The equilibrium plasma density ρe (z) in the environment of the coronal flux tube is taken to have the same structuring in z as the internal plasma density, so that at all locations z we have ρe (z) = χρ0 (z) with χ a constant. Thus, the square of the kink speed ck (z) is given by  2 2B20 2B20 4z2 2 (11.88) = 1 − (1 − κ) 2 . ck = μ(ρ0 (z) + ρe (z)) μ(1 + χ )ρ0 (0) L Accordingly, the governing ordinary differential equation for ξr is 1 ω2 (1 + χ )ρ0 (0) d 2 ξr + ξr = 0. 2 2 dz2 (2B0 /μ) [1 − (1 − κ) 4z2 ]2

(11.89)

L

We are interested in solutions of equation (11.89) that satisfy the boundary condition (11.72), representing the effect of a high plasma density at the loop footpoints. We are hereby modelling disturbances in a coronal loop that are confined by the extremely high plasma density of the loop’s photospheric (or low chromospheric) footpoints, permitting no displacement at the footpoints. Now the differential equation (11.89) has solutions of the form  1/2  1/2 4z2 4z2 cos αX, ξr = A1 1 − (1 − κ) 2 sin αX, (11.90) ξr = A0 1 − (1 − κ) 2 L L where A0 and A1 are arbitrary constants (determined by initial conditions) and the constant α is related to ω. The spatial variable X is connected to z by the relation   2z 2z x+ x− = 1 − λ , λ = (1 − κ)1/2 . , x+ = 1 + λ , X = ln x− L L At z = 0 we obtain x+ = x− = 1 and so X = ln(1) = 0. Thus, the apex z = 0 of the loop corresponds to X = 0 in the transformed variable X. The general solution of the differential equation (11.89) is a linear combination of the two solutions (11.90). The scaled variable X is an odd function of z: X(−z) = −X(z).

11.7 Standing Kink Waves of a Structured Loop: Example 1

331

Accordingly, we can discuss the above two solutions separately, choosing ξr (z) to be either an even function of z or an odd function of z. It is not immediately obvious that (11.90) provides solutions of the differential equation (11.89), so we begin by verifying directly that these expressions are indeed solutions of (11.89). Consider the expression  1/2 4z2 ξr = A0 1 − (1 − κ) 2 cos αX. (11.91) L The constant α is to be determined. The expression (11.91) gives an even function of z, for which ξr (−z) = +ξr (z). Notice first that dX 4λ 1 . = dz L (1 − 4z22λ2 ) L Then, from expression (11.91) it is straightforward to show that   −1/2  4λ2 z dξr 4z2 λ2 4λα = −A0 1 − sin αX + 2 cos αX , dz L L2 L  −3/2

2 d 2 ξr 4z2 λ2 2 4λ = −A cos αX. 1 + 4α 1 − 0 dz2 L2 L2 Substituting into equation (11.89) shows that we require  −3/2   4λ2 ω2 (1 + χ )ρ0 (0) 4z2 A0 1 − (1 − κ) 2 −(1 + 4α 2 ) 2 + cos αX = 0. L L (2B20 /μ) Thus, for expression (11.91) to provide a solution of equation (11.89) we require 4λ2 ω2 (1 + χ )ρ0 (0) = (1 + 4α 2 ) 2 , 2 L (2B0 /μ) that is, 2 ω2 2 4λ = (1 + 4α ) . L2 c2k (0)

(11.92)

This relation connects α and ω. For expression (11.91) to match the boundary condition that the displacement is zero at the base of the loop, we require that ξr = 0 at z = L/2 and so cos αxL = 0,

(11.93)

that is, αxL =

π (2n − 1), 2

(11.94)

332

Thin Flux Tubes: The Kink Mode

for integer n = 1, 2, 3, . . . . Here xL is the value of X at z = L/2, namely √     1+λ 1+ 1−κ xL = ln = ln . √ 1−λ 1− 1−κ

(11.95)

In the case of coronal loops, we expect κ < 1. The case when κ > 1 is treated in Section 14.13, Chapter 14. The case n = 1 gives the fundamental mode of the coronal flux tube as a whole, with the displacement being zero at the ends z = ±L/2 of the tube but otherwise non-zero. Thus, in the fundamental we have π (11.96) αxL = . 2 Thus, the frequency ω = ω1 of the fundamental is given by   2λ π 2 1/2 ω1 = ck (0) (11.97) 1+ 2 L xL and the period P1 (= 2π/ω1 ) of the fundamental is   πL π 2 −1/2 P1 = . 1+ 2 λck (0) xL

(11.98)

The period P1 of the fundamental mode of the structured loop is thus related to the fundamental period Pkink of a uniform (unstructured) loop through P1 = C(κ)Pkink ,

Pkink =

2L , ck (0)

(11.99)

where the multiplicative factor C(κ) is C(κ) =

  π π 2 −1/2 . 1+ 2 2λ xL

(11.100)

The factor C(κ) decreases monotonically from unity when κ = 1 (and there is no structuring along the flux tube) to π/2 when κ = 0 (and structuring is most marked). However, the approach of C(κ) to π/2 as κ → 0 is rather slow; for example, we have C(κ = 1/2) = 1.0870, and C(0.1) = 1.2530, C(0.01) = 1.3979 and C(10−4 ) = 1.5061. Accordingly, structuring of the density profile leads to a reduction in the local kink speed ck (z) and consequently an increase in the travel time along the loop and an increase in the period of the fundamental. We can examine more closely the expression xL which determines the behaviour of C(κ). In the case of an almost uniform tube, corresponding to κ being close to unity, we can expand the logarithm term in powers of λ, thus, 1 1 ln(1 + λ) ∼ λ − λ2 + λ3 − · · ·, λ → 0, 2 3 and apply the same expansion to ln(1 − λ). The result is that   1 xL ∼ 2λ 1 + λ2 + · · · , 3

11.7 Standing Kink Waves of a Structured Loop: Example 1

333

valid for κ → 1, λ → 0. In the opposite extreme, of a strongly stratified tube for which κ → 0, the behaviour is a little more complicated. Note first that √ 1+λ 1 1 (1 + λ)2 2 = = 1−κ] = (1 + λ) [2 − κ + 2 1−λ κ κ 1 − λ2 and so on expanding the square root term we obtain     4 κ κ2 xL ∼ ln + ln 1 − − −··· , κ 2 16 valid as κ → 0. These results may be used to examine the approximate behaviour of xL and associated expressions such as C(κ). Turning now to the odd modes, we note that  1/2 4z2 ξr = A1 1 − (1 − κ) 2 sin αX (11.101) L is a solution of equation (11.89). The displacement ξr (z) is an odd function of z, that is ξr (−z) = −ξr (z). Again, α is determined by equation (11.92) but now the boundary condition that ξr = 0 at the footpoint z = L/2 means that sin αxL = 0,

(11.102)

αxL = nπ

(11.103)

that is,

for integer n (= 1, 2, . . .). The case n = 1 provides the first overtone or first harmonic of the fundamental mode of oscillation of the loop as a whole; the overtone has frequency ω2 and period P2 given by     2λ πL 4π 2 1/2 4π 2 −1/2 ω2 = ck (0) , P2 = . (11.104) 1+ 2 1+ 2 L λck (0) xL xL Formulas of the kind (11.98) and (11.104), giving the fundamental and the overtones, are potentially useful as a diagnostic tool in coronal seismology (Andries et al. 2005; see also the review article Andries et al. 2009) in situations where it has been observationally possible to detect harmonics in loop oscillations (Verwichte et al. 2004; Van Doorsselaere, Nakariakov and Verwichte 2007). In particular, the ratio of the fundamental period P1 to twice the first overtone period P2 is given as   P1 1 4π 2 + xL2 1/2 = . (11.105) 2P2 2 π 2 + xL2 This period ratio falls from unity in a uniform unstructured loop (for which κ = 1) to 1/2 in a strongly structured loop (for which κ is small). The period ratio is of particular interest because it depends directly on a measure of the longitudinal stratification along a tube, here xL , unlike the separate P1 and P2 which depend upon ck as well as some measure of stratification.

334

Thin Flux Tubes: The Kink Mode

Period ratios are further discussed in Example 2 below and in Section 14.13, Chapter 14.

11.8 Standing Kink Waves of a Structured Loop: Example 2 In this second example of a structured tube we consider a case where the kink mode wave equation (11.69) may be solved in terms of special functions, namely Bessel functions. We draw on the work of McEwan, D´ıaz and Roberts (2008; see also McEwan et al. 2006). Again, we ignore the direct influence of gravity. In fact, McEwan, D´ıaz and Roberts (2008) discussed two cases where Bessel functions arise, a linear profile and an exponential profile for propagation speed ck ; here we consider the case of an exponential profile. We suppose that the equilibrium plasma density ρ0 within the tube is of the form   z ρ0 (z) = ρ0 (0) exp 0 ≤ z ≤ L/2, (11.106) = ρ0 (0)ez/c , c corresponding to an exponentially varying density that increases from ρ0 (0) at the loop apex (z = 0) to ρ0 (L/2) = ρ0 (0)eL/(2c ) at the loop footpoint (z = L/2). We assume also that the density in the environment of the loop has the same exponential profile, with the same spatial scale c . The constant c has the dimensions of length and gives the spatial scale of variation of the density; it may be related to the constant κ arising in Example 1 by noting that     L 1 ρ0 (0) 1/2 , = ln = − ln κ. κ= ρ0 (L/2) 4c κ The limit c → ∞ recovers the special case of a uniform loop (κ = 1). We assume that there is symmetry between one side of a loop and the other, so that the profile for z < 0 is simply the same as in 0 ≤ z ≤ L/2. For this assumed density profile the kink speed ck (z) is an exponentially decreasing function of z: ck (z) = ck (0)e−z/(2c ) ,

0 ≤ z ≤ L/2.

11.8.1 Dispersion Relations for an Exponential Density Profile For the above choice of profile, the governing differential equation (11.69), namely, d 2 ξr ω2 ξr = 0, + dz2 c2k (z) becomes ω2 z/c d 2 ξr e + ξr = 0, dz2 c2k (0)

0 ≤ z ≤ L/2.

(11.107)

Equation (11.107) may be solved in terms of the zeroth order Bessel functions J0 and Y0 (Abramowitz and Stegun 1965) yielding ξr (z) = AJ0 (ez/(2c ) ) + BY0 (ez/(2c ) ),

0 ≤ z ≤ L/2,

(11.108)

11.8 Standing Kink Waves of a Structured Loop: Example 2

335

where =

2c ω . ck (0)

The displacement ξr (z) must vanish at the footpoint base z = L/2, giving B = −A

J0 (eL/(4c ) ) . Y0 (eL/(4c ) )

Accordingly, a solution of (11.107) satisfying the condition that ξr (z) = 0 at z = L/2 is given by

0 ≤ z ≤ L/2, ξr (z) = A1 J0 (ez/(2c ) )Y0 (eL/(4c ) )−J0 (eL/(4c ) )Y0 (ez/(2c ) ) , (11.109) for arbitrary constant A1 . It follows from this expression that

 2z  dξr e c J0 (ez/(2c ) )Y0 (eL/(4c ) ) − J0 (eL/(4c ) )Y0  (ez/(2c ) ) , = A1 dz 2c (11.110) where a dash ( ) denotes the derivative of a Bessel function: J0  () = dJ0 (x)/dx calculated at x = , J0  (ez/(2c ) ) = dJ0 (x)/dx calculated at x = ez/(2c ) , etc. Equation (11.110) applies in 0 ≤ z ≤ L/2. As with Example 1, we consider modes that are either even or odd about the apex z = 0. Even modes have a displacement ξr that is symmetric about z = 0 and so have dξr /dz = 0 at z = 0; odd modes have zero displacement at the apex, so that ξr = 0 at z = 0. Expression (11.109) applies to both even and odd modes, but the relationship between  (and so ω) and L is different for the different modes. The even modes have dξr /dz = 0 at z = 0 and so satisfy (McEwan, D´ıaz and Roberts 2008) J0  ()Y0 (eL/(4c ) ) − J0 (eL/(4c ) )Y0  () = 0.

(11.111)

The odd modes have ξr = 0 at z = 0 and so satisfy the dispersion relation (McEwan, D´ıaz and Roberts 2008) J0 ()Y0 (eL/(4c ) ) − J0 (eL/(4c ) )Y0 () = 0.

(11.112)

These relations determine the way the dimensionless frequency  is related to the length L of a tube with an exponential density profile that is symmetric about the apex z = 0, the detailed form of the profile being determined by the value of the spatial scale c .

11.8.2 Asymptotic Behaviour of Dispersion Relations The dispersion relations (11.111) and (11.112), which determine the connection between frequency ω and the length L of standing waves in a straight magnetic tube (representing a magnetic loop), are transcendental and so it is not obvious what information they carry. However, we can examine their behaviour in the limit of large , since the case of a uniform

336

Thin Flux Tubes: The Kink Mode

tube corresponds to the limit c → ∞. Accordingly, we consider the asymptotic relations (Abramowitz and Stegun 1965) for the relevant Bessel functions, namely as x → +∞  1/2   1/2    2 2 1 1 J0 (x) ∼ sin χ , Y0 (x) ∼ cos χ , cos χ + sin χ − πx 8x πx 8x 

2 πx

1/2 

 3 cos χ , − sin χ − 8x



 3 ∼ ∼ sin χ , cos χ − 8x (11.113) where χ = x − π/4. These expressions give only the leading terms in the expansions; further terms may be carried if so desired. Substituting these expansions into the dispersion relation (11.111) for the even modes and making use of some elementary trigonometric relations, we obtain (to leading order) the result   1 3 1 (11.114) + L/(4 ) sin[(eL/(4c ) − 1)] = 0. cos[(eL/(4c ) − 1)] + c  8 8e J0 (x)

Y0 (x)

2 πx

1/2 

To leading order we have cos[(eL/(4c ) − 1)] = 0, and so (eL/(4c ) − 1) = π/2, for the first zero of the cosine function. It is evident, then, that to leading order  → ∞ and we require L/c → 0. Accordingly, for the even mode we expand  in the series     L 2c + + · · ·, L  c , + even even =π 1 2 L 2c even where the coefficients even 1 , 2 , etc. are to be determined. We may also expand the exponential function in a power series with the result that   L L L L 4 c −1= +··· ,  1, 1+ e 4c 8c c

and so L

(e 4c − 1) =

    π L 1 even 1 1+ 1 + +··· , 2 2c π 4

Also, we have 1 



3 1 + 8 8eL/(4c )



  L2 L 1 +O = . c 4π 2c

Thus, to leading order O( Lc ) the expansion (11.114) gives −

L 1 L even π  + = 0, 1 + 4c 4 c 4π

and so even 1

1 =− π



 π2 −1 . 4

L  1. c

11.8 Standing Kink Waves of a Structured Loop: Example 2

337

Thus, to leading order the first two terms in an expansion of  in powers of L/c yields   c 1 π2 L  1.  = 2π − −1 , L π 4 c Then the frequency ω1 of the even mode – the frequency of the fundamental mode – and its associated period P1 (= 2π/ω1 ) are given by    π ck (0) Pkink L 1 1

  , (11.115) ω = ω1 = 1− − 2 , P1 = L L 2c 4 π 1 − π4 − π1 2π c where Pkink is the fundamental period of the kink mode in a uniform tube. In fact, we can expand the dispersion relation (11.111) to higher orders; this was done by McEwan, D´ıaz and Roberts (2008), who obtained the result P1 =

Pkink

  2 L π 1 − π4 − π1 2π c + 48 −

1 8

−

1 π2



L 2π c

2 .

(11.116)

Notice that P1 reduces to Pkink in the limit L/c → ∞. The odd modes may be treated in a similar way, with the result that the frequency ω = ω2 of the odd mode – the first overtone of the fundamental mode of the tube as a whole – and its associated period P2 (= 2π/ω2 ) are given by (McEwan, D´ıaz and Roberts 2008)   2πck (0) (Pkink )/2 L

. ω = ω2 = 1− , P2 = (11.117) L 8c 1 − 18 Lc McEwan, D´ıaz and Roberts (2008) take the expansion one term further, obtaining P2 =

1−

π 4

(Pkink )/2

 2 L π 2π c + 48 −

1 32



L 2π c

2 .

(11.118)

Equation (11.118) gives the period of the first harmonic of the loop as a whole. The fundamental period P1 and its first overtone P2 , given by equations (11.116) and (11.118) for a tube that is exponentially stratified, determine the periods in terms of Pkink , the global period of a kink mode in a uniform tube; Pkink depends upon the loop length L, the internal and external Alfv´en speeds, and the internal and external densities. By combining P1 and P2 to form their ratio we may eliminate Pkink , obtaining a period ratio P1 /P2 or P1 /2P2 , as preferred, which depends upon L/c , the ratio of loop length L to density scale height c (McEwan, D´ıaz and Roberts 2008):    P1 1 L 2 5 L 2 = 1− 2 + − . (11.119) 2P2 2c π 2c π 4 32π 2 Formulas of this form may be used in conjunction with observations to deduce a measure of the longitudinal stratification along a loop. A discussion of various aspects of this type of analysis is presented in the review by Andries et al. (2009) and is also discussed in Section 14.13, Chapter 14.

338

Thin Flux Tubes: The Kink Mode

11.9 Non-Uniform Magnetic Field In this section we consider an equilibrium magnetic field that is no longer unidirectional, corresponding to a magnetic flux tube that expands or contracts with distance along the tube. This case has been discussed by Verth and Erd´elyi (2008)1 and Ruderman, Verth and Erd´elyi (2008). We follow closely the treatment in Ruderman, Verth and Erd´elyi (2008). The analysis we present is algebraically complicated but it leads to a simple result: under coronal conditions (plasma β = 0), a thin magnetic flux tube, with longitudinally varying internal radius a0 (z) and predominantly longitudinal magnetic field B0 = B0z ez gives rise to the one-dimensional wave with speed of propagation ck (z):   2 ∂ ∂2 1/2 − c2k 2 {B0z ξ⊥ } = 0, (11.120) 2 ∂t ∂z where ck is the usual kink speed, given here by c2k =

2B20z μ(ρ0 + ρe )

.

The wave equation is for the variable ξ⊥ /a0 (z), the transverse displacement ξ⊥ in units of the internal radius a0 (z) of the thin tube. We turn now to a derivation of this result, following closely the analysis in Ruderman, Verth and Erd´elyi (2008), save for notational changes.

11.9.1 Equilibrium We have in mind the case of an expanding magnetic flux tube such as arises in a coronal loop or in a photospheric magnetic tube. To be specific, we focus on the coronal case and again consider the β = 0 plasma. The equilibrium magnetic field B0 is taken in the form   (11.121) B0 = B0r (r, z), 0, B0z (r, z) , written in cylindrical coordinates r, φ, z. The equilibrium magnetic field is untwisted (B0φ = 0) and independent of the azimuthal angle φ. The constraint that div B0 = 0 means that the components B0r (r, z) and B0z (r, z) are coupled through ∂B0z 1 ∂B0r + B0r + = 0. ∂r r ∂z

(11.122)

Accordingly, we may express B0 in terms of a magnetic flux function ψ0 (r, z) setting B0r = −

1 ∂ψ0 , r ∂z

B0z =

1 ∂ψ0 . r ∂r

(11.123)

1 Note that Verth and Erd´elyi (2008) have erroneously omitted a term B B /(μr) arising in the φ -component of their 0r φ

momentum equation.

11.9 Non-Uniform Magnetic Field

339

The equilibrium current j0 is given by       ∂B0r ∂ 1 ∂ψ0 ∂B0z 1 ∂ 2 ψ0 − eφ = − + eφ , μj0 = curl B0 = ∂z ∂r ∂r r ∂r r ∂z2 where eφ is the unit vector in the φ-direction. We consider here the case when the equilibrium current vanishes (j0 = 0), requiring that ∂B0r ∂B0z = . ∂z ∂r For this case the flux function ψ0 is such that 1 ∂ψ0 ∂ 2 ψ0 ∂ 2 ψ0 − = 0. + r ∂r ∂r2 ∂z2 The special case ψ0 = B0 r2 /2 where B0 is a constant gives the case of a unidirectional field of strength B0 discussed in Section 11.3.

11.9.2 Perturbations Consider the linearized equations (given in Section 11.2) describing perturbations about the equilibrium (11.121), taken as current-free (so j0 = 0). The momentum equation is ρ0

∂ 2ξ = j × B0 ∂t2 = −grad pm +

1 1 (B0 · grad)B + (B · grad)B0 , μ μ

(11.124)

where the magnetic pressure perturbation pm is pm =

1 1 B0 · B = (B0r Br + B0z Bz ). μ μ

(11.125)

The perturbation magnetic field B follows from the induction equation, B = curl (ξ × B0 ),

div B = 0,

(11.126)

and perturbations in plasma density are given by ρ = −div(ρ0 ξ ),

(11.127)

determined once the displacement ξ is known. The vector  expressions arising  in the momentum equation, for an equilibrium magnetic field B0 = B0r (r, z), 0, B0z (r, z) , displacement ξ = (ξr , ξφ , ξz ) and perturbation magnetic field B = (Br , Bφ , Bz ), may be expanded thus:   ∂ ∂ (B0 · grad)B = B0r + B0z B ∂r ∂z     ∂B0r ∂B0r 1 ∂B0z ∂B0z + Bz + eφ B0r Bφ + ez Br + Bz . (B · grad)B0 = er Br ∂r ∂z r ∂r ∂z

340

Thin Flux Tubes: The Kink Mode

Accordingly, the linearized momentum equation (11.124) becomes     ∂ 2ξ ∂ ∂ 1 1 ∂B0r 1 ∂B0r B0r + B0z B + er Br + Bz ρ0 2 = −grad pm + μ ∂r ∂z μ ∂r μ ∂z ∂t     1 ∂B0z 1 B0r 1 ∂B0z + eφ Bφ + ez Br + Bz . (11.128) r μ μ ∂r μ ∂z To make further progress with the momentum equation we need to examine the magnetic field that arises through the induction equation, which here is most easily considered directly from the curl form (rather than its vector expansion), giving Br = −

B0r ∂ξφ ∂ + (B0 ξ⊥ ), r ∂φ ∂z Bz = −

Bφ =

∂ ∂ (B0r ξφ ) + (B0z ξφ ), ∂r ∂z

1 ∂ 1 ∂ (rB0 ξ⊥ ) − (B0z ξφ ), r ∂r r ∂φ

(11.129)

where we have introduced a measure of the displacement perpendicular to the applied magnetic field through writing ξ⊥ =

B0z ξr − B0r ξz . B0

(11.130)

Here B0 is the strength of the applied magnetic field and is a function of r and z; B20 = B20r + B20z . We can use the above expressions for Br and Bz together with the definition of pm to obtain the magnetic pressure perturbation: pm = −

1 B20 ∂ξφ B0z 1 ∂ B0r ∂ + (B0 ξ⊥ ) − (rB0 ξ⊥ ). r μ ∂φ μ ∂z μ r ∂r

(11.131)

Returning to the momentum equation (11.128), we find that its components yield   ∂ 2 ξr B0z ∂Br ∂Bz − , ρ0 2 = μ ∂z ∂r ∂t   ∂ 2 ξφ ∂ ∂ 1 ∂pm 1 1 + B0r + B0z Bφ + B0r Bφ , ρ0 2 = − r ∂φ μ ∂r ∂z μr ∂t   ∂ 2 ξz B0r ∂Br ∂Bz ρ0 2 = − − . (11.132) μ ∂z ∂r ∂t Notice that the component of the displacement along the applied magnetic field, ξ = ξ · Bo/B0 = (B0r ξr + B0z ξz )/B0 , satisfies ∂ 2 ξ = 0, (11.133) ∂t2 from which we take the solution ξ = 0; that is, there is no displacement in the direction of the applied magnetic field, since the magnetic force acts perpendicular to the magnetic field. Notice too that we can rewrite the equation for ξφ in the form ρ0

∂ 2 ξφ 1 ∂pm B0z ∂Bφ B0r 1 ∂ =− + (rBφ ) + . r ∂φ μ r ∂r μ ∂z ∂t2

(11.134)

11.9 Non-Uniform Magnetic Field

341

It is convenient to work in terms of variables B⊥ =

1 (B0z Br − B0r Bz ), B0

pm =

1 (B0r Br + B0z Bz ), μ

B⊥ , B0

Bz = −B0r

Q=

μpm . B20

(11.135)

Then, we have Br = B0r Q + B0z

B⊥ + B0z Q. B0

(11.136)

Accordingly,     ∂Q ∂Q ∂ B⊥ ∂ B⊥ ∂Bz ∂Br − = B0r − B0z + B0r + B0z ∂z ∂r ∂z ∂r ∂r B0 ∂z B0      ∂B0r ∂B0z ∂B0z B⊥ ∂B0r + + − Q. + ∂r ∂z B0 ∂z ∂r Thus   ∂Br ∂Q ∂Q ∂ 1 B⊥ ∂Bz − = B0r − B0z + rB0r ∂z ∂r ∂z ∂r ∂r r B0     ∂ B⊥ ∂B0z ∂B0r + B0z − Q, + ∂z B0 ∂z ∂r

(11.137)

where we have used the fact that the equilibrium magnetic field B0 must be divergence-free (div B0 = 0), ∂B0z 1 ∂B0r + B0r + = 0. ∂r r ∂z If, furthermore, the equilibrium magnetic field is also current-free (j0 = 0), so that the equilibrium magnetic field is potential, then the last term on the right of equation (11.137) vanishes and we obtain     ∂Q ∂Q ∂ 1 B⊥ ∂ B⊥ ∂Bz ∂Br − = B0r − B0z + rB0r + B0z . (11.138) ∂z ∂r ∂z ∂r ∂r r B0 ∂z B0 Also, from the r and z components of the momentum equation (11.132) we obtain μρ0 ∂ 2 ξ⊥ ∂Br ∂Bz = − 2 B0 ∂t ∂z ∂r     ∂Q ∂Q ∂ 1 B⊥ ∂ B⊥ − B0z + rB0r = B0r + B0z ∂z ∂r ∂r r B0 ∂z B0

(11.139) (11.140)

for a current-free magnetic equilibrium. We can also obtain an expression for B⊥ . From the definition in equation (11.135), we have B0 B⊥ = B0z Br − B0r Bz = B0z

∂ ∂ 1 (B0 ξ⊥ ) + B0r (rB0 ξ⊥ ), ∂z r ∂r

where we have used equation (11.129) for the field components Br and Bz .

(11.141)

342

Thin Flux Tubes: The Kink Mode

11.9.3 Transformed Coordinates Following the analysis in Ruderman, Verth and Erd´elyi (2008), we introduce new coordinates R and ζ to replace the original coordinates r and z. We set R = ψ0 (r, z),

ζ = z.

(11.142)

Then, from the chain rule for partial derivatives, we have ∂ ∂ ∂R ∂ ∂ζ ∂ ∂ψ0 ∂ = + = = rB0z ∂r ∂r ∂R ∂r ∂ζ ∂r ∂R ∂R and ∂ ∂ ∂R ∂ ∂ζ ∂ ∂ψ0 ∂ ∂ ∂ = + = + = −rB0r + . ∂z ∂z ∂R ∂z ∂ζ ∂z ∂R ∂ζ ∂R ∂ζ In the above we have used relations (11.123) connecting ψ0 and the equilibrium magnetic field components B0r and B0z . Consider equation (11.129) for the field component Bφ . In the transformed coordinates, we have Bφ =

   ∂  ∂  ∂  ∂ ∂ (B0r ξφ ) + (B0z ξφ ) = rB0z B0r ξφ + B0z ξφ − rB0r B0z ξφ , ∂r ∂z ∂R ∂ζ ∂R

which, after some algebra, reduces to ∂ Bφ = rB0z ∂ζ



 1 ξφ . r

(11.143)

In writing this result we have noted that B0r ∂r . = ∂ζ B0z Expressions of this form follow from the transformation (11.142) if we differentiate the equation R = ψ0 partially with respect to R or ζ , and use the chain rule for partial derivatives: ∂R ∂r ∂ψ0 ∂r ∂ψ0 ∂z = + = rB0z , ∂R ∂r ∂R ∂z ∂R ∂R ∂r ∂ψ0 ∂r ∂ψ0 ∂z ∂R = + = rB0z − rB0r . 0= ∂ζ ∂r ∂ζ ∂z ∂ζ ∂ζ

1=

Hence ∂r 1 , = ∂R rB0z

∂r B0r . = ∂ζ B0z

Then from equation (11.140) for ξ⊥ we obtain μρ ∂ 2 ∂Q ∂Q ∂ (B0 ξ⊥ ) = B0r − rB20 + rB0z 2 2 ∂ζ ∂R ∂ζ B0 ∂t



 1 B⊥ . r B0

(11.144)

11.9 Non-Uniform Magnetic Field

343

Note that this equation is equivalent to equation (32) in Ruderman, Verth and Erd´elyi (2008).2 From the equation for the magnetic pressure perturbation we may deduce that Q=

∂ B0z 1 ∂ξφ B0r ∂ (B0 ξ⊥ ) − r (B0 ξ⊥ ) − 2 (B0 ξ⊥ ) − . 2 ∂R r ∂φ B0 ∂ζ rB0

(11.145)

The equation for B⊥ follows from equation (11.141), transformed accordingly: B⊥ =

 B0z 1 ∂  rB0 ξ⊥ , B0 r ∂ζ

(11.146)

whilst the equation for ξφ is ∂ 2 ξφ B0z ∂ μρ0 2 = r ∂ζ ∂t



∂ r B0z ∂ζ 2



1 ξφ r

 −

B20 ∂Q . r ∂φ

If we substitute expression (11.146) for B⊥ in equation (11.144) we obtain   μρ0 ∂ 2 ∂ ∂Q ∂Q B0z ∂  (B0 ξ⊥ ) − rB0z rB0 ξ⊥ ) = B0r − rB20 . 2 2 2 2 ∂ζ ∂ζ ∂ζ ∂R ∂t B0 r B0

(11.147)

(11.148)

11.9.4 Special Case We can simplify our equations by considering the special case of a thin magnetic flux tube that corresponds to the choice 1 2 r h(z), 2 for which the equilibrium magnetic field has components ψ0 =

1 B0r = − rh (z), 2

B0z = h(z),

(11.149)

(11.150)

where the prime denotes the derivative with respect to distance z along the tube (h (z) = dh/dz). Then the magnetic field strength B0 is given by r2  (h (z))2 . 4 To leading order in r this is simply B0 = h. Magnetic flux conservation gives B0z A0 = 0 , where 0 is the constant flux within the tube of radius a0 (z) and cross-sectional area A0 (z): 0 = h(z)A0 (z) = πh(z)a20 (z) = π B0z (z)a20 (z). B20 = h2 (z) +

2 It should be note that there is a typographical error in equation (32) of Ruderman, Verth and Erd´elyi (2008): the last term on

the right-hand side of their equation (32) should involve, in their notation, an expression of the form   ∂ b⊥ , ∂z rB replacing the typographically incorrect term ∂ ∂r



b⊥ rB

 .

344

Thin Flux Tubes: The Kink Mode

Substituting the forms of B0r and B0z into the various coefficients that arise in the partial differential equation (11.148), and noting that we can express r in terms of R (= ψ0 ) and ζ through   2R 1/2 , r= h(ζ ) we obtain

    2 B ξ  ∂ 2 B0 ξ⊥ 0 ⊥ 2 ∂ 1/2 1  ∂Q 3 ∂Q hh +h , −h = −(2R) μρ0 2 √ √ 2 ∂ζ ∂R ∂t ∂ζ 2 h h

where now the expression for Q is    ∂ξφ ∂ h ∂ 1 R (B0 ξ⊥ ) + 2R (B0 ξ⊥ ) + B0 ξ⊥ + h . Q=− ∂R ∂φ (2Rh)1/2 h2 ∂ζ

(11.151)

(11.152)

Additionally, we need the equation for the azimuthal displacement ξφ , which reduces to μρ0 On multiplying by

2 ' ( ∂ 2 ξφ h5/2 ∂Q 3/2 ∂ 1/2 h − = h ξ . φ ∂t2 ∂ζ 2 (2R)1/2 ∂φ

√ h we can rewrite this equation in the form μρ0

2 '√ ( ∂ 2 '√ ( h3 ∂Q 2 ∂ − h = − hξ hξ . φ φ ∂t2 ∂ζ 2 (2R)1/2 ∂φ

(11.153)

11.9.5 Multiple Scalings Equations (11.151)–(11.153) determine our system. This is a complicated system of partial differential equations. To make progress in understanding its behaviour we now introduce scaled coordinates, following the lead suggested by the earlier treatment in Section 11.5. As earlier, we introduce scaled coordinates Z and τ where Z = ζ (= z),

τ = t.

With these coordinates we have ∂ ∂ = , ∂ζ ∂Z

2 ∂2 2 ∂ =  , ∂ζ 2 ∂Z 2

∂ ∂ = , ∂t ∂τ

2 ∂2 2 ∂ =  , ∂t2 ∂τ 2

and then equations (11.151)–(11.153) become   2 B ξ  2 B ξ  1 2 dh ∂Q 0 ⊥ 0 ⊥ 2 ∂ 2 2 ∂ 1/2 3 ∂Q μρ0  +  h , (11.154) h − h = −(2R) √ √ ∂R 2 dZ ∂Z ∂τ 2 ∂Z 2 h h Q=

  ∂ξφ −1 ∂ 2 R dh ∂ ξ ) + B ξ + h ξ ) (B +  (B 2R 0 ⊥ 0 ⊥ 0 ⊥ , ∂R ∂φ (2Rh)1/2 h2 dZ ∂Z μρ0  2

2 '√ ( ∂ 2 '√ ( h3 ∂Q 2 2 ∂  hξ − h = − hξ . φ φ ∂τ 2 ∂Z 2 (2R)1/2 ∂φ

In these equations we have noted that h (z) = dh/dZ.

(11.155)

(11.156)

11.9 Non-Uniform Magnetic Field

345

In equations (11.154)–(11.156) the parameter  is taken to be a measure of the thinness of a coronal flux tube, considered to be much longer than its internal radius. For example, in a flux tube of length L and internal radius a0 (0) at the location z = 0, we can form the measure  = 2a0 (0)/L. In thin tubes,   1 (see also Chapter 8). We regard the parameter  2 as a small quantity. Note that  enters into equations 2 (11.154)–(11.156) only in the √ form  , and therefore we may expand the three variables B0 ξ⊥ (or equivalently B0 ξ⊥ / h), Q and ξφ in power series that are in powers of  2 (rather than in powers of ). Accordingly, we may write B0 ξ⊥ = u0 +  2 u2 +  4 u4 + · · ·,

Q = Q0 +  2 Q2 + · · ·,

ξφ = ξφ0 +  2 ξφ2 + · · ·,

where the terms u0 , u2 , Q0 , . . . are in principle to be determined. In fact, it proves necessary to determine only the leading order behaviour of these expansions. We can draw an important deduction immediately from equation (11.156). Regarding ξφ as an order unity term (so ξφ0 = 0) and provided ∂Q/∂φ = 0 (which applies for all modes except the sausage mode), then the pressure term ∂Q/∂φ (and so also Q) enters as an order  2 term. That is, the first term in the expansion of Q must be zero, Q0 = 0, and thus to leading order Q =  2 Q2 . This applies for the kink mode and also the fluting modes; only for the sausage mode, for which Q is independent of φ, do we expect a different ordering. For the kink mode, then, we can examine our equations to leading order. Accordingly, to leading order in  2 equations (11.154) and (11.156) become   2 B ξ  ∂ 2 B0 ξ⊥ ∂Q2 0 ⊥ 2 ∂ , (11.157) μρ0 2 √ −h = −(2R)1/2 h3 √ 2 ∂R ∂τ ∂Z h h ∂ 2 '√ ( ∂ 2 '√ ( h3 ∂Q2 = − hξ . (11.158) μρ0 2 hξφ − h2 φ ∂τ ∂Z 2 (2R)1/2 ∂φ Since Q is of order  2 , consistency with equation (11.155) requires that to leading order in  2 ∂ξφ ∂ 2R (B0 ξ⊥ ) + B0 ξ⊥ + h = 0. ∂R ∂φ Writing the φ-dependence in the form ξφ ,

ξ⊥ ,

Q ∝ e−imφ ,

with, for the kink mode, the azimuthal order being m = 1, so that ∂ξφ /∂φ = −iξφ , then to leading order we have ∂ (11.159) (B0 ξ⊥ ). ∂R Equations (11.157), (11.158) and (11.159) define our system. We may write the equations more compactly if we introduce the wave operator ihξφ = B0 ξ⊥ + 2R

L = μρ0 (Z)

∂2 ∂2 2 − h (Z) . ∂τ 2 ∂Z 2

346

Thin Flux Tubes: The Kink Mode

Then

 L

B0 ξ⊥ √ h

 = −(2R)1/2 h3

√  hξφ = i

∂Q2 , ∂R

L

h3 Q2 , (2R)1/2

  √ 1 ∂ i hξφ = √ B0 ξ⊥ + 2R (B0 ξ⊥ ) . ∂R h

(11.160)

Set  U=

2R h



1/2 (B0 ξ⊥ ),

B0 ξ⊥ =

h 2R

1/2 U.

(11.161)

Then ∂ (B0 ξ⊥ ) = ∂R



h 2R

1/2 

1 ∂U − U ∂R 2R



and so B0 ξ⊥ + 2R Thus

∂U ∂ (B0 ξ⊥ ) = (2hR)1/2 . ∂R ∂R

  2R ∂U L ∂R h3     ∂Q2 ∂U ∂ 2U 2 2R = − 3L − 3L . ∂R ∂R h h ∂R2 Q2 = −

Finally, substituting in the first equation in (11.160) we obtain   2 ∂U 2∂ U + 4R − U = 0. L 4R ∂R ∂R2

(11.162)

(11.163)

This equation was first obtained by Ruderman, Verth and Erd´elyi (2008). One solution of equation (11.163) follows by inspection, namely that 4R2

∂ 2U ∂U + 4R − U = 0. ∂R ∂R2

(11.164)

It is easy to see that this partial differential equation has general solution U = C1 (Z, τ )R1/2 + C2 (Z, τ )R−1/2 ,

(11.165)

where C1 (Z, τ ) and C2 (Z, τ ) are arbitrary functions of the coordinates Z and τ (and we have suppressed the φ factor e−imφ ). In terms of the transverse displacement the expression for U leads to  1/2   B0z C2 + . C ξ⊥ = 1 R 2B20

11.9 Non-Uniform Magnetic Field

347

Applied to the inside of a magnetic flux tube, for ξ⊥ to be finite at the centre of the tube r = 0 (which corresponds to R = 0) we must choose C2 = 0 and then  1/2 B0z C1 , inside tube. ξ⊥ = 2B20 Outside the tube, it is reasonable to choose a solution for U that declines with R, corresponding to U = Ce /R1/2 with an associated transverse displacement being  1/2 Ce B0z , outside tube. ξ⊥ = 2 R 2B0 For a thin tube, we can take B0 ∼ B0z and then ⎧ 1/2 ⎪ ⎨ 1 C1 , inside tube, 2B ξ⊥ = 0z 1/2 ⎪ Ce ⎩ 1 outside tube. 2B0z R, For ξ⊥ to be continuous across R = R0 , corresponding to r = a0 (z) in the original coordinates, we choose Ce = R0 C1 and then ⎧ 1/2 ⎪ ⎨ 1 C1 (Z, τ ), inside tube, 2B (11.166) ξ⊥ = 0z 1/2

 ⎪ R0 ⎩ 1 C (Z, τ ) , outside tube. 1 2B0z R Finally, we require also that the magnetic perturbation pm is continuous across the tube boundary, and this implies that Q (and so Q2 ) is continuous also, given that B0z is continuous. Thus, from equation (11.162), we require   2R ∂U − 2 3 L ∂R h to be continuous. Since

 1 C1 R−1/2 , R < R0 , ∂U = 21 −3/2 ∂R − 2 C1 R0 R , R > R0 ,

this leads to     ∂2 1 −1/2 ∂2 1 ∂2 ∂2 2 2 2 2 = − μρe 2 − h (Z) 2 C1 R0 R−3/2  μρ0 2 − h (Z) 2 C1 R 2 2 ∂τ ∂Z ∂τ ∂Z with R = R0 . Hence

 ∂2 ∂2 2 μ(ρ0 + ρe ) 2 − 2h (Z) 2 C1 (Z, τ ) = 0. ∂τ ∂Z



In the original coordinates, this becomes 2 ∂ 2 1/2 1/2 2 ∂ {B (z)ξ } = c {B (z)ξ⊥ }, ⊥ k 0z ∂t2 ∂z2 0z

(11.167)

348

Thin Flux Tubes: The Kink Mode

where ck is the usual kink speed, given here by  1/2 2B20z (z) . ck (z) = μ(ρ0 (z) + ρe (z)) 1/2

Equation (11.167) is the main result of this section, namely that the quantity B0z ξ⊥ satisfies the wave equation with speed ck . Since magnetic flux conservation implies that πB0z (z)a20 (z) = constant, we can also write this result in the form    2  ξ ξ⊥ ∂2 ⊥ 2 ∂ = c , (11.168) k 2 ∂t2 a0 (z) ∂z a0 (z) where a0 (z) is the local radius (r = a0 (z)) of the expanding thin flux tube. Thus we have obtained the wave equation for the kink mode in a magnetic flux tube. The wave equation for the kink mode in an expanding magnetic tube was first obtained by Ruderman, Verth and Erd´elyi (2008). The special case of a straight (a0 (z) = constant) tube was investigated by Dymova and Ruderman (2005). Andries and Cally (2011) give an overview of the operators that arise in such problems.

11.10 Radial Expansion Equations: Effect of Gravity In Chapter 10 we showed how the sausage mode may be described using a treatment derived from an expansion procedure in which the dependent variables such as the motion or perturbed magnetic field may be expanded in a Taylor series. Such a procedure was first put forward for the sausage mode by Roberts and Webb in 1978. Curiously, the extension of the method to described the kink mode has only recently been carried out, by Lopin and Nagorny (2013) and Lopin, Nagorny and Nippolainen (2014). Their approach applied to the kink mode is very similar to that developed earlier for the sausage mode. The effect of gravity is included. 11.10.1 Equilibrium The equilibrium magnetic field is taken to be of the form   B0 = B0r (r, z), 0, B0z (r, z) .

(11.169)

Magnetostatic pressure balance gives an equilibrium plasma pressure p0 that is stratified by gravity, with   B20 1 grad p0 + div B0 = 0. (11.170) = ez gρ0 + (B0 · grad)B0 , 2μ μ This is the equilibrium investigated in Chapter 10 for the sausage mode. We may expand the equilibrium magnetic field in a Taylor series about r = 0: 2 B0r (r, z) = rB(1) 0r (z) + O(r ),

B0z (r, z) = B0z (z) + O(r),

where O(r) indicates that the next term in the expansion is of order r (see, for example, Bender and Orszag 1978). The solenoidal constraint div B0 = 0 implies that

11.10 Radial Expansion Equations: Effect of Gravity

349

B0r (z) = − 12 [B0z (z)] . Here we have used a prime ( ) to denote the derivative with respect to depth z of an equilibrium quantity (here the longitudinal magnetic field, B0z , within the tube). Then, to leading order in r, we have (1)

1 B0r (r, z) = − rB0z  (z). 2 Accordingly, to leading order in r the equilibrium magnetic field is   1  B0 = − rB0z (z), 0, B0z (z) . 2

(11.171)

(11.172)

11.10.2 Perturbations Small perturbations about this equilibrium lead to the following system of linear equations describing the displacement ξ , perturbation density ρ, plasma pressure p and perturbation magnetic field B: ρ = −div (ρ0 ξ ), ρ0

p + ξ · grad p0 = c2s (ρ + ξ · grad ρ0 ),

∂ 2ξ 1 1 = −grad pT + (B0 · grad)B + (B · grad)B0 + ρg, 2 μ μ ∂t

B = −B0 (div ξ ) + (B0 · grad)ξ − (ξ · grad)B0 ,

div B = 0,

(11.173)

where cs denotes the sound speed and pT the total pressure perturbation pT = p +

1 1 B0 · B = p + (B0r Br + B0z Bz ). μ μ

(11.174)

In cylindrical polar coordinates r, φ, z and with an equilibrium field (B0r , 0, B0z ) and perturbed field (Br , Bφ , Bz ) we have   ∂ ∂ (B0 · grad )B = B0r + B0z B ∂r ∂z   ∂Bφ ∂Bφ ∂Br ∂Br ∂Bz ∂Bz + B0z , B0r + B0z , B0r + B0z = B0r ∂r ∂z ∂r ∂z ∂r ∂z and

  ∂B0r ∂B0r 1 ∂B0z ∂B0z + Bz , B0r Bφ , Br + Bz . (B · grad )B0 = Br ∂r ∂z r ∂r ∂z

Similar expressions apply to (B0 · grad )ξ and (ξ · grad )B0 . Applied to the induction equation, we obtain the transverse and longitudinal components of the perturbation magnetic field as ∂ξφ 1 ∂B0r ∂ξr 1 ∂ − B0r ξr − B0r − ξr − (B0r ξz ), ∂z r r ∂φ ∂r ∂z ∂ξφ ∂ξφ 1 Bφ = B0r + B0z − B0r ξφ , ∂r ∂z r   ∂ξφ ∂ξz B0z ∂B0z ∂ Bz = − ξr + − (B0z ξr ) − ξz + B0r . r ∂φ ∂r ∂z ∂r Br = B0z

(11.175)

350

Thin Flux Tubes: The Kink Mode

11.10.3 Radial Expansion Introduce the following radial expansions of perturbation variables: ξφ (r, φ, z, t) = ξφ(0) + r2 ξφ(2) + · · ·,

ξr (r, φ, z, t) = ξr(0) + r2 ξr(2) + · · ·,

2 (2) Bφ (r, φ, z, t) = B(0) φ (φ, z, t) + r Bφ + · · · (11.176)

2 (2) Br (r, φ, z, t) = B(0) r + r Br + · · ·,

and ξz (r, φ, z, t) = rξz(0) + r3 ξz(3) + · · ·,

3 (3) Bz (r, φ, z, t) = rB(0) z + r Bz + · · ·, (1)

(3)

pT (r, φ, z, t) = rpT + r3 pT + · · ·.

(11.177)

Each coefficient ξr(0) , ξr(2) , etc. in these expansions is a function of φ, z and t. The form of the expansions (11.176) and (11.177) in r is physically what one would expect for the kink (m = 1) mode, since it is a global mode which displaces the central axis of the tube and so ξr is expected to be non-zero at r = 0, whereas the longitudinal displacement ξz is expected to be small. The expansions contrast with those used for the sausage mode, where the longitudinal displacements and motions dominate (see Chapter 10). As with the sausage mode, we will be content with calculating only the leading terms in the r-expansions. To leading order in r, the transverse components of the induction equation yield B(0) r = B0z

(0) (0) ∂ξr B0r ∂ξφ − + B0z  ξr(0) , ∂z r ∂φ

(0)

Bφ = B0z

∂ξφ(0) ∂z

−

B0r (0) ξ . r φ

(11.178)

However, examination of the z-component of the induction equation (11.175) shows that the term   ∂ξφ B0z ξr + − r ∂φ must be zero as r tends to zero, for otherwise Bz would be unbounded at the origin. Thus we require ξr(0)

+

∂ξφ(0) ∂φ

= 0,

(11.179)

and then the longitudinal perturbation field Bz is bounded as r → 0. Combined with the solenoidal constraint (11.171) on the magnetic field within a tube, we see that the transverse components of the perturbed magnetic field have expansion terms B(0) r = B0z

∂ξr(0) 1 + B0z  ξr(0) , ∂z 2

(0)

B(0) φ = B0z

∂ξφ

1 + B0z  ξφ(0) , ∂z 2

(11.180)

and the constraint div B = 0 becomes B(0) r + which is satisfied given (11.179).

∂B(0) φ ∂φ

= 0,

(11.181)

11.10 Radial Expansion Equations: Effect of Gravity

351

Consider the radial component of the momentum equation:   ∂ 2 ξr ∂Br ∂Br ∂pT 1 1 1 ∂B0r 1 ∂B0r ρ0 2 = − + B0z + Br + B0r + Bz . ∂r μ ∂z μ ∂r μ ∂r μ ∂z ∂t The term in { } is of order r2 or smaller, and so is neglected. Taking into account the solenoidal constraint (11.171), we then have ρ0

∂ 2 ξr(0) ∂B(0) 1 1 r (1) = −p + B − B0z  B(0) 0z r . T μ ∂z 2μ ∂t2 (0)

Hence, substituting for Br from (11.180) we obtain (0)

(1) pT

∂ 2 ξr = −ρ0 ∂t2

+

B20z ∂ 2 ξr(0) μ

∂z2

+

(B20z ) ∂ξr(0) 2μ

∂z

 +

 1 1  2  B0z B0z − (B0z ) ξr(0) . 2μ 4μ

Thus, inside the tube the total pressure perturbation pT (r, φ, z, t) (= rp(1) T ) is given, to leading order in r, by     (0) B20z ∂ 2 ξr(0) (B20z ) ∂ξr(0) ∂ 2 ξr 1 1  2  (0) pT = r −ρ0 + + B − ) + B (B ξ . 0z 0z 0z r μ ∂z2 2μ ∂z 2μ 4μ ∂t2 (11.182) The radial displacement ξr and the total pressure perturbation pT must be continuous (0) (1) across r = a, so ξr = ξre (a) and apT = pTe (a) on r = a. Here ξre and pTe denote the radial displacement and the (total) pressure perturbation outside the tube, both calculated as r → a. Thus, we need to discuss the behaviour in the environment of the tube. We consider two cases: the β = 0 plasma in an unstratified medium (g = 0), and the stratified (g = 0) tube in a field-free environment. 11.10.4 The β = 0 Plasma in an Unstratified (g = 0) Medium We consider a β = 0 plasma with the effects of gravity ignored (setting g = 0). This is the circumstances discussed earlier through a different approach in Section 11.4. We suppose that the flux tube is uniform with a magnetic field B0 ez and plasma density ρ0 . Then B0z = B0 , and expression (11.182) for pT reduces to   (0) B20 ∂ 2 ξr(0) ∂ 2 ξr pT = r −ρ0 + . (11.183) μ ∂z2 ∂t2 The environment in which the tube is embedded is taken to be a β = 0 plasma with a uniform equilibrium field Be ez and uniform density ρe . Then, the radial component ξre of the displacement ξ e is related to the total pressure perturbation pTe by (see Section 11.3)   2 ∂pTe ∂2 2 ∂ ρe − cAe 2 ξre = − . (11.184) 2 ∂r ∂t ∂z The total pressure perturbation satisfies the three-dimensional wave equation, ∂ 2 pTe = c2Ae ∇ 2 pTe , ∂t2

(11.185)

352

Thin Flux Tubes: The Kink Mode

where cAe (= Be /(μρe )1/2 ) denotes the Alfv´en speed in the environment. Note that in a β = 0 plasma the total pressure perturbation pTe consists solely of the magnetic pressure perturbation, so that pTe = pme = Be Bez /μ with Bez denoting the z-component of the perturbation field in the environment. We are interested here in the kink mode, which has a φ-dependence e−imφ with m = 1 (so that the φ-dependence is e−iφ ). Then, equation (11.185) gives ∂ 2 pTe 1 ∂pTe ∂ 2 pTe 1 ∂ 2 pTe 1 + p = − + . − Te r ∂r ∂r2 r2 ∂z2 c2Ae ∂t2

(11.186)

Ignoring for the moment the terms on the right-hand side of equation (11.186), we see by inspection that pTe is of the form Ae r or Ae r−1 . The first solution is unbounded as r → ∞, so for a bounded solution in r > a we choose 1 pTe = e−iφ Ae , r > a, r with Ae being an arbitrary function of z and t. Accordingly,   2 aAe ∂2 2 ∂ ρe − cAe 2 ξre = e−iφ 2 . 2 ∂t ∂z r Hence

 e

−iφ

Ae = aρe

∂2 ∂2 − c2Ae 2 2 ∂t ∂z

 ξre (a)

with ξre (a) being ξre evaluated as r → a from the region r > a. Continuity of pT across r = a gives     (0) 2 B20 ∂ 2 ξr(0) ∂ 2 ξr ∂2 2 ∂ pT (r = a) = a −ρ0 + − cAe 2 ξre (a). = aρe μ ∂z2 ∂t2 ∂t2 ∂z (0)

Furthermore, the radial displacement is continuous across r = a and so ξr Hence,   (0) (0) B20 ∂ 2 ξr B2e ∂ 2 ξr (ρ0 + ρe ) = . + μ μ ∂t2 ∂z2

= ξre (a).

Thus, 2 (0) ∂ 2 ξr(0) 2 ∂ ξr = c , k ∂t2 ∂z2

(11.187)

where ck denotes the usual kink mode speed given by c2k =

B20 + B2e ρ0 c2A + ρe c2Ae = . ρ0 + ρe μ(ρ0 + ρe )

This result agrees with that obtained by Ruderman, Verth and Erd´elyi (2008) for a β = 0 plasma with gravity ignored, though their derivation is based upon a different procedure (see Section 11.9) to the r-expansion presented here.

11.10 Radial Expansion Equations: Effect of Gravity

353

Finally, consider the magnitude of the term neglected in the above analysis. We neglected ∂ 2 pTe 1 ∂ 2 pTe − ∂z2 c2Ae ∂t2

compared with

1 pTe . r2

Evaluating at r = a and taking ∂/∂t ∼ iω and ∂/∂z ∼ ikz , we see this requires )  ) ) ) ω2 1 ) 2 ) pTe )  2 |pTe |, ) kz − 2 ) ) a cAe which for ω2 = kz2 c2k and ρ0 c2A = ρe c2Ae is satisfied if   ρ0 − ρe kz2 a2  1. ρ0 + ρe This is readily met in a thin tube with kz a  1. 11.10.5 The Stratified (g = 0) Tube in a Field-Free Environment We turn now to a treatment of a stratified medium, considering the case of a tube embedded in an environment that is field-free. In a field-free medium (B ≡ 0), equations (11.173) give ρ e = −div (ρe ξ e ), ρe

p + ξ e · grad pe = c2se (ρ e + ξ e · grad ρe ), ∂ 2ξ e = −grad pTe + ρ e gez . ∂t2

Here ρ e denotes the plasma density perturbation in the environment and ξ e = (ξre , ξφe , ξze ) is the displacement; the equilibrium density is ρe , the equilibrium pressure is pe , and the sound speed is cse (= (γ pe /ρe )1/2 ). These equations imply that the external pressure perturbation pe satisfies the equation (see Chapter 9, Section 9.3)     ∂2 1 ∂ 2 pe 1 2 ∂ ∂2 2 − − (ρe ξze ) (11.188) ∇ − 2 pe = 2  g g ∂z ∂t2 ∂z cse ∂t2 where g is the buoyancy frequency (see Chapter 9) applied in the environment. We argue that it is reasonable to neglect the terms on the right-hand side of the above equation. The neglect of the first term on the right-hand side of (11.188) is reasonable, in view of the treatment of the environment discussed for the β = 0 case; it is evident that a similar argument may be made here for this term. The neglect of the terms involving ξze is reasonable, on the grounds that in the kink mode displacements are predominantly horizontal, so we can expect ξze to make only a small contribution to the external pressure variations pe. Accordingly, the total pressure perturbation pTe , which for a field-free medium is simply the plasma pressure pe , in the kink mode satisfies 1 ∂pTe 1 ∂ 2 pTe + − 2 pTe = 0. 2 r ∂r ∂r r

(11.189)

354

Thin Flux Tubes: The Kink Mode

The solution of this equation is 1 r > a, pTe = e−iφ Ae , r with Ae again being an arbitrary function of z and t. The pressure perturbation pe is related to the radial displacement by ρe

(11.190)

∂ 2 ξre ∂pe = − . ∂r ∂t2

Hence e−iφ Ae = aρe

∂2 e ξ (a), ∂t2 r

(11.191)

where ξre (a) is assumed to correspond to lim ξre (r) as r → a+ (that is, the limit as we approach r = a from within the environment (r > a)); this is the case for a time dependence of the form exp(iωt). We are now in a position to evaluate the total pressure perturbation in the environment and inside the tube. For inside the tube we use the r-expansion result, equation (11.182); for the environment, we use equations (11.190) and (11.191). Thus, evaluating the total pressure perturbation as r → a− and as r → a+ , continuity of pT means that     ∂ 2 ξr(0) B20z ∂ 2 ξr(0) (B20z ) ∂ξr(0) 1 1  2  (0) + + + B0z B0z − (B0z ) ξr pT (r = a) = a −ρ0 μ ∂z2 2μ ∂z 2μ 4μ ∂t2   ∂ 2 ξre (a) = a ρe . (11.192) ∂t2 Finally, continuity of the radial component of the displacement means that ξr(0) = ξre (a), and so we find   (0) B20z ∂ 2 ξr(0) (B20z ) ∂ξr(0) ∂ 2 ξr 1 1  2  (ρ0 + ρe ) = + B − ) + B (B ξr(0) . (11.193) 0z 0z 0z μ ∂z2 2μ ∂z 2μ 2 ∂t2 This equation was first obtained by Lopin and Nagorny (2013), using the r-expansion (0) approach. Equation (11.193) governs the behaviour of the displacement (ξr = ξr , to leading order in r) of a stratified vertical magnetic flux tube embedded in a field-free environment. (0) Evidently, then, ξr does not satisfy the wave equation. Nonetheless, the wave equation still arises, as we may now show. We can transform equation (11.193) into a more convenient form by setting ξr(0) (φ, z, t) = f (z)Q(φ, z, t), (0)

choosing the function f (z) so as to remove the first order derivative ∂ξr /∂z. Much the same procedure was used in Chapter 10, in obtaining the Klein–Gordon equation. Applied here, we choose f (z) such that B0z  (z) f  (z) =− f (z) 2B0z (z)

11.10 Radial Expansion Equations: Effect of Gravity

355

and so   f (z) = exp −

z 0

   B0z  (s) B0z (0) 1/2 . ds = 2B0z (s) B0z (z)

Thus,  ξr(0) (φ, z, t) =

B0z (0) B0z (z)

1/2 Q(φ, z, t),

(11.194)

and then Q satisfies the wave equation 2 ∂ 2Q 2∂ Q = c , k ∂t2 ∂z2

(11.195)

where ck denotes the kink speed in a magnetic flux tube surrounded by a field-free environment, c2k (z) =

B20z μ(ρ0 + ρe )

.

A derivation based upon the r-expansion procedure for a fully compressible plasma with g = 0, for an isolated photospheric tube, was first given by Lopin and Nagorny (2013; see also Lopin and Nagorny 2014).

11.10.6 Implications We have seen in the previous sections that, to good approximation, the radial displacement (0) ξr (≈ ξr ) in the kink mode satisfies the equation       2 B0z (z) 1/2 B0z (z) 1/2 ∂2 2 ∂ ξr = ck 2 ξr , (11.196) B0z (0) B0z (0) ∂t2 ∂z where ck denotes the usual kink mode speed defined by c2k =

ρ0 c2A + ρe c2Ae . ρ0 + ρe

This is the case both for a β = 0 plasma in which gravity is ignored, or for a flux tube in a field-free (cAe = 0) environment in which gravity plays an important role. Of particular interest is the case of an isothermal tube stratified under gravity, a situation corresponding to a photospheric flux tube in a field-free environment. With the external Alfv´en speed zero (cAe = 0), the kink speed ck is defined by c2k =

ρ0 c2 . ρ0 + ρe A

Now the equilibrium state of the tube is such that (see Chapter 10, Section 10.7)   2 B0z (z)  = g(ρe − ρ0 ), 2μ

356

Thin Flux Tubes: The Kink Mode

which for an isothermal atmosphere with the same constant scale height 0 both inside and outside the thin tube (e = 0 ) yields 

B20z (z) 2μ

and so



=

B20z (z) 1 , 2μ 0

  z B0z (z) = exp . B0z (0) 20

Consequently, in an isothermal tube embedded in an isothermal atmosphere we have      2  z z ∂2 2 ∂ ξr exp = ck 2 ξr exp . (11.197) 40 40 ∂t2 ∂z Thus, we have obtained the wave equation and so there is no cutoff frequency. This is in sharp contrast to the sausage mode, which possesses a cutoff that is close to the cutoff frequency of a vertically propagating sound wave (see Chapter 10). The conclusion that there is no cutoff frequency for a kink wave is in contradiction with an earlier analysis by Spruit (1981), which argued for a non-zero cutoff frequency that was much smaller than that pertaining for the sausage mode or for a sound wave. The recent analysis by Lopin and Nagorny (2013) and Lopin and Nagorny (2014), however, suggests that an error in the analysis carried out by Spruit (1981) came about through the implicit neglect of a contribution from the radial field, B0r . The radial field B0r is smaller than the longitudinal field B0z but its contribution is of zeroth order when it arises in the form ∂B0r /∂r or B0r /r, and so it is necessarily part of the dynamics of a thin tube. We can examine this situation a little more closely. In the case of an isolated flux tube that is also isothermal with the same constant scale height 0 in both the tube interior and in the environment, equation (11.197) may be written in the form 2 c2k ∂ξr c2k ∂ 2 ξr 2 ∂ ξr = c + ξr . + k 20 ∂z ∂t2 ∂z2 1620

(11.198)

However, according to Spruit (1981) the transverse displacement ξr satisfies the equation (see his eqn. (29))   2 ρ0 ρ0 − ρe ∂ξr ∂ 2 ξr 2 ∂ ξr = c − g . ρ0 + ρe A ∂z2 ρ0 + ρe ∂z ∂t2 Now the equilibrium of a thin tube is such that β ρ0 = , ρe 1+β where β = 2c2s /(γ c2A ) is the plasma beta in the tube, and so ρ0 − ρe 1 =− , ρ0 + ρe 1 + 2β

c2k =

ρ0 c2 = ρ0 + ρe A



β 1 + 2β

 c2A .

11.11 Dispersive Correction for a Uniform Tube (g = 0)

357

Thus, Spruit’s equation may be rewritten as 2 c2k ∂ξr ∂ 2 ξr 2 ∂ ξr = c + . k 20 ∂z ∂t2 ∂z2

Comparing the above equation with (11.198), we see the loss of a term proportional to ξr , and it is the absence of this term that subsequently gives rise to an erroneous cutoff frequency in the transformed form of Spruit’s equation when rewritten as a Klein–Gordon equation. 11.11 Dispersive Correction for a Uniform Tube (g = 0) We end of our discussion here with a return to the case of a uniform tube in which gravity is ignored. The expansion approach developed in the previous section has the virtue that it may in principle be extended and applied to a variety of more complicated situations, incorporating a range of possible additional physical effects. But of course it is also applicable to the simpler problem of a uniform tube. This was treated at some length in Chapter 6. Nonetheless, it is of interest to revisit this problem from the viewpoint of the expansion approach. Thereby we may examine more fully the nature of the approximate theory. A similar analysis was carried out for the sausage mode in Chapter 10, where we recovered the approximate dispersion relation that described the departure of the wave speed c from the slow mode speed ct as dispersive effects arise for small kz a (see Equation (10.2) and Section 10.3.3). Consider then a uniform tube embedded in a uniform environment. Gravity is ignored. The environment of the tube may be magnetized or field-free. Accordingly, our treatment covers both the uniform coronal tube (with its strongly magnetized environment) and the uniform photospheric tube (with its field-free environment). Our starting point is the expression for the total pressure perturbation pT in a thin tube; with B0z  = 0, B0z = B0 , to leading order in r equation (11.182) gives   B20 ∂ 2 ξr(0) ∂ 2 ξr(0) + , r < a. (11.199) pT = r −ρ0 μ ∂z2 ∂t2 This expression applies for the kink (m = 1) mode. Fluting modes (m = 2, 3, . . .) require a separate treatment, not given here. Now in the environment of a uniform tube the results of a uniform medium apply and in cylindrical coordinates the kink (m = 1) mode is such that (see Chapter 6) pT (r) = Ae K1 (me r),

r > a,

(11.200)

(kz2 c2se − ω2 )(kz2 c2Ae − ω2 )

(11.201)

where m2e =

kz2 c2se c2Ae − ω2 (c2se + c2Ae )

with me > 0. We have taken the t and z dependence to be of the form exp(iωt − ikz z).

358

Thin Flux Tubes: The Kink Mode

The constant Ae is arbitrary and K1 denotes the modified Bessel function of order unity (see Abramowitz and Stegun 1965). In the environment the sound speed is cse and the Alfv´en speed is cAe . The nature of the environment, be it a field-free medium for which cAe = 0 (as for a photospheric tube) or strongly magnetized with cAe > cse (such as for a coronal loop), has a bearing on the behaviour of the wave through the value of me . The radial component ur of the motion in the environment is related to the pressure perturbation through the general relation (see Chapter 6) iω

dpT = −ρ0 (r)(kz2 c2A (r) − ω2 )ur . dr

Accordingly, with ur = ∂ξr /∂t = iωξr the radial displacement ξre in the environment of the tube is given by ξre (r) = −

1 ρe (kz2 c2Ae

− ω2 )

dpT , dr

r > a.

(11.202)

Now we require continuity of pT (r) and ξr (r) across the tube boundary r = a. Continuity of pT and ξr (r) means   B20z ∂ 2 ξr(0) ∂ 2 ξr(0) + ξr(0) = ξre (a). a −ρ0 = Ae K1 (me a), μ ∂z2 ∂t2 The constant Ae is related to ξre (a) through ξr(0) = ξre (a) = −

1 ρe (kz2 c2Ae

− ω2 )

Ae me K1  (me a),

where the prime denotes the derivative of the function, the result being evaluated at me a (see Chapter 6 for a discussion of similar expressions). (0) Eliminating Ae in preference for ξr and using the Fourier form for the t and z dependences we finally obtain ρ0 ω2 − ρ0 kz2 c2A = ρe (ω2 − kz2 c2Ae )

K1 (me a) . me aK1  (me a)

(11.203)

This is the desired dispersion relation for the kink mode, obtained through use of the expansion procedure for the region inside the uniform flux tube combined with the behaviour in a uniform medium for the region outside the flux tube. We require that me > 0. Now for small x it follows from the properties of modified Bessel functions (see Chapter 6) that xK1  (x) ∼ −1 − x2 K0 (x), K1 (x)

K1 (x) ∼ −1 + x2 K0 (x), xK1  (x)

(11.204)

where K0 denotes the Bessel function of order zero. Thus, to leading order for small x (corresponding to small me a) we immediately obtain the familiar result that ω2 = kz2 c2k ,

c2k =

ρ0 c2A + ρe c2Ae . ρ0 + ρe

(11.205)

11.11 Dispersive Correction for a Uniform Tube (g = 0)

359

We can carry this calculation a little further and obtain the correction to this result by writing ω2 = kz2 c2k (1 + ), for small . Substituting into (11.203) gives  2  ck − c2Ae ρe = m2e a2 K0 (me a), ρ0 + ρe c2k with me now calculated for ω2 = kz2 c2k . Hence we obtain the result that the speed c (= ω/kz ) of the kink wave is given approximately by ρe (c2 − c2Ae )m2e a2 K0 (me a) c2 = c2k + ρ0 + ρe k ρ0 ρe = c2k − (c2 − c2A )m2e a2 K0 (me a) (11.206) (ρ0 + ρe )2 Ae with me calculated for ω2 = kz2 c2k . These results agree with those derived in Chapter 6. It is evident that c2 ≤ c2k in the coronal case of cAe > cA , whereas c2 ≥ c2k in the photospheric case of a field-free environment (cAe = 0).

12 Damping

12.1 Introduction In this chapter we examine the role of energy loss mechanisms on wave propagation. We have in mind non-zero diffusivity η or viscosity ν or thermal conduction κ, each of which leads to energy loss and redistribution of energy. In the description of wave propagation these effects are often ignored, since in many circumstances their role is a minor one, best ignored in a first look at a problem. But in some problems such effects are both important and interesting. We set out their role here. The general set of nonlinear magnetohydrodynamic equations we examine here may be written in the form ∂u + div ρu = 0, ∂t  ρ

(12.1)

 1 ∂u + (u · grad)u = −grad p + (curl B) × B + Fvisc , ∂t μ

∂p γp + u · grad p − ∂t ρ



∂ρ + u · grad ρ ∂t

p=

(12.2)

 = −(γ − 1)L,

kB ρT, m ˆ

(12.3)

(12.4)

∂B = curl (u × B) + η∇ 2 B, ∂t

(12.5)

div B = 0.

(12.6)

We have ignored the effect of gravity. The viscous force Fvisc that arises in the momentum equation has components F1 , F2 , F3 written in tensor form using Cartesian coordinates x1 , x2 , x3 as (see, for example, Bray and Loughhead 1974, chap. 6; Whitham 1974, chap. 6; Boyd and Sanderson 2003, chap. 12)    ∂uj ∂ 2 ∂uk ∂ui Fi = + − δij μvisc , (12.7) ∂xj ∂xj ∂xi 3 ∂xk 360

12.2 Damping Coefficients

361

where u1 , u2 , u3 denote the components of the motion u and δij is the Kronecker delta (δij = 1 if i = j, δij = 0 if i = j). In writing Fi we are using the summation convention, summing over a repeated index. The term μvisc is the coefficient of shear viscosity and in general is not a constant but varies with, in particular, temperature. However, in many situations it is convenient to treat μvisc as a constant. When μvisc is treated as a constant, the viscous force Fvisc = (F1 , F2 , F3 ) simplifies to the convenient vector form (see Landau and Liftshitz 1959, chap. II; Parker 1979a, sect. 7.6; Priest 2014, chap. 2)   1 (12.8) Fvisc = ρν ∇ 2 u + grad (div u) , 3 where ν is the coefficient of kinematic viscosity (μvisc = ρν). Again, it is common to treat ν as a constant, except when it comes to providing its value and then it is usual to determine ν according to the specific temperature of the plasma (as well as other factors). In keeping with this approach, we calculate the value of the coefficient of kinematic viscosity using a temperature-dependent formula – see Section 12.2 below – though otherwise regard it as a constant. Other coefficients such as diffusivity and thermal conduction are treated similarly. In the energy equation (12.3) we take the form of L to include thermal conduction along the magnetic field, ignoring the much smaller component of thermal conduction across the magnetic field. Then we can write L = −div (κ ∇ T),

(12.9)

where κ is the coefficient of thermal conduction along the magnetic field. We have also ignored the ohmic heating term j2 /σ that arises in L and any radiative losses or applied heating. (See Priest (2014) for a discussion of the possible terms that may contribute to L.) Finally, in the induction equation (12.5) we have treated the electrical diffusivity η as a constant.

12.2 Damping Coefficients Equations (12.1)–(12.9) provide a complicated system of equations in which a number of damping effects are incorporated; these non-ideal effects are represented by the three coefficients ν, η and κ . Wave damping may be expected whenever any one of these coefficients is non-zero. Viscosity damps the motion, electrical diffusivity leads to ohmic heating and acts to redistribute the magnetic field, smoothing out any peaks or sharp gradients in the field, and thermal conduction redistributes the energy in a sound wave, smoothing out any temperature disturbances. As remarked above, although these coefficients vary with plasma conditions, especially temperature, we have found it convenient to treat them as constants as far as the evaluation of the dynamics associated with the governing system of equations (12.1)–(12.9); this assumption, common in the literature, is purely for convenience. Evidently, to illustrate the discussion of the damping of waves we need to know the values of the diffusivity η, kinematic viscosity ν and thermal conductivity κ . Although we have taken, for simplicity, each of these coefficients to be a constant, in reality they may vary with circumstances, particularly with temperature. Accordingly, we discuss the values

362

Damping

of these three coefficients that typically arise. In an incompressible liquid these parameters are generally known through experiment. For example, in liquid mercury η ≈ 0.8 m2 s−1 and ν ≈ 1.2 × 10−7 m2 s−1 (see, for example, Cowling 1976, p. 128), about one-tenth that of the kinematic diffusivity of water at room temperature (νwater ≈ 1 × 10−6 m2 s−1 ) (see Landau and Liftshitz 1959, p. 49). Thus, in laboratory liquids we typically find η  ν. For a plasma, we may use calculations of η, ν and κ that are based upon the particle view of the plasma. The diffusivity η in a fully ionized plasma at a temperature T0 is given by (see, for example, Priest 2014, p. 79) −3/2

η = 5.2 × 107 (ln ) T0

m2 s−1 ,

(12.10)

where ln  is the Coulomb logarithm (see Spitzer 1956, chap. 5; Braginskii 1965). The Coulomb logarithm term varies a little with temperature and density (see the table of values in, for example, Priest (2014, p. 79)). For coronal temperatures and densities, ln  ≈ 20 and then (see, for example, Parker 1979a, p. 115)  −3/2 T0 −3/2 9 2 −1 η ≈ 10 × T0 m s = m2 s−1 . (12.11) 106 K Thus, high temperatures lead to a low diffusivity (high electrical conductivity σ ). The kinematic viscosity ν of a plasma with density ρ0 is given by (see, for example, Chapman 1954; Priest 2014, p.81) ν = 2.21 × 10−16 (ln )−1

5/2

T0 ρ0

m2 s−1 .

(12.12)

Again, for typical coronal conditions with ln  ≈ 20, we have ν ≈ 10−17

5/2

T0 ρ0

m2 s−1 = ν0

5/2

T0 ρ0

m2 s−1 ,

(12.13)

where ν0 = 10−17 in SI units; in this expression for ν the plasma density ρ0 is in the units kg m−3 . In the corona it is common to express plasma density in terms of the number ˆ p n0 , density n0 (in units of particles per m3 ) of particles in the plasma, taking ρ0 = μm where mp is the mass of a proton (see Chapter 1). Then, with μˆ ≈ 0.6 we have  −1 5/2  5/2 T T0 n0 m2 s−1 . (12.14) ν ≈ 1 × 1010 × 0 m2 s−1 = 1 × 109 × n0 106 K 1016 m−3 A relation similar to this is given in Heyvaerts and Priest (1983). Evidently, the kinematic viscosity is raised in high temperatures and low plasma densities. Note that ν > η when the the temperature T0 (in K) and plasma density ρ0 (in kg m−3 ) satisfy (see also the discussion in Parker (1979a, sect. 7.6)) T04 > 1026 ρ0 . Thus, diffusive damping tends to be more important than viscous damping in the cooler less dense regions of the Sun (such as the photosphere). However, viscous damping is more important than diffusivity in the high temperature, low density regions such as the corona.

12.3 The Incompressible Fluid

363

To illustrate these diffusivities, we note that in a sunspot at the photospheric level with temperature T0 = 4000 K and photospheric density of ρ0 = 2 × 10−4 kg m−3 ( = 2 × 10−7 g cm−3 ), we obtain an electrical diffusivity of η = 103 m2 s−1 and a kinematic viscosity ν ≈ 10−4 m2 s−1 . Thus, in the photospheric layers of a sunspot we see that η  ν. In the corona, at a temperature of T0 = 106 K the situation changes and we obtain η = 1 m2 s−1 . With a coronal number density of n0 = 1016 m−3 (= 1010 cm−3 ) we find ν = 109 m2 s−1 ; for a particle number density of n0 = 1015 m−3 we obtain ν = 1010 m2 s−1 . Thus, in the corona ν  η. Hence, whether we are in the photospheric layers of a sunspot or the tenuous plasma of the corona, we find that one of the diffusivities dominates over the other. Finally, we have thermal conduction. Thermal conduction is operative whenever a wave creates a temperature perturbation; accordingly, it is important for fast and slow magnetoacoustic waves, but unimportant for Alfv´en waves. In a magnetized plasma thermal conduction acts primarily along the field (see, for example, Braginskii 1965), the thermal conduction across the magnetic field being substantially reduced (usually to negligible proportions). This gives rise to a thermal conduction of the form (see, for example, Priest 2014, sect. 2.3.2) κ = 1.8 × 10−10 (ln )−1 T0

5/2

W m−1 K−1 .

(12.15)

Thus, under solar plasma conditions we have 5/2

κ ≈ κ0 T0

W m−1 K−1 ,

(12.16)

with the coefficient κ0 varying slightly with local conditions, being (in SI units) κ0 = 9 × 10−12 under coronal conditions and κ0 = 40 × 10−12 under photospheric conditions (see also the discussion in Priest (2014, sect. 2.3.2)). 12.3 The Incompressible Fluid 12.3.1 Formulation We begin our examination of wave damping by considering the problem of viscous and ohmic damping in an incompressible fluid. Thermal conduction is ignored. While there are limited direct applications of this analysis to the situation in the solar atmosphere, the assumption of incompressibility allows an explicit treatment of damping by viscosity and electrical diffusivity, thus making clear the manner in which wave propagation is affected by these agencies. Our treatment draws on the presentation in Cowling (1957, 1976) and Ferraro & Plumpton (1966). Consider, then, an incompressible fluid acted upon by viscous and conductive forces. The equilibrium magnetic field is taken to be uni-directional, acting in the z-direction of a Cartesian coordinate system x, y and z: B0 = B0 (x)ez .

(12.17)

The equilibrium magnetic field may be non-uniform, its strength depending upon the transverse coordinate x. Similarly, the equilibrium plasma pressure p0 (x) and density ρ0 (x) may also be non-uniform with pressure balance maintained through

364

Damping



B2 d p0 + 0 dx 2μ

 = 0.

(12.18)

The force of gravity is ignored. Linear perturbation motions u about the equilibrium (12.18) are assumed to be incompressible, satisfying the constraint div u = 0.

(12.19)

Equation (12.19) in effect replaces the energy equation (12.3) and the ideal gas law (12.4). The linearized momentum equation links the perturbation motions u with the perturbation magnetic field B: ρ0

∂u 1 1 = −grad pT + (B0 · grad)B + (B · grad)B0 + Fvisc . ∂t μ μ

(12.20)

The viscous force Fvisc acting on the fluid is taken to be proportional to the Laplacian of the motion u, the form appropriate for an incompressible fluid (see, for example, Cowling 1976, chap. 1); for the system here this gives a viscous force Fvisc = ρ0 ν∇ 2 u,

(12.21)

where ν denotes the kinematic viscosity (assumed to be a constant) of the fluid. Also, we have introduced the perturbation in total pressure (the sum of the plasma pressure perturbation, p, and the perturbation in magnetic pressure) pT = p +

1 B0 B0 · B = p + Bz , μ μ

(12.22)

where Bz denotes the component of the perturbed field in the z-direction (the direction of the applied magnetic field). The evolution of the magnetic field B is described through the induction equation in which we retain the effect of a finite electrical conductivity σ : ∂B = curl (u × B0 ) + η∇ 2 B, ∂t

div B = 0,

(12.23)

with the diffusivity η assumed constant (η = (μσ )−1 ). As usual we may expand the magnetic force term in the momentum equation and the curl term in the induction equation, much as is described in Chapters 2 and 3. Thus, since the motion u is incompressible and the magnetic field solenoidal (so div u = 0 and div B = 0), we have ∂u − ux B0  ez , curl (u × B0 ) = B0 ∂z the prime  denoting the derivative of an equilibrium quantity with respect to x. Then we may write our linear equations in the form ρ0

∂u B0 ∂B B0  = −grad pT + + Bx ez + ρ0 ν∇ 2 u, ∂t μ ∂z μ ∂B ∂u = B0 − ux B0  ez + η∇ 2 B. ∂t ∂z

(12.24) (12.25)

12.3 The Incompressible Fluid

365

These are the basic equations we wish to consider, describing incompressible motions in a structured medium. When ν = 0 and η = 0 these equations reduce to equations (3.12) and (3.13) discussed in Chapter 3.

12.3.2 Uniform State Consider first the case when the equilibrium is a uniform state, with B0 , p0 and ρ0 constants (and B0  = 0). Taking the divergence of the linearized momentum equation (12.24), we obtain (since div grad ≡ ∇ 2 ) ∇ 2 pT = 0,

(12.26)

in view of the fact that both u and B are divergence-free. Thus we obtain Laplace’s equation in x, y and z, holding in an infinite and unbounded domain. Since there are no singularities, the appropriate solution of Laplace’s equation is here pT = constant. Since only the derivative of pT arises in (12.24) we may take (without loss of generality) pT = 0. Then, in a uniform medium, equations (12.24) and (12.25) become     ∂u ∂ ∂ B0 ∂u 2 2 − ν∇ u = , − η∇ B = B0 . ∂t μρ0 ∂z ∂t ∂z Hence, in a uniform medium    ∂ 2u ∂ ∂ − ν∇ 2 − η∇ 2 u = c2A 2 , ∂t ∂t ∂z where

 cA =

B20 μρ0

(12.27)

(12.28)

(12.29)

1/2

is the Alfv´en speed in the fluid. Equation (12.29) is the basic equation determining the behaviour of the incompressible motion u in a uniform medium; the same partial differential equation applies to B, and also to the current j and the vorticity curl u. In many circumstances it turns out that of the two diffusivities one is much larger than the other, so that either ν  η or ν  η. Then we can simplify equation (12.29) a little: if either ν = 0 or η = 0, then (12.29) reduces to   2 ∂ 2u 2 ∂ u 2 ∂u − c = ν ∇ (12.30) m A ∂t ∂t2 ∂z2 where νm denotes the larger of the two diffusivities ν and η.

12.3.3 Dispersion Relation in a Uniform Medium Consider the general partial differential equation (12.29) with both diffusivities ν and η retained. The Fourier form of (12.29) follows by writing

366

Damping

u = u0 ei(ωt−kx x−ky y−kz z) , leading to the relation (iω + νk2 )(iω + ηk2 ) + kz2 c2A = 0,

(12.31)

where k2 = kx2 + ky2 + kz2 . The dispersion relation (12.31) is a quadratic equation for ω with solutions 2ω = ik2 (ν + η) ± {4kz2 c2A − k4 (ν − η)2 }1/2 .

(12.32)

It is evident that ω is purely imaginary if 1 2 k (ν − η)2 , 4

kz2 c2A


k2 (ν − η)2 , 4c2A

(12.33)

where we have written kz = k cos θ . For θ approaching π/2, the wave cannot propagate. Even if θ = 0, so propagation is along the direction of the applied magnetic field, propagation occurs only for c2A >

k2 (ν − η)2 . 4

(12.34)

In terms of field strength B0 and wavelength λ = 2π/k, propagation occurs provided B0 > π(μρ0 )1/2

|η − ν| . λ

(12.35)

As an illustration, consider liquid mercury with a density of ρ0 = 13 593 kg m−3 and electrical diffusivity of η = 0.8 m2 s−1 , much larger than the kinematic viscosity ν. With these values of the parameters, wave propagation in liquid mercury requires the magnetic field strength to be (see also Ferraro & Plumpton 1966, p. 85) B0 >

0.3 tesla λ

(liquid mercury).

Thus, a container of typical dimension 1 m supporting waves of wavelength λ ∼ 1 m in a fluid of liquid mercury would require a field of strength 0.3 T (= 3000 G). Only a very strong magnetic field is able to support wave propagation in opposition to diffusion of any magnetic perturbation.

12.3 The Incompressible Fluid

367

In sharp contrast, a sunspot with a vastly more tenuous density of ρ0 = 2 ×10−4 kg m−3 but considerably larger electrical diffusivity of η = 103 m2 s−1 , much greater than the kinematic viscosity ν ≈ 10−4 m2 s−1 , supports wave propagation for field strengths B0 >

0.05 tesla λ

(sunspot).

A wavelength λ ∼ 107 m, comparable with the size of a sunspot, would thus support waves for B0 > 5 × 10−9 T. Diffusion is now a weak process and wave propagation readily occurs. 12.3.4 Spatial Damping in a Uniform Medium Consider the dispersion relation (12.31) for the case when k2 = kz2 and either ν or η is assumed to be zero. Then ω2 − ikz2 ωνm − kz2 c2A = 0,

(12.36)

where νm is the value of the non-zero diffusion coefficient. For spatial damping we suppose that ω is prescribed and then kz is determined by the dispersion relation. We have in mind an oscillation that is sustained at the base z = 0 at a specific frequency ω; this results in a disturbance that decays with height z. For the circumstances where ω  2c2A /νm we may expand the dispersion relation to give ωνm ω kzR ≈ ± , kzI ≈ −kzR 2 , (12.37) cA 2cA where kz = kzR + ikzI is written in terms of its real and imaginary parts. This gives a disturbance with a spatial dependence e−z/z0 . eiωz/cA , unif

indicating a spatial decline on a scale (see, for example, Alfv´en 1942b, 1947) 2c3A c3A P2 = (12.38) ω2 νm 2π 2 νm where P (= 2π/ω) denotes the period of the sustained oscillation of frequency ω. The spatial scale zunif is evidently large: Alfv´en waves are not easily damped. As an 0 illustration, consider a coronal hole with a temperature of 106 K and a coronal number density of n0 = 1014 m−3 ; this produces a kinematic viscosity of νm = ν = 1011 m2 s−1 , much larger than the diffusivity η. Suppose that the oscillation has a period of P = 102 s and that the Alfv´en speed within the coronal hole is cA = 106 m s−1 (= 103 km s−1 ). Then = 5 × 109 m (= 5 × 106 km) which, in terms of the solar radius R◦ , gives we obtain zunif 0 unif z0 ≈ 7R◦ . = zunif 0

12.3.5 Temporal Damping in a Uniform Medium Consider again the dispersion relation (12.31) for the case when k2 = kz2 and either ν or η is chosen to be zero. We now suppose that kz is fixed; solving the dispersion relation for complex ω:

368

Damping

 1/2 kz2 νm 2 1 2 ω = ikz νm ± kz cA 1 − . 2 4c2A

(12.39)

For small kz νm /cA , this yields ωR ≈ kz cA ,

ωI ≈

giving an exponential damping of the form e−t/τ0

unif

τ0unif =

1 kz νm , 2

(12.40)

where

2 kz2 νm

(12.41)

is the timescale for damping in a uniform medium. For a standing wave with fundamental wavenumber kz = π/L in a loop of length L, we have τ0unif =

2L2 . π 2 νm

(12.42)

Accordingly, for the illustration given earlier of spatial damping in the corona, yielding a damping length of zunif = 5 × 106 km, a loop of length L = 108 m gives a damping time 0 unif 4 of τ0 = 2 × 10 s.

12.4 Phase Mixing of Alfv´en Waves: Basic Aspects We return to the basic equations (12.24) and (12.25) but now consider the case of a nonuniform equilibrium. It turns out that damping, both spatial and temporal, takes place on very different scales to the traditional ones of a uniform medium discussed above. To simplify the treatment we consider the case when motions u and perturbation magnetic field B are purely in the y-direction, but have no dependence on that coordinate (and so are independent of y, ∂u/∂y = 0 and ∂B/∂y = 0). Thus, motions and perturbed magnetic field are assumed to be perpendicular to both the applied magnetic field B0 and the coordinate x of inhomogeneity:     (12.43) u = 0, uy (x, z, t), 0 = uy (x, z, t)ey , B = 0, By (x, z, t), 0 = By (x, z, t)ey . The motions are incompressible: div u = 0.

(12.44)

In contrast to the previous section, the medium is taken to be non-uniform, the equilibrium density ρ0 (x) and magnetic field strength B0 (x) being functions of x. Accordingly we are considering Alfv´en waves in a non-uniform medium. Again we take pT to be a constant. From the y-components of equations (12.24) and (12.25), we obtain   ∂uy ∂2 ∂2 B0 (x) ∂By + 2 uy (12.45) = + ρ0 (x)ν ρ0 (x) ∂t μ ∂z ∂x2 ∂z and

12.4 Phase Mixing of Alfv´en Waves: Basic Aspects

∂By ∂uy = B0 (x) +η ∂t ∂z



∂2 ∂2 + ∂x2 ∂z2

369

 By .

(12.46)

In the absence of any magnetic field, equation (12.45) corresponds to the diffusion equation in x, z and t, smoothing the motion uy . Similarly, in the absence of any motion the field By , given by equation (12.45), satisfies the diffusion equation. Thus, the diffusion equation underpins our description in two ways: through the diffusion of uy with diffusivity ν and through the diffusion of By with diffusivity η. These two processes are intermingled in equations (12.45) and (12.46). If either of the two damping coefficients η and ν is zero, so one of the diffusive processes is absent, we may introduce a new variable u defined according to which diffusivity is zero:  uy if ν = 0, η = 0, u= (12.47) By if η = 0, ν = 0. Then the variable u satisfies the propagation diffusion equation   2u 2 2 ∂ ∂ ∂ 2u ∂ ∂u − c2A (x) 2 = νm + 2 , ∂t ∂t2 ∂z ∂x2 ∂z

(12.48)

where νm denotes the larger of ν and η:  νm =

ν

if ν = 0, η = 0,

η

if η = 0, ν = 0.

(12.49)

The variable u is the motion uy if η = 0, ν = 0 (and then νm = ν), and the perturbed field By if ν = 0, νm = η = 0. Equation (12.48), which we refer to as the general phase mixing equation, represents a combination of propagation along the applied field with diffusion both along and across the field. In a non-uniform medium the diffusion term on the right-hand side of (12.48) is found to be dominated by the x-gradients, which may become much larger than the gradients along the applied magnetic field; in which case, we may consider the reduced form of equation (12.48), namely   ∂ 2 ∂u ∂ 2u ∂ 2u 2 − cA (x) 2 = νm 2 . ∂t2 ∂z ∂x ∂t

(12.50)

Equation (12.50), which we refer to as the reduced phase mixing equation, represents propagation along the magnetic field together with diffusion across the field. Equations (12.48) and (12.50) are the basic equations of phase mixing. Most attention in the literature has been given to the reduced equation (12.50), which describes the more fully developed stage of phase mixing when x-gradients across the applied magnetic field dominate over gradients along the applied field. Equation (12.50) was first discussed in detail by Heyvaerts and Priest (1983). Heyvaerts and Priest (1983) is important in that

370

Damping

it makes clear the possibility of rapid damping of an Alfv´en wave. Other authors had suggested that inhomogeneity in the equilibrium state may result in rapid damping (see, for example, Tataronis and Grossmann 1973; Chen and Hasegawa 1974; Hasegawa and Chen 1974; Ionson 1978; Lee 1980; Rae and Roberts 1981; Lee and Roberts 1986), through a process of resonant absorption, but the complexity of these analyses made the effect of a physical damping more difficult to incorporate into their work and in any case the damping that occurred corresponded to a transfer of energy from one motion to another, and not to a decay process. By contrast, in Heyvaerts and Priest (1983) the role of damping as an efficient mechanism is more transparent. However, a number of approximations are invoked in their treatment, warranting further exploration; see the discussion by Nocera, Leroy and Priest (1984) and Parker (1991). In exploring the behaviour of the phase mixing equation (12.50) it is convenient to divide attention between spatial damping and temporal damping, just as in the case of a uniform medium; we treat these topics separately.

12.5 Phase Mixing of Alfv´en Waves: Spatial Damping We wish to construct a solution of the phase mixing equation (12.48) or the reduced phase mixing equation (12.50). Notice first what happens if νm = 0, so that diffusion is absent and both equations become the one-dimensional wave equation, save that the propagation speed cA is a function of the transverse coordinate x: ∂ 2u ∂ 2u 2 = c (x) . A ∂t2 ∂z2

(12.51)

Then we can construct a D’Alembert solution of the form u = f (z − cA (x)t),

or

g(z + cA (x)t)

corresponding to a wave travelling in the positive or negative z-direction. (These functions can also depend independently on x, but we ignore that possibility.) The functions f and g may be arbitrary, save only that they may be differentiated appropriately. The interesting feature about these solutions is that while u may be bounded for all z or t, this is not the case for its x-derivatives. For example, for a solution propagating in the positive z-direction we have  ∂  ∂u = f (z − cA (x)t) = −t cA  (x)f  (z − cA (x)t), ∂x ∂x where the prime denotes the derivative of the function (so, for example, f  (z) = df /dz and cA  (x) = dcA (x)/dx). Hence, even if f  is everywhere bounded, ∂u/∂x grows in time t. In the descriptive language used by Parker (1991), ‘wave fronts become increasingly corrugated as they propagate’. Similar remarks may be made about the second derivative ∂ 2 u/∂x2 , and so no matter how small the diffusivity νm is, provided it is not zero, it eventually contributes significantly to the damping.

12.5 Phase Mixing of Alfv´en Waves: Spatial Damping

371

With these comments in mind, we turn now to the approximate solution of the phase mixing equation (12.50). Following Heyvaerts and Priest (1983), we look for a solution in the form1 u(x, z, t) = ei(ωt−κ(x)z) A(x, z),

(12.52)

where κ(x) =

ω . cA (x)

We have in mind a motion that corresponds to a sustained oscillation of fixed frequency ω maintained at the base z = 0, and we are interested to determine the amplitude function A. In the absence of damping (νm = 0), A is a constant and expression (12.52) with A = constant provides a solution of equation (12.50). With the substitution (12.52), the left-hand side of equation (12.50) evaluates to   ∂ 2u ∂ 2u ∂A ∂ 2 A iωt −iκ(x)z 2 2 − cA (x) 2 = cA (x) 2iκ(x) . − 2 e .e ∂z ∂t2 ∂z ∂z To calculate the right-hand side of (12.50) notice first that   ∂A ∂u = − izκ  (x)A eiωt .e−iκ(x)z . ∂x ∂x Now for κ  (x) = 0 we may expect that the second term in this expression dominates over the first term, provided |z| is sufficiently large. Then ∂u ≈ −izκ  (x)Aeiωt .e−iκ(x)z = −izκ  (x)u, ∂x  ∂ 2u ≈ − izκ  (x) + z2 (κ  )2 u. 2 ∂x We may further approximate this expression by neglecting the term involving izκ  compared with z2 (κ  )2 . However, the presence of the factor i suggests that these terms play different roles and therefore both terms could be informatively retained (see Section (12.5.1) below). Nonetheless, retaining the z2 term only we obtain ∂ 2u ≈ −z2 (κ  )2 u. ∂x2 Hence, the phase mixing wave equation (12.50) yields the approximate result   2A ∂A ∂ − 2 = −iωνm z2 (κ  )2 A. c2A (x) 2iκ(x) ∂z ∂z

(12.53)

Finally, assuming that the term involving the second derivative is negligible, we have a first order linear differential for A:

1 In their construction of an approximate solution describing phase mixing, Heyvaerts and Priest (1983) refer to an equation

describing a general phase mixing, equivalent to our (12.48), but in fact analyse a reduced equation, effectively equivalent to the reduced equation (12.50).

372

Damping

∂A ωνm (κ  )2 2 + 2 z A = 0, ∂z 2cA (x) κ(x) which we may rewrite as ∂A 1 (κ  )2 2 + νm z A = 0. ∂z 2 cA (x)

(12.54)

Equation (12.54) is a key result describing phase mixing. It was first obtained by Heyvaerts and Priest (1983). Equation (12.54) is a linear differential equation for A and as such may readily be integrated by noting that it may be rewritten in the form    ∂ 1 (κ  )2 3 A exp νm z = 0, ∂z 6 cA (x) and hence the solution of equation (12.54) follows as   1 (κ  )2 3 z , A = A0 exp − νm 6 cA (x)

(12.55)

where A0 is an arbitrary function of x and A = A0 when z = 0. The solution (12.55) may be rewritten in the form   z3 −( z )3 A = A0 exp − 3 = A0 e z0 , z0 mixing

revealing the spatial scale z0 ≡ z0  mixing z0

=

6 νm ω2

(12.56)

, where



cA cA 



1/3

2 c3A

=

6a2 c3A ω2 νm

1/3 .

(12.57)

Here we have introduced the characteristic transverse lengthscale a determined by the x-dependence in the Alfv´en speed, writing ) ) ) cA ) a = ))  )) . (12.58) cA mixing

may be compared either with the transverse scale a Now the phase mixing scale z0 determined by cA (x) or with the longitudinal scale λ (= 2π/κ) determined by the frequency ω of the sustained oscillation together with the Alfv´en speed cA : λ=

2π cA . ω

Introduce two dimensionless numbers R and Re by R=

ωa2 , νm

Re =

acA . νm

12.5 Phase Mixing of Alfv´en Waves: Spatial Damping

373

R is an effective Reynolds number based upon a, and Re is an effective Reynolds number based upon longitudinal wavelength λ; R, Re and a all depend upon x. Then mixing

z0

=

cA (6R)1/3 = ω



3 2π 2

1/3  a 1/3 1/3 Re λ. λ

(12.59)

mixing

Thus, the spatial scale z0 for spatial damping depends upon the cube root of the effective Reynolds number, R or Re. This result was first obtained by Heyvaerts and Priest (1983); see also Priest (2014, chap. 10). Now the solution (12.56) demonstrates two things. Firstly, it shows that damping is mixing , for then (z/z0 )3 is very small and the exponential term is negligible for |z|  z0 ≡ z0 close to unity, indicating no spatial damping. Secondly, when z exceeds z0 the cubic term (z/z0 )3 rapidly becomes large, and damping is correspondingly very rapid: for z slightly in excess of z0 damping reduces A to very small values. Thus, the cubic expression (12.56) demonstrates negligible damping for distances |z| much smaller than the vertical scale z0 but thereafter damping is effective for z ∼ z0 and finally u is reduced to negligible values for |z| much larger than z0 . To illustrate, to begin with A/A0 falls only gradually from its value of unity at z = 0, falling to 0.88 (88%) by z = 0.5 z0 ; by z = 0.9 z0 the fall has only amounted to A/A0 = 0.48 (48%), but by z = 1.2 z0 it has fallen to 18%, and by z = 1.5 z0 , A/A0 has fallen to 3% of its initial value (i.e., A/A0 = 0.034 at z = 1.5 z0 ). In phase mixing, then, we see the generation of small scales produces a more effective damping than occurs when the medium is uniform. The damping is determined by the mixing , which in turn depends upon the cube root of an effective Reynolds spatial scale z0 number. Under common solar or astrophysical conditions this gives a scale z0 which is much less than the scale that arises in a uniform medium. Cargill, De Moortel and Kiddie (2016) give a recent discussion of coronal damping and its relation to phase mixing. As an illustration of z0 , for a coronal temperature of 106 K and a coronal number density of n0 = 1014 m−3 (= 108 cm−3 ), typical of a coronal hole, we obtain a kinematic viscosity of νm = ν = 1011 m2 s−1 ; then, in a plasma with an Alfv´en speed cA = 106 m s−1 mixing and a driven oscillation of period P = 102 s, we obtain z0 = 2.477 × 107 m ≈ 4 6 2.5 × 10 km for an inhomogeneity scale of a = 10 m. This is much reduced below the damping scale in a uniform medium, where for the same conditions we found that = 5 × 106 km. zunif 0

12.5.1 Retention of Certain Terms We remarked earlier that both terms in the expression for the second derivative of u with respect to z could usefully be retained, since the occurrence of the factor of i indicated a change in behaviour in u. We return to this topic here and consider what happens if we retain all the terms in the calculation of u. Moreover, we consider at the same time the full equation of phase mixing, namely equation (12.48). This topic has not been discussed in the literature but warrants a brief treatment here. Following some straightforward algebra, we obtain

374

Damping

∂u ∂x ∂ 2u ∂x2

 ∂A  = − izκ (x)A eiωt .e−iκ(x)z , ∂x     ∂ 2A ∂A = − 2izκ  (x) − izκ  + z2 (κ  )2 A eiωt .e−iκ(x)z . ∂x ∂x2 

(12.60) (12.61)

Hence, the general phase mixing equation (12.48) gives   2A ∂A ∂ c2A (x) 2iκ(x) − 2 ∂z ∂z      ∂ 2A ∂A   ∂A ∂ 2A  2  2 2 = iωνm − 2izκ (x) − 2iκ(x) − izκ + z (κ ) A + − κ (x)A . ∂x ∂z ∂x2 ∂z2 (12.62) If now we neglect the higher order derivatives of A retaining just the lowest order term on both sides of equation (12.62), so that on the left of (12.62) we retain the first derivative of A but not the second derivative while on the right of (12.62) (which is multiplied by the diffusivity νm ) we retain only those terms that are proportional to A, then we obtain νm  2 ∂A 2 (12.63) + κ + izκ  + z2 κ  A = 0. ∂z 2cA (x) Equation (12.63) reduces to (12.54) if we neglect the κ 2 and zκ  terms and retain the z2 term only, valid for sufficiently large z. Thus, we recover the earlier result. For sufficiently large z we again obtain cubic spatial damping, but for small z or for an Alfv´en speed profile that is close to uniform the terms involving the derivatives of κ are small and the term proportional to νm is dominated by the κ 2 term, corresponding to uniform damping. The integral of equation (12.63) is readily obtained by noting that we can rewrite equation (12.63) in the form     ∂ 1 1 νm 2 A exp zκ 2 + iz2 κ  + z3 κ  = 0. ∂z 2cA (x) 2 3 Hence,

  A = A0 exp −

z zunif 0

+

z3 z30



  i κ  (x) 2 z , · exp − νm 4 cA

(12.64)

mixing

. Thus, we recover uniform where A0 denotes the value of A at z = 0 and z0 ≡ z0 damping for small z and cubic damping for moderate or large z; additionally, we see an oscillatory aspect determined by κ  . The uniform damping term and the oscillatory term were not retained in the treatment by Heyvaerts and Priest (1983).

12.5.2 Approximations Let us consider the various approximations used to obtain equation (12.63) from (12.62). To simplify the left-hand side of (12.62) we neglect the second derivative term compared with the first derivative one, requiring

12.6 Phase Mixing of Alfv´en Waves: A Similarity Solution

375

) ) ) ) ) ∂ 2A ) ) ) ∂A ) ) )2iκ(x) )  ) ), ) ) ∂z2 ) ∂z )

that is, ω

cA . 2lz

Here lz denotes a typical lengthscale in the z-direction, so that we have the estimate A ∂A ∼ . ∂z lz To simplify the right-hand side of (12.62) we neglect all but the last term on the right, requiring ) ) ) ∂ 2A ) ) ) |z2 (κ  )2 )A|  ) 2 ) , ) ∂x ) that is, |z|  Thus, altogether we require that cA ω 2lz

cA . ω

and

|z| 

cA . ω

(12.65)

For the solution (12.56), lz = z0 ; then the first of conditions (12.65) is met for 2 (6R)1/3  1, a condition satisfied for very large R. The second condition requires that we are considering large distances z far from the oscillation source at z = 0. For example, with an Alfv´en speed of 103 km s−1 and an oscillation with period P (= 2π/ω) = 102 s the condition is |z|  105 /(2π ) km, that is, |z|  1.6 × 104 km. 12.6 Phase Mixing of Alfv´en Waves: A Similarity Solution Hood, Ireland and Priest (1997) have pointed out an exact solution of the reduced phase mixing equation (12.50). Their discussion applies to the special case of an Alfv´en speed of the form

x cA (x) = cA0 exp − . (12.66) a Here a is a measure of the spatial scale of the variation in Alfv´en speed and cA0 is the value of cA at x = 0; the constant a has the same meaning as earlier. To consider the phase mixing equation (12.50), introduce a new variable s through the substitution z x/a s= (12.67) ·e H for constant scale height H (to be specified). Then ∂s d ∂ = ∂z ∂z ds and so

376

Damping

cA (x)

∂ cA0 d = , ∂z H ds

Also

c2A (x)

c2 d2 ∂2 = A02 2 . 2 ∂z H ds

    2 1 d d ∂2 d 1 2 d = 2s s = 2 s 2 +s . ds ∂x2 a ds ds a ds

∂ 1 d = s , ∂x a ds

With this change of variable and assuming a time dependence of eiωt , the phase mixing equation (12.50) becomes   c2A0 d2 u 1 d du 2 −ω u − 2 2 = iωνm 2 s s . ds H ds a ds Choosing the constant H to be cA0 /ω, this differential equation may be written in the form (1 + iδs2 ) that is, (1 + iδs2 )1/2

d2 u du + iδs + u = 0; 2 ds ds

  d du (1 + iδs2 )1/2 + u = 0. ds ds

(12.68)

The constant δ is defined by νm 1 = 2 . R a ω Following Hood, Ireland and Priest (1997), we can solve this differential equation by introducing a new variable w through  s 1 1 dw , w = dp. = 2 1/2 2 1/2 ds (1 + iδs ) 0 (1 + iδp ) δ=

Here we have written p as our integration variable, and we have chosen w = 0 at s = 0. It then follows that d d , (1 + iδs2 )1/2 = ds dw and the differential equation (12.68) becomes d2 u + u = 0. dw2

(12.69)

This equation is readily solved to give u = A1 eiw + A2 e−iw . Now the integral defining w may readily be evaluated by elementary means, yielding w=

 1 sinh−1 (iδ)1/2 s , 1/2 (iδ)

which we may write in the form w=

 1 ln (iδ)1/2 s + (1 + iδs2 )1/2 . 1/2 (iδ)

(12.70)

12.6 Phase Mixing of Alfv´en Waves: A Similarity Solution

377

Now we require u = u0 at z = 0 (corresponding to w = 0) which implies that A1 + A2 = u0 . Hence u = A1 eiw + (u0 − A1 )e−iw . To determine A1 we need to examine the behaviour of the solution as z → +∞, corresponding to s → +∞. Consider the form of w for δs2  1. Carrying out an expansion of the logarithm, we have     s s 1 1/2 w = log α + log 1 + 1 + 2 , α α α where we have written α = (iδ)1/2 s. For δ 1/2 s large we have

1  πi/4 1/2  w∼ · δ s + ln 2 . log e (iδ)1/2 Thus, for δ 1/2 s  1 we have w ∼ (1 − i)

1 ln(δ 1/2 s). (2δ)1/2

Hence the imaginary part of w is negative. Now the function eiw grows (and the function e−iw declines) with positive s if the imaginary part of w is negative. Thus, the function eiw is unbounded for large s and we must eliminate this contribution by choosing A1 = 0. Then u = u0 e−iw

(12.71)

gives a solution that has u → 0 as s → +∞. Accordingly, an exact solution of the phase mixing equation (12.50) for an exponential profile in the Alfv´en speed is given by   (1 + i)  1/2 2 1/2 u = exp − , (12.72) ln (iδ) s + (1 + iδs ) (2δ)1/2 with s=

x ωz exp , cA0 a

δ=

1 νm = 2 . R a ω

This exact solution was first obtained by Hood, Ireland and Priest (1997). Finally, we consider the behaviour of the solution (12.72) for small δs2 . Expansion of the logarithm term in (12.70) gives 1 w = s − iδs3 + O(s5 ). 6 Thus, for small δs2 we have



 1 3 u = u0 exp − δs · e−is , 6

(12.73)

showing the same cubic damping form discussed in Section 12.5. For x ≈ 0, s ≈ z/H and the cubic damping has precisely the same lengthscale z0 obtained earlier.

378

Damping

The solution (12.72) expressed in terms of a single variable s is termed a similarity solution (see, for example, Whitham 1974). It comes about because the choice of exponential profile cA (x) allows us to express the operations cA (x)

∂ ∂z

c2A (x)

and

∂2 ∂z2

entirely in terms of a single variable, the similarity variable s. Although the solution (12.72) captures the cubic damping discovered by Heyvaerts and Priest (1983), it fails to give the linear damping that arises in a uniform medium. This is because the chosen similarity variable s becomes independent of x in the uniform medium limit a−1 → 0. Another example of a similarity variable solution of equation (12.50) is given in Hood, Gonzalez-Delgado and Ireland (1997).

12.7 Phase Mixing of Alfv´en Waves: Temporal Damping In the discussion of spatial damping we regarded the frequency ω as fixed and then determined the spatial behaviour as arises according to the phase mixing equation (12.48) or its reduced form (12.50). Here we instead impose the wavenumber kz , regarding it as fixed by the imposition of boundaries such as the ends of a coronal loop. The question then is: what happens to the time dependence of u? An analysis similar to the spatial case may be given. Consider, then, the general equation of phase mixing, namely (12.48), rewritten here for convenience:   ∂ 2u ∂ 2 ∂u ∂2 ∂ 2u 2 − cA (x) 2 = νm + 2 . (12.74) ∂t ∂t2 ∂z ∂x2 ∂z We look for a solution of equation (12.74) of the form u = eikz (cA (x)t−z) · T(x, t)

(12.75)

with kz fixed. Thus, the z-dependence is regarded as fixed. (Our analysis also applies, with minor changes, to a z-dependence of the form sin kz z, appropriate for a coronal loop with ends at z = 0 and z = L (with kz L = π).) Then   ∂T ∂u = eikz (cA (x)t−z) + ikz cA (x)T , ∂t ∂t   2 2 ∂ u ∂T ikz (cA (x)t−z) ∂ T 2 2 =e + 2ikz cA (x) − kz cA (x)T , ∂t ∂t2 ∂t2 ∂ 2u = −kz2 eikz (cA (x)t−z) T. ∂z2 Hence ∂ 2u ∂ 2u − c2A (x) 2 = eikz (cA (x)t−z) 2 ∂t ∂z Also,



 ∂T ∂ 2T + 2ikz cA (x) . ∂t ∂t2

(12.76)

12.7 Phase Mixing of Alfv´en Waves: Temporal Damping

379





∂u ∂T = eikz (cA (x)t−z) + ikz t cA  (x)T ∂x ∂x   2T  ∂ 2u ∂T ∂ 2 = eikz (cA (x)t−z) + 2ikz t cA  (x) + ikz tcA  − kz2 t2 cA  T . ∂x ∂x2 ∂x2 It then follows that ∂ 2u ∂ 2u + 2 = eikz (cA (x)t−z) ∂x2 ∂z



 ∂T  ∂ 2T   2 2 2 + 2ikz cA (x)t + ikz t cA − kz (1 + t cA ) T . ∂x ∂x2 (12.77)

Hence, differentiating with respect to t yields (after some algebra) 

∂2 ∂2 + 2 2 ∂x ∂z







∂ 2T ∂ 2T ∂ 3T + ikz cA (x) 2 + 2ikz cA  (x)t 2 ∂x∂t ∂x ∂t ∂x ∂T ∂T  2 ∂T − 2kz2 cA cA  (x)t + ikz t cA  − kz2 (1 + t2 cA  ) + 2ikz cA  (x) ∂x ∂x ∂t  2 2 + ikz cA  − kz2 t(2cA  + cA cA  ) − ikz3 cA (1 + t2 cA  ) T . (12.78) ∂u ∂t

= eikz (cA (x)t−z)

Expressions (12.76) and (12.78) are now to be substituted in the general phase mixing equation (12.48). However, if now we neglect all but the lowest order terms, ignoring all the derivatives of T on the right-hand side of (12.78) and on the left-hand side of (12.76) retaining the first derivative only, we find that the general phase mixing equation (12.48) reduces to 2ikz cA (x)

 ∂T 2 2 = νm ikz cA  − kz2 t(2cA  + cA cA  ) − ikz3 cA (1 + t2 cA  ) T, ∂t

and so ∂T νm   2 2 cA − kz2 cA (1 + t2 cA  ) + ikz t(2cA  + cA cA  ) T = 0. − ∂t 2cA

(12.79)

This is a first order partial differential equation which we can readily solve. But note first that if we ignore all but the highest powers of t in the differential equation (12.79) we obtain ∂T 1 2 + νm kz2 (cA  ) t2 T = 0, ∂t 2

(12.80)

   ∂ 1  2 3 2 T exp νm kz (cA ) t = 0. ∂t 6

(12.81)

  1 3 3 2 T = T(0) exp − νm kz2 (cA  ) t3 = T(0)e−t /τ0 , 6

(12.82)

which implies

Hence

380

Damping mixing

where T(0) is the initial value of T (so T = T(0) when t = 0). The timescale τ0 ≡ τ0 a temporal measure of the scale by which the damping operates, is  mixing τ0

=

6 νm kz2 (cA  )2

,

1/3 .

(12.83)

Thus, we obtain cubic damping in time, with a timescale that depends upon the cube root of the Reynolds number. A result of this form was first obtained by Heyvaerts and Priest (1983). See Cargill, De Moortel and Kiddie (2016) for a recent discussion. mixing We can illustration the timescale τ0 for the same parameters as used in Section 12.5 to illustrate spatial damping. With a kinematic viscosity of νm = 1011 m2 s−1 , an Alfv´en speed cA = 106 m s−1 with an inhomogeneity scale of a = 106 m, a loop of length L = 108 m supports a fundamental standing Alfv´en mode (with kz = π/L) which mixing = 39.3 s. This is to be compared with the timescale has a phase mixing timescale of τ0 unif 4 τ0 = 2.03 × 10 s arising in a uniform medium. Returning to the more general first order differential equation (12.79), we can rewrite it in the form       νm 1 1 ∂ 2 2 = 0, T exp − cA  t − kz2 cA t + t3 cA  + ikz t2 (2cA  + cA cA  ) ∂t 2cA 3 2 and hence        2 νm 1 1  1 2 3  2 2 cA T = T(0) exp − + cA ikz νm t k cA t + cA t − cA t exp . 2cA z 3 2 cA 2 (12.84) Thus, we see the occurrence of a damping of the form 

     1 2 νm kz2 cA t + cA  t3 − cA  t . exp − 2cA 3 A solution of the form (12.84), though for the special case of a linear cA (x) (for which cA  = 0), was pointed out by Roberts (1988), exploiting an analogy with the flux expulsion problem and adapting a solution describing the removal of magnetic flux from within a convective cell that was put forward by Moffatt and Kamkar (1983). In the present context, when cA is linear in x, we note that the damping expression (12.84) depends upon time t only. If the medium is uniform (cA = 0, cA  = 0) then (12.84) recovers the damping scale τ0unif = 2/(kz2 νm ) of a uniform medium. If the medium is non-uniform then eventually mixing (given in equadamping is cubic in form, progressing on a timescale determined by τ0 tion (12.83)). For short times t, diffusion operates slowly√on the long timescale 2/(kz2 νm ) that holds in a uniform medium. But at a time of order 3/|cA | the slow diffusion of a uniform medium gives way to the rapid damping of phase mixing. The solution (12.84) thus exhibits both the short time behaviour for initial times and the eventual cubic damping for longer times.

12.8 Damping of the Slow Mode

381

Finally, we note that for the case cA  = 0 the expression for u that follows from (12.84) is in fact an exact solution of the equation  2  ∂u 1 ∂ u ∂ 2u ∂u + + cA (x) = νm , (12.85) ∂t ∂z 2 ∂x2 ∂z2 which may be thought of as an approximate first integral of equation (12.74).

12.7.1 Phase Mixing in the Presence of Stratification We end our discussion of the Alfv´en wave with a brief mention of the role of stratification (see Chapter 9), which has so far been ignored in our treatment of phase mixing. In fact, it is possible to incorporate the effects of stratification and indeed a two-dimensionality in the equilibrium magnetic field. This was done by Ruderman, Nakariakov and Roberts (1998) and Smith, Tsiklauri and Ruderman (2007). It turns out that phase mixing in a stratified isothermal medium with a uniform vertical magnetic field proceeds at a rate that is dependent upon the factor   3 z , 30 1 − exp − 20 where 0 is the pressure scale height in the isothermal atmosphere. In the limit of no gravity, 0 → ∞, this factor is proportional to z3 /8 and we recover the cube root damping of classical phase mixing, as determined by Heyvaerts and Priest (1983). Also, for low heights when z/0 is small the above factor again approaches z3 /8, and so phase mixing proceeds much as in the absence of gravity. However, for moderate heights the correction from stratification may be of interest. For a more detailed account of the role of stratification, we refer the reader to Ruderman, Nakariakov and Roberts (1998) and Smith, Tsiklauri and Ruderman (2007).

12.8 Damping of the Slow Mode 12.8.1 Formulation: One-Dimensional Propagation We turn now to a consideration of how the slow mode is damped, examining the roles of viscosity and thermal conduction. We begin by examining a sound wave. As far as the corona is concerned, a slow wave in the low β plasma behaves somewhat like a sound wave propagating one-dimensionally with motions purely along the direction of the applied magnetic field. Magnetic perturbations are negligible and the main role of the magnetic field is simply to guide motions, much as a train is guided by a rigid rail. The role of the magnetic field, then, is to provide a preferred direction, which we here take to be the z-axis. Aspects of this approximation are further considered in Section 12.13. Consider then the general system of equations (12.1)–(12.9) with the magnetic field now set to zero (B ≡ 0) and motions u assumed to be purely in the z-direction and to vary with z and t, so that u = uz (z, t)ez .

382

Damping

The continuity and momentum equations for a fluid in which the magnetic field is absent are (see equations (12.1) and (12.2)) simply   ∂uz ∂uz ∂ ∂p ∂ρ ρ (12.86) + (ρuz ) = 0, + uz =− + ez · Fvisc . ∂t ∂z ∂t ∂z ∂z Here ρ denotes the plasma density and p is the plasma pressure. In the momentum equation we have ignored the force of gravity. In the one-dimensional case discussed here there is a simplification in the form of the viscous force given in (12.7), which now becomes   ∂ 4 visc ∂uz μ . (12.87) Fvisc = ez ∂z 3 ∂z If, further, we treat μvisc as a constant, then we can write Fvisc =

4 visc ∂ 2 uz 4 ∂ 2 uz 4 ∂ 2 uz e = e = ez , μ ρν η0 z z 3 3 3 ∂z2 ∂z2 ∂z2

(12.88)

where η0 (= ρν = μvisc ) is sometimes introduced as the compressive viscosity. For the one-dimensional motion u = uz (z, t)ez the energy equation (12.3) now takes the form   ∂p γ p ∂ρ ∂ρ ∂p + uz − + uz = −(γ − 1)L, (12.89) ∂t ∂z ρ ∂t ∂z where L is the gain or loss function (energy per unit volume) and γ is the ratio of specific heats. The term L is taken to correspond solely to heat redistribution by thermal conduction. In the presence of a magnetic field, thermal conduction is constrained to be almost purely along the direction of the magnetic field, which here is taken to be the z-direction. Hence, thermal conduction is assumed to arise purely along the z-axis, giving for a fluid with temperature T   ∂T ∂ (12.90) −κ , κ = κ0 T 5/2 . L= ∂z ∂z The coefficient κ0 varies slightly with plasma conditions; see Section 12.2. Finally, we note the equation of state p=

kB ρT. m ˆ

(12.91)

12.8.2 Wave-Like Equations We consider a uniform equilibrium in which the plasma pressure p0 , density ρ0 and temperature T0 are all constants. Then the linear equations describing motions uz (z, t) that are purely in the z-direction and depend upon z and t only, with associated pressure perturbation p(z, t), density perturbation ρ(z, t) and temperature perturbation T(z, t), are ∂uz ∂ρ + ρ0 = 0, ∂t ∂z

ρ0

∂uz ∂ 2 uz ∂p 4 =− + ρ0 ν 2 , ∂t ∂z 3 ∂z

12.8 Damping of the Slow Mode

p ρ T = + , p0 ρ0 T0

∂ρ ∂ 2T ∂p − c2s = (γ − 1)κ 2 . ∂t ∂t ∂z

383

(12.92)

As usual, cs (= (γ p0 /ρ0 )1/2 ) denotes the sound speed in the fluid. The compressive viscosity ν and thermal conduction κ are taken as constants, but calculated according to the temperature-dependent expressions given in Section 12.2. A number of authors have considered aspects of the system of equations (12.92) and its application to solar phenomena; see, for example, Field (1965), Nakariakov et al. (2000), Ofman and Wang (2002), De Moortel and Hood (2003), Sigalotti, MendozaBrice˜no and Luna-Cardoza (2007), Macnamara and Roberts (2010), Macnamara (2011), Ruderman (2013) and Wang et al. (2015).2 We may readily eliminate all variables in favour of uz and T/T0 . Using the continuity equation, momentum equation and the linear form of the ideal gas law, we obtain 2 2 T  ∂ 2 uz 4 ∂ 3 uz 2 ∂ uz 2 ∂ − cN 2 − ν 2 = −cN , (12.93) 3 ∂z ∂t ∂z∂t T0 ∂t2 ∂z where cN denotes the Newtonian sound speed:  1/2 p0 1 cN = = √ cs . ρ0 γ Similarly, eliminating p and ρ from the energy equation we obtain     κ T0 ∂ 2 ∂ T ∂uz T . − (γ − 1) = −(γ − 1) ∂t T0 p0 ∂z2 T0 ∂z

(12.94)

Introduce3 7/2

νth =

κ0 T0 κ T0 = . p0 p0

(12.95)

The introduced coefficient νth has the same dimensions as the kinematic viscosity ν, namely m2 s−1 , and so may be viewed as representing the role of thermal conduction much as ν is a measure of the viscous force. Then equation (12.94) describing the evolution of T/T0 reads     ∂ T ∂2 T ∂uz . (12.96) − (γ − 1)νth 2 = −(γ − 1) ∂t T0 T0 ∂z ∂z The coupled differential equations (12.93) and (12.96) describe the evolution of T/T0 and the motion uz in the presence of viscosity ν and thermal conduction νth . 12.8.3 Fourier Form Introduce uz (z, t) = u0 ei(ωt−kz z) ,

T(z, t) = T00 ei(ωt−kz z) ,

2 Care needs to be taken regarding the notation for coefficients employed in the description of viscous effects. For example,

Ofman and Wang (2002) and De Moortel and Hood (2003) worked in terms of the compressive viscosity η0 = ρ0 ν ; Macnamara and Roberts (2010) used the notation ν to denote the compressive viscosity, here taken as ρ0 ν .

3 This notation does not appear to have been used in the literature, though it seems a natural step.

384

Damping

for constants u0 and T00 . Then equations (12.93) and (12.96) imply    T00  T00 4 , iω + (γ − 1)νth kz2 = i(γ − 1)kz u0 . ω2 − kz2 c2N − iνωkz2 u0 = c2N ωkz 3 T0 T0 (12.97) Eliminating the constants u0 and T00 (assumed non-zero) between these equations leads to the relation       4 4 3 2 2 2 2 2 4 γ −1 νth c2s = 0. ω − i (γ − 1)νth + ν kz ω − cs + ννth (γ − 1)kz kz ω + ikz 3 3 γ (12.98) Equation (12.98) is the dispersion relation for a longitudinally propagating sound wave in the presence of both viscous damping and thermal conduction. Thermal conduction is assumed to act purely in the direction of the applied magnetic field. A relation of the form (12.98) was first obtained by Field (1965), who was particularly interested in the topic of thermal instability in which radiative terms and a heating source arise, but viscosity is ignored. Recent applications have been discussed by De Moortel and Hood (2003), and Macnamara and Roberts (2010). De Moortel and Hood (2003) were investigating the damping of standing slow modes in coronal loops. Macnamara and Roberts (2010) employed the dispersion relation to determine period ratios in coronal loops, of interest in coronal seismology (see the review by Andries et al. 2009). These topics are further discussed in Chapter 14. In the absence of viscous dissipation and thermal conduction, corresponding to ν = 0 and νth = 0, relation (12.98) reduces to ω = 0 and ω2 = kz2 c2s . The zero mode corresponds to the thermal mode; the non-zero modes to sound waves propagating in either direction along the z-axis.

12.9 Damping of the Slow Mode by Viscosity Only (νth = 0, ν = 0) With νth = 0 (in fact if (γ − 1)νth = 0) then (12.96) implies that   ∂uz ∂ T , = −(γ − 1) ∂t T0 ∂z and so equation (12.93) becomes 2 4 ∂2 ∂ 2 uz 2 ∂ uz − c = ν s 3 ∂z2 ∂t2 ∂z2



 ∂uz . ∂t

(12.99)

An equation of this form seems first to have been discussed by G. G. Stokes in 1845, examining the damping of sound in air; see the treatment in Rayleigh (1877, sect. 346). See also the discussion in Lighthill (1978; sect. 1.13). In the context of the damping of a slow wave in the solar atmosphere, the wave-like equation arises in the work of Sigalotti, Mendoza-Brice˜no and Luna-Cardoza (2007) and Macnamara and Roberts (2010). The corresponding dispersion relation associated with equation (12.99) is 4 ω2 − i νkz2 ω − kz2 c2s = 0. 3

(12.100)

12.9 Damping of the Slow Mode by Viscosity Only (νth = 0, ν = 0)

385

We may view the dispersion relation (12.100) in two distinct ways, depending on whether we are considering standing waves with fixed spatial form (fixed kz ) but temporally dependent amplitude (corresponding to complex ω), or waves with fixed (real) frequency ω but spatially varying amplitude (corresponding to complex kz ). We treat these cases separately.

12.9.1 Temporal Behaviour: Viscosity Only Suppose that kz is real and fixed, corresponding to standing waves. Then we may regard (12.100) as a quadratic equation for complex ω, with roots   2 2 4 2 4 1/2 2 2 ω = iνkz ± kz cs − ν kz . (12.101) 3 9 The form of ω determines the nature of the disturbance. Now the nature of the roots (12.101) depends upon the magnitude of kz and ν. If |kz | > 3cs /(2ν) then ω is purely imaginary: viscosity is so strong that wave propagation is not possible and purely temporally decaying modes result. However, if |kz | < 3cs /(2ν), then ω has both real and imaginary parts: wave propagation occurs but the motion is damped. Setting ω = ωR + iωI , where ωR and ωI are the real and imaginary parts of ω, we have  1/2 4 2 ωR = ± kz2 c2s − ν 2 kz4 , ωI = νkz2 . (12.102) 9 3 Thus, damping proceeds as e−ωI t , that is as exp(−t/τ viscous ), with a viscous timescale τ viscous given by τ viscous =

1 3 = , ωI 2νkz2

|kz | < 3cs /(2ν).

(12.103)

An application of this calculation to a determination of the viscosity coefficient ν when thermal conduction is dominated by viscosity is given in Wang et al. (2015).

12.9.2 Spatial Behaviour: Viscosity Only We can also consider spatial damping for a given fixed real frequency ω. Then the dispersion relation (12.100) gives a complex kz2 for real ω: kz2 c2s =

ω2 1+

4iνω 3c2s

.

(12.104)

386

Damping

Writing kz = kzR + ikzI for real and imaginary parts kzR and kzI , then for low frequencies or weak viscous damping such that 4ων/3c2s  1, the dispersion relation (12.104) yields kzI = −

2ν 2 ω , 3c3s

ω

3c2s , 4ν

(12.105)

with a corresponding spatial damping scale Lviscous of 3c3s 3c3s 2 3cs 2 = (12.106) P = λ , 2 2νω 8π 2 ν 8π 2 ν where P = 2π/ω is the (fixed) period of the oscillation and λ = 2π/kzR is its wavelength (kzR = ω/cs ). A result of this form was first given by Stokes (1845). The damping length Lviscous is proportional to the square of the period P or wavelength λ; low period waves with short wavelengths suffer the most damping by viscosity.4 Turning now to the opposite extreme of high frequencies, we may expand (12.104) to yield  1/2 3ω 3c2 , ω s, (12.107) kzI = − 8ν 4ν Lviscous =

for which the associated spatial damping scale is  1/2  1/2 8ν 4ν = P1/2 . Lviscous = 3ω 3π Altogether, then, we have

⎧ ⎨− 2ν3 ω2 , 3cs

kzI =

⎩  3 1/2 1/2 − 8ν ω ,

ω ω

3c2s 4ν , 3c2s 4ν ,

(12.108)

(12.109)

giving a spatial damping scale Lviscous

⎧ 3  ⎨ 3cs P2 , 2 =  8π ν ⎩ 4ν 1/2 P1/2 , 3π

P  Pvisc ∗ , P  Pvisc ∗ ,

(12.110)

where = Pvisc ∗

8π ν . 3c2s

(12.111)

The timescale Pvisc ∗ provides a measure of period, for damping by viscosity alone, by which we can compare the given period P. High period (P  Pvisc ∗ ), low frequency, waves are damped on a scale that is proportional to the square of the period P, whereas low period (P  Pvisc ∗ ), high frequency, waves are damped on a scale that is proportional to the square

4 Following Rayleigh (1877, sect. 346), it is perhaps of interest to illustrate (12.106) for air at room temperature. Taking a kinematic viscosity of ν = νair = 1.5 × 10−5 m2 s−1 and a sound speed of cs = 340 m s−1 , appropriate for air at room temperature, for a sound wave of wavelength λ (in metres) we obtain a damping length of Lviscous = 8.6 × 105 λ2 m. Thus, for

a sound wave of wavelength λ = 10−2 m (= 1 cm) we obtain a damping length of Lviscous = 86 m, whereas a wavelength of λ = 10−1 m (= 10 cm) has a damping length of Lviscous = 8600 m.

12.10 Damping of the Slow Mode by Thermal Conduction Only (ν = 0, κ = 0)

387

root of the period P. High period waves, for which Lviscous is large and so damping only slight, feel almost no damping by viscosity; low period waves are effectively damped. The case P  Pvisc ∗ frequently applies in applications and then the spatial damping scale viscous may be written in the approximate form L Lviscous =

cs 1 2 P , π Pvisc ∗

P  Pvisc ∗ .

(12.112)

We illustrate numerically the above results in Section 12.11. Finally, we end our discussion of viscous damping by pointing out that the dispersion relation (12.104) can in fact be solved exactly. For most situations, the approximate results (12.109) are adequate, but it is nonetheless of interest to examine the exact treatment. In terms of the real and imaginary parts of kz2 , dispersion relation (12.104) implies that 2 2 − kzI = kzR

(ω2 /c2s ) , 1 + ω2 τν2

2kzR kzI = −

(ω2 /c2s ) ωτν , 1 + ω2 τν2

τν =

4ν , 3c2s

(12.113)

where τν (= Pvisc ∗ /(2π )) is a natural viscous timescale by which the driving frequency ω 2 , we obtain a quadratic may be compared. Equations (12.113) are exact. Eliminating kzR 2 equation for kzI :   2 /c2 1 ω ω6 τν2 s 2 2 2 (kzI ) + − = 0. (12.114) k zI 1 + ω2 τν2 4c4s (1 + ω2 τν2 )2 2 > 0) is the appropriate root (Stokes 1845): The positive solution (kzI   1 ω2 1 2 kzI − = 2 , . 2cs (1 + ω2 τν2 ) (1 + ω2 τν2 )

(12.115)

This exact result recovers the approximate results (12.109) in the extremes ωτν  1 and ωτν  1. In most cases, ωτν  1 applies. 12.10 Damping of the Slow Mode by Thermal Conduction Only (ν = 0, κ = 0) With ν = 0, so viscous effects are absent, equation (12.93) reduces to 2 2 T  ∂ 2 uz 2 ∂ uz 2 ∂ − cN 2 = −cN . ∂z∂t T0 ∂t2 ∂z

(12.116)

Equation (12.96) remains unchanged. We may eliminate uz between equations (12.96) and (12.116), with the result         2 2 ∂2 T ∂ T ∂2 ∂2 2 ∂ 2 ∂ − c − c = (γ − 1)ν . (12.117) th s N 2 ∂t2 ∂z2 ∂t T0 ∂t2 ∂z ∂z2 T0 Thus, if νth = 0 then we obtain the one-dimensional wave equation (for the quantity ∂(T/T0 )/∂t) with propagation at the sound speed cs . If νth → ∞ then we again obtain the one-dimensional wave equation, but now for the quantity ∂ 2 (T/T0 )/∂z2 and propagation is at the Newtonian sound speed cN .

388

Damping

The dispersion relation associated with (12.117) is ω(ω2 − kz2 c2s ) − i(γ − 1)νth kz2 (ω2 − kz2 c2N ) = 0,

(12.118)

which may be written out in the explicit cubic form ω3 − i(γ − 1)νth kz2 ω2 − ωkz2 c2s + i(γ − 1)νth kz4 c2N = 0.

(12.119)

Equation (12.119) is of course the reduced form of equation (12.98) with ν = 0. A relation of the form (12.119) was first obtained by Field (1965; see his eqn. (15)).

12.10.1 Temporal Behaviour: Thermal Conduction Only To explore the dispersion relation (12.118) or (12.119) it is again convenient to examine the cases of fixed real kz and fixed real ω separately, the first case corresponding to standing waves and the second case to driven propagating waves. We begin with the case of given kz and ask what is the resulting time behaviour. To answer this question it is convenient to consider the extremes of small or large kz (in some suitable) measure. To begin, notice that when kz → 0 then (12.118) implies that ω2 → c2s . Accordingly, if we set ω = kz cs (1 + ) with ||  1, then =i

(γ − 1)2 νth kz . 2γ cs

Since we have assumed that || is small this amounts to requiring that (γ − 1)2 νth |kz |  1. 2γ cs Thus, for complex frequency ω = ωR + iωI , where ωR and ωI are the real and imaginary parts of ω, we find that ωR ∼ kz cs ,

ωI ∼

(γ − 1)2 νth kz2 , 2γ

|kz | 

2γ cs . (γ − 1)2 νth

(12.120)

Thus, the damping rate ωI is small for small kz , vanishing as kz → 0. In the opposite extreme, when νth kz  cs , we can set ω = kz cN (1 + 1 ) for small |1 | and then find that cN 1 = i . 2kz νth Thus, ωR ∼ kz cN ,

ωI ∼

c2N , 2νth

cs |kz |  √ , 2 γ · νth

(12.121)

and so the speed of the wave is reduced to the Newtonian value cN and the damping rate ωI reaches a limit of c2s /(2γ νth ).

12.10 Damping of the Slow Mode by Thermal Conduction Only (ν = 0, κ = 0)

389

12.10.2 Spatial Behaviour: Thermal Conduction Only Turning to an examination of the spatial behaviour as determined by the dispersion relation (12.118) we set kz = (ω/cs )(1 + δ) for given ω and small |δ|. Then, neglecting squares of δ we readily obtain δ = −i

ω (γ − 1)2 νth 2 , 2γ cs

kz =

ω (γ − 1)2 ω2 −i νth 3 , cs 2γ cs

valid for (γ − 1)2 ω νth  1. 2γ cs Hence, writing kz = kzR + ikzI for real and imaginary parts kzR and kzI , to leading order we obtain kzR =

ω , cs

kzI = −

(γ − 1)2 ω2 νth 3 , 2γ cs

c2s 2γ . (γ − 1)2 νth

ω

(12.122)

Thus, under the action of thermal conduction alone, we obtain a damping lengthscale of Lthermal =

c3s 2γ , (γ − 1)2 ω2 νth

ω

c2s 2γ . (γ − 1)2 νth

(12.123)

P  Ptherm , ∗

(12.124)

In terms of period P (= 2π/ω) this is Lthermal =

c3s 2 γ P , 2π 2 (γ − 1)2 νth

where = 2π Ptherm ∗

(γ − 1)2 νth . γ c2s

(12.125)

provides a measure by which we can compare whether a given period The timescale Ptherm ∗ P is high or low. In this context of damping by thermal conduction alone, high period waves , whereas low period waves are ones for which P  Ptherm . are ones for which P  Ptherm ∗ ∗ ; this Purely for convenience, we have introduced a factor of 2 in our definition of Ptherm ∗ is made so as to provide a closer parallel between damping by viscosity alone and thermal is in fact satisfied when conduction alone. Accordingly, the requirement that P  Ptherm ∗ . P  12 Ptherm ∗ The case of high period arises frequently in applications, and for this case we may cast the damping length formula in the convenient form Lthermal =

cs 1 P2 , π Ptherm ∗

P  Ptherm . ∗

(12.126)

The damping length formula (12.126) is of the same form as (12.112) for viscous damping.

390

Damping

Finally, turning to the opposite extreme of high frequency and low period, a similar expansion approach as given above leads to the result that kzR =

ω , cN

kzI = −

cN , 2νth

ω

c2N , νth

(12.127)

giving a damping length of Lthermal =

2νth , cN

P

νth . c2N

(12.128)

Thus, whereas the damping length is proportional to the square of the period at high periods, at very low periods it is independent of the period. We illustrate these results in the next section. A recent discussion of observations taken with the Solar Dynamics Observatory (SDO) and showing evidence of frequency-dependent damping has been given by Krishna Prasad, Banerjee and Van Doorsselaere (2014).

12.11 Viscous and Thermal Damping of Slow Waves: Illustrations 12.11.1 Spatial Damping of Propagating Waves It is of interest to illustrate the various formulas obtained for viscous and thermal damping, applying them to typical coronal conditions. We take the adiabatic index to be γ = 5/3 throughout our illustrations. Consider first a coronal temperature of T0 = 106 K, typical of loops observed with TRACE, giving a sound speed of cs ≈ 151 km s−1 (= 1.51 × 105 m s−1 ). For a coronal number density of n0 = 1015 m−3 , the kinematic viscosity is = 3.67 s, and so (according to equation (12.14)) ν = 1010 m2 s−1 . We then find that Pvisc ∗ is easy to meet. Thus, according to (12.110), the damping length the condition P  Pvisc ∗ due to compressive viscosity is Lviscous = 1.31×104 ×P2 m (with the period P in seconds); a period of say P = 500 s then gives Lviscous = 3.27 × 109 m. Turning to damping by thermal conduction, again for conditions typical of the active = region corona, we find that the thermal coefficient is νth = 6.52× 1011 m2 s−1 and Ptherm ∗ thermal = 1.003 × 103 × P2 m; for is easy to meet), and then L 47.91 s (again P  Ptherm ∗ P = 500 s we obtain Lthermal = 2.51 × 108 m. Thus, Lviscous ≈ 13 Lthermal and consequently thermal conduction dominates over viscous damping at this temperature and density, being about an order of magnitude more effective than viscosity. As a second illustration, consider the case of hot loops. With a coronal temperature of (say) T0 = 8.7 × 106 K giving a sound speed of cs = 445 km s−1 , typical of hot loops observed with SUMER, and a number density of (say) n0 = 5 × 1015 m−3 , we obtain a kinematic viscosity of ν = 4.46 × 1011 m2 s−1 and a thermal coefficient of νth = 2.91 × = 18.86 s and a thermal critical 1013 m2 s−1 . This gives a viscous critical period of Pvisc ∗ = 245.9 s. The damping lengths for an oscillation with period P are then period of Ptherm ∗ Lviscous = 7.52 × 103 × P2 m by viscosity, and Lthermal = 5.76 × 102 × P2 m by thermal conduction. For a period of P = 500 s we obtain Lviscous = 1.88 × 109 m and Lthermal = 1.44 × 108 m. Again, Lviscous ≈ 13 Lthermal and so thermal conduction dominates over viscous damping at these high temperatures too.

12.11 Viscous and Thermal Damping of Slow Waves: Illustrations

391

12.11.2 Temporal Damping of Standing Waves Turning now to an examination of standing waves, such as might occur in a coronal loop, we consider the damping timescales that arise in response to viscosity or thermal conduction and assess these for a given wavenumber kz . We take the wavenumber kz to be determined by the length L of a loop, setting kz = Nπ /L for integer N. In the main we will illustrate results for the fundamental standing mode, corresponding to taking N = 1. The viscous timescale is given by τ viscous =

3 3 = L2 , 2 2νkz 2νπ 2 N 2

(12.129)

which for the fundamental (N = 1) mode is τ viscous =

3 L2 . 2νπ 2

(12.130)

For this expression to hold we require |kz | < 3cs /(2ν), which can be expressed in terms of loop length L for the fundamental mode as L > L∗visc ,

L∗visc ≡ 2π ν/(3cs ).

(12.131)

The timescale arising from damping by thermal conduction is τ thermal =

2γ L2 , π 2 N 2 (γ − 1)2 νth

(12.132)

requiring (for N = 1) L  L∗therm ,

L∗therm ≡

π (γ − 1)2 νth . 2 γ cs

(12.133)

For conditions typical of active region loops, as detected by TRACE, we take T0 = 106 K and a coronal number density of n0 = 1015 m−3 . Then L∗visc = 1.39 × 105 m and L∗therm = 1.81 × 106 m. For a loop length L = 100 Mm = 108 m we satisfy the conditions L > L∗visc and L  L∗therm and find that τ viscous = 15.2 × 104 s and τ thermal = 1.17 × 104 s. Thus, thermal conduction is the main process of wave damping in freely propagating waves under conditions typical of active regions. For hot loops as typically detected by SUMER, with temperature T0 = 8.7 × 106 K and number density n0 = 5 × 1015 m−3 , we obtain L∗visc = 2.1 × 106 m and L∗therm = 27.4 × 106 m. For a loop length L = 100 Mm = 108 m, satisfying the conditions L > L∗visc and L  L∗therm , we find τ viscous = 3403.8 s and τ thermal = 261.0 s. Thus, thermal conduction is the main process damping slow waves in hot coronal loops. Thus, overall the main agency bringing about the decay of slow modes under coronal conditions is thermal conductivity (Ofman and Wang 2002).

392

Damping

12.11.3 Viscosity Versus Thermal Conduction We have seen in the above illustrations of damping lengths and damping times that under a wide range of conditions thermal conduction dominates over viscosity. We can compare the two roles a little further. Consider a loop of length L, and set kz = π/L so that we are examining the fundamental standing mode. Generally, we have seen that for suitably small kz then both viscously and thermally damped modes have a kz2 dependence. For very small kz neither damping is very strong. However, we may compare the two decay rates through using equations (12.102) and (12.120): 3(γ − 1)2 νth ωI |thermal = . 4γ ν ωI |viscous Under coronal conditions with γ = 5/3 we then find that νth κ0 1 = ≈ 65, ν ν0 kB /m ˆ essentially independent of temperature T0 , and so ωI |thermal ≈ 13 ωI |viscous . Thus, for small kz thermal conduction dominates, by an order of magnitude, over viscous damping. This argument depends upon our assumptions that L > L∗visc and L  L∗therm , so that we can use the approximate expressions for ωI for both viscous and thermal modes. For typical active region loops with a temperature of say T0 = 106 K and number density n0 = 1015 m−3 we obtain L∗visc = 1.39 × 105 m and L∗therm = 1.81 × 106 m, and so the conditions L > L∗visc , L  L∗therm are readily met. Hence, for active region loops thermal conduction dominates over viscous damping. For hot SUMER loops with a temperature of T0 = 8.7 × 106 K and number density n0 = 5 × 1015 m−3 we find L∗visc = 2.1 × 106 m and L∗term = 27.4 × 106 m, and so conditions L > L∗visc , L  L∗therm are again met in all but the shortest loops. Hence, thermal conduction dominates over viscous effects in all but the shortest loops. For larger kz , the damping rate ωI due to viscous effects continues to increase, whereas thermal conduction reaches a plateau with ωI → c2N /(2νth ) as |kz | → ∞. Thus ωI |thermal 3c2s = . viscous ωI | 4γ kz2 ν νth Hence, thermal conduction dominates over viscous effects if  1/2 3c2s |kz | < . 4γ ν νth In terms of loop length L, considering the fundamental mode for which kz = π/L thermal conduction dominates over viscous damping in loops of length L for which L>

1/2 2π  γ ννth /3 . cs

12.12 Thermal Conduction and Viscosity Combined

393

For TRACE loops with a temperature of 106 K, sound speed 151 km s−1 (= 1.51 × a kinematic viscosity ν = 1.0 × 1010 m2 s−1 and thermal coefficient νth = 11 6.52 × 10 m2 s−1 , we find that thermal conduction dominates in loops longer than about 2.5 × 106 m; since TRACE loops are longer than this, we expect thermal conduction to dominate in TRACE loops. In SUMER loops, with a temperature of (say) 8.7 × 106 K, a sound speed of about 445 km s−1 , a kinematic viscosity of ν = 4.46×1011 m2 s−1 and a thermal coefficient νth = 2.91 × 1013 m2 s−1 , we find that thermal damping dominates in loops longer than 3.79 × 107 m (or about 38 Mm). Hence, only in the shortest hot loops can we expect viscosity to play a significant role; in the majority of hot loops, thermal conduction dominates. A similar conclusion on the dominance of thermal conduction was reached by Ofman and Wang (2002) and De Moortel and Hood (2003), though Sigalotti, Mendoza-Brice˜no and Luna-Cardoza (2007) argue that both viscosity and thermal conduction are important in SUMER loops. 105 m s−1 ),

12.12 Thermal Conduction and Viscosity Combined The dispersion relation (12.98) with ν = 0, κ = 0 holds when both thermal conduction and viscosity are retained. We may discuss its behaviour for temporal effects (fixed kz ) and for spatial effects (fixed ω), which we do separately.

12.12.1 Temporal Behaviour Suppose that kz is fixed, as arises for a standing wave; we ask what is the resulting behaviour in time? This is answered from the dispersion relation by determining the real and imaginary parts of complex ω (= ωR + iωI ) for a given kz . Set ω = kz cs (1 + ), for ||  1. Then expanding the dispersion relation (12.98) for a small correction to ω = kz cs , rejecting terms involving squares of , we obtain   4 1 kz 1 (γ − 1)2 νth + ν . = i 2 cs γ 3 In order that ||  1, and noting that typically under coronal conditions νth ≈ 65 ν, it is sufficient that |kz |  and then

cs 2γ , (γ − 1)2 νth

   1 1 kz 4 ν + (γ − 1)2 νth . ω = kz cs 1 + i 2 cs 3 γ

(12.134)

(12.135)

Thus, ωR = kz cs ,

ωI =

  1 2 4 1 kz ν + (γ − 1)2 νth . 2 3 γ

(12.136)

394

Damping

We can express our results in terms of the length L of a magnetic tube or loop. Set kz = Nπ/L; then, for the fundamental mode (N = 1) we obtain a damping timescale of τ (= 1/ωI ) given by τ=

4L2 1 1 2L2 = · · , π 2 [ γ1 (γ − 1)2 νth + 43 ν] πc2s [Ptherm + Pvisc ∗ ∗ ]

L  L∗therm ,

(12.137)

therm were introduced earlier: , Pvisc where the critical scales Ptherm ∗ ∗ and L∗

Ptherm = 2π ∗

(γ − 1)2 νth , γ c2s

Pvisc = ∗

8π ν , 3c2s

L∗therm =

π (γ − 1)2 νth . 2 γ cs

(12.138)

To illustrate the result (12.137), consider conditions typical of the corona detected by TRACE, with a temperature T0 = 106 K and number density n0 = 1015 m−3 . Then the = 47.91 s, Pvisc = 3.67 s and L∗therm = 1.81 × 106 m. Accordingly, critical scales are Ptherm ∗ ∗ for a TRACE loop of length L = 100 Mm = 108 m (so L  L∗therm and thus condition (12.134) is met) we find τ = 1.08 × 104 s, indicating a slow decay by thermal conduction and viscosity. In a hot corona, typical of the conditions detected by SUMER, with a temperature of = 245.9 s, T0 = 8.7 × 106 K and number density n0 = 5 × 1015 m−3 , we find Ptherm ∗ visc therm 6 = 27.4 × 10 m. Then, for a loop of length L = 100 Mm we P∗ = 18.86 s and L∗ obtain a far more rapid decay, of τ = 242.4 s. 12.12.2 Spatial Behaviour The spatial behaviour may be calculated in much the same way, setting kz = kzR + ikzI = (ω/cs )(1 + δ) for small δ. Substituting in the dispersion relation (12.98) we find   4ν ω 1 (γ − 1)2 νth + , δ = −i 2 3 2cs γ and thus kz R ≈

ω , cs

kz I ≈ −

ω2 2c3s



 4ν 1 (γ − 1)2 νth + , γ 3

(12.139)

valid for small |δ|. Equation (12.139) gives a damping length Ldamp (= 1/|kz I |), due to the combined effects of thermal conduction and viscosity: Ldamp =

ω2



2c3s 1 γ (γ

− 1)2 νth +

4ν 3

=

cs 1 P2 . π [Ptherm + Pvisc ∗ ∗ ]

(12.140)

For this result to hold we require |δ|  1, equivalent to + Pvisc P  (Ptherm ∗ ∗ )/2,

(12.141)

so the driving period P must be much larger than the average of the thermal conduction and viscosity timescales. For conditions typical of the corona detected by TRACE, with a temperature T0 = 106 K and number density n0 = 1015 m−3 , a wave with a driving period of P = 500 s, evidently

12.13 Damping of the Slow Wave: Magnetic Effects

395

much greater than the average of Pvisc and Ptherm (which is 25.79 s) has a damping length ∗ ∗ damp 8 = 2.33 × 10 m. of L In a hot corona, typical of the conditions detected by SUMER, with a temperature of T0 = 8.7 × 106 K and number density n0 = 5 × 1015 m−3 , a slow wave driven with a period of P = 500 s has a damping length of Ldamp = 1.34 × 108 m.

12.13 Damping of the Slow Wave: Magnetic Effects So far we have been content to treat the damping of a longitudinally propagating sound wave as representative of the slow mode under coronal conditions. Given a strong magnetic field typical of the corona, this seems physically to be appropriate, any magnetic field being simply a wave guide for the propagation of sound. However, we can explore this concept a little further by adding explicitly a uniform magnetic field, including thermal conduction but ignoring viscosity. In fact, the inclusion of a uniform equilibrium magnetic field B0 = B0 ez was carried out by Field (1965), who derived a quintic dispersion relation (see his eqn. (54)) describing the combined effects of a magnetic field and thermal conduction (as well as optically thin radiation losses and a heating source). In our notation, Field (1965) obtained the dispersion relation5 ω5 − iω4 kz2 (γ − 1)νth − ω3 k2 (c2s + c2A ) + iω2 kz2 k2 (c2N + c2A )(γ − 1)νth + ωc2s c2A kz2 k2 − ik2 kz4 (γ − 1)νth c2N c2A = 0,

(12.142)

where k2 = kx2 + ky2 + kz2 . Dispersion relation (12.142) describes the effect of thermal conduction on both slow and fast magnetoacoustic waves; the relation is of degree five because it describes the two magnetoacoustic waves (modified by thermal conduction) propagating either in the z-direction or in the opposite direction, and the thermal mode itself. In the absence of a magnetic field, relation (12.142) reduces to ω2 = 0 together with a cubic dispersion relation, namely equation (12.118), describing sound propagation in the ±z-directions and the thermal mode. In the absence of thermal conduction, equation (12.142) reduces to the standard dispersion relation for magnetoacoustic waves (see Chapter 2). We end our discussion here by considering how relation (12.142) offers justification for the commonly used simplification of discussing a slow mode under coronal conditions as a sound wave propagating along the applied magnetic field. Suppose then that cA  cs , appropriate to the low β corona. For the slow mode, terms involving ω4 and ω5 are small in comparison with the lower degree terms which are all multiplied by c2A . (For the fast wave, this is not the case; the terms involving ω4 and ω5 play a role.) Thus, for the slow magnetoacoustic wave the behaviour of the dispersion relation is dominated by the cubic and lower order terms in ω. Rearranging our expressions slightly and dividing out a factor of k2 , we obtain (c2s + c2A )ω3 − iω2 kz2 (c2N + c2A )(γ − 1)νth − ωc2s c2A kz2 + ikz4 c2A c2N (γ − 1)νth = 0, 5 In fact, the inclusion of radiative losses and a heating source still gives a quintic dispersion relation, but of course its

coefficients are more complicated than given by relation (12.142).

396

Damping

which may be rewritten in the form



ω(ω2 − kz2 c2t ) − i(γ − 1)νth kz2

c2N + c2A



c2s + c2A

(ω2 − kz2 c2tN ) = 0,

(12.143)

where c2t =

c2s c2A c2s + c2A

,

c2tN =

c2N c2A c2N + c2A

.

We see here the occurrence of the sub-sonic, sub-Alfv´enic cusp (or tube) speed ct and its isothermal equivalent ctN (which is less than cN and cA ). Now it is evident that the dispersion relation (12.143) which holds when cA  cs is closely similar to the relation (12.118) which holds in the absence of a magnetic field. Moreover, if we proceed to the limit cA /cs → ∞, for which ct → cs and ctN → cN , then (12.143) reduces to ω(ω2 − kz2 c2s ) − i(γ − 1)νth kz2 (ω2 − kz2 c2N ) = 0.

(12.144)

Thus, we obtain dispersion relation (12.118) which applies in the absence of a magnetic field, but is now seen to apply also in the presence of a strong magnetic field. Hence, a study of the field-free case is justified for applications in the low β conditions of the solar corona.

13 Nonlinear Aspects

13.1 Introduction We turn now to a consideration of nonlinear effects. There is no surprise that nonlinear effects are often important in wave propagation in the Sun. This is likely to be especially so in the region round the photosphere, where pressure and density scale heights are relatively small (of order 102 km) and energy sources are abundant, with granules buffeting magnetic fields. In the corona, the larger scale heights (of order 105 km) mean that nonlinear effects are perhaps less important, but nonetheless significant. Perhaps not surprising, the description of nonlinear effects on magntohydrodynamics waves is more complicated that the acoustic counterpart. Here we concentrate on the behaviour of slow sausage and kink waves in magnetic flux tubes, drawing on the work of Roberts and Mangeney (1982), Roberts (1985a), Merzljakov and Ruderman (1985), Molotovshchikov and Ruderman (1987) and Ruderman (2013). Useful recent reviews include Ballai (2006), Erd´elyi and Fedun (2006b) and Ruderman (2006). Recently, we note that Mikhalyaev and Ruderman (2015) have derived the nonlinear Schr¨odinger equation for the fast sausage wave in a tube. We discuss also the combined effects of nonlinearity and damping, drawing in particular on the work of Edwin and Roberts (1986), Nakariakov and Roberts (1999) and Nakariakov et al. (2000). We do not discuss MHD shock waves in any detail; treatments of this topic may be found in, for example, Shercliff (1965), Jeffrey (1966), Cowling (1976), Walker (2005) and Priest (2014). Nonlinear effects are typically captured in a term of the form ∂uz ∂uz + uz , ∂t ∂z which is evidently present in the momentum equation but is also typical of other terms in the full nonlinear system. A term of this form may be contrasted with the typical term that arises in linear theories, namely ∂uz ∂uz + c0 , ∂t ∂z where c0 is a constant (with the dimensions of speed). The linear equation ∂uz ∂uz + c0 =0 ∂t ∂z

(13.1)

397

398

Nonlinear Aspects

has the general solution uz = f (z − c0 t) for arbitrary function f (satisfying necessary conditions of differentiability, so that the various derivatives may be formed). Such a solution remains bounded for all time t, provided it is bounded initially (at t = 0). Also, its derivatives with respect to z or t are bounded for all time t. Contrast the linear behaviour with that arising from the nonlinear equation ∂uz ∂uz + uz = 0. ∂t ∂z

(13.2)

This equation has a solution of similar form as for the linear case, namely the form uz = f (z − uz t); that is,   uz (z, t) = f z − uz (z, t)t , (13.3) again for arbitrary function f ; this gives uz = uz (z, t) implicitly. That the above implicit form is indeed a solution of (13.2) is readily verified directly, differentiating (13.3) partially with respect to t and z to give ∂uz f  (z − uz t) = −uz , ∂t 1 + tf  (z − uz t)

∂uz f  (z − uz t) = , ∂z 1 + tf  (z − uz t)

(13.4)

where f  (z) denotes the derivative, df /dz, of the function f (z). It is evident that these expressions satisfy equation (13.2). Moreover, it is also apparent that provided f  < 0 the first order derivatives ∂uz /∂t and ∂uz /∂z may become unbounded in a finite time: provided f  < 0, gradients grow and become arbitrarily steep. This steepening occurs in a time t given implicitly by t=

−1 , f  (z − uz t)

f  < 0.

(13.5)

The condition f  < 0 is typically met at the front of a propagating wave, and so it is typically the front of a propagating wave that steepens. An illustration of the effect is provided by the example

z z , 0 ≤ ≤ π. f (z) = u0 sin a a This corresponds to a sinusoidal profile of amplitude u0 , the profile extending over 0 ≤ z ≤ πa, for spatial scale a. In linear theory, according to equation (13.1) and its solution uz = f (z − c0 t) this shape would simply propagate to the right with speed c0 and without change of shape. However, in nonlinear theory according to equation (13.2) the shape steepens at its front (where f  < 0) as it propagates, the front becoming infinitely steep in a time t = a/u0 with   z − uz t u0 cos = −1. a The front of the wave is the first to steepen, occurring at the base (where uz = 0). The above example illustrates a general feature, namely that a wave profile steepens as it propagates and may eventually form an arbitrarily steep section in a finite time, provided f  < 0 at some location; the section where f  < 0 typically corresponds to the front of a wave. Accordingly, such a deformation of a wave front is to be expected in any

13.2 The Sausage Mode

399

nonlinear system, unless some physical effect serves to limit the steepening. Damping or dispersion may act in such a way as to limit steepening, and then the possibility arises of a balance between the various effects being achievable. In this chapter we consider the role of geometric dispersion, brought about by the finite width of a magnetic slab or tube. 13.2 The Sausage Mode The starting point for our discussion is the thin flux tube equations for the sausage mode, ignoring the effect of gravity (see Chapter 10). Longitudinal motions uz (z, t) in a thin flux tube or slab with plasma pressure p(z, t), density ρ(z, t) and cross-sectional area A(z, t) with longitudinal magnetic field Bz (z, t), all varying in time t and space z, are related by   ∂ ∂uz ∂uz ∂ ∂p ρ (ρA) + (ρAuz ) = 0, + uz =− , ∂t ∂z ∂t ∂z ∂z   B2 ∂p ∂ρ ∂p γ p ∂ρ p + z = πe , Bz A = 0 , + uz = + uz . (13.6) 2μ ∂t ∂z ρ ∂t ∂z The magnetic flux 0 is a constant. The term πe represents an external pressure, calculated on the boundary of the magnetic flux tube. The pressure πe (z, t) may vary in response to motions within the magnetic slab or tube or as a result of variations in the environment. When πe departs from the uniform equilibrium pressure pe the sausage mode suffers dispersion. In the treatment to be given here dispersion is due to the geometrical nature of the slab or tube – consisting of an inner medium (the slab or tube interior) and an unbounded surrounding environment (the external medium) – which gives rise to a term that may balance the tendency for growth that typically arises when nonlinear effects are included. An extended set of thin tube equations has been determined by Zhugzhda (1996) and used to obtain possible nonlinear behaviour (Zhugzhda and Nakariakov 1997; Zhugzhda and Goossens 2001; Zhugzhda 2004, 2005; see also Ruderman 2005), but we will not pursue this topic here. The set of equations (13.6) applies to both a magnetic slab (a Cartesian geometry), described by coordinates x, y, z, and to a magnetic tube (cylindrical geometry), described by cylindrical coordinates r, φ, z. For convenience, we will describe both situations as a tube, leaving aside the distinction between the two geometries to a specific discussion when the need arises. The equilibrium state is one of a uniform tube of strength B0 and radius a (crosssectional area A0 ), with uniform pressure p0 , density ρ0 , and a uniform external pressure pe . Equilibrium is achieved through a static (uz = 0) balance of the total (plasma plus magnetic) pressure, so that p0 +

B20 = pe . 2μ

(13.7)

As discussed in Chapter 10, the linear form of the thin tube equations (13.6) leads to the wave-like equation 2 ∂ 2 uz c2t 1 ∂ 2 πe1 2 ∂ uz − c = − , t ∂t2 ∂z2 c2A ρ0 ∂z∂t

(13.8)

400

Nonlinear Aspects

where πe1 denotes the pressure variations in the environment of the tube. In the absence of such variations, when the external pressure is taken to be simply the equilibrium value pe , the right-hand side of (13.8) is zero and we obtain the one-dimensional wave equation with a propagation speed ct , with 1 1 1 = 2+ 2. 2 c ct cA s

(13.9)

The wave propagates with the subsonic and sub-Alfv´enic speed ct , a combination of the sound speed cs and Alfv´en speed cA within the tube. We may eliminate explicit mention of the longitudinal magnetic field in the thin tube equations by combining flux conservation with pressure balance, so that   20 1 = πe , (13.10) p+ 2μ A2 where 0 = B0 A0 is the constant magnetic flux (determined by equilibrium conditions). Notice that 20 /(2μ) is ρ0 c2A A20 /2. Then the nonlinear thin tube equations may be written in the form ∂ ∂ (ρA) + (ρAuz ) = 0, ∂t ∂z 

∂uz ∂uz + uz ρ ∂t ∂z 1 p + ρ0 c2A 2





A20 A2

=−

(13.11)

∂p , ∂z

(13.12)

 = πe ,

(13.13)

  ∂p ∂ρ γ p ∂ρ ∂p + uz = + uz . ∂t ∂z ρ ∂t ∂z

(13.14)

13.3 Derivation of the Evolution Equation: The Interior We are interested in the behaviour of weakly nonlinear waves, which corresponds to taking the behaviour of our system of equations to one higher order than linear theory. To be specific, we write ρ = ρ0 + ρ1 +  2 ρ2 + · · ·,

p = p0 + p1 +  2 p2 + · · ·,

A = A0 + A1 +  2 A2 + · · ·,

πe = pe + πe1 +  2 πe2 + · · ·,

uz = uz1 +  2 uz2 + · · ·. (13.15)

Each of the terms ρ1 , ρ2 , p1 , p2 , πe1 , etc. are functions of z and t. The parameter  is a dimensionless measure of amplitude, and is considered to be small;  = 0 gives the equilibrium state, whereas carrying the calculation to order  (ignoring squares and higher powers of ) gives the linear theory, and retaining terms up to order  2 (ignoring cubes and higher powers of ) gives the weakly nonlinear system.

13.3 Derivation of the Evolution Equation: The Interior

401

We introduce also a scaling of the spatial variable z and the time t, setting Z = (z − ct t),

τ =  2 t.

(13.16)

In introducing Z we have taken a moving coordinate system, which travels in the direction of positive z with a speed ct ; this is purely for convenience, knowing that in the linear theory of the sausage mode the wave has a speed close to ct . We have also scaled the longitudinal coordinate z (more specifically z − ct t) by ; this is because in the linear theory we have the approximate result that ω = kz ct for small kz (corresponding specifically to kz a small). Furthermore, we have introduced a slow timescale τ , which captures temporal changes that are in addition to the basic movement of the wave with speed ct , already captured by the coordinate Z. Then, with the transformed variables Z and τ replacing the original variables z and t, we have the corresponding operator transformations ∂ ∂ = , ∂z ∂Z

∂ ∂ ∂ = −ct + 2 . ∂t ∂Z ∂τ

(13.17)

With the transformed coordinates (13.16) and the expansions (13.15), the system of equations (13.11)–(13.14) yields (on cancellation of any common powers of ) the following results. The continuity equation (13.11) in a tube becomes   ∂ (ρ0 A1 + ρ1 A0 ) + [ρ0 A2 + ρ1 A1 + ρ2 A0 ] + · · · −ct ∂Z   ∂ + (ρ0 A1 + ρ1 A0 ) +  2 [ρ0 A2 + ρ1 A1 + ρ2 A0 ] + · · · ∂τ   ∂ + ρ0 A0 uz1 + [ρ0 A0 uz2 + (ρ0 A1 + ρ1 A0 )uz1 ] + · · · = 0, ∂Z (13.18) while the momentum equation (13.12) reads   ∂ ∂ (ρ0 + ρ1 +  2 ρ2 + · ··) −ct (uz1 + uz2 + · ··) +  (uz1 + uz2 + · ··) ∂Z ∂τ   ∂ ∂ 1 + (ρ0 + ρ1 +  2 ρ2 + · · · ) (uz1 + uz2 + · ··)2 + · ·· + (p1 + p2 + · ··) = 0. ∂Z 2 ∂Z (13.19) Also, transverse pressure balance (13.13) (combined with flux conservation) yields     A21 1 A2 A1 2 2 + 3 2−2 (p0 + p1 +  p2 + · · · ) + ρ0 cA 1 − 2 +··· 2 A0 A0 A0 2

= pe + πe1 +  2 πe2 + · · · , on expanding A−2 binomially.

(13.20)

402

Nonlinear Aspects

Finally, the isentropic condition (13.14) gives   ∂ρ1 ∂ρ2 ∂ρ1 ∂ρ1 ∂ρ1 +  −γ p0 ct + γ p0 + γ p0 uz1 − γ p1 ct +··· − γ p0 ct ∂Z ∂Z ∂τ ∂Z ∂Z   ∂p1 ∂p2 ∂p1 ∂p1 ∂p1 +  −ρ0 ct + ρ0 + ρ0 uz1 − ρ1 ct + · · ·. (13.21) = −ρ0 ct ∂Z ∂Z ∂τ ∂Z ∂Z The system of equations (13.18)–(13.21) governs the behaviour in the tube.

13.3.1 Linear Order The above system of equations governing the perturbations about a uniform equilibrium confined within a flux tube is algebraically complicated. The uniformity of the equilibrium state, with constant plasma pressure p0 and density ρ0 , has already been allowed for in the derivation of the perturbation equations, though pressure balance (13.7) remains to be imposed. The lowest order terms in the amplitude parameter  now describe the linear theory and the next order terms, corresponding to  2 , give the weakly nonlinear order. Consider, then, the linear system of first order terms which yield the following system of linear equations: ct

∂ ∂uz1 ∂uz1 ∂p1 (ρ0 A1 + ρ1 A0 ) = ρ0 A0 , ρ0 ct = , ∂Z ∂Z ∂Z ∂Z A1 ∂ρ1 ∂p1 p1 − ρ0 c2A = πe1 , = c2s . A0 ∂Z ∂Z

(13.22)

We can solve equations (13.22) subject to p1 , ρ1 , A1 , uz1 → 0 as |Z| → ∞, corresponding to the requirement that disturbances are confined in Z, with the result πe1 = 0,

p1 = c2s ρ1 = ρ0 ct uz1 ,

A1 ct 1 = 2 uz1 = p1 . A0 cA ρ0 c2A

(13.23)

Thus, external pressure variations only arise at order  2 .

13.3.2 Nonlinear Order Consider now the next order of terms, corresponding to the weakly nonlinear order. The equation of continuity yields        ∂ A2 ρ1 A1 ρ2 ρ1 A1 ∂ ∂uz2 ∂ A1 ρ1 + + + + −ct + uz1 + = 0, (13.24) ∂τ A0 ρ0 ∂Z A0 ρ0 ∂Z ∂Z A0 ρ0 A0 ρ0 while the momentum equation gives ρ1 ∂uz1 1 ∂ 2 ∂ ∂uz1 + (uz1 ) − ct + ∂τ 2 ∂Z ρ0 ∂Z ∂Z Pressure balance implies



1 p2 − ct uz2 ρ0

  A21 1 A2 2 , p2 = πe2 − ρ0 cA 3 2 − 2 2 A0 A0

 = 0.

(13.25)

(13.26)

13.3 Derivation of the Evolution Equation: The Interior

403

and the condition for isentropic energy exchange becomes simply (γ − 1)ρ0

c2t ∂uz1 ∂ uz1 − [p2 − c2s ρ2 ] = 0. 2 ∂Z ∂Z cs

(13.27)

To analyse this algebraically complicated system of equations it is convenient to express A2 and ρ2 in terms of uz1 , p2 and πe2 . Rearranging the pressure balance condition and expressing all first order perturbations in terms of uz1 , we obtain A2 1 3c2 = (p2 − πe2 ) + 4t u2z1 . 2 A0 ρ0 cA 2cA

(13.28)

Similarly, with the aid of the isentropic condition we have 1 1 c2t 2 ρ2 = p − u . (γ − 1) 2 ρ0 2 ρ0 c2s c4s z1 Thus,

  ρ2 1 1 1 2 2 3 (γ − 1) A2 + =− π + p + u − c . e2 2 A0 ρ0 2 t z1 c4A c4s ρ0 c2A ρ0 c2t

(13.29)

(13.30)

Armed with these relations we may now express our weakly nonlinear equations purely in terms of uz1 , πe2 , p2 and uz2 , with continuity giving    ∂uz1 1 4 3 (γ − 1) c2t ∂πe2 c4t 1 ∂ ∂ 2 − − ) + + 1− c (u = (p2 − ρ0 ct uz2 ), t z1 2 4 4 2 2 ∂τ ∂Z ρ0 ∂Z cs cs cA 2 cA ρ0 cA ∂Z (13.31) and momentum giving   1 ∂ 1 c2t ∂uz1 ∂ 2 (13.32) + 1− 2 (uz1 ) = − (p2 − ρ0 ct uz2 ). ∂τ 2 ρ0 ∂Z cs ∂Z Finally, adding (13.31) and (13.32) eliminates the expression involving p2 − ρ0 ct uz2 , with the result that ∂uz1 c2 1 ∂ ∂uz1 + β0 uz1 + t2 (πe2 ) = 0, ∂τ ∂Z 2cA ρ0 ∂Z

(13.33)

where the coefficient of the nonlinear term is β0 = c2A

(γ + 1)c2A + 3c2s 2(c2s + c2A )2

.

(13.34)

Equation (13.33) is the main result of this section. Equations (13.33) and (13.34) were first derived in Roberts (1985a). However, an expression of the form of β0 seems first to have arisen in Adam (1975), in an investigation of the effect of viscosity on magnetoacoustic waves. It is interesting to note that β0 → 0 in the incompressible limit of cs → ∞. In the opposite extreme of cA  cs , β0 → (γ + 1)/2. Expressed in the original coordinates and replacing uz1 by uz and writing πe = pe +  2 πe2 , to the order we are working here the evolution equation for the longitudinal velocity becomes ∂uz ∂uz ∂uz c2t ∂ + ct + β0 uz + (πe ) = 0. ∂t ∂z ∂z 2ρ0 c2A ∂z

(13.35)

404

Nonlinear Aspects

To progress further it is necessary to examine the environment of the magnetic tube or slab. Our aim is to construct a solution in the environment that is bounded at infinity and then match that solution to the tube equation (13.33). Additionally, we have to match the normal component of the velocity across the tube boundary. 13.3.3 The Environment We need to determine the pressure variations that make up the term πe in the evolution equation (13.33). To do this, we suppose that the environment is field-free. Then the motions u(e) within the environment satisfy the gas dynamic equations: ∂ρ (e) ∂u(e) ρ (e) + div ρ (e) u(e) = 0, + ρ (e) (u(e) · grad)u(e) = −grad p(e) , ∂t ∂t   γ p(e) ∂ρ (e) ∂p(e) (e) (e) (e) (e) + u · grad p = (e) + u · grad ρ , (13.36) ∂t ∂t ρ where ρ (e) and p(e) denote the density and pressure in the environment. We assume that motions are as a consequence of the disturbances within the flux tube. Since we are discussing the sausage mode, motions are principally longitudinal and so we take the motions in the environment to be of order  2 , expanding variables thus u(e) =  2 u2 (e) · · · ,

(e)

p(e) = pe +  2 p2 + · · ·,

(e)

ρ (e) = ρe +  2 ρ2 + · · ·.

(13.37)

Here pe and ρe are the equilibrium plasma pressure and density in the environment. Furthermore, we transform our coordinates so that τ = 2t

and

Ze = z − ct t,

(13.38)

with the coordinate perpendicular (x in a slab geometry, r in a cylindrical geometry) to the tube axis z remaining unchanged (no scaling). Then, the operator ∂/∂t is dominated by the movement at speed ct , becoming −ct ∂/∂Ze (the correction of order  2 is not needed in the present calculation). Also, the inertial acceleration term (u(e) · grad)u(e) is negligible. Consequently, to leading order in  we have ct

∂ρ2(e) = ρe div u(e) 2 , ∂Ze

ρe ct

∂u2 (e) = grad p(e) 2 , ∂Ze

∂ρ (e) ∂p(e) 2 = c2se 2 , ∂Ze ∂Ze

(13.39)

where cse denotes the sound speed in the environment (c2se = γ pe /ρe ). Then the system (13.39) of equations applying in the environment reduces to (e)

c2t ∂ 2 p2 (e) = ∇ 2 p2 , c2se ∂Ze2

(13.40)

where ∇ 2 ≡ div grad denotes the Laplacian operator. Similarly, the z-component of motion u2 (e) satisfies equation (13.40). It may be noted that we may rewrite (13.40) as a form of Laplace’s equation in terms of a scaled Ze , but we will not follow this path. At this stage it is convenient to treat the two geometries, slab or tube, separately. For simplicity, we will also assume that the temperature inside the slab or tube is the same as

13.4 The Magnetic Slab: The Benjamin–Ono Equation

405

that in the environment, so that cs = cse . The question of other conditions pertaining is addressed in Section 13.8.

13.4 The Magnetic Slab: The Benjamin–Ono Equation 13.4.1 Derivation of the Evolution Equation With cse = cs and Xe = x, equation (13.40) gives (e) ∂ 2 p(e) c2t ∂ 2 p2 2 + = 0. ∂Xe2 c2A ∂Ze2

We can construct a solution of (13.41) in Fourier form, taking  ∞ = f (s)e−iscA Ze /ct e−|s|(Xe −a) ds, p(e) 2 −∞

(13.41)

X > a,

(13.42)

where we have rejected a solution that has an integrand growing exponentially fast as Xe → +∞. The Fourier pair of transforms we are using are (see, for example, Bracewell 2000)  ∞  ∞ 1 F(κ) = f (s)e−isκ ds, f (s) = F(κ)eisκ dκ. 2π −∞ −∞ On the slab boundary Xe = a (i.e., on x = a) we have    ∞ cA Ze (e) p2 (Xe = a) = f (s)e−iscA Ze /ct ds = F . ct −∞ (e)

Thus, p2 (Xe = a) = F(cA Ze /ct ) and so (e)

π2 = F(cA Ze /ct ).

(13.43)

Also, from equations (13.39) we have ρe ct

∂ (e) ∂ (e) ux2 = p ∂Ze ∂Xe 2   ∞ ∂ −iscA Ze /ct −|s|(Xe −a) = f (s)e e ds ∂Xe −∞  ∞ =− |s|f (s)e−iscA Ze /ct e−|s|(Xe −a) ds −∞   ∞ ct ∂ |s| −iscA Ze /ct −|s|(Xe −a) = e ds . f (s)e cA ∂Ze −∞ is

Thus, we may take (e) ux2 |Xe =a

1 = icA ρe



∞ −∞

sgn (s) · f (s)e−iscA Ze /ct ds,

where sgn denotes the sign function (defined in Chapter 8, equation (8.53)).

(13.44)

406

Nonlinear Aspects

Equation (13.44) involves a product of two Fourier transforms, allowing us to exploit the convolution theorem (see, for example, Bracewell 2000, p. 118) and write  ∞ 1 cA Ze (e) h(κ − s)F(s)ds, κ= , ux2 |Xe =a = icA ρe −∞ ct where h(t) =

1 2π



∞ −∞

sgn (s)e−ist ds

is proportional to the Fourier transform of sgn (s). The Fourier transform h(t) may be evaluated (Lighthill 1958; Table 1, p. 43):   −2i 1 · . h(t) = 2π t Finally, we obtain (see Roberts 1985a for further details) (e)

ux2 |Xe =a =

 1  H(uz ) , ρe cA

(e)

π2 = ρe a

 c2t ∂  H(uz ) . cA ∂Ze

(13.45)

Here H denotes the Hilbert transform1 (see Erd´elyi 1954; see also Weisstein; Poularikas 1999; Bracewell 2000) defined here by  1 ∞ uz (s, t) H(uz (z, t)) = − ds. (13.46) π −∞ s − z In this expression, t is simply playing the role of a parameter. The integrals arising here are taken as Cauchy principal value integrals (see, for example, * Whittaker and Watson 1969), denoted by a horizontal dash through the integration sign, −. Thus, equation (13.33) for the longitudinal velocity perturbation within the tube becomes    3  ∂uz1 ∂uz1 ∂2  1 ρe ct ct 2 H(uz ) = 0. (13.47) + β0 uz1 + a ∂τ ∂Z 2 ρ0 cA ∂Z In the original variables, this becomes

 ∂uz ∂uz 1 ∂ 2 ∞ uz (s, t) ∂uz − + ct + β0 uz +α ds = 0, ∂t ∂z ∂z π ∂z2 −∞ s − z

where 1 α= 2



ρe ρ0



ct cA

(13.48)

3 act .

(13.49)

Equation (13.48) is the Benjamino–Ono equation, a nonlinear integro-differential equation that first arose in the context of waves in fluids (Benjamin 1967; Ono 1975). See Ablowitz and Segur (1981) and Matsuno (1984) for a general discussion of the Benjamin– Ono equation and its properties. The occurrence of the Benjamin–Ono equation in solar 1 It should be noted that some authors use a definition of the Hilbert transform that differs from that used here and in Erd´elyi (1954); their definitions differ by a minus sign, which in effect implies that the inverse Hilbert transform H−1 is being used

to that employed here. In any case, H−1 = −H.

13.4 The Magnetic Slab: The Benjamin–Ono Equation

407

magnetohydrodynamics was first demonstrated by Roberts and Mangeney (1982), and further discussed in Roberts (1984). A detailed derivation was given in Roberts (1985a). All these analyses used the thin tube approximation for the sausage mode (Roberts and Webb 1978). A derivation based directly upon the full magnetohydrodynamics equations, without invoking the thin tube equations, was given by Merzljakov and Ruderman (1985), using multiple scales. The Benjamin–Ono equation may be rewritten in the form ∂uz ∂uz ∂2 ∂uz + ct + β0 uz + α 2 H(uz (z, t)) = 0 ∂t ∂z ∂z ∂z or alternatively

  ∂uz ∂uz ∂uz ∂ 2 uz (z, t) = 0. + ct + β0 uz + αH ∂t ∂z ∂z ∂z2

(13.50)

(13.51)

13.4.2 Recovery of Linear Theory The analysis leading to equations (13.48) or (13.50) is somewhat lengthy, so it is reassuring to note that when the nonlinear term is neglected these equations recover the dispersion relation for a sausage wave propagating in the positive z-direction. To see this, note that the linear form of equation (13.50) is  ∂uz ∂uz ∂2  + ct + α 2 H uz (z, t) = 0. ∂t ∂z ∂z This equation admits of a solution proportional to

(13.52)

eiωt−ikz z provided (iω − ikz ct )eiωt−ikz z + α(−ikz )2 eiωt H(e−ikz z ) = 0. Now, noting that the Hilbert transform of the exponential e−ikz z is (compare with Bracewell 2000, p. 365)  1 ∞ e−ikz s H(e−ikz z ) = − ds π −∞ s − z = −i sgn (kz ) e−ikz z ,

we obtain ω = kz ct − αkz2 sgn (kz ),

c = ct − α|kz |.

(13.53)

Thus, we have recovered the result obtained in Chapter 5, equation (5.90), where the dispersion relation is solved for c2 (= ω2 /kz2 ). 13.4.3 The Soliton Solution Returning to the Benjamin–Ono equation (13.48), we note that it possesses a solitary wave form that preserves itself following interactions with other solitary waves, and is in fact a

408

Nonlinear Aspects

soliton; see, for example, Drazin and Johnson (1989) for a general discussion of solitons. The single soliton solution of equation (13.48) is (Benjamin 1967; Ono 1975) u0 uz (z, t) = (13.54)  z−C t 2 , 1+ l where u0 is the value of uz at the moving point z = Ct on the solitary wave (uz = u0 at the moving location z = Ct). For equation (13.54) to provide a solution of the nonlinear equation (13.48) the speed C must be related to the spatial scale l and the amplitude u0 by 1 C = ct + β0 u0 , 4

l=

4α . u0 β0

(13.55)

Thus, despite the complexity of the governing equation (13.48) – as compared with, for example, the well-known Korteweg–de Vries equation (see Drazin and Johnson 1989) – it possesses a relatively simple solitary wave solution, (13.54). In general there are two cases arising: either α > 0 or α < 0. If α > 0, then u0 > 0 and so equation (13.55) shows that C > ct : the speed C of the nonlinear solitary wave is faster than the speed ct of a linear wave. From (13.23), we see that for α > 0, u0 > 0 the soliton solution represents a swelling (or expansion) of the magnetic slab (the perturbation A1 is positive) and the swelling travels along the slab with the speed C. For low amplitudes |u0 | the solitary wave propagates with almost the slab speed ct with an extended spatial scale along the slab (l is large), whereas for stronger amplitudes (larger |u0 |) the wave moves faster than ct and has a more concentrated (l is small) and pronounced bulge in the slab. For α < 0, we find that u0 < 0 (given l > 0) and now the solitary wave travels slower than the speed ct . Moreover, the slab has now contracted, no longer a swelling but a constriction travelling slower than ct . See Section 13.8. We can obtain the solitary wave form (13.54) as follows. We begin by looking for a solution of equation (13.48) in the form uz (z, t) = f (Zˆ e ),

Zˆ e = z − Ct,

(13.56)

where the speed C and the profile shape f are to be found. Then df ∂uz , = −C ∂t dZˆ e

∂uz df . = ∂z dZˆ e

Substituting these expressions into (13.48) and integrating once with respect to Zˆ e , assuming that f and its first derivative are zero at infinity (|Zˆ e | → ∞), we obtain  1 d  (ct − C)f + β0 f 2 + α H(f ) = 0. (13.57) 2 dZˆ e We can demonstrate that the form f =

f0 Zˆ e2

+ κ2

(13.58)

provides a solution of (13.57), for an appropriate choice of the constants f0 and κ. First notice that, from the properties of the Hilbert transform (Erd´elyi 1954, p. 245), we have (for real κ)   z 1 =− . H 2 s + κ2 |κ|(z2 + κ 2 )

13.5 The Magnetic Slab: Whitham’s Evolution Equation

Accordingly, we obtain f02 1 αf0 d + β0 − (ct − C) 2 2 2 2 2 2 (Zˆ e + κ ) |κ| dZˆ e Zˆ e + κ f0



Zˆ e Zˆ e2 + κ 2

409

 = 0.

(13.59)

Aside from the trivial solution f0 = 0, this yields 1 α 2 (ct − C)(Zˆ e2 + κ 2 ) + β0 f0 − (κ − Zˆ e2 ) = 0, 2 |κ|

(13.60)

from which it follows that ct − C +

1 (ct − C)κ 2 + β0 f0 − α|κ| = 0. 2

α = 0, |κ|

Hence, |κ| =

β0 f0 , 4α

C = ct +

4α 2 . β0 f0

(13.61)

These expressions may be rearranged into the forms given in (13.54) and (13.55). If α < 0, then so also is f0 and u0 and then C < ct . However, if α > 0 then f0 > 0 and u0 > 0, giving C > ct .

13.5 The Magnetic Slab: Whitham’s Evolution Equation The recovery of the dispersion relation for the slow wave in a magnetic slab makes clear that the integro-differential term that arises in the Benjamin–Ono equation is simply a representation of the dispersive effect on the wave. In Fourier space, that dispersive effect is represented by a term proportional to kz |kz |, whereas in the space of z and t it is the term involving the Hilbert transform of uz . Whitham (1967; 1974, chap. 11) has pointed out, in a general context, that the linear evolution equation representing the dispersion relation ω = kz c(kz ) may be encapsulated by the integro-differential equation  ∞ ∂uz ∂uz + (s, t)C(z − s)ds = 0, (13.62) ∂t −∞ ∂s where 1 C(z) = 2π



∞ −∞

 ikz z

c(kz )e

dkz ,

c(kz ) =

∞

−∞

C(z)e−ikz z dz.

Here c = ω/kz ; c(kz ) is the Fourier transform of C(z), which is the inverse Fourier transform of c. To see how this works out for a magnetic slab, consider the dispersion relation (13.53) for which  ∞ 1 (ct − α|kz |)eikz z dkz C(z) = 2π −∞  ∞ α = ct δ(z) − |kz |eikz z dkz . 2π −∞

410

Nonlinear Aspects

We have evaluated the term proportional to ct by use of an integral representation of the delta function δ:  ∞ 1 δ(z) = eikz z dkz . 2π −∞ Evaluating the remaining integral (Lighthill 1958, Table I, p. 43) gives  ∞ 1 1 |kz |eikz z dkz = − 2 . 2π −∞ πz Hence, C(z) = ct δ(z) +

α . π z2

Then the evolution equation (13.62) yields  ∂uz ∂uz 1 ∂uz α ∞ + ct + − (s, t)ds = 0. ∂t ∂z π −∞ (z − s)2 ∂s Carrying out an integration by parts, we obtain  ∂uz ∂uz 1 ∂ 2 uz α ∞ (s, t)ds = 0. + ct + − ∂t ∂z π −∞ (s − z) ∂s2 Finally, noting that the Hilbert transform of a derivative is the derivative of the transform, that is (Erd´elyi 1954, p. 245)   d   duz = H uz , H ds dz we conclude that

  ∞ ∂uz ∂uz 1 α ∂2 − (s, t)ds = 0, + ct + u z ∂t ∂z π ∂z2 −∞ (s − z)

(13.63)

where 1 α= 2



ρe ρ0



ct cA

3 act .

Thus, we have obtained the linearized form (13.52) of the Benjamin–Ono equation.

13.6 The Magnetic Flux Tube 13.6.1 Whitham’s Evolution Equation for a Magnetic Tube In the case of a cylindrical flux tube it is clear that an analysis much as given for the magnetic slab applies to a magnetic flux tube. Linear theory leads to the approximate dispersion relation (see Chapter 6, Section 4) c = ct − 2α  kz2

K0 (λ|kz |a) , λ|kz |aK1 (λ|kz |a)

(13.64)

13.6 The Magnetic Flux Tube

where 1 α = 8 



ρe ρ0



 2

a ct

ct cA

4 λ=

,

411

ct . cA

(13.65)

Relation (13.64) applies for kz a  1, and in giving the form of the constant λ we have supposed, for simplicity, that the sound speed cse in the environment is equal to the sound speed cs within the tube (so that cse = cs ). We can simplify relation (13.64) a little further if we note that (Abramowitz and Stegun 1965) K1 (x) ∼ 1/x for small x. Then (13.64) can be simplified to c = ct − 2α  kz2 K0 (λ|kz |a),

(13.66)

the coefficients λ and α  remaining the same. What form does Whitham’s evolution equation (13.62) now take? Write the dispersion relation as c = ct − 2α  kz2 g(kz ).

(13.67)

In this form we can treat both relations (13.64) and (13.66) in the one formulation, noting that we have the two cases K0 (λ|kz |a) or g(kz ) = . g(kz ) = K0 (λ|kz |a) λ|kz |aK1 (λ|kz |a) For either case, g(kz ) is an even function of kz : g(−kz ) = g(kz ). Then, whichever case for g(kz ) we consider, we have  ∞ 1 c(kz )eikz z dkz C(z) = 2π −∞  ∞  1 α ∞ 2 ikz z = ct e dkz − k g(kz )eikz z dkz . 2π −∞ π −∞ z Thus, C(z) = ct δ(z) −

α π



C(z − s) = ct δ(z − s) −

∞

kz2 g(kz )eikz z dkz , −∞  α ∞ 2 k g(kz )eikz (z−s) dkz , π −∞ z

where we have used the integral definition of the delta function δ(z). Then Whitham’s equation (13.62) becomes   ∂uz ∂uz α  ∞ ∂uz (s, t) ∞ 2 kz g(kz )eikz (z−s) dkz ds = 0. + ct + ∂t ∂z π −∞ ∂s −∞ Consider the s-integral and carry out an integration by parts:  ∞ ∂uz (s, t) 2 kz g(kz )eikz (z−s) ds ∂s −∞  ∞ = kz2 g(kz )eikz (z−s) uz (s, t)|∞ + ikz3 g(kz )eikz (z−s) uz (s, t)ds s=−∞ ∂3 =− 3 ∂z



∞ −∞

−∞

g(kz )eikz (z−s) uz (s, t)ds.

412

Nonlinear Aspects

We have assumed that uz (s, t) → 0 as |s| → ∞ and noted that ∂ 3 ikz (z−s) (e ) = −ikz3 eikz (z−s) . ∂z3 Then Whitham’s equation becomes ∂uz ∂uz α ∂ 3 + ct + ∂t ∂z π ∂z3 Now  ∞  ikz (z−s) g(kz )e dkz = −∞

∞ −∞





∞ −∞

uz (s, t)



∞

−∞

g(kz )eikz (z−s) dkz ds = 0.



g(kz ) cos kz (z − s) dkz + i



∞

−∞

  g(kz ) sin kz (z − s) dkz .

Breaking these integrals into separate contributions from −∞ to 0 and from 0 to ∞, and observing that g(kz ) is an even function of kz , we find that the imaginary contribution cancels out and the real contributions combine, so that  ∞  ∞   ikz (z−s) g(kz )e dkz = 2 g(kz ) cos kz (z − s) dkz . −∞

0

It is convenient to introduce a function fg (z) defined as  2 ∞ fg (z) = g(kz ) cos(kz z)dkz . π 0 Then



∞ −∞

(13.68)

g(kz )eikz (z−s) dkz = π fg (z − s),

and Whitham’s equation reads ∂uz ∂uz ∂3 + ct + α 3 ∂t ∂z ∂z



∞

−∞

uz (s, t)fg (z − s)ds = 0.

(13.69)

Equation (13.69) is the general form of the integro-differential equation that applies for any dispersion relation that can be written in the form (13.67) with g(kz ) an even function of kz . In the case of the approximate dispersion relation (13.66), we have g(kz ) = K0 (aλ|kz |) and so fg (z) =

2 π



∞

 K0 (aλkz ) cos kz z)dkz .

0

(The modulus sign is no longer needed on kz because in the integral expression kz ranges from 0 to +∞.) This integral may be evaluated. First change the integration variable from kz to κ = λakz , the limits remaining as 0 to ∞ (given aλ > 0). Then  ∞

z  2 K0 (κ) cos dκ fg (z) = πaλ 0 aλκ 1 = 1/2 . 2 2 a λ + z2

13.7 The Magnetic Tube: The Leibovich–Roberts Equation

413

Here we have made use of Basset’s integral (see Watson 1995, p. 388):    2 ∞ 1 K0 (κ) cos zκ dκ =  1/2 . π 0 1 + z2 Hence, finally we obtain ∂uz ∂uz ∂3 + ct + α 3 ∂t ∂z ∂z



∞

−∞

uz (s, t) ds = 0. [λ2 a2 + (z − s)2 ]1/2

(13.70)

This equation expresses in z and t space what the dispersion relation (13.66) expresses in kz and ω Fourier space. This result was first obtained by Roberts (1985a). In the case K0 (aλ|kz |) g(kz ) = aλ|kz |K1 (aλ|kz |) an analytical evaluation of the integral defining the function fg (z) is not possible. Instead, following Molotovshchikov and Ruderman (1987), Ruderman (2006) and Barbulescu and Erd´elyi (2016), we can write our results in a compact form if we define a function Y(z) as the integral  2 ∞ K0 (s) cos(sz)ds. (13.71) Y(z) = π 0 sK1 (s) Then 2 fg (z) = π λa

 0

∞

sz  K0 (s) 1 z  cos ds = Y , sK1 (s) aλ λa λa

and Whitham’s equation (13.69) becomes    ∞ ∂uz ∂uz z−s α ∂ 3 u (s, t)Y + ct + ds = 0. z ∂t ∂z λa ∂z3 −∞ λa

(13.72)

13.7 The Magnetic Tube: The Leibovich–Roberts Equation The linear equation (13.70) for a sausage mode in a thin magnetic flux tube was first obtained in Roberts (1985a). What is the nonlinear form of this equation? In the case of a cylindrical flux tube it is clear that an analysis much as given earlier for the magnetic slab applies to the flux tube. The main difference lies in the dispersive behaviour of the wave, and this is encapsulated by the linear theory. The nonlinear term must be the same in a thin flux tube as in a thin magnetic slab. Hence we can conclude that in a thin isolated magnetic flux tube the slow magnetoacoustic sausage mode propagates according to 3  ∞ ∂uz ∂uz ∂uz uz (s, t)  ∂ ds = 0, (13.73) + ct + β0 uz +α 3 2 2 ∂t ∂z ∂z ∂z −∞ [λ a + (z − s)2 ]1/2 where the coefficients α  , β0 and λ are given by    4 (γ + 1)c2A + 3c2s 1 ρe ct  2 α = , β0 = c2A , a ct 8 ρ0 cA 2(c2s + c2A )2

λ=

ct . cA

(13.74)

414

Nonlinear Aspects

The parameters α  and λ are given here for the case of a field-free environment with sound speed cse equal to the sound speed cs within the tube, so that cse = cs . Notice also that λ < 1. Equation (13.73) was first derived as a description of the slow magnetoacoustic wave in a thin flux tube by Roberts (1985a; see also Roberts 1984), who obtained the coefficients α  , β0 and λ.2 The derivation in Roberts (1985a) rests upon the thin flux tube equations for the sausage mode (Roberts and Webb 1978). An equation of similar form arose earlier, though in a quite different context (namely, that of a non-magnetic, incompressible rotating fluid), and was obtained by Leibovich (1970). Accordingly, equation (13.73) is sometimes referred to as the Leibovich–Roberts equation (see, for example, Bogdan and Lerche 1988; Zhugzhda 2000, 2004; Zhugzhda and Goossens 2001; Ballai 2006; Erd´elyi and Fedun 2006b; Ruderman 2006; Barbulescu and Erd´elyi 2016). A derivation of equation (13.73) directly from the full magnetohydrodynamic equations was given by Molotovshchikov and Ruderman (1987), using a multiple scaling analysis. It is evident from the analyses leading to the two nonlinear descriptions of a slow magnetoacoustic wave in a magnetic slab or cylinder, resulting in the Benjamin–Ono equation for a slab and the Leibovich–Roberts equation for a cylindrical flux tube, that these two nonlinear equations are of similar kind. However, whereas the Benjamin–Ono equation has a known solitary wave solution – see Section 13.4 – the equivalent form for the Leibovich– Roberts equation has not so far been discovered. Numerical solutions have been presented (Molotovshchikov and Ruderman 1987; Weisshaar 1989; Erd´elyi and Fedun 2006b), in general showing features of nonlinear wave propagation that are similar to those known for the Benjamin–Ono equation. Nonetheless, an analytical solution similar to the solitary wave (13.54) of the Benjamin–Ono equation has not been obtained. Turning to the dispersion relation with both K0 and K1 Bessel functions, for a fieldfree environment with the sound speed in the tube and the environment being the same (cse = cs ), the nonlinear form of (13.72) follows immediately:    ∞ ∂uz ∂uz ∂uz z−s α ∂ 3 u (s, t)Y + ct + β0 uz + ds = 0. (13.75) z ∂t ∂z ∂z λa ∂z3 −∞ λa This result was first derived by Molotovshchikov and Ruderman (1987), who obtained their equation from a multiple scales analysis of the magnetohydrodynamic equations, without use of the thin flux tube approximation. Molotovshchikov and Ruderman (1987) and later Erd´elyi and Fedun (2006b) investigated the nonlinear equation numerically. 13.8 Effect of an External Magnetic Field (cAe = 0) and cse = cs In all of the above analysis we have chosen to present results for the specific case of a field-free environment (cAe = 0) with a sound speed in the environment that is equal to the sound speed within the slab or tube (cse = cs ). These restrictions are made for

2 Note the slight difference of notation with regard to the coefficient λ used here and that used in Roberts (1985a). Here λ is

defined to be a dimensionless ratio of two speeds, and the tube radius a arises explicitly in the final nonlinear equation. In Roberts (1985a), the tube radius was absorbed into the definition of λ.

13.8 Effect of an External Magnetic Field (cAe = 0) and cse = cs

415

simplicity. However, much the same analysis applies even if cAe = 0 and cse = cs , though some restrictions arise because of the need to maintain the external wavenumber me as positive. In fact, we can deduce the form of the final governing nonlinear equation simply by comparison with the appropriate linear theory of the corresponding waves (as described in Chapters 5 and 6). 13.8.1 Magnetic Slab To see how this works, consider first a magnetic slab and note the dispersion relation for the sausage mode as given in equation (5.76) of Chapter 5, namely, 2 −1 ρe c2Ae m−1 e + ρ0 cA m0 coth(m0 a)

c2 =

−1 ρe m−1 e + ρ0 m0 coth(m0 a)

.

(13.76)

Now we are interested in the form of this relation when |kz a|  1. Assuming that m0 → 0 as kz a → 0, we note that tanh(m0 a) ≈ m0 a and so coth(m0 a) ≈ 1/(m0 a). Then the slab dispersion relation (13.76) for the sausage mode becomes c2 =

2 ρ0 c2A + ρe c2Ae m−1 e m0 a 2 ρ0 + ρe m−1 e m0 a

Noting that we may write m20 in the form   1 m20 = kz2 μˆ 2 2 , ct − c2

μˆ 2 =

.

(13.77)

(c2s − c2 )(c2A − c2 ) (c2s + c2A )

,

we recast relation (13.77) as 2 2 2 2 ˆ kz a. ρ0 (c2t − c2 )(c2 − c2A ) = ρe m−1 e (cAe − c )μ

It may be noted that c2 → c2t as kz a → 0 for the sausage slow mode. Then, writing c2 = c2t (1 + A) for small A, it readily follows that      2 ct − c2Ae ρe c2s − c2t kz A=− kz a. 2 2 2 ρ0 m ct cs + cA e The external wavenumber me is calculated with c2 = c2t and we require that me /kz > 0. If we write c2 in the form c2 = c2t − 2αct |kz |, then we deduce that 1 α= 2



ρe ρ0



c2t − c2Ae c2t



c2t c2A

2 

kz me

 act .

(13.78)

Expression (13.78) gives the general form of the coefficient α, applying whether there is an external magnetic field or not and also whether the sound speed in the environment differs

416

Nonlinear Aspects

from the sound speed in the slab interior or not. We require me /kz > 0. Notice that α > 0 if c2Ae < c2t , whereas α < 0 if c2Ae > c2t . In the special case of a field-free environment, cAe = 0, the coefficient α reduces to 1/2    4  1 ρe ct c2se α= act . (13.79) 2 ρ0 cA c2se − c2t If also the sound speeds inside and outside the slab are equal and the environment is fieldfree, so cse = cs and cAe = 0, then    3 1 ρe ct α= act , (13.80) 2 ρ0 cA which is the form of the coefficient given earlier (in equation (13.49)). The forms (13.79) and (13.80) for α are suitable for photospheric situations. However, for coronal conditions we return to the general expression (13.78) and consider its reduction for the case of a low β-plasma, when cA , cAe  cs , cse . In such a situation, the expression for α reduces to (on use of the equilibrium constraint that ρc2A ≈ ρe c2Ae )   1 c2s (13.81) act . α=− 2 c2A Thus, under coronal conditions α < 0. Whichever form the coefficient α takes, the Benjamin–Ono equation (13.48) governs the weakly nonlinear behaviour of the sausage mode.

13.8.2 Magnetic Tube We can consider the case of a flux tube in much the same way as a slab. To begin with, note the dispersion relation for the sausage mode in a flux tube when kz a  1 is given by equation (6.56) of Chapter 6, from which we have       2 cAe − c2t 1 ρe ct 4 c ∼ ct 1 + (kz a)2 K0 (λt |kz |a) , (13.82) 4 ρ0 cA c2t where λ2t =

(c2se − c2t )(c2Ae − c2t ) (c2se + c2Ae )(c2te − c2t )

.

(13.83)

We can write the approximate relation (13.82) in the form of (13.66), obtaining c = ct − 2α  kz2 K0 (λt |kz |a), where now the coefficient α  is given by     2 cAe − c2t 1 ρe c4t  α =− a2 ct . 8 ρ0 c2t c4A

(13.84)

(13.85)

13.9 Dissipation in a Magnetic Slab: The Benjamin–Ono–Burgers Equation

417

It then follows immediately that the previous analysis of this form of dispersion relation holds, and thus we obtain the nonlinear equation (13.73) in the form  ∞ ∂uz ∂uz ∂uz ∂3 uz (s, t) ds = 0, (13.86) + ct + β0 uz + α 3 ∂t ∂z ∂z ∂z −∞ [λ2t a2 + (z − s)2 ]1/2 where now the coefficients λt and α  are given by (13.83) and (13.85). The coefficient β0 remains unchanged, as given in (13.74). The expressions λt and α  hold generally, whether the tube’s environment is magnetic or otherwise and whatever the external sound speed cse (see also Zhugzhda and Goossens 2001). Of course, in the special case of a thin flux tube in a field-free environment, for which cAe = 0 and cse = cs , we then have λt = cs /cA and (13.85) reduces to the form of α  given earlier in equation (13.65). Similarly, it follows too that the form of the nonlinear equation discussed by Molotovshchikov and Ruderman (1987) remains appropriate for a tube with a uniform external field cAe = 0 and a temperature difference (cse possibly different from cs ) between the tube and the environment. Thus,    ∞ ∂uz ∂uz z−s α ∂ 3 u (s, t)Y + ct + ds = 0. (13.87) z ∂t ∂z λt a ∂z3 −∞ λt a In fact, the derivation by multiple scales provided by Molotovshchikov and Ruderman (1987) specifically allowed for cAe = 0 and cse distinct from cs .

13.9 Dissipation in a Magnetic Slab: The Benjamin–Ono–Burgers Equation The effect of dissipation on the nonlinear wave propagation of the sausage mode in a magnetic slab was first considered by Edwin and Roberts (1986), and leads to the Benjamin– Ono–Burgers equation. Edwin and Roberts included the effect of both thermal conduction and radiative losses. Such dissipative processes as thermal conduction and viscosity were discussed in Chapter 12. For the case of thermal conduction acting solely along the magnetic field, Edwin and Roberts (1986) obtained a nonlinear equation of the form3  ∂uz ∂uz ∂uz ∂ 2 uz 1 ∂ 2 ∞ uz (s, t) − (13.88) + ct + β0 uz +α ds = μ0 2 , 2 ∂t ∂z ∂z π ∂z −∞ s − z ∂z where μ0 =

c2t (γ − 1)2 νth 2γ c2s

(13.89)

with νth being the thermal conductivity coefficient introduced in Chapter 12. The presence of a second derivative term is of course not unexpected; accordingly, a formal derivation from the thin tube equations will not be given here. Precisely the same form of equation is expected for a flux tube, of course, though no formal demonstration of that outcome is available. Nakariakov and Roberts (1999)

3 Edwin and Roberts (1986) started from a slightly different form of the energy equation than used here; their thermal

conductivity (denoted by λ) is in fact equivalent to κ used here and in Chapter 12.

418

Nonlinear Aspects

discuss the form of the evolution equation when thermal conduction and radiative effects are included. Thus, in the presence of thermal conduction and nonlinearity in a magnetic slab we obtain a combination of the Benjamin–Ono equation and a term representative of dissipation. Weakly nonlinear wave propagation in the presence of weak dissipation typically leads to Burgers’ equation, the canonical form of which is (see, for example, Whitham 1974; Billingham and King 2000) ∂uz ∂uz ∂ 2 uz + β0 uz = ν1 2 , ∂t ∂z ∂z

(13.90)

for suitable nonlinear coefficient β0 and dissipative coefficient ν1 . Burgers’ equation is named after J. M. Burgers, who proposed such an equation as a possible means of modelling shock formation and turbulence (Burgers 1948); see also Whitham (1974) for a discussion of the equation. Burgers’ equation provides one of the simplest forms that incorporate both nonlinearity and diffusion. One useful feature is that the initial value problem for Burgers’ equation can be solved in explicit form. This is made possible through the Cole–Hopf transformation which relates Burgers’ equation to the linear diffusion equation (Whitham 1974, chap. 4). We discuss Burgers’ equation further in Section 13.10. It is not possible to solve equation (13.88) exactly but Edwin and Roberts (1986) obtained an approximate solution for weak damping, following a procedure given earlier for the Korteweg–de Vries equation by Ott and Sudan (1970). In particular, for thermal conduction along the magnetic field, Edwin and Roberts (1986; see also Nakariakov and Roberts 1999) showed that the soliton would decay slowly in amplitude N(t) according to −1/2  β02 N02 N0 μ0 t = (13.91) N(t) = N0 1 + t 1/2 , 8α 2 (1 + τsol ) where N0 = N(t = 0) is the initial amplitude of the soliton and μ0 is defined in equation (13.89), with α and β0 being defined earlier. The timescale τsol is  2 (13.92) τsol = (8/μ0 ) α/(β0 N0 ) . Relation (13.91) is derived assuming μ0  act . Thermal conduction causes a slow decrease in the amplitude N(t) of the nonlinear solitary wave, and moreover the wave gradually slows down and lengthens in shape as its amplitude falls. In the limit of no damping, νth → 0 and then μ0 → 0, and so τsol tends to infinity. Note too that damping tends to be slight (τsol large) if the initial amplitude N0 is small. Other effects, such as radiative losses, may also be discussed (Edwin and Roberts 1986; Nakariakov and Roberts 1999; Kumar, Nakariakov and Moon 2016). As an illustration of solitary wave decay brought about by thermal conduction, we take a coronal slab of width 2a = 2000 km (so a = 106 m) and an Alfv´en speed cA related to the sound speed cs by c2A = 10 c2s , for a coronal sound speed cs = 1.5 × 105 m s−1 . Then, with γ = 5/3 and a density ratio of ρe /ρ0 = 1/4, the solitary wave decays on a timescale τsol = 16.39 s, for an initial amplitude of N0 = 10 km s−1 and a thermal coefficient νth = 6.52 × 1011 m2 s−1 (corresponding to a temperature of 106 K and particle number density of 1015 m−3 ). The corresponding propagation distance ct τsol is of order

13.10 Dissipative Effects: Sound Waves and Burgers’ Equation

419

2.3 × 106 m: the solitary wave would propagate a distance of 2300 km before feeling any appreciable decay brought about by thermal conduction. This is very short, indicating that under coronal conditions the soliton is likely to be dissipated shortly after being generated. In the photosphere, thermal conduction has very little effect and the solitary wave propagates relatively unscathed, though of course stratification (here ignored) may have a bearing and other dissipative effects may operate. Edwin and Roberts (1986) suggest that radiative effects in the photospheric–chromospheric region may dampen the wave in a time of 1750 s, which is sufficiently long that a solitary wave might well propagate through the region without suffering significant damping. Some evidence from high resolution observations of the lower solar atmosphere is discussed in Zaqarashvili, Kukhianidze and Khodachenko (2010). Further discussion of such damping is given in Nakariakov and Roberts (1999), who also explore the phenomenon of autosolitons which may arise under certain conditions. 13.10 Dissipative Effects: Sound Waves and Burgers’ Equation 13.10.1 Formulation We turn now to a discussion of longitudinally propagating slow magnetoacoustic waves. We consider the case of a field-free medium, so are in fact examining sound waves. However, as noted in Chapter 12, the case of propagation in the presence of a strong magnetic field is equivalent to the case of no magnetic field. Thus, although we specifically consider B0 = 0, our discussion is relevant for the solar corona where cA  cs . We show how for weakly nonlinear waves a description in terms of Burgers’ equation again arises. Our treatment follows quite closely the discussion in Nakariakov et al. (2000), where Burgers’ equation is derived and applied to propagating slow waves in coronal loops. Similar techniques were used in Nakariakov and Oraevsky (1995) in a discussion of fast magnetoacoustic waves. A derivation of Burgers’ equation for standing waves in coronal loops is given in Ruderman (2013), and followed up in an investigation by Kumar, Nakariakov and Moon (2016) which included the effect of radiative terms. The general set of nonlinear equations, with magnetic field set to zero, that we examine here may be written in the form ∂u + div ρu = 0, ∂t   ∂u + (u · grad)u = −grad p − gρ ez + Fvisc , ρ ∂t ∂p γp + u · grad p − ∂t ρ



∂ρ + u · grad ρ ∂t

(13.93)

(13.94)

 = (γ − 1)div (κ ∇ T),

(13.95)

kB ρT. (13.96) m ˆ As usual, ρ denotes the plasma density, T the temperature, p the plasma pressure and the motion is u. The viscous force is Fvisc and κ is the coefficient of thermal conduction p=

420

Nonlinear Aspects

(taken to act solely along the z-direction, representative of a magnetic field direction). We have included the effect of gravity; for ease of comparison with the literature in this area, we have taken the z-axis to point upwards (opposite to gravity). Consider then the general system of nonlinear equations (13.93)–(13.96), with motions u assumed to be purely in the z-direction and varying with z and t alone: u = uz (z, t)ez . The continuity and momentum equations for the fluid follow from equations (13.93) and (13.94) as simply     ∂ρ ∂uz ∂uz ∂ ∂p ∂ 4 ∂uz ρ + (ρuz ) = 0, + uz =− − gρ + ρν , (13.97) ∂t ∂z ∂t ∂z ∂z ∂z 3 ∂z where ν denotes the coefficient of kinematic viscosity. For the one-dimensional motion u = uz (z, t)ez the energy equation (13.95) now takes the form     ∂p ∂p γ p ∂ρ ∂ρ ∂T ∂ + uz − + uz = (γ − 1) κ , (13.98) ∂t ∂z ρ ∂t ∂z ∂z ∂z where the coefficient of thermal conduction is κ = κ0 T 5/2 ; thus, thermal conduction is taken to act purely along the z-axis. See Chapter 12 for a discussion of damping effects. Note that equation (13.98) may be combined with the equation of continuity (the first equation in (13.97)) to give   ∂p ∂p ∂T ∂uz ∂ + uz + γp = (γ − 1) κ . (13.99) ∂t ∂z ∂z ∂z ∂z This form is sometimes more convenient for analysis. Finally, we note the equation of state (13.96).

13.10.2 Nonlinear Perturbation Equations (g = 0) To investigate the system of equations (13.96), (13.97) and (13.99), for the sake of clarity it is convenient to begin with the case g = 0, setting the gravitational acceleration to zero; we return to the case g = 0 in Sub-section 13.10.5. Nakariakov et al. (2000) treated the stratified problem from the outset. Consider, then, an equilibrium in which the plasma pressure is p0 , the plasma density is ρ0 , and the temperature is T0 . For this to be an equilibrium state in system (13.96)–(13.99) we require, from the momentum equation, that in the absence of gravity p0  = 0,

(13.100)

where a prime denotes the derivative of an equilibrium quantity with respect to z. Thus, p0 is a constant. Also, we assume that the medium is isothermal (T0 = constant), which requires that ρ0 is a constant also. The equilibrium state (13.100) changes if gravity is included, though again we may consider the medium to be isothermal (constant T0 ).

13.10 Dissipative Effects: Sound Waves and Burgers’ Equation

421

Denote by p(1) (z, t) the perturbation in the pressure, so that the total fluid pressure is p = p0 + p(1) (z, t).

(13.101)

Similarly, the total fluid density ρ (equal to the equilibrium value ρ0 plus the perturbation ρ (1) ) and temperature T (equal to the equilibrium value T0 plus the perturbation T (1) ) are ρ = ρ0 + ρ (1) (z, t),

T = T0 + T (1) (z, t).

(13.102)

The motion is simply denoted as uz (z, t). At this stage of our investigation the perturbations uz (z, t), p(1) , ρ (1) and T (1) may be large or small; they are simply the departures from the equilibrium state p0 , ρ0 , T0 . Then the nonlinear equations describing the perturbations about a uniform equilibrium follow from (13.96)–(13.99) as: ∂ρ (1) ∂ + (ρ0 uz ) = N1 , ∂t ∂z

ρ0

∂uz ∂p(1) + = N2 + D ν , ∂t ∂z

∂p(1) ∂uz + γ p0 = N 3 + Dκ , ∂t ∂z

(13.103)

where N1 = −

∂ (1) (ρ uz ), ∂z

N2 = −ρ (1)

N3 = −uz

∂uz ∂uz ∂uz − ρ0 uz − ρ (1) uz , ∂t ∂z ∂z

∂p(1) ∂uz − γ p(1) ∂z ∂z

(13.104)

are the nonlinear terms arising in our equations that are not directly related to the viscous or thermal damping terms.4 In writing the expression for N3 we have used the form (13.99) for the energy equation; this simplifies the expression somewhat. Also, in writing the energy equation in (13.103) we have noted that the medium is uniform (p0  = 0). The terms arising directly from viscous and thermal damping are Dν =

4 ∂ 2 uz ρν 2 , 3 ∂z

Dκ = (γ − 1)κ

∂ 2 T (1) . ∂z2

(13.105)

In writing these terms we have treated the coefficients ν and κ as constants, though they are evaluated as a function of the equilibrium temperature T0 ; this is a common assumption made for simplicity, and unlikely to have any significant bearing on the final results. We consider this system of nonlinear equations in the weakly nonlinear limit. That is, we treat uz to be of order , where the parameter  is a measure of the amplitude of the perturbation and is taken to be small (  1). All other dependent quantities are taken to be expanded in power series in . Perturbations of order  are considered to be linear, 4 We have adopted a different notation from Nakariakov et al. (2000), here separating out damping terms from the nonlinear

terms that do not directly involve the dissipative terms; in Nakariakov et al. dissipative terms are grouped together within the nonlinear coefficients that arise. We use the notation N1 , N2 and N3 for the nonlinear terms that arise, but because of a difference in our approach – we used the alternative form (13.99) rather than (13.98) – the coefficient N3 takes a different form from that arising in Nakariakov et al. though the coefficients N1 and N2 are the same in both formulations. Any differences that arise are of course purely a matter of presentation.

422

Nonlinear Aspects

whereas perturbations of order  2 are weakly nonlinear. In our treatment here we do not consider terms that are higher than quadratic in . We neglect all terms involving  3 . To make the order of terms explicit we write uz = u1 ,

p(1) = p1 +  2 p2 + · · ·,

with similar expansions for ρ (1) and T (1) as for p(1) . In consequence of the assumption of weak nonlinearity, in N2 we neglect the expression ρ (1) uz

∂uz . ∂z

Furthermore, we assume that the dissipative terms are small; they already include a term proportional to uz (and so of order ) but we further assume that they are an order smaller, so that they do not enter at linear order but may compete with weakly nonlinear terms of order  2 . We can declare this choice explicitly by writing ν =  νˆ ,

κ =  κˆ  ,

where νˆ and κˆ  are of order unity.

13.10.3 Linear Case With the above assumptions, the ideal linear order equations are given by neglecting the nonlinear terms N1 , N2 and N3 and the damping terms involving ν and κ : ∂ρ1 ∂ + (ρ0 u1 ) = 0, ∂t ∂z ∂u1 ∂p1 + γ p0 = 0, ∂t ∂z

∂u1 ∂p1 =− , ∂t ∂z p1 ρ1 T1 = + . p0 ρ0 T0

ρ0

(13.106)

We may eliminate variables between this set of linear equations to obtain the onedimensional wave equation for u1 (and therefore also for uz ): ∂ 2 u1 ∂ 2 u1 = c2s 2 , 2 ∂t ∂z

(13.107)

where c2s = γ p0 /ρ0 denotes the square of the sound speed. Consider a solution of the wave equation giving propagation in the z-direction at the sound speed cs : u1 = u0 f (z − cs t),

(13.108)

with u0 being an arbitrary constant and f (z) is an arbitrary function of z. This is, of course, a part of D’Alembert’s general solution of the wave equation. Then the system of linear equations readily leads to relationships between the linear perturbations: for a propagating wave, u1 p1 =γ , p0 cs

ρ1 u1 = , ρ0 cs

T1 u1 = (γ − 1) . T0 cs

(13.109)

13.10 Dissipative Effects: Sound Waves and Burgers’ Equation

423

To see this, note first that from (13.106) and (13.108) we have ∂u1 ρ0 ∂u1 ∂ρ1 = −ρ0 = −ρ0 u0 f  (z − cs t) = , ∂t ∂z cs ∂t where f  (z) denotes the derivative with respect to z of the function f (z). It then follows that we may take ρ1 = ρ0 u1 /cs . Equations for p1 and T1 are treated similarly.

13.10.4 Nonlinear Evolution Returning to equations (13.103)–(13.105), we proceed to eliminate all variables except uz . Differentiating the momentum equation with respect to t gives   ∂ 2 uz ∂ ∂p(1) ∂Dν ∂N2 ρ0 2 + = + . ∂z ∂t ∂t ∂t ∂t Eliminating p(1) , we obtain     2 ∂ 2 uz 1 ∂N2 ∂N3 1 ∂Dν ∂Dκ 2 ∂ uz − c = − + − . s ρ0 ∂t ∂z ρ0 ∂t ∂z ∂t2 ∂z2

(13.110)

Equation (13.110) is the desired evolution equation; the terms on the left side involve uz and the terms on the right may also be expressed entirely in terms of uz . To analyse (13.110) further we again use the method of multiple scales (see, for example, Bender and Orszag 1978) and change coordinates from z and t to a moving coordinate ζ and a scaled coordinate Z: ζ = z − cs t,

Z = z.

(13.111)

The moving coordinate ζ travels along the z-axis with the sound speed cs . The scaled form of z is chosen because we expect that as we move with the wave the variation in the longitudinal coordinate will be slow; it is equivalent in the linear theory to taking a z-dependence e−ikz z for small wavenumber kz in suitable units of measurement. The scaling is made with , as we expect that for   1 the dependence on z will be slight for weakly nonlinear disturbances. With these new coordinates, the operators ∂/∂t, ∂ 2 /∂t2 , ∂/∂z and ∂ 2 /∂z2 in the original coordinates z and t transform to ∂ ∂ = −cs , ∂t ∂ζ ∂ ∂ ∂ = + , ∂z ∂ζ ∂Z

∂2 ∂2 = c2s 2 , 2 ∂t ∂ζ 2 2 ∂ ∂2 ∂ 2 ∂ = + 2 . +  ∂Z∂ζ ∂z2 ∂ζ 2 ∂Z 2

(13.112)

We can now transform equation (13.110) into the new coordinates. We work to order  2 , bearing in mind that uz is of order  (uz = u1 ). Consider the left side of equation (13.110): ∂ 2 uz ∂ 2 uz ∂ 2 uz − c2s 2 = −2c2s  , 2 ∂Z∂ζ ∂t ∂z

424

Nonlinear Aspects

to leading order in . For the right side of equation (13.110), we note that ∂ ∂ (1) ρ0 ∂uz (ρ uz ) = − 2 (ρ1 u1 ) = −2 uz , ∂z ∂ζ cs ∂ζ ∂uz ∂uz ∂uz N2 = −ρ (1) − ρ0 uz − ρ (1) uz , ∂t ∂z ∂z ∂p(1) ∂uz ∂uz − γ p(1) = −(γ + 1)ρ0 cs uz . N3 = −uz ∂z ∂z ∂ζ

N1 = −

(13.113)

We have used the linear relations (13.109) in reducing the expressions for N1 , N2 and N3 , and have worked out the expressions to order  2 . The coefficient N2 vanishes to order  2 , becoming non-zero only at order  3 , and so does not contribute to the weakly nonlinear behaviour of uz . The coefficients N1 and N3 are of order  2 (since proportional to u2z ). Finally, we consider the damping terms: to leading order in , we have Dν =

4 ∂ 2 uz ρ0 ν 2 , 3 ∂ζ

Dκ = (γ − 1)2

κ T0 ∂ 2 uz , cs ∂ζ 2

(13.114)

with ν and κ calculated for equilibrium values T0 . Thus, to leading order in  equation (13.110) with N2 = 0 becomes −2c2s 

∂ 2 uz 1 ∂ =− [N3 + cs Dν + Dκ ]. ∂Z∂ζ ρ0 ∂ζ

Integrating once with respect to ζ (and setting to zero the arbitrary function of integration), we obtain 

∂uz 1 [N3 + cs Dν + Dκ ]. = ∂Z 2ρ0 c2s

Finally, noting that ∂uz N3 = −(γ + 1)ρ0 cs uz , ∂ζ

 cs Dν + Dκ = ρ0 cs

(13.115)

 (γ − 1)2 4 ∂ 2 uz , ν+ νth 3 γ ∂ζ 2

where (see Chapter 12) νth = T0 κ /p0 , we obtain ∂uz ∂uz 1 1 + (γ + 1)uz = cs ∂Z 2 ∂ζ 2



 4 (γ − 1)2 ∂ 2 uz . ν+ νth 3 γ ∂ζ 2

(13.116)

This is the desired nonlinear equation governing the evolution of the longitudinal motion uz in a weakly nonlinear and weakly dissipative system. An equation of this form (with also a term arising from gravity; see Section 13.10.5 below) was obtained in Nakariakov et al. (2000; see their eqn. (27)) and applied to coronal loops. We can rewrite equation (13.116) in the original coordinates, noting first that ∂uz ∂uz ∂uz + cs = cs . ∂t ∂z ∂Z

13.10 Dissipative Effects: Sound Waves and Burgers’ Equation

Then, in the original coordinates we obtain ∂uz ∂uz ∂uz 1 1 + cs + (γ + 1)uz = ∂t ∂z 2 ∂z 2



 4 (γ − 1)2 ∂ 2 uz . ν+ νth 3 γ ∂z2

425

(13.117)

This is a form of Burgers’ equation (Nakariakov et al. 2000); see also Taniuti and Wei (1968), Kakutani et al. (1968), Ruderman (2006), and Afanasyev and Nakariakov (2015). It is interesting to observe that the coefficient arising here that determines the nonlinear term, namely (γ + 1)/2, is precisely the coefficient we would obtain from the expression (13.34) for β0 if we take the limit cA  cs . Notice too that the linear form of equation (13.117) has a solution of the form uz = u0 e(iωt−ikz z) , with corresponding dispersion relation 1 ω = kz cs + ikz2 2



 (γ − 1)2 4 ν+ νth . 3 γ

(13.118)

This result agrees with the approximate dispersion relation obtained in Chapter 12 (see equations (12.135) and (12.136)), applying for suitably small kz . 13.10.5 Effect of Gravity (g = 0) Equations (13.116) and (13.117) have been derived for a uniform equilibrium, in which ρ0 , p0 and T0 are all constants. However, in many situations it is important to include the effect of gravity. Inclusion of gravity has two significant effects: it makes the equilibrium a nonuniform one and it introduces the gravitational force into the momentum equation. Both these effects complicate the determination of the weakly nonlinear equation governing uz . However, if we assume the effect of gravity is small – this is appropriate for the corona, where the pressure scale height is large – entering at order  2 (as with weak damping), then the changes to the above argument given for g = 0 are relatively slight. Consider then the analysis leading to (13.116), but allowing now that g = 0. The equilibrium is one of hydrostatic balance in which5 p0  = −gρ0 (z).

(13.119)

We suppose again that the medium is isothermal (so that T0 and cs are constants). The nonlinear equations describing the perturbations about the stratified equilibrium given by (13.119) are the equations of continuity, the momentum equation including the gravitational force, and the energy equation: ∂uz ∂ρ (1) ∂ ∂p(1) ρ0 + (ρ0 uz ) = N1 , + + gρ (1) = N2 + Dν , ∂t ∂z ∂t ∂z ∂p(1) ∂p(1) + p0  uz + γ p0 = N3 + D κ . ∂t ∂z

(13.120)

5 Taking the z-axis to point upwards, opposite to gravity, leads to sign changes in certain terms when compared with Chapter 9.

426

Nonlinear Aspects

The equations for the nonlinear terms N1 , N2 and N3 and the damping terms Dν and Dκ remain as before. Differentiating the momentum equation with respect to t and using the equations of continuity and energy leads to     2 ∂uz 1 ∂N2 ∂N3 ∂Dκ 1 ∂Dν ∂ 2 uz 2 ∂ uz − cs 2 + γ g = − − gN1 + − . (13.121) ∂z ρ0 ∂t ∂z ρ0 ∂t ∂z ∂t2 ∂z Applying the same scaling (13.111) as for g = 0, we obtain −2c2s 

1 ∂ ∂ 2 uz ∂uz 1 ∂  + γg =− cs N2 + N3 + gN1 − [cs Dν + Dκ ]. ∂Z∂ζ ∂ζ ρ0 ∂ζ ρ0 ∂ζ

We regard the effects of gravity as a small correction to the g = 0 case, effectively scaling g by writing g =  gˆ . Moreover, we assume that gravity is sufficiently weak that we may take the equilibrium density ρ0 as effectively a constant, so that the pressure and density perturbations are as determined when g = 0. Then, to linear order, the same terms arise as before, and we find that for g = 0 p(1) = ρ0 cs uz ,

ρ (1) = ρ0 uz /cs ,

T (1) = (γ − 1)T0 uz /cs .

(13.122)

We may note that to order  2 , N2 = 0 and gN1 = 0 (for weak gravity), and so the nonlinear term arises entirely from N3 . Finally, substituting for N3 and the damping terms we obtain   ∂uz ∂uz 1 1 1 4 (γ − 1)2 ∂ 2 uz uz + (γ + 1)uz , (13.123) − = ν+ νth cs ∂Z 20 2 ∂ζ 2 3 γ ∂ζ 2 where 0 = c2s /(γ g) is the scale height in the isothermal atmosphere (see Chapter 9). This is the desired nonlinear equation governing the weakly nonlinear evolution of uz in a weakly dissipative and weakly stratified system. An equation of this form was given in Nakariakov et al. (2000; see their eqn. (27)); see also Tsiklauri and Nakariakov (2001). As before, we can rewrite the nonlinear equation (13.123) in the original coordinates, to leading order in , as   ∂uz ∂uz 1 1 1 4 (γ − 1)2 ∂ 2 uz ∂uz uz + (γ + 1)uz . (13.124) + cs − = ν+ νth ∂t ∂z 20 2 ∂z 2 3 γ ∂z2

13.11 Dissipative Effects: Slow Standing Waves and Burgers’ Equation Ruderman (2013)6 has recently addressed the weakly nonlinear behaviour of standing waves in a coronal loop, treated by use of the field-free equations discussed in Section 13.10. Standing waves are assumed to have no longitudinal motion at the two 6 It should be noted that the form of the energy equation used by Ruderman (2013) – his eqn. (3) – is not quite correct, though

the error does not affect the derived linear and nonlinear equations he subsequently presents. The right side of Ruderman’s eqn. (3) should read     ∂T ∂ ∂T 1 ∂ ρκ in place of κ . ρ ∂x ∂x ∂x ∂x

13.11 Dissipative Effects: Slow Standing Waves and Burgers’ Equation

427

ends of a loop. Then, a similar expansion method as used for propagating waves applies also for standing waves. To linear order, we again obtain the wave equation (13.107), and so ∂ 2 uz ∂ 2 uz = c2s 2 . 2 ∂t ∂z

(13.125)

The general solution of the one-dimensional wave equation can be written as uz (z, t) = f (z − cs t) + g(z + cs t)

(13.126)

for arbitrary functions f and g. Now we require uz = 0

z = 0, L

at

(13.127)

where L denotes the loop length. Thus, g(cs t) = −f (−cs t) and so g(z) = −f (−z). Write the solution in the form uz = cs [f (ξ− ) − f (ξ+ )], where

  z ξ− = ω t − , cs

(13.128)

  z ξ+ = ω t + cs

and the frequency ω may be chosen arbitrarily. This solution satisfies the condition that uz = 0 at z = 0. There remains the condition that uz = 0 at z = L, and this is satisfied when we choose f to be a periodic function with period 2ωL/cs :   2ωL f (z) = f z − . (13.129) cs If now we choose ω to be πcs /L then f is periodic with a period of 2π . Proceeding to order  2 (we omit details), we require that ∂f ∂f ∂ 2f − 2λD f − 2 = 0. ∂τ ∂y ∂y

(13.130)

This equation was first obtained by Ruderman (2013). It was further considered by Kumar, Nakariakov and Moon (2016), who additionally included radiative effects which lead to a term linear in f . Here y = πz/L and τ = t/tdl is the dimensionless time measured against the dissipative timescale tdl , where tdl =

2L2 π 2 [ 43 ν

+

1 γ (γ

− 1)2 νth ]

,

λD =

(γ + 1)cs L 2π [ 43 ν

+ γ1 (γ − 1)2 νth ]

.

(13.131)

The parameter λD is dimensionless; as noted by Ruderman (2013), it can exceed unity but is less than one for a range of typical coronal conditons. For example, for a loop of length L = 400 Mm, number density 5 × 1014 m−3 and temperature T0 = 6.3 × 106 K, for amplitude parameter  = 0.1 we obtain λD = 0.17. Thus, in the units of τ and y, the

428

Nonlinear Aspects

behaviour of equation (13.130) is determined by a single parameter, λD , which for a range of coronal loop conditions is much less than unity. Equation (13.130) is a form of Burgers’ equation. Rewritten in terms of the original variables z and t, we have   2 ∂f ∂f 1 1 4 1 ∂ f 2 , (13.132) − cs (γ + 1)f = ν + (γ − 1) νth ∂t 2 ∂z 2 3 γ ∂z2 to be compared with Burgers’ equation (13.117) obtained for propagating waves. Returning to the form (13.130), we consider its solution for periodic conditions. First note that the standard Cole–Hopf transformation may be employed here to convert Burgers’ equation to the linear diffusion equation (see, for example, the discussion by Whitham 1974). We set f =

1 ∂h λD h ∂y

(13.133)

for unknown h(y, τ ) in place of f . Then equation (13.130) becomes ∂h ∂ 2h = 2, ∂τ ∂y

(13.134)

which is the linear diffusion equation for h. We consider its solution for an initial state f (y, t = 0) = − sin y.

(13.135)

In terms of h, this initial condition corresponds to h(y, τ = 0) = eλD cos y .

(13.136)

Thus, we wish to solve the diffusion equation (13.134) subject to the initial condition (13.136). Construct a periodic solution of the form h(y, τ ) = a0 +

∞ &

e−n τ [an cos ny + bn sin ny], 2

(13.137)

n=1

where the coefficients an and bn are to be determined. Since the initial condition is an even function of y we may set bn = 0 for all n, and then h(y, τ ) = a0 +

∞ &

e−n τ an cos ny. 2

(13.138)

n=1

The function h is 2π -periodic in y (that is, h(y + 2π , τ ) = h(y, τ )). The initial condition implies that eλD cos y = a0 +

∞ &

an cos ny.

(13.139)

n=1

Now the initial form has a Fourier expansion (Abramowitz and Stegun 1965, p. 376) eλD cos y = I0 (λD ) +

∞ & n=1

2In (λD ) cos ny,

(13.140)

13.11 Dissipative Effects: Slow Standing Waves and Burgers’ Equation

429

where In (z) denotes the modified Bessel function of order n. Comparing (13.139) and (13.140) we deduce immediately that a0 = I0 (λD ),

an = 2In (λD ).

(13.141)

Moreover, from the properties of the modified Bessel functions for small argument, we may conclude that an = O(λnD ) as λD → 0. It now follows that h(y, τ ) = I0 (λD ) + 2

∞ &

e−n τ In (λD ) cos ny, 2

(13.142)

n=1

from which f may be deduced through (13.133). Following Ruderman (2013), we may expand the series (13.142) for h and the deduced expression for f in powers of λD (assumed small), using the well-known expansions for the modified Bessel functions In (Abramowitz and Stegun 1965). The result is 1 h(y, τ ) = 1 + λD e−τ cos y + λ2D (1 + e−4τ cos 2y) 4   1 1 + λ3D e−τ cos y + e−9τ cos 3y + O(λ4D ). 8 3

(13.143)

The expansion for f is algebraically more involved but leads finally to7   1 1 f (y, τ ) = − 1 − λ2D (1 − e−2τ )2 e−τ sin y + λD (1 − e−2τ )e−2τ sin 2y 8 2 1 (13.144) − λ2D (1 − e−2τ )2 (2 + e−2τ )e−3τ sin 3y + O(λ3D ). 8 Notice that this expansion gives f = − sin y when τ = 0, as required by the initial condition. If we consider the expansion for f to the lowest order in λD we obtain f (y, τ ) = −e−τ sin y. Then f (ξ− , τ ) = −e−τ sin ξ− and f (ξ+ , τ ) = −e−τ sin ξ+ , and so   ωz −τ uz = cs [f (ξ− , τ ) − f (ξ+ , τ )] = 2cs e cos(ωτ ) sin . cs In the original coordinates, this is −t/tdl

uz = 2cs e

(13.145)

(13.146)



  π cs t πz cos sin . L L

(13.147)

We may use expression (13.144) to determine how nonlinearity modifies the damping time of f . In the linear theory f (and also uz ) damp as exp(−t/tdl ), giving an exponential damping on the timescale tdl . In terms of the dimensionless variable τ , this corresponds to 7 Comparing our expansion (13.144) with expression (52) in Ruderman (2013) suggests that there is a typographical error in Ruderman, whereby a minus sign is omitted in front of the e−τ sin y and e−3τ sin 3y terms (but not the e−2τ sin 2y term)

arising in his eqn. (52). This error does not impact on the subsequent determination of the nonlinear damping rate.

430

Nonlinear Aspects

a damping time of unity. With weak nonlinearity taken into account, Ruderman considers the energy in the oscillation integrated over a period 2π/ω, leading to an integral of the form  2π f 2 (y, τ )dy. E(τ ) ≡ 0

The nonlinear damping time τnonlin is then defined as the dimensionless time over which the square root of the energy integral, E1/2 (τ ), falls to 1/e of its initial value (so that E(τ ) falls to e−2 of its initial value), which is given by setting E(τ ) = e−2 E(0)

when

τ = τnonlin .

Using the expansion (13.144) and performing the integration that arises, we obtain   1 e−2τ 1 − λ2D (1 − e−2τ )3 = e−2 , 4 from which it readily follows that 1 (13.148) τ = τnonlin = 1 − (1 − e−2 )3 λ2D . 8 This expression, first obtained by Ruderman (2013), shows that the nonlinearity causes a decrease in the nonlinear damping time, but the effect is slight; for example, for an observation reported in Ofman and Wang (2002) the value of λD is estimated to be 0.4, giving a nonlinear damping time that is only 1.3% shorter than the linear damping time. Thus, at least for small λD we see that the nonlinear effect on the damping rate is slight, the linear damping rate providing a good guide. Of course, for values of λD exceeding unity, nonlinear effects may be important, especially for amplitude parameter  that is not small; Ruderman (2013) estimates  > 0.2 for nonlinearity to be important.

13.12 The Kink Mode It would be natural to present a discussion of the kink mode that had close parallels with that given earlier for the sausage mode. However, there is a sharp distinction between the two modes that prevents such an analysis: whereas the sausage mode (m = 0) propagates basically with the speed ct , in an essentially one-dimensional manner, the kink mode (m = 1) cannot be separated readily from the fluting modes (m = 2, 3, 4, . . .) since it is apparent from linear theory (see Chapter 6) that all the m ≥ 1 modes propagate with the same speed ck in the thin tube extreme of kz a  1. In consequence, this suggests that the kink mode and the fluting modes are coupled together and that a nonlinear description can be expected to involve the angular dependence φ, since the kink and fluting modes have a different dependence upon φ. A first analysis of this situation was presented in Ruderman (1992), who considered a tube with a sharp boundary in an incompressible medium and a field-free environment. Ruderman (1992) obtained a complicated nonlinear evolution equation in terms of φ, z and t that governed the evolution of all non-axisymmetric (m = 0) modes simultaneously. This work was extended in Ruderman, Goossens and Andries (2010), who again considered an incompressible

13.12 The Kink Mode

431

medium but allowed for the additional effect of a transition layer smoothly connecting the inside of the tube and its environment; thus resonant absorption effects are incorporated in this nonlinear analysis (see Chapter 11 for a discussion of similar linear problems). An early analysis by Ryutova and Sakai (1993) argued that the kink mode could be described by the well-known modified Korteweg–de Vries equation (see, for example, Drazin and Johnson 1989). However, the analysis by Ruderman (1992) and Ruderman, Goossens and Andries (2010) argues that such a description is not appropriate here (and stems from an incorrect treatment of the boundary conditions connecting the interior and exterior regions of the tube). Instead of the modified Korteweg–de Vries equation, the tube motion is described by a two-dimensional wave-like equation involving the spatial coordinates φ and z as well as time t. The Ruderman, Goossens and Andries (2010) analysis is based upon a multiple scales treatment of the interior of the tube and its environment. The medium is assumed incompressible and the environment of the tube is taken to be field-free (cAe = 0). The analysis leading to the resulting nonlinear equation is too lengthy to present here and so we will be content with presenting their result. After considerable analysis, Ruderman, Goossens and Andries (2010) obtain a nonlinear evolution equation of the form8     2   ∂ 2ψ ρe l ρe ∂ ψ 1 ∂3 C + Hφ γ0 Ht (ψ) − γ1 ψ + 1+ ∂t∂z 2a ρ0 + ρe ρ0 + ρe a ∂φ∂t2 ∂t2           ∂ψ ρe ∂ψ ∂ ∂ ∂ψ ∂ψ 1 Hφ ψ − ψHφ − Hφ . (13.149) = a ρ0 + ρe ∂φ ∂t ∂t ∂t ∂t ∂t Equation (13.149) describes the nonlinear evolution of the kink mode oscillation of a magnetic flux tube in an incompressible medium. Here ψ(φ, z, t) is a measure of the radial displacement of the tube (of undisturbed radius a) and l is the thickness of the transition layer that connects the interior of the tube (of plasma density ρ0 ) with the field-free environment (with plasma density ρe ); the special case l = 0 corresponds to a sharp (discontinuous) change at the tube boundary. The speed C arising in (13.149) is defined by C2 =

ρ0 c2A , ρ0 + ρe + (l/a)ρe

and so depends upon the thickness of the transition layer; for a thin layer, C ≈ ck . The operators that arise in equation (13.149) correspond to the Hilbert transform of ψ = ψ(φ, z, t) with respect to time t or periodic variable φ:  1 ∞ ψ(φ, z, s) ds, Ht (ψ) = − π −∞ s − t  2π 1 1 [ψ(φ  , z, t) − ψ(φ, z, t)] cot (φ  − φ) dφ  . (13.150) Hφ = 2π 0 2

8 There are a number of notational differences between their work and that presented here.

432

Nonlinear Aspects

The coefficients γ0 and γ1 occurring in (13.149) are determined by the details of the transition layer (Ruderman, Goossens and Andries 2010): πρe C2

γ0 =

,  2ρ0 (rA )|(c2A ) (rA )|     ρ0 (r) C2 − c2A (r) 1 a+l ρe C 2  − γ1 = − dr, 2 a ρe C2 ρ0 (r) C2 − c2A (r)

(13.151)

where the prime  denotes the derivative with respect to r. Both ρ0 and (c2A ) are calculated at the resonance point r = rA in the transition layer. See Chapter 8 for a fuller discussion of these aspects related to resonant absorption. In the absence of a transition layer, when l = 0, equation (13.149) may be related to an equation of similar form obtained earlier by Ruderman (1992). It is of interest to examine the linear limit of equation (13.149), when the right-hand side of (13.149) is negligible and the system is governed by     2   ρe l ρe ∂ ψ 1 ∂3 ∂ 2ψ + Hφ γ0 Ht (ψ) − γ1 ψ = 0. + 1+ C ∂t∂z 2a ρ0 + ρe ρ0 + ρe a ∂φ∂t2 ∂t2 (13.152) Adopting the same Fourier notation for the perturbation as used in Chapter 8, we seek a solution in the form ψ(φ, z, t) = ψ0 e−iωt+imφ+ikz z . Noting that ∂ψ = −iωψ, ∂t

∂ψ = imψ, ∂φ

∂ψ = ikz ψ, ∂z

Ht (e−iωt ) = −i sgn (ω) · e−iωt ,

we have Hφ (eimφ ) = i sgn (m) · eimφ , Hφ γ0 Ht (ψ) − γ1 ψ = −i sgn (m)(iγ0 sgn (ω) + γ1 )ψ. 

Substituting these expressions in (13.152) we obtain the dispersion relation       ρe γ1 m γ0 ρe l + +i |ω||m|. Ckz = ω 1 + ρ0 + ρe 2a a a ρ0 + ρe Expanding C−1 binomially for small l/a, we obtain     ω γ1 γ0 ρe l |ω||m|. + |m| +i kz ≈ 1+ ck ρ0 + ρe a a ack

(13.153)

(13.154)

In writing this form we have taken γ0 /a and γ1 /a to be of order l/a, and retained terms to within order l/a and neglected terms of order (l/a)2 . Writing kz = kz real + i kz imag , it follows from the real and imaginary parts of (13.154) that       ω γ0 γ1 ρe l ρe real imag = = + |m| , kz kz 1+ |ω||m|, ck ρ0 + ρe a a ack ρ0 + ρe (13.155)

13.12 The Kink Mode

433

where we have taken only the leading order terms (for small l/a) in the real and imaginary parts of kz . Thus, to leading order we have ω = kz real ck , corresponding to wave propagation at the kink speed ck . The wave is subject to damping, caused by the thin transition layer in the Alfv´en speed connecting the inner part of the tube with its field-free environment. The damping corresponds to an exponential decline in the amplitude of the wave, ψ ∼ e−z/ , with the spatial scale  given by =

1 kz

imag

=

ack . ρe ( ρ0 +ρe )|ω||m|γ0

In the absence of a layer, l = 0 and so γ0 = 0 and the wave propagates without damping ( → ∞ as l → 0). In the presence of a transition layer, high m modes damp more strongly than the kink mode: fluting modes (|m| ≥ 2) are damped more strongly than the |m| = 1 kink mode. Using the definition of the coefficient γ0 , we may write   ρ0 1 2a ρ0 (rA )  (13.156) |(c2 ) (rA )|. 1+ = πck ρe ρe |ω||m| A We can write the expression (13.156) for the damping length and an associated timescale τ = /ck in terms of a wavelength Lwave ≡ 2π/kz real = 2π ck /ω:    ρ0 1 a ρ0 (rA )  τ= . (13.157) |(c2A ) (rA )|Lwave , 1+ = 2 2 ρ ρ |m| c π ck e e k These expressions agree with results presented in Ruderman, Goossens and Andries (2010) and with the dispersion relation9 for an incompressible medium derived by Goossens, Hollweg and Sakurai (1992; see their eqn. (77)).

9 Note the typographical error in Goossens, Hollweg and Sakurai (1992) reported in Chapter 8.

14 Solar Applications of MHD Wave Theory

14.1 Introduction In this chapter we present examples of the theory as developed in the earlier discourse. Although we of course draw on the developments in Chapters 6–13, in the main our discussion here is presented as self-contained as possible. We draw on examples of observations in the literature, examining how they illustrate the general theoretical concepts developed earlier. Where possible, we present both historical examples and more recent ones that have arisen from the latest space or ground-based observations of oscillating solar structures. By its very nature, our treatment of applications is necessarily more speculative than the theories developed in earlier chapters. This is at the heart of the mathematical modelling we present. In essence, we are choosing a relatively simple model to represent a complex physical feature; whether our choices are helpful in the understanding of those physical features remains to be seen by developments in the subject yet to occur. A convenient starting point for our discussion is the set of linear equations for an inhomogeneous plasma in a unidirectional equilibrium magnetic field. Working in a cylindrical coordinate system r, φ, z, the equilibrium magnetic field is B0 = B0 (r)ez ,

(14.1)

a radially structured field that is aligned with the z-axis in the cylindrical polar coordinates. The equilibrium plasma pressure p0 (r) and density ρ0 (r) are also structured in radius r, with total pressure balance (see equation (3.2), Chapter 3) requiring p0 (r) +

B20 (r) = a constant. 2μ

(14.2)

Linear perturbations about this equilibrium lead to the components of the disturbance u(r, φ, z) = (ur , uφ , uz ) satisfying a set of coupled partial differential equations (see equations (6.11)–(6.14), Chapter 6):  ρ0 (r)

434

   ∂2 ∂2 ∂ ∂pT 2 − c (r) = − , u r A ∂r ∂t ∂t2 ∂z2

(14.3)

14.2 Radial Pulsations of a Coronal Tube: Rosenberg’s Approach







435



∂2 ∂2 1 ∂ ∂pT − c2A (r) 2 uφ = − , 2 r ∂φ ∂t ∂t ∂z  2    ∂ ∂2 c2s (r) ∂ ∂pT 2 ρ0 (r) − ct (r) 2 uz = − 2 , ∂t2 ∂z cf (r) ∂z ∂t

ρ0 (r)

(14.4) (14.5)

with the perturbation pT in total (plasma plus magnetic) pressure satisfying   ∂pT 1 ∂ 1 ∂uφ ∂uz 2 = −ρ0 (r)cf (r) (rur ) + −ρ0 (r)c2s (r) . ∂t r ∂r r ∂φ ∂z

(14.6)

As usual cs denotes the sound speed and cA the Alfv´en speed, from which we form the slow (tube) speed ct and the fast magnetoacoustic speed cf defined by c2t =

c2s c2A c2s + c2A

,

c2f = c2s + c2A .

(14.7)

The set of partial differential equations (14.3)–(14.6) underpins much of the development in recent wave studies, treating in particular the case of a magnetic flux tube. Various degrees of complexity may be studied with this system but generally the picture that emerges from a study of a simple flux tube with uniform interior and exterior, though perhaps also with a smooth layer of transition between inside and outside the tube, has proved remarkably robust: the notion of a simple flux tube acting as a waveguide is a fruitful mathematical model (Roberts 2004, 2008). The gravitational force is ignored in equations (14.3)–(14.6), though it is undoubtedly important in some applications, especially to photospheric tubes including sunspots and to certain modes of oscillation in the corona including in prominences. Gravity is included in our description of longitudinal oscillations in prominences (see Section 14.12) and in sunspot oscillations (Section 14.14).

14.2 Radial Pulsations of a Coronal Tube: Rosenberg’s Approach One of the earliest applications of magnetoacoustic wave theory applied to the interpretation of oscillations in the corona was made by Rosenberg (1970, 1972). To understand his contribution it is convenient to place it within our theoretical description as represented by (14.3)–(14.6) and outlined in detail in Chapter 6. Rosenberg (1970) did not consider a system as general as (14.3)–(14.6), but instead examined from the outset the case of radial motions that were independent of the azimuthal angle φ and the longitudinal coordinate z. Consider then equations (14.3)–(14.6) with the assumption that ∂/∂φ = 0 and ∂/∂z = 0; the motion is assumed to be purely radial, so that uφ = 0 and uz = 0, and ur and the associated total pressure perturbation pT are functions of radius r and time t alone. Such an oscillation is symmetric about the z-axis and the same at all locations along the z-axis: it represents a breathing in and out of a magnetic flux tube. Equations (14.3)–(14.6) are satisfied by requiring that the radial motion ur (r, t) and pressure variations pT (r, t) are related by

436

Solar Applications of MHD Wave Theory

∂ 2 ur ∂ 2 pT 1 ∂ ∂pT =− , = −ρ0 (r)c2f (r) (rur ). 2 ∂r∂t ∂t r ∂r ∂t Thus, eliminating pT , radial motions ur satisfy   ∂ 2 ur ∂ 1 ∂ 2 ρ0 (r) 2 = ρ0 (r)cf (r) (rur ) . ∂r r ∂r ∂t ρ0 (r)

(14.8)

(14.9)

Consider equation (14.9) for a uniform plasma (ρ0 , cs and cA all constants): then   ∂ 2 ur 1 ∂ 2 ∂ = c ) (ru r , f ∂r r ∂r ∂t2  2  1 ∂ur 1 2 ∂ ur = cf + − 2 ur . (14.10) r ∂r ∂r2 r This is the equation used by Rosenberg (1970) in his discussion of radio pulsations in the coronal plasma.1 Looking for a Fourier time dependence of the form ur (r, t) = ur (r)eiωt , we obtain d2 ur 1 dur + + 2 r dr dr



1 ω2 − 2 2 r cf

(14.11)

 ur = 0.

Thus we obtain a form of Bessel’s equation, the appropriate solution of which is   ω r ur (r) = u0 J1 cf

(14.12)

(14.13)

where J1 denotes the Bessel function of the first kind (Abramowitz and Stegun 1965, chap. 9) and u0 is an arbitrary constant (with the dimensions of speed). In writing this solution we have rejected a solution Y1 (ωr/cf ), involving the Bessel function Y1 of the second kind, which produces an unbounded motion ur at r = 0. Rosenberg (1970) proposed (somewhat arbitrarily) that the radial motion ur (r) reaches a local spatial maximum on the tube boundary at r = a, requiring that J1 (ωa/cf ) reach a maximum at r = a, and so    ω a = 0, (14.14) J1 cf where J1  denotes the derivative of the function J1 . Thus cf ω = j1,s , a

(14.15)

where j1,s denotes the zeros of the Bessel function J1  (x). The first two zeros are j1,1 = 1.8412 and j1,2 = 5.3314 (Abramowitz and Stegun 1965, Table 9.5). If instead we work in terms of the total pressure perturbation pT (r, t), then in place of (14.9) we obtain the wave equation 1 See the wave equation following Rosenberg’s eqn. (5).

14.3 Radial Pulsations of a Coronal Tube: The Role of an Environment



437



∂ 2 pT 1 ∂ 1 ∂pT = ρ0 (r)c2f (r) r . r ∂r ρ0 (r) ∂r ∂t2

(14.16)

In a uniform medium this becomes

  2 1 ∂pT ∂ 2 pT 2 ∂ pT = ρ0 cf + . r ∂r ∂t2 ∂r2

(14.17)

Again we obtain a form of Bessel’s equation, save now it is for the Bessel functions J0 (ωr/cf ) and Y0 (ωr/cf ) of zero order. Rejecting the function Y0 (ωr/cf ) as being singular at r = 0, we obtain   ωr pT (r) = pT0 J0 , (14.18) cf where pT0 is an arbitrary constant (with the dimensions of pressure). Similarly to the motion ur , we may now suppose that pT reaches a maximum pT (r) on the tube boundary r = a, requiring (since J0  = −J1 )   ωa J1 = 0, (14.19) cf giving cf , (14.20) a where j1,s denotes the zeros of the Bessel function J1 . The first two zeros of J1 are at j1,1 = 3.8317 and j1,2 = 7.0156 (Abramowitz and Stegun 1965, Table 9.5). Now neither of the expressions (14.15) and (14.20) are dispersion relations, since there seems to be no compelling reason why ur or pT should happen to have a local maximum on the tube boundary. Moreover, Rosenberg (1970, 1972) never gave (14.20) but simply considered ur . Nonetheless, Rosenberg (1970) is important in that it suggested, early in the development of coronal oscillation theory, that timescales associated with magnetohydrodynamic oscillations may be determined by the radius a of a coronal flux tube, and such spatial scales are short. For example, with a tube radius a ∼ 103 km, say, and a magnetoacoustic fast speed of the order of the Alfv´en speed in a low β corona, say cf ∼ 103 km s−1 , from the maximum in |ur (r)| we obtain ω ∼ 1.8 s−1 , with an associated period P (= 2π/ω) of P ∼ 3.5 s; from the maximum in |pT (r)|, we obtain ω ∼ 3.8 s−1 and P ∼ 1.6 s. ω = j1,s

14.3 Radial Pulsations of a Coronal Tube: The Role of an Environment The idea of a magnetoacoustically oscillating tube was developed much further in independent works by Zaitsev and Stepanov (1975) and Edwin and Roberts (1983). These authors constructed and investigated a basic model of a magnetic flux tube which included the role of an external medium. The work of Zaitsev and Stepanov (1975) directly addressed the question of what happens if an environment is different from the core of a magnetic flux tube, removing also the assumptions of ∂/∂φ = 0 and ∂/∂z = 0 (equivalent in Fourier space to requiring m = 0 and kz = 0). The basic dispersion relation they derived

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was formulated in terms of Bessel functions and Hankel functions, appropriate for examining the role of wave leakage in the environment of the tube. Even with m = 0 and kz = 0, the model leads to radiative decay as vibrations inside the tube disturb the environment of the tube and leak their energy radially outwards to the far environment (see Section 6.6 of Chapter 6). The topic of wave leakage was further explored in Spruit (1982) and Cally (1986). The model of Edwin and Roberts (1983), which coincided with that of Zaitsev and Stepanov (1975) in the way in which an environment was included, aimed principally at exploring the general nature of magnetoacoustic waves in a magnetic flux tube such as arises in the corona or photosphere. The work of Edwin and Roberts (1983) concentrated on trapped waves, oscillations for which the flux tube is a natural wave guide, and was cast in terms of Bessel functions and modified Bessel functions (representing a decaying function in the tube’s environment) rather than Hankel functions. The analysis of Edwin and Roberts (1983) grew naturally out of earlier work by Roberts and Webb (1978, 1979) on the sausage mode (m = 0) oscillations of a flux tube embedded in a field-free environment (cAe = 0), as well as similar work on a magnetic slab for the cases cAe = 0 (Roberts 1980, 1981a–c) and cAe = 0 (Edwin and Roberts 1982). The work of Edwin and Roberts gives rise to a number of diagrams that display the behaviour of the phase speed of a wave as a function of the longitudinal wavenumber. These diagrams, constructed using both analytical and numerical work, make clear the dispersive nature of magnetoacoustic waves in structured atmospheres. Zaitsev and Stepanov’s (1975) work placed Rosenberg’s idea on a firmer footing through their exploration of the modes of oscillation of a flux tube of radius a, but they concluded that decay due to radiative leakage of the oscillation – a leaky mode – was too rapid to allow an explanation along the lines outlined by Rosenberg (1970). Maintaining the assumptions ∂/∂φ = 0 and ∂/∂z = 0 (corresponding to m = 0 and kz = 0), in the extreme of a dense tube (ρ0  ρe ) the sausage mode has a frequency ωR and decay rate ωI (so ω = ωR + iωI ) given by (Zaitsev and Stepanov 1975; Meerson, Sasorov and Stepanov 1978; see Section 6.6, Chapter 6)      1 ρe ρe cA ωR = j0,1 1 + , ln 2 ρ0 ρ0 a

ωI = j0,1

π 2



ρe ρ0



cA , a

ρe  ρ0 . (14.21)

It follows that π ωI ∼ ωR 2



 ρe , ρ0

ρe  ρ0 .

(14.22)

Here j0,1 (≈ 2.4048) denotes the first zero of the Bessel function J0 . It is interesting to see that although neither of the zeros, j1,1 and j1,1 , of a Bessel function that might be expected to arise from the work of Rosenberg (1970), there nonetheless does arise a Bessel function zero, namely j0,1 .

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Under coronal conditions, the associated period Psaus (= 2π/ωR ) of the sausage mode, for kz a  1 and ρe  ρ0 , follows from equation (14.21): 1 2π a . 1 j(0,1) cA [1 + 2 (ρe /ρ0 ) ln(ρe /ρ0 )]

Psaus =

(14.23)

Strictly, Psaus is the period of the sausage mode only for kz = 0, when ρe /ρ0  1, under coronal conditions; but expression (14.23) serves as a useful guide to the period even if the mode is trapped. It turns out – see Section 14.5.3 – that expression (14.23) provides an upper bound to the actual sausage mode period for all kz , whether the mode is leaky or trapped. With this understanding, we use Psaus for the sausage mode period generally, under coronal conditions. For very large ρ0 /ρe the period Psaus reduces to Psaus =

2π a a ≈ 2.61 , j0,1 cA cA

(14.24)

with an associated decay time τsaus (= 1/ωI ) given by   1 ρ0 τsaus = 2 ρe  ρ0 . Psaus , ρe π Thus, τsaus =

2 πj0,1



ρ0 ρe



a ≈ 0.26 cA



ρ0 ρe



a , cA

(14.25)

ρe  ρ0 .

(14.26)

It follows that for moderate density coronal tubes the sausage mode (with kz = 0) is all but damped out in a single period. For example, with a = 103 km and cA = 103 km s−1 , appropriate for the corona, the sausage mode in a tube with density ratio ρ0 /ρe = 10 has a period Psaus ≈ 2.95 s according to equation (14.23) (or Psaus ≈ 2.61 s if the approximation (14.24) is used), but a decay time of τsaus ≈ 2.99 s (or τsaus ≈ 2.65 s if equation (14.26) is used). The oscillation would be quenched before it had hardly started. However, for a density ratio ρ0 /ρe = 102 the period is little changed, Psaus = 2.67 s, but now the decay time is τsaus = 27.10 s, and so the oscillation may persist for several periods without suffering undue decay by radiative leakage.

14.4 Coronal Oscillations and Coronal Seismology In earlier chapters, much of our discussion of waves in a magnetic flux tube was theoretical, with its applications only tentative. Historically, this was the situation generally, the theory of coronal oscillations being developed (see, for example, Zaitsev and Stepanov 1975; Spruit 1982; Edwin and Roberts 1982, 1983; Roberts, Edwin and Benz 1983, 1984) with only limited and indirect observational support. The situation changed dramatically in 1999 with the discovery by the spacecraft TRACE (Transition Region And Coronal Explorer) that coronal loops exhibited global oscillations that could be directly imaged and measured (Aschwanden et al. 1999; Nakariakov et al. 1999). The discovery of waves is of interest in its own right, but the topic acquired an additional significance because of its role in the development of coronal seismology (Roberts, Edwin

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and Benz 1984; Roberts 2000, 2008). The idea that waves could be used to unravel seismological information about the solar atmosphere is of course not surprising, the study of the Sun’s interior through the use of sound and gravity waves – helioseismology – providing a natural impetus, just as the study of terrestrial seismology of the Earth’s interior provided a natural spur to the development of helioseismology. Indeed, Uchida (1970) proposed a ‘seismological diagnosis’ of a global corona using large-scale wave disturbances known as Moreton waves. Also, there were hints of a local seismology in Rosenberg’s explanation of a periodic signal in the corona (Rosenberg 1970) detected in radio wavelengths. However, the notion of a local seismology was not properly explored until the work of Roberts, Edwin and Benz (1984), which exploited the wave diagrams of Edwin and Roberts (1983) and suggested explicitly that ‘magnetoacoustic oscillations provide a potentially useful diagnostic tool for determining physical conditions in the inhomogeneous corona’. Roberts, Edwin and Benz (1984) argued that the combination of theory and observations provides ‘a valuable diagnostic tool for in situ conditions in the corona’, allowing determination of the local Alfv´en speed and spatial dimension of the coronal inhomogeneity that forms a loop (Roberts 2008). There is of course no reason why such concepts cannot be equally applied to parts of the Sun other than the corona, leading to an MHD seismology (see, for example, Roberts and Nakariakov 2003; Erd´elyi 2006a); sunspots are one obvious application. Consider then how the theory of coronal oscillations may be applied to solar observations. We follow the approach used in Roberts, Edwin and Benz (1984). A convenient starting point is the set of wave equations (14.3)–(14.6). Consider a uniform magnetic flux tube of strength B0 and radius a within which the sound speed is cs , the Alfv´en speed is cA , and the plasma density is ρ0 , the whole tube being embedded in a uniform magnetic environment with field strength Be , sound speed is cse , Alfv´en speed cAe , and plasma density ρe . Disturbances of the form ur (r) exp i(ωt − mφ − kz z), representing a wave with frequency ω, azimuthal order m and longitudinal wavenumber kz , are examined, with similar expressions for other perturbations (e.g. uz , pT ). After an appropriate Fourier analysis this model of a magnetic flux tube leads, through equations (14.3)–(14.6), to the complicated dispersion relation (Edwin and Roberts 1983; for details, see Chapter 6) 1 ρ0 (kz2 c2A

− ω2 )

n0 a

Jm  (n0 a) Km  (me a) 1 me a = , 2 Jm (n0 a) Km (me a) ρe (kz2 cAe − ω2 )

(14.27)

with n20 =

(ω2 − kz2 c2A )(ω2 − kz2 c2s ) (c2s + c2A )(ω2 − kz2 c2t )

,

m2e =

(kz2 c2se − ω2 )(kz2 c2Ae − ω2 ) (c2se + c2Ae )(kz2 c2te − ω2 )

.

(14.28)

Equation (14.27) is the dispersion relation for trapped (wave-guided) magnetoacoustic waves in a magnetic flux tube; it is subject to the constraint that me > 0.

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441

The complexity of the transcendental relation (14.27) means that a numerical investigation is needed, as well as any supporting analytical cases. This is the approach taken in Edwin and Roberts (1983), who produced a number of dispersion diagrams displaying the expected behaviour of modes as predicted by the dispersion relation (14.27). In particular, as shown in greater detail in Chapter 6 and reproduced for convenience here, Figure 14.1 provides a guide to the various modes that may be expected to arise under coronal conditions. We may use this diagram to aid the interpretation of observations, discussing both standing and propagating modes. Standing waves occur in closed structures such as loops, provided the wave has had time to travel the length of the loop and back again; for shorter times, the wave has not reached the ends of the loop, where line-tying in the dense lower atmosphere causes reflection, and so the wave propagates freely as if the structure were open. Recent reviews of observational and theoretical aspects are available in Aschwanden (2004), Nakariakov and Verwichte (2005), Wang (2016) and Nakariakov et al. (2016). Reviews that give emphasis to theoretical developments include Roberts (2000, 2004, 2008) and De Moortel and Nakariakov (2012). Pascoe (2014) provides an overview of theoretical aspects and how numerical simulations are used to better understand and interpret observed oscillations.

Figure 14.1 The dispersion diagram of Edwin and Roberts (1983), showing solutions of the dispersion relation (14.27) for a magnetic flux tube under coronal conditions. Speeds, densities and notation are as in Figure 6.6 of Chapter 6. The diagram gives the phase speed c (= ω/kz ) as a function of the dimensionless wavenumber kz a for fast and slow magnetoacoustic body modes in a coronal magnetic flux tube. Solid curves correspond to sausage waves and dashed curves to kink waves. (After Edwin and Roberts 1983.)

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Solar Applications of MHD Wave Theory

14.5 Standing Waves The dispersion relation (14.27) applies to both propagating modes and standing modes. Consider a coronal magnetic flux tube of total length L, with the ends of the tube fixed. We have in mind a coronal loop of length L with its footpoints embedded in the lower photospheric-chromospheric atmosphere. Standing waves are characterized by having a wavelength that fits into the anchored loop. This corresponds to taking kz as determined by kz = Nπ/L,

(14.29)

with the integer N describing the oscillations along the loop. The case N = 1 gives the fundamental mode (which disturbs globally the whole of the loop, save its anchored ends, and there are no nodes along the tube); the case N = 2 gives the first harmonic of the fundamental and has a node at the mid-point z = L/2 of the tube (corresponding to the apex of the loop). Now dispersion relations for many types of wave end up relating the frequency ω to the wavenumber kz through a relation of the form ω ∼ kz c,

(14.30)

for an appropriate propagation speed c. Such a relation may apply at least for some range of kz (typically small kz a). Combining (14.29) and (14.30), we may write our relation for standing waves in terms of the period P and its relationship with loop length L and speed c, namely 2L 2 × loop length P= = . (14.31) Nc N × mode speed As a relation for speed c, we have 2L . (14.32) NP When coupled with observations, a formula of the form (14.32) may be used directly to determine c; with assumptions about the mode of oscillation, this may then lead to (say) the propagation speed of a particular mode. With further assumptions about the medium, this may also lead to the Alfv´en speed cA , and ultimately to the magnetic field strength in the medium. This was done in Roberts, Edwin and Benz (1984), using the then available data. But with the discovery of TRACE oscillations it proved possible to take the idea much further, and this was done by Nakariakov and Ofman (2001) and Verwichte et al. (2004). Interestingly, some of the examples explored by Verwichte et al. (2004) were interpreted as loop oscillations in the N = 2 harmonic rather than the fundamental (N = 1) mode. The relation (14.32) is useful for interpreting observations in a number of ways, including its use as a means to rule out waves that do not accord with this expression. For example, in the corona where normally the Alfv´en speed is far in excess of the sound speed cs , an observation with say a period of P = 256 s in a loop of length L = 1.3 × 108 m – as discussed in more detail below – suggests a speed for the fundamental (N = 1) of 1.016 × 106 m s−1 , or about 1000 km s−1 . Such a speed is far in excess of the sound speed in the corona (of about 200 km s−1 ) so we can immediately rule out sound waves or slow magnetoacoustic waves. This leaves Alfv´en waves or fast magnetoacoustic waves. Of course, to progress further we need more detailed observational information, but at least c=

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443

Figure 14.2 The temporal evolution of the loop displacement calculated as an average coordinate of the loop position for four neighbouring perpendicular cuts through the loop apex (diamonds), with error bars (≈ 0.5 pixel) starting at 13:13:51 UT on 14 July 1998. The solid curve is the best fit function. The effect of the image motion through the field of view was subtracted in this analysis. (From Nakariakov et al. 1999.)

we have readily narrowed down the type of wave expected. We should note, however, that these conclusions on which type of wave may be being detected is more difficult to carry out for photospheric waves, since propagation speeds in the photosphere are all of a similar magnitude. To discuss the various waves that may arise we now focus on the classification of Figure 14.1, which in turn is a graphical representation of the solutions of the dispersion relation (written in the form appropriate for trapped body waves).

14.5.1 Fast Kink Standing Modes The first directly imaged observations of global oscillations in coronal loops were reported by Aschwanden et al. (1999) and Nakariakov et al. (1999), the waves being directly imaged by the instrument TRACE; see also Aschwanden et al. (2002) and Schrijver, Aschwanden and Title (2002). That such oscillations generally decay, rapidly, was brought out by Nakariakov et al. (1999). The oscillations were generally interpreted in terms of the kink mode of oscillation of a magnetic flux tube (Aschwanden et al. 1999; Nakariakov et al. 1999). Figure 14.2 and the sketch in Figure 14.3, both taken from Nakariakov et al. (1999), show the typical behaviour. As discussed in detail in Chapter 6, for a thin coronal loop the fast kink mode propagates with a speed c = ck given in the long-wavelength limit of kz a  1 by ω ∼ kz ck . The period

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Solar Applications of MHD Wave Theory

Figure 14.3 A sketch of the vibrating loop, exhibiting a transverse oscillation. (From Nakariakov et al. 1999.)

of each of these oscillations is given by the relation (14.31). Further progress depends upon us being able to identify the mode of oscillation that is being detected. In cases where the kink mode is evident, we may set c = ck , the usual kink speed in a medium. Then (14.32) with c = ck and P = Pkink implies ck =

2L , NPkink

(14.33)

where Pkink is the period of the kink mode. Now the speed ck is determined by c2k =

ρ0 c2A + ρe c2Ae . ρ0 + ρe

(14.34)

If we assume that the magnetic field strength B0 inside the loop is much the same as the field strength in the environment (reasonable in a low β medium), so that B0 ≈ Be , then we have c2k =

2B20 . μ(ρ0 + ρe )

If now we combine (14.35) with (14.33), we obtain   ρe 1/2 L 1/2 1/2 . 1+ B0 = (2μ) ρ0 ρ0 NPkink

(14.35)

(14.36)

Equation (14.36) gives the field strength B0 in tesla, for a loop length L in metres, kink mode period Pkink in seconds, and plasma densities ρ0 and ρe in kg m−3 . A formula of the form (14.36) was first given by Nakariakov and Ofman (2001).2 2 In fact, there is an ambiguity in Nakariakov and Ofman (2001) where their distance L is not explicitly defined. We may

interpret their formula (6) in terms of footpoint separation distance Lfootpt . Supposing a loop to be a semi-circle standing vertically, the loop length L and footpoint separation distance Lfootpt are related by 2L = π Lfootpt , and then in terms of Lfootpt we have for the fundamental (N = 1) the relation (Roberts and Nakariakov 2003) B0 = π(μ/2)1/2 ρ0 1/2 (1 + ρe /ρ0 )1/2 (Lfootpt /Pkink ),

giving the magnetic field strength in tesla. In cgs units (with distances in cm and densities in g cm−3 ) the equivalent formula is B0 = (2π 3 )1/2 ρ0 1/2 (1 + ρe /ρ0 )1/2 (Lfootpt /Pkink ),

giving the field strength in gauss. This is effectively formula (6) of Nakariakov and Ofman (2001).

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445

Expressed in terms of the total number density N0 of particles per cubic metre inside ˆ p N0 , with mp being the mass of a proton and μm ˆ p the the flux tube (where ρ0 = μm mean particle mass of the plasma) and the density ratio ρe /ρ0 of the plasma density in the environment to the plasma density inside the flux tube, the Nakariakov and Ofman formula (14.36) becomes   ρe 1/2 L , (14.37) B0 = CSI N0 1/2 1 + ρ0 NPkink ˆ p )1/2 . Expression (14.37) gives the magnetic field strength with the constant CSI = (2μμm in tesla for loop length L in metres, kink period Pkink in seconds, internal particle total number density N0 in m−3 and harmonic number N (for the fundamental, N = 1). With a proton mass mp = 1.673 × 10−27 kg and the mean atomic weight μˆ = 0.6 appropriate for the corona, we obtain CSI = 0.502 × 10−16 in SI (mks) units.3 We can write (14.37) in a more convenient form as   1/2   N0 m−3 ρe 1/2 L [Mm] . (14.38) 1 + B0 [G] = 15.88 ρ0 NPkink [s] 1015 This gives the field strength B0 in gauss for total number density N0 in units of m−3 , loop length L in Mm, and period Pkink in seconds. Consider the TRACE observations discussed by Nakariakov and Ofman (2001). They focussed on two events reported by Aschwanden et al. (1999), Nakariakov et al. (1999) and Schrijver and Brown (2000), where standing global oscillations of a loop were detected and could be modelled as the fundamental mode N = 1. In the event of 14 July 1998 they recorded a period of P = 256 s in a loop with estimated length L = 1.3 × 108 m (with a footpoint separation distance of Lfootpt = 8.3 × 107 m, the loop being treated as a semicircle standing vertically). Thus, for this event we have a propagation speed of 2L/(NP) which for the fundamental of N = 1 is 1.016 × 106 m s−1 . The second event reported, for 4 July 1999, had a period of P = 360 s in a loop of length L = 1.9 × 108 m (with a footpoint separation distance of Lfootpt = 1.2 × 108 m). The propagation speed of 2L/P for the fundamental is then 1.055 × 106 m s−1 . The propagation speeds for these two events are thus far in excess of the sound speed cs in the corona (at a temperature of 2 × 106 K we find that cs = 2.1 × 105 m s−1 ), suggesting that a fast magnetacoustic or Alfv´en wave is involved. Moreover, the observations make clear that the loop as a whole is disturbed, in a manner akin to the kink mode rather than an Alfv´en wave. Thus, the observations are in accord with the kink mode of oscillation. To illustrate (14.37) in the event of 14 July 1998, a plasma number density in the range N0 = (1–6) × 1015 m−3 is assumed, together with an assumed density ratio of

3 Expressed in the cgs Gaussian system of units, formula (14.37) becomes

  ρe 1/2 L B0 = Ccgs N0 1/2 1 + , ρ0 NPkink

where the constant Ccgs = (8π μm ˆ p )1/2 and now L is in cm, Pkink in s, and N0 in cm−3 ; with a proton mass mp = 1.673 × 10−24 g, Ccgs = 5.02 × 10−12 in cgs units and B0 is in gauss (G).

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ρe /ρ0 = 0.1. Then, we find that equation (14.37) yields a field strength in the range B0 = 0.00085–0.00207 tesla (= 8.5–20.7 G). For the event of 4 July 1999, the same density range and ratio give B0 = 0.00088–0.00215 tesla (= 8.8–21.5 G). We end the discussion of the kink mode by examining three recent examples of such oscillations. These illustrations are based upon data taken with the Atmospheric Imaging Assembly onboard the Solar Dynamics Observatory (SDO/AIA). The first case pertains to a detailed analysis of a coronal loop that was observed to oscillate vertically. Conducted by Aschwanden and Schrijver (2011), this was the first loop oscillation observed with SDO/AIA and it pertained to an event recorded on 16 October 2010. Aschwanden and Schrijver (2011) report a period of P = 375.6 s in a loop structure of length L = 143 Mm; the interior of the loop is determined to have an electron number density of 1.9 × 108 cm−3 (giving an internal total number density of N0 = 3.8 × 1014 m−3 ). The density ratio ρe /ρ0 is determined to be 0.08. Then, application of (14.37) gives a magnetic field strength of B0 = 0.00039 tesla (= 3.9 G), with a corresponding Alfv´en speed of cA = 560 km s−1 ; the density ratio of 0.08 implies an external Alfv´en speed of cAe = 1978 km s−1 and a kink speed of ck = 761 km s−1 . Our second recent application of the kink mode formula is provided by Zhang et al. (2016), who investigated the evolution of two prominences and two bundles of coronal loops, also observed with SDO/AIA. Interestingly, it turned out that one of the prominences and one of the loop bundles supported large amplitude oscillations, while in the other prominence and loop bundle no oscillations were detectable. For a loop oscillation, Zhang et al. (2016)4 reported a period of P = 960 s in a loop of length L = 430 Mm, with an electron number density of 9.0 × 1014 m−3 (with corresponding total number density N0 = 1.8. × 1015 m−3 and plasma density of ρ0 = 1.81 × 10−12 kg m−3 ). It is assumed that ρe /ρ0 = 0.1. Then from equation (14.37) we determine a magnetic field strength of B0 = 0.001 tesla (10 G) with an associated Alfv´en speed of cA = 664 km s−1 and a kink speed of ck = 896 km s−1 . For a temperature of T0 = 2 × 106 K we determine a sound speed of cs = 214 km s−1 , yielding a plasma β of 0.12. Finally, we consider the results of Long et al. (2017a), who used SDO and CoMP to examine an oscillation in a trans-equatorial loop system. They report a period of 17.45 minutes in a loop of length L = 711 Mm. Densities are determined as N0 = 1.1 × 1014 m−3 inside the loop and Ne = 0.25×1014 m−3 outside, producing a ratio of ρe /ρ0 = 0.227. The loop is observed to oscillate in the second harmonic, not the fundamental. Then, application of (14.37) with N = 2 produces a field strength of B0 = 0.000198 tesla or close to 2 G (see Long et al. 2017b). 14.5.2 Slow Standing Modes The discovery of acoustic-like longitudinal oscillations in hot flare loops (Wang et al. 2002, 2003a, b; Ofman and Wang 2002) raises their interpretation in terms of slow mode waves. The first direct detection of SUMER oscillations in imaging data was made by Kumar, Innes and Inhester (2013). Just as with the fast kink modes, slow modes may be 4

Zhang et al. (2016) wrongly apply formula (14.36), so their numbers needed recomputing; this is carried out here.

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447

used as a means to determine the magnetic field strength in a loop (Roberts, Edwin and Benz 1984; Roberts 2004, 2006; Wang, Innes and Qiu 2007). To see how this comes about consider as an illustration the observation reported as having a period of 17.6 min in a loop of length L = 191 Mm. This has an associated speed of 2L/P, which for this illustration is 3.62 × 105 m s−1 , or 362 km s−1 . This immediately suggests a slow wave in a hot loop, for otherwise an Alfv´enic or fast wave would be expected to have a faster speed than this, which is close to the expected sound speed in a hot loop. Accordingly, we take c = ct , the propagation speed of a slow magnetoacoustic wave in a thin (kz a  1) tube. Then with P = Pslow equation (14.32) gives 1/2  2L 2L 1 Pslow = = , (14.39) 1 + γβ Nct Ncs 2 which may be rearranged to give the plasma β (= p0 /(B20 /2μ) = 2c2s /γ c2A ):   2 N 2 P2slow −1 , β= 2 γ τac

(14.40)

where τac = 2L/cs is the travel time for a sound wave along the length of the flux tube and back again. Rearranged to give the Alfv´en speed cA , equation (14.40) gives   2L 1 cA = (14.41)  1/2 , NPslow 4L2 1− 2 2 2 cs N Pslow

from which the magnetic field strength B0 follows:  √  2L 1/2 B0 = CSI N0  NPslow 1−

1 4L2 c2s N 2 P2slow

1/2 .

(14.42)

Here (as earlier) CSI = (2μμm ˆ p )1/2 . Formula (14.42) gives the magnetic field strength B0 in tesla, with loop length L in m, slow mode period Pslow in s for harmonic number N, and sound speed cs in m s−1 . A formula of this form was first given in Wang, Innes and Qiu (2007), who analysed seven SUMER Doppler events, finding that β is in the range 0.15–0.91, with a magnetic field strength 12–61 G. The equivalent formula in the cgs system is  √  2L 1 1/2 B0 = Ccgs N0 (14.43)  1/2 NPslow 4L2 1− 2 2 2 cs N Pslow

where (as earlier) Ccgs = (8π μm ˆ p )1/2 . This gives the magnetic field strength B0 in gauss, with loop length L in cm and loop total number density N0 in particles per cm3 and proton mass mp in grams (g). For the observation reported by Wang et al. (2003a, b), of an oscillation on 15 April 2002 with a period of 17.6 minutes in a loop of length L = 191 Mm, this corresponds to

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a slow tube speed of ct = 362 km s−1 . At a temperature of 6.3 × 106 K the sound speed is cs = 380 km s−1 ; for a total number density of 1016 m−3 , application of (14.42) produces a magnetic field strength of B0 = 41.9 G, an Alfv´en speed of cA = 1179 km s−1 and a plasma beta of β = 0.12. Finally, turning to Wang, Innes and Qiu (2007), who analysed seven events, we note that their loop 1 has a length L = 134 Mm and exhibited an oscillation with a period of 12.9 minutes (i.e., P = Pslow = 774 s), with a plasma temperature of T0 = 6.6 MK and number density N0 = 10.4 × 1015 m−3 ; application of formula (14.42) produces a field of B0 = 27.5 G, with associated sound speed cs = 389 km s−1 , Alfv´en speed cA = 759 km s−1 , tube speed ct = 346 km s−1 , and a plasma beta of β = 0.31. As a second illustration, for their loop 5, for which L = 98 Mm, Pslow = 558 s, T0 = 6.3 MK and N0 = 22.2 × 1015 m−3 , using (14.42) we find that B0 = 48.7 G, with an associated sound speed cs = 380 km s−1 , Alfv´en speed cA = 920 km s−1 , tube speed ct = 351 km s−1 and a plasma beta of β = 0.20. Similar values are calculated by Wang, Innes and Qiu (2007).

14.5.3 Fast Sausage Standing Modes The two cases of standing waves that we have discussed above share in common the dispersive behaviour that ω ∼ kz c as kz → 0. But not all modes have this property. In particular, the fast sausage waves exhibit a low wavenumber cutoff. Cutoff arises when ω = kz cAe , c = cAe . Figure 14.1 shows the behaviour: for the fast sausage mode to be a trapped mode of the tube it is necessary that kz > kzcutoff ,

(14.44)

where the cutoff wavenumber kzcutoff is given by (Roberts, Edwin and Benz 1984) kzcutoff

1 1 = j0,s a



(c2s + c2A )(c2Ae − c2t ) (c2Ae − c2A )(c2Ae − c2s )

1/2 ,

(14.45)

with j0,s denoting the sth zero of the Bessel function J0 . There are an infinite number of zeros of the function J0 ; the first (s = 1) zero is j0,1 = 2.4048, the second zero is j0,2 = 5.5201, etc. If kz > kzcutoff , then the fast sausage mode is trapped within the tube, which acts as a waveguide for such oscillations. However, if kz < kzcutoff , then the waveguide is leaky and energy is leaked to the environment from the interior of the tube. This is the case discussed earlier in Section 14.2. Consider the case of a β = 0 plasma, corresponding to a uniform magnetic field (B0 = Be ) within which a flux tube is defined by being a region of high (or low) plasma density; then equilibrium requires that ρ0 c2A = ρe c2Ae , with sound speeds effectively set to zero. Then the expression for kzcutoff simplifies to (Roberts, Edwin and Benz 1984; Nakariakov, Melnikov and Reznikova 2003; Aschwanden, Nakariakov and Melnikov 2004) kzcutoff

= j0,1

1 a



c2A c2Ae − c2A

1/2 = j0,1

1 a



ρe ρ0 − ρe

1/2 ,

(14.46)

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449

Figure 14.4 The dependence of the period P of oscillations on the wavelength for different measures α of steepnesses of the radial profile in Alfv´en speed cA (r). The thick solid curve corresponds to the parameter α → ∞; other curves are for various α ranging from 2 to 8 (the higher the value of α, the steeper is the profile, with α → ∞, giving a step function in Alfv´en speed (or density ρ0 (r)). The Alfv´en speed at the centre of the tube is here related to the Alfv´en speed cAe in the far environment (as r → ∞) through cA (r = ∞) = 5cA (r = 0). The period is measured in units of a/cA (∞) and the wavelength λ = 2π/kz is given in units of a. The straight lines correspond to the cutoffs P = 2π/kz cAe and P = 2π/kz cA (r = 0). For the step function case, here cAe = 5cA , P/(a/cAe ) ≈ 13.9627 as λ → ∞ and Pcutoff /(a/cAe ) ≈ 12.7999. (From Nakariakov, Hornsey and Melnikov 2012.)

where to be specific we have selected the first radial mode (so s = 1 and j0,s = j0,1 ≈ 2.4048). Note that akzcutoff is small for high density (ρ0  ρe ) tubes. Now when kz < kzcutoff , the sausage mode is leaky and the frequency ω is complex. With ω = ωR + iωI for real part ωR and imaginary part ωI , the period Psaus (= 2π/ωR ) associated with the frequency ωR is given (for kz = 0, ρe /ρ0  1) by equation (14.23). Psaus is proportional to the travel time a/cA across the tube. When kz > kzcutoff , the wave is trapped within the tube and no leakage occurs (ω = ωR , ωI = 0). With kz quantized (by (14.29)) to fit into a loop of length L, the cutoff condition kz > kzcutoff for trapped sausage waves requires that πN L< j0,1



c2Ae c2A

1/2 −1

a.

(14.47)

It is apparent that this inequality favours dense (ρ0  ρe ) loops or short, broad loops (a/L of order unity) or modes with many nodes along the tube (so N is large).

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The timescale Pcutoff (= 2π/ωcutoff ) associated with the cutoff frequency ωcutoff = is given by (for β = 0)  1/2   2πa 1 1 2π a ρ0 − ρe 1/2 cutoff = − 2 = . (14.48) P j0,1 c2A j0,1 cA ρ0 cAe

kzcutoff cAe

Note that Psaus > Pcutoff , for ρe < ρ0 . In the extreme of a very high density loop, ρ0  ρe , Psaus ≈ Pcutoff and Psaus ∼ 2.6 a/cA . This timescale is very short. For example, for a loop with cross-sectional radius a ∼ 103 km and an Alfv´en speed of order cA ∼ 103 km s−1 we obtain a timescale of 2.6 s. The frequency and decay of fast sausage waves has recently been discussed by Vasheghani Farahani et al. (2014), determining the behaviour of the dispersion relation near the cutoff locations of a fast wave in a β = 0 plasma. We can ask what is the behaviour of the period P(kz ) of the sausage mode as a function of wavenumber kz ? Equation (14.23) gives the behaviour as kz → 0 in the leaky zone: P → Psaus as kz → 0. In the trapped zone, when kz > kzcutoff , Figure 14.1 shows that ω ∼ kz cA for kz → ∞, and so the period P = 2π/ω ∼ 2π/(kz cA ). Thus P(kz ) → 0 as kz → ∞; in terms of wavelength λ (= 2π/kz ), this gives P(λ) → 0 as λ → 0. Now for finite values of kz within the trapped mode zone we need to use the dispersion relation (14.27) to determine the behaviour, but in fact we know that when kz → kzcutoff then ω ∼ kz cAe , as illustrated in the dispersion diagram 14.1. Then, for a β = 0 plasma, expression (14.48) gives the cutoff period. In a plot of period P(λ) versus wavelength λ (which is proportional to loop length for a standing wave), we see that Psaus is the asymptotic limit as wavelengths approach infinity: P → Psaus as λ → ∞ or kz → 0, with Psaus given by Psaus =

1 2π a . j(0,1) cA [1 + 12 (ρe /ρ0 ) ln(ρe /ρ0 )]

(14.49)

Moreover, Psaus is the largest period attained. This may be inferred from the numerical results presented in Kopylova et al. (2007) for the step function profile, and is made explicit in Nakariakov, Hornsey and Melnikov (2012) who consider a range of smooth profiles (including the step function as a parametric extreme); see also Chen et al. (2015a, b). Figure 14.4, from Nakariakov, Hornsey and Melnikov (2012), shows the behaviour. As a function of kz the sausage mode period P(kz ) falls off from a high of Psaus , passing through the cutoff period Pcutoff as kz reaches kzcutoff , and then tends to zero at large kz a. As a function of wavelength λ (or equivalently of loop length L), the period P(λ) of the sausage mode climbs from zero for λ ≈ 0, passes through the cutoff value Pcutoff and finally asymptotes to Psaus for large λ (or large loop length L). Thus, under coronal conditions the period Psaus is the largest period associated with the sausage mode.

14.6 Fast Kink Modes and Resonant Absorption: The Seismology of Small Scales The observation that many coronal oscillations are subject to decay has raised the question of what is the cause of that decay. Is it due to natural loss mechanisms such as viscosity or thermal conduction? Or is it due to other processes? One strong contender as an explanation

14.6 Fast Kink Modes and Resonant Absorption: The Seismology of Small Scales

451

of the observed rapid decay in kink mode oscillations is resonant absorption. This is not a decay mechanism such as viscosity or thermal conduction whereby the energy in an oscillation is subject to viscous damping and heating, but rather it is a process by which the energy in one mode of oscillation is transferred into another mode of oscillation, building up the new mode of oscillation at the expense of the initial oscillation. Of course other loss mechanisms such as viscosity may contribute to this picture, but the essential process of resonant absorption is one in which no loss of energy actually arises. In consequence of this process it is natural to examine the timescale by which it may act, and from this we may deduce information about the process that operates on small scales. Ultimately, we may be able to deduce the thickness of a layer in which resonant absorption is considered to act, providing a seismology of a small-scale process. To see how this might work, recall a result obtained in Chapter 6 and further discussed in Chapter 8 which describes the timescale expected from the resonant absorption process. It was shown that for the two density profiles, the sinusoidal and linear ones, the damping timescale τdamping is related to the kink mode period Pkink through   a ρ0 + ρe τdamping = C (14.50) Pkink . l ρ0 − ρe The constant C depends upon the specific details of the density profile in the thin layer connecting the boundary of the tube to the environment. In particular, for the linear and sinusoidal profiles we have  4/π 2 , linear profile, C= (14.51) 2/π , sinusoidal profile. With observational measurements of the kink mode period Pkink and the damping timescale τdamping , we can rewrite equation (14.50) to determine the width of the thin transition layer thickness l as a fraction of the tube radius a:   ρ0 + ρe Pkink l . (14.52) =C a ρ0 − ρe τdamping In the extreme ρ0  ρe , we obtain Pkink l . =C a τdamping

(14.53)

Formulas such as these may be used to determine l/a. They were first applied directly to TRACE observations of a kink loop oscillation by Ruderman and Roberts (2002), who deduced that the observed damping and period of a kink oscillation in a coronal loop was consistent with a loop flux tube having a thin density transition layer of width l = 0.23 a. Ruderman and Roberts (2002) used the observation first studied by Nakariakov et al. (1999). At about the same time, Goossens, Andries and Aschwanden (2002) examined 11 oscillating loops and concluded that the observed damping was consistent with a layer ranging from l = 0.16 a to l = 0.46 a. Aschwanden et al. (2003), Roberts (2008) and Goossens (2008) give discussions.

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In a transitional layer that is no longer thin, the thin boundary approximation (l  a) is no longer applicable and a numerical investigation becomes necessary, though the analytical results holding for a thin boundary provide a useful guide. Van Doorsselaere et al. (2004), Terradas, Oliver and Ballester (2006), Arregui et al. (2007), Terradas, Goossens and Verth (2010) and Soler et al. (2013) provide a detailed appraisal of the situation. Finally, we note that these ideas also apply, with little change, to prominences; this is discussed briefly in Section 14.11.1.

14.7 Fast Kink Modes: Spatial and Temporal Damping Coronal Multi-Channel Polarimeter (CoMP) observations have revealed periodic Doppler shift oscillations propagating along large off-limb coronal loops (Tomczyk et al. 2007; Tomczyk and McIntosh 2009). The waves exhibit a strong decay in amplitude as they propagate along a loop. These oscillations, which are Alfv´enic in nature, appear to be a ubiquitous coronal phenomenon. Motivated by the observations, Pascoe et al. (2012) carried out a numerical study of the propagation of kink oscillations in a coronal tube with an inhomogeneous density profile of the form studied above. The oscillations exhibited decay, but that decay was described by a Gaussian form followed by an exponential form. In certain circumstances, the decay might appear to be entirely Gaussian, involving an expression of the form 2 ) exp(−z2 /LG

rather than the exponential

  exp −|z|/LE .

Here LE and LG are appropriate damping scales or distances. Motivated by the observational and numerical studies, Hood et al. (2013) showed analytically that a Gaussian decay may indeed arise in the first stages (low z) of a driven oscillation and thus in some circumstances this is likely to be the observed behaviour, rather than the exponential decay behaviour that occurs at high z. In fact, a more complicated spatial decay, involving both Gaussian and exponential elements, can be expected. The analysis by Hood et al. (2013) is for a thin transition layer with a linear density profile (though we can expect other profiles to give similar results), and is based upon the β = 0 equations for a structured magnetic flux tube. The β = 0 linear equations follow immediately from equations (14.3)–(14.6) with cs = 0, uz = 0 and the equilibrium such that ρ0 (r)c2A (r) = a constant: ∂ 2 ur ∂ 2 ur ∂ − c2A (r) 2 = c2A (r) , 2 ∂r ∂t ∂z

∂ 2 uφ ∂ 2 uφ 1 ∂ − c2A (r) 2 = c2A (r) , 2 r ∂φ ∂t ∂z

(14.54)

where  = div u =

∂ur 1 ∂uφ 1 + ur + . ∂r r r ∂φ

(14.55)

These equations were discussed in Chapter 11, Section 11.3. The analysis by Hood et al. (2013) is for a density ρ0 (r) that has a linear transition region of width l on a tube of

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453

radius a (see equation (8.111) in Chapter 8) and shows that the amplitude A(z) of a kink oscillation with driven frequency ω is given approximately by    ⎧ ⎨ 1 A 1 + exp − z2 , 0 ≤ z < zh , 0 2 LG (14.56) A(z) = 2    ⎩ h , , z > z A1 exp − z−z h LE where

) ) 4 a )) ρ0 + ρe )) LE = 2 ) ck P, π l ρ0 − ρe )

) ) 2 a 1/2 )) ρ0 + ρe )) LG = ) ρ − ρ ) ck P, π l 0 e

(14.57)

for a driver of period P (= 2π/ω) (see also Pascoe et al. 2013). The constant A0 is the amplitude at z = 0 and A1 may be chosen to give continuity of A(z) at z = zh ; zh is an appropriate spatial scale, typically a few wavelengths. Notice that as the transition layer thickness l tends to zero, LE → ∞ and LG → ∞ (implying no decay). Notice also that the Gaussian spatial scale LG and the exponential spatial scale LE are related, for a linear density transition layer, by LG = (π/2)(l/a)1/2 LE . Furthermore, the scale LE is related to the exponential damping timescale τdamping given in equation (14.50); specifically, LE = ck τdamping . To illustrate these scales, suppose l = a/10 and ck = 103 km s−1 and a period P = 102 s. Then for a density enhancement ρ0 = 3ρe we obtain LG = 4.0 × 105 km and LE = 8.1 × 105 km. A related analysis can be carried out for standing waves in a coronal loop, as shown by Ruderman and Terradas (2013); this suggests a temporal dependence of the form (Pascoe et al. 2018)   ⎧ 2 ⎪ ⎨A0 exp − t 2 , 0 ≤ t < th , τG (14.58) A(t) =

 ⎪ ⎩A exp − t−th , , t > t 1 h τE where τE = τdamping is the exponential damping time and th is the time to transition from Gaussian to exponential temporal behaviour. Results of this type, describing either spatial decay or temporal decay, offer a further opportunity for the seismology of coronal flux tubes (Pascoe et al. 2018).

14.8 Fast Sausage Modes and Quasi-Periodic Pulsations Solar flares have revealed themselves as a source of oscillations that offer the potential for developments in the seismology of the corona. Almost all solar flares exhibit quasiperiodic pulsations, often with multiple periods evident. Similar events are also detected in stellar flares. Aschwanden (1987, 2004) provides summaries of the observations and possible interpretations. More recent reviews are given in Nakariakov and Melnikov (2009) and Van Doorsselaere, Kupriyanova and Yuan (2016). Short periods (typically of seconds or a few tens of seconds) and long periods (typically several minutes) may be present in

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a single oscillating structure or loop, providing constraints on the seismology of a loop. The greater the number of modes that can be detected, and identified, the more fruitful the seismology of a loop is likely to prove. A simple illustration of the use of two modes is provided by Kolotkov et al. (2015), who reported oscillations detected by the Nobeyama Radioheliograph of period 15 s and 100 s (there is also a 45 s period detected). Kolotkov et al. propose that the short period oscillation is due to the fast sausage mode and that the long period oscillation is due to the fast kink mode, all taking place simultaneously in the same structure. The intensity oscillations seem to be associated with a loop of width 2a = 8 Mm and length L = 100 Mm, suggesting a tube with ratio a/L = 1/25. If we take the short period as being due to the fast sausage mode with period Psaus (which we take to be given by the expression (14.23) or its extreme form in (14.24) and the long period as being due to the fast kink mode with period Pkink given by equation (14.33) for the fundamental (N = 1) kink mode, then we have Pkink =

2L , ck

Psaus =

2π a . j0,1 cA

(14.59)

In a β = 0 medium the kink speed ck is related to the Alfv´en speed cA inside the tube through  ck =

2ρ0 ρ0 + ρe

1/2 cA ,

(14.60)

and so   a ρe 1/2 Psaus j0,1 . = √ 1+ L ρ0 Pkink π 2

(14.61)

For ρe  ρ0 and observed periods of Psaus = 15 s and Pkink = 100 s, we obtain a/L ≈ 0.08, which compares well with the observed ratio of 0.1. Both modes were observed to decay, the sausage mode on a timescale τsaus = 90 s and the kink mode on a timescale τkink = 250 s. With τsaus

1 = 2 π



ρ0 ρe

 Psaus ,

(14.62)

we obtain ρ0 /ρe ≈ 59.2 (and so ρe /ρ0 ≈ 0.017). Turning to the decay of the kink mode, if we again assume that this is due to a layer of inhomogeneity of spatial scale l on the boundary between the inside of the tube and its surroundings, then (14.50) applies and gives a transition layer of thickness l ≈ 0.25 a for a sinusoidal profile, l ≈ 0.16 a for a linear profile. A more systematic approach to this seismology has been given by Chen et al. (2015a, b), who also considered the event reported by Kolotkov et al. (2015). The effect of a non-zero plasma β is investigated in Chen et al. (2016).

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455

14.9 Propagating Waves 14.9.1 Fast Kink Mode: Propagating Waves In contrast to the standing waves, there has been far less observational information about propagating kink waves. Nonetheless, there have been some observations which seem to have a natural interpretation in terms of propagating waves. Propagating transverse waves have been detected in an open magnetic field structure by TRACE. Verwichte, Nakariakov and Cooper (2005) report dark ‘tadpole’-like structures moving downward in a post-flare arcade. Quasi-periodic transverse displacements of dark tadpole tails are seen, with periods in the range of 90–220 s. Phase speeds are in the range 200–700 km s−1 at a height of 90 Mm above the arcade, falling to 90–200 km s−1 at a height of 60 Mm. This phenomenon has been interpreted as fast magnetoacoustic kink waves propagating in and guided by an evolving open magnetic structure.

14.9.2 Fast Sausage Mode: Propagating Waves The impulsive excitation of fast waves in a magnetic slab has been discussed in Chapter 5, where it is explained that there are close analogies between the behaviour of fast magnetoacoustic body modes in a strongly magnetized plasma and Love waves in the Earth’s crust and Pekeris sound waves in an internal ocean layer. This close analogy was noted in Edwin and Roberts (1983) and the analogy was exploited in Roberts, Edwin and Benz (1983, 1984). We may expect that much the same behaviour that occurs in a magnetic slab also arises in a magnetic flux tube. Indeed, for a low β plasma comparing results for the two geometries, a slab and a cylinder, it is evident that the fast sausage mode behaves much the same in the two geometries. This is also the case for the harmonics of the fast kink waves. Only the fundamental kink wave behaves significantly different in the two geometries, in that in a cylindrical flux tube the fundamental kink mode in a thin tube has a phase speed close to the kink speed ck , whereas the same wave in a magnetic slab has a speed that is close to the external Alfv´en speed cAe . Nonetheless, despite this marked difference the results for the two geometries are closely similar. As noted in Chapter 5, the cutoff frequency also plays an important role in the behaviour of propagating fast waves. Indeed, the full form of the dispersion relation (14.27) comes into force when we consider impulsively excited waves, for then the whole spectrum of frequencies is involved in the determination of the generated wave. Impulsively excited fast sausage waves were discussed in some detail by Roberts, Edwin and Benz (1983, 1984), drawing on an analogy with sound waves in an ocean layer (Pekeris 1948). An impulsively excited wave exhibits a three-phase temporal signal, consisting of a periodic phase, a quasi-periodic phase, and a decay phase. The cutoff frequency determines the largest period of oscillation carried by the propagating wave. It is interesting to note that Williams et al. (2001, 2002) have observed the rapid propagation of an oscillatory disturbance in a loop. This remarkable result was achieved by the deployment of ground-based rapid photography using the camera instrument SECIS, taking 44 frames per second of the eclipsed corona. Comparisons with SoHO observations of the corona then allowed a loop to be identified as a candidate for supporting the oscillation. The loop, of length 200 Mm and

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radius a = 5 Mm, supported a propagating wave travelling along the loop with a speed of 2100 km s−1 and carrying a periodicity of 6 s. Using the similarity in the dispersion relations for a slab and a tube, we can explain this observation in terms of the theory of an impulsively excited fast sausage mode. Dispersion in the mode means that a wave packet, generated impulsively so that all frequencies are excited, propagates with the group velocity of the wave and carries an oscillation with a period of order Pcutoff , the cutoff period being determined for a tube through equation (14.48). Numerical simulations of this effect have been given in Nakariakov et al. (2004) and Pascoe, Nakariakov and Arber (2007) using a smooth cross-field density profile. In fact, the timescale Pcutoff is the longest timescale in the impulsively excited wave packet. The densities reported by Williams et al. give a ratio ρe /ρ0 = 0.4. We associate the observed speed of 2100 km s−1 with the group velocity cg of a fast wave, determined numerically from the dispersion relation (14.27). Then, taking Pcutoff = 6 s and ρe /ρ0 = 0.4, equation (14.48) implies a tube radius of a = 6216 (cA /cg ) km. So with cg ≈ cA we obtain a tube radius of approximately 6000 km, consistent with the observationally determined radius. 14.9.3 Slow Modes: Propagating Waves The slow waves (with phase speeds between the slow speed ct and the tube’s sound speed) are only weakly dispersive. In a strongly magnetized plasma, their speed is close to the sound speed in the tube. The coronal loop provides an almost rigid tube for the onedimensional ducting of sound waves. Propagating longitudinal waves have been detected in the legs of loops (Berghmans and Clette 1999; De Moortel, Ireland and Walsh 2000; Robbrecht et al. 2001; De Moortel et al. 2002a–c) and seem to have a natural interpretation in terms of the slow mode. Figure 14.1 shows that under coronal conditions the slow mode has only a weak dispersion, and propagates sub-sonically with the tube speed ct . Under typical TRACE loop conditions this gives a speed only very slightly below the sound speed inside the tube; for example, a sound speed of cs = 200 km s−1 and an Alfv´en speed of cA = 1000 km s−1 produce a tube speed of ct = 196 km s−1 . The observed waves penetrate only some 10% into the loop before they are lost from view and are presumably damped. 14.10 Prominence Oscillations Prominences are dense, cool structures observed in the corona and may last for days or even months. They are complex structures with temperatures and densities about two orders of magnitude different from the coronal environment of the prominence (see Mackay et al. (2010) and Priest (2014) for recent discussions of their general features). Gravity and magnetism are expected to play an important role in maintaining their basic state. Because even their equilibrium structure is uncertain, though a number of models exist, analysing the modes of oscillation of a prominence is complicated. Nonetheless, it seems likely that a number of generic features may be deduced from the study of relatively simple models. This is the viewpoint taken in Roberts (1991a), Roberts and Joarder (1994) and Anzer (2009), and explored in greater detail in Joarder and Roberts (1992, 1993) and Oliver et al. (1992, 1993). Observations and theories of oscillations in prominences are reviewed

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in Oliver (2001, 2009), Oliver and Ballester (2002), Tripathi, Isobe and Jain (2009), Arregui, Oliver and Ballester (2012), and Ballester (2013). Here we point out some aspects of oscillations in such structures that may hopefully prove to be of generic interest. It should be noted that simple models do not necessarily mean easy to calculate models; it means only that the equilibrium state is simple, defined by at most a few basic physical processes and a simple geometry. Such models have one very important feature: they tend to generate a specific formula for basic physical quantities such as the oscillation period. Such formulas may be tested against refinements in the model, improvements that are perhaps best treated by numerical means or through statistical evidence from multiple observations of prominence oscillations. The observational classification of prominence oscillations remains at a basic stage of development, though that is likely to change as new observations are analysed. Early investigations suggested a classification into short, intermediate or long period oscillations, ranging from a few minutes to an hour or more, but this does not currently appear helpful. Oscillations are instead classified as small amplitude or large amplitude, which although helpful in terms of what sort of analysis (linear or nonlinear) may be needed to understand the oscillations, it remains only a partial classification. Small amplitude oscillations often occupy only a part of the prominence, and not the whole, and typically have velocity amplitudes of up to a few km s−1 . Large amplitude oscillations usually involve the vibration of the whole prominence, with velocity amplitudes in tens of km s−1 ; the whole prominence may suffer large amplitude (ranging from a few Mm to 40 Mm) displacements. Periods range from tens of minutes to 1–3 hours. Transverse oscillations are perpendicular to the long axis of the prominence, longitudinal oscillations are along the long axis of the filament (prominence). Damping times are typically several periods. The trigger for the oscillation may be a large-scale event, such as a Moreton wave, or it may be a more local event such as a reconnection jet. Also, prominences are observed to have a fibril nature, being made up of many thin magnetic threads which are filled with cold dense plasma. Threads sometimes oscillate together, sometimes in isolation. Periods of thread oscillations are typically 135– 250 s, with thread lengths up to 16 000 km, the threads occupying a much longer magnetic flux tube (length of order 102 Mm). In any case, on simple theoretical grounds based upon linear theories, it seems natural to distinguish modes as transverse or longitudinal or smallscale (thread) oscillations. Prominence oscillations may also be used for prominence seismology. An early suggestion of using oscillations for the determination of prominence properties was made by Hyder (1966), who modelled the oscillation as a damped harmonic oscillator with the frequency related to the strength of the local radial magnetic field (and damping proportional to viscosity). Somewhat later, Roberts and Joarder (1994) opened their discussion with the comment ‘Prominences oscillate. That simple fact is potentially very important for it affords us with the possibility of using data on the modes of oscillation of a prominence to derive seismic information about the structure of the prominence and its coronal environment’ (see also the review by Ballester 2013). More recently, prominence seismology has been carried out by a number of authors. For example, R´egnier, Solomon and Vial (2001) and R´egnier et al. (2002) used space (SoHO) and ground-based (the Tenerife operated telescope Themis) observations combined with the model calculations of Joarder

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and Roberts (1993) to deduce the angle that the magnetic field makes with the prominence, finding 18◦ in one case and 31◦ in another. Very recently, large amplitude longitudinal oscillations observed by SoHO and SDO have been used to place a lower limit on the magnetic field strength that supports a particular prominence (Luna et al. 2017); this topic is treated in Section 14.12 below. There is one other aspect worth emphasizing regarding prominence oscillations. At least for some of the oscillations, in the words used by Jing et al. (2003) ‘the trigger for these large amplitude oscillatory motions in filaments is highly visible, unlike other types of solar oscillations’. In other words, they are likely to repay careful study.

14.10.1 Equilibrium Force Balance It is reasonable to assume that the mass of the prominence is supported against gravity by the tension force in the magnetic field, forming a slight dip in the field where the bulk of the prominence resides – much like a load in a sailor’s hammock. For an equilibrium magnetic field B0 = (Bx , 0, Bz ), with the x-axis being perpendicular to the prominence sheet and the z-axis pointing vertically upwards, the field produces an equilibrium current density j0 in the y-direction, with   1 ∂Bx ∂Bz j0 = − . μ ∂z ∂x There is thus an equilibrium magnetic force j0 × B0 = (j0 Bz , 0, −j0 Bx ). The vertical component of the magnetic force balances the gravitational force on the prominence, so that (ignoring the plasma pressure force) we have   Bx ∂Bz ∂Bx (14.63) − = gρ0 (x). μ ∂x ∂z The main variation in the equilibrium magnetic field arises in x, through the vertical component Bz (x). With the x-component of field through the prominence assumed to be constant, the divergence condition div B0 = 0 is met for Bz = Bz (x). Then force balance gives Bx ∂Bz = gρ0 (x). μ ∂x

(14.64)

Integrate this equation across the prominence width, taken to be a slab of width 2lp located in the region −lp < x < lp . Then 2 g Bx Bz (lp ) = Mprom , μ A where Mprom denotes the prominence mass for a sheet or slab with area A:  Mprom = A

lp −lp

ρ0 (x)dx ∼ 2lp Aρprom

(14.65)

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459

for a mean prominence plasma density ρprom . Thus, the ratio of the vertical field strength Bz (lp ) to the horizontal field strength Bx is Bz (lp ) gMprom 1 = . Bx 2A B2x /μ

(14.66)

Introduce the angle θ0 (in radians) that the equilibrium magnetic field makes with the horizontal on the outer surface of the prominence (at x = ±lp ); write Bz (lp ) = tan θ0 . Bx Then the equilibrium balance (14.66) may be rewritten in the form   gMprom 1 2 Bx = cot θ0 . μ 2A

(14.67)

Equation (14.67) gives us a means of estimating the horizontal magnetic field component Bx that threads through the prominence structure, giving Bx in terms of the prominence mass Mprom and the angle θ0 that the dipped field Bz makes with the horizontal as it leaves the prominence slab. To illustrate such an estimate, suppose the prominence has a mean density of ρprom = 10−10 kg m−3 in a slab of width 2lp = 5 × 103 m, the prominence sheet being of length 200 000 km and height 40 000 km with an area A = 8 × 1015 m2 . Then Mprom = 4.0 × 1012 kg and gMprom /(2A) = 0.0685 Pa. We have taken g = 274 m s−2 for the solar gravitational acceleration. Suppose first that the dip angle θ0 is π/4 radians (or 45◦ ). Then Bz (lp ) = Bx and (14.67) gives Bx = 0.00029 tesla (or 2.9 G); the field strength B0 = (B2x + B2z )1/2 is thus 4.1 G. Cowling (1976, p. 30)5 gives a similar estimate, pointing out that such a field strength is comparable with observations. For a perhaps more realistic dip angle of θ0 = π/36 radians (or 5◦ ), for which Bz (lp ) = 0.0875 Bx , equation (14.67) gives Bx = 0.00099 tesla (or 9.9 G), and then B0 = 9.96 G.

14.10.2 Transverse Oscillations: String (or Hybrid) Modes The contrast in density between a prominence and its environment allows the existence of what are variously called string (Roberts 1991a; Joarder and Roberts 1992; Roberts and Joarder 1994) or hybrid (Oliver et al. 1993) modes. These are transverse oscillations with a characteristic frequency ω = ωstring and period P (= 2π/ω) = Pstring given by (Roberts 1991a; Joarder and Roberts 1992, 1993; Roberts and Joarder 1994) ωstring =

cp , (Lp lp )1/2

Pstring =

2π (Lp lp )1/2 , cp

(14.68)

where cp is the elastic speed in a prominence sheet of width 2lp . Gravity is neglected, the oscillation resulting from the tension in the string (or magnetic field). Various string modes arise, corresponding to different propagation speeds cp . The field lines threading the prominence are taken to be anchored at locations a distance Lp either side of the prominence sheet, modelling the effect of photospheric or chromospheric line-tying at 5 Note that Cowling (1976) uses M to denote mass per unit area, equivalent here to M prom /A.

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footpoints a distance 2Lp apart. For an anchoring distance of 2Lp = 105 km, a prominence width of 2lp = 5000 km and a propagation speed of 27 km s−1 (corresponding to a fast magnetoacoustic speed in a medium with sound speed cs = 15 km s−1 and an Alfv´en speed cA = 23 km s−1 ), we obtain a period of about 43 minutes. To see how formulas such as (14.68) come about, consider the one-dimensional equation for a vibrating string (see, for example, Coulson 1955, chap. 2), with transverse motions u⊥ (z, t) satisfying 2 ∂ 2 u⊥ 2 ∂ u⊥ = c . ∂t2 ∂z2

(14.69)

Here c is the wave speed of the string; in terms of the tension T0 along the string and the string’s line density ρˆ0 we have c2 =

T0 . ρˆ0

Note that we have used line density in describing the motion of a string; when considering a prominence with plasma density ρp , the line density is just ρˆ0 = ρp A, where A is the cross-sectional area of the string. When the ratio of two densities arises, the area factor cancels out and we have ρˆ0 /ρˆe = ρp /ρe . Also, several possible speeds arise in an magnetohydrodynamic treatment based upon an analogy with a string: the fast magnetoacoustic speed, the slow speed, and the Alfv´en speed (with an angle reflecting the propagation direction with respect to the applied magnetic field). Such speeds are treated more fully in Joarder and Roberts (1992, 1993), Oliver et al. (1992, 1993) and Roberts and Joarder (1994). With a time dependence of the form eiωt , the wave equation gives ω2 d2 u⊥ + u⊥ = 0. dz2 c2

(14.70)

We are particularly interested in oscillations that disturb the string centre, corresponding to u⊥ = 0 at z = 0. Consider a section of string with line density ρˆ0 and length 2lp extending over −lp < z < lp , with wave speed cp (for which T0 = ρˆp c2p ). Then  u⊥ = A0 cos

 ωz . cp

(14.71)

Similarly, in an outer section of string where the line density is ρˆe and the wave speed is ce (so T0 = ρˆe c2e ) we take our solution of the wave equation in the form  u⊥ = Ae sin

 ω (Lp − z) . ce

(14.72)

Here we have chosen the solution of (14.70) that gives u⊥ = 0 at z = Lp , corresponding to the motions being anchored at z = Lp . The total field line (or string) length is thus 2Lp , with the dense (prominence) material occupying the central region −lp < z < lp . We assume

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symmetry about the centre point z = 0. Altogether, then, we may write our solution in z ≥ 0 as    Ae sin cωe (Lp − z) , lp < z ≤ Lp , u⊥ = (14.73) ωz A0 cos( cp ), 0 ≤ z < lp . We require that u⊥ and ∂u⊥ /∂z are continuous across z = lp , giving         ωlp ωlp ω ω ω ω (Lp − lp ) , A0 sin (Lp − lp ) . = Ae sin = Ae cos A0 cos cp ce cp cp ce ce Thus, eliminating the arbitrary constants A0 and Ae we obtain       ωlp cp ω tan (Lp − lp ) . = cot cp ce ce

(14.74)

This is the dispersion relation describing even modes (u⊥ = 0 at z = 0) on an elastic string with sections of differing densities joined together. The centre of the string, at z = 0, is displaced by this mode of oscillation. The equilibrium configuration is one for which the tension T0 in the string satisfies T0 = ρˆp c2p = ρˆe c2e . Then, for a plasma configuration with ρp c2p = ρe c2e , we can rewrite relation (14.74) in the form      1/2 ωlp ω ρe cot (Lp − lp ) . (14.75) tan = cp ρp ce Relations of these forms were first discussed in the context of prominence oscillations by Joarder and Roberts (1992, 1993), Joarder (1993), Oliver et al. (1993) and Roberts and Joarder (1994).

14.10.3 Heavy Load on a String It is interesting to note the extreme of lp → 0,

ρp → ∞,

taken in such a way that 2lp Aρp → Mprom , the (finite) mass representing the prominence. This limiting case corresponds to a mass Mprom attached to a vibrating string. In the current context, it corresponds to a narrow prominence mass vibrating transversely, with the whole prominence and its environment oscillating. In this limit we find that ωlp ω(Mprom /A) ∼ → 0. 1/2 cp 2T0 ρp 1/2 Accordingly, tan(ωlp /cp ) ≈ ωlp /cp and the dispersion relation (14.75) becomes     ωLp ωLp 2ρe Lp . tan = ce ce (Mprom /A)

(14.76)

This is the well-known dispersion relation for a heavy load on a vibrating string (Rayleigh 1877, sect. 136; Coulson 1955, sect. 23).

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Expanding the tan function in its power series, the first term in the expansion gives ω2 and the dispersion relation (14.76) then yields ω2 , from which we may deduce the frequency ω = ωstring and the associated period Pstring (= 2π/ωstring ): ωstring =

cp , (Lp lp )1/2

Pstring =

2π (Lp lp )1/2 . cp

(14.77)

Note that we have used 2ρe Lp /(Mprom /A) = Lp ρe /(lp ρp ). Relations of the form (14.77) arise whenever ρp  ρe and lp  Lp apply together. In the context of prominence oscillations, relations of the form (14.77) were first given by Roberts (1991a), and discussed in further detail in Roberts and Joarder (1994), Anzer (2009) and Arregui, Oliver and Ballester (2012). It is interesting to note that the frequency varies inversely with the geometric mean of two distances, the prominence half-width lp and the half-length Lp of the field lines between the anchor points. Finally, we note that we can take the approximate result (14.77) further by including the next term in the expansion of the tan function. The result for the period is      2π 1 Lp ρe 1/2 Pstring = (Lp lp ) 1+ cp 6 lp ρp     2  cp Lp 2π 1 = (Lp lp )1/2 1 + . (14.78) cp 6 lp c2e A similar result for a string was given by Rayleigh (1877). 14.11 Prominence Oscillations: Vibrations in Prominence Threads Observations indicate that prominences are often made up of fibril magnetic fields, termed threads, that appear to be thin (diameter about 200 km) flux tubes, part of which is filled with cold, dense prominence matter; the threads may extend over several thousand kilometers (3500–28000 km) (see Engvold 2001a, b; Martin 2001). An early model of such fibril structure in prominences was put forward by Ballester and Priest (1989). Waves have been detected in prominence threads (Lin et al. 2005). In particular, transverse swayings of the threads have been reported with periods of several minutes, both from high resolution ground-based observations (Lin et al. 2005, 2007, 2009) and from space observations such as Hinode (Okamoto et al. 2007; Ning et al. 2009). Such oscillations can be used to carry out a prominence seismology. In some respects the oscillations in prominence threads are akin to the oscillations in coronal loops: the periods are typically a few minutes or tens of minutes, with strong damping also observed. In other respects, there are significant differences: strong (two orders of magnitude) density enhancement in prominence threads compared with their coronal surroundings, whereas coronal loops have a less extreme (one order of magnitude) density enhancement. An early model of thread oscillations was proposed by Joarder, Nakariakov and Roberts (1997), and analysed further in D´ıaz et al. (2001), D´ıaz, Oliver and Ballester (2002, 2003) and Terradas et al. (2008). See Figure 14.5. Joarder, Nakariakov and Roberts (1997) examined the magnetoacoustic waves in a magnetic flux tube modelled as

14.11 Prominence Oscillations: Vibrations in Prominence Threads

463

Figure 14.5 A sketch of a prominence geometry, showing a prominence fibril or thread partially filled with cool dense matter. (From Joarder, Nakariakov and Roberts 1997.)

a Cartesian slab within which is a plasma density enhancement confined to a longitudinal section of the slab. The results for slab and cylinder geometry are similar. Of particular interest here is the kink mode in a thin tube. Consider, then, a thin magnetic flux tube, ignoring gravity. For the kink mode in a thin tube the radial motions ur (z, t) satisfy the wave equation (Dymova and Ruderman 2005; see Chapter 11) ∂ 2 ur ∂ 2 ur 2 = c (z) , k ∂t2 ∂z2

(14.79)

where ck is the kink speed. In a uniform tube for which ck is a constant, the frequency ω and longitudinal wavenumber kz are related by ω2 = kz2 c2k

(14.80)

with the kink speed given by c2k =

ρp c2Ap + ρe c2Ae ρp + ρe

.

(14.81)

Here cAp (= B0 /(μρp )1/2 ) is the Alfv´en speed inside the prominence thread where the plasma density is ρp ; cAe is the Alfv´en speed in the region everywhere outside of the dense part of the thread, where the plasma density is ρe . If we assume that the equilibrium magnetic field is uniform, as required in a low β plasma, then we can write the general kink speed squared as c2k =

2B20 /μ . ρp + ρe

(14.82)

If, further, the density contrast between the dense part of the thread and everywhere else outside the dense part is large, so that ρp  ρe (as appropriate under prominence conditions), then √ ck ∼ 2 · cAp , ρp  ρe . (14.83)

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Figure 14.6 A model of a prominence thread, showing a magnetic flux tube of total length 2Lp (anchored at the ends, representing the photosphere) with a dense plasma in the central region −lp < z < lp . The magnetic field is everywhere uniform and of strength B0 . The density in the central region of the thread is ρp , considered much larger than the uniform density ρe in the thread’s legs or in the external corona environment. [Note: in the text, 2W ≡ 2lp , 2L ≡ 2Lp , ρc ≡ ρe ; we have ignored the flow v0 .] (From Joarder, Nakariakov and Roberts 1997; see also Terradas et al. 2008.)

Thus, with a period P = 2π/ω and longitudinal wavelength λ = 2π/kz , we have λ , cAp = √ 2·P

ρp  ρe .

(14.84)

As an illustration, taking a period of P = Pkink = 3 minutes for the kink mode and a wavelength of λ = 3000 km (Lin et al. 2007, 2009; see also Arregui, Oliver and Ballester 2012) we obtain a thread Alfv´en speed of cAp = 11.8 km s−1 . The magnetic field strength then follows once we specify the density ρp in the relation B0 = (μρp )1/2 cAp . For the estimate ρp = 5 × 10−11 kg m−3 , we obtain B0 = 0.92 × 10−4 tesla or almost 1 G. This seems rather low. However, the interpretation depends upon taking a wavelength of λ = 3000 km, which in a tube of length 2Lp would correspond (for the fundamental) to Lp = 1500 km, which is rather short. If instead we take a tube of length Lp = 105 km (as suggested by the observations reported in Okamoto et al. 2007) we obtain cAp = 786 km s−1 and a field strength of B0 = 62.3 G. To model the prominence thread in more specific detail, we consider a thin flux tube within which is a dense plasma (of density ρp ) occupying the region −lp < z < lp . See Figure 14.6. The governing equation for the kink mode in a thin tube is precisely the wave equation (14.69) discussed earlier, save now the wave speed c is the kink speed ck . Moreover, the boundary conditions of continuity of the motion ur and the derivative ∂ur /∂z across z = ±lp remain the same. Thus, it follows immediately from our earlier discussion of an elastic string, leading to the relation (14.74) describing the mode that disturbs the prominence centre (z = 0), that this same mode in a prominence thread is described by the relation       ckp ωlp ω(Lp − lp ) tan = cot . (14.85) ckp cAe cAe Equation (14.85) is the dispersion relation for prominence kink modes in a thin thread. In fact, it was first obtained, using a slightly different approach, by Dymova and Ruderman (2005; see their eqn. (27)); see also Terradas et al. 2008 and D´ıaz, Oliver and

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465

Ballester 2010). The kink speed ckp that arises here is the kink speed in the dense prominence thread, given by c2kp =

2c2Ae (1 +

c2Ae =

ρp , ρe )

B20 , μρe

where again cAe denotes the corona Alfv´en speed in a region with density ρe . For convenience, we have supposed that the density outside the thread is ρe . Notice in passing the reductions of the dispersion relation (14.85) in the two extremes lp = 0 and lp = Lp . The case lp = 0 corresponds to the absence of plasma with prominence densities, and then   ωLp = 0, cot cAe yielding ωlp /cAe = π/2 for the first root; thus ω = π cAe /(2Lp ) and P = 4Lp /cAe (i.e., the period is the travel time, with speed cAe , back and forth along the tube of length 2Lp ). In the opposite extreme of lp = Lp , corresponding to the tube being all prominence matter, then (14.85) yields   ωlp = 0; cot ckp thus the principal mode is ω = πckp /(2Lp ) and then P = 4Lp /ckp (i.e., the period is again the travel time back and forth along the tube of length 2Lp , but now at the speed ckp ). Continuing our discussion of the dispersion relation (14.85), we note that, just as with an elastic string, we can expand the dispersion relation using the standard power series for the tan and cot functions; to leading order, we obtain ω2 = Thus,

 ω=

2 1 + (ρe /ρp )

c2kp lp (Lp − lp ) 1/2

.

1 cAp , [lp (Lp − lp )]1/2

(14.86)

(14.87)

where cAp is the Alfv´en speed in the dense prominence thread: cAp =

B0 . (μρp )1/2

In terms of period P (= 2π/ω) we have √ π  1/2 [lp (Lp − lp )]1/2 1 + (ρe /ρp ) . P= 2 cAp

(14.88)

If we suppose that ρe  ρp , appropriate for a prominence thread, then these relations simplify a little and we obtain √ √ π 1 ω= 2 cAp , P= 2 [lp (Lp − lp )]1/2 . (14.89) 1/2 cAp [lp (Lp − lp )] These expressions agree with D´ıaz, Oliver and Ballester (2010; see their eqn. (15)); see also Arregui, Oliver and Ballester (2012, eqn. (45)).

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We can illustrate the use of such relations by considering the observations of Lin et al. (2007) and Okamoto et al. (2007). The flux tube length 2Lp in which the threads reside is uncertain; we assume 2Lp = 100 Mm in each of our illustrations. Lin et al. observes an oscillation with a period of 16 minutes (P = 960 s) in a thread of length 2lp = 13000 km; then (14.89) implies a prominence Alfv´en speed of cAp = 78 km s−1 . Turning to the observations of Okamoto et al. , they report oscillations in six threads of various lengths, with periods ranging from 135 s to 250 s. In particular, in thread 1 of length 2lp = 3600 km, the period is P = 174 s; application of (14.89) gives a speed of cAp = 238 km s−1 . Finally, in thread 6 (the thread with longest period) Okamoto et al. report an oscillation period of P = 250 s in a thread of length 2lp = 1700 km, for which application of (14.89) gives a prominence Alfv´en speed of cAp = 115 km s−1 .

14.11.1 Resonant Damping of Thread Oscillations So far we have used only the fact that a transverse wave might be a fast magnetoacoustic wave propagating at the kink speed. But observations indicate strong damping of the modes, and it is natural to suppose that this is due to resonant absorption, just as in a coronal loop (see Section 14.6). Under prominence conditions, we may assume that ρp  ρe and then we obtain Pkink l . =C a τdamping

(14.90)

Thus, for a period of Pkink = 3 minutes and a damping time of τdamping = 9 minutes, we obtain l ≈ 0.1 a in the linear case and l ≈ 0.2 a in the sinusoidal case. Similar estimates to these were first given by Arregui et al. (2008) and Soler et al. (2009).

14.12 Prominence Oscillations: Longitudinal Vibrations Large amplitude longitudinal oscillations have been detected in prominences and appear to be excited by nearby impulsive events (see, for example, Jing et al. 2003; Luna et al. 2014, 2017). They typically have periods of about an hour, with a range of 40 to 100 minutes noted by Luna et al. (2017). Luna and Karpen (2012) used one-dimensional simulations to argue that many such longitudinal oscillations can be understood in terms of a simple pendulum model, in which gravity provides the principal restoring force and the length of the pendulum is determined by the radius of curvature of the equilibrium magnetic field lines. In this model the magnetic field provides a channel that guides longitudinal motions in a sloshing motion, back and forth, along the field. An analytical model that gives a dispersion relation that includes the pendulum frequency describing the oscillation has been developed by Luna, D´ıaz and Karpen (2012). We treat that model here. A related model, which aims to incorporate both nonlinearity and damping due to mass exchange (or accretion) has been put forward by Ruderman and Luna (2016), and the model recovers the pendulum frequency as a special case.

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14.12.1 Model Formulation Consider an equilibrium magnetic field that consists of a horizontal thin flux tube that dips down in its central region where the prominence mass is located. The dip is modelled as a sector of a circle of radius Rc , with the centre of the circle on the vertical axis of symmetry in a Cartesian coordinate system. Gravity and plasma pressure are included. The whole configuration is symmetric about the vertical axis. The plasma in the dipped section of tube is stratified by gravity; in the horizontal section, the tube is unstratified. Within each section of tube the usual thin tube equations apply (see Chapter 10). The magnetic field is taken as so strong that it may be regarded as rigid and any perturbation motions are therefore directed along the rigid field. The governing system of thin tube equations consists of: the equation of continuity ∂ρ ∂ + (ρu) = 0, ∂t ∂s the equation of momentum

 ρ

∂u ∂u +u ∂t ∂s



∂p − g ρ, ∂s

(14.92)

 ∂ρ ∂ρ +u . ∂t ∂s

(14.93)

=−

and the adiabatic relation ∂p γp ∂p +u = ∂t ∂s ρ



(14.91)

Here u(s, t) denotes the longitudinal motion along the tube; distance s is measured as arc length along the tube, with s = 0 taken to be at the lowest part of the dip. The gravitational force is represented by g , the component of gravity acting along the tube. These equations coincide with the system discussed in Chapter 10, except for a sign change in the gravity term; here the coordinate s is increasing as we move along the tube from the low point at s = 0, but gravity points downward (opposite to s increasing), whereas in Chapter 10 the z-axis acts down (and so in the same direction as gravity). Note that g (s) varies with s in the curved section of tube and is zero in the horizontal section of tube. In the equilibrium state (u = 0) the plasma is stratified according to p0  = −g ρ0 .

(14.94)

The prime ( ) here denotes differentiation (of an equilibrium quantity) with respect to arc length s. It is convenient to suppose that the curved section of the tube is the arc of a circle of radius Rc , so that in the curved section we have s = Rc θ and g = g sin θ , where g (= 274 m s−2 ) is the usual gravitational acceleration and θ (s) is the angle that the radius vector makes with the vertical; at the centre of the dipped field, θ = 0. Figure 14.7 shows the geometry. Consider equations (14.91)–(14.93) linearized about the equilibrium (14.94): ∂ρ ∂ + (ρ0 (s)u) = 0, ∂t ∂s

∂u ∂p =− − g ρ, ∂t ∂s   ∂p ∂ρ + p0  u = c2s (s) + ρ0  u , ∂t ∂t ρ0 (s)

(14.95)

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z s

r

x Figure 14.7 The geometry of a prominence thin thread (magnetic flux tube). The dense prominence plasma (shown shaded) is at the lowest part of the curved tube. Arc length s is taken to be measured from the lowest point s = 0. The tube of dense material is modelled as a circular section of radius Rc (shown as r in the figure). (From Ruderman and Luna 2016.)

where cs (s) denotes the sound speed in the tube (c2s = γ p0 (s)/ρ0 (s)). Here ρ and p denote the perturbation density and pressure, and u is now the perturbation motion along the tube. The set of linear equations (14.95) may be readily manipulated to yield a wave-like equation for u: ∂ 2u ∂u ∂ 2u 2 = c (s) − γ g (s) − g  (s)u. s ∂s ∂t2 ∂s2

(14.96)

In the horizontal section of the tube, g = 0 and the one-dimensional wave equation describes the motion: ∂ 2u ∂ 2u 2 = c (s) . s ∂t2 ∂s2

(14.97)

In the curved section, where the tube outlines the arc of a circle of radius Rc , equation (14.96) leads to   ∂ 2u ∂u g ∂ 2u 2 = c (s) − (γ g sin θ ) cos θ u. (14.98) − s ∂s Rc ∂t2 ∂s2 Recall that s = Rc θ for the circular section of tube. Notice too that in the limit of Rc → ∞, which corresponds to the circle being arbitrarily large (and then any particular section of the circumference approaches a straight line), equation (14.98) approximates the one-dimensional wave equation. Equation (14.98) was first given by Luna, D´ıaz and Karpen (2012; see their eqn. (6)).

14.12.2 The Klein–Gordon Equation It is interesting to note that equation (14.98) may also be cast in the form of the Klein– Gordon equation. To see this, introduce u(s, t) = X(s)Q(s, t).

(14.99)

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Then the function X(s) must be chosen such that γ g X = 2, X 2cs and then no term involving ∂Q/∂s arises. Consequently,   s  g γ ds X = X0 exp 2 0 c2s

(14.100)

for arbitrary constant X0 . With this choice of X(s), we obtain ∂ 2Q ∂ 2Q 2 = c (s) − 2g Q, s ∂t2 ∂s2 where 2g =

γ 2 g2 4c2s

 − c2s

γ g 2c2s



+ g  ,

(14.101)

(14.102)

For the arc of a circle for which g = g sin θ , and furthermore supposing the medium to be isothermal (cs = constant), then the cutoff frequency g is given by     γ 2 g2 s γ g s 2 2g = sin cos + 1 − . (14.103) Rc 2 Rc Rc 4c2s In the limit Rc → ∞, g → 0, and there is no cutoff frequency; we recover the onedimensional wave equation for a horizontal thin flux tube. However, we note that at the lowest point of the field-line dip, where θ = 0 (and s = 0), Equation (14.103) gives

γ g 2g = 1 − . 2 Rc 14.12.3 The Dispersion Relation Returning to the velocity equation (14.98), we suppose that the curved arc is not too long so that sin θ ≈ θ and cos θ ≈ 1. Then   ∂ 2u ∂ 2u g ∂u g 2 = cs (s) 2 − γ s − u. (14.104) 2 Rc ∂s Rc ∂t ∂s We may solve this equation as follows. Suppose first that the medium is isothermal in the curved tube region, so that cs = constant in that region. Then, for a time dependence of the form eiωt for frequency ω, equation (14.104) becomes d2 u c2s 2 ds

  γ g du g 2 − s + ω − u = 0. Rc ds Rc

Introduce a variable change by taking x = Cs2

(14.105)

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where the constant C is to be chosen. Then differential operators transform according to   d2 1 d d2 d 1/2 1/2 d = 4C x 2 + = 2C x , . ds dx 2 dx ds2 dx Hence the differential equation (14.105) leads to x

d2 u + dx2



 g 2 1 du (ω − Rc ) −x + u = 0. 2 dx 4c2s C

We now choose the constant C: γg . 2c2s Rc

C=

(14.106)

Then equation (14.105) becomes d2 u x 2 + dx



 1 du −x − λu = 0, 2 dx

where 1 λ= 2γ

(14.107)



 Rc 2 1− ω . g

(14.108)

Equation (14.107) is a form of Kummer’s equation for the confluent hypergeometric functions (see Abramowitz and Stegun 1965, chap. 13). Now the confluent hypergeometric equation (14.107) has solutions     1 3 1 and x1/2 M + λ, , x , M λ, , x 2 2 2 where M denotes the Kummer function. The general solution of the differential equation is a linear combination of these two functions. Now the model of a curved flux tube that we are examining is symmetric about the vertical axis corresponding to s = 0. So, without loss of generality, we may examine even or odd solutions separately. Thus, in terms of the original variable s, we consider the mode that disturbs the whole prominence (u = 0 at s = 0):   1 2 (14.109) u = u0 M λ, , Cs , 0 ≤ s < lp , 2 where u0 is an arbitrary constant (with the dimensions of velocity). This solution of (14.107) is such that u = 0 at s = 0; in fact, u = u0 at s = 0. Thus, we are examining a solution for which the curved thread region moves longitudinally as a whole during the oscillation. This section of curved tube is taken to be occupying the region −lp < s < lp ; through symmetry, we treat the section 0 ≤ s < lp . Outside of the circular (dipped) tube section we assume the tube to be horizontal, and then the one-dimensional wave equation (14.97) applies. We assume that the horizontal tube section is also an isothermal region, though with a different temperature (and density)

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to the dipped field region. Then we may write the solution of (14.97) in the form (suppressing the eiωt dependence)     ωs ωs + u2 sin , (14.110) u = u1 cos cse cse where cse denotes the constant sound speed in the external (horizontal) tube region and u1 and u2 are arbitrary constants. Now we require that the motion u vanishes at the locations s = ±Lp , modelling the role of the dense chromosphere and photosphere which remains undisturbed by motions within the prominence. Thus, 2Lp represents the length of the field lines passing through the prominence that are anchored either side of the structure in the dense lower solar atmosphere. Accordingly, with u = 0 at s = Lp , we take   ω u = ue sin (Lp − s) , lp < s ≤ Lp , (14.111) cse where ue is an arbitrary constant. Note that this is precisely of the form that arises in an elastic string (see equation (14.72)). Altogether, then, we have ⎧   ⎨ue sin cω (Lp − s) , lp < s ≤ Lp , se (14.112) u= ⎩u M(λ, 1 , Cs2 ), 0 ≤ s < l , 0 p 2 where C=

1 . 2Rc p

Here p denotes the pressure scale height in the prominence region (0 ≤ s < lp ), related to the prominence sound speed csp through c2sp = γ gp . The solution (14.112) corresponds to a motion u that is non-zero at the mid-point (s = 0) of a curved thin tube region extending from −lp ≤ s < lp , coupled to a horizontal thin tube extending (in s > 0) from lp < s ≤ Lp . There is symmetry about s = 0. In the curved tube region 0 ≤ s < lp , the plasma is dense and cool (with sound speed csp ) and also stratified by gravity; this region corresponds to the prominence plasma. Outside of this region, in |s| > lp , the plasma is uniform and represents the ambient corona with a sound speed cse . Now we require that both u and ∂u/∂s are continuous across s = lp . These conditions imply a relationship between the constants ue and u0 , and lead to the relation   M(λ, 12 , Clp2 ) ω ω . (14.113) cot (Lp − lp ) = lp λC cse cse M(1 + λ, 32 , Clp2 ) This is the dispersion relation describing longitudinal oscillations of a prominence comprising thin flux tubes (or threads). It was first given by Luna, D´ıaz and Karpen (2012; see their eqn. (20)6 ). The mode described disturbs the prominence as a whole (u = 0 at s = 0). 6 Note the different notation for λ used by Luna, D´ıaz and Karpen (2012).

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Solar Applications of MHD Wave Theory

Now in general terms the dispersion relation (14.113) requires a numerical treatment (given in Luna, D´ıaz and Karpen 2012). However, there is an important analytical reduction of the equation which is illuminating. Suppose Clp2  1,

that is lp2  2Rc p ,

a circumstance achievable whenever there is a thin dense prominence region or a large Rc . Then the confluent M function may be approximated by the first term in its series expansion, namely unity, and we obtain   ω ω cot (Lp − lp ) = lp λC. (14.114) cse cse Write =

ω (Lp − lp ). cse

Then, expressing the parameter λ in terms of  we can rewrite (14.114) as   lp (Lp − lp ) Rc c2se 2  −1 .  cot  = γ Rc p g (Lp − lp )2

(14.115)

Finally, we can expand  cot  through its Taylor series: 1  cot  = 1 − 2 + · · ·, 3

|| < π .

If we retain only the first term (unity) in the expansion of the  cot , then   c2sp (Lp − lp )2 g 2  = + . · Rc lp (Lp − lp ) c2se

(14.116)

Hence, to leading order we obtain 2 = ω2 = ωlong

c2sp g + . Rc lp (Lp − lp )

(14.117)

This relation was first obtained by Luna, D´ıaz and Karpen (2012; see their eqn. (25)). It 2 shows that the frequency squared ωlong of longitudinal oscillations is determined by two sources, a gravitational term and an acoustic term. For typical solar conditions the gravity term dominates over the acoustic term, and so the frequency and associated period are given (to leading order) by  1/2  1/2 g Rc ω = ωgravity = , P = Pgravity = 2π . (14.118) Rc g In fact, the gravity term dominates whenever Rc 

lp (Lp − lp ) , γ p

a condition readily met. For example, for a prominence with a line-tied field line of length 2Lp = 2 × 102 Mm, with a dense region of length 2lp = 2 × 5 Mm and a prominence sound

14.12 Prominence Oscillations: Longitudinal Vibrations

473

speed csp = 20 km s−1 (with a corresponding pressure scale height of p = 876 km), gravity dominates over the acoustic effects for Rc  325 Mm. In the opposite extreme, when gravity is negligible or Rc is very large, then the frequency is dominated by the acoustic effects and we get ω2 =

c2sp lp (Lp − lp )

.

(14.119)

Thus we obtain a relation closely similar to the string mode, even though here the mode is longitudinal and not transverse. In the general model developed above, Rc is the radius of the circle that defines the dipped magnetic field in which the dense prominence material resides. However, it seems reasonable to take Rc to be the radius of curvature of the equilibrium magnetic field that maintains the prominence, and this approach is lent support by the numerical simulations presented by Luna, D´ıaz and Karpen (2012) which were in fact developed before the longitudinal model. Finally, we note that we can improve on the result (14.117) simply by retaining the first two non-zero terms in the expansion of the  cot . After a little algebra, ω2 =

g Rc

1+

1 3

c2

+ lp (Lpsp−lp )

 , p e

Lp −lp lp

with an associated period P (= 2π/ω) given by ⎡

  ⎤1/2 Lp −lp 1 p  1/2 1 + 3 e lp Rc ⎢ ⎥ P = 2π ·⎣ ⎦ . 2 R c g sp c 1 + lp (Lp −lp )g

(14.120)

(14.121)

Relation (14.121) has not as yet been explored in the literature, so we content ourselves with a simple illustration. We take our basic figures from the parameters used in Luna, D´ıaz and Karpen (2012; see their Fig. 3). Setting Lp = 100 Mm, lp = 5 Mm and cse = 200 km s−1 with a density contrast of ρp /ρe = 102 , and a radius Rc = 100 Mm, we find a prominence scale height of p = 876 km (compared with a coronal scale height of e = 8.76 × 104 km). Then Clp2 = 0.1427 (so the assumption Clp2  1 is satisfied) and we find from (14.121) that P = 3423 s (or 57 minutes and 3 seconds). This may be compared with the zeroth order result of P = 3796 s that follows from (14.118).

14.12.4 The Pendulum Mode and Prominence Seismology We end this section with a recent illustration of prominence seismology. Luna et al. (2017) report the detection of large amplitude longitudinal oscillations in a prominence. It seems that the eruption of one prominence sets off the longitudinal oscillations in another prominence. Two phases are observed; first there is a 69 minute oscillation, and later there is a 54 minute period. We can use these measurements to put a constraint on the strength of the magnetic field. Since the theory of longitudinal oscillations developed so far assumes a rigid

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magnetic structure, we cannot determine the magnetic field strength but we can determine a lower limit. The field must be sufficiently strong so as to support the prominence, requiring that the magnetic tension force supports the dense material (Luna and Karpen 2012):7 B20 > gρp , μRc

(14.122)

where ρp denotes the density of prominence material. This inequality can be rewritten in the form cAp > (gRc )1/2 ,

(14.123)

where cAp = (B20 /μρp )1/2 denotes the Alfv´en speed in the prominence. Suppose now that the period of the oscillation is determined to reasonable accuracy by the gravitational term in (14.121), so that  1/2 Rc . (14.124) P = 2π g The form of this relation for the period of the mode suggests the name pendulum mode. Note that typically the pendulum mode has a long period, roughly an hour. For example, with Rc = 108 m and g = 274 m s−2 relation (14.124) gives a period of P = 3795.8 s, or 63.26 minutes. Returning to expression (14.122), the inequality implies that g g B0 > (μρp )1/2 P, or cAp > P, (14.125) 2π 2π with P in seconds and B0 in tesla. Take the plasma density in the prominence to be ρp = 2μm ˆ p ne , where ne is the prominence electron density (in m−3 ), mp = 1.673 × 10−27 kg is the mass of a proton, and we set μˆ = 0.635 (Aschwanden 2004, sect. 3.1). Then, in the perhaps slightly more convenient units of B0 in gauss and P in minutes, we have (Luna et al. 2014, 2017)  1/2 ne [m−3 ] . (14.126) B0 [G] > 0.43 P[minutes] 1017 Thus, for an assumed prominence electron density of ne = 1016 m−3 (so total number density 2ne = 2 × 1016 m−3 ) and a period of P = 69 minutes, we obtain B0 > 9.4 G; for the shorter period of P = 54 minutes, we obtain B0 > 7.3 G. For denser prominence material, these lower bounds are correspondingly increased; for ne = 1017 m−3 , we obtain 7 There is a typographical error in inequality (6) in Luna and Karpen (2012), where the factor 4π is omitted. In cgs Gaussian

units, the inequality should read 1 1/2 B0 > √ gP ρp , π

yielding the inequality for the magnetic field strength B0 (in gauss) as B0 > 2.25 × 10−8 P ne

1/2

where P is in seconds and the prominence electron density ne is in particles per cm3 .

14.13 Period Ratios in Coronal Loops and Prominence Threads

475

B0 > 29.7 G for the 69 minute oscillation, and B0 > 23.2 G for the 54 minute oscillation. Overall, this suggests a magnetic field strength of at least 7–30 G, with the uncertainty mainly coming from the equilibrium density. Estimates of lower bounds of this form were first given in Luna et al. (2017).

14.13 Period Ratios in Coronal Loops and Prominence Threads In a uniform elastic string, the modes of oscillation satisfy a simple relation such as ω2 = kz2 c2 = N 2 c2 /L2 for propagation speed c in a string of length L; accordingly, the period P1 (= 2π/ω) of the fundamental (N = 1) mode is P1 = 2L/c and the period P2 of its first harmonic is half of this value, P2 = L/c. Thus, the ratio of the two periods is P1 /P2 = 2, and so P1 /(2P2 ) = 1. This result is independent of c, the speed of propagation of the wave. Such ratios as these – both are referred to as period ratios – are potential seismological tools, of use in determining the physical conditions in the solar atmosphere (see Chapter 11). The important point to note is that such period ratios may depart from the simple values of a uniform elastic string, and any such departures reflect non-uniformity in the plasma and so become a seismological tool for the probing of such structures. The suggestion that the period ratio is a useful tool for loop oscillations was first made by Andries et al. (2005), and further explored by a number of authors (see, for example, Dymova and Ruderman 2006, 2007; McEwan et al. 2006; McEwan, D´ıaz and Roberts 2008; Macnamara and Roberts 2010; Soler, Goossens and Ballester 2015). A review of the topic is provided in Andries et al. (2009). In our treatment here we discuss the simple case of a β = 0 plasma in a thin flux tube. The tube is taken to be stratified by gravity, resulting in a kink mode speed ck that varies with distance z along the tube. It turns out that the period ratio for a coronal loop in which the kink speed decreases from its high value at the coronal apex to a low value at the base of the loop has a period ratio P1 /P2 that is less than two and so a period ratio P1 /(2P2 ) that is less than unity. By contrast, a thin flux tube with a kink speed that increases from a low value inside a prominence thread, where the plasma density is high, to a high value outside the thread, results in a period ratio P1 /(2P2 ) that is greater than unity. We examine analytically the case of a density profile that produces these opposing results in the two situations. In fact, the case illustrates a general feature of wider validity recently demonstrated analytically by Ruderman, Petrukhin and Pelinovsky (2016). The starting point for our investigation is the wave equation for the kink wave in a β = 0 plasma (Dymova and Ruderman 2005, 2006; see Chapter 11): ω2 d 2 ξr ξr = 0. + dz2 c2k (z)

(14.127)

We consider an equilibrium density ρ0 (z) of the form ρ0 (z) =

ρ0 (0) 2

[1 − (1 − κ) Lz 2 ]2

.

(14.128)

p

The constant κ is taken to be positive. The plasma density is ρ0 (0) at the apex point z = 0 and ρ0 (z = ±Lp ) = ρ0 (0)/κ 2 at the ends z = ±Lp of the thin tube. Then, in a β = 0

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Solar Applications of MHD Wave Theory

plasma with a uniform magnetic field strength B0 , the square of the kink wave speed ck is given as c2k (z) =

2B20 ρ0 c2A + ρe c2Ae 2c2 (z) = A = , ρ0 + ρe 1+χ μρ0 (z)(1 + χ )

(14.129)

where cA (z) = B0 /(μρ0 (z))1/2 is the Alfv´en speed within the tube and χ=

ρe (z) ρ0 (z)

is a measure of the density contrast between the inside of the tube and its environment. For convenience, we assume that the external density ρe (z) has the same z-dependence as the internal density ρ0 (z), and then the factor χ is a constant. The governing differential equation (14.127) becomes 1 ω2 d 2 ξr + ξr = 0. 2 2 dz ck (0) [1 − (1 − κ) z22 ]2

(14.130)

Lp

It is convenient to introduce a scaled form of ξr , writing  1/2 z2 ξr = 1 − (1 − κ) 2 F = fF, Lp where



z2 f (z) = 1 − (1 − κ) 2 Lp

(14.131)

1/2 .

Then the thin tube equation (14.130) for the kink mode yields    μρ0 (0)ω2 (1 + χ ) f d2 F 2f  dF + + + F = 0, f dz f dz2 2B20 f 4

(14.132)

the prime  denoting differentiation of an equilibrium quantity with respect to z. There are two cases to discuss: κ < 1, for which the plasma density at the base (z = Lp ) is greater than the density at the apex (z = 0), and the kink speed ck is greater at the apex than at the base; and κ > 1, for which the plasma density at the base is less than the density at the apex, and the kink speed at the apex is less than at the base. The case κ < 1 is typical of a coronal loop, whereas the case κ > 1 corresponds to a prominence thread, though it might also arise in a coronal loop if magnetic field expansion dominates over stratification. The case when the tube is uniform, corresponding to κ = 1, may be deduced as a limiting case of κ → 1, the limit being either from below or above one, or of course treated separately by setting κ = 1 at the outset. 14.13.1 Case κ < 1 Let X+ = 1 +

z (1 − κ)1/2 , Lp

X− = 1 −

z (1 − κ)1/2 Lp

14.13 Period Ratios in Coronal Loops and Prominence Threads

and set

 x = ln

X+ X−

477

 = ln(X+ ) − ln(X− ).

Note that z = 0 corresponds to x = 0. Then dF 2(1 − κ)1/2 dF = , dz dx Lp f 2   2(1 − κ)1/2 2(1 − κ)1/2 d2 F 2f  dF d2 F = − , f dx dz2 Lp f 2 Lp f 2 dx2 and equation (14.130) reads d2 F + α 2 F = 0, dx2

(14.133)

where 2 2 ω2 Lp2 1 1 μρ0 (0)ω (1 + χ )Lp = − . + α2 = − + 4 4 4c2k (0)(1 − κ) 8B20 (1 − κ)

(14.134)

Thus, we obtain an equation with constant coefficients with solution F = C1 cos αx + C2 sin αx,

(14.135)

where C1 and C2 are arbitrary constants. Fundamental Mode: κ < 1 Consider the solution with F = 0 at x = 0, corresponding to F = 0 at z = 0. We choose C2 = 0; then F = C1 cos αx. Let xL denote the value of x when z = Lp . Then requiring that F = 0 at x = xL (corresponding to ξr = 0 at z = Lp ) we require cos αxL = 0, with solution αxL = π/2 (for the first root). Hence, the principal mode has   πx (14.136) F = C1 cos 2xL with dispersion relation



 π2 = (1 − κ) 1 + 2 , c2k (0) xL ω2 Lp2

κ < 1.

(14.137)

Here 1 + x = ln 1−

z 1/2  Lp (1 − κ) , z 1/2 Lp (1 − κ)

  1 + (1 − κ)1/2 xL = ln , 1 − (1 − κ)1/2

κ < 1.

(14.138)

The period P1 (= 2π/ω) of the fundamental mode is accordingly P1 =

2πLp  ck (0) (1 − κ) 1 + 

π2 xL2

1/2 = Pkink 

π/2  (1 − κ) 1 +

π2 xL2

1/2 ,

κ < 1. (14.139)

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Solar Applications of MHD Wave Theory

Here Pkink = 4Lp /ck (0) is the fundamental kink period in a uniform tube. For small (1 − κ), we can expand to obtain     1 2 − (1 − κ) + · · · . P1 = Pkink 1 − 3 π2

(14.140)

First Harmonic: κ < 1 The first harmonic is given by taking the solution F = C2 sin αx,

(14.141)

which has F = 0 at x = 0 (and so ξr = 0 at the apex z = 0). Additionally, we require that F = 0 at the end z = Lp and so sin αxL = 0. Thus, the first harmonic is given by requiring that αxL = π . The frequency ω then satisfies 



4π 2 ω = (1 − κ) 1 + 2 xL

(14.142)

1/2

ck (0) , Lp

κ < 1,

(14.143)

κ < 1.

(14.144)

and the period P2 (= 2π/ω) of the first harmonic is given by P2 =

2πLp  ck (0) (1 − κ) 1 + 

4π 2 xL2

1/2 ,

Again, we can expand for small (1 − κ) to obtain     1 1 1 P2 = Pkink 1 − − (1 − κ) + · · · . 2 3 2π 2

(14.145)

Period Ratio, κ < 1 Consequently, the period ratio is given by   4π 2 + xL2 1/2 P1 = , 2P2 4π 2 + 4xL2

κ < 1.

(14.146)

It follows immediately that P1 /(2P2 ) is less than unity. A result equivalent to (14.146) was first given by Dymova and Ruderman (2006; see their eqn. (92)). The√case of a uniform (κ = 1) tube follows on taking the limit κ → 1, for which xL ∼ 2 1 − κ, resulting in P1 ∼

4Lp , ck (0)

P2 ∼

2Lp , ck (0)

κ → 1,

(14.147)

with a period ratio P1 /(2P2 ) of unity. It is of interest to give a simple illustration. For κ = 3/4, corresponding to ρ0 (0) = (16/9)ρ0 (Lp ) (so the density at the apex of the loop is a little higher than at the loop base), we obtain xL = ln 3, and then the period ratio is P1 /(2P2 ) = 0.9583.

14.13 Period Ratios in Coronal Loops and Prominence Threads

479

We end this sub-section with an illustration from a recent detection of a fundamental and its first harmonic in a coronal loop. Duckenfield et al. (2018) report the presence of a spatially resolved oscillation in a loop of length L = 292 Mm, finding periods of P1 = 10.3 minutes and P2 = 7.4 minutes. The oscillations were detected with AIA on SDO, and occurred in a loop that supported non-decaying oscillations (see also Anfinogentov, Nistic´o and Nakariakov 2013; Nistic´o, Anfinogentov and Nakariakov 2014). Duckenfield et al. conclude that P1 is the fundamental of a kink mode standing in the coronal loop, and P2 is its harmonic (see also De Moortel and Brady 2007; Pascoe, Goddard and Nakariakov 2016). If we use (14.146) to relate to the determination of a period ratio of P1 /(2P2 ) ≈ 0.7, we deduce that xL = 4.58. For small κ we have the approximation xL ≈ ln(4/κ) from which we deduce κ ≈ 0.04. Such a value of κ is associated with a density at the loop apex that is some κ 2 = 1.6 × 10−3 smaller than the density at the base of the loop. If we associate this stratification with an exponential falloff of the form exp(−z/c ) for a spatial scale c (so that exp(−z/c ) = 1.6 × 10−3 with z = Lp and Lp = L/2), then we find that c = 22.7 Mm. The kink mode speed is correspondingly κ smaller at the base than at the apex; for a speed ck = 103 km s−1 at the apex, say, this corresponds to around ck = 40 km s−1 at the base. Similar estimates are given by Duckenfield et al. (2018), though employing a different model density profile.

14.13.2 Case κ > 1 When κ > 1 a form of the above treatment still applies but is better reworked in terms that are directly applicable. The governing equation for F is now d2 F − α12 F = 0, dx2

(14.148)

where α12 =

1 ω 2 L2 . + 2 4 4ck (0)(κ − 1)

(14.149)

Fundamental Mode: κ > 1 The fundamental mode is now given by F = C1 cosh α1 x,

(14.150)

with x = log

1 + i

z Lp (κ 1 − i Lzp (κ

− 1)1/2  − 1)1/2

= 2iθ ,

κ > 1,

(14.151)

and tan θ =

z (κ − 1)1/2 , Lp

κ > 1.

(14.152)

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Solar Applications of MHD Wave Theory

Thus x is purely imaginary for the case κ > 1. Then F = C1 cos(2α1 θ ).

(14.153)

For F = 0 at z = Lp we have (for the first zero) π 2α1 θL = , 2 where θ = θL when z = Lp , and so tan θL = (κ − 1)1/2 ,

(14.154)

κ > 1.

(14.155)

The dispersion relation for the fundamental mode follows from expressions (14.149) and (14.154), giving  2  ω2 Lp2 π = (κ − 1) −1 , κ > 1, (14.156) c2k (0) 4θL2 with an associated fundamental period P1 =

2π Lp  ck (0)

 (κ − 1)

1 π2 4θL2

π/2 

1/2 = Pkink  −1

(κ − 1)

π2 4θL2

1/2 ,

κ > 1.

−1 (14.157)

For small (κ − 1), we can expand to obtain     2 1 P1 = Pkink 1 − − 2 (κ − 1) + · · · , 3 π

κ > 1.

(14.158)

First Harmonic: κ > 1 Turning to the first harmonic, we take F = C2 sin(−2α1 θ )

(14.159)

2α1 θL = π .

(14.160)

and require (for the first zero) that

This leads to the dispersion relation 1/2   ck (0) π2 ω= −1 , (κ − 1) Lp θL2

κ > 1,

(14.161)

κ > 1.

(14.162)

with the period of the first harmonic being P2 =

2πLp  ck (0)

 (κ − 1)

1 π2 θL2

1/2 , −1

For small (κ − 1), we can expand to obtain     1 1 1 − (κ − 1) + · · · , P2 = Pkink 1 − 2 3 2π 2

κ > 1.

(14.163)

14.14 Sunspots, Pores and Photospheric Flux Tubes

481

Period Ratio, κ > 1 The period ratio follows from the above:   (π/θL )2 − 1 1/2 P1 = , 2P2 (π/θL )2 − 4

κ > 1.

(14.164)

It follows from this expression that when κ > 1 the period ratio P1 /(2P2 ) is greater than unity. For small (κ − 1), expanding gives P1 3 =1+ (κ − 1) + · · ·, 2P2 2π 2

κ > 1.

(14.165)

A result of this form was first obtained by Verth and Erd´elyi (2008; see their eqn. (96)), in their discussion of the role of magnetic field expansion: the tube expansion acts in opposition to density stratification, in that it leads to a decrease in the wave speed ck . When magnetic expansion overcomes density stratification, so that the kink speed ck decreases from tube base to apex, the result is a period ratio that exceeds unity. Thus, the cases discussed here whereby κ may be less than unity or greater than unity provides a useful illustration of a general result established analytically by Ruderman and Luna (2016), namely that for a tube with a kink speed ck that increases from the base of a tube to its apex then the period ratio P1 /(2P2 ) is less than unity, whereas if ck decreases from base to apex then the period ratio is greater than unity. We note too that, as observed from the results for κ < 1, the case of a uniform tube may be deduced from the behaviour of our expressions in the limit κ → 1; observing first that √ θL ∼ κ − 1 as κ → 1, the limiting behaviour in a uniform tube is readily recovered. Finally, we give a simple illustration of the solution for a specific κ. For κ = 4, corresponding to ρ0 (0) = 16 ρ0 (Lp ) (so the density √ at the apex of the loop is 16 times the density at the loop base), we obtain tan θL = 3 and so θL = π/3. Then the period √ ratio is P1 /(2P2 ) = 8/5 and evidently larger than unity.

14.14 Sunspots, Pores and Photospheric Flux Tubes Sunspots are extensive regions of strong (about 3000 G) magnetic field, visible in the photosphere as dark regions. They are cooler than their surroundings, typically 4000 K in the photosphere compared with their surroundings at around 6000 K. The coolness of sunspots is generally attributed to the ability of the mainly vertical magnetic field to inhibit convective processes. The magnetic field of a sunspot typically expands outward with height, so that a more horizontal field is apparent as we move away from the central part of the sunspot. The coolest central part of the sunspot is the umbra, and this is typically surrounded by a somewhat warmer region known as the penumbra. Pores are mini versions of sunspots, dark regions in the photosphere that lack any penumbra; their magnetic fields are in the kG range. There are also small concentrations of magnetic field that occupy thin intense magnetic flux tubes of kilogauss field strength. These flux tubes are typically located in the regions between granules where downdraughts between the convective cells occur. From the point of view of mathematical modelling, we may regard pores and intense magnetic flux tubes

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as much the same object, though perhaps pores are cooler in temperature and larger in size than the intense flux tubes. The internal structure of a sunspot in the layers below the photosphere is uncertain; there are two commonly discussed views. Either a sunspot is better considered as a monolithic structure, viewed as a single unified magnetic flux tube, or it is better considered as a collection of more or less separate magnetic flux tubes, pushed close together in the visible layers but nonetheless retaining some identity. The clustering of flux tubes model was first put forward by Parker (1979b). An extensive general discussion of sunspots and pores is given in Solanki (2003) and Thomas and Weiss (2008). The first observation of oscillations in sunspots was reported by Beckers and Tallant (1969), who noted the occurrence in the chromospheric layers of what they termed umbral flashes which seemed to occur repeatedly, with a period in the range 110–190 s; they tentatively suggested the flashes were associated with wave activity and that magnetoacoustic waves (the fast mode) might be responsible. Slightly later, Beckers and Schultz (1972) identified umbral oscillations in the photospheric layers of a sunspot, and these oscillations appeared to fall into two distinct ranges in period, of around 3 minutes and around 5 minutes. This indicated that two distinct sets of oscillation were taking place. We now believe that the 5-minute band is associated with p-modes which are studied in detail in the subject of helioseismology, and that the 3-minute band is associated with slow magnetoacoustic waves (see the reviews in Thomas 1981, 1985; Bogdan and Judge 2006; Khomenko and Collados 2015), and so is intrinsic to the sunspot’s magnetic field. The 5-minute band of p-modes is in fact a distribution of modes centred around a cyclic frequency of roughly ν = 3 mHz, and the 3-minute band of umbral oscillations is a distribution of modes centred around a cyclic frequency of about ν = 5 mHz.8 We now believe that the 3-minute oscillations are the cause of the flashes, probably when they develop nonlinearly into strong disturbances or shocks. The 3-minute oscillations are generally interpreted to be acoustic-like disturbances propagating vertically upwards and guided by the sunspot’s magnetic field; this is consistent with the waves being the slow magnetoacoustic mode. The oscillations are best understood as local disturbances in the sunspot, rather than as a global oscillation of the spot as a whole (Zhugzhda and Sych 2014, 2018; Zhugzhda 2018). As well as umbral oscillations, sunspots support running penumbral waves. Running penumbral waves were first detected by Giovanelli (1972) and Zirin and Stein (1972). As their name suggests, they appear to be waves that expand radially outwards from the inner penumbra to the outer penumbra. At the umbral–penumbral boundary, phase speeds are 15–20 km s−1 and frequencies are 4–5 mHz; at the outer edge of the penumbra, propagation speeds have dropped to 4–7 km s−1 and frequencies to 0.7–1.5 mHz; see the review by Bogdan and Judge (2006). In many ways we might have expected sunspots to be the ideal object for the study of magnetohydrodynamic waves (see, for example, Roberts 1992; Evans and Roberts 1990a; Zhugzhda and Sych 2014). However, perhaps because of their complexity and gravitational stratification, this has not so far proved to be the case. Nonetheless, recently 8 Recall that a cyclic frequency ν is the reciprocal of period P, so that ν = 1/P ≈ 5.5 mHz for umbral oscillations with period

P = 180 s.

14.14 Sunspots, Pores and Photospheric Flux Tubes

483

progress is being made using the concepts of MHD seismology (see, for example, Zhugzhda and Sych 2014; Jess et al. 2016; Cho et al. 2017; Deres and Anfinogentov 2018; Zhugzhda 2018). See also the review by Sych (2016). For example, Jess et al. (2016) were able to map the magnetic field of a sunspot, in its atmosphere, by using observations of the spot’s wave modes, interpreted as slow magnetoacoustic waves. They were able to deduce the Alfv´en speed and the magnetic field strength of a spot and its chromospheric–coronal atmosphere. (A somewhat related approach, though for standing waves in a coronal loop, was presented by Wang, Innes and Qiu (2007); see Section 14.5.2.)

Figure 14.8 The dispersion diagram of Edwin and Roberts (1983), showing solutions of the dispersion relation (6.42) for a magnetic flux tube under photospheric conditions, namely cAe < cs < cse < cA . Parameters are the same as given in Figure 6.4 of Chapter 6. The diagram displays the phase speed c (= ω/kz ) as a function of kz a for fast and slow magnetoacoustic waves in a magnetic flux tube in an environment with low Alfv´en speed. [Notational changes from the present text: VA ≡ cA , VAe ≡ cAe , C0 ≡ cs , CT ≡ ct , CTe ≡ cte , Ce ≡ cse , and k ≡ kz .] (After Edwin and Roberts 1983.)

484

Solar Applications of MHD Wave Theory

To see how the approach of Jess et al. works, we first recall the dispersion diagram for wave propagation. For convenience, this is shown here in Figure 14.8, reproduced from Chapter 6. The diagram displays the speed of waves in a magnetic flux tube under photosphere conditions (Edwin and Roberts 1983). Of particular note is the presence of slow magnetoacoustic waves propagating, with only slight dispersion, at the speed ct . Jess et al. (2016) focus on the slow modes. Recall from (14.7) that the speed ct of a slow mode in an atmosphere with sound speed cs and Alfv´en speed cA is given by 1 1 1 = 2+ 2, 2 c ct cA s which may be rearranged to give the Alfv´en speed:  cA =

c2s c2t c2s − c2t

1/2 .

(14.166)

Jess et al. (2016) use the relation (14.166) to deduce cA from temperature maps of the sunspot region (giving cs ) and wave information (giving ct ). Given the definition of the Alfv´en speed, we then have a magnetic field strength of B0 =

√

 μρ0

c2s c2t c2s − c2t

1/2 ,

(14.167)

where ρ0 is the plasma density and μ the magnetic permeability. The sound speed high in the corona above the spot does not vary much when we travel radially from a high location directly above the umbral centre to the outer zone of the spot; Jess et al. determine a range cs = 160–210 km s−1 . For the purposes of illustration here, we assume that cs = 185 km s−1 , say. Then a tube speed of ct = 184 km s−1 , determined from observed ubiquitous upwardly propagating waves – interpreted as slow Alfv´en waves from their typical speed – from the lower reaches of the spot, gives rise to an Alfv´en speed of cA = 1772 km s−1 , with a corresponding magnetic field strength of B0 = 28.8 G for a plasma density of ρ0 = 2.1 × 10−12 kg m−3 . The associated plasma β is 0.013, for γ = 5/3. If we examine the spot’s atmosphere radially outwards from the umbral centre but high in its atmosphere, say at a distance of around 3.8 Mm corresponding to the umbral– penumbral boundary in the photosphere, then the determined propagation speed falls to ct = 80 km s−1 ; with the same sound speed of cs = 185 km s−1 , the calculated Alfv´en speed now falls to cA = 89 km s−1 and the determined magnetic field strength is the rather low 1.4 G. The plasma β is now 5.2. These calculations illustrate the typical numbers found by Jess et al. (2016) in their more systematic determination of the magnetic field and associated Alfv´en speed in the sunspot’s upper atmosphere. So far little has been made of stratification, which must enter at some level in theoretical models of photospheric phenomena. One approach has been to introduce the Klein–Gordon equation which arises naturally in thin tubes (Rae and Roberts 1982) and in a uniform

14.14 Sunspots, Pores and Photospheric Flux Tubes

485

magnetic field (Roberts 2004, 2006). The Klein–Gordon equation is discussed in Chapters 9 and 10. To make clear this aspect, recall that the Klein–Gordon equation in a uniform magnetic field aligned with gravity in an otherwise arbitrarily stratified atmosphere is (Roberts 2006; see equations (10.92) and (10.93) in Chapter 10)9 ∂ 2Q ∂ 2Q 2 − c (z) + 2str (z)Q = 0, t ∂t2 ∂z2 where 2str (z)

 =

c2t (z)

1 420



ct cs

4

    1 1 g2 g c2t c2t g + γg 4 + 2 − 2 + . 2 0 c2s cs cs cA H0

(14.168)

(14.169)

As usual, the prime  denotes the derivative with respect to z of an equilibrium quantity (e.g., the ratio c2t /c4s ). The z-axis points downwards. Equation (14.168) displays a characteristic speed ct (z) and a characteristic frequency str (z), both of which vary with depth z. In the two extremes of cA  cs (corresponding to the upper layers of a sunspot where magnetism dominates over acoustic effects) and cA  cs (corresponding to the sub-photospheric layers of a sunspot, where magnetism starts to become unimportant) the speed and cutoff frequency simplify a little (Roberts 2006), giving  cA  cs , cs , (14.170) ct = cA , cA  cs and 2str =

⎧ ⎨ ⎩

c2s (1 − 20  ), 420

2g ,

cA  cs ,

(14.171)

cA  cs .

Here g is the buoyancy frequency (see Chapter 9). The approach from the Klein–Gordon equation has been used recently by Zhugzhda and Sych (2014, 2018) and Cho et al. (2017). Zhugzhda and Sych (2014) examine the cutoff frequency str , noting in particular its behaviour in the two extremes in (14.171) and emphasizing the variation of str over an extensive range of heights varying from the sub-photosphere to the chromosphere. Another illustration of the employment of the Klein–Gordon equation has been offered by Cho et al. (2017), who use seismology to deduce the Alfv´en speed and plasma β in 478 sunspots observed in the optical continuum using SDO. Interestingly, Cho et al. (2017) have pointed out that the general expression (14.169) for the square of the cutoff frequency str may be written entirely in terms of the ratio c2t /c2s , representing the magnetic effects, and the pressure scale height 0 and the density scale height H0 representing the equilibrium stratification. After some algebra, we can rearrange expression (14.169) to read (Cho et al. 2017)

9 In Roberts (2006), the z-axis was taken to point upwards leading to certain terms changing sign.

486

Solar Applications of MHD Wave Theory



1 1 2str (z) = g − H0 γ 0





 c2t c2s   3   2 2 γ 1 γ 3γ g c2t ct +g − − + . 2H0 0 0 40 c2s c2s 

1 1 1 + − +g 0 γ 0 H0

(14.172)

This expression is exact, a rewrite of the form (14.169) convenient for the case cA  cs . The two scale heights arising here are the pressure scale 0 and the density scale H0 defined by (see Chapter 9) p0 1 = , 0 p0

1 ρ0 = , H0 ρ0

(14.173)

and the equilibrium constraint (p0  = gρ0 ) implies that 0  = 1 −

0 . H0

(14.174)

Notice also that the magnetic factor c2A c2t = c2s c2s + c2A arising in (14.172) is less than unity and becomes small in the sub-photospheric layers of a sunspot (where cA  cs ), but approaches 1 in the strong field limit cA  cs appropriate for the upper layers of a sunspot, pore (or coronal loop). The expression (14.172) holds for arbitrary values of the sound and Alfv´en speeds but it is not particularly convenient for the case cA  cs . For this case, it is better to work in terms of c2t /c2A rather than c2t /c2s , since while both ratios are less than unity the first form is small when cA  cs whereas the latter form tends to unity when cA  cs . After a little algebra, we may rewrite (14.172) in the form 2str (z)

     2  γg 2 g 1 1 g γ ct = + + (γ − 1) − + −1 − 4 0 H0 4 γ 0 H0 c2A  3      2 2 γg 5 3γ g c2t g ct + + − . γ −1 2H0 4 0 40 c2A c2A

(14.175)

This expression is exact, a rewrite of the forms (14.169) or (14.172) convenient for the case cA  cs . It is of interest to examine the extreme cA  cs ; this is the expected situation in the higher atmosphere above the umbra of a sunspot, and more generally high in the corona. Then c2t /c2A is small and so to leading order in this small parameter, we obtain       γg g 1 g 0 γ c2s 2 str (z) ≈ + (γ − 1) + −1 −1 + 2 − , c2A  c2s . 40 H0 4 γ 0 H0 c2s + c2A (14.176)

14.14 Sunspots, Pores and Photospheric Flux Tubes

487

Given that 0 and H0 are positive, it is evident that   γg 0 2 str (z) < −1 + 2 , 40 H0 the value of the cutoff squared in a rigid (cA → ∞) magnetic field or in the absence of a magnetic field (when the square of the acoustic cutoff is obtained). It is of interest to see the form of equation (14.176) in two specific cases. In an isothermal atmosphere, for which 0  = 0 and H0 = 0 , equation (14.176) reduces to (Roberts 2006)       c2s 2 2 c2s 2 str ≈ (14.177) 1− 1+ 2− , c2A  c2s , γ 420 c2s + c2A showing that the slow mode cutoff frequency str (now a constant) in a strong magnetic field is less than the acoustic cutoff frequency cs /(20 ) of an isothermal field-free atmosphere. Turning to the second case, in a thermally stratified medium the value of 2str (z) depends upon the temperature gradient in the medium. The special case of a neutrally stratified medium is of interest. This case gives a buoyancy frequency g that is zero; this is achieved if H0 = γ 0 . Then (14.176) reduces to    c2s c2s 2 2 str (z) ≈ (14.178) −1 − , c2A  c2s . γ 420 c2s + c2A In the photospheric layers of a sunspot, the sound and Alfv´en speeds are roughly comparable, with the Alfv´en speed expected to be higher; speeds of cs = 7 km s−1 and cA ∼ 10 km s−1 are expected. Thus the ratio of squared speeds c2t /c2s arising in (14.172) is broadly of order unity, and so we need to retain all the terms in the expression (14.172) for the cutoff frequency. Cho et al. (2017) use this relation to map the cutoff frequency in the umbra of sunspots, using the measured power spectrum of the 3-minute modes as well as temperature measurements obtained using the Heliospheric Magnetic Imager onboard SDO. Sound speeds close to 7 km s−1 and Alfv´en speeds mainly in the range 6–11 km s−1 are obtained. To obtain an estimate of the magnetic field strength, we need a determination of the plasma density ρ0 . For example, with ρ0 = 1.84 × 10−4 kg m−3 , sound speed cs = 7.23 km s−1 and an Alfv´en speed of cA = 8.94 km s−1 we obtain a magnetic field strength of B0 = 1359 G and a plasma β of β = 0.78. Potentially, this is a fruitful area of development in the seismology of sunspots. Finally, turning to pores and intense magnetic flux tubes, we note that in principle their waves ought to be easier to model than the waves in the more complex sunspots, though stratification must play an important role. Pores typically have diameters of 1–6 Mm and magnetic field strengths of about 1.5 kG (no more than 1700 G according to Grant et al. 2015) and are often modelled as a single magnetic flux tube (e.g., Evans and Roberts 1990a). A first detection of sausage modes in pores, in the lower atmosphere, is reported by Dorotovi˜c, Erd´elyi and Karlovsk´y (2008) and a detailed study of sausage modes in pores is provided by Morton et al. (2011), Dorotovi˜c et al. (2014) and Grant et al. (2015). Grant et al. (2015) reported speeds of 15 km s−1 , rather higher than expected, but this is in part connected with a detected upflow which adds to the propagation speed;

488

Solar Applications of MHD Wave Theory

they conclude that the actual propagation speed of the sausage mode is about 5 km s−1 . Many of these observations are made using ground-based instruments, such as the Dutch Open Telescope (DOT) operating on the island of La Palma in the Canary Islands and the Rapid Oscillations in the Solar Atmosphere (ROSA) instrument installed in Sacramento Peak Observatory, New Mexico. A very recent study by Keys et al. (2018) uses ROSA and SDO data to explore the presence of surface and body slow sausage modes in pores. They find that both modes may be present but surface modes arise more frequently than body waves. They explore the modes by examining both area and intensity variations in seven data sets of pores. Intensity variations tend to be in phase with area variations, suggesting the slow sausage mode (Moreels, Goossens and Van Doorsselaere 2013). Moreover, the oscillations in pores appear to be in phase across the whole structure of the pore, suggesting a monolithic flux tube. It is interesting to note that such coherent behaviour is not found in the oscillatory signal outside of the pore (Keys et al. 2018) and nor is it seen in sunspots. Photospheric flux tubes are difficult to study, partly because stratification is likely to play a significant part and also their smallness makes observations of their vibrations difficult to carry out. There is little distinction here between small-scale magnetic flux tubes and pores, save perhaps the size of pores makes them more readily observed, so many studies of the oscillations in pores are also relevant to the oscillations of isolated flux tubes. Despite obvious difficulties, attempts to carry out a seismology of photospheric flux tubes have been made, either from purely a modelling viewpoint (Evans and Roberts 1990a) or more recently from a combination of modelling and high resolution space or ground-based observations (Fujimura and Tsuneta 2009; Moreels and Van Doorsselaere 2013; Moreels, Goossens and Van Doorsselaere 2013; Moreels et al. 2015; Freij et al. 2016). So far, the effects of gravity have not been properly taken into account. The main effort has been to calculate the dispersion relation in a form suitable for photospheric conditions (see, for example, Edwin and Roberts 1983; Evans and Roberts 1990; Moreels, Goossens and Van Doorsselaere 2013; see also Chapter 10) and furthermore to determine (numerically or otherwise) the behaviour of observables such as velocity, area change and intensity (Morton et al. 2011; Moreels, Goossens and Van Doorsselaere 2013; Freij et al. 2016).

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Index

acoustic cutoff frequency, 251, 262, 265, 306 coronal, 252 photospheric, 252 terrestrial, 252 acoustic-gravity waves, 253, 257 confluent hypergeometric functions, 262 linear temperature profile, 261 Adam, J. A., 277 adiabatic stratification, 263 advection of magnetic field, 18 Alfv´en frequency, 186, 210 Alfv´en singularity, 211 local behaviour, 211 Alfv´en speed, 10, 12, 27, 28, 69, 140 in corona, 12 in photosphere, 12 in sunspot, 484–487 Alfv´en wave, 2, 28, 71, 315 equipartition of energy, 42 in stratified loop, 272 in stratified medium, 271 phase mixing, 78, 370 phase speed and group velocity, 38 similarity solution of phase mixing, 375 Alfv´en, H., 2, 7 Ampere’s law, 8 Andries, J., 180, 239, 333, 475 angle φ that cg makes with B0 , 47 angle  that motion makes with B0 , 53, 54 angle ψ in fast wave that cg makes with direction of propagation k, 49 angle ψ that cg makes with k, 47 angular frequency ω, 23 anisotropic wave propagation, 3, 24 anisotropy in Alfv´en wave, 38 in MHD, 27, 33, 143 in slow wave, 46 Anzer, U., 456, 462 Appert, K., 77, 182, 189 Arber, T. D., 133 Arregui, I., 457, 462, 466

Aschwanden, M. J., 6, 7, 180, 239 associated Legendre functions, 134 autosolitons, 419 Ballester, J. L., 129, 457, 462, 465 Basset’s integral, 413 Beckers, J. M., 482 Benjamin–Ono equation, 405, 406 for magnetic slab, 405 linearized form, 410 soliton in magnetic slab, 407 verification of soliton solution, 408 Benjamin–Ono soliton spatial scale l, 408 speed C, 408 Benjamin–Ono–Burgers equation, 417 Bennett, K., 182, 192, 193, 195, 196 Benz, A. O., 131, 132, 440 β = 0 plasma, 58, 63, 80, 312 dispersion relation for kink mode, 318 equations, 312 linear system of equations, 313 spatial damping of kink wave, 452 temporal damping of kink wave, 452 wave-like equation for total pressure perturbation, 315 Bessel differential equation, 149, 159 twisted, purely azimuthal, field, 203 Bessel equation, 77, 436, 437 leaky waves, 159 uniform tube, 149, 150 Bessel functions, 139, 149–151, 153, 154, 156, 159, 194, 203–205, 282, 436 behaviour at infinity, 157 expansion, 154, 176 recurrence relations, 153 small argument behaviour, 154 body wave, 116, 140 Bogdan, T. J., 482 boundary conditions across interface, 86 boundary conditions for uniform tube, 150 Boyd, T. J., 7

501

502

Index

Bray, R. J., 7 Brunt–V¨ais¨al¨a frequency, see buoyancy frequency buoyancy frequency, 251, 263, 275, 485 coronal, 252 photospheric, 252 terrestrial, 252 Cally, P. S., 151, 159, 161, 163 Campbell, W. R., 279 Cargill, P. J., 373, 380 Cauchy principal value, 406 Chandrasekhar, S., 279 Chen, L., 73, 99, 101 Cho, I.-H., 485 Choudhuri, A. R., 7 chromosphere, 4 cold plasma, 58 equations, 312 Colladas, M., 482 CoMP, see Coronal Multi-channel Polarimeter complex variable representation of perturbations, 34 compression, 31, 185, 254, 283 confluent hypergeometric functions, 197, 262, 280, 470 connection formula, 179, 180, 207 Alfv´en singularity, 228, 230 β = 0 plasma, 234 linear density profile, 237 sinusoidal density profile, 238 cusp singularity, 230 derivation of dispersion relations, 231 for Alfv´enic point, 229 general formalism, 231 no twist, 230 photospheric tube, 240 slow mode, 230 untwisted tube, 231 twisted tube, 228 untwisted limit, 230 continuity equation, 7 convection zone, 4 convective collapse, 302 convective operator in cylindrical coordinates, 185 Cooper, F. C., 133 corona, 4 coronal loop magnetic field strength, 446 Coronal Multi-channel Polarimeter (CoMP), 446, 452 coronal oscillations, 439 coronal seismology, 6, 439, 440, 444, 453, 454 Cowling, T. G., 2, 7, 256 cusp frequency, 210 cusp singularity, 211 cusp speed, 207 cutoff frequency for kink mode, 356, 357 for sausage mode in thin tube, 301 in isothermal loop, 307 in straight field, 306

in straight tube, 306 in sunspot, 484–487 in uniform vertical magnetic field, 485 in vertical magnetic field, 275 of sound, 265 slow mode in a neutral atmosphere, 487 slow mode in an isothermal atmosphere, 487 cutoff wavenumber: Epstein profile, 138 cyclic frequency, 252 cylindrical coordinates convective operator (A · grad) B, 185 D’Alembert’s solution of the wave equation, 71, 78 D´ıaz, A. J., 334, 337, 465, 466, 468, 471, 472 damping exponential, 452 Gaussian, 452 slow wave, 381 damping scale for Alfv´en wave in uniform medium, 367 phase mixing wave, 372–374 damping scale for slow mode coronal illustrations, 390 thermal conduction alone, 389 thermal conduction and viscosity, 394 viscosity alone, 386 damping time for Alfv´en wave in uniform medium, 368 phase mixing wave, 380 damping time for slow wave thermal conduction alone, 387 thermal conduction and viscosity, 394 viscosity alone, 385 De Moortel, I., 309, 383, 393 decay rate in β = 0 plasma, 236 resonant absorption at interface of incompressible plasma, 247 resonant absorption in β = 0 plasma, 180 decay time of leaky sausage wave, 165 Defouw, R. J., 292, 301 delta function, 219, 266 Fourier representation, 219, 266 density scale height, 250, 262 differential equations for perturbations in β = 0 twisted plasma, 201 for sausage mode in β = 0 twisted plasma, 202 in cylindrical coordinates, 145 diffusion of magnetic field, 17 diffusive decay time, 17 diffusive spatial scale, 216 diffusivity, 14 dispersion diagram coronal conditions, 441 for Alfv´en wave, 39, 40 for magnetoacoustic waves, 45 for sound wave, 39 photospheric conditions, 483, 484

Index dispersion relations, 24 Alfv´en wave, 38 in stratified loop, 272 β = 0 surface wave, 98 dispersive correction in a thin tube, 319, 321 for magnetic flux tube with thin transition layer, 234 in uniform medium, 43 incompressible surface waves, 97 incompressible tube, 167 kink mode in β = 0 plasma, 318 kink modes in slab, 108, 109 leaky kink waves, 130 leaky sausage waves, 128 magnetoacoustic waves, 42, 44 p and f modes, 280 photospheric tube, 168 principal kink mode, Epstein profile, 137 principal sausage mode, Epstein profile, 137 sausage modes in slab, 108, 109 sausage modes in thin tube, 282 sausage waves in twisted annular region of β = 0 plasma, 204, 206 small twist, 197 sound and gravity waves, 258 sound wave, 36 with thermal conduction and viscosity, 384 standing Alfv´en waves in stratified loop, 272 surface waves, 88, 96 thin tube fluting modes, 157 kink modes, 156 sausage modes, 154 thin tube relation for kink mode, 319 uniform tube, 151 body waves, 152 connection with surface waves, 152 leaky waves, 161 short (kz a1) wavelength behaviour, 157 small kz a expansion, 153 surface waves, 151 waves in compressible tube with small twist, 200 waves in incompressible tube, 167 waves in uniformly twisted tube, 195 dispersive and non-dispersive waves, 24 dispersive wave propagation, 103 Dorotovi˜c, I., 487 DOT, see Dutch Open Telescope Dungey, J. W., 2, 182, 192, 195, 196 Dutch Open Telescope (DOT), 488 Dymova, M. V., 329, 464 Edwin and Roberts dispersion diagram coronal conditions, 441 photospheric conditions, 483 Edwin and Roberts dispersion relation, 437, 439 Edwin, P. M., 110, 121, 127, 128, 131, 132, 148, 151, 168, 171, 279, 437, 440, 483 energetics, 21 energy balance equation, 9

503

Epstein profile, 133 Erd´elyi, R., 183, 195–199, 201, 207, 279, 440 Evans, D. J., 279, 488 exponential damping spatial scale, 452 temporal scale, 453 f mode, 280 Faraday’s law of induction, 1, 8 fast kink mode, 443 fast kink standing modes, 443 fast magnetoacoustic wave, 29 fast mode, 448–456 cutoff wavenumber, 448 fast speed, 69, 140, 435 fast surface wave, 88, 90 fast waves angle ψ that cg makes with k, 49 in presence of stratification, 271 speed cf , 29 Fedun, V., 183, 195–199, 201 Ferraro, V. C. A., 2, 7, 271, 272 Ferriz-Mas, A., 283 Field, G. B., 395 fluting modes, 283 force-free equilibrium, 313 Foullon, C., 279 Fourier representation, 23 complex variables form, 34 in cylindrical coordinates, 144 of delta function, 219, 266, 410, 411 of operators, 24 of plane wave, 23 of scalar and vector operators, 34 Fourier transform, 218, 405 of sgn function, 406 of wave speed c, 409 Friedrichs, K. O., 3, 39, 45 frozen-in flux, 34 g modes, 256 Gaussian damping, 452 spatial scale, 452 temporal scale, 453 general phase mixing equation, 369 geometric dispersion, 399 geometrical shapes of an oscillating tube, 139, 145 Giovanelli, R. G., 482 Goedbloed, J. P., 7, 73, 277 Goossens, M., 7, 159, 163, 171, 177, 180, 207, 212, 215, 217, 218, 229, 239, 243, 283, 417 Grant, S. D. T., 487 granules and supergranules, 5, 281 gravity waves, 258 Grossmann, W., 99 group speed, 26 group velocity, 24, 25, 36 Gruber, R., 77, 182, 189

504 Hain, K., 76 Hain–L¨ust equation, 76, 148 Hale, G. E., 5 Hankel functions, 151, 160 and leaky waves, 160 Hasegawa, A., 73, 99, 101 helioseismology, 440, 482 Herlofson, N., 3 Heyvaerts, J., 78, 369, 380 Hilbert transform, 406 Hollweg, J. V., 101, 159, 163, 180, 207, 212, 229, 238, 239 Hood, A. W., 375, 383, 393, 452 hydromagnetic surface wave, 89, 103, 277 in presence of gravity, 279 hypergeometric equation, 134, 135 hypergeometric functions, 135 ideal conductor, 18 ideal gas law, 9 impulsive timescale for slab, 132 impulsively generated waves, 131 incompressible fluid, 58, 60, 80 induction equation, 9, 14 initial value problem β = 0 tube, 179 for kink oscillations, 180 incompressible surface waves, 100, 179 intense flux tubes, 5, 481, 487 interface boundary conditions, 86 Ionson, J. A., 99, 101 Ireland, J., 375 isentropic balance equation, 9 Isobe, H., 457 isothermal atmosphere, 251, 257 j × B force, 1, 19 Jain, Rekha, 84, 98, 279, 457 Jeffrey, A., 7 Jess, D. B., 483 Joarder, P. S., 456, 457, 459–464 Joule heating, 9 Judge, P. G., 482 jump conditions, 228 at single interface, 244 incompressible medium, 245 linear c2A , 247 Karpen, J. T., 466, 468, 471, 472 Kendall, P. C., 7 Keppens, R., 7 Khomenko, E., 482 Khongorova, O. V., 204–206 kinematic viscosity, 15 kink mode, 106, 316 effect of gravity, 348 in loop period ratio, 333, 337 in non-uniform magnetic field, 338

Index low compression, 322 magnetic pressure force, 322 magnetic tension force, 322 perturbations in β = 0 plasma, 321 standing wave in structured loop, 329, 334 use of multiple scales, 323 kink speed ck , 100, 141, 279, 311, 319 kink wave compression, 327 Klein–Gordon equation, 264, 307–309 for acoustic-gravity waves, 264 for slow mode in stratified medium, 275 for vertically propagating sound waves, 264 Fourier solution, 265 in vertically stratified medium, 275 initial value solution, 266 integral form of solution, 267 magnetic flux tube, 301 oscillating wake effect, 269 photospheric tube, 300, 301 properties, 265 recovery of wave equation, 269 reduction of solution to D’Alembert’s form, 267 rigid tube, 295 slow mode, 485 solution of initial value problem, 268 straight magnetic field, 306 straight tube, 306 Kranzer, H., 39, 45 Kruskal, M., 279 Kummer equation, 262, 470 Kummer function, 470 L¨ust, R., 76 Lamb, H., 254, 255, 264 leakage time, 129 leakage versus resonant absorption, 159, 239 leaky waves, 116, 151, 159, 438 kink mode in β = 0 plasma, 163 kink modes, 129 sausage modes, 128 Lee, M. A., 99–101 Legendre’s equation, 134 Leibovich–Roberts equation, 414 magnetic tube, 413 Leroy, B., 370 Lighthill, M. J., 3 Lin, Y., 462 linear vs nonlinear behaviour, 398 linear wave equation including pressure forcing, 400 linearization, 22 linearized equations, 30 Lopin, I., 354 Lorentz force, 1, 28 Loughhead, R. E., 7, 182, 192, 195, 196 Love, A. E. H., 128 Luna, M., 466–468, 471, 472 Luna-Cardozo, M., 384, 393

Index Macnamara, C. K., 384, 475 magnetic diffusivity η, 9, 14 magnetic field advection, 18 diffusion, 17 magnetic pressure, 19 magnetic tension, 20 radius of curvature, 21 strength, 3, 444–446 Gaussian units, 445 in a coronal loop, 446 in coronal loop, 444 Nakariakov and Ofman formula, 444 SI units, 444, 445 magnetic flux tube, 13, 148 equilibrium, 148 magnetic force, j × B, 19, 28 magnetic helioseismology, 263, 279 magnetic interface, 83 magnetic Love waves, 128 magnetic Pekeris waves, 128 magnetic permeability μ, 8 magnetic pressure, 19, 28, 68 magnetic reconnection, 16 magnetic Reynolds number, 15 Earth’s core, 16 molten iron, 16 solar, 16 magnetic slab, Whitham evolution equation, 409 magnetic tension, 20, 28 magnetoacoustic dispersion diagram Alfv´en speed greater than sound speed, 45 Alfv´en speed less than sound speed, 45 magnetoacoustic dispersion relation, 43 compact form, 44 magnetoacoustic waves close alignment of fast wave’s group velocity with k, 48 close alignment of slow wave’s group velocity with B0 , 48 fourth order wave equation, 33 in extremes cA  cs or cA  cs , 44 in stratified medium, 273 parallel propagation, 44 perpendicular propagation, 44 phase speed, 42 phase speed diagram when cA = cs , 58 phase speed diagrams, 45 special case cA = cs , 57 surface waves, 88 uniform medium dispersion relation, 42 magnetohydrodynamics (MHD), 1 Mann, I. R., 4 MAPLE, 262 mathematical modelling, 434, 435 Maxwell, James Clerk, 1 McEwan, M. P., 334, 337 McKenzie, J. F., 151

505

Meerson, B. I., 151 Mendoza-Brice˜no, C., 384, 393 MHD equations (linear) in terms of displacement, 184 MHD induction equation, 9 MHD seismology, 440, 483 Mikhalyaev, B. B., 204–206 Miles, J. A., 277 modified Bessel functions, 149 momentum equation, 7 Morton, R. J., 487 moving coordinates, 401 multiple scales, 323, 423 Murawski, K., 132, 144 Nagorny, I., 354 Nakariakov and Ofman formula, 444 Nakariakov, V. M., 7, 132, 133, 144, 180, 329, 333, 381, 419, 426, 440, 443, 462–464 Narain, U., 182, 192, 193, 195, 196 Narayanan, A. S., 7 neutral atmosphere, 263 Newtonian sound speed, 383 Ning, Z., 462 Nocera, L., 370 nonlinear coefficient β0 , 403 nonlinear effects importance in Sun, 397 wave steepening time, 398 nonlinear waves Benjamin–Ono equation for magnetic slab, 405 weakly nonlinear waves, 400 Ofman, L., 391, 393 Ohm’s law, 1, 8 Okamoto, T. J., 462 Oliver, R., 129, 132, 456, 457, 460–462, 465 p modes, 256, 280 Parker, E. N., 7, 9, 101, 279, 292, 303, 370, 482 Pascoe, D. J., 133, 452, 453 Pekeris diagram, 132 Pekeris, C. L., 128, 131 perfect conductor, 18 period of leaky sausage wave, 165 of slow mode in loop, 308 of slow mode in SUMER loop, 308 period ratio, 329, 333 in coronal loops, 475–481 in prominence threads, 475–481 kink mode in loop, 333, 337 perturbations in Alfv´en wave, 40 in magnetoacoustic wave, 50 phase, 23 phase mixing, 78, 79, 144, 370 effect of stratification, 381 general equation, 369 in coronal loop, 79

506 phase mixing (Cont.) in sunspot, 79 of Alfv´en wave, 78 of propagating wave, 78 of standing wave, 79 reduced equation, 369 secular growth timescale, 78 spatial scale, 367, 373 temporal scale, 380 phase speed, 23, 24, 36 phase velocity, 23 photosphere, 4, 5 Pinter, B., 279 plane wave, 23, 33 plane wave representation, 36 plasma β, 20 in corona, 312 in sunspots, 484 Plumpton, C., 7, 271, 272 Poedts, S., 7 pores, 481, 487 pressure scale height, 250, 262 pressure variations magnetic, 68 total, 69 pressureless equations, 312 Priest, E. R., 6, 7, 78, 279, 369, 370, 375, 380, 462 principal kink mode, Epstein profile, 137 principal modes of Epstein profile, 136 principal sausage mode, Epstein profile, 137 prominence oscillations hybrid modes, 459 load on a string analogy, 461 string modes, 459 prominence seismology, 7, 457, 462, 473 prominences, 7 propagating slow waves in loop, 309 propagation speeds of tube, 140 quasi-mode, 229 R´egnier, S., 457 radial pulsations, 435, 437 Rae, I. C., 99–101, 301, 309, 485 Rapid Oscillations in the Solar Atmosphere (ROSA), 488 ratio of specific heats, γ , 9 Rayleigh–Taylor instability, 279 reduced phase mixing equation, 369 refraction, 260 p modes, 261 resonance point, 179 resonant absorption, 99, 180 at single interface, 99 decay rate in β = 0 flux tube, 179 decay time in β = 0 plasma, 181 decay timescale, 101 in β = 0 flux tube, 179 in incompressible fluid, 99, 101

Index timescales, 238 versus leakage, 239 resonant layers F function, 223 G function, 223 Roberts, B., 73, 84, 96, 98–101, 110, 121, 127, 128, 131–133, 144, 148, 151, 155, 168, 171, 178–180, 182, 192, 193, 195, 196, 238, 239, 275, 277, 279, 283, 288, 290, 291, 301–306, 309, 334, 337, 381, 384, 403, 413, 414, 419, 437, 440, 456, 457, 459–464, 483, 485, 487, 488 Roberts, P. H., 7 ROSA, see Rapid Oscillations in the Solar Atmosphere Rosenberg, H., 435–437, 440 Ruderman, M. S., 132, 171, 179, 180, 204–207, 229, 238, 329, 381, 414, 427, 453, 464, 466, 467 running penumbral waves, 482 Ryutov, D. D., 279 Ryutova, M. P., 7, 279 Sakurai, T., 180, 207, 212, 229 Sanderson, J. J., 7 sausage mode, 106, 283, 439 decay rate, 438, 439 dispersive correction in thin tube, 282 effects of stratification, 291 general dispersion relation in thin tube, 290 in β = 0 plasma, 144 in slab, recovery of linear theory, 407 nonlinear thin tube equations, 399, 400 period, 439, 450 thin tube wave equation, 288 wave speed in thin tube, 282 scale height coronal, 252 photospheric, 252 scaled coordinates, 324, 401 operator transformations, 401 Schultz, R. B., 482 Schussler, M., 283 Schwarzschild, M., 279 SDO, see Solar Dynamics Observatory Sedlacek, Z., 99, 101 seismology of photospheric tubes, 488 sgnsgn function, 217 shear , 32 Sigalotti, L. D. G., 384, 393 singularities Alfv´en, 77 slow (or cusp), 77 slab sausage wave, effect of environment, 404 slow body wave, 155 slow frequency, 187, 210 slow magnetoacoustic wave, 28 slow magnetoacoustic waves in stratified medium, 274 slow mode, 446–448 in coronal loops, 306 in stratified medium, 304

Index Klein–Gordon equation, 485 period, 446 propagating waves, 456 singularity, 211 slow speed, 69, 140, 435 slow surface wave, 88, 90, 155 slow timescale, 401 slow waves angle φ that cg makes with B0 , 49 damping, 381 in magnetic tube, 296 in photospheric flux tube, 298 in presence of stratification, 271 in rigid tube, 294 speed ct , 29, 33 Smith, P. D., 381 SoHO, see Solar and Heliospheric Observatory Solar and Heliospheric Observatory (SoHO), 455 Solar Dynamics Observatory (SDO), 446 solar radius, 4 solenoidal constraint, 8, 30 Soler, R., 211, 234, 240, 243, 466 soliton, 419 in magnetic slab, 407 sound speed, 10, 11, 27, 69, 140, 251 sound waves, 35, 70, 258 Southwood, D. J., 4, 73 spatial and temporal behaviour thermal conduction and viscosity, 393 spatial damping of kink wave in β = 0 plasma, 452 speed ck for interface, 279 Spruit, H. C., 151, 159, 161, 163, 178, 279, 303, 304 standing Alfv´en waves in stratified loop, 272 standing waves, 441, 442 in coronal loop, 308 Stepanov, A. V., 151, 159, 161, 163, 437 Stix, M., 283 stratification relative unimportance in corona, 308 string modes, 461 Sturm–Liouville formulation, 302 Sun’s magnetic field, 5 sunspots, 2, 5, 481 Alfv´en speed, 487 magnetic field strength, 5, 484 plasma β, 484, 487 running penumbral waves, 482 seismology, 485 sound speed, 487 umbral flashes, 482 umbral oscillations, 482 superadiabatic effect, 302 surface wave, 116, 140 surface waves basic properties, 94 β = 0 medium, 93, 98 dispersion relation, 88, 96 non-parallel propagation, 96 fast wave, 90

507

graphical approach, 90 hydromagnetic, 98, 277 incompressible case in stratified medium, 277 incompressible medium, 97 non-parallel propagation, 95 on magnetic–nonmagnetic interface, 89 parallel propagation, 85 slow wave, 90 wave speed, 88 Sych, R. A., 482, 485 Tallant, P. E., 482 Tataronis, J., 99 temporal damping of kink wave in β = 0 plasma, 452 Terradas, J., 129, 132, 453, 464, 465 thin flux tube theory: sausage mode, 282 thin tube decay rate from resonant absorption in β = 0 plasma, 236 decay rate from resonant absorption in incompressible plasma, 243 thin tube equations for sausage mode, 287 straight tube, 304 with gravity, 291 zeroth order continuity equation, 284 zeroth order equations, 287 thin tube limit fluting modes, 162 kink mode, 162 thin tube, thin boundary approximation, 231 thin tube, thin boundary dispersion relation incompressible tube, 241 Thomas, J. H., 482 Tirry, W., 215, 217, 218, 229 torsional Alfv´en waves, 79, 140, 144, 316 total pressure changes, 69 total pressure perturbation pT , 30, 56 TRACE, see Transition Region And Coronal Explorer, see Transition Region And Coronal Explorer Transition Region And Coronal Explorer (TRACE), 6, 439 Tripathi, D., 457 Tsiklauri, D., 381 tube shape dependence on azimuthal wavenumber m, 145 tube speed ct , 29, 140, 288, 400, 435 in sunspot, 484–487 twisted magnetic flux tube, 76, 182, 183, 208 boundary conditions, 189 coefficients D, C1 , C2 and C3 of fundamental differential equations, 188 confluent hypergeometric functions, 197 connection formulas, 229 differential equation for pT and ξr , 189 fundamental differential equations for perturbations, 188 jump relations, 229 Whittaker equation, 199

508 umbral flashes, 482 umbral oscillations, 482 uniform medium, 73 Bessel equation, 77 uniform tube β = 0 plasma, 316 incompressible medium, 167 isolated tube, 168 units for magnetic field, 3 Vaclavik, J., 77, 182, 189 van de Hulst, H. C., 3 Van Doorsselaere, T., 329, 333 velocity field in magnetoacoustic waves, 52 vertical propagation of sound waves, 263 Verwichte, E., 168, 171, 329, 333 viscous damping of sound wave, exact solution, 387 vorticity ωz along the magnetic field, 33 Walker, A. D. M., 4 Wang, T. J., 391, 393 wave differential equations in Cartesian coordinates, 72 in cylindrical polar coordinates, 75 wave equation for sound, 35 wave equations for a β = 0 plasma, 312 wave guides, 3 wave packet, 26 wave speeds, 3 wave vector, 23

Index waves in incompressible fluid, 103, 104 waves in vertical magnetic field, 269 weakly nonlinear waves, 400 Webb, A. R., 151, 155, 178, 275, 283, 288, 290, 291, 301–304 Weiss, N. O., 482 Wentzel, D. G., 96 Whitham equation, 409 for magnetic tube, 410 Williams, D. R., 131, 133 Wilson, P. R., 151 Wright, A. N., 4 Yang, G., 238 Zaitsev and Stepanov dispersion relation, 437, 439 Zaitsev, V. V., 161, 163, 437 Zaqarashvil, T., 419 zero β case, 58 zeroth order equations continuity equation, 289 isentropic equation, 286 second order differential equation, 302 stratification, 291 uniform tube, 286 zeroth order thin tube, 284 Zhugzhda, Y. D., 283, 417, 482, 485 Zirin, H., 482 Zweibel, E. G., 303, 304