Energy Transfer and Dissipation in Plasma Turbulence: From Compressible MHD to Collisionless Plasma [1st ed.]
 978-981-13-8148-5;978-981-13-8149-2

Table of contents :
Front Matter ....Pages i-xix
Introduction (Yan Yang)....Pages 1-21
Theoretical Modelling (Yan Yang)....Pages 23-33
Hybrid Scheme for Compressible MHD Turbulence (Yan Yang)....Pages 35-67
Energy Cascade in Compressible MHD Turbulence (Yan Yang)....Pages 69-90
Energy Transfer and Dissipation in Collisionless Plasma Turbulence (Yan Yang)....Pages 91-110
Conclusions and Discussion (Yan Yang)....Pages 111-113
Back Matter ....Pages 115-134

Citation preview

Springer Theses Recognizing Outstanding Ph.D. Research

Yan Yang

Energy Transfer and Dissipation in Plasma Turbulence From Compressible MHD to Collisionless Plasma

Springer Theses Recognizing Outstanding Ph.D. Research

Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists.

Theses are accepted into the series by invited nomination only and must fulfill all of the following criteria • They must be written in good English. • The topic should fall within the confines of Chemistry, Physics, Earth Sciences, Engineering and related interdisciplinary fields such as Materials, Nanoscience, Chemical Engineering, Complex Systems and Biophysics. • The work reported in the thesis must represent a significant scientific advance. • If the thesis includes previously published material, permission to reproduce this must be gained from the respective copyright holder. • They must have been examined and passed during the 12 months prior to nomination. • Each thesis should include a foreword by the supervisor outlining the significance of its content. • The theses should have a clearly defined structure including an introduction accessible to scientists not expert in that particular field.

More information about this series at http://www.springer.com/series/8790

Yan Yang

Energy Transfer and Dissipation in Plasma Turbulence From Compressible MHD to Collisionless Plasma Doctoral Thesis accepted by the Peking University, Beijing, China

123

Author Dr. Yan Yang Peking University Beijing, China

Supervisors Prof. Yipeng Shi Peking University Beijing, China Prof. William H. Matthaeus University of Delaware Newark, Delaware, USA Prof. Minping Wan Southern University of Science and Technology Shenzhen, Guangdong, China Prof. Shiyi Chen Southern University of Science and Technology Shenzhen, Guangdong, China

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-981-13-8148-5 ISBN 978-981-13-8149-2 (eBook) https://doi.org/10.1007/978-981-13-8149-2 © Springer Nature Singapore Pte Ltd. 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

To my husband Weiwei and to my supervisors Yipeng Shi, William H. Matthaeus, Minping Wan and Shiyi Chen

Supervisors’ Foreword

Foreword I Several decades of development—one might even say, evolution—of two subjects, and of several generations of physical scientists, come to fruition in the present volume. This is a personal observation on both counts, but I believe that in a very real sense one may trace a valuable increase in the maturity in the study of nonlinear plasma turbulence, through tracking the academic leanings of a group of scientists with whom I have had the good fortune of collegial interactions over this time period. Plasma physics in the late 1960s to mid-1970s was a subject dominated by linear instabilities and small amplitude wave theory. So deeply ingrained was this “linear theory” approach, that the surge of papers on magnetic reconnection, a manifestly nonlinear phenomenon, did not emerge until the early 1990s, even though the foundational works of Sweet and Parker were already known in the late 50s. Instead, the 70s was an era of discussion of tearing modes and other attempts to describe explosive nonlinear activity using a small amplitude vocabulary. I was not yet in the subject, but my Ph.D. thesis advisor David Montgomery was, and by the mid-60s had established himself, at a young age, as an eminent plasma kinetic theorist, having published, among other things, a definitive text on electrostatic plasma kinetic theory by his mid-20s. As the story has been related to me, Prof. Montgomery was puzzled by the enigmatic phenomenon of Bohm diffusion, and was thus deeply inspired by the explanation given Taylor and McNamara in 1971 based on self-organization of guiding center fluid, a direct analog of two-dimensional hydrodynamics. Montgomery’s subsequent excursion into hydrodynamic and magnetohydrodynamic (MHD) turbulence theory led him to close associations with Robert Kraichnan, Uriel Frisch, Keith Moffatt, and other established leaders in fluid turbulence theory. Several years later, when Montgomery's own contributions to theories of diffusion, selective decay, and Gibbsian statistical mechanics were already well on their way, I joined his group as a graduate student, and learned to

vii

viii

Supervisors’ Foreword

make contributions in this fertile field of study—not plasma physics per se, but rather MHD turbulence theory applied to laboratory and space plasmas. The following years were kept busy by developing MHD turbulence ideas in the solar wind. This approach had been recommended by the NASA Plasma Turbulence Explorer Panel, chaired by David Montgomery and co-authored by Mel Goldstein, who taught me space plasma physics while supervising my postdoc at Goddard Space Flight Center. This was an interesting period, extending well into my years at Bartol Research Institute at the University of Delaware. Even while space plasma observations provided a seemingly endless potential for application of MHD turbulence theory, many questions were also arising, particularly with regard to the nature of cascade and dissipation processes when the medium is a collisionless plasma. MHD and hydrodynamic computer simulations of turbulence were coming into their own, but equivalent advances in plasma simulations would await another 20 years of development of computational capabilities. In the late 80s, I was privileged to work almost simultaneously, with four postdoctoral fellows whose subsequent paths and prominent careers in turbulence serve to well define the era. Ye Zhou worked with me on solar wind transport and went on to have a successful career at Stanford and Livermore. Gary Zank worked with me on weakly compressible MHD and launched a successful and diverse career in outer heliospheric plasma physics. Hudong Chen worked on lattice gases and lattice boltzmann hydrodynamic numerical methods and now is chief scientist at EXA Corporation. Shiyi Chen, worked on turbulence and lattice Boltzmann methods with Hudong and me, prior to becoming a scientist at IBM Yorktown Heights, and then Chair of Mechanical Engineering at Johns Hopkins. A pivotal moment, both for my group and for the emerging field of kinetic plasma turbulence, came when Shiyi accepted the position of founding Dean of Engineering at Peking University. Prior to departing Hopkins, in 2008, Shiyi called and asked if I would be interested in hiring Minping Wan, his newly graduated turbulence student, a Ph.D. in Mechanical Engineering. Minping came as a postdoc to Delaware and worked with me for the next 7 years, producing an array of fundamental papers on MHD, numerical methods, and notably, turbulence theory applied to kinetic plasmas. This work meshed seamlessly with pioneering kinetic turbulence simulation work being done at Delaware by Tulasi Parashar and by collaborator and former postdoc Sergio Servidio and his colleagues in Calabria. Minping added a healthy dose of pure turbulence methods to all the work in the Delaware group. During this period came the important recognition that Particle-in-cell and Eulerian Vlasov simulations had become powerful enough to probe larger kinetic systems, that is, those having high enough effective Reynolds numbers to meaningfully probe turbulence effects. As we redoubled our pursuit of understanding plasma dissipation in turbulence, we were informed that Shiyi Chen would leave Beijing to become President at Southern University of Science and Technology in Shenzhen, leaving his new Ph.D. student under the direction of a young professor, Yipeng Shi. We were asked if we would host the new student, who was well-schooled in fluid mechanics and hydrodynamic turbulence, at Delaware for a

Supervisors’ Foreword

ix

2-year period to broaden her experience for the Ph.D. degree in Mechanical Engineering. We agreed and this student, Yan Yang, came to Delaware and asked to do research on a new subject for her, plasma turbulence. After just a few months of study, Yan announced that she was ready for research projects, and so began the series of studies that comprise the substantial amount of material contained in the present volume. It is my belief that this approach to understanding plasma turbulence through identifying and quantifying the channels of energy transfer and conversion is a valuable new direction for understanding complex nonlinear behavior of space plasma turbulence. The assembly of this material along with appropriate introductory and supplementary material will be a valuable and unique resource for researchers and graduate students entering the field of turbulence in magnetized space plasmas. Above all, the merger of the approaches of hydrodynamics and plasma physics that can be seen in this formulation bodes well, in this writer’s opinion, for the future understanding of plasma turbulence in space and its many applications. Newark, Delaware, USA February 2019

William H. Matthaeus

Foreword II Turbulence is a state of fluid motion prevalent in nature and various engineering applications, such as air flow, river rapids, flow around the surface of an aircraft, flow in fluid machinery, and combustion. An essential feature of turbulence is the randomness of fluid motion, characterized by chaotic changes in flow velocity. Unlike laminar flow, the trajectories of fluid parcels are extremely turbulent, change rapidly with time, and are difficult to predict. Another feature of turbulent flow is the existence of many vortices or eddies of different sizes that oscillate, squeeze, collide, connect, and merge. In this nonlinear interaction, the kinetic energy is transferred from the large eddies to the small eddies, and then to the smaller eddies until being dissipated at the smallest eddies by the viscosity. This transfer of energy across scales is called the energy cascade, another important feature of turbulence first proposed by L. F. Richardson in the 1920s. Energy cascades also exist in turbulent flows in magnetized fluids and plasmas. The energy cascades in MHD turbulence is more complicated than its counterpart in hydrodynamic turbulence, including magnetic–magnetic and kinetic–kinetic energy transfers, as well as energy conversion between magnetic and kinetic energies. Nevertheless, great progress has been achieved in the past 20 years in understanding energy cascades in incompressible MHD turbulence. For the cascades in compressible MHD turbulence, more efforts are needed. This book provides such

x

Supervisors’ Foreword

an effort for a deep understanding of the energy transfers in compressible MHD with numerical simulations. For turbulent plasma flows, such as the solar wind, the importance of energy cascades has been recognized from the 1980s, as they were regarded as an important heating source of solar wind plasma. However, little is known about how energies are transferred across scales and dissipated, as solar wind plasma is collisionless or nearly collisionless. In the past several years, this topic has attracted lots of attentions, as large-scale simulations of collisionless plasma turbulence and high-resolution satellite data from new missions provided much-needed tools for this kind of studies. From 2017, we have been working with the author, Dr. Yan Yang, on this topic. Yan, then a Ph.D. student majoring in fluid mechanics at Peking University, developed a way to study this problem, with the idea borrowed from HD turbulence studies. A filter approach is applied to the Vlasov equation— the governing equation of kinetic plasma. Such an approach brings up several terms associated with different physical processes responds for the energy transfers and dissipations. A first of its kind in this field, this work opens the door to understand the complicated energy transfers and dissipation in collisionless plasma turbulence, from a point view of classic turbulence theories. Shenzhen, China March 2019

Minping Wan Yipeng Shi Shiyi Chen

Preface

The material presented in this book grew out of a doctoral dissertation, written by the same author. Then one might think that it is more of a research monograph. But it is worth emphasizing first that this book is not designed to be too dense. Sufficient introductory text, although not as extensive as several excellent textbooks on turbulence and plasma physics, is provided. Sufficient references are given as well to provide an entry into the research literature. Most of the material is presented including the critical steps of derivation so that the reader can follow without need to do extensive algebra. We presuppose an average knowledge of turbulence and plasma physics. So this work is designed to be a useful reference, not only for professional researchers, but also for students first encountering the field and desiring a qualitative introduction to the field. There is a vast range of scales in kinetic plasma, spanning from macroscopic fluid scales to sub-electron scales. Magnetohydrodynamic (MHD) model remains a credible approximation for a kinetic plasma at scales large enough to be well separated from kinetic effects, while more refined kinetic description is required at kinetic scales. Many plasma physics researchers are proficient in the kinetic theory, but unfamiliar with fluid mechanics. Engineering researchers often have the opposite bias. Then it is natural to imagine that there exists a gap between MHD theory and kinetic theory, yet each of them has its adherents. The turbulence, in particular the energy transfer scenario, can eliminate the gap and tie them all together. Meanwhile, the energy transfer process is of great importance in studying energy dissipation mechanism for weakly collisional or collisionless plasma, thus might explaining the heating of collisionless plasmas such as solar corona and solar wind. In this book, we voyage from MHD scales to kinetic scales in a hope to clarify the key steps of energy transfer. This book consists of three parts. Chapters 1 and 2 are provided to serve as introductory text. Chapters 3–5, on the other hand, are more of professional researches. Chap. 6 is the conclusion. In Chap. 1, we give a general overview of energy transfer properties in plasma turbulence and their relevant state of the art, in particular at MHD scales and kinetic scales. Chapter 2 presents a theoretical

xi

xii

Preface

foundation of this work, which is based primarily on scales in plasmas, the objective being to develop in the reader an understanding for the tractable models of plasmas (e.g., kinetic theory, multi-fluid theory, and MHD theory), and how they are related to each other. Chapter 3 gives the detailed numerical methods for compressible MHD turbulence, including the hybrid compact-weighted essentially non-oscillatory (WENO) scheme, the approaches to maintain the divergence-free property of the magnetic field, and the derivation of an eighth-order pentadiagonal compact filter. We then present a series of one-dimensional and two-dimensional benchmark tests to verify the capability of the hybrid scheme. Finally, simulations of three-dimensional compressible MHD turbulence are carried out and some statistical results are illustrated, which are the basic of Chap. 4. Several aspects of compressible MHD turbulence are studied in Chap. 4, including the effect of forcing mechanisms, compressibility effect on scaling laws, and compressibility effect on energy transfer. The MHD description is at the macroscopic fluid level. Its validity is thus broken as moving into smaller scales. Scales small enough require consideration of, for example, the Hall effect or kinetic plasma effect. In Chap. 5, we analyze ideal energy transfer for collisionless plasma through the full Vlasov– Maxwell equation. Several energy transfer channels, including the electromagnetic work and the pressure-stress interaction, are studied in detail. We finally end, in Chap. 6, with a summary of this work and possible perspectives. Throughout this book, emphasis is placed on providing an instructive picture of energy transfer process in plasma turbulence, rather than rigorous mathematical deduction, or engineering applications. Although only simulation results are included in this book, they have provided a better understanding of plasma turbulence and inspired further simulation and observational studies. I am profoundly grateful to many people for their help in the research presented in this book and in the preparation of this book. I am grateful to my supervisors, Yipeng Shi (Peking Univeristy), William H. Matthaeus (University of Delaware), Minping Wan (Southern University of Science and Technology), and Shiyi Chen (Southern University of Science and Technology), with whom I have worked and learned about this subject. Everything I know about turbulence and plasma physics, I owe to them. I have benefitted from the help of many colleagues at the Peking University, University of Delaware, and Southern University of Science and Technology—I cannot list them all here. But I would like to thank the following people: Kun Yang (Peking University) for helping with the hybrid scheme, Tulasi N. Parashar (University of Delaware) for providing the PIC simulation data, and Jianchun Wang (Southern University of Science and Technology) for valuable discussion on compressible turbulence. I would like to thank the colleagues who read the manuscript carefully and many suggestions were very valuable to the development of this book. I acknowledge the support provided by the China Scholarship Council during my 2-year visit to the University of Delaware. I am particularly grateful to Minping Wan for his support, when I worked on this book instead of finishing other projects in time. I also would like to thank the staff at Springer, in particular Li Shen and Saranya Kalidoss, for their patience and

Preface

xiii

assistance in the production process of this book. Finally, I wish to thank my husband, Weiwei Jin, and my friends, Qing, Sufen, and Lulu for their support and encouragement over the years. Shenzhen, China February 2019

Yan Yang

Contents

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

1 1 2 4 5 7 8 9 9 11 12 12 13 13 15

2 Theoretical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Electromagnetic Field Equations . . . . . . . . . . . . . . . 2.2 Charged Particle Motion . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Kinetic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Two-Fluid Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Single-Fluid Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 The Reduced Form of Electrodynamic Equations for MHD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Magnetic Induction Equation . . . . . . . . . . . . . . . 2.5.3 The Full Equations of MHD . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

23 23 24 25 27 29

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

30 32 33 33

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Plasma Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 An Overview of Approaches . . . . . . . . . . . . . . . . . . . . 1.3 Incompressible MHD Turbulence . . . . . . . . . . . . . . . . 1.3.1 Energy Cascade and Spectra . . . . . . . . . . . . . . . 1.3.2 Intermittency . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Localness of Energy Transfer . . . . . . . . . . . . . . 1.4 Compressible MHD Turbulence . . . . . . . . . . . . . . . . . 1.4.1 MHD Shock Waves . . . . . . . . . . . . . . . . . . . . . 1.4.2 Numerical Simulation . . . . . . . . . . . . . . . . . . . 1.4.3 Possible Effects of Compressibility and Shocks . 1.5 Plasma Turbulence at Kinetic Scales . . . . . . . . . . . . . . 1.5.1 Energy Spectra at Kinetic Scales . . . . . . . . . . . 1.5.2 Kinetic Dissipation . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

xvi

Contents

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

35 35 35 37 40 40 45 48 52 52 55 62 66 66

4 Energy Cascade in Compressible MHD Turbulence . . . . . . . . 4.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Large-Scale Forcing Mechanism . . . . . . . . . . . . . . . 4.1.2 Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Filtered Energy Equations . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Effect of Forcing Mechanisms . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Statistical Quantities . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Turbulent Structures . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Energy Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Compressibility Effect on Scaling Laws . . . . . . . . . . . . . . . 4.4.1 Four-Thirds Law . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Density Structure Functions . . . . . . . . . . . . . . . . . . 4.5 Compressibility Effect on the Localness of Energy Transfer 4.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

69 69 70 71 72 75 75 77 79 80 82 82 83 88 88

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

91 91 93 95 96 98

3 Hybrid Scheme for Compressible MHD Turbulence . . . . . . 3.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Ideal MHD Equations . . . . . . . . . . . . . . . . . . . . 3.1.2 Dimensionless MHD Equations . . . . . . . . . . . . . 3.2 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 The Hybrid Compact-WENO Scheme . . . . . . . . . 3.2.2 Pentadiagonal Filter . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Divergence-Free Constraint of the Magnetic Field 3.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Accuracy Analysis of the Hybrid Scheme . . . . . . 3.3.2 1D and 2D Numerical Tests . . . . . . . . . . . . . . . . 3.3.3 Isotropic MHD Turbulence . . . . . . . . . . . . . . . . . 3.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

5 Energy Transfer and Dissipation in Collisionless Plasma Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 PIC Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Role of Pressure Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Intermittency of Pressure-Strain Interaction . . . . . . . 5.3.2 Coherent Structures in Plasma Turbulence . . . . . . . . 5.3.3 Correlation Between Energy Transfer and Coherent Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Energy Transfer Channels . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . 101 . . . . . 102

Contents

5.4.1 Correlation Between Pressure-Stress Interaction and Electromagnetic Work . . . . . . . . . . . . . . . . . . . . 5.4.2 Filtered Fluid Kinetic Energy Equation . . . . . . . . . . . 5.4.3 Energy Transfer Across Scales and Between Different Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xvii

. . . . 103 . . . . 103 . . . . 106 . . . . 107 . . . . 109

6 Conclusions and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Appendix A: Fifth-Order WENO Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 115 Appendix B: Eigen-System in MHD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Appendix C: A Parallel Algorithm for Pentadiagonal Systems . . . . . . . . 125 Appendix D: Filtered Electromagnetic Energy Equation . . . . . . . . . . . . . 131 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Acronyms

1D 2D 2.5D 3D CFL CT K41 MHD PDF PIC Pi-D R-H r.m.s. SIC0 SIC2 WENO

One-dimensional Two-dimensional Three components of dependent field vectors and a two-dimensional spatial grid Three-dimensional Courant–Friedrichs–Lewy Constrained transport A. N. Kolmogorov’s work in 1941 Magnetohydrodynamics Probability density function Particle in cell Pressure–strain interation Rankine-Hugoniot Root-mean-square Purely solenoidal forcing Injecting energy into solenoidal and compressible parts at a ratio 1/2 Weighted essentially non-oscillatory

xix

Chapter 1

Introduction

1.1 Plasma Turbulence Plasma turbulence is ubiquitous in space and astrophysical flows. For example, the solar wind emitting from the sun into interplanetary space, one of the most studied natural plasmas, is in a turbulent state [30, 60]. Plasma turbulence is a multiscale phenomenon, involving structures across a wide range of spatial scales. Richness of scales in plasma turbulence makes the wealth of physical behaviors that take place in a plasma. Broadly these scales can be divided into two categories–(i) characteristic plasma parameters and (ii) scales of turbulent motion. There are a number of characteristic plasma parameters that can be found in any standard textbook on plasma physics. We only list two of themhere. One basic parameter for the description of a plasma is the Debye length, λ D = (ε0 kT )/(n e e2 ), with ε0 being the free space permittivity, k the Boltzmann constant, T the temperature, n e the number density of electrons, and e the electron charge. The Debye length is a measure of the distance of influence of an individual charged particle in a plasma. An important plasma property is to restore its charge neutrality. If the overall quasineutrality is disturbed, the electrons, due to their inertia, react in such a way as to give rise to collective oscillations to restore the violated charge  neutrality. The characteristic oscillation frequency is the plasma frequency, ω pe = (n e e2 )/(m e ε0 ), with m e being the electron mass. Since the ions are much heavier than the electrons, ω pi is smaller than ω pe . The various lengthscales of turbulent motion are sketched in the energy spectrum E(k) in Fourier space, see Fig. 1.1. Lengthscale  corresponds to wavenumber k ∼ −1 . In the fluid description of plasmas, say magnetohydrodynamic (MHD) theory that we describe in detail in Chap. 2, only large scales are resolved, and these scales are split into three ranges: the energy-containing range, the inertial range and the dissipation range. The bulk of the energy is in the large-scale range, which is therefore called the energy-containing range, whereas the bulk of the dissipation is in the smallscale range, which is therefore called the dissipation range. The range between the energy-containing and dissipation ranges is the inertial range. Below the MHD scales, © Springer Nature Singapore Pte Ltd. 2019 Y. Yang, Energy Transfer and Dissipation in Plasma Turbulence, Springer Theses, https://doi.org/10.1007/978-981-13-8149-2_1

1

2

1 Introduction

Fig. 1.1 Sketch showing the spectra of magnetic energy

10

0

E(k)

di

10

10

λD

de

-6

-12

10

-1

0

1

10

10

k

the kinetic scales (e.g., the ion inertial scale di = c/ω pi and the electron inertial scale de = c/ω pe , with c being the speed of light) can be resolved via the kinetic description of plasmas (see Chap. 2). Given the vast range of scales in plasma turbulence, the energy transfer process can tie these scales all together. One direct support of this idea is the observations of power-law spectra of magnetic energy in the solar wind [5, 90, 114, 129], which we describe in detail in Sects. 1.3.1 and 1.5.1. The energy spectrum has a power law in the inertial range, the slope steepens near the ion inertial scale and then steepens again neat the electron inertial scale. The main objective of this book is to investigate the energy transfer process in plasma turbulence from MHD scales well into kinetic scales by sophisticated analyses.

1.2 An Overview of Approaches In a plasma, the coupling between the electromagnetic fields and charged particles are two-way. The electromagnetic fields are determined by the positions and motions of charged particles, while these fields in turn cause charged particles to move along their orbits. Therefore, one must solve a self-consistent problem in a time-dependent manner. Various numerical approaches are used by researchers to study the system of their interest. Below we briefly summarize the approaches and their pros and cons. More details about their theoretical foundation will come later in Chap. 2. MHD–The dynamics of plasmas at scales large enough (see Fig. 1.1) can be described by fluid equations: the continuity equation, the momentum equation and the energy equation. They are called MHD equations, in which the plasma is taken as a conducting fluid. The set of equations is in three spatial dimensions. So it is feasible to simulate very large systems for a long time using MHD modeling. However, MHD being applicable to large scales is not suitable to study kinetic effects.

1.2 An Overview of Approaches

3

Fig. 1.2 Sketch of a spatial grid (blue diamonds) and particle distribution (orange circles) in PIC code

Vlasov–The Vlasov equation is an appropriate limit of the Boltzmann equation for a collisionless plasma. When solving the Vlasov equation, there is no need to model the collision term. This method can resolve down to sub-electron scales (see Fig. 1.1). However, the six-dimensional phase space of the equation requires extensive computations. The best of our knowledge, there currently exist no fully kinetic Vlasov codes capable of conducting high-resolution simulations of reasonably big systems. Also the solution of the Vlasov equation suffers from velocity space filamentation. Particle In Cell (PIC)–Instead of solving the Vlasov equation, the PIC method simulates a plasma system by following a number of particle trajectories and uses “pseudo particles” to sample the phase space distribution. The electromagnetic fields are defined on a spatial grid, and a large number of particles are located within each grid cell, see Fig. 1.2. The program solves the fields from the particles and then moves the particles; this cycle is shown in Fig. 1.3. The charge and current densities, (q, j), on the grid are computed from the particle position and velocities, (x, v) using some weighting. Then solve the Maxwell’s equations to obtain the electromagnetic fields on the grid. The electromagnetic fields are interpolated from the grid to the particles in order to apply force and advance the particles. The reader can find further details of this method in the book by Birdsall and Langdon [19]. In comparison with Vlasov, this method is much less computationally expensive. But the finite number of particles in PIC method introduces counting statistics noise. Hybrid code–Given the limitations of MHD at kinetic scales and fully kinetic models at large scales, the hybrid code is a compromise that treats ions kinetically and electrons as a fluid. It is natural to expect that the hybrid code is much less expensive than Vlasov and PIC, but it discards electron kinetic physics.

4

1 Introduction

Fig. 1.3 The computational cycle in PIC code

A more refined model can provide a more complete characterization of the plasma. The suitability of a particular model for a particular plasma turbulence problem depends on the relative weighting of the importance of various criteria, like level of description, range of applicability and available computer power. In this book, we apply the MHD and PIC methods.

1.3 Incompressible MHD Turbulence The development of turbulence theory in MHD flows lags significantly behind that in neutral fluids. Then one might inquire whether the theory for MHD turbulence can be developed on the same lines as hydrodynamic turbulence. Caution is required in doing this, since MHD flows differ from neutral fluids in that MHD equations couple velocity with magnetic field. The mutual interaction between the velocity and magnetic fields introduces many more degrees of freedom into the system, for example, both field-line deformation by turbulent eddy motions and energy transfer by shear Alfvén waves coexist. This section are provided to briefly review incompressible MHD turbulence, the aim being to give a qualitative understanding of certain ideas that are involved for energy transfer process.

1.3 Incompressible MHD Turbulence

5

Fig. 1.4 Spectral diagram of turbulence suggesting the energy cascade

1.3.1 Energy Cascade and Spectra A. N. Kolmogorov’s work in 1941 (in short K41) [78, 79] on hydrodynamic turbulence is the basis for most turbulence research. The classical theory [79] suggests an energy cascade (introduced first by Richardson [112]) where energy is transferred from large to small scales at a constant rate, i.e., Π ∼ ε (see Fig. 1.4), until at the smallest scales this energy is dissipated by viscous action. MHD theory based on this picture has led to numerous advances, including the well-known Kolmogorov spectrum [79] and the Iroshnikov-Kraichnan spectrum [70, 81]. In Richardson’s notion, energy is cascading via successive generations of eddies of various sizes. Eddies of size  have a characteristic velocity u  and energy of order u 2 . The eddy turnover time associated with the scale  is τ ∼

 . u

(1.1)

This timescale can be considered as the typical time for the eddy of size  to undergo a significant distortion, but also the typical time taken for the transfer of energy from scales ∼ to smaller ones. So the energy transfer rate can be estimated as Π ∼

u 2 u3 ∼ . τ 

(1.2)

Imposing a constant energy flux Π ∼ ε in the inertial range, u 3 ∼ ε, 

(1.3)

u  ∼ ε1/3 1/3 .

(1.4)

Π ∼ leads to the scaling

6

1 Introduction

We denote the energy spectrum function by E(k) and the wavenumber by k. The energy in the wavenumber range k ≤ |k| < k + dk is E(k)dk. The lengthscale  corresponds to wavenumber k ∼ −1 . By Eq. 1.4, the Kolmogorov spectrum is E(k) = C K ε2/3 k −5/3 ,

(1.5)

where C K is a dimensionless constant. The Kolmogorov spectrum can be obtained also from dimensional analysis by applying the Kolmogorov hypotheses directly to the spectrum. Several results in numerical simulations [95, 96] and in observations [60, 90] support the Kolmogorov-like spectrum. Besides the eddy turnover time, the incompressible MHD turbulence is also characterized by the period of Alfvén waves, τa ∼ (ca k)−1 ,

(1.6)

where ca is the Alfvén speed. Iroshnikov [70] and Kraichnan [81] assumed that small scales can interact not only through eddies but also through Alfvén waves. In that case, small-scale fluctuations are viewed as wave packets propagating along the large-scale magnetic field and those traveling in opposite directions interact during which energy transfer occurs. By dimensional arguments, one can obtain ε = CτT k 4 E 2 (k),

(1.7)

where C is a dimensionless constant. The Kolmogorov spectrum is recovered if τT is taken to be the eddy turnover time τT = τ ∼

 −1/2  ∼ k 3 E(k) . u

(1.8)

Iroshnikov and Kraichnan suggested to take the Alfvén timescale, τT = τa ∼ (ca k)−1 ,

(1.9)

which gives the Iroshnikov-Kraichnan spectrum E(k) = C I K (εca )1/2 k −3/2 ,

(1.10)

where C I K is a dimensionless constant. The variety of timescales gives rise to other models, e.g., [91, 147] take the combination of the eddy turnover and Alfvén timescales 1/τT = 1/τ + 1/τa . Several groups have also discussed the anisotropy of MHD turbulence [21, 27, 59, 92, 94, 118]. Spectra following Kolmogorov-like, Iroshnikov-Kraichnan or even other laws, were observed, and the existence of a universal power law in MHD has been controversial.

1.3 Incompressible MHD Turbulence

7

1.3.2 Intermittency According to the self-similarity hypothesis, the K41 theory predicts that the structure function of order p exhibits a scaling behavior at inertial-range separations,  p S p (r ) =  δu  (r)  ∼ r ζ p ,

(1.11)

where · · ·  denotes the ensemble average, δu(x, r) = u(x + r) − u(x) is the increment between two points separated p by r, δu  p (x, r) = δu(x, r) · r/r is the longitudinal increment,  δu  (x, r)  =  δu  (r)  since the dependence on x vanishes after averaging because of homogeneity, and the scaling exponent scaling exponent ζ p is linear in p, p (1.12) ζp = . 3 The experimental data on structure functions [9] gives a power-law dependence on r as well, but the measured scaling exponents deviate from the Kolmogorov prediction Eq. 1.12, which is attributed to the intermittency. The fluctuation of a quantity is said to be intermittent: it is distributed sparsely in time or space, indicative of occasional occurrence of very strong fluctuations. For instance, the dissipation occurs in a highly intermittent manner, i.e., the local and instantaneous dissipation only attains very large values in a part of the space and during a fraction of the time. Therefore, several intermittency models, such as the β-model [52], the bifractal model [51], the multifractal model [51] and the log-normal model [80, 98], have taken into consideration intermittency corrections to the K41 theory. The celebrated SheLévêque formula [117] achieves a major success, which shows excellent agreement with experimental and numerical results. These models foster the intermittency models for MHD turbulence [61, 67, 95, 104]. But the proper choice of variables is now the two Elsasser fields z+ = u + B and z− = u − B in Alfvén speed units. Intermittency invalidates the K41 theory for structure functions of order p ≥ 4. There are, however, a few exact relations for low-order structure functions derived from the dynamic equations. A well known result in hydrodynamic turbulence is the Kolmogorov’s four-fifths law [51, 78], which relates the third-order moment of velocity to the energy dissipation rate. The central step of its derivation is the Kármán-Howarth-Monin relation [93, 132], ∇r · δu(r)|δu(r)|2  = −4ε,

(1.13)

where ∇r denotes derivative with respect to the separation vector r. This relation requires homogeneity (but isotropy is not necessary) and hypothesis about the finiteness of the energy dissipation in the limit of infinite Reynolds number. With further isotropy assumption, it follows 4 δu  (r)|δu(r)|2  = − εr, 3

(1.14)

8

1 Introduction

which is equivalent to the Kolmogorov’s four-fifths law, 3  4  δu  (r)  = − εr, 5

(1.15)

see the detailed derivation by Monin and Yaglom [93]. A four-thirds law analogous to the Kolmogorov’s four-fifths law was established by Yaglom [93, 140] for the transport of a passive quantity. Several attempts have also been made to extend the Kolmogorov phenomenology to MHD. Following the similar procedure to the Yaglom’s four-thirds law, but without assuming isotropy, a relation for mixed third-order moment of Elsasser fields increments was obtained, ∇r · δz∓ (r)|δz± (r)|2  = −4ε± ,

(1.16)

where ε± are the dissipation rates of the Elsasser fields z± = u ± B. Assuming also that isotropy applies, this relation [105, 106] reads 4 δz ∓ (r)|δz± |2  = − ε±r. 3

(1.17)

1.3.3 Localness of Energy Transfer The Kolmogorov phenomenology of incompressible hydrodynamic turbulence [79] assumes the locality of energy transfer across scales, which means that nonlinear interactions occur predominately between comparable scales, thus giving rise to an inertial range over which flow motions are independent of the large and small scales in the system and therefore have universal statistics in high Reynolds number range. In particular, Eyink [48] gives a useful definition: for the energy flux at scales ∼, if the contribution from scales Δ  is negligible, the energy flux is infrared local; if the contribution from scales δ  is negligible, the energy flux is ultraviolet local. We should keep in mind that the locality of energy transfer discussed here seems to be reminiscent of, but not equivalent to, the locality of interactions. As suggested by Kraichnan [82, 83], three wavenumbers (k, p, q) are involved in any mode-to-mode interaction. It is called triadic interaction with the triad of wavenumbers satisfying k + p + k = 0. If the three wavenumbers involved in the interaction are of comparable size, the interaction is local, and nonlocal, if otherwise. Therefore, locality of interactions indicates locality of transfer, but not the other way round. We will focus on the locality of energy transfer without being sidetracked by the more complicated triadic interactions. Although the properties of energy transfer across scales in hydrodynamic turbulence have been studied in great details, and its locality for very large Reynolds numbers has been verified, see [7, 39–42, 47–49, 77, 99, 145, 146], this issue in MHD turbulence is less clear. The complexity of energy transfer process in MHD turbulence

1.3 Incompressible MHD Turbulence

9

lies in the fact that several energy transfer channels between different fields are involved. Meanwhile, a mean magnetic field cannot be removed by a Galilean transformation. As a result small scales may interact directly with large scales. The energy transfer in incompressible MHD turbulence has been theoretically and numerically investigated by [1, 26, 34, 115, 116, 130, 144]. A shell-to-shell model in Fourier space has been used to measure energy fluxes in incompressible MHD turbulence [2, 3, 24, 35, 43, 126, 131]. Their results showed that energy transfer between the same fields is fairly local, while the transfer between the velocity and the magnetic field can have varying degrees of nonlocality depending on whether the turbulence is forced and depending on the presence of an external magnetic field. Aluie and Eyink [8] used a coarse-graining approach to investigate the scale locality of energy cascades of incompressible MHD turbulence. They proved the scale locality of flux of total energy and conversion of kinetic and magnetic energy, and explained the disagreement with the result obtained with the shell-to-shell model [2].

1.4 Compressible MHD Turbulence In most treatises on turbulence theory, turbulence is assumed incompressible; it is indeed fortunate that the turbulence theory under this assumption is sufficient to explain the majority of observed phenomena. However, there are some cases lying outside the region of the validity of incompressibility. For example, some astrophysical plasmas are compressible, in which the Mach number tends to be large. Then one must evidently appeal to compressible MHD turbulence to understand it.

1.4.1 MHD Shock Waves An immediate consequence of compressibility is the formation of shocks. The simplest model of compressible turbulence is Burgers’ turbulence following ∂u ∂ 2u ∂u +u = ν 2, ∂t ∂x ∂x

(1.18)

where ν is the diffusion coefficient. This equation generates discontinuities from continuous initial conditions, see, for example, Fig. 1.5, with an initial condition u(x, t = 0) = − sin(x), periodic boundary condtion in the domain −1 ≤ x ≤ 1 and ν = 0.01/π . The corresponding analytical solution [139] is ∞ u(x, t) =

−∞

sin[π(x − ξ )] f (x − ξ ) exp[−ξ 2 /(4νt)] dξ ∞ , 2 −∞ f (x − ξ ) exp[−ξ /(4νt)] dξ

(1.19)

10

1 Introduction

Fig. 1.5 Discontinuity formation in Burgers’ equation

where f (x − ξ ) = exp[− cos(π(x − ξ ))/(2π ν)]. The importance of this simplied model cannot be enough underlined since it embodies the full convective nonlinearity of the flow equations and develops discontinuities in a process very similar to the formation of shock. Shocks are thin transition layers, which could be infinitely thin in the Euler limit. Although the differential forms of the fundamental equations are no longer applicable at shocks, the properties of these discontinuous solutions can yet be obtained from the integral form of the conservation equations, which is written in a relation between variable jumps (or called Rankine–Hugoniot (R–H) relation) [20, 71, 97], 

 1 pτ + τ p¯ + [τ ] [Bt ]2 = 0, γ −1 4

(1.20)

where τ = 1/ρ is the specific volume and Bt is the magnetic vector component vertical to the shock normal direction. The brackets [ f ] = f 2 − f 1 is the jump of f between the downstream state of a shock “2” and its upstream state “1”, and the bar f¯ = ( f 1 + f 2 )/2 is the arithmetic average of f on two sides of a shock. After some algebraic manipulations, the pressure ratio p2 / p1 can be deduced from Eq. 1.20 as

prRH

 γ + 1 ρ2 [Bt ]2 ρ2 −1+ −1 p2 γ − 1 ρ1 2 p1 ρ1 = = . γ + 1 ρ2 p1 − γ − 1 ρ1

(1.21)

1.4 Compressible MHD Turbulence

11

1.4.2 Numerical Simulation A large amount of algorithms have been proposed to obtain accuracy, robustness and efficiency in MHD problems, including upwind schemes [23, 107, 108], flux corrected transport schemes [37], piecewise parabolic method [33], discontinuous Galerkin method [138], convective upwind and split pressure schemes [119], and so on. Previous schemes are mainly low-order to guarantee the stability. Moving into the realm of high-order schemes, pioneered by Lele [85], high-order compact finite difference schemes have gained popularity. A number of high-order finite difference schemes were proposed. Many of them are based on central differences which are non-dissipative so that they are not suitable for flows with strong shocks since they yield spurious oscillations. A central scheme, in general, should be equipped with filters or artificial diffusivity to stabilize the numerical solution. A high-order compact scheme in conjunction with filters were developed by Gaitonde [54], wherein the filter algorithm is designed to eliminate oscillations at high wavenumber range without affecting lower-wavenumber resolved waves. The filter algorithm could be, for example, the linear filter [55, 76] or the adaptive characteristic-based filter [121, 142, 143]. Apart from filters, another approach is introducing artificial diffusivity [75] or hyperdiffusivity [31, 32]. For example, Kawai [75] designed a localized artificial diffusivity method that regularized the governing equations by adding non-linear artificial diffusivity in the regions with discontinuous. Instead of introducing artificial dissipation, weighted essentially non-oscillatory (WENO) scheme, which is an alternative to compact scheme, is considered as a powerful tool to capture shock robustly. It is therefore employed to compressible hydrodynamic [13–15, 72, 120] and MHD [11, 13, 14, 28, 73] turbulence. However, WENO scheme suffers from successive dissipation at small scales for the detailed simulation of turbulent flows [84]. Note that high-order compact finite difference scheme has spectral-like spatial resolution in the smooth region. Whether there might be schemes that draw on the strengths of both approaches? The hybrid scheme could be the answer, in which a shock detector restricts the use of shock-capturing to regions near shocks in order to cancel the dissipation in smooth regions. There are several studies on hybrid schemes for hydrodynamic turbulence [103, 109, 136], while to my knowledge it has not been widely used for MHD turbulence. Controlling the numerical error of the divergence-free constraint of the magnetic field for high order schemes has been a bottleneck in MHD numerical simulations. Violating the constraint can cause nonphysical plasma transport as well as the alteration of the topological structure. The projection method [22], which is adopted in hydrodynamic turbulence to preserve the incompressibility condition, is proved to be accurate and efficient in all geometries as long as the Poisson equation is solved efficiently [12, 128]. Powell [108] proposed a non-conservative formulation, also called eight-wave formulation. It adds a source term, which corresponds to an additional wave. Therefore, rather than accumulated in a fixed point, the nonzero divergence is convected with the wave. However, this method is found to spoil conservation and produce incorrect jump conditions across discontinuities [128]. In order to avoid a

12

1 Introduction

loss of conservation, Dedner et al. [36] developed the generalized Lagrange multiplier method which might be viewed as an extension of the eight-wave formulation. Another popular method is constrained transport (CT) method [16, 46]. If a particular finite difference discretization is used on a staggered grid, it maintains the divergence-free property of the magnetic field in a specific discretization. Central difference type CT method introduced in Ref. [128] is analogous to CT method, but there is no need for staggered mesh and spatial interpolation.

1.4.3 Possible Effects of Compressibility and Shocks The Helmholtz decomposition of the velocity field u = us + uc

(1.22)

with ∇ · us = 0 and ∇ × uc = 0, gives the intrinsic constituents of u in terms of shearing and compressing processes. The solenoidal velocity, the compressive velocity and the magnetic field corresponds to vortex, shock and current structures, respectively. Shocks emerge as a result of compressibility, while vortex and current structures arise even in default of this. We expect that, to some extent, the flow behaves differently with the influence of shocks and compression. In particular, there is growing evidence that strongly compressible turbulence plays a role in amplification of magnetic fields [58], and possibly the production of energetic particles [10, 18, 44, 50, 74], especially near shocks [89]. For compressible hydrodynamic turbulence, it is found that strong shocks lead to the k −2 power-law energy spectrum and the saturation of scaling exponents of velocity structure functions [135]. Aluie et al. [6] and Wang et al. [137] analyzed the cross-scale transfer of kinetic energy for compressible hydrodynamic turbulence. However, in MHD community, the most common investigated is for incompressible MHD turbulence. No extensive studies on the energy transfer and its related properties exist for compressible MHD turbulence where shock are pervasive. It is however, an important step towards understanding the compressible MHD turbulence in astrophysical plasmas.

1.5 Plasma Turbulence at Kinetic Scales Depending on the nature of the turbulent cascade in MHD inertial range, the nature of fluctuations available at kinetic scales could be very different. Kinetic processes may lead to phenomena such as temperature anisotropy, heating, particle energization, entropy cascade, and so on. Out of many questions that arise from kinetic processes, the ones that here emphasis is placed on are the heating of collisionless plasma and particle energization. One relevant observation is the non-adiabatic radial

1.5 Plasma Turbulence at Kinetic Scales

13

temperature profile [111] of the solar wind. The solar wind temperature falls off as the wind expands in the solar system, but it is higher than the temperature expected in an expanding system with no obvious source or sink of energy. A mystery that pervades decades of studies without a consensus solution has been to identify the dissipation mechanism in weakly collisional or collisionless plasma by which heat is deposited to accelerated solar wind and corona.

1.5.1 Energy Spectra at Kinetic Scales Measurement of energy spectra in the vicinity of ion and electron plasma scales has attracted a lot of attention since it may help to clarify our understanding of the processes of dissipation of turbulent energy in collisionless plasmas. It is well established that at MHD scales, the solar-wind turbulent spectrum of magnetic fluctuations follows Kolmogorov’ scaling k −5/3 . Combining high resolution magnetic and electric field data of the Cluster spacecraft, Sahraoui et al. [113, 114] and Alexandrova et al. [4, 5] found the magnetic spectrum from MHD to electron scales. Sahraoui et al. [114] used a double power-law model with a break to fit the observations in the electron inertial and dissipation ranges (see Fig. 1.6a). Unlike the double power-law spectrum, Alexandrova et al. [4] proposed an exponential with a power-law prefactor down to electron scales (see Fig. 1.6b). While there might be debate as to whether it should follow a power law or an exponential function, it does demonstrate that the spectrum steepens as approaching smaller scales. This steepening has many implications, for example, it is interpreted as a signature of energy being deposited into thermal energy and the energy dissipation rate depends on the slope of the energy spectrum.

1.5.2 Kinetic Dissipation In studying energy dissipation, the most satisfactory way to proceed would be to directly study the explicit dissipation expression. MHD model adopt an ad hoc closure, without engaging the details of the small-scale dynamics that make up the plasma dissipation range, to get the viscous and resistive dissipation. The turbulence in most astrophysical contexts, however, is typically of weak collisionality, and frequently modeled as collisionless, thus collisional (viscous and resistive) dissipation at small scales cannot emerge immediately. Therefore, energy dissipation mechanism for weakly collisional or collisionless plasma is of principal importance for addressing long-standing puzzles like the acceleration of energetic particles, and related questions that arise in space and astrophysical applications such as the solar wind. One popular approach to explain dissipation is to recourse to wave-particle interactions [56, 57, 62, 63, 65, 66, 68, 88], with instability regulating the dynamics of

14 Fig. 1.6 Magnetic spectra measured by Cluster in the solar wind [4, 114]. a Reprinted figure with permission from F. Sahraoui et al., Physical Review Letters, 102, 231102, 2009. Copyright (2009) by the American Physical Society. b Reproduced figure with permission from O. Alexandrova et al., The Astrophysical Journal, 760:121, 2012. Copyright (2012) by the American Astronomical Society

1 Introduction

(a)

(b)

extreme distortions of the distribution function. The other category is the heating by coherent structures and reconnections [38, 64, 101, 102, 110, 125, 127]. Whatever the dissipative mechanism is, we can focus on the energy transfer process. In this process, one can envision that significant amounts of energy reside in large-scale fluctuations. Through the cascade, these are transported to smaller scales where dissipation occurs and finally drives kinetic processes that absorb these energy fluxes and energize charged particles. The dissipation rate can be computed from third-order moment [17, 29, 86, 87, 122–124]. Since particles in collisionless plasma interact only through the electromagnetic fields and collisions are too infrequent to provide the necessary dissipation, the

1.5 Plasma Turbulence at Kinetic Scales

15

work done by electromagnetic fields on particles, j · E, as Wan et al. [133, 134] suggested, might be a plausible estimation of the dissipation rate. A number of studies have investigated the electromagnetic work [25, 45, 53, 69, 100, 133, 134, 141] and found that it is highly localized in association with current sheets [25, 100, 133, 134]. Although the electromagnetic work contains the necessary energy supply from electromagnetic fields that ultimately goes into the thermal energy reservoir, a question emerges is that what fraction of the electromagnetic energy released through the electromagnetic work ends up as ion and electron random motion as opposed to fluid flow. With this caveat in mind, in this work we will try to find other pathways that are responsible for the generation of plasma thermal energy.

References 1. Alexakis A, Bigot B, Politano H, Galtier S (2007) Anisotropic fluxes and nonlocal interactions in magnetohydrodynamic turbulence. Phys Rev E 76:056,313 2. Alexakis A, Mininni PD, Pouquet A (2005) Shell-to-shell energy transfer in magnetohydrodynamics. I. Steady state turbulence. Phys Rev E 72:046,301 3. Alexakis A, Mininni PD, Pouquet A (2007) Turbulent cascades, transfer, and scale interactions in magnetohydrodynamics. New J Phys 9:298 4. Alexandrova O, Lacombe C, Mangeney A, Grappin R, Maksimovic M (2012) Solar wind turbulent spectrum at plasma kinetic scales. Astrophys J 760(2):121 5. Alexandrova O, Saur J, Lacombe C, Mangeney A, Mitchell J, Schwartz SJ, Robert P (2009) University of solar wind turbulent spectrum from MHD to electron scales. Phys Rev Lett 103:165,003 6. Aluie H (2011) Compressible turbulence: the cascade and its locality. Phys Rev Lett 106:174,502 7. Aluie H, Eyink GL (2009) Localness of energy cascade in hydrodynamic turbulence. II. Sharp-spectral filter. Phys Fluids 21:115,108 8. Aluie H, Eyink GL (2010) Scale locality of magnetohydrodynamic turbulence. Phys Rev Lett 104:081,101 9. Anselmet F, Gagne Y, Hopfinger E, Antonia R (1984) High-order velocity structure functions in turbulent shear flows. J Fluid Mech 140:63–89 10. Axford W (1969) Acceleration of cosmic rays by shock waves. In: Invited Papers. Springer, pp 155–203 11. Balsara DS (2009) Divergence-free reconstruction of magnetic fields and WENO schemes for magnetohydrodynamics. J Comput Phys 228:5040–5056 12. Balsara DS, Kim J (2004) A comparison between divergence-cleaning and staggered-mesh formulations for numerical magnetohydrodynamics. Astrophys J 602:1079–1090 13. Balsara DS, Meyer C, Dumbser M, Du H, Xu Z (2013) Efficient implementation of ADER schemes for Euler and magnetohydrodynamical flows on structured meshes—speed comparisons with runge-kutta methods. J Comput Phys 235:934–969 14. Balsara DS, Rumpf T, Dumbser M, Munz CD (2009) Efficient, high accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics. J Comput Phys 228:2480–2516 15. Balsara DS, Shu CW (2000) Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J Comput Phys 169:405–452 16. Balsara DS, Spicer DS (1999) A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations. J Comput Phys 149:270–292

16

1 Introduction

17. Bandyopadhyay R, Chasapis A, Chhiber R, Parashar T, Maruca B, Matthaeus W, Schwartz S, Eriksson S, Le Contel O, Breuillard H et al (2018) Solar wind turbulence studies using mms fast plasma investigation data. Astrophys J 866(2):81 18. Bell A (1978) The acceleration of cosmic rays in shock fronts-i. Month Not Royal Astron Soc 182(2):147–156 19. Birdsall CK, Langdon AB (2004) Plasma physics via computer simulation. CRC Press 20. Biskamp D (2003) Magnetohydrodynamic turbulence. Cambridge University Press, Cambridge 21. Boldyrev S (2005) On the spectrum of magnetohydrodynamic turbulence. Astrophys J Lett 626(1):L37 22. Brackbill JU, Barnes DC (1980) The effect of nonzero ∇ · B on the numerical solution of the magnetohydrodynamic equations. J Comput Phys 35:426–430 23. Brio M, Wu CC (1988) An upwind differencing scheme for the equations of ideal magnetohydrodynamics. J Comput Phys 75:400–422 24. Carati D, Debliquy O, Knaepen B, Teaca B, Verma M (2006) Energy transfers in forced MHD turbulence. J Turbul 7:51 25. Chasapis A, Matthaeus W, Parashar T, Wan M, Haggerty C, Pollock C, Giles B, Paterson W, Dorelli J, Gershman D et al (2018) In situ observation of intermittent dissipation at kinetic scales in the earth’s magnetosheath. Astrophys J Lett 856(1):L19 26. Cho J (2010) Non-locality of hydrodynamic and magnetohydrodynamic turbulence. Astrophys J 725:1786–1791 27. Cho J, Vishniac ET (2000) The anisotropy of magnetohydrodynamic Alfvénic turbulence. Astrophys J 539:273–282 28. Christlieb AJ, Rossmanith JA, Tang Q (2014) Finite difference weighted essentially nonoscillatory schemes with constrained transport for ideal magnetohydrodynamics. J Comput Phys 268:302–325 29. Coburn JT, Forman MA, Smith CW, Vasquez BJ, Stawarz JE (2015) Third-moment descriptions of the interplanetary turbulent cascade, intermittency and back transfer. Phil Trans R Soc A 373:20140,150 30. Coleman PJ Jr (1968) Turbulence, viscosity, and dissipation in the solar-wind plasma. Astrophys J 153:371–388 31. Cook AW, Cabot WH (2004) A high-wavenumber viscosity for high-resolution numerical methods. J Comput Phys 195:594–601 32. Cook AW, Cabot WH (2005) Hyperviscosity for shock-turbulence interactions. J Comput Phys 203:379–385 33. Dai W, Woodward PR (1994) Extension of the piecewise parabolic method to multidimensional ideal magnetohydrodynamics. J Comput Phys 115:485–514 34. Dar G, Verma MK, Eswaran V (2001) Energy transfer in two-dimensional magnetohydrodynamic turbulence: formalism and numerical results. Phys D 157:207–225 35. Debliquy O, Verma MK, Carati D (2005) Energy fluxes and shell-to-shell transfers in threedimensional decaying magnetohydrodynamics turbulence. Phys Plasmas 12:042,309 36. Dedner A, Kemm E, Kröner D, Munz CD, Schnitzer T, Wesenberg M (2002) Hyperbolic divergence cleaning for the MHD equations. J Comput Phys 175:645–673 37. Devore CR (1991) Flux-corrected transport techniques for multidimensional compressible magnetohydrodynamics. J Comput Phys 92:142–160 38. Dmitruk P, Matthaeus WH, Seenu N (2004) Test particle energization by current sheets and nonuniform fields in magnetohydrodynamic turbulence. Astrophys J 617:667–679 39. Domaradzki JA (1988) Analysis of energy transfer in direct numerical simulations of isotropic turbulence. Phys Fluids 31:2747 40. Domaradzki JA, Carati D (2007) An analysis of the energy transfer and the locality of nonlinear interactions in turbulence. Phys Fluids 19:085,112 41. Domaradzki JA, Carati D (2007) A comparison of spectral sharp and smooth filters in the analysis of nonlinear interactions and energy transfer in turbulence. Phys Fluids 19:085,111

References

17

42. Domaradzki JA, Rogallo RS (1990) Local energy transfer and nonlocal interactions in homogeneous, isotropic turbulence. Phys Fluids A 2:413 43. Domaradzki JA, Teaca B, Carati D (2010) Locality properties of the energy flux in magnetohydrodynamic turbulence. Phys Fluids 22:051,702 44. Drury LO (1983) An introduction to the theory of diffusive shock acceleration of energetic particles in tenuous plasmas. Reports Progress Phys 46(8):973 45. Ergun R, Goodrich K, Wilder F, Ahmadi N, Holmes J, Eriksson S, Stawarz J, Nakamura R, Genestreti K, Hesse M et al (2018) Magnetic reconnection, turbulence, and particle acceleration: Observations in the earth’s magnetotail. Geophys Res Lett 45(8):3338–3347 46. Evans CR, Hawley JF (1988) Simulation of magnetohydrodynamic flows: a constrained transport method. Astrophys J 332:659–677 47. Eyink GL (1994) Energy dissipation without viscosity in ideal hydrodynamics I. Fourier analysis and local energy transfer. Phys D 78:222–240 48. Eyink GL (2005) Locality of turbulent cascades. Phys D 207:91–116 49. Eyink GL, Aluie H (2009) Localness of energy cascade in hydrodynamic turbulence. I. Smooth coarse graining. Phys Fluids 21:115,107 50. Fisk LA, Gloeckler G (2012) Particle acceleration in the heliosphere: implications for astrophysics. Space Sci Rev 173:433–458 51. Frisch U (1995) Turbulence. The legacy of A. N, Kolmogorov 52. Frisch U, Sulem PL, Nelkin M (1978) A simple dynamical model of intermittent fully developed turbulence. J Fluid Mech 87(4):719–736 53. Fu H, Vaivads A, Khotyaintsev YV, André M, Cao J, Olshevsky V, Eastwood J, Retino A (2017) Intermittent energy dissipation by turbulent reconnection. Geophys Res Lett 44(1):37–43 54. Gaitonde DV (2001) Higher-order solution procedure for three-dimensional nonideal magnetogasdynamics. AIAA J 39:2111–2120 55. Gaitonde DV, Visbal MR (1998) High-order schemes for Navier-Stokes equations: algorithm and implementation into FDL3DI. Technical Report AFRL-VA-WP-TR-1998-3060, US Air Force Research Laboratory, Wright-Patterson AFB 56. Gary SP, Saito S (2003) Particle-in-cell simulations of alfvén-cyclotron wave scattering: proton velocity distributions. J Geophys Res 108:1194 57. Gary SP, Saito S, Li H (2008) Cascade of whistler turbulence: particle-in-cell simulations. Geophys Res Lett 35:L02,104 58. Giacalone J, Jokipii JR (2007) Magnetic field amplification by shocks in turbulent fluids. Astrophys J 663:L41–L44 59. Goldreich P, Sridhar S (1995) Toward a theory of interstellar turbulence. 2: Strong alfvenic turbulence. Astrophys J 438:763–775 60. Goldstein ML, Roberts DA, Matthaeus W (1995) Magnetohydrodynamic turbulence in the solar wind. Ann Rev Astron Astrophys 33(1):283–325 61. Grauer R, Krug J, Marliani C (1994) Scaling of high-order structure functions in magnetohydrodynamic turbulence. Phys Lett A 195(5–6):335–338 62. He J, Tu C, Marsch E, Chen CH, Wang L, Pei Z, Zhang L, Salem CS, Bale SD (2015) Proton heating in solar wind compressible turbulence with collisions between counter-propagating waves. Astrophys J Lett 813(2):L30 63. He J, Wang L, Tu C, Marsch E, Zong Q (2015) Evidence of landau and cyclotron resonance between protons and kinetic waves in solar wind turbulence. Astrophys J Lett 800(2):L31 64. He J, Zhu X, Chen Y, Salem C, Stevens M, Li H, Ruan W, Zhang L, Tu C (2018) Plasma heating and alfvénic turbulence enhancement during two steps of energy conversion in magnetic reconnection exhaust region of solar wind. Astrophys J 856(2):148 65. Hollweg JV (1986) Transition region, corona, and solar wind in coronal holes. J Geophys Res 91:4111–4125 66. Hollweg JV, Isenberg PA (2002) Generation of the fast solar wind: a review with emphasis on the resonant cyclotron interaction. J Geophys Res 107:1147 67. Horbury T, Balogh A (1997) Structure function measurements of the intermittent mhd turbulent cascade. Nonlinear Process Geophys 4(3):185–199

18

1 Introduction

68. Howes G, Dorland W, Cowley S, Hammett G, Quataert E, Schekochihin A, Tatsuno T (2008) Kinetic simulations of magnetized turbulence in astrophysical plasmas. Phys Rev Lett 100(6):065,004 69. Howes GG, McCubbin AJ, Klein KG (2018) Spatially localized particle energization by landau damping in current sheets produced by strong alfvén wave collisions. J Plasma Phys 84(1) 70. Iroshnikov P (1964) Turbulence of a conducting fluid in a strong magnetic field. Sov Astron 7:566 71. Jeffrey A, Taniuti T (1964) Nonlinear wave propagation. ACADEMIC PR, New York 72. Jiang GS, Shu CW (1996) Efficient implementation of weighted ENO schemes. J Comput Phys 126:202–228 73. Jiang GS, Wu CC (1999) A high-order WENO finite difference scheme for the equations of ideal magnetohydrodynamics. J Comput Phys 150:561–594 74. Jokipii JR, Lee MA (2010) Compression acceleration in astrophysical plasmas and the production of f (v) ∝ v −5 spectra in the heliosphere. Astrophys J 713:475–483 75. Kawai S (2013) Divergence-free-preserving high-order schemes for magnetohydrodynamics: an artificial magnetic resistivity method. J Comput Phys 251:292–318 76. Kim JW (2010) High-order compact filters with variable cut-off wavenumber and stable boundary treatment. Comput Fluids 39:1168–1182 77. Kishida K, Araki K, Kishiba S, Suzuki K (1999) Local or nonlocal? Orthonormal divergencefree wavelet analysis of nonlinear interactions in turbulence. Phys Rev Lett 83:5487 78. Kolmogorov AN (1941) Dissipation of energy in locally isotropic turbulence. Dokl Akad Nauk SSSR 32:16–18 79. Kolmogorov AN (1941) The local structure of turbulence in incompressible viscous fluid for very large reynolds numbers. Dokl Akad Nauk SSSR 30:299–303 80. Kolmogorov AN (1962) A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high reynolds number. J Fluid Mech 13(1):82– 85 81. Kraichnan RH (1965) Inertial-range spectrum of hydromagnetic turbulence. Phys Fluids 8(7):1385–1387 82. Kraichnan RH (1967) Inertial ranges in two-dimensional turbulence. Phys Fluids 10(7):1417– 1423 83. Kraichnan RH (1971) Inertial-range transfer in two- and three-dimensional turbulence. J Fluid Mech 47(3):525–535 84. Larsson J, Cook A, Lele SK, Moin P, Cabot B, Sjögreen B, Yee H, Zhong X (2007) Computational issues and algorithm assessment for shock/turbulence interaction problems. J Phys Conf Ser 78:012,014 85. Lele SK (1992) Compact finite difference schemes with spectral-like resolution. J Comput Phys 103:16–42 86. MacBride BT, Smith CW (2008) The turbulent cascade at 1 AU: energy transfer and the third-order scaling for MHD. Astrophys J 679:1644–1660 87. Marino R, Sorrisovalvo L, Carbone V, Noullez A, Bruno R, Bavassano B (2008) Heating the solar wind by a magnetohydrodynamic turbulent energy cascade. Astrophys J 677(1):L71 88. Markovskii SA, Vasquez BJ, Smith CW, Hollweg JV (2006) Dissipation of th perpendicular turbulent cascade in the solar wind. Astrophys J 639:1177–1185 89. Matsumoto Y, Amano T, Kato TN, Hoshino M (2015) Stochastic electron acceleration during spontaneous turbulent reconnection in a strong shock wave. Science 347:974–978 90. Matthaeus WH, Goldstein ML (1982) Measurement of the rugged invariants of magnetohydrodynamic turbulence in the solar wind. J Geophys Res Space Phys 87(A8):6011–6028 91. Matthaeus WH, Zhou Y (1989) Extended inertial range phenomenology of magnetohydrodynamic turbulence. Phys Fluids B: Plasma Phys 1(9):1929–1931 92. Milano LJ, Matthaeus WH, Dmitruk P, Montgomery DC (2001) Local anisotropy in incompressible magnetohydrodynamic turbulence. Phys Plasmas 8:2673–2681

References

19

93. Monin AS, Yaglom AM (1975) Statistical fluid mechanics: machanics of turbulence, vol 2. MIT Press, Cambridge, MA 94. Montgomery D, Matthaeus WH (1995) Anisotropic modal energy transfer in interstellar turbulence. Astrophys J 447:706 95. Müller WC, Biskamp D (2000) Scaling properties of three-dimensional magnetohydrodynmaic turbulence. Phys Rev Lett 84:475 96. Müller WC, Grappin R (2005) Spectral energy dynamics in magnetohydrodynamic turbulence. Phys Rev Lett 95(11):114,502 97. Myong RS, Roe PL (1997) Shock waves and rarefaction waves in magnetohydrodynamics. Part 2. The MHD system. J Plasmas Phys 58:521–552 98. Obukhov A (1962) Some specific features of atmospheric tubulence. J Fluid Mech 13(1):77– 81 99. Ohkitani K, Kida S (1992) Triad interactions in a forced turbulence. Phys Fluids A 4:794 100. Osman K, Kiyani K, Matthaeus W, Hnat B, Chapman S, Khotyaintsev YV (2015) Multispacecraft measurement of turbulence within a magnetic reconnection jet. Astrophys J Lett 815(2):L24 101. Parashar TN, Servidio S, Shay MA, Breech B, Matthaeus WH (2011) Effect of driving frequency on excitation of turbulence in a kinetic plasma. Phys Plasmas 18:092,302 102. Perri S, Goldstein ML, Dorelli JC, Sahraoui F (2012) Detection of small-scale structures in the dissipation regime of solar-wind turbulence. Phys Rev Lett 109:191,101 103. Pirozzoli S (2002) Conservative hybrid compact-WENO schemes for shock-turbulence interaction. J Comput Phys 178:81–117 104. Politano H, Pouquet A (1995) Model of intermittency in magnetohydrodynamic turbulence. Phys Rev E 52(1):636 105. Politano H, Pouquet A (1998) Dynamical length scales for turbulent magnetized flow. Geophys Res Lett 25:273–276 106. Politano H, Pouquet A (1998) Von Kármán-Howarth equation for magnetohydrodynamics and its consequences on third-order longitudinal structure and correlation functions. Phys Rev E 57:R21–R24 107. Powell K, Roe PL, Linde TJ, Gombosi TI, Zeeuw DLD (1999) A solution-adaptive upwind scheme for ideal magnetohydrodynamics. J Comput Phys 154:284–309 108. Powell K, Roe PL, Myong RS (1995) An upwind scheme for magnetohydrodynamics. AIAA Paper 95–1704 109. Ren YX, Liu M, Zhang H (2003) A characteristic-wise hybrid compact-WENO scheme for solving hyperbolic conservation laws. J Comput Phys 192:365–386 110. Retinò A, Sundkvist D, Vaivads A, Mozer F, André M, Owen CJ (2007) In situ evidence of magnetic reconnection in turbulent plasma. Nature Phys 3:235–238 111. Richardson JD, Paularena KI, Lazarus AJ, Belcher JW (1995) Radial evolution of the solar wind from imp 8 to voyager 2. Geophys Res Lett 22(4):325–328 112. Richardson LF (1922) Weather prediction by numerical process. Cambridge University Press, Cambridge 113. Sahraoui F, Goldstein ML, Belmont G, Canu P, Rezeau L (2010) Three dimensional anisotropic k spectra of turbulence at subproton scales in the solar wind. Phys Rev Lett 105(13):131,101 114. Sahraoui F, Goldstein ML, Robert P, Khotyaintsev YV (2009) Evidence of a cascade and dissipation of solar-wind turbulence at the electron gyroscale. Phys Rev Lett 102:231,102 115. Schekochihin AA, Cowley SC, Taylor SF, Maron JL, McWilliams JC (2004) Simulations of the small-scale turbulent dynamo. Astrophys J 612:276–307 116. Schilling O, Zhou Y (2002) Triadic energy transfers in non-helical magnetohydrodynamic turbulence. J Plasma Phys 68:389–406 117. She ZS, Lévêque E (1994) Universal scaling laws in fully developed turbulence. Phys Rev Lett 72:336 118. Shebalin JV, Matthaeus WH, Montgomery D (1983) Anisotropy in mhd turbulence due to a mean magnetic field. J Plasma Phys 29:525–547

20

1 Introduction

119. Shen Y, Zha G, Huerta MA (2012) E-CUSP scheme for the equations of ideal magnetohydrodynamics with high order WENO scheme. J Comput Phys 231:6233–6247 120. Shu CW (2009) High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Rev 51:82–126 121. Sjögreen B, Yee HC (2004) Multiresolution wavelet based adaptive numerical dissipation control for high order methods. J Sci Comput 20:211–255 122. Sorriso-Valvo L, Carbone V, Noullez A, Politano H, Pouquet A, Veltri P (2002) Analysis of cancellation in two-dimensional magnetohydrodynamic turbulence. Phys Plasmas 9:89–95 123. Sorriso-Valvo L, Marino R, Carbone V, Noullez A, Lepreti F, Veltri P, Bruno R, Bavassano B, Pietropaolo E (2007) Observation of inertial energy cascade in interplanetary space plasma. Phys Rev Lett 99:115,001 124. Stawarz JE, Smith CW, Vasquez BJ, Forman MA, MacBride BT (2009) The turbulent cascade and proton heating in the solar wind at 1 AU. Astrophys J 697:1119–1127 125. Sundkvist D, Retinò A, Vaivads A, Bale SD (2007) Dissipation in turbulent plasma due to reconnection in thin current sheets. Phys Rev Lett 99:025,004 126. Teaca B, Carati D, Domaradzki JA (2011) On the locality of magnetohydrodynamic turbulence scale fluxes. Phys Plasmas 18:112,307 127. TenBarge JM, Howes GG (2013) Current sheets and collisionless damping in kinetic plasma turbulence. Astrophys J Lett 771:L27 128. Tóth G (2000) The ∇ · B = 0 constraint in shock-capturing magnetohydrodynamics codes. J Comput Phys 161:605–652 129. Tu CY, Marsch E (1995) Mhd structures, waves and turbulence in the solar wind: observations and theories. Space Sci Rev 73:1–210 130. Verma MK (2004) Statistical theory of magnetohydrodynamic turbulence: recent results. Phys Rep 401:229–380 131. Verma MK, Ayyer A, Chandra AV (2005) Energy transfers and locality in magnetohydrodynamic turbulence. Phys Plasmas 12:082,307 132. Von Kármán T, Howarth L (1938) On the statistical theory of isotropic turbulence. Proc Roy Soc London Ser A 164:192 133. Wan M, Matthaeus WH, Karimabadi H, Roytershteyn V, Shay M, Wu P, Daughton W, Loring B, Chapman SC (2012) Intermittent dissipation at kinetic scales in collisionless plasma turbulence. Phys Rev Lett 109:195,001 134. Wan M, Matthaeus WH, Roytershteyn V, Karimabadi H, Parashar T, Wu P, Shay M (2015) Intermittent dissipation and heating in 3d kinetic plasma turbulence. Phys Rev Lett 114:175,002 135. Wang J, Shi Y, Wang LP, Xiao Z, He XT, Chen S (2012) Scaling and statistics in threedimensional compressible turbulence. Phys Rev Lett 108:214,505 136. Wang J, Wang LP, Xiao Z, Shi Y, Chen S (2010) A hybrid numerical simulation of isotropic compressible turbulence. J Comput Phys 229:5257–5279 137. Wang J, Yang Y, Shi Y, Xiao Z, He XT, Chen S (2013) Cascade of kinetic energy in threedimensional compressible turbulence. Phys Rev Lett 110:214,505 138. Warburton TC, Karniadakis GE (1999) A discontinuous Galerkin method for the viscous MHD equations. J Comput Phys 152:608–641 139. Whitham GB (2011) Linear and nonlinear waves, vol 42. John Wiley & Sons 140. Yaglom A (1949) On the local structure of a temperature field in a turbulent flow. Dokl Akad Nauk SSSR 69:743–746 141. Yao Z, Rae I, Guo R, Fazakerley A, Owen C, Nakamura R, Baumjohann W, Watt CE, Hwang K, Giles B et al (2017) A direct examination of the dynamics of dipolarization fronts using mms. J Geophys Res Space Phys 122(4):4335–4347 142. Yee HC, Sandham ND, Djomehri MJ (1999) Low-dissipative high-order shock-capturing methods using characteristic-based filters. J Comput Phys 150:199–238 143. Yee HC, Sjögreen B (2005) Efficient low dissipative high order schemes for multiscale MHD flows, II: minimization of ∇ · B numerical error. J Sci Comput 29:115–164

References

21

144. Yousef TA, Rincon F, Schekochihin AA (2007) Exact scaling laws and the local structure of isotropic magnetohydrodynamic turbulence. J Fluid Mech 575:111–120 145. Zhou Y (1993) Degrees of locality of energy transfer in the inertial range. Phys Fluids A 5:1092 146. Zhou Y (1993) Interacting scales and energy transfer in isotropic turbulence. Phys Fluids A 5:2511 147. Zhou Y, Matthaeus WH (1990) Models of inertial range spectra of interplanetary magnetohydrodynamic turbulence. J Geophys Res Space Phys 95(A9):14881–14892

Chapter 2

Theoretical Modelling

For plasmas, the Liouville equation of many-body distribution functions is the complete statistical description of a many-body system and describes every detail of plasmas from the largest to the smallest spatial and temporal scales. The amount of information contained in the many-body distribution function is vast, and as a consequence it is very hard to directly deal with the Liouville equation. Based on what plasma properties we are interested in studying, be they dominant at small or large scales, a plasma can be treated as tractable models in various limits, such as the kinetic theory (Sect. 2.3), the multi-fluid theory (Sect. 2.4) and the single-fluid theory (Sect. 2.5). In this chapter, we briefly review the reduced descriptions of plasmas. More comprehensive material can be found in the books of Krall and Trivelpiece [3], Bittencourt [2], and Baumjohann and Treumann [1].

2.1 The Electromagnetic Field Equations Motion of charged particles builds heavily on knowledge of electrodynamics. The electromagnetic fields obey Maxwell’s equations which are composed of formulae as follows. 1. Gauss’s law ∇ ·E=

q , ε0

(2.1)

 where q = α qα n α , is the total charge density, the summation is over the different species of type α with charge, qα , and number density, n α , and ε0 is the electric permittivity of free space. 2. Faraday’s law ∂B ∇ ×E=− . (2.2) ∂t © Springer Nature Singapore Pte Ltd. 2019 Y. Yang, Energy Transfer and Dissipation in Plasma Turbulence, Springer Theses, https://doi.org/10.1007/978-981-13-8149-2_2

23

24

2 Theoretical Modelling

3. Ampère-Maxwell equation   ∂E , ∇ × B = μe0 j + ε0 ∂t

(2.3)

 where μe0 is the magnetic permeability of free space, j = α qα n α uα , is the total electric current density, and uα is the average velocity of the charges of species α. The second term on the right-hand side was introduced by Maxwell as a correction to Ampère’s circuital law and is called the displacement current. 4. Solenoidal nature of magnetic field ∇ · B = 0,

(2.4)

which indicates that magnetic manopoles do not exist. Since neither charges nor particles can be destroyed in electrodynamics, the rate at which charge is changing inside a small volume must equal the rate at which charge flows in or out across the surface of that volume. Therefore, conservation of charge requires that ∂q + ∇ · j = 0. (2.5) ∂t Note that the conservation of electric charge is already implied by Maxwell’s equations. By taking the divergence of Eq. 2.3 and using Eq. 2.1, we can derive the equation ε0

∂q ∂ ∇ ·E+∇ ·j= + ∇ · j = 0. ∂t ∂t

Meanwhile, taking the divergence of Eq. 2.2 yields ∂ ∇ · B = 0. ∂t

(2.6)

Therefore, Eq. 2.4 can be considered as an initial condition for Eq. 2.2 since, if ∇ · B = 0 at initial time, then Eq. 2.2 implies that this condition will remain satisfied for all subsequent times.

2.2 Charged Particle Motion A particle moving with velocity v and carrying a charge q is, in general, subject to forces due to electric and magnetic fields, F = q(E + v × B).

(2.7)

2.2 Charged Particle Motion

25

The equation of motion for a particle, under these forces, can be written as m

dv = F = q(E + v × B), dt dx = v, dt

(2.8) (2.9)

where m is the particle mass and x its position. In the absence of an electric field, the equation of motion reduces to m

dv = q(v × B). dt

(2.10)

In a uniform magnetostatic field, we can get that the velocity component parallel to the magnetic field is constant, while the particle gyrates in the plane perpendicular to the magnetic field with constant angular velocity, ωc =

|q|B , m

(2.11)

which is also called the gyrofrequency or cyclotron frequency.

2.3 Kinetic Theory A typical plasma might be composed of trillions of ion-electron pairs per cm 3 . All particles do not move with the same velocity, nor does the velocity of a given particle remain constant with time. It is obvious that predicting the plasma’s behavior is hopeless in the way following the complicated trajectory of each particle. Otherwise necessary, some crude models are sufficient to describe the majority of the plasma’s behaviors. For example, kinetic theory is not interested in the motion of each particle in detail, but instead, it counts on the distribution function of particles of species α, f α (x, v, t), in phase space (x, v, t). To make the definition of f α (x, v, t) more precise, let us consider the sixdimensional phase space (x, v), spanned by the position coordinates (x, y, z) and the velocity coordinates (vx , v y , vz ). A volume element around x in configuration space is d 3 r = d x d y dz and a volume element around v in velocity space is d 3 v = dvx dv y dvz . They are schematically represented in Fig. 2.1. We denote by d 6 Nα (x, v, t) the number of particles of species α which have position lying inside d 3r about x and velocity lying inside d 3 v about v at the instant t. Then the distribution function f α (x, v, t) is defined by f α (x, v, t) =

d 6 Nα (x, v, t) , d 3r d 3 v

(2.12)

26

2 Theoretical Modelling

(a)

(b)

Fig. 2.1 Volume elements: a d 3 r = dx dy dz around the point x in configuration space; b d 3 v = dvx dv y dvz around the point v in velocity space

which is interpreted as the density of particles of species α in the phase space volume element d 3r d 3 v. The distribution function f α (x, v, t) can be also expressed as an ensemble average of the exact particle density function Fα (x, v, t), that is, f α (x, v, t) = Fα (x, v, t).

(2.13)

The exact particle density function Fα (x, v, t) is given by Fα (x, v, t) =



δ(x − xi (t))δ(v − vi (t)),

(2.14)

i

where the sum is over all particles of species α, δ(x − xi ) and δ(v − vi ) are threedimensional Dirac delta functions and xi (t) and vi (t) are the position and velocity, respectively, of the i-th particle at time t. It is obvious that Fα (x, v, t) depends on the exact position of all particles of species α, but f α (x, v, t) only depends on the phase space coordinates (x, v, t) and can be interpreted as the probability to find particles of species α in a certain phase space volume element. The Boltzmann equation  ∂t f α + v · ∇ f α + a · ∇v f α =

δ fα δt

 ,

(2.15)

c

describes the evolution of f α (x, v, t) under the action of average fields. In this equation, the ∇ operator is the differentiation with respect to position, while the ∇v operator is the differentiation with respect to velocity. It is assumed that the force on particles is entirely electromagnetic. The acceleration a thus becomes a = qα /m α (E + v × B). The remaining term that needs to be specified is the collision term on the right-hand side. Although it is impossible to have an exact functional form of the collision term that can accommodate all kinds of collisions, several fea-

2.3 Kinetic Theory

27

sible expressions for the collision term arise, such as the Krook collision term, the Boltzmann collision integral and the Fokker-Planck collsion term. The simplest possible form of the Boltzmann equation is the Vlasov equation, in which the collision term is entirely neglected. Although the distribution function is not a macroscopic observable, its velocity moments give macroscopically observable quantities. The number density of particles of species α is given by  n α (x, t) =

f α (x, v, t)d 3 v.

(2.16)

v

The bulk flow velocity of particles of species α is defined by 1 uα (x, t) = n α (x, t)

 v f α (x, v, t)d 3 v.

(2.17)

v

The pressure tensor is  Pα = m α

(v − uα )(v − uα ) f α (x, v, t)d 3 v.

(2.18)

v

One-third the trace of the pressure tensor gives the scalar pressure pα =

mα 3

 (v − uα ) · (v − uα ) f α (x, v, t)d 3 v.

(2.19)

(v − uα ) · (v − uα )(v − uα ) f α (x, v, t)d 3 v.

(2.20)

v

The heat flux vector is hα =

1 mα 2

 v

2.4 Two-Fluid Theory In the fluid approximation, particle species in plasmas are interpenetrating fluids containing electrical charges, in which the identity of the individual particle is neglected, and only the fluid element is taken into account. These fluid elements are large enough to contain a large number of particles and yet small enough in comparison with characteristic macroscopic scales. So now it is not necessary to know the exact evolution of the distribution function but it is sufficient to investigate the evolution of macroscopic variables. Just as the velocity moments of the distribution function give the macroscopic variables, so do the velocity moments of the Boltzmann equation describe the time evolution of the plasma from a macroscopic point of view. The theory using the macroscopic equations are called multi-fluid theory, wherein the simplest one is the two-fluid theory, which treats ions and electrons each as conduct-

28

2 Theoretical Modelling

ing fluids that are coupled through momentum transfer collisions and by Maxwell’s equations. In order to derive fluid-like equations from the Boltzmann equation, we denote by χ (v) a function of v. Let us multiply the Boltzmann equation by χ (v) and integrate it over the entire velocity space 

 χ ∂t f α d v + v







χ v · ∇ fα d v +

3

χ a · ∇v f α d v =

3

χ

3

v

v

v

δ fα δt

 d 3 v. (2.21) c

Note that χ (v) is independent from x and t. Since the spatial coordinate x, the velocity coordinate v and the time t are independent variables, the time derivative ∂t and the spatial differentiation ∇ can be exchanged with the integration over d 3 v. In addition, ∇v · a = qα /m α ∇v · (E + v × B) = 0. So Eq. 2.21 can be rewritten as   δ ∂ (n α χ α ) + ∇ · (n α χ vα ) − n α a · ∇v χ α = (n α χ α ) , ∂t δt c

(2.22)

where the symbol  α denotes the average over velocity space 1 χ α = n α (x, t) and



δ (n α χ α ) δt

 χ (v) f α (x, v, t)d 3 v,

(2.23)

v





 = c

χ v

δ fα δt

 d 3 v.

(2.24)

c

By taking χ = m α , we obtain the continuity equation of the conducting fluid of particle species α ∂ρα + ∇ · (ρα uα ) = Sα , (2.25) ∂t where ρα = n α m α is the mass density and Sα represents particle production or destruction due to collisions. By taking χ = m α v, Eq. 2.22 reduces to the momentum equation ∂(ρα uα ) + ∇ · (ρα uα uα ) + ∇ · (ρα cα cα α ) − n α qα (E + uα × B) = Aα , (2.26) ∂t where cα = v − uα and Aα denotes the collision term. According to the expression of pressure tensor Eq. 2.18, the momentum equation can be written as ∂(ρα uα ) + ∇ · (ρα uα uα ) + ∇ · Pα − n α qα (E + uα × B) = Aα . ∂t By taking χ = m α v2 /2, we have the energy equation

(2.27)

2.4 Two-Fluid Theory

∂ Wα +∇ · ∂t

29



 1 ρα v2 vα − n α qα uα · E = Mα , 2

(2.28)

1 where Wα = m α v v2 f α (x, v, t) d 3 v and Mα represents the rate of energy density 2 change as a result of collisions. Substituting the expressions of pressure tensor Eq. 2.18 and heat flux vector Eq. 2.20 into the energy equation gives ∂ Wα + ∇ · (Wα uα + Pα · uα + hα ) − n α qα uα · E = Mα . ∂t

(2.29)

In summary, the multi-fluid equations that are satisfied by the mass density, momentum and energy are ∂ρα + ∇ · (ρα uα ) = Sα , ∂t

∂(ρα uα ) + ∇ · (ρα uα uα ) + ∇ · Pα − n α qα (E + uα × B) = Aα , ∂t ∂ Wα + ∇ · (Wα uα + Pα · uα + hα ) − n α qα uα · E = Mα . ∂t In the simplest case, when the gas is fully ionized and there is only one species of ion, we shall need two sets of equations, one for the ion fluid and one for the electron fluid. In a partially ionized gas, we shall also need an extra sets of equations for the neutral fluid. The above equations are the first three moments of the Boltzmann equation. Note that any finite set of transport equations, is insufficient to form a closed system of equations. Consequently, it is necessary to truncate the system of transport equations at some point in the hierarchy of moments of the Boltzmann equation, such as the set of Eqs. 2.25, 2.27 and 2.29, and consider some approximation for the highest moment of the distribution function appearing in the system, say, the pressure tensor Pα and the heat flux vector hα . The collision terms Sα , Aα and Mα are still to be specified as well. They are related to both inelastic and elastic collisions. But the elastic collisions, in which there is conservation of mass, of momentum and of energy, are usually taken into account.

2.5 Single-Fluid Theory Without separately describing the behavior of ions and electrons, the two-fluid equations can be further simplified by combining the density and velocity of ions and electrons. The single-fluid theory so obtained, which is also called MHD theory, is based on the total macroscopic variables, such as the total mass density ρ, the total charge density q, the center-of-mass velocity u, the total electric current density j,

30

2 Theoretical Modelling

the total pressure tensor P, the total kinetic energy W and the total heat flux vector h. They are defined as follows: ρ=



ρα ,

(2.30)

n α qα ,

(2.31)

ρα uα ,

(2.32)

n α qα u α ,

(2.33)

α

q=

 α

ρu =

 α

j=

 α

P=





 α

(2.34)

v

α

W =

(v − u)(v − u) f α (x, v, t)d 3 v,

mα Wα ,

1 h= mα 2 α

(2.35)  (v − u) · (v − u)(v − u) f α (x, v, t)d 3 v.

(2.36)

v

Note that u in Eqs. 2.34 and 2.36 is the center-of-mass velocity and not uα . Equations satisfied by the total macroscopic variables are simply obtained by summing the two-fluid Eqs. 2.25 to 2.29 over all particle species, which gives ∂ρ + ∇ · (ρu) = 0, ∂t ∂(ρu) + ∇ · (ρuu) = −∇ · P + j × B, ∂t ∂W + ∇ · (W u) = −∇ · (P · u) − ∇ · h + j · E. ∂t

(2.37) (2.38) (2.39)

We have used the assumption that macroscopic charge neutrality is maintained and therefore the total charge density q is setequal to zero. terms vanish  The collision  when summed over all particle species, α Sα = 0, α Aα = 0, and α Mα = 0, by conservation of the total mass, momentum and energy, respectively.

2.5.1 The Reduced Form of Electrodynamic Equations for MHD Several approximations allow the elimination of some terms in the electrodynamic equations for MHD. Let us say L 0 , T0 and U0 are the characteristic length, time and velocity, respectively, and U0 = L 0 /T0 . From Eq. 2.2, it is easy to get

2.5 Single-Fluid Theory

31

|∇ × E| = O(1) = O |∂B/∂t| So it indicates



E T0 B L0



 =O

E BU0

 .

(2.40)

E ∼ U0 . B

(2.41)

The contribution of the displacement current to Eq. 2.3 is neglected in MHD. The validity of this approximation is roughly confirmed as follows. For most of the fluids normally considered in MHD problems, the sort of speeds we are interested in is much less than the speed of light, i.e., U0  c. Therefore, the ratio of the two terms in Eq. 2.3 is |μe0 ε0 ∂E/∂t| =O |∇ × B|



ε0 μe0 E/T0 B/L 0



 =O

E L0 ε0 μe0 B T0



 =O

U02 c2

  1,

(2.42) where c2 = 1/(ε0 μe0 ) is the square of the light velocity. Consequently, the displacement current is negligible, and the Ampère-Maxwell equation reduces to ∇ × B = μe0 j.

(2.43)

Under the assumptions that the plasma is completely ionized containing only electrons and one type of ions that are singly charged and the macroscopic charge neutrality of the plasma is preserved, i.e., n e = n i = n and the approximation m e  m i , the generalized Ohm’s law reads 1 1 m e ∂j 1 − ∇ · Pe = E + u × B − j × B − j, 2 ne ∂t ne ne σ

(2.44)

where σ = (ne2 )/(m e νei ) is the electrical conductivity and νei is the collision frequency between electrons and ions. We can estimate the orders of magnitude of different terms in Eq. 2.44. For example, the ratio of the time derivative term on the left-hand side of Eq. 2.44 and the electric field on the right-hand side of Eq. 2.44 is

ω2 c2 =O =O , ω2pe U02 (2.45) where ω = T0−1 is the characteristic frequency. To obtain the orders of other terms, we proceed in a similar way, which leads to  2  1 |∇ · Pe | ω Vthe , (2.46) =O ne |E| ωce U02 |u × B| = O(1), (2.47) |E| m e |∂j/∂t| =O ne2 |E|



B m e ε0 1 2 ne ε0 μe0 T0 L 0 E



c2 1 2 ω pe U0 L 0 T0





32

2 Theoretical Modelling

1 |j × B| =O ne |E| 1 |j| =O σ |E|

 

ω ωce c2 ω pe ω pe U02 ω νei c2 ω pe ω pe U02

 ,

(2.48)

,

(2.49)



where for MHD, the assumption of isotropic pressure, ∇ · Pe ∼ ∇ pe , is used in deriving Eq. 2.46, and Vthe is the electron thermal speed. The common approximations of MHD theory are valid under the following conditions: ω2 c2 m e ∂j can be neglected if  1. ne2 ∂t ω2pe U02 2 1 ω Vthe • The pressure gradient term ∇ · Pe can be neglected if  1. ne ωce U02 ω ωce c2 j×B can be neglected if • The Hall effect term  1. ne ω pe ω pe U02

• The time derivative term

It is often assumed that collisions are sufficient frequent, that is, the collision frequency νei is very large so the last term j/σ in Eq. 2.44 could be important. Therefore, when the terms ∂j/∂t, ∇ · Pe , and j × B are all neglected, Eq. 2.44 reduces to Ohm’s law, j = σ (E + u × B). (2.50)

2.5.2 Magnetic Induction Equation The magnetic induction equation follows from a combination of Ohm’s law (i.e., Eq. 2.50), Faraday’s law (i.e., Eq. 2.2) and Ampère’s law (i.e., Eq. 2.43). They serve to replace the electric field and electric current density with the velocity and magnetic fields, ∂B = −∇ × E ∂t 

(2.51) 

j −u×B σ   1 ∇ ×B−u×B = −∇ × μe0 σ

= −∇ ×

= ∇ × (u × B) + η∇ 2 B,

(2.52) (2.53) (2.54)

where η = 1/(μe0 σ ) is the magnetic resistivity. So the magnetic induction equation is ∂B − ∇ × (u × B) = η∇ 2 B. (2.55) ∂t

2.5 Single-Fluid Theory

33

2.5.3 The Full Equations of MHD In summary, the full equations of MHD are composed of ∂ρ + ∇ · (ρu) = 0,   ∂t ∂u + u · ∇u = −∇ p − ∇ · Π + j × B, ρ ∂t ∂W + ∇ · (W u) = −∇ · ( pu) − ∇ · (Π · u) − ∇ · h + j · E, ∂t ∂B − ∇ × (u × B) = η∇ 2 B, ∂t μe0 j = ∇ × B, j = σ (E + u × B),

(2.56) (2.57) (2.58) (2.59) (2.60) (2.61)

where P = pI + Π and I is the unity tensor. The set of Eqs. 2.56 to 2.61 are not closed; i.e., there are more unknowns than equations. This situation is generally remedied by taking approximations of some unknowns. For example, the scalar pressure p is related to the density by an equation of state. The choice of equation of state could be to use the assumption of adiabatic model, isothermal model, or ideal gas model. The deviatoric pressure tensor Π is the viscous stress, which can be written in terms of velocity gradients,

2 Π = −μ ∇u + (∇u)T + μθ I, 3 where the superscript T denotes transpose, θ = ∇ · u is the divergence of velocity, and μ is the viscosity. The heat flux vector h could result from heat flow due to temperature gradients, diffusion processes in gas mixtures and radiation. Here we only consider the effect caused by temperature gradients. One can relate h to gradients of temperature via Fourier’s heat conduction law h = −κ∇T,

(2.62)

where κ is the thermal conductivity.

References 1. Baumjohann W, Treumann RA (2012) Basic space plasma physics. World Scientific Publishing Company 2. Bittencourt JA (2004) Fundamentals of plasma physics, 3rd edn. Springer, New York 3. Krall N, Trivelpiece A (1973) Principles of plasma physics. International series in pure and applied physics. McGraw-Hill

Chapter 3

Hybrid Scheme for Compressible MHD Turbulence

In this chapter, we describe in detail an efficient, high-resolution and oscillation-free hybrid scheme for shock-turbulence interactions in compressible MHD problems. The hybrid scheme couples compact finite difference scheme with WENO scheme, wherein compact finite difference scheme used in the smooth region ensures the accuracy, while WENO scheme applied in the shock region captures discontinuities robustly. In order to maintain numerical stability and eliminate spurious oscillations, we utilize a pentadiagonal filter. Various benchmarks are shown to verify the capability of the hybrid method. We expect that the hybrid scheme can be applied to the numerical simulations of compressible MHD turbulence with high Reynolds number and Mach number.

3.1 Governing Equations Ideal MHD is the most basic single-fluid model for determining the macroscopic equilibrium and stability properties of a plasma. However, the real cases do not always satisfy all the criteria of its validity. Therefore, the numerical examples presented here are based on ideal MHD equations, while dissipation terms are embedded as simulating MHD turbulence.

3.1.1 Ideal MHD Equations Let us start by considering the ideal extreme, where there is no viscosity (μ = 0), no resistivity (η = 0) and no thermal conductivity (κ = 0). The set of Eqs. 2.56 to 2.59 becomes © Springer Nature Singapore Pte Ltd. 2019 Y. Yang, Energy Transfer and Dissipation in Plasma Turbulence, Springer Theses, https://doi.org/10.1007/978-981-13-8149-2_3

35

36

3 Hybrid Scheme for Compressible MHD Turbulence

∂ρ + ∇ · (ρu) =  ∂t  ∂u ρ + u · ∇u = ∂t ∂W + ∇ · (W u) = ∂t ∂B − ∇ × (u × B) = ∂t

0,

(3.1)

−∇ p + j × B,

(3.2)

−∇ · ( pu) + (j × B) · u,

(3.3)

0.

(3.4)

They can be written in conservative flux vector form as ∂U ∂E ∂F ∂G + + + = 0, ∂t ∂x ∂y ∂z

(3.5)

where U = (ρ, ρu, ρv, ρw, Bx , B y , Bz , E )T is the vector of conservative variables, E = W + B 2 /2 = ρu 2 /2 + p/(γ − 1) + B 2 /2 is the total energy, and γ is the adiabatic index. The inviscid fluxes E, F, G can be written as ⎤ ⎡ ρu ⎥ ⎢ ρu 2 + pt − Bx2 ⎥ ⎢ ⎥ ⎢ ρuv − Bx B y ⎥ ⎢ ⎥ ⎢ ρuw − Bx Bz ⎥, (3.6) E=⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ −Ωz ⎥ ⎢ ⎦ ⎣ Ωy (E + pt )u − Bx (u · B) ⎡ ⎤ ρv ⎢ ⎥ ρvu − B y Bx ⎢ ⎥ 2 2 ⎢ ⎥ ρv + pt − B y ⎢ ⎥ ⎢ ⎥ ρvw − B y Bz ⎥, (3.7) F=⎢ ⎢ ⎥ Ωz ⎢ ⎥ ⎢ ⎥ 0 ⎢ ⎥ ⎣ ⎦ −Ωx (E + pt )v − B y (u · B) ⎡ ⎤ ρw ⎢ ⎥ ρwu − Bz Bx ⎢ ⎥ ⎢ ⎥ ρwv − Bz B y ⎢ ⎥ 2 2 ⎢ ⎥ ρw + pt − Bz ⎢ ⎥, G=⎢ (3.8) ⎥ −Ω y ⎢ ⎥ ⎢ ⎥ Ωx ⎢ ⎥ ⎣ ⎦ 0 (E + pt )w − Bz (u · B)

3.1 Governing Equations

37

where the thermal pressure is combined with the magnetic pressure to give the total pressure, pt = p + B 2 /2. The electric field vector is defined as  = −u × B. Note that there exist symmetries between the flux components, Ωx = −F7 = G 6 , Ω y = E 8 = −G 5 , Ωz = −E 6 = F5 ,

(3.9)

which facilitate the numerical calculation.

3.1.2 Dimensionless MHD Equations The full MHD equations (see Eqs. 2.56 to 2.59) can be rearranged in the form of Eq. 3.5 as well, ∂U ∂E ∂F ∂G + + + = Hν . (3.10) ∂t ∂x ∂y ∂z In comparison with the ideal MHD Eq. 3.5, the above equation has an extra term Hν on the right hand which accounts for dissipative effects, ⎡

⎤ 0 ⎢ ⎥ −∇ · ⎥. Hv = ⎢ ⎣ ⎦ −∇ × (ηj) −∇ · ( · u) + ∇ · (κ∇T ) + ∇ · (B × ηj)

(3.11)

The MHD equations are non-dimensionalized by introducing several reference scales. For example, length is normalized to L 0 , velocity to U0 density to ρ0 , magnetic field to B0 , temperature to T0 , viscosity to μ0 , thermal conductivity to κ0 , and resistivity to η0 . The dimensionless variables are then t ∗ = t L 0 /U0 , x∗ = x/L 0 , u∗ = u/U0 , ρ ∗ = ρ/ρ0 , B∗ = B/B0 , T ∗ = T /T0 , μ∗ = μ/μ0 , κ ∗ = κ/κ0 , η∗ = η/η0 ,

(3.12)

and the dimensionless MHD equations read ∂U∗ ∂E∗ ∂F∗ ∂G∗ + + + = H∗ν . ∂t ∗ ∂x∗ ∂ y∗ ∂z ∗

(3.13)

Here for notational convenience, we drop the superscript ∗ without confusion. The dimensionless equivalents of quantities are E = ρu 2 /2 + p/(γ − 1) + B 2 /(2Mm 2 ), pt = p + B 2 /(2Mm 2 ),

38

3 Hybrid Scheme for Compressible MHD Turbulence

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ E=⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ F=⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ G=⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣



ρu

Bx2 Mm2 Bx B y ρuv − Mm2 B x Bz ρuw − Mm2

ρu 2 + pt −

0 −Ωz Ωy Bx (E + pt )u − 2 (u · B) Mm ρv B y Bx ρvu − Mm2 ρv2 + pt −

B y2

Mm2 B y Bz ρvw − Mm2

Ωz 0 −Ωx By (E + pt )v − 2 (u · B) Mm

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

ρw Bz B x ρwu − Mm2 Bz B y ρwv − Mm2 ρw2 + pt −

Bz2 Mm2

−Ω y Ωx 0 (E + pt )w −

(3.14)

Bz (u · B) Mm2

(3.15)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(3.16)

3.1 Governing Equations

and

39



⎤ 0 ∇ · ⎢ ⎥ ⎢ ⎥ − ⎢ ⎥ Re ⎢ ⎥ ⎢ ⎥. ∇ × (ηj) Hv = ⎢ ⎥ − ⎢ ⎥ Rem ⎢ ⎥ ⎣ ∇ · ( · u) ∇ · (κ∇T ) ∇ · (B × ηj) ⎦ + + − Re α Rem Mm2

(3.17)

The coefficient α is equal to Pr Re (γ − 1) Ma 2 . There are six dimensionless parameters which regularly appear in the numerical simulation for MHD turbulence. The Reynolds number, ρ0 U0 L 0 , (3.18) Re = μ0 as in conventional fluid mechanics, is representative of the ratio of inertia to viscous forces. The magnetic Reynolds number, Rem =

U0 L 0 , η0

(3.19)

indicates the relative strengths of advection and diffusion in the magnetic induction equation. The Mach number, U0 M= , (3.20) a0 √ where a0 = γ RT0 is the sound speed, and R is the universal gas constant, assesses the compressibility of mean flows. The Alfvén Mach number is defined as Mm =

U0 , A0

(3.21)

√ where A0 = B0 / ρ0 is the Alfvén speed. The Prandtl number is defined as Pr = μ0 C p /κ0 , where C p is the specific heat capacity at constant pressure, and the magnetic Prandtl number is a hybrid of Re and Rem , Prm = μ0 /(ρ0 η0 ) = Rem /Re. Throughout the book the ideal gas law, p = ρT /(γ M 2 ), is applied in the numerical simulations of MHD problems. For simplicity, we set Pr = 0.7 and Prm = 1, and the dimensionless μ, κ and η are assumed to be described by the Sutherland’s law as used by [16], 1.4042T 3/2 μ=κ=η= . (3.22) T + 0.40417

40

3 Hybrid Scheme for Compressible MHD Turbulence

3.2 Numerical Method Here we intend to design a hybrid scheme with high-order accuracy in smooth regions, and with the capability of capturing discontinuities robustly and maintaining a divergence-free magnetic field. The basic points are essentially three: shock detector implemented in the hybrid scheme to transit automatically between compact and WENO schemes, filters to eliminate spurious oscillations and stabilize numerical solutions, and strategies to preserve the divergence-free constraint of the magnetic field.

3.2.1 The Hybrid Compact-WENO Scheme To describe the basic formulation of the hybrid scheme, we consider that the spacing between the discrete points is constant and indicate by (· · · )i the variable on the i-th grid point.

3.2.1.1

Shock Detector

The first step in the procedure of the scheme is shock detector, which distinguishes shock regions from smooth regions. The shock detector used here is consist of two sensors. Shocks are thin transitional layers, where flow variables undergo a discontinuous variation. Therefore, the first sensor is based on the pressure jump from its upstream value to its downstream value,







S1 = pi+1, j,k − pi−1, j,k + pi, j+1,k − pi, j−1,k + pi, j,k+1 − pi, j,k−1

−β min( pi, j,k , pi+1, j,k , pi−1, j,k , pi, j+1,k , pi, j−1,k , pi, j,k+1 , pi, j,k−1 ), (3.23) where β > 0 is an adaptive parameter. In conjunction with the pressure jump, the dilatation is applied as well, S2 = −δ max(Ci, j,k , Ci+1, j,k , Ci−1, j,k , Ci, j+1,k , Ci, j−1,k , Ci, j,k+1 , Ci, j,k−1 ) − h(∇ · u)i, j,k ,

(3.24) where Ci, j,k = γ pi, j,k /ρi, j,k + Bi,2 j,k /ρi, j,k is the local fast speed, h = (Δx + Δy + Δz)/3 is the averaged grid spacing, and δ > 0 is an adaptive parameter. We set here β = 0.25 and δ = 0.05, whose small variations give rise to negligible effects on the performance of the scheme, as shown in Sect. 3.3.3. The two sensors are accomplished by ensuring that the local minima and maxima of pressure are not far apart, and the divergence of the velocity does not greatly exceed the local fastest signal speed. In general, the sharp difference and the highly negative dilatation are associated with discontinuities. Therefore, the local regions

3.2 Numerical Method

41

Fig. 3.1 The diagram of the partition of regions in the hybrid scheme. The black solid circle is the shock front identified by the shock detector [18]. Reprinted from Journal of Computational Physics, 306, Yan Yang et al., A hybrid scheme for compressible magnetohydrodynamic turbulence, 73–91, Copyright (2015), with the permission from Elsevier

with S1 > 0 or S2 > 0 are identified as shock fronts that require WENO scheme. The shock regions include the shock fronts identified by the shock sensor and two additional grid points on both left and right in each spatial direction around the fronts. Two transitional points adjoining with the shock regions are referred to as the buffer region. Remaining gird points are the smooth regions. Figure 3.1 illustrates the partition of regions.

3.2.1.2

Scalar Conservation Law in 1D

We consider one-dimensional (1D) conservation law, first for a scalar equation, ∂ f (u) ∂u + = 0. ∂t ∂x

(3.25)

The most important is the scheme for approximating the spatial derivative entering in the conservation law. We denote by f i the approximation to the first derivative ∂ f /∂ x at the i-th point. The compact finite difference approximation to the first derivative is generalized in the form [10]: f f f − f i−3 − f i−2 − f i−1     β f i−2 + α f i−1 + f i + α f i+1 + β f i+2 = c i+3 + b i+2 + a i+1 , 6h 4h 2h

(3.26)

where h = Δx and the relations between the coefficients a, b, c and α, β are derived by matching the Taylor series coefficients of various orders. Here, in the smooth region, the sixth-order compact finite difference scheme is employed:

42

3 Hybrid Scheme for Compressible MHD Turbulence   α f i−1 + f i + α f i+1 =b

f i+1 − f i−1 f i+2 − f i−2 +a , 4h 2h

(3.27)

where α = 1/3, b = 1/9 and a = 14/9. It can be written in the conservative form as Compact

  α f i−1 + f i + α f i+1 =

Fi+1/2

Compact

− Fi−1/2

,

h

(3.28)

where the numerical flux is Compact Fi+1/2

b = ( f i+2 + f i−1 ) + 4



 b a + ( f i+1 + f i ) . 2 4

(3.29)

Although the compact central finite difference scheme has improved resolution (spectral-like behaviour), high-frequency waves are not damped by the scheme, which creates a risk for instabilities generated by the nonlinearity of the convective terms, for example, oscillatory behaviour near discontinuities (Gibbs’ phenomenon). Therefore, in the shock region, we resort to the WENO scheme that controls such instabilities. The scheme uses a conservative approximation to the spatial derivative, f i =

w w − f i−1/2 f i+1/2

h

,

(3.30)

w with the numerical flux f i+1/2 yet to be defined by the WENO scheme. The fifthorder WENO scheme is used here. See Appendix A for details. In analogy with Eq. 3.28, it can be rewritten as

  + f i + α f i+1 = α f i−1

WENO WENO − Fi−1/2 Fi+1/2

h

,

(3.31)

where the numerical flux is WENO w w w Fi+1/2 = α f i+3/2 + f i+1/2 + α f i−1/2 .

(3.32)

Similarly, in the buffer region, we have   + f i + α f i+1 = α f i−1

J oint J oint − Fi−1/2 Fi+1/2

h

.

(3.33)

J oint The numerical flux Fi+1/2 could be a combination of the compact and WENO fluxes, like Compact J oint WENO = (1 − ηi )Fi+1/2 + ηi Fi+1/2 , (3.34) Fi+1/2

where 0 ≤ ηi ≤ 1 is a flattener, which guarantees that one can smoothly go from one type of flux to another. For simplicity, we directly take the arithmetic average

3.2 Numerical Method

43

Compact J oint J oint of the compact flux and WENO flux as Fi+1/2 here, that is Fi+1/2 = Fi+1/2 +

WENO /2. Fi+1/2 In summary, the approximation to the convective term by the hybrid scheme is H ybrid

  + f i + α f i+1 = α f i−1

where H ybrid

Fi+1/2

3.2.1.3

H ybrid

Fi+1/2 − Fi−1/2 h

,

⎧ Compact ⎪ ⎨ Fi+1/2 , in the smooth region, WENO = Fi+1/2 , in the shock region, ⎪ ⎩ J oint Fi+1/2 , in the buffer region.

(3.35)

(3.36)

System of Conservation Laws in 1D

Let us consider 1D systems of conservation laws, ∂U ∂F(U) + = 0, ∂t ∂x

(3.37)

where the conserved property is described by a vector quantity U and a vector flux F. For the compact finite difference scheme, the only thing that we need to do is solving the system of conversation laws component by component. Therefore, here we will focus on the WENO scheme. The WENO scheme again approximates the spatial derivative as Fi =

w w − Fi−1/2 Fi+1/2

h

.

(3.38)

Generalization of the WENO scheme to systems of conservation laws is based on a local characteristic decomposition of waves. The eigenvalues λs of the Jacobian matrix A = (∂F/∂U) correspond to the wave speeds of the system, which form a diagonal matrix . The corresponding right eigenvectors are column vectors and the left eigenvectors are row vectors. Let us denote the matrices of the right and left eigenvectors by R and L, respectively. It is obvious that A = RL.

(3.39)

The eigen-system in MHD is given in Appendix B. Based on an average of Ui and Ui+1 , Ui+1/2 = (Ui + Ui+1 )/2, a mean Jacobian of the flux vector Ai+1/2 and the matrices Ri+1/2 and Li+1/2 at the interface xi+1/2 are defined.

44

3 Hybrid Scheme for Compressible MHD Turbulence

Firstly, both the fluxes and the variables at the k-th grid point on stencils are projected into the right eigenvector space, Fk = Uk =

m 

s gks Ri+1/2 ,

s=1 m 

s vks Ri+1/2 ,

(3.40) (3.41)

s=1 s s · Fk , vks = Li+1/2 · Uk , and Rs and Ls are the s-th right and left where gks = Li+1/2 eigenvectors, respectively. Secondly, the fluxes gks are split into positive and negative parts by Lax-Friedrichs flux splitting,  1 s gk ± α s vks . (3.42) gk s,± = 2

There are several ways to obtain α s . For example, one can use the global LaxFriedrichs flux splitting,



α s = max λls

l

(3.43)

1,± 2,± where the maximum is taken over the entire range of l. Given g± k = gk , gk , · · · ,  gks,± , the WENO reconstruction is performed on each component of g± k to obtain w,± . the corresponding numerical flux gi+1/2 w as We finally obtain the numerical flux Fi+1/2 w,+ w,− w Fi+1/2 = Ri+1/2 (gi+1/2 + gi+1/2 ).

(3.44)

The scheme can be applied to two-dimensional (2D) and three-dimensional (3D) problems in a simple way: approximating the spatial derivative dimension by dimension. The dissipative and diffusive terms on the right hand side of Eq. 3.10 are discretized by the explicit sixth-order central finite difference scheme.

3.2.1.4

Time Integration

For an ordinary differential equation, du = R(u), dt

(3.45)

an important family of explicit time integration techniques is Runge-Kutta method, which combines information at multiple stages, such that achieving a high-order approximation of u n+1 at time (n + 1)Δt. A detailed description of the Runge-Kutta

3.2 Numerical Method

45

method can be found in [4, 5, 9, 13, 15]. We apply the third-order Runge-Kutta method, u (1) = u n + Δt R(u n ), 3 1 u (2) = u n + u (1) + 4 4 1 n 2 (2) n+1 u = u + u + 3 3

1 Δt R(u (1) ), 4 2 Δt R(u (2) ). 3

(3.46)

The time step Δt is given by Δt = CFL ×



max |u| + c xf

h



, y + max |v| + c f + max |w| + c zf

(3.47)

y

where c xf , c f , c zf are the fast wave speeds in the x, y and z directions, respectively, and the maxima are taken over all computational grids. The dimensionless quantity, Courant-Friedrichs-Lewy (CFL) number, can be regarded as the ratio of two speeds, namely the possible fastest wave propagation speed and the grid speed h/Δt defined by the discretization of the domain. The stability condition of the time-integration scheme makes a restriction on the CFL number (or Δt). We set the CFL number to be 0.4 for all runs.

3.2.2 Pentadiagonal Filter Since the central finite difference scheme is non-dissipative, high-frequency modes could grow due to nonlinear effects. Meanwhile, the transition between compact scheme and WENO scheme might also generate artificial oscillations at the interface between shock and smooth regions. Filtering the numerical solution is commonly used to cure nonlinear instabilities of central schemes, while retaining high-order accuracy. A good filter should provide adequate dissipation at high wavenumber range to eliminate high-frequency oscillations and at the same time, reserve the well-resolved solution at low wavenumber range.

3.2.2.1

Derivation of the Filter Formula

In view of the fact that compact formulas have greater accuracy over their noncompact counterparts, Gaitonde et al. [3] derived a series of tridiagonal filters, α f f˜i−1 + f˜i + α f f˜i+1 =

N  an n=0

2

( f i+n + f i−n ) ,

(3.48)

46

3 Hybrid Scheme for Compressible MHD Turbulence

where f˜ is the filtered value of f and α f is a free parameter in the range 0 < α f < 0.5. Under the restrictions that the highest frequency mode must be eliminated and the filter is designed to be certain order of accuracy, the coefficients a0 , a1 , · · · a N are obtained. For example, the coefficients of the eighth-order tridiagonal filter are a0 =

93 + 70α f 7 + 18α f −7 + 14α f 1 − 2α f −1 + 2α f , a1 = , a2 = , a3 = , a4 = . 128 16 32 16 128

The damping effect decreases with increase in order of accuracy and α f . Following the same guideline of the sixth-order pentadiagonal filter in [8], we derive an eighth-order pentadiagonal filter. The general formula can be written as α2 f˜i−2 + α1 f˜i−1 + f˜i + α1 f˜i+1 + α2 f˜i+2 = a f i + b ( f i+1 + f i−1 ) + c ( f i+2 + f i−2 ) + d ( f i+3 + f i−3 ) + e ( f i+4 + f i−4 ) , (3.49) where the unknowns α1 , α2 , a, b, c, d, e can be derived by matching the Taylor series coefficients of various orders and other constraints. The first unmatched Taylor series coefficients of the left and right sides expanded about the i-th grid point determines the formal truncation error of the filter, i.e., the order of accuracy. For the eighth-order filter, the second-order error has to vanish, which yields 1 + 2 (α1 + α2 ) = a + 2b + 2c + 2d + 2e.

(3.50)

On plugging the above equation into Eq. 3.49, we have α2 Δ f˜i−2 + α1 Δ f˜i−1 + Δ f˜i + α1 Δ f˜i+1 + α2 Δ f˜i+2 =

4 

an ( f i+n − 2 f i + f i−n ) ,

n=1

(3.51) where a1 = b − α1 , a2 = c − α2 , a3 = d, a4 = e, and Δ f˜i = f˜i − f i . The remaining equations arising from the matching procedure are a1 + 22 a2 + 32 a3 + 42 a4 = 0, a1 + 24 a2 + 34 a3 + 44 a4 = 0,

(3.52)

a1 + 2 a2 + 3 a3 + 4 a4 = 0. 6

6

6

Apart from the accuracy, three additional constraints lie in the performance of the filter at high wavenumber range. The Fourier analysis provides an effective way to quantify the resolution characteristics. Let us consider the problem with a domain [0, L] equally discretized with N0 grid points and h = L/N0 . The variable f is assumed to be periodic over the domain, i.e., f 1 = f N0 +1 . Its Fourier coefficient is fˆ(k) =

N 0 −1  j=0

 N  0 −1   2πikx j = f (x j )exp − f (x j )exp −iωs j , L j=0

(3.53)

3.2 Numerical Method

47

√ where i = −1, ω = 2π k/N0 , and s j = x j / h. Equation 3.51 can be transformed into the Fourier space, 

4 





 1 + α1 eiω + e−iω + α2 e2iω + e−2iω = fˆ(k) an e−niω − 2 + eniω .

fˆ˜(k) − fˆ(k)

n=1

(3.54) The corresponding spectral transfer function is  2 4n=1 an [cos(nω) − 1] fˆ˜(k) . T (ω) = =1+ 1 + 2α1 cos(ω) + 2α2 cos(2ω) fˆ(k)

(3.55)

When T (ω) = 1, the filter has no impact on the k-th wave mode. Otherwise, when T (ω) = 0, all information of the k-th wave mode is eliminated. To ensure the performance of the filter at high wavenumber range, the constraints T (π ) = 0, dT /dω(π ) = 0 (this constraint is automatically satisfied), and d2 T /dω2 (π ) = 0 (as used by Lele [10]) are also imposed, resulting in 22 a1 + 22 a3 = 1 − 2(α1 − α2 ), a1 − 2 a2 + 3 a3 − 4 a4 = −(α1 − 4α2 ). 2

2

2

(3.56) (3.57)

The crux is to define a cut-off wavenumber kc , which provides some control on the “degree” of filtering, just like the role of α f in the tridiagonal filters. We define T (kc ) = R (0 < kc < π and 0 < R < 1), and the corresponding constraint is 2

4 

an [cos (nkc ) − 1] = (R − 1)(1 + 2α1 cos kc + 2α2 cos 2kc ).

(3.58)

n=1

Solving Eqs. 3.52, 3.56, 3.57 and 3.58, we obtain 2 + 256a4 1 − 256a4 , α2 = , 3 6 a1 = −56a4 , a2 = 28a4 , 3 + 4 cos kc + cos 2kc , a3 = −8a4 , a4 = A (kc ) α1 =

(3.59)

where A (kc ) =

− (512R − 176) cos kc + (256R − 88) cos 2kc − 48 cos 3kc + 6 cos 4kc + 210 . R−1

48

3 Hybrid Scheme for Compressible MHD Turbulence

Fig. 3.2 Filtering effect variation with kc for the eighth-order pentadiagonal filter

1.2 1

T(ω)

0.8

kc=0.60π 0.6

kc=0.70π

0.4

kc=0.75π kc=0.80π

0.2

kc=0.85π kc=0.90π

0

0

0.5

1

1.5

2

2.5

3

ω

3.2.2.2

Fourier Analysis

In this work, we set R = 0.5. The cut-off wavenumber kc is therefore the only parameter yet to be defined. An effective wavenumber is defined as ωeff = sup{ω| |T (ω) − 1| < ε}, where ε is small error tolerance. As mentioned at the onset, the larger ωeff and at the same time, at ω → π , T (ω) → 0, the better performance of the filter. The spectral transfer functions of the eighth-order pentadiagonal filters with different kc are shown in Fig. 3.2. One can see that the damping effect decreases with increase in kc , while the effective wavenumber ωeff increases with increase in kc . Comparison is also made with other filters, such as the eighth-order tridiagonal filter of Gaitonde [3] at α f = 0.49 and the fourth-order pendidiagonal filters of Lele [10], as shown in Fig. 3.3. The newly derived pentadiagonal filter at kc = 0.9π and R = 0.5 has better performance. Therefore, for all applications in this book, we will use this eighth-order pentadiagonal filter. In fact, results presented here are not sensitive to the small variations of kc and R, which will be discussed in Sect. 3.3.3.

3.2.3 Divergence-Free Constraint of the Magnetic Field We adopt the unstaggered central difference type constrained transport (CT) method [14] to prevent the divergence of the magnetic field from increasing too fast. Meanwhile, the projection method [1, 14] is also applied to the magnetic field to eliminate additional numerical errors that might lead to the violation of the divergence-free condition, such as those introduced by the filtering process. Note that the divergencecleaning steps are required in 2D and 3D problems, while they are not necessary in 1D since Bx ≡ const is automatically satisfied.

3.2 Numerical Method

49

1.2 1

T(ω)

0.8 0.6 0.4

Lele1992(1) Lele1992(2) Gaitonde1999 Pentadiagonal

0.2 0

0

0.5

1

1.5

ω

2

2.5

3

Fig. 3.3 Spectral transfer functions of different filters. Dashed line: fourth-order filter of Lele [10] at T (1.5) = 0.95 and T (2.0) = 0.5. Dash-dotted line: fourth-order filter of Lele [10] at T (2.0) = 0.95 and T (2.5) = 0.5. Dash-dot-dotted line: eighth-order tridiagonal filter of Gaitonde [3] at α f = 0.49. Solid line: newly derived eighth-order pentadiagonal filter at kc = 0.9π and R = 0.5 [18]. Reprinted from Journal of Computational Physics, 306, Yan Yang et al., A hybrid scheme for compressible magnetohydrodynamic turbulence, 73–91, Copyright (2015), with the permission from Elsevier

3.2.3.1

Constrained Transport Method

The magnetic induction equation can be cast in the form, ∂B + ∇ × ∗ = 0, ∂t

(3.60)

where ∗ =  for the ideal MHD system, and ∗ =  + ηj/Rem for the viscous MHD system. We could select for instance, a central difference formula for the discretization of the spatial derivative at the (i, j, k)-th point. This leads to the semidiscrete form,  ∗  ∗ ∗ ∗ Ωz,i, j+1,k − Ωz,i, Ω y,i, dBx,i, j,k j−1,k j,k−1 − Ω y,i, j,k+1 =− + , dt 2Δy 2Δz  ∗  ∗ ∗ ∗ Ωz,i−1, j,k − Ωz,i+1, Ωx,i, dB y,i, j,k j,k j,k+1 − Ωx,i, j,k−1 =− + , dt 2Δx 2Δz  ∗  ∗ ∗ ∗ Ω y,i+1, j,k − Ω y,i−1, Ωx,i, dBz,i, j,k j,k j−1,k − Ωx,i, j+1,k =− + . dt 2Δx 2Δy

(3.61)

As noted earlier (see Eq. 3.9), there are symmetries among the flux components. Therefore, i, j,k can be expressed by

50

3 Hybrid Scheme for Compressible MHD Turbulence

Fig. 3.4 The central difference type CT method

 1 G 6,i, j,k+1/2 + G 6,i, j,k−1/2 − F7,i, j+1/2,k − F7,i, j−1/2,k , 4  1 Ω y,i, j,k = E 7,i+1/2, j,k + E 7,i−1/2, j,k − G 5,i, j,k+1/2 − G 5,i, j,k−1/2 , (3.62) 4  1 Ωz,i, j,k = F5,i, j+1/2,k + F5,i, j−1/2,k − E 6,i+1/2, j,k − E 6,i−1/2, j,k . 4 Ωx,i, j,k =

The term ηj/Rem is constructed from the viscous flux terms in analogy with Eq. 3.62. The procedure of the central difference type CT method can be easily read from Fig. 3.4. Consequently, if ∇ · B is discretized as (∇ · B)i, j,k =

Bx,i+1, j,k − Bx,i−1, j,k B y,i, j+1,k − B y,i, j−1,k Bz,i, j,k+1 − Bz,i, j,k−1 + + , 2Δx 2Δy 2Δz

(3.63) it is compatible with Eq. 3.61 that gives rise to d (∇ · B)i, j,k = 0. dt

(3.64)

Therefore, the divergence-free constraint of the magnetic field will be maintained in this discretization as long as ∇ · B = 0 initially.

3.2 Numerical Method

3.2.3.2

51

Projection Method

Projection method introduces correction to the magnetic field to prevent divergence errors due to the filter step from increasing. The critical step in the projection method is projecting the magnetic field onto its solenoidal part. The filtered magnetic field ˜ It is decomposed as is denoted by B. B˜ = ∇φ + ∇ × ψ,

(3.65)

where φ and ψ are the scalar and vector potentials of the magnetic field, respectively. Taking the divergence of both sides, we obtain ˜ ∇ 2 φ = ∇ · B.

(3.66)

Then the magnetic field is corrected by Bc = B˜ − ∇φ.

(3.67)

The scalar potential φ is obtained by solving the Poisson equation Eq. 3.66. We can discretize the Poisson equation as φi, j+2,k − 2φi, j,k + φi, j−2,k φi, j,k+2 − 2φi, j,k + φi, j,k−2 φi+2, j,k − 2φi, j,k + φi−2, j,k + + 4Δx 2 4Δy 2 4Δz 2 ˜ ˜ ˜ ˜ ˜ ˜ Bx,i+1, j,k − Bx,i−1, j,k B y,i, j+1,k − B y,i, j−1,k Bz,i, j,k+1 − Bz,i, j,k−1 = (3.68) + + , 2Δx 2Δy 2Δz

and the correction step is discretized as φi+1, j,k − φi−1, j,k c ˜ , Bx,i, j,k = Bx,i, j,k − 2Δx φi, j+1,k − φi, j−1,k c ˜ B y,i, , j,k = B y,i, j,k − 2Δy φi, j,k+1 − φi, j,k−1 c ˜ Bz,i, . j,k = Bz,i, j,k − 2Δz

(3.69)

One can verify that the corrected magnetic field based on Eqs. 3.68 and 3.69 has zero divergence in the discretization as Eq. 3.63. In order to solve for the unknowns in Eq. 3.68, we apply the successive overrelaxation iterative method, wherein the overrelaxation coefficient is 1.49. If we indicate ˜ i, j,k the sum of the by φi,(n)j,k the approximation to φi, j,k at iteration n and (∇ · B) (n+1) right-hand side terms in Eq. 3.68, φi, j,k is obtained in the following steps: 1. n → n + 1/3,

52

3 Hybrid Scheme for Compressible MHD Turbulence (n+1/3)

(n+1/3)

(n+1/3)

φi+2, j,k − 2φi, j,k

+ φi−2, j,k

4Δx 2 (n+1/3)

+

φi,(n)j,k+2 − 2φi, j,k

(n+1/3)

+

+ φi,(n)j,k−2

φi,(n)j+2,k − 2φi, j,k

+ φi,(n)j−2,k

4Δy 2

˜ i, j,k . = (∇ · B)

4Δz 2

(3.70)

2. n + 1/3 → n + 2/3, (n+1/3)

(n+2/3)

φi+2, j,k − 2φi, j,k

(n+1/3)

+ φi−2, j,k

4Δx 2 (n+1/3)

+

(n+2/3)

φi, j,k+2 − 2φi, j,k

(n+2/3)

+

(n+1/3)

+ φi, j,k−2

4Δz 2

(n+2/3)

φi, j+2,k − 2φi, j,k

(n+2/3)

+ φi, j−2,k

4Δy 2 ˜ i, j,k . = (∇ · B)

(3.71)

3. n + 2/3 → n + 1, (n+2/3)

(n+2/3)

φi+2, j,k − 2φi,(n+1) j,k + φi−2, j,k 4Δx 2 +

(n+2/3)

+

(n+1) (n+1) φi,(n+1) j,k+2 − 2φi, j,k + φi, j,k−2

4Δz 2

(n+2/3)

φi, j+2,k − 2φi,(n+1) j,k + φi, j−2,k 4Δy 2

˜ i, j,k . = (∇ · B)

(3.72)

The above equations are pentadiagonal systems. How to solve the pentadiagonal systems is one of the key issues in numerical mathematics. There are various serial algorithms. However, they will be extremely time-consuming when used to solve massive pentadiagonal systems. Therefore, we apply a parallel pentadiagonal algorithm (see Appendix C).

3.3 Numerical Results We present in this section the results of the hybrid scheme for several 1D and 2D test problems. Its application to 3D MHD turbulence is presented as well.

3.3.1 Accuracy Analysis of the Hybrid Scheme 3.3.1.1

Alfvén Wave Problem

The Alfvén wave problem is given by (ρ, u, v, w, Bx , B y , Bz , p) =

1 (10, 0, sin(2π x), cos(2π x), 10, sin(2π x), cos(2π x), 1) , 10

(3.73)

3.3 Numerical Results

53

with γ = 5/3. The solution consists of a sinusoidal wave propagating at constant speed without changing shape, and the wave propagation is sensitive to the numerical dissipation and dispersion errors, thus making it a prime candidate to verify order of accuracy. It is also of particular relevance to astrophysical phenomena because the propagation of Alfvén waves in the solar wind is thought to be a possible source for the heating of the solar corona. The Alfvén wave propagates at the Alfvén speed,  ca = Bx2 /ρ = 1, thus by the time t = 1 the wave convects the distance of a unit wave length and returns to its initial state. The computational domain is 0  x  1 with N grid points. Periodic boundary condition is used. No filters are used in this case. The solutions by the sixth-order compact central finite difference scheme, hybrid compact-WENO scheme and fifth-order WENO scheme at t = 5 are shown in Fig. 3.5. All the results simulated by the compact scheme and the hybrid scheme are in excellent agreement with the exact solution, and there are no visible numerical (a) N=8

(b) N=16

0.05

0.05

By

0.1

By

0.1

0 Exact solution Compact WENO Hybrid

-0.05 -0.1 0

0.2

0.4

-0.05

0.6

x

0

0.8

-0.1 0

1

0.2

0.4

x

0.6

0.8

1

(c) N=32 0.1

By

0.05 0

-0.05 -0.1 0

0.2

0.4

x

0.6

0.8

1

Fig. 3.5 Plots of B y for Alfvén wave problem with varying grid points at t = 5 [18]. Red solid lines: exact initial wave; green dash-dotted lines with squares: the solution by the 6th-order compact central finite difference scheme; grey dash-dotted lines with circles: the solution by the 5th-order WENO scheme; blue dash-dotted lines with triangles: the solution by the hybrid scheme. a N = 8; b N = 16; c N = 32. Reprinted from Journal of Computational Physics, 306, Yan Yang et al., A hybrid scheme for compressible magnetohydrodynamic turbulence, 73–91, Copyright (2015), with the permission from Elsevier

54

3 Hybrid Scheme for Compressible MHD Turbulence

Table 3.1 Accuracy of the hybrid scheme for Alfvén wave problem at t = 5 [18]. Reprinted from Journal of Computational Physics, 306, Yan Yang et al., A hybrid scheme for compressible magnetohydrodynamic turbulence, 73–91, Copyright (2015), with the permission from Elsevier N L 1 error L 1 order L ∞ error L ∞ order 8 16 32 64

2.28e-04 3.51e-06 5.47e-08 8.67e-10

– 6.02 6.01 5.98

3.78e-04 5.59e-06 8.61e-08 1.36e-09

– 6.08 6.02 5.98

dissipation and dispersion errors. However, the result of the WENO scheme with N = 8 is far from the exact solution, which is improved on finer grids as shown in Fig. 3.5b, c. These results confirm that the hybrid method automatically turn off the WENO scheme for the smooth flow, thus preserving the non-dissipative and highresolution characteristics of the compact central finite difference scheme in smooth regions. In order to measure errors of the hybrid scheme, we compute L 1 and L ∞ norms of B y and the corresponding convergence rates, as shown in Table 3.1. N L 1 = N1 i=1 |u i − u 0 (xi )| and L ∞ = max{|u i − u 0 (xi )|, i = 1, · · · , N }, where u i and u 0 are numerical and exact solutions, respectively. The hybrid scheme has asymptotic 6th order of accuracy, even on very coarse grids.

3.3.1.2

Intermediate Shock Waves

The initial conditions of intermediate shock wave problem are Bx = 1, B y = 0.5 sin 2π x, Bz = 0, w = 0, and other variables ρ, u, v, p are obtained by solving the relation between dependent variables and B y numerically, ⎛

⎞ ρ ⎟ d ⎜ 1 ⎜ ρu ⎟ = ⎝ dB y ρv ⎠ cs2 − a 2 ρp



⎞ By ⎜ (u + cs )B y ⎟ ⎜ 2 ⎟ ⎝ a Bx /cs − Bx cs + v B y ⎠ , a2 By

(3.74)

where cs is the slow wave speed and a is the sound speed. Nonlinear steepening builds intermediate shocks from these smooth initial conditions. The computational domain is [0, 1] with a periodic boundary condition, and γ = 5/3 is used. Here the solution by the WENO scheme on 2560 grid points is taken as a reference. Figure 3.6 shows the time evolution of B y solved with the hybrid method on a coarse grid points of N = 160. These results are in excellent agreement with the reference. At t = 1, the intermediate shocks occur at x ∼ 0.493 and x ∼ 0.993. We should notice that across the intermediate shock waves, the transverse component of the magnetic field changes its sign, which is a property of intermediate shock.

3.3 Numerical Results

55

(b) t=0.50

(a) t=0.25 0.6

0.6 N=2560

0.4

0.4

N=160

0.2

By

By

0.2 0

0

-0.2

-0.2

-0.4

-0.4

-0.6 0

0.2

0.4

x

0.6

0.8

-0.6 0

1

0.6

0.4

0.4

0.2

0.2

0

-0.2

-0.4

-0.4 0.2

x

0.6

0.8

1

0.6

0.8

1

0

-0.2

-0.6 0

0.4

(d) t=1.0

0.6

By

By

(c) t=0.60

0.2

0.4

x

0.6

0.8

1

-0.6 0

0.2

0.4

x

Fig. 3.6 Plots of B y at t = 0.25, 0.5, 0.6, 1 to show the formation of intermediate shocks [18]. The profiles by the WENO scheme on 2560 grid points are for reference. Reprinted from Journal of Computational Physics, 306, Yan Yang et al., A hybrid scheme for compressible magnetohydrodynamic turbulence, 73–91, Copyright (2015), with the permission from Elsevier

We also calculate B y deviations away from the reference. Since a pair of shocks begin to be formed at t ∼ 0.54, two time snapshots, i.e., t = 0.25 before the shock formation and t = 0.6 after the shock formation, are considered. The L 1 errors and L ∞ errors at t = 0.25 are shown in Table 3.2. As grid is refined, the order can reach about 6. In Table 3.3, we show the L 1 errors at t = 0.6. The accuracy of the hybrid scheme reduces to 1st order due to discontinuities, while in the smooth region 0.3 ≤ x ≤ 0.4, high order of accuracy is recovered. This demonstrates that the hybrid scheme can reserve both the shock capturing capability and good resolution in smooth regions simultaneously.

3.3.2 1D and 2D Numerical Tests 3.3.2.1

Magnetic Sod Shock-Tube Problem

Magnetic Sod shock-tube problem is a standard test to demonstrate the capability of a numerical MHD scheme to handle discontinuities. The problem is advanced up

56

3 Hybrid Scheme for Compressible MHD Turbulence

Table 3.2 Accuracy of the hybrid scheme for intermediate shock wave problem at t = 0.25 [18]. Reprinted from Journal of Computational Physics, 306, Yan Yang et al., A hybrid scheme for compressible magnetohydrodynamic turbulence, 73–91, Copyright (2015), with the permission from Elsevier N L 1 error L 1 order L ∞ error L ∞ order 20 40 80 160 320

1.44e-03 6.64e-05 8.24e-07 1.01e-08 2.12e-10

– 4.44 6.33 6.35 5.58

4.08e-03 4.28e-04 1.07e-05 1.41e-07 2.11e-09

– 3.26 5.32 6.24 6.00

Table 3.3 Accuracy of the hybrid scheme for intermediate shock wave problem at t = 0.6 [18]. Reprinted from Journal of Computational Physics, 306, Yan Yang et al., A hybrid scheme for compressible magnetohydrodynamic turbulence, 73–91, Copyright (2015), with the permission from Elsevier N L 1 error L 1 order L ∗1 errora L ∗1 order 20 40 80 160 320 a L∗ 1

3.85E-02 1.54E-02 7.03E-03 3.36E-03 1.80E-03

– 1.32 1.13 1.06 0.90

5.45E-03 1.05E-03 1.30E-04 1.38E-05 5.69E-07

– 2.37 3.02 3.23 4.60

covers only 0.3 ≤ x ≤ 0.4

to t = 0.1 on the computational domain x ∈ [0, 1]. γ is taken to be 5/3. The initial condition is given by  (1.000, 0, 0, 0, 0.75, +1.0, 0, 1.0) for x < 0.5, (ρ, u, v, w, Bx , B y , Bz , p) = (0.125, 0, 0, 0, 0.75, −1.0, 0, 0.1) for x > 0.5. (3.75) The solution at t = 0.1 on 400 grid points is shown in Fig. 3.7. The solution obtained with the WENO scheme on 4000 grid points is taken as a reference, which includes a left fast rarefaction wave, a left intermediate shock, a left slow rarefaction wave, a right contact discontinuity, a right slow shock, and a right fast rarefaction wave. The hybrid method robustly captures discontinuities without oscillations, whose performance is as good as the WENO scheme. The velocity profile shows a small undershoot error on the trailing end of the right fast rarefaction wave, which may arise from the initial start-up error.

3.3.2.2

Orszag-Tang Problem

Orszag-Tang problem is widely used in the literature and serves as a good verification test where nonlinear steepening builds strong discontinuities from smooth

3.3 Numerical Results

57 1.2

1+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++ ++ ++ ++ ++ ++ ++ ++ ++ ++ + +++++++ +++++++++ +++ +++ + +

ρ

0.6

0 0

0.6

+

0.2

0.4

0.4

+ +++++++++++ +

Ref. Hybrid WENO

0.2

++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

x

0.6

0.8

0 0

1

+++++++ + ++ + ++++++++++++++++++++++++++ + + 0.6 + + ++ + + ++ + + + 0.4 + + + + + + 0.2 + + + + + 0+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ +

0.8 0.4

0.2

0.4

x

0.6

0.8

1

+++++++++++++++++++++++++++ ++ ++ ++ + ++++++++++++++++++++++++++++++++++++++ +

-0.2 0.2

0.4

x

0.6

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ +++ +++ +++ +++ +++ +++++++++ ++ +

By

u

+ +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

1.2

0.8

-0.4 0

++ ++ ++ ++ ++ ++ ++ + ++ ++ ++ ++ + +++++++++++++++++++++++++++ ++++++++ +

0.8

+

0.4 0.2

1++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

p

0.8

0.8

1

0 + + ++ +++++++++++++++++++++++++++ +

-0.4 -0.8 -1.2 0

++++++++++++++++++++++++++++++++++++++++ +++++ +++++++++++++++++++++++++++++

0.2

0.4

x

0.6

0.8

1

Fig. 3.7 Profiles of density, pressure, velocity, and magnetic field for magnetic Sod shock-tube problem at t = 0.1 [18]. Black solid lines: reference by the WENO scheme on 4000 grid points; blue circles: results by the hybrid scheme on 400 grid points; red crosses: results by the WENO scheme on 400 grid points. Reprinted from Journal of Computational Physics, 306, Yan Yang et al., A hybrid scheme for compressible magnetohydrodynamic turbulence, 73–91, Copyright (2015), with the permission from Elsevier

initial conditions in 2D MHD system and these discontinuities interact with coherent structures. The initial condition is given by (ρ, u, v, w, Bx , B y , Bz , p) = (γ 2 , − sin y, sin x, 0, − sin y, sin 2x, 0, γ ). (3.76) γ = 5/3 is used. The computational domain is [0, 2π ] × [0, 2π ]. We perform the simulation with the hybrid scheme on 200 × 200 zone meshes. Periodic boundary conditions are imposed in both x and y directions. As time advances, the discontinuities are developed from the initial smooth condition. Figure 3.8 shows the contours of density, pressure, velocity magnitude and magnitude of magnetic field at t = 3.14, which well agree with those presented in previous studies [6, 7, 12]. In order to figure out the performance of the scheme, we take a slice of the density field along y = 0.5π in Fig. 3.9. Also shown are the result by the WENO scheme on 200 × 200 zone meshes and the reference by the WENO scheme on 512 × 512 zone meshes. Although the hybrid scheme causes a lit-

58

3 Hybrid Scheme for Compressible MHD Turbulence

(a) ρ

1.10 1.64 2.18 2.71 3.25 3.79 4.33 4.87 5.40 5.94

(b) p

0.30 0.93 1.56 2.19 2.82 3.46 4.09 4.72 5.35 5.98

(c) |u|

0.00 0.17 0.35 0.52 0.69 0.86 1.04 1.21 1.38 1.55

(d) |B|

0.00 0.31 0.61 0.92 1.23 1.53 1.84 2.14 2.45 2.76

Fig. 3.8 Contours of density, pressure, magnitude of velocity and magnitude of magnetic field for Orszag-Tang problem on 200 × 200 zone meshes at t = 3.14 [18]. Reprinted from Journal of Computational Physics, 306, Yan Yang et al., A hybrid scheme for compressible magnetohydrodynamic turbulence, 73–91, Copyright (2015), with the permission from Elsevier

tle smearing around discontinuities (e.g. at x ∼ 0.5 and x ∼ 0.8), it gets an accurate solution in other regions.

3.3.2.3

Rotor Problem

Rotor problem consists of the propagation of strong Alfvén waves. The resultant flow field involves a large pressure jump with a very low-pressure region around the

3.3 Numerical Results

59

5 +

Ref. Hybrid WENO

++ ++++++ ++ + ++ + + ++ ++++ ++ ++++++++++++ ++++ + + ++++ + + + + + + + + + + + +++++++++ +++ + + 3 ++ + ++ + ++ + + +++++ ++ + + ++ + +++ +++ + + + + + ++++++++ + + + + +++++ ++++ + + + + ++ ++++++++++++ + + + 2 ++ + + + + + + + + ++ ++ + +++++ +++++++++ +++ + ++ +++++++ ++++++++ +++++

ρ

4

1

0

1

2

3

4

5

6

x Fig. 3.9 Profile of density for Orszag-Tang problem along y = 0.5π at t = 3.14 [18]. Black solid lines: reference by the WENO scheme on 512 × 512 zone meshes; blue circles: results by the hybrid scheme on 200 × 200 zone meshes; red crosses: results by the WENO scheme on 200 × 200 zone meshes. Reprinted from Journal of Computational Physics, 306, Yan Yang et al., A hybrid scheme for compressible magnetohydrodynamic turbulence, 73–91, Copyright (2015), with the permission from Elsevier

center of the rotor, and thus, the robustness of the hybrid method can be assessed through this problem. It is initialized by a dense rotating disk at the center, ⎧  y − 0.5 x − 0.5 ⎪ ⎪ , u0 for r  r0 , ⎪ 10, − u 0 ⎪ ⎨ r0 r0  (y − 0.5) f (r ) (x − 0.5) f (r ) (ρ, u, v) = 1 + 9 f (r ), − u 0 for r0 < r < r1 , , u0 ⎪ ⎪ ⎪ r r ⎪ ⎩ for r  r1 , (1, 0, 0)

(3.77)

 where u 0 = 2, r0 = 0.1, r1 = 0.115, r = (x − 0.5)2 + (y − 0.5)2 √ and f (r ) = (r1 − r )/(r1 − r0 ). Other variables are initialized by u z = 0, Bx = 5/ 4π , B y = Bz = 0 and p = 1. γ = 1.4 is used. The computational domain is [0, 1] × [0, 1]. We perform the simulation with the hybrid scheme on 200 × 200 zone meshes. Periodic boundary conditions are imposed in both x and y directions. Shown in Fig. 3.10 are the contours of density, thermal pressure, magnetic pressure and Mach number at t = 0.15. The rapidly spinning rotor causes Alfvén waves to be launched into the ambient fluid. Within the rotor the fluid is still in uniform rotation out to a certain radial distance, but beyond that distance the angular momentum of the rotor is diminished because it exchanges angular momentum with the ambient fluid. Figure 3.10b presents a large pressure jump with very low pressure at the center. Because of the large pressure jump, many conventional algorithms fail owing to negative pressures [12, 14]. The hybrid method robustly simulates the propagation of strong Alfvén waves and the large pressure jump without the negative pressure issue. The rapid buildup of magnetic pressure around the rotor’s boundary (see Fig. 3.10c)

60

3 Hybrid Scheme for Compressible MHD Turbulence

(a) ρ

(b) pgas 1.85

10.19 9.12 8.04

1.64 1.44 1.24

6.97 5.90

1.04

4.83

0.84

3.76

0.64 0.43

2.69

0.23 0.03

1.62 0.55

(c) pmag

(d) Ma

2.35

3.17

2.09 1.84 1.58

2.81 2.46 2.11

1.33 1.08

1.76 1.41

0.82

1.06

0.57 0.31

0.70 0.35

0.06

0.00

Fig. 3.10 Contours of density, pressure, magnetic pressure and Mach number for rotor problem on 200 × 200 zone meshes at t = 0.15 [18]. Reprinted from Journal of Computational Physics, 306, Yan Yang et al., A hybrid scheme for compressible magnetohydrodynamic turbulence, 73–91, Copyright (2015), with the permission from Elsevier

compresses the fluid in the rotor, giving it an oblong shape. Meanwhile, the hybrid method successfully maintains an uniform distribution of Mach number near the center (see Fig. 3.10d), which indicates that this region hasn’t been affected by the rotation. The local Mach number can reach about 4.

3.3.2.4

Blast Wave Problem

Blast wave problem, instead of a density jump as rotor problem, consists of a large pressure jump. This problem leads to the formation and propagation of strong MHD discontinuities relevant to astrophysical phenomena. It is initialized by a high pressure at the center,

3.3 Numerical Results

61

⎧ for r  r0 , ⎨1000 p = 10 + 990 f (r ) for r0 < r < r1 , ⎩ 10 for r  r1 ,

(3.78)

 where r0 = 0.1, r1 = 0.115, r = (x − 0.5)2 + (y − 0.5)2 and √ f (r ) = (r1 − r )/ (r1 − r0 ). Other variables are initialized by ρ = 1, Bx = 100/ 4π , B y = Bz = 0 and u = v = w = 0. γ = 1.4 is used. The computational domain is [0, 1] × [0, 1]. We perform the simulation with the hybrid scheme on 200 × 200 zone meshes. Periodic boundary conditions are imposed in both x and y directions. Shown in Fig. 3.11 are the contours of density, thermal pressure, velocity magnitude and magnetic pressure at t ∼ 0.01. One can see an anisotropic explosion behavior because of the existence of the non-zero magnetic field strength in x direction. When Bx = 0, an isotropic explosion is formed. For the strong magnetic field (a) ρ

(b) p gas

3.74 3.34 2.95 2.56 2.17 1.77

251.77 224.43 197.09 169.75 142.41 115.07

1.38 0.99 0.59 0.20

87.72 60.38 33.04 5.70

(c) |u|

(d) p mag

14.90 13.24 11.59 9.93 8.28 6.62

570.83

4.97 3.31 1.66 0.00

326.28 285.52 244.76 204.00

530.07 489.31 448.55 407.79 367.03

Fig. 3.11 Contours of density, pressure, velocity magnitude and magnetic pressure for blast wave problem on 200 × 200 zone meshes at t ∼ 0.01

62

3 Hybrid Scheme for Compressible MHD Turbulence

√ case, like Bx = 100/ 4π here, the explosion becomes highly anisotropic. Stronger Bx , higher anisotropy, i.e., the motion in the transverse y direction is increasingly inhibited. An important parameter is defined as β = p/ pmag , which is about 0.025 in this case. For low-β state, many schemes fail due to negative pressure, which, however, does not occur here in the β ∼ 0.02 case solved by the proposed hybrid scheme.

3.3.3 Isotropic MHD Turbulence MHD system is highly nonlinear. A direct consequence of the nonlinearity is being turbulence, which is a dominant element of most of the flows encountered in nature and in technology. The ultimate objective here is to obtain a tractable algorithm that can be used to simulate MHD turbulence, in particular, with strong compression. Here we focus on homogeneous and isotropic MHD turbulence without accommodating any effects of complex geometries, mean flows, boundary conditions, and so on. An external force F is imposed at large scales to maintain the stationary state of the turbulent field. There are coexistence and interaction of two fundamental dynamic processes, i.e., shearing and compressing. The large-scale force is applied to the solenoidal part and the compressive part of the velocity field with the ratio rs /rc , of which a detail description will be given in Sect. 4.1. A useful quantity, especially for qualitative discussions, is the energy spectrum  E(k) =

k

2 ˆ |u(k)| , 2

(3.79)

ˆ where k = |k| is the wavenumber, and u(k) is the Fourier transform of the velocity field. Then E(k)dk represents the contribution to the turbulent kinetic energy from modes in the range k ≤ |k| < k + dk. Two important statistical parameters characterizing the compressible MHD turbulence , i.e., the Taylor-scale Reynolds number Reλ and the turbulent Mach number Mt , are defined as u rms λ ρ Reλ = Re , μ

√ 3u rms Mt = M √ , T

(3.80)

where · · · denotes the ensemble average, the root-mean-square (r.m.s.) velocity u rms is defined as  ! 1/2 2 u · u = u rms = , (3.81) E(k)dk 3 3 and the Taylor scale λ is defined as λ = "

u rms

# . (∂u/∂ x)2 + (∂v/∂ y)2 + (∂w/∂z)2 /3

(3.82)

3.3 Numerical Results

3.3.3.1

63

Nearly Incompressible Limit

In order to validate the accuracy of the hybrid scheme in 3D, we carry out numerical simulations of nearly incompressible MHD turbulence and compare the results with those from pseudo-spectral method. The numerical simulation is done in a [0, 2π ] × [0, 2π ] × [0, 2π ] domain at 1283 resolution with periodic boundary conditions. The large-scale force is only applied to the solenoidal part of the velocity field (i.e., rs = 1 and rc = 0), Re = 120, and M = 0.1. The fields are initialized with Gaussian random phases and fluctuation amplitudes. We carry out our analysis on snapshots after the flow reaches a statistically steady state. The resultant steady-state field has Mt = 0.098, which indicates that the compressibility effect can be neglected and the flow can be considered as incompressible. The energy spectra of the velocity and magnetic field are shown in Fig. 3.12. One can see that the results obtained with the hybrid scheme almost overlap with those from pseudo-spetral method. This validates that the accuracy of the 3D hybrid code is near that of the pseudo-spectral method for flow fields without shocks.

3.3.3.2

Parametric Dependence of the Filter and Shock Sensors

In this section, we shall discuss the effects of the cut-off wavenumber kc in the pentadiagonal filter (see Sect. 3.2.2) and the parameters β and δ in the shock detector (see Sect. 3.2.1). Table 3.4 shows seven cases applied to demonstrate the sensitivity of kc , β and δ, wherein Run1∼Run3 have different kc , Run3∼Run5 have different β

100

E(k)

10

-2

10-4 10

-6

10-8 10

Spectral E(k) Hybrid E(k) Spectral Em(k) Hybrid Em(k)

-10

k

20

40

60

 2 /2 ˆ Fig. 3.12 Energy spectra of the velocity field E(k) and the magnetic field E m (k) = k | B(k)| obtained with hybrid method and pseudo-spectral method [18]. Energy spectra of the magnetic field E m (k) are shifted vertically for better recognition. Reprinted from Journal of Computational Physics, 306, Yan Yang et al., A hybrid scheme for compressible magnetohydrodynamic turbulence, 73–91, Copyright (2015), with the permission from Elsevier

64

3 Hybrid Scheme for Compressible MHD Turbulence

Table 3.4 Description of the simulations to test the effects of kc , β and δ in the filter and shock sensors [18]. Reprinted from Journal of Computational Physics, 306, Yan Yang et al., A hybrid scheme for compressible magnetohydrodynamic turbulence, 73-91, Copyright (2015), with the permission from Elsevier Runs kc β δ Reλ Mt Run1 Run2 Run3 Run4 Run5 Run6 Run7

0.80π 0.85π 0.90π 0.90π 0.90π 0.90π 0.90π

0.25 0.25 0.25 0.15 0.35 0.25 0.25

0.05 0.05 0.05 0.05 0.05 0.03 0.07

86.5 87.4 87.6 89.1 81.5 86.3 83.2

0.61 0.61 0.62 0.61 0.61 0.62 0.62

and Run3, Run6 and Run7 have different δ. The numerical experiments are conducted in a periodic cube of 1283 grid cells in size of [0, 2π ] × [0, 2π ] × [0, 2π ]. The largescale force is only applied to the solenoidal part of the velocity field (i.e., rs = 1 and rc = 0). We initialize the calculations with random Gaussian fields for the velocity and magnetic fields, and Re = 100 and M = 0.3. We take the time average over 3 eddy turnover time to obtain the following statistical properties. Firstly, the variations of Reλ and Mt over different kc are vanishingly small and at the same time, only small deviations are observed for E(k) of different kc in the high wavenumber range, as shown in Fig. 3.13a. The same conclution is drawn as β and δ vary. Therefore, the statistics of MHD turbulence could be insensitive to the small variations of kc , β and δ values.

3.3.3.3

Application to Compressible MHD Turbulence

The MHD system allows discontinuous solutions in certain cases, such as shock waves occuring in compressible flows. To demonstrate the robustness of the hybrid scheme for shock-turbulence interaction problems, we carry out a simulation on 5123 grid points with Re = 500 and Ma = 0.3. The input energy for the compressive part through large-scale force is twice of that for the solenoidal part, i.e. rs /rc = 1/2, such that compressive energy is available to facilitate the formation of shocks. The computational domain is a periodic cube in size of [0, 2π ] × [0, 2π ] × [0, 2π ]. The fields are initialized with Gaussian random phases and fluctuation amplitudes. The resultant steady-state field has Reλ = 150 and Mt = 0.68. Seen in Fig. 3.14, both the kinetic energy of the solenoidal part E s (k) and the magnetic energy E m (k) exhibit a k −5/3 power law [2, 11], which is expainlable in term of the classical Kolmogorov theory. However, the energy spectrum of the compressive part E c (k) fits well with a k −2 power law due to the formation of shocks, which has also been observed in compressible hydrodynamic turbulence [17].

3.3 Numerical Results

65

100

100

10-1

10-1

E(k)k5/3

(b) 101

E(k)k5/3

(a) 101

10-2 10

-3

kc=0.80π (Run1) kc=0.85π (Run2)

10-4

10

0

10

10-3

β=0.15 (Run4) β=0.25 (Run3)

10-4

kc=0.90π (Run3)

10-5

10-2

β=0.35 (Run5)

10-5

1

2

10

10

0

1

2

10

k

10

k

(c) 101

E(k)k5/3

100 10-1 10-2 10-3 10

δ=0.03 (Run6) δ=0.05 (Run3)

-4

δ=0.07 (Run7)

10-5 0 10

1

2

10

10

k

Fig. 3.14 Energy spectra of the solenoidal velocity E s (k), the compressive velocity E c (k) and the magnetic field E m (k) compensated with k 5/3 [18]. Reprinted from Journal of Computational Physics, 306, Yan Yang et al., A hybrid scheme for compressible magnetohydrodynamic turbulence, 73–91, Copyright (2015), with the permission from Elsevier

E(k)k5/3

Fig. 3.13 Kinetic energy spectra compensated with k 5/3 for runs with different kc , β and δ values [18]. Reprinted from Journal of Computational Physics, 306, Yan Yang et al., A hybrid scheme for compressible magnetohydrodynamic turbulence, 73–91, Copyright (2015), with the permission from Elsevier 10

2

10

1

10

0

10

-1

10

-2

10

-3

10

-4

10

-5

10

-6

10

-7

10

k-5/3 k-5/3 -2 k

Es(k) Ec(k) Em(k) 0

1

10

10

k

2

66 5 4

RH

3

pr

Fig. 3.15 Pressure ratio across shocks [18]. Reprinted from Journal of Computational Physics, 306, Yan Yang et al., A hybrid scheme for compressible magnetohydrodynamic turbulence, 73–91, Copyright (2015), with the permission from Elsevier

3 Hybrid Scheme for Compressible MHD Turbulence

2

1

1

2

pr

3

4

5

We also test the satisfaction of the R-H relation (see Eq. 1.20) across shocks in Fig. 3.15. Indeed, by comparing the pressure ratio pr = p2 / p1 of the simulated field with the pressure ratio prRH (see Eq. 1.21) based on the R-H relation, it is obvious that only small derivations from the ideal line pr = prRH are observed.

3.4 Concluding Remarks The hybrid scheme couples sixth-order compact central finite difference scheme with fifth-order WENO scheme. It can achieve high-order accuracy with fewer grid points in smooth regions and at the same time, capture discontinuities robustly. The shock regions marked by shock sensors only occupy a small amount of the whole volume, which guarantees the efficiency. The proposed eighth-order pentadiagonal filter eliminates spurious oscillations effectively yet has little effect on the resolution of the hybrid scheme. The combination of central difference type CT method and projection method maintains the divergence-free constraint of the magnetic field. A variety of test problems verify the capability of the hybrid method. A start-up analysis of the compressible MHD turbulence at moderate Mach number gives us confidence in applying the hybrid scheme to MHD turbulence with larger Reynolds number and Mach number. Statistical properties of compressible MHD turbulence will be examined in more detail in Chap. 4.

References 1. Balsara DS, Kim J (2004) A comparison between divergence-cleaning and staggered-mesh formulations for numerical magnetohydrodynamics. Astrophys J 602:1079–1090 2. Biskamp D (2003) Magnetohydrodynamic turbulence. Cambridge University Press, Cambridge

References

67

3. Gaitonde DV, Visbal MR (1998) High-order schemes for Navier-Stokes equations: algorithm and implementation into FDL3DI. Technical Report AFRL-VA-WP-TR-1998-3060, US Air Force Research Laboratory, Wright-Patterson AFB 4. Gear CW (1971) Numerical initial value problems in ordinary differential equations. Prentice Hall PTR 5. Hirsch C (2007) Numerical computation of internal and external flows: fundamentals of computational fluid dynamics. Butterworth-Heinemann 6. Jiang GS, Wu CC (1999) A high-order WENO finite difference scheme for the equations of ideal magnetohydrodynamics. J Comput Phys 150:561–594 7. Kawai S (2013) Divergence-free-preserving high-order schemes for magnetohydrodynamics: an artificial magnetic resistivity method. J Comput Phys 251:292–318 8. Kim JW (2010) High-order compact filters with variable cut-off wavenumber and stable boundary treatment. Comput Fluids 39:1168–1182 9. Lambert JD (1973) Computational methods in ordinary differential equations. Wiley 10. Lele SK (1992) Compact finite difference schemes with spectral-like resolution. J Comput Phys 103:16–42 11. Müller WC, Biskamp D (2000) Scaling properties of three-dimensional magnetohydrodynmaic turbulence. Phys Rev Lett 84:475 12. Shen Y, Zha G, Huerta MA (2012) E-CUSP scheme for the equations of ideal magnetohydrodynamics with high order WENO scheme. J Comput Phys 231:6233–6247 13. Shu CW, Osher S (1988) Efficient implementation of essentially non-oscillatory shockcapturing schemes. J Comput Phys 77:439–471 14. Tóth G (2000) The ∇ · B = 0 constraint in shock-capturing magnetohydrodynamics codes. J Comput Phys 161:605–652 15. Van Der Houwen PJ (2012) Construction of integration formulas for initial value problems, vol 19. Elsevier 16. Wang J, Wang LP, Xiao Z, Shi Y, Chen S (2010) A hybrid numerical simulation of isotropic compressible turbulence. J Comput Phys 229:5257–5279 17. Wang J, Yang Y, Shi Y, Xiao Z, He XT, Chen S (2013) Cascade of kinetic energy in threedimensional compressible turbulence. Phys Rev Lett 110:214–505 18. Yang Y, Wan M, Shi Y, Yang K, Chen S (2016) A hybrid scheme for compressible magnetohydrodynamic turbulence. J Comput Phys 306:73–91

Chapter 4

Energy Cascade in Compressible MHD Turbulence

The most commonly studied turbulent flows are assumed that the incompressibility holds. As mentioned at the outset, for some cases, the validity of incompressibility might be deemed questionable, but rather, compressibility effects are invoked. In this chapter, we shall be concerned with statistical properties of energy transfer across scales in compressible MHD turbulence, recognizing that this approach is much less developed than related incompressible and hydrodynamic models. The basic ideas are essentially three: 1. forcing mechanisms to achieve a statistically steady state and their effects on flow properties, 2. compressibility effects on scaling laws, 3. compressibility effects on energy transfer.

4.1 Simulation Setup We study the properties of turbulence in a compressible MHD medium. According to Eq. 3.13, the dimensionless equations are: ∂t ρ + ∇ · (ρu) = 0,   ∇ · BB + F, ∂t (ρu) + ∇ · ρuu + pt I − 2 = − Mm Re η∇ 2 B ∂t B + ∇ · (uB − Bu) = , Rem   ∇ · ( · u) ∇ · (κ∇T ) B + ∂t E + ∇ · (E + pt ) u − 2 (u · B) = − Mm Re α ∇ · (B × ηj) + − Λ + F · u, Rem Mm2 © Springer Nature Singapore Pte Ltd. 2019 Y. Yang, Energy Transfer and Dissipation in Plasma Turbulence, Springer Theses, https://doi.org/10.1007/978-981-13-8149-2_4

69

70

4 Energy Cascade in Compressible MHD Turbulence

where I is the unity tensor and a mechanical force F is imposed at large scales to arrive at a stationary state. Part of the kinetic and magnetic energies is dissipated into thermal energy. This energy loss is supplied by the external force, and a cooling function Λ is introduced as well to prevent the thermal energy from increasing sequentially.

4.1.1 Large-Scale Forcing Mechanism ˆ The velocity field is transformed into the Fourier space as u(k), and decomposed into a solenoidal velocity uˆ s (k) and a compressive velocity uˆ c (k) as ˆ u(k) = uˆ s (k) + uˆ c (k),

(4.1)

where uˆ c (k) =

 k  ˆ k · u(k) . k2

2 ˆ /2 at wavenumber k is decomposed into Correspondingly, the kinetic energy |u(k)| two parts as well: 2 ˆ |u(k)| |uˆ s (k)|2 |uˆ c (k)|2 = + . (4.2) 2 2 2

It yields E(k) = E s (k) + E c (k),

(4.3)

where 

2 ˆ |u(k)|

E(k) =

k−0.5|k| , the boundaries of infrared scale bands are (4.21) Δ(n) = (χ )n , for n = 0, 1, 2, . . .. The coefficient χ can be any value > 1. Here χ = 1.25, ensuring enough scale bands (e.g., n in our case can be up to 7). Similarly, for scales < , the boundaries of ultraviolet scale bands are δ (n) = (χ )−n .

(4.22)

¯ Δ(n) the low-pass filtered velocity and magnetic field, respecLet us define u¯ Δ(n) and B tively, with the filtering scale Δ(n) . We then obtain the contribution from all scales > Δ(n) to the flux Π b , Π b,>Δ

(n)

   = −¯ε u¯ Δ(n) , B¯ Δ(n) · ∇ × (B¯ Δ(n) ) ,

(4.23)

  ¯ ¯ where ε¯ u¯ Δ(n) , B¯ Δ (n) = (u¯ Δ(n) × BΔ(n) ) − (u¯ Δ(n) ) × (BΔ(n) ) . The scales > con(n) (n+1) (n = 0, 1, 2, . . .), and the contribution of the band sist of scale bands Δ ,Δ

(n) Δ , Δ(n+1) to the flux Π b is Π b,Δ

(n)

(n)

= Π b,>Δ − Π b,>Δ

(n+1)

.

(4.24)

86

4 Energy Cascade in Compressible MHD Turbulence

Fig. 4.14 Contributions of scale bands to energy transfer fluxes across scale [54]. The left panel is contour maps with their upper left parts representing scale bands larger than and their (n) lower right parts representing scale bands smaller than . (a.1) upper left: Π u,Δ /Π u ; lower (n)

(n)

(n)

δ right: Π u,δ /Π u ; (b.1) upper left: ΛΔ /Λ ; lower right: Λ /Λ ; (c.1) upper left: b,Δ(n)

b,δ (n)

Π /Π b ; lower right: Π /Π b . The right panel is plots at the scale /ηk ∼ 50. (c.1) and (c.2) are reprinted figures with permission from Yan Yang et al., Physical Review E, 93, 061102, 2016. Copyright (2016) by the American Physical Society

Similar manipulation is carried out for scales < . u δ(n) = u − u¯ δ(n) and B δ(n) = B − B¯ δ(n) are the high-pass filtered velocity and magnetic field, respectively, with the filtering scale δ (n) . The contribution from all scales < δ (n) to the flux Π b can be written as:     (n) (4.25) Π b, . Note that uT α  is vanishingly small at large scales, and increases as the filtering scale is approaching small scales, indicating that the conversion between fluid kinetic and thermal energy by uT α  is dominated by the contribution from small scales. In contrast, Λub α  is fairly constant over kinetic scales, and the increases are concentrated at large scales, thus contributions to the conversion between fluid kinetic and electromagnetic energy mainly result from large scales. To quantify the contribution from different scales, a scale-band decomposition, similar to the technique used in Sect. 4.5, is introduced as

5.4 Energy Transfer Channels

107

(a)

(b)

0.0003

0.0003

uT e ub e



0.0000

0.0000

uu

de -0.0003

10

-1

de

di 0

10

-0.0003

1

l(di)

10

-1

10



di 0

10

1

l(di)

10

Fig. 5.14 Energy transfer fluxes for a electrons and b ions, Παuu  (red solid line), uT α  (green dashed line) and Λub α  (blue dash-dotted line), varying with filtering scale 

a(x) =

a[n] (x),

(5.34)

n

where

a[n] (x) = a¯ n (x) − a¯ n+1 (x) .

(5.35)

The band-filtered field a[n] is therefore the contribution of the band (n , n+1 ] to the field a. The boundaries of these bands are defined with a logarithmic function, n = γ n 0 ,

(5.36)

where γ = 1.5 is used and 0 is taken as the grid spacing of the simulation δx ∼ 0.0365di . Then the contribution to the pressure-stress interaction − (Pα · ∇) · uα  and the electromagnetic work j · E from different scale bands is quantified  [n]  [m] by  [m] [n] · ∇ · u  and j · E , respectively. Figure 5.15 shows − Pα · ∇ · − P[m] α α u[n] α /− (Pα · ∇) · uα  from different scale bands. The most intense (dark blue) contribution is confined to a small region near the origin (< 6di in the present simulation) for both electrons and ions. For the contribution j[m] · E[n] /j · E from different scale bands shown in Fig. 5.16, it is maximized at relatively large scales (∼ [6di , 16di ] in the present simulation). All these results again confirm the point that the pressure-stress interaction mainly operates at small scales, while the electromagnetic work acts primarily at relatively large scales.

5.5 Concluding Remarks Global energy equations derived from the Vlasov-Maxwell system indicate the crucial role of the pressure-stress interaction in transforming fluid kinetic energy into thermal (random) energy. In particular, the interaction between deviatoric pressure

108

5 Energy Transfer and Dissipation in Collisionless Plasma Turbulence

Fig. 5.15 Normalized contribution to the pressure-stress interaction from different scale bands, [n] − P[m] α · ∇ · uα /− (Pα · ∇) · uα , for electrons and ions [45]. Reprinted from Yan Yang et al., Monthly Notices of the Royal Astronomical Society, 482, 4933–4940, 2018 Fig. 5.16 Normalized contribution to the electromagnetic work from different scale bands, j[m] · E[n] /j · E [45]. Reprinted from Yan Yang et al., Monthly Notices of the Royal Astronomical Society, 482, 4933–4940, 2018

tensor and traceless strain-rate tensor (Pi-D) is found quantitatively to be very strong in collisionless plasma turbulence. Pi-D shares several properties with the viscous dissipation in collisional fluid, such as spatial localizaiton and strong correlation with velocity gradients. As we have seen, it is typical to have the exchange between fluid kinetic and electromagnetic energy through j · E confined to large scales and the exchange between fluid kinetic and thermal energy through − (Pα · ∇) · uα  confined to small scales. These two conversions are bridged by the fluid kinetic energy cascade at moderate scales. Here we used a single large PIC simulation as an example case to illustrate the energy transfer channels in collisionless plasma. There is growing evidence in numerical simulations [33, 43, 44] and in direct observational analysis [4] that elucidates the role of pressure-strain interaction.

References

109

References 1. Blackburn HM, Mansour NN, Cantwell BJ (1996) Topology of fine-scale motions in turbulent channel flow. J Fluid Mech 310:269–292 2. Braginskii SI (1965) Transport Processes in a Plasma. Rev Plasma Phys 1:205 3. Carbone V, Veltri P, Mangeney A (1990) Coherent structure formation and magnetic field line reconnection in magnetohydrodynamic turbulence. Phys Fluids A: Fluid Dyn 2(8):1487–1496 4. Chasapis A, Yang Y, Matthaeus W, Parashar T, Haggerty C, Burch J, Moore T, Pollock C, Dorelli J, Gershman D, Torbert J, Russell C (2018) Energy conversion and collisionless plasma dissipation channels in the turbulent magnetosheath observed by the magnetospheric multiscale mission. Astrophys J Lett 862(32): 5. Chong MS, Perry AE, Cantwell BJ (1990) A general classification of three-dimensional flow fields. Phys Fluids A: Fluid Dyn 2:765 6. Chong MS, Soria J, Perry AE, Chacin J, Cantwell BJ, Na Y (1998) Turbulence structures of wall-bounded shear flows found using DNS data. J Fluid Mech 357:225–247 7. Del Sarto D, Pegoraro F, Califano F (2016) Pressure anisotropy and small spatial scales induced by velocity shear. Phys Rev E 93(053):203 8. Dallas V, Alexakis A (2013) Structures and dynamics of small scales in decaying magnetohydrodynamic turbulence. Phys Fluids 25(105):106 9. Del Sarto D, Pegoraro F (2017) Shear-induced pressure anisotropization and correlation with fluid vorticity in a low collisionality plasma. Mon Notices Royal Astron Soc 475(1):181–192 10. Dmitruk P, Matthaeus WH, Seenu N (2004) Test particle energization by current sheets and nonuniform fields in magnetohydrodynamic turbulence. Astrophys J 617:667–679 11. Franci L, Hellinger P, Matteini L, Verdini A, Landi S (2016) Two-dimensional hybrid simulations of kinetic plasma turbulence: Current and vorticity vs proton temperature. In: American Institute of Physics conference series, American Institute of Physics Conference Series, vol 1720, p 040003 12. Greco A, Valentini F, Servidio S, Matthaeus WH (2012) Inhomogeneous kinetic effects related to intermittent magnetic discontinuities. Phys Rev E 86(066):405 13. Haggerty CC, Parashar TN, Matthaeus WH, Shay MA, Yang Y, Wan M, Wu P, Servidio S (2017) Exploring the statistics of magnetic reconnection x-points in kinetic particle-in-cell turbulence. Phys Plasmas 24(10):102,308 14. Huba JD (1996) The Kelvin-Helmholtz instability: finite Larmor radius magnetohydrodynamics. Geophys Res Lett 23:2907–2910 15. Jiménez J, Wary AA, Saffman PG, Rogallo RS (1993) The structure of intense vorticity in isotropic turbulence. J Fluid Mech 255:65–90 16. Karimabadi H, Roytershteyn V, Wan M, Matthaeus WH, Daughton W, Wu P, Shay M, Loring B, Borovsky J, Leonardis E, Chapman SC (2013) Coherent structures, intermittent turbulence, and dissipation in high-temperature plasmas. Phys Plasmas 20(012):303 17. Kolmogorov AN (1941) The local structure of turbulence in incompressible viscous fluid for very large reynolds numbers. Dokl Akad Nauk SSSR 30:299–303 18. Markovskii SA, Vasquez BJ, Smith CW, Hollweg JV (2006) Dissipation of th perpendicular turbulent cascade in the solar wind. Astrophys J 639:1177–1185 19. Matthaeus WH (1982) Reconnection in two dimensions: localization of vorticity and current near magnetic X-points. Geophys Res Lett 9:660–663 20. Matthaeus WH, Montgomery D (1980) Selective decay hypothesis at high mechanical and magnetic reynolds numbers. Ann New York Acad Sci 357(1):203–222 21. Montgomery D, Nielson C (1970) Thermal relaxation in one-and two-dimensional plasma models. Phys Fluids 13(5):1405–1407 22. Ooi A, Martin J, Soria J, Chong MS (1999) A study of the evolution and characteristics of the invariants of the velocity-gradient tensor in isotropic turbulence. J Fluid Mech 381:141–174 23. Osman KT, Matthaeus WH, Greco A, Servidio S (2011) Evidence for inhomogeneous heating in the solar wind. Astrophys J Lett 727:L11

110

5 Energy Transfer and Dissipation in Collisionless Plasma Turbulence

24. Osman KT, Matthaeus WH, Hnat B, Chapman SC (2012) Kinetic signatures and intermittent turbulence in the solar wind plasma. Phys Rev Lett 108(261):103 25. Osman KT, Matthaeus WH, Wan M, Rappazzo AF (2012) Intermittency and local heating in the solar wind. Phys Rev Lett 108(261):102 26. Parashar TN, Matthaeus WH (2016) Propinquity of current and vortex structures: effects on collisionless plasma heating. Astrophys J 832:57 27. Parashar TN, Servidio S, Shay MA, Breech B, Matthaeus WH (2011) Effect of driving frequency on excitation of turbulence in a kinetic plasma. Phys Plasmas 18(092):302 28. Perri S, Goldstein ML, Dorelli JC, Sahraoui F (2012) Detection of small-scale structures in the dissipation regime of solar-wind turbulence. Phys Rev Lett 109(191):101 29. Retinò A, Sundkvist D, Vaivads A, Mozer F, André M, Owen CJ (2007) In situ evidence of magnetic reconnection in turbulent plasma. Nature Phys 3:235–238 30. Sahoo G, Perlekar P, Pandit R (2011) Systematics of the magnetic-Prandtl-number dependence of homogenous, isotropic magnetohydrodynamic turbulence. New J Phys 13(013):036 31. Servidio S, Valentini F, Califano F, Veltri P (2012) Local kinetic effects in two-dimensional plasma turbulence. Phys Rev Lett 108(045):001 32. Servidio S, Valentini F, Perrone D, Greco A, Califano F, Matthaeus WH, Veltri P (2015) A kinetic model of plasma turbulence. J Plasma Phys 81(325810):107 33. Sitnov M, Merkin V, Roytershteyn V, Swisdak M (2018) Kinetic dissipation around a dipolarization front. Geophys Res Lett 45(10):4639–4647 34. Soria J, Sondergaard R, Cantwell BJ, Chong MS, Perry AE (1994) A study of the fine-scale motions of incompressible time-developing mixing layers. Phys Fluids 6:871 35. Sundkvist D, Retinò A, Vaivads A, Bale SD (2007) Dissipation in turbulent plasma due to reconnection in thin current sheets. Phys Rev Lett 99(025):004 36. TenBarge JM, Howes GG (2013) Current sheets and collisionless damping in kinetic plasma turbulence. Astrophys J Lett 771:L27 37. Vasquez BJ, Markovskii SA (2012) Velocity power spectra from cross-field turbulence in the proton kinetic regime. Astrophys J 747:19 38. Vincenti WG, Kruger CH (1965) Introduction to physical gas dynamics 39. Wan M, Matthaeus WH, Karimabadi H, Roytershteyn V, Shay M, Wu P, Daughton W, Loring B, Chapman SC (2012) Intermittent dissipation at kinetic scales in collisionless plasma turbulence. Phys Rev Lett 109(195):001 40. Wan M, Matthaeus WH, Roytershteyn V, Karimabadi H, Parashar T, Wu P, Shay M (2015) Intermittent dissipation and heating in 3d kinetic plasma turbulence. Phys Rev Lett 114(175):002 41. Wan M, Matthaeus WH, Roytershteyn V, Parashar TN, Wu P, Karimabadi H (2016) Intermittency, coherent structures and dissipation in plasma turbulence. Phys Plasmas 23(042):307 42. Wu P, Perri S, Osman K, Wan M, Matthaeus WH, Shay MA, Goldstein ML, Karimabadi H, Chapman S (2013) Intermittent heating in solar wind and kinetic simulations. Astrophys J Lett 763:L30 43. Yang Y, Matthaeus WH, Parashar TN, Haggerty CC, Roytershteyn V, Daughton W, Wan M, Shi Y, Chen S (2017) Energy transfer, pressure tensor, and heating of kinetic plasma. Phys Plasmas 24(072):306 44. Yang Y, Matthaeus WH, Parashar TN, Wu P, Wan M, Shi Y, Chen S, Roytershteyn V, Daughton W (2017) Energy transfer channels and turbulence cascade in Vlasov-Maxwell turbulence. Phys Rev E 95(061):201 45. Yang Y, Wan M, Matthaeus WH, Sorriso-Valvo L, Parashar TN, Lu Q, Shi Y, Chen S (2018) Scale dependence of energy transfer in turbulent plasma. Mon Notices Royal Astron Soc 482(4):4933–4940 46. Zeiler A, Biskamp D, Drake JF, Rogers BN, Shay MA, Scholer M (2002) Three-dimensional particle simulations of collisionless magnetic reconnection. J Geophys Res 107:1230

Chapter 6

Conclusions and Discussion

In this chapter, the interpretation on energy transfer and dissipation spanning from fluid scales to kinetic scales are summarized based on the results in MHD model and in kinetic model; and then a final perspective is given.

6.1 Conclusions In the context of plasma turbulence, many models are applicable to particular problems. As mentioned in Chap. 1, now and into the future, there is no one “best” model, but rather there is a range of models, relying on the scales and problems concerned, that are adequate in a broad range of applicability. In this book, we have applied two methods, MHD simulation and PIC simulation, and investigated the energy transfer and dissipation process in plasma turbulence. The basic results of this book are essentially three: 1. For the study of MHD turbulence, a continuing major challenge is to develop numerical schemes to calculate the fields and turbulence properties of practical relevance, e.g., accounting for compressibility. In Chap. 3, we have developed an efficient, high-resolution and oscillation-free hybrid scheme for shockturbulence interactions in compressible MHD problems. The hybrid scheme couples a sixth-order compact finite difference scheme in smooth regions with a fifth-order WENO scheme in shock regions, which has been proved to be accurate for smooth solutions and be able to capture discontinuities robustly. 2. Compressible MHD turbulence differs from incompressible MHD and hydrodynamic turbulence in many ways. In Chap. 4, we have studied the degree to which some turbulence theories proposed in the incompressible case remain applicable in the compressible one. The flows with varying forcing mechanisms display different features, such as compression and coherent structures. The © Springer Nature Singapore Pte Ltd. 2019 Y. Yang, Energy Transfer and Dissipation in Plasma Turbulence, Springer Theses, https://doi.org/10.1007/978-981-13-8149-2_6

111

112

6 Conclusions and Discussion

shear-dominated simulation behaves like a nearly incompressible flow, while the more compressive simulation is characterized by shocks, which lead to saturated scaling exponents of high-order structure functions of density and enhanced energy cascade. In addition, the conversion between kinetic and thermal energy by pressure dilatation is augmented in the strongly compressed case and dominated by the largest scale contribution. 3. For collisionless plasma, we analyzed ideal energy transfer in the full VlasovMaxwell system, which clarifies several energy transfer channels. The electromagnetic work, j · E, represents the conversion between fluid kinetic and electromagnetic energy, while the pressure-stress interaction, − (Pα · ∇) · uα , is the term converting fluid kinetic energy into thermal (random) energy. In particular, the pressure-strain part “Pi-D” is of particular importance for both electrons and ions. The Pi-D, highly localized in spatial regions, is strongly correlated with velocity gradients, indicating that enhanced energy transfer occurs preferentially in strong velocity gradients. By performing scale-dependent spatial filtering on the Vlasov equation, we have quantified the highly inhomogeneous energy cascade as it proceeds from macroscopic fluid flow scales down to kinetic scales. The pressure-stress interaction terminates the cascade and dissipates kinetic energy into thermal energy at kinetic scales. Taken together, we can sketch out a picture: energy mainly resides in largescale fluctuations and can be rearranged in space due to transport terms; nonlinear interactions drive a cascade that transfers energy from large to small scales and the dynamics progressively generates coherent structures; within these structures, intermittent distributions of several channels of energy conversion provide the dominate dissipation mechanism; the electromagnetic work on particles drives flows and the pressure-stress interaction increases thermal energy.

6.2 Perspectives There is a broad range of turbulent plasma systems, varying in characteristic parameters, many involving additional physical processes, for which valuable extensions of present study are sought. Our results in compressible MHD turbulence are based on the numerical simulations with no background magnetic field which, however, exists in many astrophysical plasmas. Therefore we did not address the well known property that MHD energy transfer becomes anisotropic relative to a large-scale imposed magnetic field. The theoretical framework employed in Chap. 5 is based on the full, multi-species Vlasov-Maxwell system, but it lacks collisional effects. We therefore did not address whether the energy conversion to thermal energy is able to produce irreversible heating through collisions. We have found that the pressure-strain interaction is concentrated at some locations. Then such issue should be investigated: Are hot spots of Pi-D associated with certain mechanisms such as magnetic reconnection?

6.2 Perspectives

113

The energy cascade within inertial range satisfies the Politano-Pouquet law that describes a linear scaling relation between the mixed third-order moments of Elsasser fields increments and mean dissipation rate. Therefore instead of studying specific dissipation mechanisms at kinetic scales, one can appeal to the third-order law. It is promising to apply more sophisticated forms of third-order moments, which take into account more real effects, such as anisotropy, solar wind expansion and Hall effect. For plasma turbulence, there is a very large set of possibilities for looking at energy dissipation mechanism, and the results in this book are hinting at a few of them. For many others, substantial work are required.

Appendix A

Fifth-Order WENO Scheme

The purpose of this appendix is to briefly review the fifth-order WENO scheme proposed by Jiang and Shu [4]. The research literature in this area is vast. Useful details are provided by [1–8]. The basic idea of the WENO scheme is to construct a high-order numerical flux from a convex combination of low-order polynomial reconstructions over a set of stencils. The weights of candidate stencils, which determine the contribution of each stencil to the final approximation of the numerical flux, approach optimal weights to achieve higher-order accuracy in smooth regions, while are assigned to be nearly zero on the stencil which contains discontinuities. We use 1D scalar conservation law as an example: ∂ f (u) ∂u + = 0. ∂t ∂x

(A.1)

As shown in Eq. 3.30, the spatial derivative is approximated as f i =

w w − f i−1/2 f i+1/2

h

.

(A.2)

A general flux can be decomposed into two parts, f (u) = f + (u) + f − (u),

(A.3)

with d f + /du ≥ 0 and d f − /du ≤ 0. The numerical fluxes for the positive and negw,+ w,− and f i+1/2 , are computed separately. Then we have ative parts, f i+1/2 w,+ w,− w f i+1/2 = f i+1/2 + f i+1/2 .

(A.4)

w,+ Firstly, we describe how f i+1/2 is computed. As illustrated in Fig. A.1, three stencils, S1 = {xi−2 , xi−1 , xi }, S2 = {xi−1 , xi , xi+1 } and S3 = {xi , xi+1 , xi+2 }, are chosen to approximate the numerical flux of the fifth-order WENO scheme. Each of

© Springer Nature Singapore Pte Ltd. 2019 Y. Yang, Energy Transfer and Dissipation in Plasma Turbulence, Springer Theses, https://doi.org/10.1007/978-981-13-8149-2

115

116

Appendix A: Fifth-Order WENO Scheme

Fig. A.1 Illustration of stencils for fifth-order WENO reconstruction

the stencils can render an approximation of the flux. Let us say q1 , q2 and q3 are the third-order accurate reconstruction of the flux in the three different stencils, 1 7 11 f i−2 − f i−1 + fi , 3 6 6 1 5 1 q2 = − f i−1 + f i + f i+1 , 6 6 3 1 5 1 q3 = f i + f i+1 − f i+2 . 3 6 6 q1 =

(A.5)

Then the fifth-order WENO numerical flux is constructed as w,+ = ω1 q1 + ω2 q2 + ω3 q3 , f i+1/2

(A.6)

where ω1 , ω2 and ω3 are the weights to stencils. The weights are required to adapt to the relative smoothness of f on each stencil, such that if the stencil contains a discontinuity, the weight is nearly zero. In [4], the weight ωk is defined as αk , α1 + α2 + α3

(A.7)

Ck , k = 1, 2, 3, (ε + I Sk )2

(A.8)

ωk = and αk =

where ε is introduced to avoid the denominator becoming zero. It is suggested to be in the range of 10−5 –10−7 . The coefficients Ck are properly selected with values C1 =

1 6 3 , C2 = , C3 = , 10 10 10

(A.9)

to achieve maximum order of accuracy. The I Sk is the smoothness measurement of f on the k-th stencil 13 ( f i−2 − 2 f i−1 + f i )2 + 12 13 I S2 = ( f i−1 − 2 f i + f i+1 )2 + 12 13 I S3 = ( f i − 2 f i+1 + f i+2 )2 + 12 I S1 =

1 ( f i−2 − 4 f i−1 + 3 f i )2 , 4 1 ( f i−1 − f i+1 )2 , 4 1 (3 f i − 4 f i+1 + f i+2 )2 . 4

(A.10) (A.11) (A.12)

Appendix A: Fifth-Order WENO Scheme

117

The formulas for the negative part of the split flux are easily obtained following the symmetry with respect to xi+1/2 . That is w,− = ω1 q1 + ω2 q2 + ω3 q3 , f i+1/2

(A.13)

1 7 11 f i+3 − f i+2 + f i+1 , 3 6 6 1 5 1 q2 = − f i+2 + f i+1 + f i , 6 6 3 1 5 1 q3 = f i+1 + f i − f i−1 , 3 6 6

(A.14)

where q1 =

and the definition of the weight is the same as Eq. A.7, while the smoothness estimators herein are given as 13 1 ( f i+3 − 2 f i+2 + f i+1 )2 + ( f i+3 − 4 f i+2 + 3 f i+1 )2 , 12 4 13 1 2 I S2 = ( f i+2 − 2 f i+1 + f i ) + ( f i+2 − f i )2 , 12 4 13 1 2 I S3 = ( f i+1 − 2 f i + f i−1 ) + (3 f i+1 − 4 f i + f i−1 )2 . 12 4 I S1 =

(A.15) (A.16) (A.17)

References 1. Borges R, Carmona M, Costa B, Don WS (2008) An improved weighted essentially nonoscillatory scheme for hyperbolic conservation laws. J Comput Phys 227(6):3191–3211 2. Harten A, Engquist B, Osher S, Chakravarthy SR (1987) Uniformly high order accurate essentially non-oscillatory schemes, 111. J Comput Phys 71(2):231–303 3. Henrick AK, Aslam TD, Powers JM (2005) Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points. J Comput Phys 207(2):542–567 4. Jiang GS, Shu CW (1996) Efficient implementation of weighted ENO schemes. J Comput Phys 126:202–228 5. Liu XD, Osher S, Chan T (1994) Weighted essentially non-oscillatory schemes. J Comput Phys 115(1):200–212 6. Martín MP, Taylor EM, Wu M, Weirs VG (2006) A bandwidth-optimized WENO scheme for the effective direct numerical simulation of compressible turbulence. J Comput Phys 220:270–289 7. Shu CW, Osher S (1998) Efficient implementation of essentially non-oscillatory shock-capturing schemes. J Comput Phys 77:439–471 8. Shu CW, Osher S (1989) Efficient implementation of essentially non-oscillatory shock-capturing schemes, II. J Comput Phys 83:32–78

Appendix B

Eigen-System in MHD

The eigenvalues and eigenvectors are necessary ingredients for the construction of the hybrid scheme in MHD. For the MHD system, there are seven waves in all, including one entropy wave, two fast magneto-acoustic waves, two slow magnetoacoustic waves, and two Alfvén waves. There are cases where these eigenvalues may coincide. So the MHD system is non-strictly hyperbolic. Details on the MHD eigen-system can be found in [1, 2, 4].

B.1

1D MHD System

We first present the eigen-system for 1D MHD system, ∂U ∂E + = 0, ∂t ∂x

(B.1)

where U = (ρ, ρu, ρv, ρw, B y , Bz , E )T and ⎡

⎤ ρu ⎢ ⎥ ρu 2 + pt − Bx2 ⎢ ⎥ ⎢ ⎥ ρuv − B B x y ⎢ ⎥ ⎥. ρuw − B B E=⎢ x z ⎢ ⎥ ⎢ ⎥ u B − v B y x ⎢ ⎥ ⎣ ⎦ u Bz − w B x (E + pt )u − Bx (u · B) Note that Bx ≡ const. The Jacobian A = ∂E/∂U of the system is

© Springer Nature Singapore Pte Ltd. 2019 Y. Yang, Energy Transfer and Dissipation in Plasma Turbulence, Springer Theses, https://doi.org/10.1007/978-981-13-8149-2

119

120

Appendix B: Eigen-System in MHD ⎡

⎤ 0 1 0 0 0 0 0 ⎢ γ −3 2 γ −1 2 ⎥ ⎢ u + (v + w2 ) (3 − γ )u (1 − γ )v (1 − γ )w (2 − γ )B y (2 − γ )Bz γ − 1 ⎥ ⎢ 2 ⎥ 2 ⎢ −uv v u 0 −Bx 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ −uw w 0 u 0 −Bx 0 ⎥ ⎢ A=⎢ ⎥, By By Bx Bx ⎢ ⎥ u+ v − 0 u 0 0 ⎥ − ⎢ ⎢ ⎥ ρ ρ ρ ρ ⎢ ⎥ Bz Bx Bz Bx ⎢ − u+ w 0 − 0 u 0 ⎥ ⎣ ⎦ ρ ρ ρ ρ α1 α2 α3 α4 α5 α6 α7

(B.2) where 

α1 α2 α3 α4 α5 α6

γ −1 2 Bx 2 2 (u + v + w ) + (u · B), = −u H − 2 ρ B2 = H − x − (γ − 1)u 2 , ρ Bx B y , = (1 − γ )uv − ρ B x Bz , = (1 − γ )uw − ρ = (2 − γ )B y u − Bx v, = (2 − γ )Bz u − Bx w,

α7 = γ u, E + pt . H= ρ The eigenvalues are λ1,7 = u ∓ c f , λ2,6 = u ∓ ca , λ3,5 = u ∓ cs , λ4 = u, and λ1 ≤ λ2 ≤ λ3 ≤ λ4 ≤ λ5 ≤ λ6 ≤ λ7 . They correspond to • two fast magneto-acoustic waves traveling with speed u ∓ c f , where



2 1 2 2 2 2 2 2 cf = a + b + a + b − 4a bx ; 2 • two Alfvén waves traveling with speed u ∓ ca , where

ca =

Bx2 = |bx |; ρ

• two slow magneto-acoustic waves traveling with speed u ∓ cs , where

(B.3)

Appendix B: Eigen-System in MHD

121

 2 1 2 a + b2 − a 2 + b2 − 4a 2 bx2 ; cs = 2 • one entropy wave traveling with speed u, √ with the notations bx , b y , bz = Bx , B y , Bz / ρ and b2 = bx2 + b2y + bz2 and a = √ γ RT is the sound speed. Note that there are cases where these eigenvalues may coincide, for example, • At bx = 0, cs = ca = 0, then λ2 = λ3 = λ4 = λ5 = λ6 = u is an eigenvalue of multiplicity 5; • At b y = bz = 0, cs2 = min(a 2 , bx2 ), and c2f = max(a 2 , bx2 ). If a 2 = bx2 , then λ2,6 = u ∓ ca are eigenvalues of multiplicity 2. If a 2 = bx2 , then λ2,6 = u ∓ ca are eigenvalues of multiplicity 3. Given the eigenvalues, the matrix A has a complete set of right and left eigenvectors. The set of eigenvectors given in the book by Jeffrey and Taniuti [2] is not well defined and becomes singular at the points where the eigenvalues degenerate. Therefore, the re-normalization of these eigenvectors was introduced by Brio and Wu [1] to avoid singularities and utilized in [3, 5]. Let ⎧ (B y , Bz ) ⎪ ⎪ , if B y2 + Bz2 = 0, ⎪ ⎨ 2 B y + Bz2 (β y , βz ) =   ⎪ 1 1 ⎪ ⎪ ⎩ √ , √ , if B y2 + Bz2 = 0, 2 2

⎧  2 ⎪ ( a − cs2 , c2f − a 2 ) ⎪ ⎪ ⎪

, if B y2 + Bz2 = 0 or a 2 = ca2 , ⎨ c2f − cs2 (α f , αs ) =   ⎪ ⎪ ⎪ 1 1 ⎪ ⎩ √ ,√ , ifB y2 + Bz2 = 0 and a 2 = ca2 , 2 2  1, if Bx ≥ 0, sgn(Bx ) = −1, if Bx < 0.

(B.4)

(B.5)

(B.6)

The right and left eigenvectors are given by ⎡

r1,7

⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎣



⎡ ⎤T γ1 α f (u 2 + v 2 + w2 ) ± Γ f ⎥ ⎢ ⎥ −2γ1 α f u ∓ α f c f ⎥ ⎢ ⎥ ⎥ ⎢ −2γ1 α f v ± cs αs β y sgn(Bx ) ⎥ ⎥ ⎢ ⎥ 1 ⎥ ⎢ ⎥ , l1,7 = 2 ⎢ −2γ1 α f w ± cs αs βz sgn(Bx ) ⎥ ⎥ , √ ⎥ 2a ⎢ ⎥ ⎥ ⎢ −2γ1 α f B y + √ρaαs β y ⎥ ⎥ ⎣ −2γ1 α f Bz + ρaαs βz ⎦ ⎦   α f (u 2 + v 2 + w2 )/2 + c2f − γ2 a 2 ∓ Γ f 2γ1 α f αf α f (u ∓ c f ) α f v ± cs αs β y sgn(Bx ) α f w ± cs αs βz sgn(Bx ) √ aαs β y / ρ √ aαs βz / ρ

(B.7)

122

Appendix B: Eigen-System in MHD

⎤T ⎤ ⎡ Γa 0 ⎥ ⎥ ⎢ ⎢ 0 0 ⎥ ⎥ ⎢ ⎢ ⎢ −βz sgn(Bx ) ⎥ ⎢ −βz sgn(Bx ) ⎥ ⎥ ⎥ ⎢ ⎢ 1⎢ ⎥ β y sgn(Bx ) ⎥ =⎢ , x ) ⎥ , l2,6 = ⎢ ⎢ β y sgn(B 2 ⎢ ∓√ρβ ⎥ ⎥ ⎢ ∓βz /√ρ ⎥ z ⎥ ⎥ ⎢ ⎢ ⎣ ±√ρβ y ⎦ ⎣ ±β y /√ρ ⎦ 0 −Γa ⎡

r2,6

(B.8)



r3,5

⎤ ⎡ ⎤T αs γ1 αs (u 2 + v2 + w2 ) ± Γs ⎢ ⎥ ⎢ ⎥ αs (u ∓ cs ) −2γ1 αs u ∓ αs cs ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −2γ1 αs v ∓ c f α f β y sgn(Bx ) ⎥ αs v ∓ c f α f β y sgn(Bx ) ⎢ ⎥ ⎢ ⎥ 1 ⎥ , l3,5 = ⎢ −2γ1 αs w ∓ c f α f βz sgn(Bx ) ⎥ , =⎢ αs w ∓ c f α f βz sgn(Bx ) ⎢ ⎥ ⎢ ⎥ 2 √ √ 2a ⎢ ⎢ ⎥ ⎥ −aα f β y / ρ ⎢ ⎥ ⎢ −2γ1 αs B y − √ρaα f β y ⎥ √ ⎣ ⎦ ⎣ ⎦ β / ρ −2γ α B − ρaα β −aα f z 1 s z f z  2  2 2 2 2 2γ1 αs αs (u + v + w )/2 + cs − γ2 a ∓ Γs

(B.9)



⎡ ⎤ ⎤T 1 1 − τ (u 2 + v2 + w2 )/2 ⎢ ⎢ ⎥ ⎥ u τu ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ ⎥ v τv ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ ⎥ , w τw r4 = ⎢ ⎥ , l4 = ⎢ ⎥ ⎢ ⎢ ⎥ ⎥ 0 τ By ⎢ ⎢ ⎥ ⎥ ⎣ ⎣ ⎦ ⎦ 0 τ Bz 2 2 2 (u + v + w )/2 −τ

(B.10)

where γ1 = (γ − 1)/2, γ2 = (γ − 2)/(γ − 1), τ = (γ − 1)/a 2 , Γ f = α f c f u − cs αs sgn(Bx )(β y v + βz w), Γa = sgn(Bx )(βz v − β y w), Γs = αs cs u + c f α f sgn(Bx )(β y v + βz w).

B.2

Extension to 2D and 3D

Since the similarity among the flux vectors E, F and G (see Eqs. 3.6, 3.7 and 3.8), the eigen-system for 1D MHD system can be easily extended to 2D and 3D MHD systems. If we swap the 2nd component of E with its 3rd component, swap its 5th component with its 6th component, replace u and v with v and u, respectively, and replace Bx and B y with B y and Bx , respectively, the final expression is the flux vector F. Let us define a swap matrix S, which swaps the 2nd component of a vector with

Appendix B: Eigen-System in MHD

123

its 3rd component and swaps its 5th component with its 6th component, following SS = I. It is straightforward that F(U) = SE(SU). Then B=

∂ ∂F = SE(SU) = SA(SU)S. ∂U ∂U

(B.11)

(B.12)

On denoting the eigenvalues of the Jacobian B by λ B , the right eigenvectors by r B and the left eigenvectors by l B , we have Br B = SA(SU)Sr B = λ B r B , l B B = l B SA(SU)S = λ B l B ,

(B.13) (B.14)

A(SU)Sr B = λ B Sr B , l B SA(SU) = λ B l B S.

(B.15) (B.16)

which can be written as

According to Eqs. B.15 and B.16, we obtain λ B = λ A (SU), r B = Sr A (SU), l B = l A (SU)S.

(B.17) (B.18) (B.19)

Similarly, the eigenvectors of the Jacobian of the flux vector G can be found by slightly modifying those for the flux vector F. Note that the 5th component of the flux vector E, the 6th component of F and the 7th component of G are zero, leading to zero eigenvalues. The eight-wave eigensystem with a new eighth wave traveling with the flow speed was proposed by Powell et al. [4], which could spoil conservation and produce incorrect jump conditions across discontinuities [6]. We apply the seven-wave eigen-system in our work. References 1. Brio M, Wu CC (1988) An upwind differencing scheme for the equations of ideal magnetohydrodynamics. J Comput Phys 75:400–422 2. Jeffrey A, Taniuti T (1964) Nonlinear wave propagation. ACADEMIC PR, New York 3. Jiang GS, Wu CC (1999) A high-order WENO finite difference scheme for the equations of ideal magnetohydrodynamics. J Comput Phys 150:561–594 4. Powell K, Roe PL, Myong RS (1995) An upwind scheme for magnetohydrodynamics. AIAA paper 95–1704 (1995)

124

Appendix B: Eigen-System in MHD

5. Roe PL, Balsara DS (1996) Notes on the eigensystem of magnetohydrodynamics. SIAM J Appl Math 56:57–67 6. Tóth G (2000) The ∇ · B = 0 constraint in shock-capturing magnetohydrodynamics codes. J Comput Phys 161:605–652

Appendix C

A Parallel Algorithm for Pentadiagonal Systems

Massive pentadiagonal system is a hard task to solve due to inherent data dependence and large computational load. Therefore, an efficient parallel pentadiagonal algorithm is necessary. Various parallel pentadiagonal algorithms have been studied extensively, such as the divide and conquer algorithm [1, 4], the single-width separator algorithm [3], the double-width separator algorithm [2], to name a few. The parallel algorithm used here divides the massive system into submatrices and an additional low-order pentadiagonal matirx, and has small computational cost and data communication, which is beneficial to solve massive systems. Without loss of generality, the pentadiagonal matrix can be assumed to be a cyclic matrix. For a non-cyclic one, we can simply take α1 = a1 = α2 = βn−1 = cn = βn = 0 in Eq. C.2. Thus, a parallel algorithm for solving cyclic pentadiagonal system is presented. We consider the pentadiagonal linear system Ax = d

(C.1)

where A is a pentadiagonal matrix of order n ⎡

b1 c1 β1 a2 b2 c2 β2 α3 a3 b3 c3 · · · · · ·

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ A=⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎣ βn−1 cn βn

α1 β3 · · · ·

· · · · ·

· · · · ·

· · · ·

· · ·

· ·

· αn−2 an−2 bn−2 cn−2 αn−1 an−1 bn−1 αn an

© Springer Nature Singapore Pte Ltd. 2019 Y. Yang, Energy Transfer and Dissipation in Plasma Turbulence, Springer Theses, https://doi.org/10.1007/978-981-13-8149-2

a1 α2 0



⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ βn−2 ⎥ ⎥ cn−1 ⎦ bn

(C.2)

125

126

Appendix C: A Parallel Algorithm for Pentadiagonal Systems

x = (x1 , . . . , xn )T and d = (d1 , . . . , dn )T . When solved on multicomputers, the matrix A is divided into submatrices. If Eq. C.1 is solved in p processors, then n = mp, that is, the matrix is partitioned into m × m matrices. Here we take m = 3 as example, then ⎡

b1 c1 β1 a2 b2 c2 β2 α3 a3 b3 c3 · · · · · ·

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎣ βn−1 cn βn

α1 β3 · · · ·

· · · · ·

· · · · ·

· · · ·

· · ·

· ·

· αn−2 an−2 bn−2 cn−2 αn−1 an−1 bn−1 αn an

⎤⎛

x1 x2 x3 · · · · · ·





⎞ d1 ⎥⎜ ⎟ ⎜ d2 ⎟ ⎥⎜ ⎟ ⎜ ⎟ ⎥⎜ ⎟ ⎜ d3 ⎟ ⎥⎜ ⎟ ⎜ ⎟ ⎥⎜ ⎟ ⎜ · ⎟ ⎥⎜ ⎟ ⎜ ⎟ ⎥⎜ ⎟ ⎜ · ⎟ ⎥⎜ ⎟ ⎜ ⎟ ⎥⎜ ⎟ ⎜ · ⎟ ⎥⎜ ⎟=⎜ ⎟ ⎥⎜ ⎟ ⎜ · ⎟ (.C.3) ⎥⎜ ⎟ ⎜ ⎟ ⎥⎜ ⎟ ⎜ · ⎟ ⎥⎜ ⎟ ⎜ ⎟ ⎥⎜ ⎟ ⎜ · ⎟ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ βn−2 ⎥ ⎥ ⎜ xn−2 ⎟ ⎜ dn−2 ⎟ cn−1 ⎦ ⎝ xn−1 ⎠ ⎝ dn−1 ⎠ bn xn dn a1 α2 0

So the partitioned matrix above can be written as ⎡

⎤⎛ ⎞ ⎛ ⎞ T0 S0 x0 d0 L0 ⎢ L1 T1 S1 ⎥ ⎜ x 1 ⎟ ⎜ d1 ⎟ ⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎢ ⎥⎜ · ⎟ ⎜ · ⎟ · · · ⎢ ⎥⎜ ⎟=⎜ ⎟ ⎢ ⎥⎜ · ⎟ ⎜ · ⎟, · · · ⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎣ L p−2 T p−2 S p−2 ⎦ ⎝ x p−2 ⎠ ⎝ d p−2 ⎠ S p−1 L p−1 T p−1 x p−1 d p−1 where T Li = αim+1 e1 em−1 + aim+1 e1 emT + αim+2 e2 emT , Si = βim+m−1 em−1 e1T + cim+m em e1T + βim+m em e2T , ⎤ ⎡ bim+1 cim+1 βim+1 ⎥ ⎢ aim+2 bim+2 cim+2 βim+2 ⎥ ⎢ ⎥ ⎢ αim+3 aim+3 bim+3 cim+3 βim+3 ⎥ ⎢ ⎥ ⎢ · · · · · ⎥ ⎢ ⎥, ⎢ · · · · Ti = ⎢ ⎥ ⎥ ⎢ · · · · ⎥ ⎢ ⎥ ⎢ a b c β α im+m−2 im+m−2 im+m−2 im+m−2 im+m−2 ⎥ ⎢ ⎣ αim+m−1 aim+m−1 bim+m−1 cim+m−1 ⎦ αim+m aim+m bim+m xi = (xim+1 , xim+2 , xim+3 , · · · , xim+m )T , di = (dim+1 , dim+2 , dim+3 , · · · , dim+m )T ,

(C.4)

Appendix C: A Parallel Algorithm for Pentadiagonal Systems

127

and ei is a unity column with its ith element being the other elements one and all −1 , . . . , T being zero. Then, we can multiply Eq. C.4 by diag T−1 0 p−1 and the equation becomes ⎧ x0 ⎪ ⎪ ⎪ ⎪ x1 ⎪ ⎪ ⎪ ⎪ ⎨ ··· xi ⎪ ⎪ · ·· ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ ⎩ p−2 x p−1

+ T−1 0 L0 x p−1 + T−1 1 L1 x 0 + ··· + Ti−1 Li xi−1 + ··· + T−1 L p−2 p−2 x p−3 + T−1 p−1 L p−1 x p−2

+ T−1 0 S0 x1 + T−1 1 S1 x2 + ··· + Ti−1 Si xi+1 + ··· + T−1 S p−2 p−2 x p−1 + T−1 p−1 S p−1 x0

= T−1 0 d0 = T−1 1 d1 = ··· = Ti−1 di = ··· = T−1 p−2 d p−2 = T−1 p−1 d p−1

(C.5)

According to the definitions of Li and Si , the above equation can be written as ⎧ x0 ⎪ ⎪ ⎪ ⎪ x1 ⎪ ⎪ ⎪ ⎪ ⎨ ··· xi ⎪ ⎪ · ·· ⎪ ⎪ ⎪ ⎪ x ⎪ p−2 ⎪ ⎩ x p−1

+ ω1 u0 + ω5 u1 + ··· + ω4i+1 ui + ··· + ω4 p−7 u p−2 + ω4 p−3 u p−1

+ ω2 v0 + ω6 v1 + ··· + ω4i+2 vi + ··· + ω4 p−6 v p−2 + ω4 p−2 v p−1

+ ω3 r0 + ω7 r1 + ··· + ω4i+3 ri + ··· + ω4 p−5 r p−2 + ω4 p−1 r p−1

+ ω4 t0 + ω8 t1 + ··· + ω4i+4 ti + ··· + ω4 p−4 t p−2 + ω4 p t p−1

= = = = = = =

y0 y1 ··· yi ··· y p−2 y p−1 (C.6)

where ⎧ ui ⎪ ⎪ ⎪ ⎪ ⎨ vi ri ⎪ ⎪ ⎪ t ⎪ ⎩ i yi

= Ti−1 e2 , = Ti−1 e1 , = Ti−1 em , = Ti−1 em−1 , = Ti−1 di ,

⎧ ω4i+1 =αim+2 emT · xi−1 , ⎪ ⎪ T ⎪ ⎨ω T 4i+2 =αim+1 em−1 · xi−1 + aim+1 em · xi−1 , ⎪ ω4i+3 =cim+m e1T · xi+1 + βim+m e2T · xi+1 , ⎪ ⎪ ⎩ ω4i+4 =βim+m−1 e1T · xi+1 , and 

xi−1 = x p−1 , if i − 1 < 0, xi+1 = x0 , if i + 1 > p − 1.

So as long as we can get ω4i+1 , ω4i+2 , ω4i+3 , ω4i+4 , it is trivial to obtain the solution T , and emT from Eq. C.6. From the definition of ω, we multiply Eq. C.6 by e1T , e2T , em−1 to eliminate xi . Finally, we only need to solve a cyclic pentadiagonal system with

128

Appendix C: A Parallel Algorithm for Pentadiagonal Systems

4 p order which is shown as ⎡

bˆ1 aˆ 2 αˆ 3

cˆ1 bˆ2 aˆ 3 ·

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎣ βˆ4 p−1 cˆ4 p βˆ4 p

βˆ1 cˆ2 bˆ3 · ·

βˆ2 cˆ3 · ·

αˆ 1 βˆ3 · ·

· ·

· αˆ 4 p−2 aˆ 4 p−2 bˆ4 p−2 cˆ4 p−2 αˆ 4 p−1 aˆ 4 p−1 bˆ4 p−1 αˆ 4 p aˆ 4 p

aˆ 1 αˆ 2 0

⎤⎛

ω1 ω2 ω3 · ·





dˆ1 dˆ2 dˆ3 · ·



⎥⎜ ⎟ ⎟ ⎜ ⎥⎜ ⎟ ⎟ ⎜ ⎜ ⎥⎜ ⎟ ⎟ ⎜ ⎥⎜ ⎟ ⎟ ⎜ ⎥⎜ ⎟ ⎟ ⎜ ⎥⎜ ⎟ ⎟=⎜ ⎥⎜ ⎟ (C.7) ⎟ ⎜ ⎥⎜ ⎟ ⎟ ⎥ ⎟ ⎟ ⎜ ˆ ω ⎜ βˆ4 p−2 ⎥ ⎜ d 4 p−2 ⎟ ⎜ 4 p−2 ⎟ ⎥⎜ ⎟ ⎝ ⎠ ⎝ dˆ4 p−1 ⎠ cˆ4 p−1 ⎦ ω4 p−1 ω4 p bˆ4 p dˆ4 p

where ⎧ ⎧ αˆ 4i+1 = −(t αˆ = 1/βim−1 i )1 /βim ⎪ ⎪   ⎪ ⎪ 4i+2 ⎪ ⎪ aˆ 4i+1 = (ti )2 + (ti )1 cim /βim /βim−1 ⎪ ⎪ aˆ 4i+2 = (ui )1 ⎪ ⎪ ⎪ ⎪ ⎨ bˆ ⎨ bˆ 4i+2 = (vi )1 4i+1 = (t i )2 (ui )1 − (t i )1 (ui )2 , , c ˆ = (ri )1 c ˆ = − (t (t ) (v ) ) (v ) ⎪ ⎪ 4i+2 4i+1 i i i i 2 1 1 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ βˆ = (ti )1 ⎪ ⎪ = (ti )2 (ri )1 − (ti )1 (ri )2 βˆ ⎪ ⎪ ⎩ 4i+1 ⎩ 4i+2 ˆ ˆ d4i+2 = (yi )1 d4i+1 = (ti )2 (yi )1 − (ti )1 (yi )2 ⎧ ⎧ αˆ 4i+4 = (ui )m (vi )m−1 − (ui )m−1 (vi )m αˆ 4i+3 = (ui )m ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ aˆ 4i+4 = (ui )m (ri )m−1 − (ui )m−1 (ri )m a ˆ = (v ) 4i+3 i m ⎪ ⎪ ⎪ ⎪ ⎨ bˆ ⎨ bˆ = (r ) 4i+4 = (ui)m (t i )m−1 − (ui )m−1 (t i )m 4i+3 i m  , , c ˆ = c ˆ (t ) ⎪ ⎪ 4i+4 = − (ui )m aim+m+1 /αim+m+2 + (ui )m−1 /αim+m+2 4i+3 i m ⎪ ⎪ ⎪ ⎪ ⎪ βˆ = (ui )m /αim+m+1 ⎪ βˆ4i+3 = 1/αim+m+2 ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ 4i+4 ˆ d4i+3 = (yi )m dˆ4i+4 = (ui )m (yi )m−1 − (ui )m−1 (yi )m i = 0, 1, . . . , p − 1, (·)1,··· ,m indicate the 1, . . . , m-th elements of vectors, and the periodic boundary condition is applied. For the case of 4 p n, Eq. C.7 is a loworder cyclic pentadiagonal system, which can be solved with any serial pentadiagonal algorithm efficiently. In summary, the parallel algorithm to solve Eq. C.1 consists of the following steps: 1. Allocate and initialize Ti , Li , Si , di on the i-th node, where 0 ≤ i ≤ p − 1. 2. Use algorithms to compute ui , vi , ri , ti , yi . All computations can be executed parallelly and independently on p processors. 3. Send (ui , vi , ri , ti , yi )1 , (ui , vi , ri , ti , yi )2 , (ui , vi , ri , ti , yi )m−1 , (ui , vi , ri , ti , yi )m and αi·m+m+1 , αi·m+m+2 , ai·m+m+1 , ci·m , βi·m−1 , βi·m on the ith node to main node to form Eq. C.7. 4. Use algorithms to solve Eq. C.7 on the main node. Then send ω4i+1 , ω4i+2 , ω4i+3 , ω4i+4 on the main node to all other nodes. 5. Obtain the solution by computing Eq. C.6.

Appendix C: A Parallel Algorithm for Pentadiagonal Systems

129

References 1. Bondeli S (1991) Divide and conquer: a parallel algorithm for the solution of a tridiagonal linear system of equations. Parallel Comput 17(4–5):419–434 2. Conroy JM (1989) Parallel algorithms for the solution of narrow banded systems. Appl Numerical Math 5(5):409–421 3. Dongarra JJ, Johnsson L (1987) Solving banded systems on a parallel processor. Parallel Comput 5(1–2):219–246 4. Dongarra JJ, Sameh AH (1984) On some parallel banded system solvers. Parallel Comput 1(3–4), 223–235

Appendix D

Filtered Electromagnetic Energy Equation

After filtering Maxwell’s equations, we can obtain the equation of filtered electro! 1 m 2 2 ε0 E + B /μe0 , magnetic energy E = 2 " # B m ∂t E + ∇ · E × (D.1) = −¯j · E. μe0 Separating out Λub α in Eq. 5.33 from the above equation yields "

∂t E

m

B +∇ · E× μe0

# =

$

$ E − E · u˜ α − qα n¯ α % qα n¯ α % E · u˜ α ,

α

(D.2)

α

which can be written in a simple form, ∂t E

m

+ ∇ · Jb = −

$ α

Παbb +

$ α

Λub α ,

(D.3)

where B , is the spatial transport flux of large-scale electromagnetic energy. μe0 bb • Πα = −qα n¯ α % E − E · u˜ α , is the flux of large-scale electromagnetic energy transferred to sub-scale electromagnetic energy. ¯ α% E · u˜ α , is the same as the one in Eq. 5.33. • Λub α = −qα n • Jb = E ×

The equation for filtered total fluid kinetic and electromagnetic energy takes the form " # " # $ $ $ $ $ m f u b ∂t J +J =− Π uu − Π bb − Φ uT . E% + E +∇ · α

α

α

α

α

© Springer Nature Singapore Pte Ltd. 2019 Y. Yang, Energy Transfer and Dissipation in Plasma Turbulence, Springer Theses, https://doi.org/10.1007/978-981-13-8149-2

α

α

α

α

α

(D.4) 131

Index

A Alfvén Mach number, 39 Alfvén waves, 4, 6, 53, 58, 119 Ampère-Maxwell equation, 24, 31 Anisotropy, 6, 98, 113 Artificial diffusivity, 11

B Boltzmann equation, 3, 26, 27

C Charge neutrality, 1, 30, 31 Coherent structures, 14, 98, 101 Collisionless plasma, 3, 14, 91, 96, 112 Collision term, 3, 26, 28 Compact finite difference, 11, 41 Compressibility, 9, 12, 75, 80 Compressible magnetohydrodynamic turbulence, 9, 64, 111 Compressive velocity, 12, 70 Continuity equation, 2, 28 Correlation, 99, 101, 103 CT method, 12, 50 Current sheets, 15, 78 Cyclotron frequency, 25

D Debye length, 1, 94 Deviatoric pressure tensor, 33, 95 Dissipation, 13, 111 Dissipation range, 1 Distribution function, 25 Divergence-free constraint, 11, 50

E Eddies, 5 Eddy turnover time, 5 Eigen-system, 43, 119 Electromagnetic work, 15, 92, 96, 103, 107, 112 Electron inertial scale, 2 Elsasser fields, 7 Energy cascade, 5 Energy-containing range, 1 Energy equation, 2, 28, 92 Energy flux, 5, 8, 74, 79 Energy spectrum, 1, 2, 6, 13, 62 Energy transfer, 2, 4, 8, 74, 79, 85, 102, 106, 111

F Filter, 11, 45, 63, 72 Filtered energy equation, 72, 105 Four-thirds law, 8, 82

G Geometric invariants, 98

H Heat flux vector, 27, 29, 33 Hybrid code, 3 Hybrid scheme, 11, 40, 43, 111 Hyperdiffusivity, 11

I Incompressible magnetohydrodynamic turbulence, 4

© Springer Nature Singapore Pte Ltd. 2019 Y. Yang, Energy Transfer and Dissipation in Plasma Turbulence, Springer Theses, https://doi.org/10.1007/978-981-13-8149-2

133

134 Inertial range, 1, 79 Intermittency, 7, 82 Ion inertial scale, 2 Iroshnikov-Kraichnan spectrum, 6

K K41, 5, 7 Kármán-Howarth-Monin relation, 7 Kinetic scales, 2, 12, 91 Kinetic theory, 25 Kolmogorov’s four-fifths law, 7 Kolmogorov spectrum, 6

L Large-scale force, 70 Locality, 8, 86

M Mach number, 39 Magnetic induction equation, 32 Magnetic Reynolds number, 39 Maxwell’s equations, 23 MHD scales, 1 MHD theory, 29 Momentum equation, 2, 28

O Ohm’s law, 31

P Parallel pentadiagonal algorithm, 52, 125 Particle in cell, 3, 93 Phase space, 3, 25 Plasma frequency, 1 Plasma parameter, 1 Pressure dilatation, 74, 79, 87, 96, 112

Index Pressure-strain interaction, 96, 102 Pressure-stress interaction, 92, 95, 103, 107, 112 Pressure tensor, 27, 29 Projection method, 11, 51

R Rankine-Hugoniot relation, 10, 66 Reynolds number, 39 Runge-Kutta method, 44

S Scale bands, 85, 107 Self-similarity hypothesis, 7 Shock detector, 11, 40, 63 Shock regions, 41 Shocks, 9, 12, 64, 77 Smooth regions, 41 Solenoidal velocity, 12, 70 Structure function, 7, 82

T Third-order moment, 7, 14, 82 Topology, 98 Triadic interaction, 8 Two-fluid theory, 27

V Velocity gradient, 95, 98 Vlasov equation, 3, 27, 91 Vortex structures, 77

W Wave-particle interactions, 13 Weighted Essentially Non-Oscillatory (WENO), 11, 42, 115