Lightning Electromagnetics, Volume 2: Return Electrical processes and effects [2 ed.] 1785615394, 9781785615399

Table of contents :
Cover
Contents
About the editors
Acknowledgements
1 Basic discharge processes in the atmosphere
1.1 Introduction
1.2 Electron avalanche
1.3 Streamer discharges
1.4 Corona discharges
1.5 Thermalization or heating of air by a discharge
1.6 Low-pressure electrical discharges
1.7 Leader discharges
1.8 Some features of mathematical modelling of positive leader discharges
1.9 Leader inception based on thermalization of the discharge channel
References
2 Modelling of charging processes in clouds
2.1 Introduction
2.2 Definitions of some model descriptors
2.2.1 Basic terminology
2.2.2 Terms related to microphysics
2.2.3 Categories of electrification mechanisms
2.2.4 Other categorizations of cloud models
2.3 Brief history of electrification modelling
2.4 Parameterization of electrical processes
2.4.1 Calculating the electric field
2.4.2 Charge continuity
2.4.3 The non-inductive graupel–ice collision mechanism
2.4.4 The inductive charging mechanism
2.4.5 Small ion processes
2.5 Lightning parameterizations
2.5.1 Stochastic lightning model
2.5.2 Pseudo-fractal lightning
2.6 Some applications of models
2.6.1 Ion and inductive mechanisms
2.6.2 Non-inductive graupel–ice sensitivity
2.6.3 Charge structure and lightning type
2.6.4 Concluding remarks
References
3 Numerical simulations of non-thermal electrical discharges in air
3.1 Introduction
3.2 Outline of electro-physical processes in gaseous medium under electric fields
3.2.1 Generation of charged species in gas
3.2.2 Losses of charged species in gas
3.2.3 Dynamics of densities of charge carriers in discharge plasma
3.2.4 Concepts of electron avalanche and streamer
3.3 Hydrodynamic description of gas discharge plasma
3.4 Solving gas discharge problems
3.4.1 Simulations of corona in air
3.4.2 Computer implementation of corona model
3.4.3 Study case: positive corona between coaxial cylinders
3.4.4 Study case: positive corona in rod-plane electrode system
3.5 Simulations of streamer discharges in air
3.5.1 Study case: positive streamer in a weak homogeneous background field
3.5.2 Study case: negative streamer in weak homogeneous background fields
References
4 Attachment of lightning flashes to grounded structures
4.1 Introduction
4.2 Striking distance
4.3 Leader inception models
4.3.1 Critical radius and critical streamer length concepts
4.3.2 Rizk’s generalized leader inception equation
4.3.3 Lalande’s stabilization field equation
4.3.4 Leader inception model of Becerra and Cooray (SLIM)
4.4 Leader progression and attachment models
4.5 The potential of the stepped leader channel and the striking distance
4.5.1 Armstrong and Whitehead
4.5.2 Leader potential extracted from the charge neutralized by the return stroke
4.5.3 Striking distance based on the leader tip potential
4.6 Comparison of EGM against SLIM
4.7 Points where more investigations are needed
4.7.1 Orientation of the stepped leader
4.7.2 The orientation of the connecting leader
4.7.3 The connection between the leader potential and the return stroke current
4.7.4 Inclination of the leader channel
4.7.5 Main assumptions of SLIM
4.8 Concluding remarks
References
5 Modeling lightning strikes to tall towers
5.1 Introduction
5.2 Modeling lightning strikes to tall structures
5.2.1 Engineering models
5.2.2 Electromagnetic models
5.2.3 Hybrid electromagnetic model (HEM)
5.3 Electromagnetic field computation
5.3.1 Electromagnetic field expressions for a perfectly conducting ground
5.3.2 Electromagnetic field computation for a finitely conducting ground
5.4 Review of lightning current data and associated electromagnetic fields
5.4.1 Experimental data
5.4.2 Data from short towers
5.4.3 Summary of Berger’s data
5.4.4 Other data obtained using short towers (≤100 m)
5.4.5 Data from tall towers
5.5 Summary
References
6 Lightning electromagnetic field calculations in the presence of a conducting ground: the numerical treatment of Sommerfeld’s integrals
6.1 Introduction
6.2 Lightning electromagnetic field calculation in presence of a lossy ground with constant electrical parameters
6.2.1 Over-ground electromagnetic field
6.2.2 Underground electromagnetic field
6.3 Lightning electromagnetic field calculation in presence of a lossy ground with frequency-dependent electrical parameters
6.3.1 The dependence of soil conductivity and permittivity on the frequency
6.3.2 Numerical simulation of over-ground and underground lightning electromagnetic field
6.4 Lightning electromagnetic field calculation in presence of a lossy and horizontally stratified ground
6.4.1 Statement of the problem and derivation of the Green’s functions for the electromagnetic field
6.4.2 Derivation of the lightning electromagnetic field
6.4.3 The reflection coefficient R
6.5 Conclusions
References
7 Lightning electromagnetic field propagation: a survey on the available approximate expressions
7.1 Lightning electromagnetic fields over a homogeneous soil
7.1.1 Horizontal electric field – Cooray–Rubinstein (CR) formula
7.1.2 Vertical electric field and azimuthal magnetic field
7.1.3 Lightning electromagnetic fields under the groundCooray formula
7.2 Electromagnetic fields propagation along a horizontally stratified ground
7.2.1 Lightning electromagnetic fields for a two-layer horizontally stratified ground: a simplified formulation
7.2.2 Validation of the simplified formula
7.3 Electromagnetic fields propagation along a vertically stratified ground
7.3.1 Lightning electromagnetic fields for a two-layer vertically stratified ground: a simplified formulation
7.3.2 Validation of the simplified formula
7.4 Summary
References
8 Interaction of lightning-generated electromagnetic fields with overhead and underground cables
8.1 Introduction
8.2 Transmission line theory
8.3 Electromagnetic field interaction with overhead lines
8.3.1 Single-wire line above a perfectly conducting ground
8.3.2 Taylor, Satterwhite, and Harrison model
8.3.3 Agrawal, Price, and Gurbaxani model
8.3.4 Rachidi model
8.3.5 Rusck model and its extensions
8.3.6 Inclusion of losses
8.3.7 Multiconductor lines
8.3.8 Coupling to complex networks
8.3.9 Frequency-domain solutions
8.3.10 Time-domain solutions
8.3.11 Analytical solutions
8.3.12 Application to lightning-induced voltages
8.4 Electromagnetic field interaction with buried cables
8.4.1 Field-to-buried cables coupling equations
8.4.2 Frequency-domain solutions
8.4.3 Time-domain solutions
8.4.4 Lightning-induced disturbances in a buried cable
8.5 Conclusions
Acknowledgments
References
9 Application of scale models to the study of lightning transients in power transmission and distribution systems
9.1 Introduction
9.2 Basis of scale modeling
9.3 Simulation of the electromagnetic environment
9.3.1 Lightning channel
9.3.2 Ground
9.3.3 Overhead lines
9.3.4 Transformers
9.3.5 Surge arresters
9.3.6 Buildings
9.3.7 Transmission line towers
9.4 Evaluation of lightning surges in power lines
9.4.1 Investigations associated with direct strokes
9.4.2 Investigations associated with indirect strokes
9.5 Conclusions
Acknowledgments
References
10 Lightning interaction with the ionosphere
10.1 Introduction
10.2 The full-wave FDTD model of lightning EMPs interaction with the D-region ionosphere
10.2.1 The parameterization of the lower D-region ionosphere
10.2.2 3D spherical model
10.2.3 2D symmetric polar model
10.3 VLF/LF signal of lightning EM fields propagation through the EIWG
10.3.1 The effect of Earth’s curvature
10.3.2 The effect of the ground conductivity
10.3.3 The effect of different D-region ionospheric profiles
10.4 Application to the propagation of NBEs at different distances in the EIWG
10.5 Application to lightning EM field propagation over a mountainous terrain
10.6 Application to the optical emissions of lightninginduced transient luminous events in the nonlinear D-region ionosphere
10.7 Summary
References
11 Lightning effects in the mesosphere
11.1 Introduction
11.2 Sprites
11.2.1 Basic properties and morphology of sprites
11.2.2 Mechanism of the sprite nucleation
11.2.3 Sprite development
11.2.4 Sprite models
11.2.5 Inner structure and color of sprites
11.2.6 ELF/VLF electromagnetic fields produced by sprites
11.2.7 Effects of sprites on the ionosphere
11.3 Blue jet, blue starter, and gigantic jet
11.3.1 Basic properties and morphology of blue and gigantic jets
11.3.2 Development of gigantic jet
11.3.3 Models of gigantic jet
11.4 Elves
11.5 Other transient atmospheric phenomena possibly related to lightning activity
11.5.1 Gnomes and Pixies
11.5.2 Transient atmospheric events
11.5.3 Terrestrial gamma-ray flashes
References
12 The effects of lightning on the ionosphere/magnetosphere: whistlers and ionospheric Alfven resonator
12.1 Introduction
12.2 Lightning-induced whistlers in the ionosphere/ magnetosphere
12.2.1 General description of whistlers
12.2.2 Theoretical background of plasma waves
12.2.3 Use of whistlers as a diagnostic tool of the ionosphere/magnetosphere
12.3 Ionospheric Alfve´n resonator (IAR)
12.3.1 Brief history and general introduction of IAR
12.3.2 Ground-based observations of IARs at middle latitude
12.3.3 Generation mechanisms of IAR
12.3.4 Excitation of IAR by nearby thunderstorms
12.4 Summary of lightning effects on the ionosphere/ magnetosphere
References
13 On the NOx generation in corona, streamer and low-pressure electrical discharges
13.1 Introduction
13.2 Testing the theory using corona discharges
13.3 NOx generation in electron avalanches and its relationship to energy dissipation
13.4 NOx production in streamer discharges
13.5 Discussion and conclusions
References
14 On the NOx production by laboratory electrical discharges and lightning
14.1 Introduction
14.2 NOx production by laboratory sparks
14.2.1 Radius of spark channels
14.2.2 The volume of air heated in a spark channel and its internal energy
14.2.3 NOx production in spark channels
14.2.4 Efficiency of NOx production in sparks with different current wave-shapes
14.2.5 NOx production in sparks as a function of energy
14.3 NOx production in discharges containing long-duration currents
14.4 NOx production in streamer discharges
14.5 NOx production in ground lightning flashes
14.5.1 The model of a ground lightning flash
14.5.2 NOx production in different processes in ground flashes
14.6 NOx production by cloud flashes
14.7 Global production of NOx by lightning flashes
14.8 Conclusions
Appendix 1
References
15 Lightning and climate change
15.1 Introduction
15.2 Basics of thunderstorm electrification and lightning
15.3 Thermodynamic control on lightning activity
15.3.1 Temperature
15.3.2 Dew point temperature
15.3.3 Water vapor and the Clausius–Clapeyron relationship
15.3.4 Convective available potential energy and its temperature dependence
15.3.5 Cloud base height and its influence on cloud microphysics
15.3.6 Balance level considerations in deep convection
15.3.7 Baroclinicity
15.4 Global lightning response to temperature on different time scales
15.4.1 Diurnal variation
15.4.2 Semiannual variation
15.4.3 Annual variation
15.4.4 ENSO
15.4.5 Decadal time scale
15.4.6 Multi-decadal time scale
15.4.7 Hiatus in global warming and “warming hole”
15.5 Aerosol influence on moist convection and lightning activity
15.5.1 Basic concepts
15.5.2 Observational support
15.5.3 Lightning response to the COVID-19 pandemic
15.5.4 Work of Wang et al. (2018) on the global aerosollightning relationship
15.6 Lightning as a climate variable
15.7 Lightning activity at high latitude
15.7.1 The Arctic
15.7.2 Alaska
15.8 Winter-type thunderstorms and lightning
15.8.1 Effects of global warming on winter thunderstorms
15.9 Storms at the mesoscale
15.10 Tropical cyclones
15.11 Cloud-to-ocean lightning
15.12 Lightning superbolts and megaflashes
15.13 Nocturnal thunderstorms
15.14 Meteorological control on lightning type
15.15 The global circuits as monitors for destructive lightning and climate change
15.16 Expectations for the future
References
Index
Back Cover

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IET ENERGY ENGINEERING SERIES 127

Lightning Electromagnetics

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Power Circuit Breaker Theory and Design C.H. Flurscheim (Editor) Industrial Microwave Heating A.C. Metaxas and R.J. Meredith Insulators for High Voltages J.S.T. Looms Variable Frequency AC Motor Drive Systems D. Finney SF6 Switchgear H.M. Ryan and G.R. Jones Conduction and Induction Heating E.J. Davies Statistical Techniques for High Voltage Engineering W. Hauschild and W. Mosch Uninterruptible Power Supplies J. Platts and J.D. St Aubyn (Editors) Digital Protection for Power Systems A.T. Johns and S.K. Salman Electricity Economics and Planning T.W. Berrie Vacuum Switchgear A. Greenwood Electrical Safety: a guide to causes and prevention of hazards J. Maxwell Adams Electricity Distribution Network Design, 2nd Edition E. Lakervi and E.J. Holmes Artificial Intelligence Techniques in Power Systems K. Warwick, A.O. Ekwue and R. Aggarwal (Editors) Power System Commissioning and Maintenance Practice K. Harker Engineers’ Handbook of Industrial Microwave Heating R.J. Meredith Small Electric Motors H. Moczala et al. AC-DC Power System Analysis J. Arrillaga and B.C. Smith High Voltage Direct Current Transmission, 2nd Edition J. Arrillaga Flexible AC Transmission Systems (FACTS) Y-H. Song (Editor) Embedded generation N. Jenkins et al. High Voltage Engineering and Testing, 2nd Edition H.M. Ryan (Editor) Overvoltage Protection of Low-Voltage Systems, Revised Edition P. Hasse Voltage Quality in Electrical Power Systems J. Schlabbach et al. Electrical Steels for Rotating Machines P. Beckley The Electric Car: Development and future of battery, hybrid and fuel-cell cars M. Westbrook Power Systems Electromagnetic Transients Simulation J. Arrillaga and N. Watson Advances in High Voltage Engineering M. Haddad and D. Warne Electrical Operation of Electrostatic Precipitators K. Parker Thermal Power Plant Simulation and Control D. Flynn Economic Evaluation of Projects in the Electricity Supply Industry H. Khatib Propulsion Systems for Hybrid Vehicles J. Miller Distribution Switchgear S. Stewart Protection of Electricity Distribution Networks, 2nd Edition J. Gers and E. Holmes Wood Pole Overhead Lines B. Wareing Electric Fuses, 3rd Edition A. Wright and G. Newbery Wind Power Integration: Connection and system operational aspects B. Fox et al. Short Circuit Currents J. Schlabbach Nuclear Power J. Wood Condition Assessment of High Voltage Insulation in Power System Equipment R.E. James and Q. Su Local Energy: Distributed generation of heat and power J. Wood Condition Monitoring of Rotating Electrical Machines P. Tavner, L. Ran, J. Penman and H. Sedding The Control Techniques Drives and Controls Handbook, 2nd Edition B. Drury Lightning Protection V. Cooray (Editor) Ultracapacitor Applications J.M. Miller Lightning Electromagnetics V. Cooray

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Lightning Electromagnetics Volume 2: Electrical processes and effects 2nd Edition Edited by Vernon Cooray, Farhad Rachidi and Marcos Rubinstein

The Institution of Engineering and Technology

Published by The Institution of Engineering and Technology, London, United Kingdom The Institution of Engineering and Technology is registered as a Charity in England & Wales (no. 211014) and Scotland (no. SC038698). † The Institution of Engineering and Technology 2022 First published 2012 2nd Edition published 2022 This publication is copyright under the Berne Convention and the Universal Copyright Convention. All rights reserved. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may be reproduced, stored or transmitted, in any form or by any means, only with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publisher at the undermentioned address: The Institution of Engineering and Technology Futures Place Kings Way, Stevenage Hertfordshire SG1 2UA, United Kingdom www.theiet.org While the authors and publisher believe that the information and guidance given in this work are correct, all parties must rely upon their own skill and judgement when making use of them. Neither the author nor publisher assumes any liability to anyone for any loss or damage caused by any error or omission in the work, whether such an error or omission is the result of negligence or any other cause. Any and all such liability is disclaimed. The moral rights of the author to be identified as author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

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Contents

About the editors Acknowledgements

1 Basic discharge processes in the atmosphere Vernon Cooray and Liliana Arevalo 1.1 Introduction 1.2 Electron avalanche 1.3 Streamer discharges 1.4 Corona discharges 1.5 Thermalization or heating of air by a discharge 1.6 Low-pressure electrical discharges 1.7 Leader discharges 1.8 Some features of mathematical modelling of positive leader discharges 1.9 Leader inception based on thermalization of the discharge channel References 2 Modelling of charging processes in clouds Edward R. Mansell and Donald R. MacGorman 2.1 Introduction 2.2 Definitions of some model descriptors 2.2.1 Basic terminology 2.2.2 Terms related to microphysics 2.2.3 Categories of electrification mechanisms 2.2.4 Other categorizations of cloud models 2.3 Brief history of electrification modelling 2.4 Parameterization of electrical processes 2.4.1 Calculating the electric field 2.4.2 Charge continuity 2.4.3 The non-inductive graupel–ice collision mechanism 2.4.4 The inductive charging mechanism 2.4.5 Small ion processes 2.5 Lightning parameterizations 2.5.1 Stochastic lightning model 2.5.2 Pseudo-fractal lightning 2.6 Some applications of models 2.6.1 Ion and inductive mechanisms

xix xxi

1 1 2 4 9 10 11 11 13 18 19 23 23 24 24 26 30 31 33 37 37 40 42 51 54 58 60 63 64 64

x

3

4

Lightning electromagnetics: Volume 2 2.6.2 Non-inductive graupel–ice sensitivity 2.6.3 Charge structure and lightning type 2.6.4 Concluding remarks References

65 66 70 71

Numerical simulations of non-thermal electrical discharges in air Y. V. Serdyuk 3.1 Introduction 3.2 Outline of electro-physical processes in gaseous medium under electric fields 3.2.1 Generation of charged species in gas 3.2.2 Losses of charged species in gas 3.2.3 Dynamics of densities of charge carriers in discharge plasma 3.2.4 Concepts of electron avalanche and streamer 3.3 Hydrodynamic description of gas discharge plasma 3.4 Solving gas discharge problems 3.4.1 Simulations of corona in air 3.4.2 Computer implementation of corona model 3.4.3 Study case: positive corona between coaxial cylinders 3.4.4 Study case: positive corona in rod-plane electrode system 3.5 Simulations of streamer discharges in air 3.5.1 Study case: positive streamer in a weak homogeneous background field 3.5.2 Study case: negative streamer in weak homogeneous background fields References

77

Attachment of lightning flashes to grounded structures Vernon Cooray 4.1 Introduction 4.2 Striking distance 4.3 Leader inception models 4.3.1 Critical radius and critical streamer length concepts 4.3.2 Rizk’s generalized leader inception equation 4.3.3 Lalande’s stabilization field equation 4.3.4 Leader inception model of Becerra and Cooray (SLIM) 4.4 Leader progression and attachment models 4.5 The potential of the stepped leader channel and the striking distance 4.5.1 Armstrong and Whitehead 4.5.2 Leader potential extracted from the charge neutralized by the return stroke 4.5.3 Striking distance based on the leader tip potential 4.6 Comparison of EGM against SLIM

77 77 78 79 80 80 82 86 86 87 99 104 109 117 123 127 133 133 135 137 137 138 138 139 139 142 142 142 144 145

Contents 4.7

Points where more investigations are needed 4.7.1 Orientation of the stepped leader 4.7.2 The orientation of the connecting leader 4.7.3 The connection between the leader potential and the return stroke current 4.7.4 Inclination of the leader channel 4.7.5 Main assumptions of SLIM 4.8 Concluding remarks References 5 Modeling lightning strikes to tall towers Farhad Rachidi and Marcos Rubinstein 5.1 Introduction 5.2 Modeling lightning strikes to tall structures 5.2.1 Engineering models 5.2.2 Electromagnetic models 5.2.3 Hybrid electromagnetic model (HEM) 5.3 Electromagnetic field computation 5.3.1 Electromagnetic field expressions for a perfectly conducting ground 5.3.2 Electromagnetic field computation for a finitely conducting ground 5.4 Review of lightning current data and associated electromagnetic fields 5.4.1 Experimental data 5.4.2 Data from short towers 5.4.3 Summary of Berger’s data 5.4.4 Other data obtained using short towers (  100 m) 5.4.5 Data from tall towers 5.5 Summary References 6 Lightning electromagnetic field calculations in the presence of a conducting ground: the numerical treatment of Sommerfeld’s integrals Federico Delfino, Renato Procopio, Mansueto Rossi, Daniele Mestriner and Massimo Brignone 6.1 Introduction 6.2 Lightning electromagnetic field calculation in presence of a lossy ground with constant electrical parameters 6.2.1 Over-ground electromagnetic field 6.2.2 Underground electromagnetic field 6.3 Lightning electromagnetic field calculation in presence of a lossy ground with frequency-dependent electrical parameters

xi 148 148 150 150 152 152 153 153 157 157 157 158 164 164 165 166 171 174 175 176 177 178 181 191 191

201

201 202 203 218 221

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Lightning electromagnetics: Volume 2 6.3.1 The dependence of soil conductivity and permittivity on the frequency 6.3.2 Numerical simulation of over-ground and underground lightning electromagnetic field 6.4 Lightning electromagnetic field calculation in presence of a lossy and horizontally stratified ground 6.4.1 Statement of the problem and derivation of the Green’s functions for the electromagnetic field 6.4.2 Derivation of the lightning electromagnetic field 6.4.3 The reflection coefficient R 6.5 Conclusions References

7

8

Lightning electromagnetic field propagation: a survey on the available approximate expressions Daniele Mestriner, Renato Procopio, Massimo Brignone and Federico Delfino 7.1 Lightning electromagnetic fields over a homogeneous soil 7.1.1 Horizontal electric field – Cooray–Rubinstein (CR) formula 7.1.2 Vertical electric field and azimuthal magnetic field 7.1.3 Lightning electromagnetic fields under the ground-Cooray formula 7.2 Electromagnetic fields propagation along a horizontally stratified ground 7.2.1 Lightning electromagnetic fields for a two-layer horizontally stratified ground: a simplified formulation 7.2.2 Validation of the simplified formula 7.3 Electromagnetic fields propagation along a vertically stratified ground 7.3.1 Lightning electromagnetic fields for a two-layer vertically stratified ground: a simplified formulation 7.3.2 Validation of the simplified formula 7.4 Summary References Interaction of lightning-generated electromagnetic fields with overhead and underground cables Carlo Alberto Nucci, Farhad Rachidi and Marcos Rubinstein 8.1 Introduction 8.2 Transmission line theory 8.3 Electromagnetic field interaction with overhead lines 8.3.1 Single-wire line above a perfectly conducting ground 8.3.2 Taylor, Satterwhite, and Harrison model

222 226 228 228 232 233 236 236

243

243 244 256 259 262 263 265 277 277 278 284 285

291 291 292 294 294 294

Contents 8.3.3 Agrawal, Price, and Gurbaxani model 8.3.4 Rachidi model 8.3.5 Rusck model and its extensions 8.3.6 Inclusion of losses 8.3.7 Multiconductor lines 8.3.8 Coupling to complex networks 8.3.9 Frequency-domain solutions 8.3.10 Time-domain solutions 8.3.11 Analytical solutions 8.3.12 Application to lightning-induced voltages 8.4 Electromagnetic field interaction with buried cables 8.4.1 Field-to-buried cables coupling equations 8.4.2 Frequency-domain solutions 8.4.3 Time-domain solutions 8.4.4 Lightning-induced disturbances in a buried cable 8.5 Conclusions Acknowledgments References 9 Application of scale models to the study of lightning transients in power transmission and distribution systems Alexandre Piantini and Jorge M. Janiszewski 9.1 Introduction 9.2 Basis of scale modeling 9.3 Simulation of the electromagnetic environment 9.3.1 Lightning channel 9.3.2 Ground 9.3.3 Overhead lines 9.3.4 Transformers 9.3.5 Surge arresters 9.3.6 Buildings 9.3.7 Transmission line towers 9.4 Evaluation of lightning surges in power lines 9.4.1 Investigations associated with direct strokes 9.4.2 Investigations associated with indirect strokes 9.5 Conclusions Acknowledgments References 10 Lightning interaction with the ionosphere Dongshuai Li, Alejandro Luque, Marcos Rubinstein and Farhad Rachidi 10.1 Introduction

xiii 295 296 297 298 299 302 302 303 305 306 311 311 314 315 315 316 318 318

325 325 327 329 331 333 333 334 335 338 340 343 343 347 363 364 364 375

375

xiv

Lightning electromagnetics: Volume 2 10.2 The full-wave FDTD model of lightning EMPs interaction with the D-region ionosphere 10.2.1 The parameterization of the lower D-region ionosphere 10.2.2 3D spherical model 10.2.3 2D symmetric polar model 10.3 VLF/LF signal of lightning EM fields propagation through the EIWG 10.3.1 The effect of Earth’s curvature 10.3.2 The effect of the ground conductivity 10.3.3 The effect of different D-region ionospheric profiles 10.4 Application to the propagation of NBEs at different distances in the EIWG 10.5 Application to lightning EM field propagation over a mountainous terrain 10.6 Application to the optical emissions of lightning-induced transient luminous events in the nonlinear D-region ionosphere 10.7 Summary References

11 Lightning effects in the mesosphere Vadim V. Surkov and Masashi Hayakawa 11.1 Introduction 11.2 Sprites 11.2.1 Basic properties and morphology of sprites 11.2.2 Mechanism of the sprite nucleation 11.2.3 Sprite development 11.2.4 Sprite models 11.2.5 Inner structure and color of sprites 11.2.6 ELF/VLF electromagnetic fields produced by sprites 11.2.7 Effects of sprites on the ionosphere 11.3 Blue jet, blue starter, and gigantic jet 11.3.1 Basic properties and morphology of blue and gigantic jets 11.3.2 Development of gigantic jet 11.3.3 Models of gigantic jet 11.4 Elves 11.5 Other transient atmospheric phenomena possibly related to lightning activity 11.5.1 Gnomes and Pixies 11.5.2 Transient atmospheric events 11.5.3 Terrestrial gamma-ray flashes References

376 376 378 381 384 384 386 389 396 400 403 416 417 425 425 429 429 430 433 435 441 443 446 448 448 451 453 458 460 460 460 461 463

Contents 12 The effects of lightning on the ionosphere/magnetosphere: whistlers and ionospheric Alfve´n resonator M. Hayakawa and Y. Hobara 12.1 Introduction 12.2 Lightning-induced whistlers in the ionosphere/magnetosphere 12.2.1 General description of whistlers 12.2.2 Theoretical background of plasma waves 12.2.3 Use of whistlers as a diagnostic tool of the ionosphere/magnetosphere 12.3 Ionospheric Alfve´n resonator (IAR) 12.3.1 Brief history and general introduction of IAR 12.3.2 Ground-based observations of IARs at middle latitude 12.3.3 Generation mechanisms of IAR 12.3.4 Excitation of IAR by nearby thunderstorms 12.4 Summary of lightning effects on the ionosphere/magnetosphere References 13 On the NOx generation in corona, streamer and low-pressure electrical discharges Vernon Cooray, Marley Becerra and Mahbubur Rahman 13.1 Introduction 13.2 Testing the theory using corona discharges 13.3 NOx generation in electron avalanches and its relationship to energy dissipation 13.4 NOx production in streamer discharges 13.5 Discussion and conclusions References 14 On the NOx production by laboratory electrical discharges and lightning Vernon Cooray, Mahbubur Rahman and Vladimir Rakov 14.1 Introduction 14.2 NOx production by laboratory sparks 14.2.1 Radius of spark channels 14.2.2 The volume of air heated in a spark channel and its internal energy 14.2.3 NOx production in spark channels 14.2.4 Efficiency of NOx production in sparks with different current wave-shapes 14.2.5 NOx production in sparks as a function of energy 14.3 NOx production in discharges containing long-duration currents 14.4 NOx production in streamer discharges 14.5 NOx production in ground lightning flashes 14.5.1 The model of a ground lightning flash 14.5.2 NOx production in different processes in ground flashes

xv

475 476 477 477 479 491 496 496 498 507 508 516 517

527 527 529 529 530 532 534

537 537 538 538 540 541 542 544 546 546 547 547 548

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Lightning electromagnetics: Volume 2 14.6 NOx production by cloud flashes 14.7 Global production of NOx by lightning flashes 14.8 Conclusions Appendix 1 References

15 Lightning and climate change Earle R. Williams, Joan Montanya, Joydeb Saha and Anirban Guha 15.1 Introduction 15.2 Basics of thunderstorm electrification and lightning 15.3 Thermodynamic control on lightning activity 15.3.1 Temperature 15.3.2 Dew point temperature 15.3.3 Water vapor and the Clausius–Clapeyron relationship 15.3.4 Convective available potential energy and its temperature dependence 15.3.5 Cloud base height and its influence on cloud microphysics 15.3.6 Balance level considerations in deep convection 15.3.7 Baroclinicity 15.4 Global lightning response to temperature on different time scales 15.4.1 Diurnal variation 15.4.2 Semiannual variation 15.4.3 Annual variation 15.4.4 ENSO 15.4.5 Decadal time scale 15.4.6 Multi-decadal time scale 15.4.7 Hiatus in global warming and “warming hole” 15.5 Aerosol influence on moist convection and lightning activity 15.5.1 Basic concepts 15.5.2 Observational support 15.5.3 Lightning response to the COVID-19 pandemic 15.5.4 Work of Wang et al. (2018) on the global aerosol-lightning relationship 15.6 Lightning as a climate variable 15.7 Lightning activity at high latitude 15.7.1 The Arctic 15.7.2 Alaska 15.8 Winter-type thunderstorms and lightning 15.8.1 Effects of global warming on winter thunderstorms 15.9 Storms at the mesoscale 15.10 Tropical cyclones 15.11 Cloud-to-ocean lightning 15.12 Lightning superbolts and megaflashes 15.13 Nocturnal thunderstorms

556 558 560 561 562 569 569 572 573 573 573 573 575 577 581 582 582 583 583 585 586 588 588 590 590 590 591 593 593 593 594 594 595 595 596 597 597 599 600 602

Contents 15.14 15.15

Meteorological control on lightning type The global circuits as monitors for destructive lightning and climate change 15.16 Expectations for the future References Index

xvii 604 604 606 608 627

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About the editors

Vernon Cooray is a professor emeritus at the Department of Electrical Engineering of Uppsala University, Sweden. A fellow of the IEEE and recipient of the Berger award, he is also in charge of the HV Laboratory at Uppsala University. He has authored and co-authored about 350 scientific papers and books, served as keynote speaker and session convener at various international conferences, on journal boards, and as president of ICLP. Farhad Rachidi is a professor at Ecole Polytechnique Federale de Lausanne, Switzerland. A Fellow of IEEE, he is head of the EMC Laboratory at the Swiss Federal Institute of Technology. His research focus includes lightning electromagnetics, and EMP interaction with transmission lines. Prior assignments include the NASA Kennedy Space Centre. He has served on key journal boards and as chairman or convener to key events and working groups, and has published 150 papers in peer-reviewed journals. Marcos Rubinstein is a professor in telecommunications at the University of Applied Sciences of Western Switzerland. He is an IEEE fellow and member of the Institute for Information and Communication Technologies, and serves on key positions such as head of the Applied Electromagnetics Group and chairman of the International Project on Electromagnetic Radiation from Lightning to Tall structures. He has authored or co-authored over 200 scientific publications in journals and conferences, and received several prestigious awards.

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Acknowledgements

We wish to thank all our colleagues who have spent a good deal of their free time writing the chapters of this book. We wish to express our sincere thanks to Ms. Olivia Wilkins and Ms. Nikki Tarplett from the IET publishers and Mr. N. Srinivasan from MPS Limited for their outstanding support throughout the publishing project. Despite unexpected delays in our submissions, they remained patient and accommodating, always willing to listen to our suggestions and provide valuable feedback. We are truly grateful for their professionalism and dedication.

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Chapter 1

Basic discharge processes in the atmosphere Vernon Cooray1 and Liliana Arevalo2

1.1 Introduction The main constituents of air in the Earth’s atmosphere are nitrogen (78%), oxygen (20%), noble gases (1%), water vapour (0.03%), carbon dioxide (0.97%), and other trace gas species. In general, air is a good insulator and it can maintain its insulating properties until the applied electric field exceeds about 2.8
104 V/cm at standard atmospheric conditions (i.e. T = 293 K and P = 1 atm). When the background electric field exceeds this critical value, the free electrons in air, generated mainly by the high energetic radiation of cosmic rays and radio active gases generated from the Earth, start accelerating in this electric field and gain enough energy between collisions with atoms and molecules to ionize other atoms. This cumulative ionization leads to an increase in the number of electrons initiating the electrical breakdown of air. The threshold electric field necessary for electrical breakdown of air is a function of atmospheric density. For example, the critical electric field, E, necessary for electrical breakdown of air of density d is given by E ¼ E0

d d0

(1.1)

where d0 is the density of air at sea level at standard atmospheric conditions and E0 is the corresponding critical electric field necessary for electrical breakdown under the same conditions. Since the density of air in the Earth’s atmosphere decreases with height z (in m) as d ¼ d0 ez=lp with lp  7:64
103 m, the critical electric field necessary to cause electrical breakdown in the atmosphere decreases with height as E ¼ E0 ez=le

(1.2)

When the electric field in the atmosphere increases beyond this critical value, the appearance of the resulting electrical discharge depends on the pressure and the spatial variation in the electric field. Irrespective of its apparent features the basic constituents of an electrical discharge in air can be separated into four parts. These 1 2

Department of Electrical Engineering, Uppsala University, Sweden Hitachi Energy Hr Dc Insulation System, Ludvika, Sweden

2

Lightning electromagnetics: Volume 2

are electron avalanches, streamer discharges, corona discharges, and leaders. When the leaders reach an electrode of opposite polarity or a region of opposite charge density, a rapid neutralization of the charge on the leader takes place. This neutralization process is called a return stroke. The exact mechanism of the return stroke is not yet known, but different types of models have been developed to describe them. These models are described in several chapters of this book. Here, we will concentrate on the four discharge processes mentioned above. Some parts of this chapter are adopted and summarized from Ref. [1] where an extensive description of basic physics of discharges is given.

1.2 Electron avalanche Consider a free electron originated at x = 0 in space and moving under the influence of a background electric field directed in the negative x direction. If the background electric field is larger than the critical value necessary for cumulative ionization, the electron may produce another electron through ionization collisions and these two electrons in turn will give rise to two more electrons. In this way the number of electrons increases with increasing x. Assume that the number of electrons at a distance x from the origin is nx. Let a be the number of ionizing collisions per unit length made by an electron travelling in the direction of the electric field. As the ionization processes increase the number of electrons in air, some of the electrons will get attached to electronegative gases such as oxygen in air. Let h be the number of electron attachments per unit length. The parameter a is called the Townsend’s first ionization coefficient and the parameter h is called the attachment coefficient. Consider an elementary length of width dx located at a distance x from the origin. In travelling across the length dx, nx number of electrons will give rise to dn additional electrons dn ¼ nx ða  hÞdx

(1.3)

The solution of this equation is nx ¼ eðahÞx

(1.4)

This equation shows that the number of electrons increases exponentially with distance. This exponential growth of electrons with distance is called an electron avalanche. The equation also shows that cumulative ionization is possible only if ða  hÞ > 0. The magnitude of both a and h depends on the background electric field and the air density. The quantity ða  hÞ is known as the effective ionization  . For electric field values less than the critical value coefficient and denoted by a necessary for electrical breakdown, ða  hÞ < 0 and for higher electric fields ða  hÞ > 0. This explains the reason for the existence of a critical electric field beyond which the air breaks down electrically. As one can see from the above equation whether the avalanche will continue to grow (i.e. nx continue to increase with distance) or whether it will start to decay after an initial growth (i.e. nx will decrease with distance) depends on the spatial distribution of the electric field. As long as the background electric field is such that

Basic discharge processes in the atmosphere 8 × 107

Number of electrons

300

Electric field (kV/cm)

3

200

100

6 × 107

4 × 107

2 × 107

0 × 100

0 0

100

200

300

400

0

Distance (µm)

100

200

300

400

Distance (µm)

(a)

(b)

Figure 1.1 (a) The electric field used in demonstrating the spatial variation in the number of electrons in the avalanche head as the avalanche advances in an electric field. (b) The variation in the number of electrons in the avalanche head as a function of distance as the avalanche propagates in the electric field shown in (a). Note that the origin of the avalanche is at the point corresponding to zero distance

ða  hÞ > 0, the avalanche continues to grow while it starts to decay when ða  hÞ < 0. In order to illustrate this, consider an electric field that originates from a pointed source and decreases exponentially with distance. The electric field at the source is larger than the critical value necessary for electrical breakdown in standard atmosphere (i.e. 2.8
104 V/cm). An example is shown in Figure 1.1(a). Consider an electron avalanche that originates at the source and moves into the low field region. Figure 1.1(b) shows how the number of electrons at the avalanche head varies as the avalanche extends into the low field region. Observe that the number of electrons increases initially, but it will start to decrease when the electric field goes below the critical value necessary for electrical breakdown. In calculating the electron number in the avalanche, it is necessary to evaluate a and h as a function of the background electric field. Denoting the gas density by N (in cm3) and the background electric field by E (in V/cm), these dependencies can be described by the following equations [2,3]: 2
3 15 7:248
10 a 5cm2 for E > 1:5
1015 V=cm2 (1.5)   ¼ 2:0
1016 exp4 E N N N

2
3 15 5:593
10 a 5cm2 for E  1:5
1015 V=cm2
E  ¼ 6:619
1017 exp4 N N N

(1.6)

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Lightning electromagnetics: Volume 2

  h E E ¼ 8:889
105 þ 2:567
1019 cm2 for > 1:05
1015 V=cm2 N N N

(1.7)

  h E E ¼ 6:089
104  2:893
1019 cm2 for  1:05
1015 V=cm2 N N N

(1.8)

As the avalanche grows the number of electrons at the head increases. These electrons will spread out due to random diffusion causing the avalanche head to expand. The p average ffiffiffiffiffiffiffiffi radius of the avalanche head can be calculated from the equation r ¼ 4Dt where t ¼ x=vd is the time of advance of the avalanche, D is the coefficient of diffusion and vd is the drift velocity of the electrons in the electric field. The drift velocity of the electrons can be obtained from the equation:  0:6064 16 E cm=s (1.9) vd ¼ 2:157
10 N and the coefficient of diffusion D can be obtained from the following expressions [2]:  0:3441 D E E ¼ 5:645
104 V for < 2:0
1017 V=cm2 (1.10) me N N and  0:46113 D E 7 E ¼ 2:173
10 V 2:0
1017 < < 1:23
1016 V=cm2 me N N (1.11) In the above equations me is the electron mobility that can be extracted from (1.9).

1.3 Streamer discharges The analysis given in the previous section shows that as the avalanche increases in length, the charge accumulated at the head of the avalanche increases. As a result, the electric field produced by this charge located at the head of the avalanche also increases as the avalanche moves forward. If the background electric field supports the growth of the avalanche, a situation will be reached that the electric field produced by the charge located at the avalanche head will overwhelm the critical electric field necessary for electrical breakdown in the medium. At this stage the electric field produced by the charges located at the avalanche head, i.e. the space charge electric field, starts influencing the ionization processes taking place in the vicinity of the avalanche head. When this stage is reached, the avalanche will convert itself to a streamer discharge. The exact mechanism of the formation of a streamer discharge from an avalanche depends on the polarity of the source that

Basic discharge processes in the atmosphere

5

generates the background electric field. First consider a source at positive polarity. The source could be a charged graupel particle (in a thundercloud) in a background electric field, a Franklin rod exposed to the background electric field of a thundercloud or a high-voltage electrode. For clarity we will refer to it as the anode. If the electric field in front of the anode is high enough, a photoelectron generated at a point located in front of the anode will initiate an avalanche that propagates towards the anode. The process is depicted in Figure 1.2. As the electron avalanche propagates towards the anode, mobile positive space charge accumulates at the avalanche head. When the avalanche reaches the anode, the electrons will be absorbed into it leaving behind the net positive space charge. Because of the

Figure 1.2 Mechanism of positive streamers. A photoelectron generated at a point located in front of the anode will initiate an avalanche that propagates towards the anode. When the avalanche reaches the anode, the electrons will be absorbed into it leaving behind the net positive space charge. If the number of positive ions in the avalanche head is larger than a critical value, secondary avalanches created by the photons will be attracted towards the positive space charge. The positive space charge will be neutralized by the electrons in the secondary avalanches creating a weakly conducting channel. Consequently, a part of the anode potential will be transferred to the channel making it positively charged and increasing the electric field at the tip. The high electric field at the tip attracts more electron avalanches towards it and the channel grows as a consequence

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Lightning electromagnetics: Volume 2

recombination of positive ions and electrons, avalanche head is a strong source of high energetic photons. These photons will create other avalanches in the vicinity of the positive space charge. If the number of positive ions in the avalanche head is larger than a critical value, the electric field created by the space charge becomes comparable or overwhelms the critical electric field. As a result the secondary avalanches created by the photons will be attracted towards the positive space charge. The electrons in the secondary avalanches will be neutralized by the positive space charge of the primary avalanche leaving behind a new positive space charge, little bit away from the anode. Furthermore, the neutralization process leads to the creation of a weakly conducting channel and a part of the anode potential will be transferred to this channel making it positively charged and increasing the electric field at the tip. The high electric field at the tip of this weakly conducting channel attracts more electron avalanches towards it and the resulting neutralization process causes the weakly conducting channel to extend in a direction away from the anode. This discharge that travels away from the anode is called a positive streamer. Now let us consider a source of negative polarity, i.e. a cathode. A photoelectron generated close to the cathode will generate an avalanche (primary avalanche) that moves away from the cathode leaving behind positive charge close to it. The process is depicted in Figure 1.3. When the avalanche reaches a critical size, the positive charge of the avalanche starts attracting secondary avalanches towards it. Like in the case of a positive streamer the electrons in the secondary avalanches neutralize this positive charge effectively moving it towards the cathode. When the positive charge reaches the cathode, the field enhancement associated with the proximity of positive space charge to the cathode leads to the emission of electrons from the latter. These electrons will neutralize the positive space charge creating a weakly conducting channel that connects the negative head of the electron avalanche to the cathode. As a consequence, a part of the cathode potential will be transferred to the head of this weakly ionized channel (i.e. negative streamer) increasing the electric field at its head. This streamer head will now act as a virtual cathode and the process is repeated. Repetition of this process leads to the propagation of the negative streamer away from the cathode. If the background electric field is very high, the positive space charge of the primary avalanche may reach the critical size necessary for streamer formation before reaching the anode. This may lead to the formation of a bi-directional discharge, the two ends of which travel towards the anode and the cathode, former as a negative streamer and the latter as a positive streamer. Such a discharge is called a mid gap streamer. So far we have not discussed the exact condition under which an avalanche will be converted to a streamer. As mentioned earlier, the avalanche to streamer transition takes place when the number of charged particles at the avalanche head exceeds a critical value, Nc. From cloud chamber photographs of the avalanches and streamers, Raether [4] estimated that an avalanche will be converted to a streamer when the number of positive ions in the avalanche head reaches a critical value of about 108. A similar conclusion is also reached independently by Meek [5]. On the other hand

Basic discharge processes in the atmosphere

7

Figure 1.3 Mechanism of negative streamers. An photoelectron generated close to the cathode will generate an avalanche that moves away from the cathode leaving behind positive charge close to it. When the avalanche reaches a critical size, the positive charge of the avalanche starts attracting secondary avalanches towards it. Like in the case of a positive streamer, the electrons in the secondary avalanches neutralize this positive charge effectively moving it towards the cathode. When the positive charge reaches the cathode, the field enhancement associated with the proximity of positive space charge to the cathode leads to the emission of electrons from the latter. These electrons will neutralize the positive space charge creating a weakly conducting channel that connects the negative head of the electron avalanche to the cathode. A part of the cathode potential will be transferred to the head of this weakly ionized channel (i.e. negative streamer) increasing the electric field at its head. This streamer head will now act as a virtual cathode and the process is repeated. Repetition of this process leads to the propagation of the negative streamer away from the cathode

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Lightning electromagnetics: Volume 2

Bazelyan and Raizer [6] suggest 109 as a reasonable value for this transformation. Thus, the condition for the transformation of an avalanche to a streamer can be written as Ð xc ½aðxÞhðxÞdx ¼ 108  109 (1.12) e0 Note that in writing down the above equation, it is assumed that the electric field is not uniform and therefore both a and h are a function of distance. Moreover, in the above equation the distance x is measured from the origin of the avalanche and xc is the distance from the origin of the avalanche where the background electric field goes below the critical value necessary for electrical breakdown. The advancement of the streamer in a given background electric field is facilitated by the large increase in the local electric field in the vicinity of the streamer head and by the enhanced production of the photons from the streamer head. The photons create secondary electrons in front of the streamer head and these secondary electrons give rise to secondary avalanches that will move, in the case of positive streamers, towards the streamer head. Once initiated, the streamers have been observed to travel in background electric fields that itself cannot support avalanche formation. Thus, the secondary avalanche formation in the streamer is confined to a very small region around the streamer head where the electric field exceeds 2.8
104 V/cm, the minimum electric field required for the cumulative ionization in air at atmospheric pressure. This region is called the active region. The dimension of the active region is about 200 mm and the streamer radius was found to be on the order of 10–50 mm [7,8]. This value, however, may correspond to short streamers. Since the electron multiplication in the active region is supported by the space charge electric field of the streamer head, the streamer can propagate in electric fields that are much smaller than the critical electric field necessary for cumulative electron ionization. In air, the background electric field necessary for positive streamer propagation lies in the range of 4.5–5
103 V/cm [9–11]. For negative streamers it lies in the range of 1–2
104 V/cm. Any variation in the electron loss processes can change this electric field. For example, when air is saturated with water vapour, the critical electric field for positive streamer propagation grows from 4.7
103 V/cm at humidity of 3 g/m3 to 5.6
103 V/cm at 18 g/cm3 [12,13]. The critical electric field necessary for streamer propagation decreases approximately linearly with decreasing air density [6]. In background electric fields close to the critical value necessary for streamer propagation, the speed of streamers is about 107 cm/s. However, the streamer speed increases with increasing background electric field. No direct measurements are available today on the potential gradient of the streamer channels. Experiments conducted with long sparks show that the average potential gradient of the electrode gap when the positive streamers bridges the gap between the two electrodes is about 5
103 V/cm [14]. This indicates that the potential gradient of the positive streamer channels in air at atmospheric pressure is close to this value. Note that this value is approximately the same as the critical electric field necessary for the propagation of positive streamers.

Basic discharge processes in the atmosphere

9

1.4 Corona discharges In many situations, the electric field in air in the vicinity of objects exposed to high external electric fields may overwhelm the critical electric field necessary for the formation of electron avalanches in air. Moreover, the extent of the volume in which this high electric field exist may confine to a very small region around the object (i.e. the electric field is strongly non-uniform) so that it would not lead to any electrical breakdown between the object under consideration and another one in its vicinity. In this case the electrical activity will be concentrated and confined to a small volume around the object. These types of discharge activity are called corona discharges. Corona discharge consists of either electron avalanches, streamers or both. During corona discharges ionic space charge of both polarities accumulate near the highly stressed electrode, thus modifying the electric field distribution. The equilibrium between accumulation and removal of space charge causes several modes of corona discharges. Moreover, the physical nature of these corona discharges is affected by the electronegativity of the gas under consideration. Consider first the application of a positive voltage to a point, i.e. an anode. Initially, electron avalanches start from a certain distance from the anode and start moving towards it. If the electric field is high enough, some of these avalanches may reach a critical size necessary for avalanche to streamer conversion at the anode. As a consequence, the positive space charge left behind by the avalanches (note that the electrons will be absorbed into the anode) may give rise to positive streamers. These streamers may propagate a short distance into the gap (i.e. away from the anode) and stop. The streamers will leave behind positive space charge close to the anode and this positive charge reduces the electric field at the anode leading to the cessation of streamer formation until the space charge is removed. As the space charge moves away from the anode, the electric field recovers and a new set of streamers may start from the anode. Moreover, as the space charge screens one region of the anode, streamers may develop from another region. Thus, the discharge activity spreads across the anode. This discharge activity is called onset streamers. As the electric field increases further the activity increases. However, as the electrons are not absorbed readily at the anode, they will form a negative ion sheath close to the anode and between the anode and the positive space charge. This ion sheath is called Hermitian sheath. The sheath increases the anode field but reduces the field outside so that streamer formation is quenched. The length of the high field region, i.e. the one between the anode and the negative sheath, is not long enough to give rise to streamers, but formation of electron avalanches takes place in there. Moreover, the electric field inside this zone is high enough to detach negative ions. This discharge is called glow corona or Hermitian glow. As the field increases further, the electric field outside the sheath becomes large enough and the streamer formation starts again. They are called breakdown streamers and they extend into the gap a distance depending on the applied voltage. Now let us consider the case in which the source is of negative polarity, i.e. a cathode. As the electric field increases, electron avalanches start from the cathode. The resulting positive ions will travel towards the cathode. The collision of these

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positive ions with the cathode gives out electrons that support the formation of avalanches. The electron avalanches will move out a certain distance from the cathode, but as they move away from the cathode the electric field decreases and the electrons will be attached to oxygen molecules in air. This gives rise to a negative space charge region. This negative space charge will reduce the electric field and choke off the discharge activity. However, as the negative space charge moves away due to the action of the electric field, the electric field at the cathode recovers and a new discharge activity starts from the cathode. This mode of discharge, relaxation and continuation, continues for a considerable potential range. This oscillating corona discharge is called Trichel pulses because the current consists of pulses separated in time. The repetition frequency of the pulses depends on the applied voltage but can reach values about 106 s1. As the electric field continues to increase, the negative ions are created too far from the cathode to choke off the discharge activity and a pulseless discharge activity, i.e. a glow discharge, starts from the cathode. This will continue for some range of potentials and with increasing electric fields negative streamers start to form from the cathode. Initially, they do not propagate far into the gap but with increasing voltage they move further and further into the gap.

1.5 Thermalization or heating of air by a discharge In the streamer phase of the discharge, many free electrons are lost due to attachment to electronegative oxygen in air. Furthermore, a considerable amount of energy gained by electrons from the electric field is used in exciting molecular vibrations. Since the electrons can transfer only a small fraction of their energy to neutral atoms during elastic collisions, the electrons have a higher temperature than the neutrals. That is, the gas and the electrons are not in thermal equilibrium. As the gas temperature rises to about 1,600–2,000 K, rapid detachment of the electrons from oxygen negative ions supply the discharge with a copious amount of electrons thus enhancing the ionization [15]. As the temperature rises, the time necessary to convert the energy stored in the molecules as vibrational energy to thermal or translational energy decreases and the vibrational energy converts back to translational energy thus accelerating the heating process. As the ionization process continues, the electron density in the channel continues to increase. When the electron density increases to about 1017 cm3, a new process starts in the discharge channel. This is the strong interaction of electrons with each other and with positive ions through long-range Coulomb forces [15]. This leads to a rapid transfer of the energy of electrons to positive ions causing the electron temperature to decrease, while the ion temperature increases. The positive ions, having the same mass as the neutrals, transfer their energy very quickly, in a time on the order of 108 or less (depending on the volume of the discharge) to neutrals. This results in a rapid heating of the gas. At this stage, the thermal ionization (ionization caused by the impact of ions and neutrals) sets in causing a rapid increase in the ionization and the conductivity of the channel. This process is called thermalization. During thermalization as the electron temperature

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decreases, the gas temperature increases and very quickly all the components of the discharge namely, electrons, ions and neutrals, will achieve the same temperature and the discharge will reach local thermodynamic equilibrium.

1.6 Low-pressure electrical discharges As described previously, avalanche to streamer transition requires that the avalanche grows to about 108–109 electrons and the space charge in the avalanche tip creates an electric field that is capable of attracting electron avalanches towards it. As the pressure decreases, the avalanche has to grow to longer and longer lengths before it can accumulate enough space charge at its head to modify the background electric field. This could be the case in the case of low-pressure discharges taking place in the upper atmosphere (i.e. sprites and elves) during thunderstorms. In these, the length of streamer like discharges may exceed hundreds of metres to kilometres. Another interesting feature of these low-pressure discharges is the lack of thermalization process. As mentioned earlier, the thermalization requires increasing the electron density beyond a certain limit. In low-pressure discharges, the density of molecules and atoms is such that the electron densities never reach the critical values necessary for thermalization. In these discharges, the electron temperature remains very high while the gas temperature remains close to ambient. The electron impacts are the dominant mechanism of ionization in these discharges.

1.7 Leader discharges In Section 1.3, we have considered the conditions necessary for the initiation of streamer discharges in a given electric field configuration. The streamer is a cold discharge (i.e. the gas temperature in the channel is close to ambient) and the conductivity of the streamer channel is rather small. However, a leader discharge is a hot discharge and the conductivity of the discharge channel is high. Let us now consider how the streamers will be transformed into a leader and how the leader propagates in a background electric field. Let us first consider a positive leader discharge. Consider an anode whose potential rises rapidly in time. When the electric field at the surface and in the vicinity of the anode increases to a value large enough to convert avalanches to streamers (i.e. (1.12) is satisfied), a bust of streamers is generated from the anode. Many of these streamers have their origin in a common channel called the streamer stem. The streamers stop when the electric field decreases below the critical value necessary for their propagation. Each individual streamer is a cold discharge and the current associated with this cannot heat the air sufficiently to make it highly conducting. However, the combined current of all streamers flowing through the stem causes this common region to heat up increasing the conductivity of the stem. The increase in the temperature causes the gas to expand making the E/N (E is the electric field and N is the gas density) ratio to increase leading to an increase in ionization and electron production. Since the current will be concentrated into a thin channel (i.e. the streamer stem), this in turn

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produce more heating and accelerate the ionization process. With increasing ionization, the process of thermalization sets in transforming the stem into a hot and conducting channel called the leader. Owing to its high conductivity, most of the voltage of the anode will be transferred to the head of the leader channel resulting in a high electric field there. This high electric field leads to the production of streamer discharges now from a common stem located at the head of the leader channel. With the aid of cumulative streamer currents, the new stem gradually transforms itself to a newly created leader section with the streamer process now repeating at the new leader head. The streamer system located in front of the leader is the source of current that heats the air and makes possible the elongation of the leader. The main sequences of the propagation of a positive leader are shown in Figure 1.4. T1

T2

Streamer stem

T3

T4

T5

Thermalized leader channel

Streamer bursts

Figure 1.4 Mechanism of positive leaders. When the electric field at the surface and in the vicinity of the anode increases to a value large enough to convert avalanches to streamers, a bust of streamers is generated from the anode (T1). Many of these streamers have their origin in a common channel called the streamer stem. The combined current of all streamers flowing through the stem causes this common region to heat up and, as a result, the stem will be transformed into a hot and conducting channel called the leader (T2). Owing to its high conductivity, most of the voltage of the anode will be transferred to the head of the leader channel resulting in a high electric field there. This high electric field leads to the production of streamer discharges now from a common stem located at the head of the leader channel (T2). With the aid of cumulative streamer currents, the new stem gradually transforms itself to a newly created leader section with the streamer process now repeating at the new leader head (T3, T4, T5)

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The leader usually supports a current of about 1 A at a relatively low longitudinal electric field of about 103 V/cm. The speed of propagation of the leader is about a few centimetres per second. The spectroscopic measurements show that the air in the leader channel is heated to about 5,000 K [9,10]. The development of the negative leader discharge is more complicated. As in the case of positive leaders, a negative leader also originates with a streamer burst issued from the high voltage electrode i.e. the cathode in this case. It also maintains its propagation with the aid of negative streamers generated from its head. However, the detailed mechanism of its propagation is different to that of positive leaders. A simplified schematic diagram giving the main features of propagation of a negative leader is shown in Figure 1.5. Once a negative streamer burst is generated from the leader head, a unique feature, called, a pilot system, that does not exist in the positive leaders manifests in the system. The pilot system consists of a bright spot called space stem, from which streamers of both polarity develops in opposite directions. The location of the space stem is usually at the edge of the negative streamer system. The action of these streamers heats the space stem and converts it to a hot channel. This is called a space leader. The positive streamers from the space leader propagate towards the head of the negative leader and the negative streamers generated from the other end of the space leader propagate in the opposite direction. Indeed, the positive streamers of the space stem propagate in the region previously covered by negative streamers. The space leader lengthens with a higher velocity towards the cathode (3 cm/ms) than towards the anode (1 cm/ ms). As the space leader approaches the main leader, the velocity of both increases exponentially. The connection of the two leaders is accompanied by a simultaneous illumination of the whole channel starting from the meeting point. During this process, the space leader acquires the potential of the negative leader, and the negative end of the space leader becomes the new tip of the negative leader. In photographs it appears as if the negative leader extends itself abruptly in a leader step. The change in the potential of the previous space leader generates an intense burst of negative corona streamers from its negative end that has now become the new head of the negative leader. Now a new space stem appears at the edge of the new streamer system and the process repeats itself. Recent evidence shows that repeated interaction of the negative leader with the space leader is the reason for the stepwise elongation of the negative leaders as observed in negative stepped leaders in lightning flashes [16]. Models that describe the propagation of negative leaders taking into account the space leaders were published by Mazur et al. [17] and Arevalo and Cooray [18].

1.8 Some features of mathematical modelling of positive leader discharges Consider a grounded object that is exposed to an electric field. An example being a Franklin rod exposed to the electric field generated by a downward moving negative stepped leader. The goal is to simulate the initiation and propagation of the

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T3

T2

Streamer bursts

T4

T5

T6

T7

T8

Negative leader channel

New leader step

Space stem

Space leader

Figure 1.5 Propagation of negative leaders. Once a negative streamer burst is generated from the leader head, a unique feature, called, a pilot system, that does not exist in the positive leaders manifest in the system. The pilot system consists of a bright spot called space stem, from which streamer of both polarity develop in opposite directions (T2–T3). The location of the space stem is usually at the edge of the negative streamer system. The action of these streamers heats the space stem and converts it to a hot channel. This is called a space leader. The space leader advances in both direction (the speed of extension of the positive end is generally higher than that of the negative end) through the cumulative action of positive streamers (generated from the side facing the negative leader) and negative streamers (generated from the opposite side) (T4–T5). The connection of the two leaders is accompanied by a simultaneous illumination of the whole channel stating from the meeting point (T6–T7). During this process the space leader acquires the potential of the negative leader. In fact, during this process (i.e. stepping process) the space leader becomes the new step or the new section of the negative leader channel and the negative end of the space leader becomes the new tip of the negative leader. During the formation of the step, a new streamer burst is generated from the new leader head and the process is repeated (T8). Note that while the space leader travels towards the negative leader, the latter itself may continue to grow in length as shown in the diagram. (The processes associated with the origin of the negative leader, which are almost identical to that of positive leaders, are not shown in the diagram.)

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connecting leader (a positive leader discharge) from the Franklin rod. A brief description of how this could be achieved is given below. The description is based on the work published previously by Becerra and Cooray [19–21]. Assume that the electric field at ground level as a function of time generated by the down coming stepped leader is known. This can be calculated, e.g. by using the leader charge distribution as extracted by Cooray et al. [22]. The simulation consists of several main steps and let us take them one by one. a) The first step is to extract the time or the height of the stepped leader when streamers are incepted from the grounded rod. Since the background electric field is known, the electric field at the tip of the grounded rod can be calculated, e.g. by using charge simulation method. This field is used together with the avalanche to streamer conversion criterion given in (1.12) to investigate whether the electric field at the conductor tip is large enough to convert avalanches to streamers. The simulation continues using the time-varying electric field of the stepped leader until the streamer inception criterion is satisfied. b) The moment the streamer inception criterion is satisfied a burst of streamers will be generated by the tip of the rod. The next task is to calculate the charge in this streamer burst. The charge associated with the streamer burst is calculated using a distance–voltage diagram with the origin at the tip of the grounded conductor as follows (see Figure 1.6). The streamer zone is assumed to maintain a constant potential gradient Estr. In the distance–voltage diagram, this is represented by a straight line (note that the point of zero distance corresponds to the tip of the conductor). On the same diagram the background potential produced by the thundercloud and the down-coming stepped leader (taking also into account the presence of the lightning conductor) at the current time is depicted. If the area between the two curves up to the point where they cross is A, the charge in the streamer zone is given by Q0  KQ A

(1.13)

where KQ is a geometrical factor. Becerra and Cooray [20] estimated its value to be about 3.5
10-11 C/V m. c) The next task is to investigate whether this streamer burst is capable of generating a leader. This decision is based on the fact that in order to generate a leader a minimum of 1 mC is required in the charge generated by the streamers. If the charge in the streamer zone is less than this value, then the procedure is repeated after a small time interval. Note that with increasing time the electric field generated by the stepped leader increases and, consequently, the charge in the streamer bursts increases. Thus at a certain time the condition necessary for the leader inception will be fulfilled. d) Assume that at time t, the condition necessary for leader inception is satisfied. The next task is to estimate the length and the radius of this initial leader section. In doing this it is assumed that the amount of charge you need to create a unit length of positive leader is ql. The value of ql is about 40–65 mC/m. With this the initial length of the leader section L1 is given by Q0/ql. It is important to point out here

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that Becerra and Cooray [21] utilized a more rigorous condition in which the charge necessary to thermalize a unit leader section depends on the speed of the leader. The initial radius of the leader, aL (t), is assumed to be 103 m and the initial potential gradient of the leader section, EL1 ðtÞ, is assumed to be equal to the potential gradient of the streamer region, i.e. 5.0
103 V/cm. Now we proceed to the next time step, i.e. t ¼ t þ Dt. e) During the time interval Dt the background potential is changed and we also have a small leader section of length L1. Now the new charge in the streamer zone generated from the head of the new leader section is calculated as before but now including both the leader and its streamer zone in the distance–voltage diagram. The leader is represented by a line with a potential gradient EL1 ðtÞ (see Figure 1.6). The total charge is calculated from the area between this new curve and the background potential. The charge generated in the current time step is obtained by subtracting from this the charge obtained in the previous time step. Let the charge obtained thus be Q1. This charge is used to evaluate the length of the new leader section L2. Moreover, the flow of this charge through the Background potential

Potential

Streamer potential gradient

(a)

Distance

Background potential Potential Streamer potential gradient Leader potential gradient (b)

Distance

Figure 1.6 The use of distance–voltage diagrams to calculate the streamer charge. (a) The charge in the first streamer burst is given by the area between the two curves representing the background potential and the streamer potential gradient. (b) To calculate the charge in subsequent streamer burst one has to include both the leader and the streamer region in a distance–voltage diagram

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leader channel changes the potential gradient and the radius of the older leader section L1. The new potential gradient and the radius of L1 are given by EL1 ðt þ DtÞ and aL1 ðt þ DtÞ. Now let us consider the nth time step. There are n leader sections and they have there respective potential gradients and radii. The radius and the potential gradient of ith leader section are obtained from p  a2Li ðt þ DtÞ ¼ p  a2Li ðtÞ þ ELi ðt þ DtÞ ¼

g1 EL ðtÞ  ILi ðtÞ  Dt g  p0 i

(1.14)

a2Li ðtÞ E L ðt Þ 2 aLi ðt þ DtÞ i

(1.15)

In the above equation ELi ðtÞ is the internal electric field and ILi ðtÞ is the current of the leader section Li at time t. With these, it is possible to calculate the time evolution of the internal electric field for each segment and the potential drop along the

Q3 Q2 L4

Streamer zone

Q1 L3

L3

L2

L2

L2

L1

L1

L1

Q0

L1

Streamer stem

Grounded structure

Figure 1.7 Pictorial definition of the parameters used in the mathematical modeling of positive leader discharges as described in Section 1.8. Note that Q0, Q1, Q2 , etc. are the charges in the streamer zones. The flow of charge across the streamer stem makes it conducting and converts it to a leader section

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leader channel (at a given time, with k denoting the number of leader sections) as follows: DUL ¼

k X

ELi ðtÞ  Li

(1.16)

i¼1

The steps described above can be used to simulate the inception and propagation of positive leaders. Figure 1.7 describes the basics of the process schematically. The calculation can be simplified if, instead of calculating the time evolution of leader potential gradient in each segment as above, one uses the expression derived by Rizk [23] for the potential of the tip of the leader channel that is given by

Estr Estr  E1 flLðiÞ =x0 g ði Þ ði Þ (1.17)  e Utip ¼ lL E1 þ x0 E1 ln E1 E1 ði Þ

In the above equation lL is the total leader length at the current simulation step, E1 is the final quasi-stationary leader gradient and x0 is a constant parameter given by the product vq, where v is the ascending positive leader speed and q is the leader time constant.

1.9 Leader inception based on thermalization of the discharge channel The thermo-hydrodynamic model for the leader channel proposed by Gallimberti has been used in the leader channel modeling for lightning and laboratory long spark gaps in Refs. [19,20,24–30]. The physical principles of the model are based on the fact that the current of the first corona streamers passing through the stem of the corona burst will cause heating of the gas. The model assumes that for leader inception, the plasma of the stem has to be raised to a temperature of around 1,500 K. The model assumes that only translational, rotational, and electronic excitation can contribute directly to temperature increase. Chemical energy (dissociation and ionization) of the gas is neglected while the vibrational energy is relaxed on a time scale comparable or longer than that of the leader channel formation. The model neglects the electric field increase due to rise of applied voltage and the chocking effect of streamer space charge. Recently Arevalo and Cooray [31] presented a two-dimensional model based on the gas-dynamic equations with a set of kinetic reactions including the main processes responsible for gas heating such as vibrational excitation and transfer of energy into electronic, rotational, and translational excitation, coupled with Poisson equation. The condition for streamer-to-leader transition is built on three principal assumptions: (1) the axial variations of parameters along the channel are negligible in comparison to the radial ones and, therefore, the leader head can be represented by a one-dimensional (1-D) radial system, (2) the electrical current of a propagating leader is produced in the streamer zone and injected into the leader head, and (3)

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the inception condition of the leader is attained if the gas temperature reaches a critical temperature of 1,500 K in the streamer stem which was considered as a cylindrical plasma channel. Based on this study, they concluded that (i)

(ii)

(iii)

(iv)

(v)

Photoionization is essential for the development and propagation of positive streamers. It enables the seed electrons in the high electric field region at the head of the streamer. The most important mechanisms to increase the electron density, and consequently incept a leader, were the fast electron detachment from negative ions caused by oxygen atoms and the acceleration of the electron impact ionization due to NO molecules. The rise of temperature on the leader depends directly on the energy available in the streamer channels, the vibrational energy relaxation, and the recombination of particles. Consequently, it is incorrect to assume that all streamer energy is directly used for heating and a unique amount of charge is required to heat the channel and incept a leader. Calculations indicate that sharp tips allow more charge to flow into the streamer channels before leader inception takes place than blunt tips. Therefore, the amount of electrical charge required to achieve leader inception depends on the electric field distribution of the electrode arrangement i.e., geometry, applied voltage, the space charge, and environmental conditions, among others. The study shows that the criterion of a constant minimum electrical charge of 1 mC to incept a leader channel, used in lightning attachment and long gap discharge models, is not well-founded. Even though the critical charge necessary for leader inception for rods of 1 cm radius is about 1 mC the value of this critical charge decreases as the radius of the conductor increases.

References [1] Cooray, V., 2003. Mechanism of electrical discharges, in Cooray, V., (Ed.), The Lightning Flash, The Institution of Electrical Engineers, London, UK. [2] Morrow, R., 1985. Theory of negative corona in oxygen, Phys. Rev. A, 32, 1799–1809. [3] Morrow, R. and Lowke, J. J., 1997. Streamer propagation in air, J. Phys. D: Appl. Phys., 30, 614–627. [4] Raether, H., 1940. Zur Entwicklung von Kanalentladungen, Arch. Eleektrotech., 34, 49–56. [5] Meek, J.M., 1940. A theory of spark discharge, Phys. Rev., 57, 722–728. [6] Bazelyan, E.M. and Raizer, Yu.P., 1998. Spark Discharge, CRC Press, New York. [7] Marode, E., 1983. The glow to arc transition, in Kunhardt, E. and Larssen, L. (Eds.), Electrical Breakdown and Discharges in Gases, Plenum Press, New York.

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[8] Marode, E., 1975. The mechanism of spark breakdown in air at atmospheric pressure between a positive point and a plane, I. Experimental: Nature of the streamer track, II. Theoretical: Computer simulation of the streamer track, J. Appl. Phys., 46, 2005–2020. [9] Les Renardie´res Group, 1977. Positive discharges in long air gaps – 1975 results and conclusions, Electra, 53, 31–152. [10] Les Renardie´res Group, 1981. Negative discharges in long air gaps, Electra, 74, 67–216. [11] Gao, L., Larsson, A., Cooray, V. and Scuka, V., 1999. Simulation of streamer discharges as finitely conducting channels, IEEE Trans. Dielectr. Electr. Insul., 6, 1, 35–42. [12] Griffiths, R.F., and Phelps, C.T., 1976. The effects of air pressure and water vapour content on the propagation of positive corona streamers, Quart. J.R. Mat. Soc., 102, 419–426. [13] Griffiths, R.F., and Phelps, C.T., 1976. Dependence of positive corona streamer propagation on air pressure and water vapour content, J. Appl. Phys., 47, 2929–2934. [14] Paris, L. and Cortina, R., 1968. Switching and lightning impulse discharge characteristics of large air gaps and long insulator strings, IEEE Trans., PAS98, pp. 947–957. [15] Gallimberti, I., 1979. The mechanism of long spark formation, J. de Physique., 40, 7, C7–193–250. [16] Biagi, C.J., Uman, M.A., Hill, J.D., Jordan, D.M., Rakov, V.A. and Dwyer, J., 2010. Observations of stepping mechanisms in a rocket-and-wire triggered lightning flash, J. Geophys. Res., 115, D23215, doi:10.1029/2010JD014616. [17] Mazur, V., Ruhnke, L., Bondiou-Clergerie, A. and Lalande, P., 2000. Computer simulation of a downward negative stepped leader and its interaction with a grounded structure, J. Geophys. Res., 105, D17, 22361–22369. [18] Arevalo, L. and Cooray, V., 2011. Preliminary study on the modeling of negative leader discharges, J. Phys. D: Appl. Phys., 44, 31, doi:10.1088/ 0022-3727/44/31/315204. [19] Becerra, M. and Cooray, V., 2006. A self-consistent upward leader propagation model, J. Phys. D: Appl. Phys., 39, 3708–3715. [20] Becerra, M. and Cooray, V., 2006. A simplified physical model to determine the lightning upward connecting leader inception, IEEE Trans. Power Delivery, 21, 2, 897–908. [21] Becerra, M. and Cooray, V., 2006. Time dependent evaluation of the lightning upward connecting leader inception, J. Phys. D: Appl. Phys., 39, 4695–4702. [22] Cooray, V., Rakov, V. and Theethayi, N., 2007. The lightning striking distance—Revisited, J. Electrostat., 65, 296–306. [23] Rizk, F., 1989. A model for switching impulse leader inception and breakdown of long air-gaps, IEEE Trans. Power Delivery, 4, 1, 596–603. [24] Gallimberti, I., Bacchiega, G., Bondiou-Clergerie, A., and Lalande P., 2002. Fundamental processes in long air gap discharges, C.R. Physique applique´e/ Appl. Phys., 3, 1335–1359.

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[25] Bondiou, A. and Gallimberti, I., 1994. Theoretical modeling of the development of the positive spark in long air gaps, Phys. D Appl. Phys., 27(6), 1252–1266. [26] Lalande, P., 1996. Etude des conditions de foudroiement d’une structure au soil, Ph.D. thesis, Universite de Paris-Sud U.F.R. Scientifique d’Orsay. [27] Goelian, N., Lalande, P., Bondiou-Clergerie, A., Bacchiega, G.L., A. Gazzani, I. Gallimberti, 1997. A simplified model for the simulation of positive-spark development in long air gaps, J. Phys. D: Appl. Phys., 30, 2441–2452. [28] Becerra, M. and Cooray, V., 2006, Time dependent evaluation of the lightning upward connecting leader inception, J. Phys. D: Appl. Phys., 39, pp. 4695–4702. [29] Popov, N., 2009, Study of the formation and propagation of a leader channel in air, Plasma Phys. Rep., 35, pp. 785–793. [30] Fofana, I. and Beroual, A., 1997, A predictive model of the positive discharge in long air gaps under pure and oscillating impulse shapes, J. Phys. D: Appl. Phys., 30, 1653–1667. [31] Arevalo, L. and Cooray, V., 2017, Unstable leader inception criteria of atmospheric discharges, Atmosphere, 8(9), 156.

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Chapter 2

Modelling of charging processes in clouds Edward R. Mansell1 and Donald R. MacGorman1

2.1 Introduction Our goal in this chapter is to introduce the treatments of electrical processes used by numerical cloud models that integrate dynamic, microphysical, thermodynamic and electric processes to track what happens to several classes of water particles as they move through the cloud and interact with each other and with the environment as the cloud evolves. We consider primarily models that treat ice particles, as well as liquid water, because ice is now commonly recognized as a necessary ingredient for strong electrification. We ignore models whose winds or particle spectra are unchanging and models that treat electrification as interactions of electric circuit elements driven by a current, charge or voltage source unrelated to microphysics and dynamics. When using cloud models, it is prudent to keep their limitations in mind, as no numerical model can reasonably be expected to replicate a storm exactly. Our computer resources and our knowledge of many relevant processes and of the environmental state are almost certainly never adequate to do that. What is done, therefore, is to use simplified mathematical descriptions called parameterizations to deal with the spectrum of particle types, to estimate the effects of poorly understood physics and to incorporate the effects of processes that occur on temporal or spatial scales too small to be included directly. The goal is for the parameterizations to incorporate enough physics, to treat a broad enough range of time and distance scales and to retain enough detail in the microphysical, electric, dynamic and thermodynamic fields that the model can simulate accurately the aspect of the phenomenon we are studying. To some extent, success is judged by examining how well the model simulates observations of related storm properties, such as the distribution and evolution of precipitation, cloud particles and electric field magnitudes. However, it often is difficult to assess how well a particular model has succeeded in simulating a given phenomenon. Relevant properties often are not observed well enough to judge simulations, and even if they have been observed well, it is often difficult to determine how good a match with observations is needed to establish that the model is adequately simulating the targeted behaviour. 1

National Severe Storms Laboratory, USA

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Although any model involves uncertainties, models also have strengths that help circumvent the serious difficulties one faces trying to improve understanding of storm electrification by analysing measurements alone. A sensor may itself distort what is being measured, and ambiguities can arise when interpreting measurements by a particular sensor in terms of desired storm parameters. Furthermore, no combination of technologies is likely ever to be able to observe simultaneously all of the thermodynamic, kinematic, electric and hydrometeor fields in evolving clouds with enough temporal and spatial resolution to describe all their significant behaviours and interactions. Modelling addresses these shortcomings by attempting to calculate all of the relevant fields in a physically consistent way from as close to first principles as is feasible. By providing a complete set of simulated observations of the very complex system that is a thunderstorm, modelling provides a useful means for testing the plausibility and implications of ideas gained from theory and observations. Thus, observations and models often complement each other in our efforts to advance understanding: (1) laboratory, theory and field observations provide the knowledge needed to build a model. (2) Modelling experiments provide insight into storm processes and lead to predictions and model sensitivities that can be tested with observations. (3) New laboratory and field observations are acquired under model guidance to examine the predictions and sensitivities and refine the model. (4) The refined model is then used either to probe previously defined issues more deeply or to begin investigating new issues not possible to address with the previous version of the model. Note that progress in model capabilities is also often a result of increased computer memory and processing speed. In this chapter, we limit ourselves to explaining the basics of the techniques involved in modelling electrification, as well as giving an overview of some of the electrification research pursued through modelling studies. We attempt to cover most electrical processes but present only selected examples of the treatments of each one. Beyond defining some terms, we do not review the microphysical and dynamical frameworks of cloud models, because doing so would make this chapter far too lengthy. An overview of microphysical and dynamical treatments is given by MacGorman and Rust (1998). More in-depth information about cloud models is available in several books, including Pruppacher and Klett (1997), Cotton and Anthes (1989), Houze (1993), Stensrud (2007), Straka (2009) and in many of the publications referenced in this chapter.

2.2 Definitions of some model descriptors 2.2.1

Basic terminology

Several basic terms are used to describe models. A full simulation model, the only kind we consider in this chapter, produces clouds through appropriate initial conditions in an ambient environment defined by one or more atmospheric soundings and has equations that govern the subsequent evolution of thermodynamic fields, wind fields and microphysics. A kinematic model (e.g. Ziegler, 1985, 1988) still

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must have an ambient environment defined by soundings but has no equation for dynamics to govern the development of the wind field. Instead, the model ingests a cloud wind field obtained from Doppler radars. (The wind field either can be a single wind field, assumed to be steady state, or can vary in time from an early stage of the storm.) It then uses the thermodynamic equation of state and continuity equations for water substance to retrieve temperature, water vapour mixing ratio and hydrometeor mixing ratios throughout the model domain. Models also are classified by the number of spatial dimensions of the model grid. To study storms in which three-dimensional structure and circulations are important, it obviously is necessary to use a three-dimensional model. However, for simpler situations or more limited investigations, modellers can reduce the number of spatial dimensions that they use. Because adding a spatial dimension to a model typically increases computer memory and storage requirements by more than an order of magnitude, reducing the number of dimensions greatly reduces the computer resources required to run a model. Some electrification models are essentially zero-dimensional cloud models because most parameters do not vary with either height or horizontal distance. These models normally assume that the upper and lower boundaries are infinite horizontal planes, so that the ambient electric field is constant with the distance between them. Microphysics and vertical winds also are kept uniform between the horizontal planes: on each boundary there is a source of hydrometeors, usually a source of small particles at the bottom and a source of large particles at the top. Particles and charge can vary with time, and charges are collected at the boundaries as particles reach them, thereby changing the ambient electric field. The model domain normally is considered to represent only part of a cloud, with the upper plate corresponding to the centre of the upper positive charge, and the lower plate the centre of the lower negative charge in a thunderstorm charge distribution. Illingworth and Latham (1977) pointed out that a model with infinite planes will overestimate thunderstorm electric field magnitudes in most situations. In a one-dimensional model, height is the only spatial coordinate that is retained. Cloud properties can vary with height but not with horizontal position. The model domain normally is defined as a cylinder whose radius R(z) is specified. The use of a finite horizontal extent makes it possible to incorporate parameterizations of entrainment and turbulent eddy fluxes and makes electric field magnitudes more realistic. In a two-dimensional model, height and one horizontal coordinate are retained, and cloud properties over the remaining horizontal coordinate are constant. Twodimensional models can be either slab-symmetric or axisymmetric. A slabsymmetric model uses Cartesian coordinates and keeps cloud properties constant along x or y. This symmetry is applied sometimes to squall lines. An axisymmetric model uses cylindrical coordinates and sets azimuthal variations to zero. This symmetry is useful for small thunderstorms and some aspects of hurricanes. Twodimensional models have a particular limitation, however, in that turbulence incorrectly feeds upscale growth. Proper treatment of turbulence requires a threedimensional model.

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Unlike a one-dimensional model, two- and three-dimensional models do not need to specify cloud boundaries or entrainment. Instead, the model is provided the specified environmental thermodynamic and water vapour fields, and clouds form in the model wherever these fields interact to create them. (Slab-symmetric models also can use one horizontal component of environmental winds, but axisymmetric models can treat only radially converging or diverging horizontal wind.) Modelled air motions may be adequate to resolve entrainment, but eddy fluxes must be treated by parameterization of subgrid-scale turbulence.

2.2.2

Terms related to microphysics

Temporal and spatial variations in water vapour and water substance are defined by continuity equations for water vapour and water substance expressed in terms of mixing ratios. The mixing ratio q for the Nth category of water substance is defined as the mass of the Nth category per unit mass of dry air (often expressed as kilograms or grams of water per kilogram of dry air). The mixing ratio for total water substance then is the sum of the mixing ratios for water vapour and for all types of hydrometeors. For each category of water substance, there is a continuity equation of the form dqN ¼ TransportðqN Þ þ SourceðqN Þ þ SinkðqN Þ dt

(2.1)

where transport is the net transport of qN into the volume by advection, turbulence and diffusion; source is the sum of all sources of qN in the volume and sink is the sum of all loss mechanisms for qN. Much of the challenge in parameterization is to develop physically realistic expressions for sources and sinks, once a partition of water into various categories is chosen. The water substance in a cloud normally is partitioned into several categories. For example, the following categories are described by Houze (1993): 1. 2. 3.

4. 5.

Water vapour (qv) is water in the gaseous phase. Cloud liquid water (qc) consists of liquid droplets that are too small to have appreciable terminal fall speed (droplet radius less than roughly 100 mm). Precipitation liquid water consists of liquid drops that are large enough to have an appreciable terminal fall speed. This category sometimes is divided by terminal fall speed into drizzle (qDr) (radius roughly 0.1–0.25 mm) and rain (qr) (radius > 0.25 mm). Cloud ice (qi) indicates ice particles that are too small to have an appreciable terminal fall speed. Precipitation ice consists of ice particles that have a terminal fall speed of  0.3 m/s. This category often is subdivided by density and fall speed. For example, snow (qs) has lower density and fall speeds of 0.3–1.5 m/s, graupel (qg) is denser and falls at  1–10 m/s and hail (qh) is larger and still denser, with fall speeds up to 50 m/s.

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Models that omit all forms of ice are referred to as warm cloud models. Models that include equations for the ice phase are referred to as cold cloud models or mixed-phase models. Each category of water substance interacts with the other categories to create sources and sinks. For example, cloud water droplets coalesce to form drizzle, a process that is a source for drizzle and a sink for cloud liquid water. There are several types of basic interactions: 1.

2. 3.

4.

5.

6. 7. 8.

Condensation or deposition of water vapour onto cloud nuclei (called nucleation) to form cloud liquid water droplets and cloud ice particles, respectively, having size spectra characteristic of the model cloud’s environmental conditions Hydrometeor growth through vapour condensation or deposition Collection of particles to form larger particles (Collection of the various types of particles consists of collisions followed by sticking together. The ways in which two particles stick together are labelled by specific terms depending on the types of particles involved: coalescence involves two or more liquid water particles; aggregation involves two or more ice particles; and riming is the process of cloud water droplets freezing into ice particles on contact. Accretion is used broadly to indicate collection of liquid particles by ice particles but often has a connotation of larger liquid particles sticking to larger ice particles.) Breakup of drops or splintering of ice (These processes can increase the number of particles beyond what would be expected from nucleation. The increase from splintering is referred to as ice multiplication.) Freezing of liquid water (Liquid water exists as supercooled water at heights above the 0  C isotherm, but all liquid water usually is assumed to have frozen by the time a parcel reaches the 40  C isotherm.) Evaporation or sublimation of water vapour from hydrometeors Melting of ice Precipitation reaching the ground

It is possible to expand the number of categories and subcategories of hydrometeors considerably. Typical categories are cloud droplets (D < 50 mm), rain (D > 50 mm), small ice crystals, snow (or aggregated crystals), graupel and hail. To expand categories, e.g. a modeller might want to track different shapes (called habits) of ice particles separately (such as columns, plates and dendrites). However, as more categories of water are used, the number of interactions that must be considered increases rapidly. For that reason, modellers usually use only the categories and subcategories that are essential to simulating the particular phenomenon being studied. To parameterize microphysics, models handle the size distributions of the various categories in one of two ways, referred to as bulk microphysics and bin (or spectral) microphysics. In bulk microphysics, the size distribution of particles is described by some simple function. The amount in a given category can be tracked at each grid point by a single parameter, such as the mixing ratio of water substance in that category, or by two or more parameters, such as the mixing ratio and number

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density of particles in a category. Distributions using two or more parameters provide a more versatile description than distributions using one parameter. In bulk microphysics, the size distribution of precipitating liquid often is described as a simplified gamma function of the form nðDÞ ¼ n0 Da expðlDÞ

(2.2)

where n(D) is the number of particles per unit volume between D and D + dD and a is the shape parameter, which controls the width of the spectrum. The intercept parameter n0 is a function of the total number concentration nT and the slope parameter l: n0 ¼

nT laþ1 Gða þ 1Þ

(2.3)

where l is defined as
l¼

Gða þ 1 þ dÞcnT Gða þ 1Þrair q

1=d (2.4)

where the constants c and d come from the mass–diameter relationship m(D) = cDd. The moments M(j) of the distribution are given by ð1 MðjÞ ¼ AðjÞ Dj nðDÞdD (2.5) 0

where A(j) is some constant factor. For the simplified gamma distribution, the moments are MðjÞ ¼

AðjÞnT Gða þ 1 þ jÞ lj Gða þ 1Þ

(2.6)

For j = 0 and A(0) = 1, M(0) = nT, so the zeroth moment is the number concentration. If the mass of a particle is m(D) = cD3, then the third moment gives the total mass of the distribution. Single-moment schemes predict q and usually diagnose nT by setting assuming value for n0 (the intercept parameter). Models with two-moment bulk microphysics generally predict the mixing ratio q (third moment) and number concentration nT (zeroth moment). Three-moment schemes can also diagnose the shape parameter a. Setting the shape parameter a equal to zero results in the inverse exponential (IE) distribution. This form of the size distribution is often referred to as a Marshall– Palmer distribution (Marshall and Palmer, 1948). In bulk microphysics, each property of the category has a single representative value at a given grid point in the model domain that somehow averages the values for all of the category’s particles within the corresponding volume. For example, consider the sedimentation of liquid precipitation particles that have a terminal fall

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speed relationship v(D) = aDb. The moment-weighted fall speed is used for sedimentation of each bulk moment. For example, the mass-weighted fall speed VT,m is given by Ð1 vðDÞmðDÞnðDÞdD (2.7) VT;m ¼ 0 Ð 1 0 mðDÞnðDÞdD which is used for sedimentation of the total particle mass in a grid volume. Note that the denominator in (2.7) is the mass content (mass of condensate per volume). Then C, the net rate of change in a grid point’s mass mixing ratio q due to sedimentation, is given by C¼

d ðVT ;m qÞ dz

(2.8)

Moments of the distribution in a spectral bin model are calculated by an explicit summation over the range of sizes for the water substance category being considered. For example, the mixing ratio qx of particle type x, modelled by J size bins and having mass increments of mx(j), is given by qx ¼

J 1 X mx ðjÞnx ðjÞ rair j¼1

(2.9)

where nx(j) is the number density of particles in mass bin j. To represent the size distribution of particles accurately, at least ten size bins must be included for each hydrometeor category being parameterized. With bin microphysics, it is unnecessary to assume that the size distribution has a particular form. The overall size distribution is allowed to evolve naturally as particles of different sizes gain and lose mass through their interactions. This is a more direct approach than bulk microphysics, but even with many size bins, a bin parameterization is not an exact treatment. There are gaps in our knowledge of particle interactions, particularly for some size ranges and particle types, and these gaps introduce uncertainties in bin parameterizations. Furthermore, although particles within a given size bin are more uniform than in a category that spans all sizes in a bulk treatment, variations in properties, such as size, shape or density may occur within an individual bin being treated explicitly and can affect interactions involving this bin. When such variations are significant, spectral parameterizations need to avoid treating all particles in a size bin identically, because it would incorrectly cause all particles to transfer into a new size bin at the same time. To avoid this, a spectral parameterization can treat particles in a given size bin as having a distribution of sizes across the range of the bin and can use statistical techniques to govern gradual changes in the population. The main disadvantage of bin microphysical treatments compared with bulk microphysics is that bin treatments require considerably more computational resources: Each bin must satisfy its own continuity equation, and interactions must be included between size bins, as well as between water substance categories.

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Thus, bin microphysics usually is reserved for addressing questions that cannot be addressed by bulk microphysics. When specific microphysical treatments are used in the rest of this chapter, bulk microphysics is used instead of bin microphysics.

2.2.3

Categories of electrification mechanisms

Electrification processes are broadly categorized as either inductive or non-inductive. An inductive process is driven by an ambient electric field. Hydrometeors have naturally occurring free ions that can move enough in response to an electric field to polarize the particle. Figure 2.1 depicts a possible inductive (or polarization) charge separation process in which a small liquid water droplet collides with and rebounds from a graupel pellet. Other examples of possible inductive charge transfer include the shedding of liquid water from hail (melting or undergoing wet growth) and breakoff of ice branches as snow melts (assuming the electric field forces free ions towards the material that breaks away from the larger ice particle). Research has suggested that inductive processes alone are not capable of providing strong electrification. However, they may provide secondary effects important in some regions, such as in melting layers and in regions having preferred attachment of one polarity of ion driven by the electric field (e.g. screening layer charge). Non-inductive charging processes are defined as charge transfer that is independent of an external electric field. The collisional ice–ice mechanism is generally regarded as the primary electrification mechanism, although much about the mechanism is still not well understood. The non-inductive ice–ice mechanism has been best described by the relative growth rate theory of Baker et al. (1987), which states simply that when two ice surfaces collide and rebound, the surface that is growing faster by vapour deposition will gain positive charge (by losing negative charge). Supersaturation with respect to ice is required for deposition growth and significant charge transfer. One explanation for the mechanism is that partial melting occurs around the impact site, which mixes the (negative) charges from the two surfaces (Emersic, 2006). As the two particles separate, some of the melted mass is carried off by the smaller particle, resulting in a net charge transfer (Figure 2.2). Other proposed non-inductive mechanisms involve thermoelectric

– +

–– – +++

– +

E

θ

– +

–––

+ –

+++

–

+

Figure 2.1 Example of an electric field-dependent inductive process. Here, a graupel particle and water droplet are polarized by an electric field (left). A rebounding collision transfers some negative charge from the droplet to the graupel particle (right)

Modelling of charging processes in clouds

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+ –

Figure 2.2 Example of an electric field-independent non-inductive process. Here, a graupel particle and ice crystal are initially uncharged in a cloud of small droplets (left). A rebounding collision transfers some charge between the ice and graupel particles (right). The sign of charge transfer could be positive or negative, depending on the vapour deposition growth of the interactive surfaces effects, contrasting surface potentials and electrical double layers arising from physical properties of ice surfaces.

2.2.4 Other categorizations of cloud models Besides the dynamic, thermodynamic and microphysical considerations discussed in the previous section, other choices also influence the complexity and computational requirements of models. Most cloud models are time dependent, meaning that cloud properties are allowed to evolve, but some simpler models use steady-state dynamics. Many one-dimensional models and some two- and three-dimensional models use the anelastic form of the continuity equation. The more sophisticated two- and three-dimensional cloud models usually are fully compressible. The anelastic approximation eliminates sound waves, which often are considered undesirable because their magnitude can be large and other processes are of more interest in modelling studies, but doing so requires solution of an elliptic equation for the pressure field. Elliptic equations require iterative or implicit solvers, which can be inefficient. The fully compressible form retains sound waves and allows use of explicit (forward-in-time) numerical solvers. Efficiency is maintained by integrating sound waves separately (‘split explicit’ method). A full simulation model produces clouds through appropriate initial conditions and simulates the subsequent evolution of all model fields. To simulate a cloud with a given one-dimensional, time-dependent model, a vertical profile of each relevant parameter of the ambient environment is specified, vertical forcing (e.g. a thermal perturbation or vertical velocity source) is applied at a low height and the model then computes how deviations from environmental values evolve. Similarly, a two-dimensional, axisymmetric model generally applies forcing centred on r = 0. Usually, two-dimensional, slab-symmetric models and three-dimensional models are initiated by introducing some type of forcing with a more complicated geometry into the specified environmental conditions. This forcing may be, e.g., in the

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form of convergence in some region at low levels on which are superimposed thermal bubbles, whose size, location and temperature excess above ambient conditions must be chosen (either given specific values or chosen randomly from some range). Storms that result from a given model experiment can be influenced by the form and magnitude of forcing that is applied, as well as by assumed environmental conditions; this is especially true of small thunderstorms driven primarily by local heating. Besides full simulation models, which simulate the dynamics, thermodynamics and microphysics of storms simultaneously beginning with initial forcing in a prestorm atmospheric environment, another type of model, called a kinematic model, use observed wind fields of storms to estimate the accompanying temperature and water vapour perturbations and microphysics in a particular atmospheric environment. In a kinematic model (e.g. Ziegler, 1985, 1988), the thermodynamic equation of state and continuity equations for water substance are the same as in a full simulation model, but there is no equation for dynamics to govern the development of the wind field. Instead, the model ingests an observed cloud wind field from Doppler radars and ambient environmental conditions from an atmospheric sounding. Then, the equation of state and continuity equations for water substance are solved to retrieve temperature, water vapour mixing ratio and hydrometeor mixing ratios throughout the model domain. In a time-dependent, kinematic model, the various fields are allowed to evolve from their state at an early stage of the cloud. Changes in the wind field are calculated by interpolating between observed wind fields at each time step. Changes in the retrieved fields then are calculated from changes in the wind fields. Kinematic models have the disadvantage that, because the modelled cloud does not begin with initial cloud formation, there may be significant errors in fields that are sensitive to the cloud’s history of evolution. Furthermore, the wind field of the model is not influenced by microphysics and thermodynamics, and kinematic models can be used only for periods when Doppler wind fields are available. However, a kinematic model has the advantage that it produces a model storm consistent with both the observed storm wind field and the complete set of thermodynamic and continuity equations. It can be difficult to use full simulation models to investigate some storms, especially small storms, because of model sensitivities to the form of initial forcing, to boundary conditions or to minor changes in the storm environment. Most cloud models that have been used in electrification studies have tracked the properties of water substance categories at grid points fixed with respect to the Earth (these are sometimes called Eulerian models). Particle tracing or Lagrangian models (e.g. Kuettner et al., 1981), however, compute changes to the properties of individual particles or groups of identical particles along trajectories that follow the particles through the storm, instead of at fixed grid points. Model fields related to the particles are derived after each model time step by interpolating particle parameters from particle locations to the model grid. This type of treatment typically is used with simplifications to other aspects of the model to keep computations tractable. Sometimes an Eulerian model uses statistical techniques to mimic a Lagrangian treatment for a particular type of particle.

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2.3 Brief history of electrification modelling In the 1950s and 1960s, there were several calculations of electrification rates from particle interactions that considered neither cloud dynamics nor the microphysics of the particles. The first attempt to include electrification in a numerical cloud model that considered these factors was by Pringle et al. (1973), who used a crude parameterization of charge separation that did not attempt to mimic any particular microphysical charging mechanism. Takahashi (1974) allowed hydrometeors to capture space charge in his one-dimensional, time-dependent cloud model but did not include charge exchange between particles. Ziv and Levin (1974), Scott and Levin (1975) and Levin (1976) studied inductive charging by considering hydrometeors that moved vertically and interacted between the two plates of an infinite horizontal capacitor but did not include any cloud dynamics. Illingworth and Latham (1977) included explicit parameterizations of several microphysical charging mechanisms in a one-dimensional model with steady-state dynamics. Kuettner et al. (1981) incorporated parameterizations of inductive and non-inductive charging in a two-dimensional Lagrangian model with steady-state dynamics. Although these last two studies were able to estimate the relative contributions to cloud electrification from the various mechanisms they modelled, the use of specific steady-state dynamics did not allow them to consider how electrification varied with evolving cloud dynamics. Relatively simple models continue to be used for tests of specific hypotheses (e.g. Mathpal and Varshneya, 1982; Singh et al., 1986; Canosa et al., 1993). The next step in the evolution of electrification modelling was to add electrification processes to cloud simulation models that coupled electrification with both microphysics and dynamics. Early attempts at this coupling included only warm rain processes (i.e. there was no freezing or ice). Takahashi (1979) used a two-dimensional, time-dependent axisymmetric model with bin microphysics to study electrification of shallow, warm clouds. Chiu (1978) also developed a two-dimensional, time-dependent axisymmetric model, although with bulk microphysics, and was the first to include a parameterization of small ions and their interactions with particles in a cloud simulation model. By including ion– hydrometeor interactions, simulated clouds were able to form screening layers at cloud boundaries, and mechanisms such as the Wilson’s selective ion capture mechanism could affect the charge on particles in appropriate regions. Helsdon (1980) used a similar model that was slab-symmetric instead of axisymmetric to study whether cloud electrification could be modified by injecting metal-coated chaff fibres into a warm cloud. Since several scientists already had suggested that ice–particle interactions were important for thunderstorm electrification by the non-inductive mechanism, applications of warm cloud models were considered extremely limited, and there was considerable interest in developing electrification models that included ice processes. Rawlins (1982) included electrification in a cloud simulation model that parameterized ice processes in a three-dimensional model. The model used bulk

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microphysics, a simple parameterization of the non-inductive mechanism and a traditional parameterization of the inductive mechanism. Electrification occurred only when graupel or hail interacted with snow or cloud ice, so no charge was generated by interactions involving liquid hydrometeors. Takahashi (1983, 1984) included ice processes in a two-dimensional, time-dependent axisymmetric model with bin microphysics. Parameterized charging mechanisms included inductive charging, ion–hydrometeor interactions and non-inductive charging based on the laboratory data of Takahashi (1978). Takahashi’s model had much better grid resolution than Rawlins’s but had a domain height of only 8 km, and so was limited to small storms. Helsdon and Farley (1987b) added a simple non-inductive charging parameterization to the Chiu (1978) ion capture and inductive charging parameterizations in a two-dimensional, time-dependent, slab-symmetric model with bulk ice microphysics. They used the model to simulate a storm that was observed to produce a single lightning flash during the Cooperative Convective Precipitation Experiment in Montana. Modelled space charge and electric field distributions were similar to those observed by two aircrafts that penetrated the storm only in simulations in which both the non-inductive and inductive mechanisms operated together. In that case, the time required for model electrification to increase to the point that lightning occurred was comparable to that required by the observed storm. After publication of Helsdon and Farley, an error was found in the formulation of the non-inductive parameterization for experiments in which it was used alone. Wojcik (1994) repeated the investigation with the corrected formulation of the non-inductive mechanism acting alone and found that it produced an electric field consistent with the aircraft observations. Randell et al. (1994) used a non-inductive parameterization similar to Takahashi (1984) in a configuration of the Helsdon and Farley (1987b) model that omitted the inductive mechanism (and used the corrected non-inductive formulation). They simulated storms in three different environments to examine conditions under which the non-inductive mechanism could produce a thunderstorm. Electrification processes have been incorporated into another cloud simulation model based on a simpler geometry. Mitzeva and Saunders (1990) developed a onedimensional model that included no inductive mechanism or ion capture but used a sophisticated parameterization of non-inductive charging based on laboratory studies of Jayaratne et al. (1983) and Keith and Saunders (1989), instead of Takahashi (1978). Their model employed bulk microphysics and was used primarily to examine the evolution of non-inductive charging rates as a function of the intensity of precipitation produced by three storms. The cloud models summarized above were simulation models, but some models have been kinematic models. Ziegler et al. (1986) used bulk microphysics in a one-dimensional, kinematic model whose domain consisted of a cylindrical cloud with a fixed radius. The only electrification mechanism was a parameterization of the non-inductive mechanism suggested by Gardiner et al. (1985). Ziegler et al. (1991) expanded the kinematic model to be three dimensional and modified the kinematic retrieval process to enable the model to assimilate

Modelling of charging processes in clouds

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radar-derived wind fields that were available every three minutes, from early in the storm’s lifetime throughout much of its life. They also added parameterizations of the inductive charging mechanism and of screening layer charge on the cloud boundary. Ziegler and MacGorman (1994) used the three-dimensional model to study a supercell storm (features that are not slab- or axisymmetric are critical to an adequate treatment of supercell storms). Norville et al. (1991) developed a kinematic model that used bin microphysics. The only electrification mechanism that they included was a non-inductive charging mechanism based on the laboratory work of Jayaratne et al. (1983), Baker et al. (1987) and Keith and Saunders (1989). The geometry of their model cloud consisted of two concentric cylinders, in order to be able to model coexisting updrafts and downdrafts. Conditions were horizontally uniform inside the inner cylinder and different, but again uniform, between the inner and outer cylinders. This configuration has been called a one-and-one-half dimensional model. Norville et al. simulated the same storm studied by Helsdon and Farley (1987b). Their model was able to produce electric field magnitudes comparable to observed values by using only the non-inductive mechanism. Most modelling studies prior to the 1990s, whether they used simulation or kinematic models, have examined electrification only in the absence of lightning. Without a lightning parameterization, models could simulate only the initial electrification of thunderstorms, because lightning modifies the charge distribution and limits the maximum magnitude of the electric field. Rawlins (1982) used a threshold electric field of 500 kV m1 to initiate a lightning flash and a simple charge neutralization scheme. He found that the electric field regenerated quickly after a lightning flash. Helsdon et al. (1992) developed a detailed two-dimensional unbranched lightning flash parameterization that calculated the neutralized charge from the ambient electric field. It was used successfully to simulate a storm that produced a single lightning flash, and later was expanded to a three-dimensional unbranched channel (Helsdon et al., 2002). Takahashi (1987) incorporated a simple parameterization of a lightning flash in his model to examine factors influencing the height and location of lightning. Ziegler and MacGorman (1994) developed a simple three-dimensional parameterization of the net effect of several flashes per time step to simulate a supercell storm. Baker et al. (1995) added a simple parameterization of lightning to their one-and-one-half dimensional, kinematic model with bin microphysics and Solomon and Baker (1996) developed an analytical treatment of a vertical, one-dimensional lightning channel for the same model. MacGorman et al. (2001) developed a three-dimenstional treatment to simulate discreet flashes using the crude parameterization of neutralized charge by Ziegler and MacGorman (1994). Mansell et al. (2002) developed a detailed threedimensional fractal lightning scheme that used Gauss’s law to estimate the charge neutralized by branched equipotential lightning channels. These last two lightning parameterizations are described in detail in Section 2.5. The two-dimensional simulation study of Helsdon et al. (2001) broke ground in comparing results from multiple non-inductive graupel–ice schemes. Helsdon et al. (2001) did not include lightning, however, so results were obtained only up to

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the time of the first lightning flash. Helsdon et al. (2002), using a three-dimensional version of their model and a lightning parameterization, tested the so-called ‘convective’ charging hypothesis, which relies on screening layers formed by ion attachment to be advected into the storm updraft to generate electrification via inductive charging. Helsdon et al. (2002) found only weak, disorganized electrification that could not generate electric fields sufficient for lightning without inclusion of a non-inductive graupel–ice mechanism. A study by Mansell et al. (2005) investigated differences in electrification from multiple graupel–ice parameterizations as well as inductive graupel–droplet charge separation in a threedimensional model with lightning. Sun et al. (2002) employed a three-dimensional model with two-moment microphysics (mass and number concentration) to examine the effects of electrification on the microphysics and dynamics of a small storm. Feedback to dynamics was achieved through an electrical drag term in the momentum equations using the net charge density and electric field and by electric force adjustments to hydrometeor terminal speeds. The results with electrification and feedback differed appreciably from the non-electrical simulation. An increase in latent heating was found when electrical forces were enabled. The increased latent heating was attributed to reductions in graupel fall speeds at mid-levels of the storm, which increased its residence time and total riming growth. The model almost certainly overestimated electric force effects, however, in the assumption that hydrometeors of a given class are uniformly charged, which disagrees with in situ particle charge measurements. The maximum electric field also was allowed to increase to 250 kV m1 before activating the lightning scheme (a three-dimensional adaptation of the Helsdon et al., 1992, scheme). Sun et al. (2002) incorporated somewhat out-of-date parameterizations of electrification, including drop–droplet inductive charge separation (Chiu, 1978), which is widely considered to be ineffectual due to enhanced drop coalescence even in relatively weak electric fields. Mansell et al. (2002) simulated lightning behaviour in two different storms using the same charge separation schemes. A low-shear thunderstorm had a normal tripole charge structure (main negative charge with main positive charge above and lower positive charge below). A high-shear supercell simulation, however, exhibited an inverted tripole structure. The low-shear storm produced negative cloud-toground (CG) lightning that was initiated between the main negative and lower positive charge regions. The supercell storm simulation had positive CG lightning that was initiated between a main positive region and lower negative charge. A new electrified model was described by Barthe et al. (2005), who presented initial results from a version of the Meso-NH model that included electrification and lightning parameterizations. They also included a sensitivity study of two graupel–ice schemes. Barthe and Pinty (2007) presented more details of the parameterizations used in Barthe et al. (2005), especially details concerning the lightning scheme. The model used in Mansell et al. (2002) has been used in a number of later studies. Kuhlman et al. (2006) presented simulations of a severe supercell storm and compared results to observations of that storm. Fierro et al. (2006) investigated

Modelling of charging processes in clouds

37

the effects of an inhomogeneous environment on the simulated kinematics, electrification and lightning of a supercell storm. Fierro et al. (2007) investigated electrification of an idealized tropical cyclone. Mansell et al. (2010) used a twomoment microphysics scheme to compare simulated electrification and lightning of a small thunderstorm with observed lightning evolution. A few recent cloud model studies have started examining lightning production of nitrogen oxides (NOx). Zhang et al. (2003) combined a chemistry model with an electrification and lightning model (Helsdon et al., 2002) to study production, transport and subsequent chemical reactions of NOx. Barthe et al. (2007) simulated the electrification and NOx production in a strong storm but did not include a chemistry module. Results from both of these models were included in an intercomparison study (Barth et al., 2007). The rest of this chapter describes some of the parameterizations of electrical processes used by the above cloud models.

2.4 Parameterization of electrical processes To include electrification in numerical models, the various electrical processes of clouds must be parameterized and integrated with parameterizations of dynamical, thermodynamic and microphysical processes. In this section, we discuss how numerical models produce and transport charge on hydrometeors, how the electric field is calculated and how lightning is parameterized. We present only processes that laboratory studies suggest will have appreciable effect in storm electrification. In most cases, we give only one or two examples of how a particular process has been parameterized.

2.4.1 Calculating the electric field Once a modelled storm has produced charged hydrometeors, it is necessary to calculate the resulting electric field, both for analysis of model results and for use in the next time step to calculate the results of processes dependent on the electric field. Electric field calculations in one-dimensional models are much simpler than in two- and three-dimensional models. For one-dimensional, infinite layer models, the only non-zero component of the electric field is the vertical component, Ez, which can be calculated from Gauss’s law. If the contribution of the image charge below the conducting Earth is included, Ez at height z0 (above ground level) due to charge at the kth grid level at height zk is 8 rk Dz > > for z0 < zk >

for z0 ¼ zk  k > > 2e : 0 for z0 > zk where Dz is the height increment between grid levels, rk is the charge density at the kth grid level and e is the electrical permittivity of air (8.859  1012 Fm1).

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Lightning electromagnetics: Volume 2 z

φ

zk r

dz dr

r dφ

z0

Figure 2.3 Geometry for calculating E at height z0 on the z axis from an infinitesimal element of a disk of charge at height zk

Because the electric field from an infinite layer is constant with height above and below the layer, this geometry is likely to give electric field profiles considerably different from profiles appropriate to most physically realistic thunderstorm geometries (although it might be suitable for extensive stratiform clouds). More realistic electric field profiles usually can be obtained in a onedimensional model in which the size of the cloud is limited to some radius. On the axis of a cloud with cylindrical symmetry, the only non-zero component of the electric field is again the vertical component. The contribution of an infinitesimal charge (including its image charge) to the vertical electric field on the axis at height z0 (Figure 2.3) is given by " # rk zk  z0 zk þ z0 dEz ðz0 Þ ¼  rdrdjdz þ 4pe ½r2 þ ðzk  z0 Þ2 3=2 ½r2 þ ðzk þ z0 Þ2 3=2 (2.11) The electric field on the axis due to a thin disk of charge of radius R(z) (i.e. R can vary with height) at the kth grid level then is obtained by integrating over r and j. Some care is needed in determining the constants of integration to get " # rk Dz zk  z0 zk  z0 Ez ðZ0 Þ ¼ þ C 2e ½R2 þ ðzk  z0 Þ2 1=2 ½R2 þ ðzk  z0 Þ2 1=2 where C = 0 for zk < z0, C = 1 for zk = z0 and C = 2 for zk > z0.

(2.12)

Modelling of charging processes in clouds

39

For an arbitrary charge distribution in two or three dimensions, there are no simple expressions for the electric field. It would be possible to compute the electric potential or field at an arbitrary point by adding the contributions from the charge at every grid point (from the superposition principle), but computing the potential or field this way at every grid point would be too computationally intensive in many cases. Instead, what is done usually is to use standard numerical algorithms to solve for f at all grid points by inverting the Poisson equation r2 f ¼ 

r e

(2.13)

where r is the total space charge density at the point being evaluated. Then, the electric field is computed from the potential by using the relationship E = rf. Note that, in a two-dimensional model, the symmetry of the model requires the component of the electric field perpendicular to the plane of the model to be zero. Each point in a two-dimensional (slab-symmetric) model physically represents an infinite line charge, which distorts the electric field relative to a threedimensional model. To use numerical Poisson solvers (e.g. multi-grid iteration) to determine f, it is necessary to specify boundary conditions for f on all sides of the grid. As an elliptic equation, boundary conditions specify the value of f, the normal derivative @f/@n or a linear combination of the two. Since the ground is a good conductor, hence an equipotential, it is obvious that the potential of the bottom layer of the grid at the ground should be set to a constant. The constant usually is chosen to be zero, although any constant could be chosen without affecting the resulting electric fields. It is generally inappropriate to set the potential to a constant at the lateral grid boundaries. The upper boundary can be set to a fair-weather value if it is high enough to approximate the electrosphere, as a constant potential at the top seems to result in more stable solutions than setting the normal derivative. Choosing boundary conditions for the other sides is not as straightforward as for the ground, and choices can depend on the cloud size and geometry being modelled. At r = 0 in an axisymmetric, two-dimensional model, for example, Ez is the only non-zero component of E, so the boundary condition there would be @f/@r = 0. In three-dimensional or slabsymmetric, two-dimensional models, however, Ex might well be large if the storm is near the boundary at x = 0 (or x = X), so @f/@x = 0 would not be an accurate boundary condition on that side. It often is impossible to find simple boundary conditions that are exactly correct. However, if the cloud is completely contained by the grid, suitable a priori boundary conditions can be used to produce reasonably accurate calculations of f. At least two strategies have been used to improve the solution of f when charges get close to the domain boundaries. One method is to use brute force to calculate f at the boundaries (e.g. Riousset et al., 2007); this increases solution accuracy in the interior but is computationally expensive. An alternative method is to solve for f in an enlarged domain that has fair-weather charge densities between the model boundaries and the enlarged domain (e.g. Mansell et al., 2005). Pushing

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Lightning electromagnetics: Volume 2

out the lateral and upper potential boundaries has the effect of pushing the mirror charges farther away, thereby reducing the error they cause in the solution.

2.4.2

Charge continuity

Continuity equations for charge are similar to the continuity equations for air and for water substance categories discussed earlier in this chapter. As for the hydrometeor mixing ratios, there must be a continuity equation for the charge on each water substance category. These equations govern how each water category gains and loses charge at a given grid point of a model. They express what in physics texts is called conservation of charge. The total charge at a grid point is then the sum of the charges on all the water substance categories that exist there, i.e., if rt is the total charge density at a grid point, and rn, the charge density on the nth water substance category at that location, then rt ¼

X

rn

(2.14)

The charge density from ions is included as a category if a model treats ion processes. To derive expressions for the continuity equations, first consider charged particles in a parcel of air. Here, a parcel is a fixed mass of air but with a volume that can change (e.g. can expand when rising and compress when sinking). If no charge enters or leaves the parcel, then the total charge Qt in the parcel is invariant to parcel motion and volume changes, but the total charge density rt = Qt/V is not invariant (where V is the parcel volume at a given time). Thus, we can temporarily define a ‘charge mixing ratio’ e r ¼ r=rair as the charge per mass of air. Then the parcel-following (Lagrangian) advection equation for charge on hydrometeor type n is de rn ¼ Ser n dt

(2.15)

where Ser n represents processes that can change the charge on the nth hydrometeor type, such as collisional charge separation and ion attachment. Ser n also includes transfer of charge in and out of the parcel by sedimentation (for precipitation particles) and turbulent mixing. The Lagrangian reference is then transformed to the fixed Eulerian grid volume reference frame by the coordinate transformation d=dt @=@t þ V  r as @e rn ¼ V  re r n þ Ser n @t

(2.16)

For consistency with typical cloud model equations, we multiply (2.16) by air density rair. rair ¼

@e rn ¼ rair V  re r n þ rair Ser n @t

(2.17)

Modelling of charging processes in clouds

41

Expanding the divergence term using a vector identity and dividing again by rair gives @e rn 1 ¼ ½r  ðe r n rair VÞ  e r n r  ðrair VÞ þ Ser n @t rair

(2.18)

Note that in the incompressible case we have rair = constant and r (rair V) = 0. Substituting e r n ¼ rn =rair and cancelling the factors of rair, one recovers the more familiar expression of local charge continuity, @rn =@t ¼ r  J n , with the current density Jn replacing the charge motion rnV. Equations (2.17) and (2.18) are mathematically equivalent and known as the advective and flux forms, respectively. The flux form is generally preferred in finite difference models because it has immediate conservation properties that are not guaranteed in an advective form. To (2.18), we now add the turbulent mixing and sedimentation terms (second and third terms, respectively, on the right-hand side of the following): @e rn 1 1 ¼  ½r  ðe r n ra VÞ  e r n r  ðra VÞ þ r  ðra Kh re rnÞ @t ra ra r n ra V
er Þ 1 @ðe n þ þ Ser n ra @z

(2.19)

where Kh is a mixing coefficient determined by the closure scheme employed by a given cloud model. The fall speed, V
er , is usually the mass-weighted fall speed. n For single-moment bulk microphysics, it is best to use the mass-weighted fall speed to prevent the particle mass from getting out of phase with the charge. This prevents unrealistically large particle charges from occurring during sedimentation. The term Ser in (2.19) now represents charge sources and sinks through parn ticle interactions. Examples of sources and sinks for the nth category include ion capture, charge exchange during collisions with particles in another category, and mass loss or gain as particles are transferred from one category to another. When mass is lost from one category to another, the charge carried by the mass also must be transferred to the new category, thereby decreasing the magnitude of charge in the category losing mass. However, the magnitude of charge in the category gaining mass can either decrease or increase, because the polarity of the charge gained with the new mass can be either the same as or opposite to the polarity of charge already on particles in the category. Since most processes can be either a source or a sink of charge for a given category of particle, it is not worthwhile to try to distinguish sources from sinks. Thus, we group all such processes together. Note that the storm as a whole does not lose or gain charge unless charge is transferred to or from a region outside the storm. When the charge on particles in one category is altered by interactions with particles in other categories, a compensating change (equal in magnitude, opposite in sign) must occur in the charge of the other categories, so that the net charge summed over all categories remains the same. There are no compensating changes in the charge within the storm when the

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Lightning electromagnetics: Volume 2

charge on particles is altered by a process involving charge from outside regions, such as ground flashes, cloud-to-air flashes, capture of ions emitted by point discharge from the ground and screening layer formation at cloud boundaries. Equation (2.19) can be rewritten in terms of the charge density by substituting e r n ¼ rn =rair and expanding the left-hand side: @rn @ðrn V
n Þ ¼ r  ðrn VÞ þ r  ½ra Kh rðrn= ra Þ þ þ Sn @t @z

(2.20)

Note that the advection part has been reduced to one term and that the second term effectively retains e r n . In many models, however, the scalar advection formulation is designed for parcel invariants (such as mass mixing ratio), so the formulation of (2.19) can use the same model code without modification. The state r n , so a compromise is to variable rn has more physical meaning, however, than e r n for advection and turbulent mixing, and then back to rn for transform rn to e electrification processes. As discussed by Pruppacher and Klett (1997), V
n can be generalized to include the effect of electrostatic force in addition to gravity. If this is done, the third term r n Vn Þ, where Vn is on the right of (2.19) should be replaced by ð1=rair Þr  ðrair e now a vector, not a scalar. Many microphysical processes are capable of placing charge on particles. The main processes that have been included in cloud models are the non-inductive graupel–ice mechanism, the inductive mechanism and ion capture. The rest of this section considers treatments of each of these. Electrification mechanisms that have not yet been used in models are not considered here, although other mechanisms may be significant in some situations (e.g. melting processes may play a role in electrifying the stratiform precipitation region of storm systems).

2.4.3 2.4.3.1

The non-inductive graupel–ice collision mechanism Parameterized laboratory results

Parameterizing the non-inductive collision charge separation mechanism requires that the charge per hydrometeor collision be determined and then included in a bin or a bulk parameterization of all collisions involving a particular water substance category at a grid point during a time step of the model. Laboratory studies have found (1) that the non-inductive mechanism appears to be most effective when rimed graupel collides with cloud-ice particles or snow in a region that also has liquid water (i.e. the mixed-phase region) and (2) that the sign and magnitude of the charge that is transferred depend on ambient temperature, liquid water content and impact speed. The simplest parameterization of the non-inductive mechanism assumes that a constant value of charge is transferred per collision between particular hydrometeor types, as was done by Rawlins (1982) and Helsdon and Farley (1987b), both of whom assumed that the sign of charge transfer reversed polarity at –10 C (see Table 2.1).

Modelling of charging processes in clouds

43

Table 2.1 Non-inductive parameterization of Helsdon and Farley (1987b) Growth mode

Interacting types

dq*(fC ¼ 1015 C)

Temperature range

Dry

Graupel/cloud ice

2 2 200 200 100 0 0 0

T < 10 C T > 10 C T < 10 C T > 10 C All T All T All T All T

Graupel/snow Graupel/rain† Graupel/cloud water‡ Graupel/rain‡ Rain/cloud water

Wet Other

* Polarity is for charge transferred to graupel (to rain in growth mode ‘Other’); charge transferred to the particle colliding with graupel is of equal magnitude, but opposite polarity; †Splashing interactions; ‡ Shedding or limited accretion.

Takahashi (1983, 1984) used the laboratory non-inductive charging data directly from Takahashi (1978). A lookup table of Takahashi’s results (shown contoured in Figure 2.4) was developed by Randell et al. (1994) and was used in that and later studies (e.g. Helsdon et al., 2001; Mansell et al., 2005). The base charge per collision dq0 was interpolated (or extrapolated) from the lookup table. Note that Takahashi (1978) did not factor in a collection efficiency, so the actual charge per rebounding ice collision would be higher than the charge per collision 30 20

Cloud water content (g/m3)

10

–20

1

–10 40 30

0 20

0.10

10

0.01 0

–5

–10

–15

–20

–25

–30

Temperature (°C)

Figure 2.4 Takahashi (TAK) charging diagram (Takahashi, 1978), contoured from the tabulated data in Wojcik (1994), with charge separation in units of fC

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Lightning electromagnetics: Volume 2

(which includes sticking collisions). The table covers a temperature range of 0 C to 30 C and cloud water content from 0.01 to 30 g/m3. For temperatures lower than 30 C, the charge separation values at 30 C were used. Takahashi (1984) accounted for charging dependence on crystal size and fall speed by multiplying the value obtained from the table by a factor a:  2
DI V g (2.21) a ¼ 5:0 DI V0 where DI is the diameter of the ice crystal or snow particle, V
g is the massweighted mean terminal fall speed of graupel and D0 = 100 mm and V0 = 8 m/s. This factor was based on the work of Marshall et al. (1978). In Takahashi (1984), the value of a was not allowed to be greater than 10.0 (i.e. a 10.0). Thus, the final charge per collision is dq = adq0 . Tsenova and Mitzeva (2009) tested the Takahashi scheme but used the results of Keith and Saunders (1990) regarding dependence on crystal size and impact speed, which significantly affected charge separation rates. Tsenova and Mitzeva (2009) also developed an equation set to parameterize the lookup table of the data from Takahashi (1978). Based on the work of Jayaratne et al. (1983), Gardiner et al. (1985) suggested that the charge transferred to a rimed graupel/hail particle when it collides with a cloud-ice/snow particle could be parameterized by n @q ¼ kq Dm i ðDvgi Þ ðLWC  LWCcrit Þ  f ðDT Þ

(2.22)

where kq is a constant of proportionality approximately equal to 73, Di is the diameter of the cloud-ice particle in centimeters, Dvgi is the relative impact speed (in cm/s) between the graupel particle and ice crystal given by the difference in their terminal velocities, m  4, n  3, LWC is the liquid (cloud) water content (in g/m3), LWCcrit is the value of the liquid water content below which the sign of dq reverses (a plot of LWCcrit as a function of T is given by Jayaratne et al., 1983) and DT is the degree of supercooling (DT = 273.15  T for T < 273.15 K and is 0 otherwise). The function f(DT) was a polynomial fit to the laboratory data of Jayaratne et al.: f ðDT Þ ¼ aDT 3 þ bDT 2 þ cDT þ d

(2.23)

5

where a = 17 10 , b = 0.003, c = 0.05, and d = 0.13 and dq is in fC (1015C). Subsequent laboratory experiments showed that the increase in charge with ice-crystal diameter levelled off at large values of diameter, so the D4i dependence in (2.22) overestimated the charge transferred for large Di. Saunders et al. (1991) suggested a new, more complicated parameterization for dq that was based on laboratory experiments over a broader range of cloud-ice size, liquid water content and temperature. Their expression for charge (in fC), similar in functional form to that used by Gardiner et al. (1985), was dq ¼ BDai ðDvgi Þb qðEW ; TÞ

(2.24)

Modelling of charging processes in clouds

45

where EW is effective liquid water content, a parameter defined by Saunders et al. Unlike f(DT) in (2.22) and (2.23), however, f (T, EW) had different functional forms in different regimes of temperature and effective liquid water content. Furthermore, kq, m, and n depended on the size of the ice crystal and the polarity of charge transferred. These dependencies are shown in Table 2.2. Because data did not extend to temperatures greater than 7.4 C, Helsdon et al. (2001) linearly extra polated q(EW, T) at a particular EW from the value given by the expression in Table 2.2 for T = 7.4 C to zero at T = 0 C. Saunders et al. (1991) used effective liquid water EW content instead of LWC, the liquid water content measured by in situ instruments and determined by most numerical cloud models, because their observations suggested that EW was more relevant to riming-based non-inductive charging. EW is a modification of LWC that includes only the accreted fraction of liquid water content in the path of graupel. Therefore, it is given by the product of the ambient liquid water content and the collection efficiency (EW = LWC  Ecollect). The collection efficiency Ecollect for graupel and water particles is equal to the product Ecollect Estick, where Ecolli is the collision efficiency, a factor 1 that reduces the geometric cross section of graupel Table 2.2 Non-inductive parameterization of Saunders et al. (1991) Valid for T ( C)

And for dq Sign* q(EW,T)* (fC)

EW g/m3

1.1k –

2042EW – 129 (for 0.06 < EW < 0.12)

Di (mm) B 452 –314EW þ 7.9 (for 0.026 < EW < 0.14) 23.9 C: EWcrit ¼ 0:49  6:64  102 T

(2.25)

An equation for the reversal temperature Tr can be obtained by inverting this equation (see footnote in Table 2.2). Helsdon et al. (2001) noted that the expressions given in Table 2.2 resulted in much more charge per collision than observed for some particle sizes and collision speeds at small values of EW ( RARcrit), q þ ðRARÞ ¼ 6:74ðRAR  RARcrit Þ

(2.29)

For negative charging (0.1 gm2 s1 < RAR < RARcrit), q  ðRARÞ ¼ 3:9ðRARcrit  0:1Þ !
 RAR  ðRARcrit þ 0:1Þ=2 2 1  4 RARcrit  0:1

(2.30)

Note that there is an implicit temperature dependence since RARcrit varies with temperature. The negative charging equation (2.30) shifts the parabolic function given in Brooks et al. (1997) to fit between the limits of 0.1 gm2 s1 and RARcrit, removing the discontinuity at RARcrit in the original formulation. Charging is assumed to be zero for RAR < 0.1 gm2 s1. Few quantitative results are available on charge separation at temperatures less than 30 C. (One example is Saunders and Peck, 1998, which examined the sign of charging, but not the quantity, at lower temperature.) Therefore, lacking experimental guidance, Mansell et al. (2005) suggested limiting charging at low temperature by an arbitrary factor b, such as 8 : T > 30 C T < 30 C : 0 : T < Thom The low-temperature cut-off is made at Thom  –38 C because all cloud droplets are homogeneously frozen by that temperature, so no further riming occurs. Deep convective updrafts, however, may maintain ice supersaturation for T < Thom, which might support appreciable charge separation but no laboratory data exist. Mitzeva et al. (2006) used a one-dimensional cloud model to explore possible

Modelling of charging processes in clouds

49

consequences of allowing collisional non-inductive charge separation in regions with supersaturation with respect to ice but without any liquid hydrometeors.

2.4.3.2 General formulation In models with bin microphysics for graupel and cloud ice, the derivation of an expression for the rate at which graupel and cloud-ice charge densities (rg and ri, respectively) build due to the non-inductive mechanism is fairly simple. Per unit time, the volume in which a graupel particle of diameter Dg collides with cloud-ice particles of diameter Di is just the product of the cross-sectional area p(Dg + Di)2/4 (the area of a circle in which graupel and ice particles just touch) times the vertical fall speed of graupel relative to cloud-ice particles, DVgi = |VgT  ViT|. To compute a charging rate, the volume swept out by a graupel particle per unit time must be multiplied by the collision separation efficiency for graupel and ice (Egi), which is the fraction of ice particles in this volume that collide with the graupel and separate from it. Egi is equal to the product Ecolli Esep, where Ecolli is a factor that accounts for aerodynamic effects, as discussed previously, and Esep is the fraction of colliding particles that separate (Esep = 1  Estick, where Estick is the fraction accreted by graupel). The modified volume per unit time is called the collision kernel Kgi and can be expressed as Kgi ¼

p ðDg þ Di Þ2 DVgi Egi 4

(2.32)

If ng is the number density of graupel particles of diameter Dg, then the rate at which ng graupel particles collide and separate from ni cloud-ice particles is given by Kgi ng ni, and the rate at which this process charges graupel of diameter Dg and cloud ice of diameter Di is @rg ¼ Kgi ng ni dq @t @r ¼ i @t

(2.33)

The expression for non-inductive charging between graupel and snow is the same as (2.32) and (2.33), except that the parameters for cloud ice are replaced by those for snow (ns, Ds, Egs and DVgs). Similarly for hail and snow, graupel para meters are changed to the values for hail (nh, Dh, Ehs and DVhs), although dq has not yet been determined by laboratory experiments specifically for hail. To determine the rate of change in charge density for graupel in a particular size category of a model with bin microphysics, it is necessary to add together the contributions given by (2.33) for every size category of snow and cloud ice that interacts with the graupel. Likewise, the rate of change in charge density on a particular size category of cloud ice or snow must include the contribution from all size categories of graupel and hail. To determine the rate of change in charge density on a particular type of hydrometeor in a model with bulk microphysics, two approaches have been used.

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Lightning electromagnetics: Volume 2

The first is to evaluate (2.33) at a grid point by determining mean values of dq and Kgi and the total concentrations across all sizes for Ng and Ni at that grid point. The second approach is to integrate the right-hand side of (2.33) across all sizes both of that hydrometeor category and of every other category with which that category interacts. For example, the change in charge density of graupel due to collisions with snow or cloud ice is ðð @rg p ¼ (2.34) ðDg þ D1 Þ2 DVgi Egi ng ðDg Þni ðDi Þdq dDi dDg @t 4 where the subscript i refers to either snow or cloud ice. The size distributions n(D) often are assumed to have the form of the inverse exponential distribution (2.2 with a = 0) with parameters n0 and L. For cloud ice, because Di Dg and ViT VgT, the magnitude of sums and differences of these quantities can be approximated as Dg and VgT in (2.34). To evaluate this integral for cloud ice, Ziegler et al. (1986) assumed further that since cloud ice typically has a narrow size distribution, it can
i . If Ni is the total be approximated as a population with a single average diameter D concentration of cloud-ice particles, this implies that ð @rg p
i Þ2 DVgi ng ðDg Þdq dDg ¼ Ni Egi ðDg þ D (2.35) @t 4 The general formulation of (2.34) for the non-inductive charge (density) separation rate @rxy/@t between ice hydrometeor classes x and y is ð1 ð1 @rxy p 0 ¼ dqxy ð1  Exy ÞDVxy j @t 0 0 4  ðDx þ Dy Þ2 nx ðDx Þny ðDy ÞdDx dDy

(2.36)

where Dx and Dy are the diameters of the colliding particles, Exy is the collection efficiency, DExy is the relative fall speed, nx and ny are number concentrations and dq0 xy is the charge separated per rebounding collision. In general, dq0 xy may be a function of ice-crystal diameter, impact speed, cloud water content and temperature. The collection efficiency Exy is the product of the collision efficiency (Ecolli, assumed to be unity) and the probability of sticking given a collision (Estick). In wet-growth mode, one may assume Ex,y = 1, and no charge separation occurs. As it stands, (2.36) is not a tractable integrand. The equation can be approximated and simplified by assuming a form for dq0 xy that can be pulled out of the integral. Also, the fall speed difference is approximated by the difference of massweighted mean fall speeds. The collection efficiency is assumed to be constant. Multiplying and dividing by Exy then isolates the number concentration collection rate integral (nxacy): @rxy 1 ¼ bdqxy ð1  Exy ÞExy ðnxacy Þ @t

(2.37)

Modelling of charging processes in clouds

51

Where nxacy ¼ Exy DV xy ð1 ð1 p ðDx þ Dy Þ2 nx ny dDx dDy  0 0 4

(2.38)

and dqxy is now a representative (weighted average) separated charge per rebounding collision and (1  Exy) represents the rebound probability. Each of the non-inductive charging schemes uses the monodisperse diameter D for pristine ice crystals (plates and solid columns) but the characteristic diameter Dn = 1/ln to represent the average size of an inverse exponential (IE) distribution (e.g. rimed cloud ice and ice aggregates). The number concentration collection rate nxacy is calculated by an analytical approximation. For an inverse exponential distribution category (xe) interacting with a monodisperse distribution (ymym) the number concentration collection rate is nxe acym ¼

p Exy ny nx DV xy 4  ½Gð3ÞD2n;x þ 2Gð2ÞDn;x Dy þ Gð1ÞD2y 

(2.39)

Similarly, for an inverse exponential distribution (xe) interacting with another inverse exponential distribution (ye), the number concentration collection rate is nxe acym ¼

p Exy ny nx DV xy ½Gð3ÞGð1ÞD2n;x 4 þ2Gð2ÞGð2ÞDn;x Dn;y þ Gð1ÞGð3ÞD2n;y 

(2.40)

Collision rates for other size distribution functions (e.g. a general Gamma function) are beyond the scope of this chapter. For other collision rates, readers are referred to studies of microphysics parameterizations such as Milbrandt and Yau (2005) and Seifert and Beheng (2006).

2.4.4 The inductive charging mechanism As defined above, the inductive mechanism occurs in the presence of an electric field. Laboratory studies (e.g. Aufdermaur and Johnson, 1972; Aufdermaur and Johnson, 1972; Gaskell, 1981; Brooks and Saunders, 1994) and theory (e.g. Mason, 1988) suggest (1) the magnitude of charge transferred is a function of both the magnitude of the electric field and the angular distance of the impact point from the electric field vector through the centre of the particle, (2) inductive charging during collisions of ice particles appears to be negligible because charge transfer is too slow in ice (Latham and Mason, 1962), (3) collisions of rain drops and cloud droplets do not contribute significant charge because essentially no cloud droplets separate after they collide with rain drops in an electric field, and (4) only a small fraction of colliding rain drops or colliding graupel and cloud droplets subsequently separate, but enough charge is separated to be significant (Aufdermaur and Johnson, 1972).

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Lightning electromagnetics: Volume 2

The expression for the induced charge gained by a spherical graupel or hail particle in a rebounding collision with a cloud droplet in an electric field is 2  AQg þ BQcld DQg ¼ 4pe0 g1 jEj cos qE;r rcld

(2.41)

where rcld is the radius of the cloud droplet; qE,r is the angle between the impact point and the electric field vector through the centre of the graupel/hail particle (shown in Figure 2.1); Qg and Qcld are the charge already on the graupel/hail particle and cloud droplet, respectively; g1, A, and B are dimensionless functions of rcld/rg the ratio of the radii of the two particles: A¼

B¼

g2 ðrcld =rg Þ2 1 þ g2 ðrcld =rg Þ2 1 1 þ g2 ðrcld =rg Þ2

(2.42)

(2.43)

Parameterizing the inductive mechanism requires procedures similar to those used in parameterizing the non-inductive mechanism, except it is necessary to take into account that the mechanism’s effectiveness is a function of where on their surfaces two particles collide. The charge produced by the inductive mechanism is strongly dependent on the angle between the impact point and the electric field vector. As shown in Figure 2.1 for a spherical graupel particle, the magnitude of the induced surface charge density is the largest at the two ends of the diameter that parallels the electric field vector: As the angle from the electric field increases to 90 , the surface charge density decreases to zero. Besides this effect, the probability that colliding graupel and cloud droplets will separate can vary with the angle of the impact point from the vertical axis. Moore (1975) suggested that the separation probability is much larger for glancing collisions than for head-on collisions. Since colliding particles must separate for the inductive mechanism to work, the angular dependence of charge transfer and separation probability can interact in complicated ways. The mechanism will be most effective at the location on the graupel surface where the product of the induced surface charge density and the collision separation probability is largest. Regardless of whether a model uses bulk or bin microphysics, the inductive parameterization must handle the complication of these interacting factors. Chiu (1978) treated them by defining a mean separation probability hSi and a mean impact cosine hcos ji ð 1 p=2 SðjÞ2prg2 sin j cos j dj (2.44) hSi ¼ 2 prg 0 ð 1 p=2 2SðjÞsin j cos2 j dj (2.45) hcos ji ¼ hSi 0

Modelling of charging processes in clouds

53

where the weighting factor under the first integral is an infinitesimal area in the horizontal cross section of the graupel particle. Then the mean charge transferred to a graupel particle by colliding with and separating from cloud droplets is 2 hcos ji  AhQg i þ BhQcld i dQg ¼ hSi½4pe0 g1 jEj cos qE;z rcld

(2.46)

where qE,z is the angle between the electric field vector and the lower vertical axis and hQg i and hQcld i are the mean charge per particle on graupel and cloud droplets, respectively. The rate at which this process charges a group of graupel particles of radius rg due to collisions with cloud droplets of radius rcld is found by multiplying the charge transferred per collision by the sweep-out volume of the graupel particle and the number concentrations of graupel and cloud particles:     @rg rg ; rcld ¼ prg2 Dvg;cld Ecolli ng rg ncld ðrcld ÞdQg (2.47) @t where ng(rg) and ncld(rcld) are the number density of graupel and cloud liquid water particles of radius rg and rcld, respectively, and Ecolli in this case is the collision efficiency for graupel and cloud particles (i.e. the fraction of cloud particles in the volume swept out by the graupel particle that actually collide with it). This expression neglects rcld in the sum rcld + rg for the cross-sectional area prg2 . The rate at which charge is generated by a parameterization of the inductive mechanism depends strongly on the values selected for hSi and hcos ji. If the mean separation probability is 1 (i.e. all colliding particles separate) and hcos ji ¼ 0:67, then the mechanism’s effectiveness is maximized. This is almost certainly an overestimate. Some studies, such as Moore (1975) and Aufdermaur and Johnson (1972), suggested that it is orders-of-magnitude too large. Helsdon and Farley (1987b) used a mean separation probability of 0.015 in their model experiment and a mean cosine of 0.5. Mansell et al. (2005) tested a range of values and found that higher efficiencies could have a significant effect on simulated charge structure. As is the case for the graupel–ice non-inductive mechanism, models with bulk microphysics must integrate the charge produced by the inductive mechanism across all sizes of interacting hydrometeors, so the total rate of change in the charge density of graupel would be given by integrating (2.47) over all values for cloud radius and graupel radius. Helsdon and Farley (1987b) approximated the integration by replacing rcld, rg, Dvg,cld, and Ecolli with their mean values at a grid point, thereby treating them as constants in the integral. The remaining integration of ng (rg)ncld(rcld) over all radii gave simply the total number density of graupel times the total number density of cloud droplets at the grid point being considered. Ziegler et al. (1991) developed a bulk parameterization based on several simplifying assumptions. Consistent with their parameterization of the non-inductive mechanism discussed above, they considered only Dcld Dg and vcldT vgT and so ignored the cloud droplet term in sums and differences of these quantities. Furthermore, they assumed that the narrow droplet size spectrum could be approximated as a population having a single diameter Dcld, and they used values of

54

Lightning electromagnetics: Volume 2  2 g1  p2 =2; g2  p2 =6 and A  g2 Dcld =Dg , which were appropriate for Dcld =Dg 1. In considering the collision process for inductive charging, it was assumed that only a small fraction of graupel and cloud particles separates after they collide, as found by Aufdermaur and Johnson (1972), and that rebounding occurs only during glancing collisions, as suggested by Moore (1975. Since the probability that any one cloud droplet will experience two rebounding collisions with graupel is much lower than the probability that it will experience one, the ‘B’ term involving pre-existing droplet charge (2.47) was omitted. (Droplets could not gain charge by any other mechanism in their model, which is not the case for models that treat small ion attachment or screening layers.) Furthermore, they considered only a vertical electric field, since it was the vertical electric field that had been hypothesized to contribute to thunderstorm electrification by the inductive mechanism. The equation for the rate at which charge was gained by a graupel particle of diameter Dg from the inductive mechanism then became "  #  2 @Qg p p3 2 p D2cld 2 ¼ Dcld e0 Ez hcos ji  hQg i 2 Egc Er Dg vg ncld a @t 2 6 Dg 4 (2.48) where a is the fraction of collisions that have glancing trajectories, ncld is the total number concentration of cloud particles, Egc and Er are the collision and rebound probabilities, and Ez is the vertical electric field component. Ziegler et al. (1991) integrated this equation for single-moment graupel (inverse exponential distribution with intercept n0g) to get @rg  3  ¼ p =8 egc er ncld n0g aD2cld ð4grI =3CD rair Þ1=2 @t h  i  pGð3:5ÞeEz hcos filg7=2  Gð1:5Þrg lg3=2 = 3ng

(2.49)

where G (1.5) = 0.886 and G (3.5) = 3.323. They chose Egc = 0.84, Er = 0.1, a = 0.022 and hcos ji ¼ 0:1, which gave a probability of rebounding collisions prbnd = EgcEra near the lower end of the range found by Aufdermaur and Johnson (1972).

2.4.5

Small ion processes

Some models (e.g. Chiu, 1978; Takahashi, 1979; Helsdon and Farley, 1987a; Mansell et al., 2005) explicitly treat space charge on free ions (as opposed to charge carried by hydrometeors) and incorporate ion capture by hydrometeors in order to examine how ion capture affects cloud electrification. Although an explicit treatment of ions creates difficulties, it also enables a model to deal with important phenomena, including the early stages of electrification (when charging by other electrification processes is small or non-existent), precipitation capture of ions emitted as corona beneath a storm, screening layer charge at cloud boundaries, and the dispersal and capture of ions from a lightning channel.

Modelling of charging processes in clouds

55

Cosmic rays are the source of most ions in the fair-weather troposphere, except near the ground; radioactive decay from the surface contributes up to half of the ions found near ground. In fair weather, the number of positive ions is roughly equal to the number of negative ions, so the net charge density results from small differences of large numbers of ions. Ions move under the influence of the electric field E and their average drift velocity in a given E is the mobility times the electric field m  E, where subscript gives the polarity of the ions. The mobility m increases with decreasing pressure and so increases with height (i.e. the mean free path increases). Each polarity of ion must obey its own continuity equation, which is similar to (2.20) for charge attached to hydrometeors. For ions, however, a term must be added to account for average charge motion under the influence of the electric field, since the resulting ion drift velocity can easily be much different than wind velocities. Also, the source/sink term often is split into sources and sinks occurring in fair weather and those requiring clouds or thunderstorms (e.g. Chiu, 1978; Helsdon and Farley, 1987b). The continuity equation for free ions can be written as   @n 1 n ¼ r  ½n V  n m E þ r  Km r þG (2.50) rair @t rair  anþ n  Satt þ Spd þ Sevap where n+ is the number density of positive ions and n–, the number density of negative ions. Advection (the first term in the brackets) and turbulent mixing (the second full term) are treated the same as for the other scalar variables. The second term in the brackets is the ion drift motion, which can be treated similarly to the advection term. G is the background cosmic ray ion generation rate; an+n– is the ion recombination rate and the last three terms, respectively, are ion attachment to hydrometeors (sink), point discharge current from the surface (source) and release of any charge as ions from hydrometeors that evaporate completely (source). If the ion drift speeds exceed the maximum for stable transport at the time step of the model, then the ion processes (except advection and turbulent mixing) can be performed on a subdivided time step, leaving the dynamical time step unchanged. The fair-weather state can be defined as in Gish (1944), using the modified coefficients of Helsdon and Farley (1987b): Ez;FW ¼ E0 ðb1 ea1 z þ b2 ea2 z þ b3 ea3 z Þ

(2.51)

where E0 = –80.0 V/m, b1 = 0.5, a1 = 4.5  103, b2 = 0.65, a2 = 3.8 104, b3 = 0.1, and a3 = 1.0104. At steady state in a fair-weather condition, the vertical positive and negative ion fluxes are  d nþ mþ EZ;FW ¼ GðzÞ  anþ n dz 

 d n m Ez;FW ¼ GðzÞ  anþ n dz

(2.52) (2.53)

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Lightning electromagnetics: Volume 2

where G(z) is the ion generation rate by cosmic rays (held constant as a function of altitude) and a = 1.6 1012 m3 s is the ionic recombination coefficient (Chiu, 1978). The ion mobilities m from Shreve (1970) are given by m ¼ b e1:410

4

z

(2.54)

where b+= 1.4  10 m V s and b– = 1.9 10 m V s Diffusivity is derived from mobility by the Einstein relation –4

D ¼

2

–1 –1

–4

2

–1 –1

and z is in metres.

kT m e 

(2.55)

where k is Boltzmann’s constant, T is the temperature (in Kelvin), and e is the electron charge magnitude (ions are assumed to be singly charged). Under steady-state (i.e. fair-weather) conditions, one may assume that the ion currents and charge densities vary negligibly from constant values, so that rj= 0 Therefore, from (2.52) we get GðzÞ ¼ anþ;FW ðzÞn;FW ðzÞ ðsteady stateÞ

(2.56)

(as in Takahashi, 1979) and the cosmic ray generation rate can be held constant in time throughout a simulation. Ion attachment to hydrometeors is a combination of diffusion, Sdiff, and conduction, Scond. As in Chiu Chiu (1978), the two terms are calculated separately and added (Satt = Sdiff + Scond). The equations from Chiu (1978) (based on Whipple and Chalmers, 1944)) for attachment by conduction can be found in Table 2.3. Table 2.3 Expressions for ion capture via conduction Qj Qj > Qj < –QM‡ 0 < Qj < QM –QM < Qj < 0 QM†

E||VT or E||VT m|E|

@nþ/@t*

@n–/@t*

Either Either

0 –nþnjmþQje–1 0 –nþnjmþQje–1

n–njm–Qje–1 0

Parallel

–QM < Qj < QM 0 < Qj < QM –QM < Qj 0





(6.25)

< 0. So, indiit readily follows that Im Recalling the definition one has that p < f < 2p s a consequence, if q is cating with f the phase angle the phase angle of mE, one has that either q ¼ 2f or q ¼ 2f þ p can be chosen. In the p second case 3p 2 < q < 2p, while in the first one 2 < q < p. So, to meet the f requirement (6.25), q ¼ 2 þ p must be chosen and, as a consequence, it results: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   f f 1 þ cos f þ p ¼  cos ¼ þ (6.26) cos q ¼ cos 2 2 2 of m2E , of mE2,

m2E

and 

f sin q ¼ sin þp 2



f ¼  sin ¼  2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  cos f 2

Now, expressing mE in polar form, it follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ cos f 1  cos f j Þ mE ¼ jmE jð 2 2

(6.27)

(6.28)

being l2  ee0 k2 ffi cos f ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

e e0

l  k 2

2

2

þ

s2 k2 me00

(6.29)

The numerical treatment of Sommerfeld’s integrals

211

1.2 1.0

real part

0.8 0.6 0.4 0.2 0.0 –0.2 –0.95

2.00

2.05

2.10

2.15

2.20

O(1/m)

Figure 6.5 Real part of the s-dependent term for k = 2.09  105 and ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s 2 4 e m l2  k2 þ s2 k2 0 jmE j ¼ e0 e0

(6.30)

where the square roots are all real functions. As far as m is concerned, one can observe that m ¼ lims!0þ mE , and thus it results: ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2  k2 for l > k ; (6.31) m¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j k2  l2 for l < k :

6.2.1.4.2 The s–dependent term

mE The term gs ðlÞ ¼ n2 mþm would be zero if the ground were a perfect conductor. As a E matter of fact, for fixed k, it is a function of l, and, as can be seen from Figures 6.5 and 6.6, its behavior shows some sort of “resonance” for l = k. As a consequence, the integrand of (6.24), in the neighborhood of k, in spite of being continuous, is not, so to speak, “smooth.” This fact has an important consequence: the integral (6.24) is a Hankel transform, but the above mentioned property makes it impossible for the traditional Hankel transform algorithms [46] (which basically are Gaussian quadrature methods) to correctly evaluate the integral in (6.24), as shown in Figure 6.7, where the real part of the spectrum is depicted (the same holds for the imaginary one).

Lightning electromagnetics: Volume 2 1.5

1.0

imaginary part

0.5

0.0

–0.5

–1.0

–1.5 2.0942

2.0944

2.0943 O(1/m)

2.0945 X 105

Figure 6.6 Imaginary part of the s-dependent term for k = 2.09  105

0.05 0.04 0.03 real part of the spectrum

212

0.02 0.01 0 –0.01 –0.02 –0.03 0

1

2

3

4 5 6 frequency [Hz]

7

8

9

10 ×105

Figure 6.7 Failure of the classical routine for the Hankel transform

The numerical treatment of Sommerfeld’s integrals

213

6.2.1.5 The vertical component of the electric field The inverse Fourier transform of the ideal term (6.21) can be performed analytically, thus obtaining:   ð " 1 vt 2ðz  z0 Þ2  r2 R jz 0 j eziL ðtÞ ¼ i 0; t   v 4pe0 vt c cR4  3 R jz 0 j   ð @i 0; t   2ðz  z0 Þ2  r2 t R jz0 j r2 c v 7 7Pðz0 Þdz0 ds  þ  i 0; s  5 3 5 2 @t v c R c R 0 (6.32) having indicated with lower case letters the time domain functions. Here, the range of integration is (–vt, vt) instead of (–H, H), since the current i (z0 ,t) is identically zero for z0 >vt. This is the reason why, if one performs the fields calculations in the time domain and is interested in the first microseconds of the transient, the channel height is of no use. As far as the Sommerfeld term is concerned, the inverse Fourier transform must be carried out numerically. This requires to evaluate numerically the corresponding Sommerfeld integral appearing in the first of system (6.19) for many different values of frequency in an assigned range (i.e. [0 Hz, 107 Hz]). Moreover, if one observes that, for each frequency: the integrand contains both the Bessel function (which is highly oscillating) and the s-dependent term (whose behaviour is shown in Figures 6.5 and 6.6); the integrand is singular for l = k; the integral must be carried out over a semi-infinite domain,

●

● ●

one is easily convinced that such calculation requires a huge computational effort. To overcome the first problem, a Romberg method [47–49] has been used, dividing the interval of integration into sub-intervals, with particular attention to the neighborhood of k, namely:  ð 0:99  ð1 ð 1:01 ð1 ð1 f ðkuÞdu þ k f ðkuÞdu þ k f ðkuÞdu þ k f ðkuÞdu k f ðkuÞdu ¼ k 0

0

0:99

1

1:01

(6.33) having posed u ¼ lnkand having indicated with f the integrand function of the first integral of system (6.19). As far as the singularity is concerned, let us consider the integral I1 between 0.99 and 1: ð1 u pffiffiffiffiffiffiffiffiffiffiffiffiffi h1 ðuÞdu (6.34) I1 ¼ 0:99 1  u2

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Lightning electromagnetics: Volume 2

with pffiffiffiffiffiffiffiffi 2 h1 ðuÞ ¼ J0 ðkruÞk3 u2 gs ðkuÞejk 1u z QðkuÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi Setting s ¼ 1  u2 , one has that: pffiffiffiffiffiffiffiffiffiffiffiffi ð 10:992 pffiffiffiffiffiffiffiffiffiffiffiffiffi

h1 1  s2 ds I1 ¼

(6.35)

(6.36)

0

being the integrand not singular in the range of integration. Indicating with I2 the integral between 1 and 1.01, one has that: ð 1:01 u pffiffiffiffiffiffiffiffiffiffiffiffiffi h2 ðuÞdu I2 ¼ 2 u 1 1

(6.37)

being pffiffiffiffiffiffiffiffi 2 h2 ðuÞ ¼ J0 ðkruÞk3 u2 gs ðkuÞek u 1z QðkuÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi Setting now s ¼ u2  1, one has that: pffiffiffiffiffiffiffiffiffiffiffiffi ð 1:012 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ s2 ds h2 I2 ¼

(6.38)

(6.39)

0

being again the integrand not singular in the range of integration. The integral between 1.01 and ? requires to “approximate the infinity,” that is to say to find out a number M such that the difference between the original integral and the one between 1.01 and M is sufficiently small. Typically the number M is searched with iterative procedures; here it is possible to derive an upper bound for the error Err in the so-called integral tail for each frequency and so to estimate M in an analytical way, thus reducing the computational costs. Let ð 1  pffiffiffiffiffiffiffiffi   k3 u3 k u2 1z  QðkuÞdu (6.40) ErrðMÞ ¼  J0 ðkruÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi gs ðkuÞe 2 u 1 M it readily follows that:  ð1  3 3 pffiffiffiffiffiffiffiffi   u k u2 1z J0 ðkruÞ pkffiffiffiffiffiffiffiffiffiffiffiffiffi du g ð ku Þe Q ð ku Þ ErrðMÞ  s   u2  1 M Observing that: ●

●

●

the absolute values of both Q and gs are decreasing functions for u>1; pffiffiffiffiffiffiffiffiffiffiffiffiffi if M>4/3, u2  1  u2 qffiffiffiffiffiffiffi 2 [41] jJ0 ðkruÞj < pkru

(6.41)

The numerical treatment of Sommerfeld’s integrals 3.5

215

×10–11

3

Error/Integral

2.5 2 1.5 1 0.5 0

0

1

2

3

4 5 6 Frequency [Hz]

7

8

9

10 ×10–6

Figure 6.8 Ratio between the upper bound of the tail and the absolute value of the integral between 0 and M as a function of the frequency one has pffiffiffi ð 2 2k3 1 kuz 3 ErrðMÞ  jgs ðkMÞQðkMÞj pffiffiffiffiffiffiffi e 2 u2 du pkr M

(6.42)

Finally, since the integral in relation (6.42) is known analytically [41], it follows: pffiffiffi   5   2 2k3 kz 2 5 kMz (6.43) G ; ErrðMÞ  jgs ðkMÞQðkMÞj pffiffiffiffiffiffiffi 2 2 pkr 2 being G the incomplete Gamma function [41]. In Figure 6.8, the ratio between the upper bound of the tail and the absolute value of the integral between 0 and M has been plotted as a function of the frequency, having set M = 4p/5k, showing that our goal has been achieved. The position u ¼ lnk is not possible for f = 0 Hz. This implies that another method must be used to perform the so-called static term. Let us reconsider the Sommerfeld integral appearing in the first of (6.17); observing that, if k approaches 0: 8 js > > ; n2 ! > > we0 > > > > < m 2 ! l2 ; E (6.44) > 2 > ! l2 ; m > > > > we0 > mE > ; ! : 2 n m þ mE js

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Lightning electromagnetics: Volume 2

one has that [41]: ð 3 0 j þ1 mE mðzþz Þ l dl  J ð lr Þ  e  0 m 2pwe0 0 n2 m þ mE   ð 0 1 þ1 1 3 r2 ; 1; 1; J0 ðlrÞ  elðzþz Þ l2 dl ¼ F ! 2ps 0 2 R0 2 psR0 3

(6.45)

F being the Hypergeometric function [41]. As a consequence, the static term for the Sommerfeld integral appearing in the first of (6.19) is given by: jIð0; wÞ limk!0  2pwe0

þ1 ð

0

3 0 mE mz l QðlÞdl  J ðlrÞ  e  0 n2 m þ mE m

  Ið0; 0Þ 1 3 r2 ¼ F ; 1; 1; 0 2 Pðz0 Þdz0 ps 2 R R0 3 ðH

(6.46)

0

6.2.1.6

The radial component of the electric field

The inverse Fourier transform of the ideal term (6.22) can be performed analytically, thus obtaining: vt    ð" ðt  1 3rðz  z0 Þ R jz0 j 3rðz  z0 Þ R jz0 j þ ds i 0; t   i 0; s   eriL ðtÞ ¼ 4pe0 c v c v R5 cR4 vt 0  3 R jz0 j @i 0; t   rðz  z0 Þ c v 7 7Pðz0 Þdz0 þ 2 3 5 @t c R

(6.47) having indicated with lower case letters the time domain functions. As far as the Sommerfeld term is concerned, again the inverse Fourier transform must be carried out numerically. This requires to numerically evaluate the corresponding Sommerfeld integral appearing in the second of (6.19) for many different values of frequency in an assigned range (i.e. [0 Hz, 107 Hz]). From a numerical point of view, here things go better, since the integral function is not singular for l = k. Therefore, again a Romberg method is used, splitting the range of integration into many sub-intervals, as done for the z-component of the electric field, but the neighborhood of k does not require a ‘special treatment’ as before. The only problems to solve are the ones relevant to the integral tail and the static term. Indicating again with Err the upper-bound for the integral tail and with

The numerical treatment of Sommerfeld’s integrals M the last point of the interval on which the integral is taken, one has: ð 1  pffiffiffiffiffiffiffiffi   3 2 k u2 1z  QðkuÞdu ErrðMÞ ¼  J1 ðkruÞk u gs ðkuÞe

217

(6.48)

M

With considerations similar to the ones done in the previous section, it follows that: pffiffiffi 3 ð 1 2k u 3 ek2z u2 du ErrðMÞ  jgs ðkMÞQðkMÞj pffiffiffiffiffiffiffi pkr M

(6.49)

and again [41]: pffiffiffi 3  5   2k kz 2 5 kMz G ; ErrðMÞ  jgs ðkMÞQðkMÞj pffiffiffiffiffiffiffi 2 2 pkr 2 As far as the static term is concerned, now one has that: ð 0 j þ1 mE  J1 ðlrÞ  emðzþz Þ  l2 dl 2 2pwe0 0 n m þ mE   ð 0 1 þ1 3r 1 r2 ; 2; J1 ðlrÞ  elðzþz Þ l2 dl ¼ F 2;  ! R02 2ps 0 2psR04 2

(6.50)

(6.51)

As a consequence, the static term for the Sommerfeld integral appearing in the second of (6.19) is given by: jIð0; wÞ limk!0  2pwe0 3rI ð0; 0Þ ¼ 2ps

ðH 0

ð þ1 0

0 mE  J1 ðlrÞ  emz  l2 QðlÞdl þ mE

n2 m

  1 1 r2 F 2;  ; 2; 02 Pðz0 Þdz0 R04 2 R

(6.52)

6.2.1.7 The azimuthal component of the magnetic field The inverse Fourier transform of the ideal term (6.23) can be performed analytically, thus obtaining:

2 3 jz0 j   R ð 1 vt 4 r @i 0; t  c  v r R jz 0 j 5 0 0 Pðz Þdz þ 3 i 0; t   hjiL ðtÞ ¼ v 4p vt cR2 c @t R (6.53) having indicated with lower case letters the time domain functions.

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Lightning electromagnetics: Volume 2

As far as the Sommerfeld term is concerned, in the numerical evaluation of the integral (carried out again with a Romberg method for a number of frequencies ranging between 0 and 107 Hz), one has to face: ● ● ●

the singularity in the integrand function for l = k; the integral tail; the static term.

The first problem can be solved in the same way as for the vertical component of the electric field, with the only difference that now: pffiffiffiffiffiffiffiffi 2 (6.54) h1 ðuÞ ¼ J1 ðkruÞk2 ugs ðkuÞejk 1u z QðkuÞ and pffiffiffiffiffiffiffiffi u2 1z

h2 ðuÞ ¼ J1 ðkruÞk2 ugs ðkuÞek

QðkuÞ

The quantity Err is now: ð 1  pffiffiffiffiffiffiffiffi   k2 u2 k u2 1z  QðkuÞdu ErrðMÞ ¼  J1 ðkruÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi gs ðkuÞe 2 u 1 M In this case, the upper-bound is given by: pffiffiffi ð 2 2k2 1 kuz 1 e 2 u2 du ErrðMÞ  jgs ðkMÞQðkMÞj pffiffiffiffiffiffiffi pkr M

(6.55)

(6.56)

(6.57)

and, finally [41], pffiffiffi   3   2 2k2 kz 2 3 kMz G ; ErrðMÞ  jgs ðkMÞQðkMÞj pffiffiffiffiffiffiffi 2 2 pkr 2

(6.58)

As far as the static term is concerned, recalling the third of (6.19), one can observe that such term must vanish, since: mE we0 ! n 2 m þ mE js

(6.59)

and the coefficient before the Sommerfeld integral in the third of system (6.19) does not depend on the frequency.

6.2.2

Underground electromagnetic field

In this section, the derivation of the underground lightning electromagnetic fields expression is presented, starting from the expression for their Green’s functions. The situation is now the one depicted in Figure 6.9, in which the vertical dipole is placed at source point P0 (0, 0, z0 ), while the observation point is P(r, f, z), with z < 0.

The numerical treatment of Sommerfeld’s integrals

219

az P

z z P' (0,0,z')

ay

+ ϕ

–

ax

r

aΦ ar

R Air ε0, μ0, σ = 0

z=0 P (r, ϕ, z)

Ground ε, μ0, σ

Figure 6.9 The electric dipole radiation in the subsoil

Combining (6.11) and (6.13), one can easily gets:   ð þ1 m0 l 2 ðmz0 mE zÞ 2n  J0 ðlrÞ  e dl AE ¼ 4p n2 m þ mE 0

(6.60)

Furthermore, the three nonzero components of the fields are related to the vector potential in the earth by the following: 8   jw @ 2 AE > 2 > ; E ¼ þ k A > z E E > > kE 2 @z2 > > >   < jw @ 2 AE (6.61) ; Er ¼ 2 > @z@r > k E > > > > > 1 @AE > : Hj ¼  : m0 @r Now, inserting (6.60) into (6.61), it follows: 8 ð þ1 j l3 0 > > E ¼  J0 ðlrÞ  emz emE z dl; > z 2m þ m > n 2pwe > 0 0 E > > > ð þ1 < j l2 m 0 Er ¼  J1 ðlrÞemz emE z 2 E dl; > n m þ mE 2pwe0 0 > > > ð þ1 > 2 2 > > n l 0 > : Hj ¼  J1 ðlrÞ  emz emE z dl: 2 2p 0 n m þ mE

(6.62)

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Lightning electromagnetics: Volume 2

Assuming now the same expression for the current distribution in the lightning channel used in the previous case (see (6.18)), with the same attenuation function P (z0 ), the lightning fields are given by: 8 ð jIð0; wÞ þ1 l3 > >  J0 ðlrÞ  QðlÞemE z dl; EzL ¼ > > 2m þ m > n 2pwe 0 0 > E > > > ð < jIð0; wÞ þ1 l2 m J1 ðlrÞQðlÞemE z 2 E dl; ErL ¼  > n m þ mE 2pwe0 0 > > > > ð > > n2 Ið0; wÞ þ1 l2 > > : HjL ¼   J1 ðlrÞ  QðlÞemE z dl: n2 m þ mE 2p 0

(6.63)

From a numerical point of view, the same considerations can be done as in the previous subsection, concerning the troubles in the evaluation of the integrals, due to the characteristics of the integrand functions. Here, for the sake of brevity, we simply give the expressions for the upper bound of the error that is made truncating the integral tail and for the static term for the three nonzero components of the fields. Vertical electric field: –

upper bound: pffiffiffi 4   7   kz 2 7 kMz 2k EðMÞ  jgsz ðkMÞQðkMÞj pffiffiffiffiffiffiffi  G ; 2 2 2 pkr

–

1 being gsz ðlÞ ¼ n2 mþm E static term:

ð jIð0; wÞ þ1 l3  J0 ðlrÞ  emE z QðlÞdl limk!0 2pwe0 0 n2 m þ mE   ð Ið0; 0Þ H 1 3 r2 ; 1; 1; F ¼ Pðz0 Þdz0 ps 0 R3 2 R2 Radial electric field: –

upper bound: pffiffiffi 3   5   kz 2 5 kMz 2k G ; EðMÞ  jgsr ðkMÞQðkMÞj pffiffiffiffiffiffiffi  2 2 2 pkr mE being gsr ðlÞ ¼ n2 mþm

E

The numerical treatment of Sommerfeld’s integrals –

221

static term: ð jIð0; wÞ þ1 mE  J1 ðlrÞ  emE z  l2 QðlÞdl limk!0  2pwe0 0 n2 m þ mE   ð 3rIð0; 0Þ H 1 1 r2 Pðz0 Þdz0 ; 2; ¼ F 2;  4 2 2ps 2 R R 0

–

Azimuthal magnetic field: upper bound: pffiffiffi   3   2 2k3 kz 2 5 kMz G ; EðMÞ  jgsz ðkMÞQðkMÞj pffiffiffiffiffiffiffi 2 2 2 pkr

–

static term: ð n2 Ið0; wÞ þ1 l2  J1 ðlrÞ  emE z QðlÞdl limk!0 n2 m þ mE 2p 0   ð rIð0; 0Þ H 1 3 r2 ; 0; 2; ¼ F Pðz0 Þdz0 3 2 2p 2 R R 0

6.3 Lightning electromagnetic field calculation in presence of a lossy ground with frequency-dependent electrical parameters In the previous paragraphs, even though no mathematical approximations were used in the theory underlying the outlined work, one major, yet commonly adopted, simplifying assumption was made on the considered model: the relevant soil electrical parameters, namely conductivity and permittivity, were assumed to be frequency independent. The validity of this hypothesis could be questionable, as extensive research on the subject, conducted since the early years of the last century [50], shows a non-negligible dependence of both permittivity and conductivity on the frequency [51,52]. As a matter of fact, ground electrical parameters are heavily affected by soil heterogeneous components and structure and, remarkably, by its water content (distinction between “wet ground” and “dry ground” is typically made [51]). Furthermore, a number of different phenomena take place over the frequency range of interest for lightning electromagnetic field evaluation (up to a few MHz), including dipolar molecules polarization, counter-ion diffusion polarization (due to separation of cations and anions), interfacial (Maxwell–Wagner) polarization, other polarization effects, and various conduction and loss mechanisms, each acting on a specific frequency interval [53,54], thus making conductivity and permittivity behave as frequency-dependent functions. As an example,

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Lightning electromagnetics: Volume 2

according to [51], for frequencies f up to 1 MHz, wet rocks conductivity slowly varies with f and typically lies in the range between 101 and 105 S/m, with the 1 MHz value seldom exceeding the low frequency one by more than one order of magnitude, while the relative dielectric permittivity asymptotically tends to a value less than that of the water for higher frequencies and is usually inversely proportional to f in the lower portion of the range (where for some minerals and mixtures its value can be as high as 104 and even 108 below 1 Hz). In the same frequency range, the conductivity of dry rocks is proportional to f, with starting values (i.e., at low frequencies) lying in the range from 106 to 1012 S/m, while their relative permittivity seldom exceeds 10, slowly varies with f, and is characterized by a ratio between low frequencies and 1 MHz values seldom exceeding one order of magnitude. In light of these considerations, our aim here is to investigate how the actual frequency-dependent characteristics of both permittivity and conductivity affect the electromagnetic fields radiated by a cloud-to-ground lightning return stroke. To do this, we make use of suitable relations, which have been proven effective to represent the frequency dependence of ground conductivity and permittivity for a number of soil types over the frequency range of interest [55,56]. Experimental measurements of the electrical parameters for various soils, mixtures, and minerals, as well as theoretical explanations and proposed representations for their frequency behavior can be found, for instance, in [51,53,54].

6.3.1

The dependence of soil conductivity and permittivity on the frequency

In 1966, Scott [56] reported the results of measurements of the dielectric constant e and of the conductivity s of many samples of soil, over the frequency range [102– 106] Hz and noted that the results for the many samples could be correlated quite well in terms of just one parameter, the water content p. By averaging these data, the author produced a set of curves s(w) and e(w) as functions of the angular frequency w for different values of the water content. This means that, if one knows the water content of the soil, one can obtain a sufficiently accurate waveform expressing the functional dependence of the ground conductivity and permittivity on the angular frequency. In 1975, Longmire and Smith [55] proposed a model for the soil in order to provide an analytical expression for its conductivity and permittivity. Starting from the well-known relationship, which states that J ¼ ðjwe þ sÞE;

(6.64)

J being the (total) current density in the soil and E the electric field, it is possible to define an admittance Y ¼ ðjwe þ sÞ which allows to assimilate a cubic meter of soil to a network of resistor and capacitors (Figure 6.10). Once the parameters of such a network are determined, it is possible to infer the conductivity and the permittivity of the ground from the real and the imaginary parts of the admittance, respectively.

The numerical treatment of Sommerfeld’s integrals

R0

C00

R1

Rn–1

Rn

C1

Cn–1

Cn

223

Figure 6.10 Soil equivalent network Specifically, with reference to the network of Figure 6.10, the admittance reads: Y¼

N X 1 1 þ jwC1 þ 1 R0 R þ jwC n¼1 n n

(6.65)

Therefore, it follows that e ¼ C1 þ

N X

Cn

2 n¼1 1 þ ðw=bn Þ

(6.66)

and s¼

N X 1 Cn bn ðw=bn Þ2 þ R0 n¼1 1 þ ðw=bn Þ2

(6.67)

with bn ¼ Rn1Cn . Now, defining 8 C1 > ; e1 ¼ > > > e0 > > > > 1 > > ; < s0 ¼ R0 > C > > an ¼ n ; > > e0 > > > > > : f ¼ bn ; n 2p

(6.68)

The relative permittivity er and the conductivity can be written, respectively, in the following form e r ðf Þ ¼ e 1 þ

N X

an

n¼1

1 þ ðf =f n Þ2

sðf Þ ¼ s0 þ 2pe0

N X an f n ðf =f n Þ2 n¼1

1 þ ðf =f n Þ2

(6.69)

(6.70)

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Lightning electromagnetics: Volume 2

Finally, in order to have good fitting between (6.69) and (6.70) and the experimental curves obtained by Scott, Longmire and Smith [55] have shown that it is sufficient to set: e1 ¼ 5 s1 ¼ 8  103 fn ¼

p 1:28 10

(6.71)

p 1:54  S  10

(6.72)

m

10n1 ½Hz

(6.73)

where p is the percent water volume, which typically ranges between 2 and 30. Lower values imply lots of “rock-like” content. The values for the coefficients an are reported in Table 6.1. Our analysis assumes radially homogenous soil type and moisture content. Otherwise, it would be necessary to take into account the fact that, for any given location, the moisture content in the top meter varies seasonally by at least a factor of two, leading to a (roughly) factor-of-four gradient in electrical conductivity in the top two meters of soil. This creates an additional frequencydependent behavior at higher frequencies due to dependence of skin depth on frequency. The hypothesis of radially homogenous soil justifies (6.72) and (6.73) and allows us not to consider such an effect. As an example, Figures 6.11 and 6.12 present the variation of conductivity and relative permittivity as a function of the frequency in the range [0–5106] Hz and for water content p equal to 0.2%, 1% (to simulate rock-like ground), 2%, 10%, and 30% respectively. Others formulations for the frequency-dependence of the soil conductivity and permittivity have been proposed by Portela in [57] and Visacro et al. in [58]. Portela carried out a series of measurements which comprises experimental data obtained in several geological areas in Brazil and considers soil samples measured from 100 Hz up to 2 MHz. The value of the effective conductivity s, as well as relative permittivity, is expressed as a function of the low frequency conductivity s0 obtained from the measured 100 Hz soil resistivity according to (6.74)  p

 w a  (6.74) s þ jwe s0 þ Di cot ang a þ j 2 2p  106

Table 6.1 Coefficients an n

1

an 3.410

2 6

3 5

2.7410

2.5810

4 4

5 3

3.3810

5.2610

6 2

1.3310

7 2

8 1

2.7210

1.2510

1

9

10

11

12

13

4.8

2.17 0.98 0.392 0.173

The numerical treatment of Sommerfeld’s integrals

225

10–1

Sigma [S/m]

10–2

10–3

10–4

10–5 102

p 103

104 105 Frequency [Hz]

106

107

Figure 6.11 Ground conductivity as a function of the frequency for water content p equal to 0.2%, 1%, 2%, 10%, and 30%

105

relative permittivity

104

103

102

101 P 100 102

103

104 105 Frequency [Hz]

106

107

Figure 6.12 Ground relative permittivity as a function of the frequency for water content p equal to 0.2%, 1%, 2%, 10%, and 30%

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Lightning electromagnetics: Volume 2

where w is the angular frequency, s0 ¼ r1 (where r0 is the low frequency ground 0 resistivity), Di and a are statistical parameters, which express the frequency dependence of soil conductivity and permittivity. Weibull distributions are adopted for Di and a and, according to [57], their median values can be assumed as 11.71 S/m and 0.706, respectively. More recently, the authors of [58] proposed another expression based on a large number of field measurements:  s ¼ s0 þ s0  hðs0 Þ

er ¼ er1 þ

tan

pg

f 1MHz

g

 103

(6.75)

2 s0  hðs0 Þf g1 2pe0 ð1MHzÞg

where s0 is the low-frequency conductivity (100 Hz) in mS/m, er1 is the relative permittivity at higher frequencies and f is measured in Hz. The parameters hðs0 Þ, g and er1 are: hðs0 Þ ¼ 1:26  s0 0:73 g ¼ 0:54

(6.76)

er1 ¼ 12

6.3.2

Numerical simulation of over-ground and underground lightning electromagnetic field

Let us reconsider (6.18) and (6.62) that express respectively the over-ground and underground lightning electromagnetic fields: since they are carried out in the frequency domain, it is apparent that, if one is interested to consider the effect of the frequency dependent ground parameters on the w  w fields, it is sufficient to insert in and s 2p defined in (6.72) and (6.73) those expressions the functions er 2p respectively, thus obtaining: 8 ð 3 0 > jIð0; wÞ þ1 mE mz l > > QðlÞdl; ¼ E   J ð lr Þ  e  E zL ziL 0 > > m 2pwe0 0 n2 m þ mE > > > > > ð < jIð0; wÞ þ1 2 m l J1 ðlrÞemz 2 E QðlÞdl; ErL ¼ EriL  > n m þ mE 2pwe 0 0 > > > > > ð > 2 > Ið0; wÞ þ1 mE > mz l > QðlÞdl:  J ð lr Þ  e  : HjL ¼ HjiL  1 m 2p 0 n2 m þ mE

(6.77)

The numerical treatment of Sommerfeld’s integrals

227

for the over-ground lightning electromagnetic field and: 8 ð jIð0; wÞ þ1 l3 > > ¼  J0 ðlrÞ  QðlÞemE z dl; E > zL > 2pwe0 0 n2 m þ mE > > > > > ð < jIð0; wÞ þ1 l2 m J1 ðlrÞQðlÞemE z 2 E dl; ErL ¼  > n m þ mE 2pwe0 0 > > > > ð > > > n2 Ið0; wÞ þ1 l2 > : HjL ¼   J1 ðlrÞ  QðlÞemE z dl: 2 n m þ m 2p 0 E

(6.78)

for the underground one. Equations (6.77) and (6.78) are formally identical to (6.19) and (6.63), since the dependence of the soil parameters on the frequency is hidden in the definition of the quantities n2 and mE. In [59], a detailed analysis on the effects of taking into account the frequency dependent behavior of the soil characteristics in the evaluation of the electromagnetic fields is presented. Here, we simply limit our discussion to summarize in Table 6.2 the main results.

Table 6.2 Summary of the results and recommended models for the calculation of electromagnetic fields generated by lightning return-strokes Field component

Above-ground

Underground

Er

For low percent water volume (p  1%), it is necessary to include the frequency dependence of ground electrical parameters

Ez

The Sommerfeld term is negligible, no matter the model adopted for s and e

Hf

The Sommerfeld term is negligible, no matter the model adopted for s and e

Significant differences between the model with constant soil parameters and the one with s and e variable with w only for very low (p  1%) and very high percent (p  30%) water volume Significant differences between the model with constant soil parameters and the one with s and e variable with w for very low (p  1%) and very high percent (p  30%) water volume Significant differences between the model with constant soil parameters and the one with s and e variable with w only for very high percent (p  30%) water volume

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6.4 Lightning electromagnetic field calculation in presence of a lossy and horizontally stratified ground In the previous sections, the ground parameters have been considered either constant or frequency dependent; in both cases, they have been treated as homogeneous. The aim of the present section is to investigate the effect of the soil stratification on the evaluation of the lightning electromagnetic fields. One of the first studies on the propagation of electromagnetic waves along a stratified medium is due to Wait, who showed that the concepts of ground surface impedance and attenuation function can be used to represent the effect of a multilayered soil [60]. For the case of the lightning radiation over a stratified conducting ground, a recent literature review can be found in [61]. Here, the formulation proposed by Wait in [60] for a dipole is extended to account for the presence of the lightning channel and the derivation of the final formulas is presented in details starting from the field problem, consisting of one Helmholtz equation for each layer and the suitable boundary conditions [5,42].

6.4.1

Statement of the problem and derivation of the Green’s functions for the electromagnetic field

Here, we are interested in determining the field radiated by a vertical dipole located at a height z0 over a lossy stratified ground. The geometry of the problem is shown in Figure 6.13. The upper half-space is air, which is assumed to be lossless and characterized by a magnetic permeability m0 and an electric permittivity e0. The lossy ground (lower half space) has two layers, with conductivities s1 and s2 and relative electric permittivities er1 and er2 The depth of the first layer is h1. The observation point is P(r,f,z). The following set of equations applies to this problem (in the frequency domain and assuming, as before, that the vector potential in all media has only vertical component, due to the symmetry of the problem) [5]: !

!

!

DA þ k2 A ¼ m0 dðP  P0 Þe 3

z>0

(6.79)

Air ε0,μ0,σ=0 H dz' z' h1

i(z',t) R P(r,ϕ,z) Layer 1

ε1,μ 0,σ 1

Layer 2

ε2,μ0,σ 2

Figure 6.13 Model geometry: the lightning radiation over a multilayered ground

The numerical treatment of Sommerfeld’s integrals !

!

!

!

DA E1 þ kE1 2 A E1 ¼ 0 DA E2 þ kE2 2 A E2 ¼ 0

 h1 < z < 0

(6.80)

z < h1

(6.81)

  pffiffi @A r  jkA ¼ 0 r!1 @r

(6.82)

lim

A ¼ AE1

229

z¼0

(6.83)

1 @A 1 @AE z¼0 ¼ k 2 @z kE1 2 @z

(6.84)

AE1 ¼ AE2

(6.85)

z ¼ h1

1 @AE1 1 @AE2 ¼ 2 @z kE1 kE2 2 @z

z ¼ h1

(6.86)

!  ! ! where A is the vector potential in the upper half-space (air), A E1 A E2 is the vector potential in the first (second) ground layer, k 2 ¼ w2 m0 e0 is the wave number in air 2 (w being the angular frequency), kE1ð2Þ ¼ w2 e0 er1ð2Þ m0 þ jwm0 s1ð2Þ is the wave number in the first (second) ground layer and d is the Dirac distribution. Equations (6.79)–( 6.81) are Helmholtz equations that hold respectively in air and in the two earth layers, while (6.83)–(6.86) are the interface conditions, which must be satisfied in order to ensure the continuity of the tangential magnetic and electric fields at ground level and at the transition between the first and the second layers. Finally, as shown by Sommerfeld [5], with the addition of the well-known radiation condition (6.82), the problem has a unique solution. It should be observed that the set of (6.79)–(6.86) has been derived assuming, as in the previous sections, for all variables a time-harmonic dependence of the kind ejwt . In order to solve these equations [5], one can first solve (82) and then modify the obtained solution in order to meet the interface conditions. Sommerfeld himself qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m0 ejkR showed that Ap ¼ 4p with R ¼ r2 þ ðz  z0 Þ2 is the solution of (6.79) and R (6.82). Moreover, he proved that ð m0 þ1 l mjzz0 jJ0 ðlrÞdl Ap ¼ e (6.87) 4p 0 m where m2 ¼ l2  k2 and J0 is the Bessel function of the first kind and zeroth order. Thus, it is sufficient to find a solution of the homogeneous Helmholtz equation, which, added to Ap, meets the interface conditions to find the final solution of the problem. It can be easily proven that the function u defined as: uðr; zÞ ¼ emz J0 ðlrÞ

(6.88)

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Lightning electromagnetics: Volume 2

is a solution of the problem u þ h2 u ¼ 0 in cylindrical coordinates for any l and with l2 = m2 + h2 [5]. As a consequence, also the function ð m0 þ1 0 b0 ðlÞJ0 ðlrÞemðzþz Þ dl 4p 0 is a solution of the homogeneous Helmholtz problem in air for any function b0. This means that, in order to meet the interface conditions, one can look for a solution of the kind: ð  m0 þ1 mðzþz0 Þ b0 ðlÞJ0 ðlrÞe dl (6.89) A ¼ Ap þ 4p 0 ð  m0 þ1 mz0 mE1 z mE1 z AE1 ¼ e J0 ðlrÞða1 ðlÞe þ b1 ðlÞe Þdl (6.90) 4p 0 and AE2

m ¼ 0 4p

ð þ1 e

mz0

J0 ðlrÞb2 ðlÞe

mE2 z

 dl

(6.91)

0

where the functions b0, a1, b1 and b2 can be determined imposing the four interface 2 conditions and m2Ei ¼ l2  kEi ; i ¼ 1; 2. It should be noticed that the coefficient of the term emE2 z is set to zero in order to ensure the convergence of the integral when z ! 1. Imposing the interface conditions, one obtains that: b0 ðlÞ ¼ RðlÞ

l m

(6.92)

R being the reflection coefficient [60], defined as: R ðl Þ ¼

w0 ðlÞ  z1 ðlÞ w0 ðlÞ þ z1 ðlÞ

(6.93)

where w0 ðlÞ ¼

mE1 kE1 2

and z1 is the surface impedance given by     w2 ðlÞ emE1 h1 þ emE1 h1 þ w1 ðlÞ emE1 h1  emE1 h1 z1 ðlÞ ¼ w1 ðlÞ w1 ðlÞðemE1 h1 þ emE1 h1 Þ þ w2 ðlÞðemE1 h1  emE1 h1 Þ

(6.94)

(6.95)

where the quantities w1 and w2 are defined as w1 ðlÞ ¼

mE1 kE1 2

(6.96)

The numerical treatment of Sommerfeld’s integrals

231

and w2 ðlÞ ¼

mE2 kE2 2

(6.97)

A brief discussion on the physical interpretation of the reflection coefficient is in order: (6.87) states that, for each frequency, the vector potential Ap due to the vertical dipole in the free space is a superposition of waves (belonging to a continuous spectrum) of the kind defined in (88), each characterized by a different value of l and whose amplitude is l/m. The presence of the soil generates, for each value of l, a reflected wave, whose amplitude is related to the corresponding incident wave by the coefficient R. Once the vector potential spatial distribution has been obtained, the expression of the three non-zero components of the over-ground field can be easily derived according to the following relationships:   jw @ 2 A 2 þ k A ; Ez ¼ 2 k @z2   jw @ 2 A (6.98) ; Er ¼ 2 k @z@r Hj ¼ 

1 @A : m0 @r

Substituting (6.89) and (6.92) into (6.98), one obtains the expression of the fields which read: 2 3 2 ðz  z 0 Þ2  r 2 6 7   cR4 6 7 R 6 7 ð þ1 3 0 2 2 7 jw 0 1 6 j 1  RðlÞ 2 ð z  z Þ  r mðzþz Þ l c 6 7 Ez ¼ dl; J ð lr Þe  e þ 0 7 m 4pwe0 6 2pwe0 0 2 jwR5 6 7 6 7 4 5 r2 þjw 2 3 c R 2 3rðz  z0 Þ 3 6 cR4 7   R 6 7 ð þ1 6 jw 0 1  R ðlÞ 1 6 3rðz  z0 Þ 7 7 c  j dl; l2 J1 ðlrÞemðzþz Þ Er ¼ e 6þ 7 5 4pe0 6 jwR 7 2pwe0 0 2 6 7 4 5 rðz  z0 Þ jw 2 3 c R   0   jw R þ jz j ð 2 0 1 r r 1 þ1 1  RðlÞ mðzþz Þ l v c e Hj ¼ dl: J  jw ð lr Þe  1 m 4p R3 2p 0 2 cR2 (6.99)

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Lightning electromagnetics: Volume 2

The first terms in the above expressions of the fields are as in the previous sections called “ideal fields,” since they would be the only nonzero terms if the ground was a perfect conductor. In a similar way, one can easily determine the underground fields first by obtaining the following expressions of the coefficients a1, b1, and b2 [which can be done imposing again the interface conditions (6.83)–(6.86)],     8 l l > > þ b0 ðlÞ  w0  b0 ðlÞ w1 > > m m > > ð l Þ ¼ ; a > 1 > > >  2w2   < l l (6.100) þ b0 ðlÞ þ w0  b0 ðlÞ w1 > m m > > b1 ðlÞ ¼ ; > > 2w2 > > > > b ð l Þw 1 1 > : b2 ðlÞ ¼ eh1 ðmE2 mE1 Þ ; w1 þ w2 then inserting them in (6.90) and (6.91) and finally recalling the relationships between the vector potential and the fields that state: ! 8 2 @ A > jw E1ð2Þ > > Ez1ð2Þ ¼ þ kE1ð2Þ 2 AE1ð2Þ ; > > @z2 > kE1ð2Þ 2 > > ! < @ 2 AE1ð2Þ jw (6.101) ; Er1ð2Þ ¼ > > @z@r kE1ð2Þ 2 > > > > > 1 @AE1ð2Þ > : Hj1ð2Þ ¼  : m0 @r

6.4.2

Derivation of the lightning electromagnetic field

The Green’s functions (6.99) are now utilized to obtain the expressions for the electromagnetic field due to a lightning return-stroke. In order to reach this goal, it is sufficient to do as in Section 6.2.1.3. Here, as an example, we report the expression for the over-ground field, that is to say: EzL ¼ EziL  ð 0 l3 jIð0; wÞ þ1 1  RðlÞ J0 ðlrÞemz QðlÞdl; 2pwe0 0 2 m ErL ¼ EriL  ð jIð0; wÞ þ1 2 1  R ðl Þ QðlÞdl; l J1 ðlrÞemz 2pwe0 0 2 HjL ¼ HjiL  ð Ið0; wÞ þ1 1  RðlÞ l2 J1 ðlrÞemz QðlÞdl: m 2p 0 2

(6.102)

The numerical treatment of Sommerfeld’s integrals

233

where the expressions of the ideal fields EziL, EriL, and HfiL are those defined in (6.21)–(6.23).

6.4.3 The reflection coefficient R We will discuss now the main mathematical properties of the reflection coefficient R, defined in (6.93). First of all, it can be shown that the absolute values of both R and (1R)/2 are decreasing as l increases for l >> k. This observation will be useful in the numerical treatment of the integrals in order to find an upper bound for the integral tail. Second, one can easily verify that 1  R ðl Þ m ¼ 2 E ; h1 !1 n m þ mE 2

(6.103)

lim

which confirms the fact that (6.102) reduces to the expression of the field in the case of a one-layer soil. Then, in order to evaluate the static limit of the Sommerfeld integrals (i.e., the DC fields), it is necessary to analyze the behavior of the function (1R)/2 when w approaches zero. Since it is apparent that, for small values of w jsi l ; w0 ! 2 we0 k l we0 mEi 2 ! l2 ; wi ! 2 k jsi m2 ! l2 ;

ni 2 !

(6.104)

it readily follows that lh

lh

lh

lh

1 e 1 e 1 þ e 1 þe 1  RðlÞ we0 s1 s2 ¼ lh1 lh lh1 ; w!0 js1 elh1 þe 2 þ e 1 e s1 s2

lim

(6.105)

which states that the limit is zero and the function (1R)/2 is linear in the neighborhood of w = 0. After some algebraic manipulations, (6.105) can be re-written as   1  R ðl Þ we0 ¼ f ðlÞ; (6.106) lim k!0 2 js1 in which f ðl Þ ¼

ðs2 þ s1 Þelh1 þ ðs1  s2 Þelh1 ðs2 þ s1 Þelh1  ðs1  s2 Þelh1

(6.107)

It can be observed that the function f is decreasing with l if the conductivity of the first layer is greater than the conductivity of the second one; in the other case the function is increasing with l but always remaining smaller than one. This means

234

Lightning electromagnetics: Volume 2

that in any case an upper bound for the function f is given by jf ðlÞj  maxf1; f ðMÞg

lM

(6.108)

Such bound will be useful when evaluating the upper bound for the static term. Figures 6.14–6.18 illustrate some of the most important physical properties of the reflection coefficient R, obtained considering for the first layer s1 = 0.002 S/m and er1 = 5 and for the second s2 = 0.1 S/m and er2 = 80. In Figures 6.14 and 6.15,

0.6

0.5 h1=1 [m] h1=10 [m] h1=100 [m]

ratio

0.4

0.3

0.2

0.1

0

2

4

6

8

10

Iamda/k

Figure 6.14 Ratio between the function (1R)/2 and the function gs for f = 1 kHz

1.8 h1=1 [m] h1=10 [m] h1=100 [m]

1.6 1.4

ratio

1.2 1 0.8 0.6 0.4 0.2 0

2

4

6

8

10

Iamda/k

Figure 6.15 Ratio between the function (1R)/2 and the function gs for f = 1 MHz

The numerical treatment of Sommerfeld’s integrals

235

the ratios between the function (1R)/2 and the equivalent function for the case of homogenous ground gs [62] are plotted for two different values of the frequency, namely 1 kHz and 1 MHz. As can be seen from the figures, if the frequency increases from 1 kHz to 1 MHz, it is sufficient to have a smaller depth of the first layer to make the ratio become closer to one and therefore to make the corresponding fields become closer one to each other. In other words, the more high frequencies are “present” in the spectrum of the lightning channel current, the closer are the corresponding fields to the ones which would be present in the case of a homogeneous ground. Figure 6.16 shows the absolute value of the reflection coefficient for four different values of the frequency and in the case of a first layer depth of 1 m. As can be seen, at higher frequencies the reflected wave is larger. The same observation applies to the case of larger first layer depth, namely 10 m (Figure 6.17) and 100 m

102

R

100 10–2 10–4 Frequency 10–6

0

1

2

3

4

5 6 Iamda/k

7

8

9

10

Figure 6.16 Reflection coefficient for first layer depth of 1 m

102

R

100 10–2 10–4 Frequency 10–6

0

1

2

3

4

5 6 Iamda/k

7

8

9

10

Figure 6.17 Reflection coefficient for first layer depth of 10 m

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Lightning electromagnetics: Volume 2 102

R

100

10–2 10–4 Frequency 10–6

0

1

2

3

4

5 6 Iamda/k

7

8

9

10

Figure 6.18 Reflection coefficient for first layer depth of 100 m

(Figure 6.18), with the only difference that, for each frequency, the corresponding values are higher. Extrapolating this result, we can conclude that the maximum magnitude of the reflected wave is obtained for a homogeneous single-layer ground.

6.5 Conclusions The electromagnetic field due to a lightning event in presence of a lossy ground has been studied considering three main situations: 1. 2. 3.

homogeneous lossy ground with constant conductivity and permittivity; homogeneous lossy ground with frequency-dependent soil electrical parameters; stratified lossy ground.

In all the cases, the exact field expressions have been derived starting from the Maxwell equations both in air and in the ground. Particular attention has been devoted to the numerical treatment of the Sommerfeld integrals involved in all the field expressions and a suitable algorithm has been proposed in order to evaluate them fastly and efficiently.

References [1] C. R. Paul, Analysis of Multiconductor Transmission Lines. New York, NY: John Wiley & Sons, 2007. [2] F. M. Tesche, M. Ianoz, and T. Karlsson, EMC Analysis Methods and Computational Models. New York, NY: John Wiley & Sons, 1996.

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[3] F. Rachidi, C. A. Nucci, M. Ianoz, and C. Mazzetti, “Influence of a lossy ground on lightning-induced voltages on overhead lines,” IEEE Transactions on Electromagnetic Compatibility, vol. 38, no. 3, pp. 250–264, 1996, doi: 10.1109/15.536054. ¨ ber die Ausbreitung der Wellen in der drahtlosen [4] A. Sommerfeld, “U Telegraphie,” Annalen der Physik, vol. 386, no. 25, pp. 1135–1153, 1926. [5] A. Sommerfeld, Partial Differential Equations in Physics. London: Academic Press, 1949. [6] K. A. Norton, “The propagation of radio waves over the surface of the earth and in the upper atmosphere,” Proceedings of the Institute of Radio Engineers, vol. 25, no. 9, pp. 1203–1236, 1937, doi: 10.1109/ JRPROC.1937.228544. [7] I. M. Longman, “Note on a method for computing infinite integrals of oscillatory functions,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 52, no. 4, pp. 764–768, Oct. 1956, doi: 10.1017/ S030500410003187X. [8] V. V. Novikov and G. I. Makarov, “Propagation of pulsed signals over a homogeneous flat earth,” Radio Eng. Electron. Phys, vol. 5, pp. 644–652, 1961. [9] J. R. Wait, “Electromagnetic surface waves,” Advances in Radio Research, vol. 1, pp. 157–217, 1964. [10] A. Ban˜os, Dipole Radiation in the Presence of a Conducting Halfspace, vol. 9. New York, NY: Pergamon, 1966. [11] D. Bubenik, “A practical method for the numerical evaluation of Sommerfeld integrals,” IEEE Transactions on Antennas and Propagation, vol. 25, no. 6, pp. 904–906, 1977. [12] R. Mittra, P. Parhami, and Y. Rahmat-Samii, “Solving the current element problem over lossy half-space without Sommerfeld integrals,” IEEE Transactions on Antennas and Propagation, vol. 27, no. 6, pp. 778–782, 1979, doi: 10.1109/TAP.1979.1142177. [13] P. Parhami and R. Mittra, “Wire antennas over a lossy half-space,” IEEE Transactions on Antennas and Propagation, vol. 28, no. 3, pp. 397–403, 1980, doi: 10.1109/TAP.1980.1142339. [14] P. Parhami, Y. Rahmat-Samii, and R. Mittra, “An efficient approach for evaluating Sommerfeld integrals encountered in the problem of a current element radiating over lossy ground,” IEEE Transactions on Antennas and Propagation, vol. 28, no. 1, pp. 100–104, 1980, doi: 10.1109/ TAP.1980.1142273. [15] J. Maclean, Radiowave Propagation Over Ground Software. New York, NY: Springer Science & Business Media, 2013. [16] T. J. Cui and W. C. Chew, “Modeling of arbitrary wire antennas above ground,” IEEE Transactions on Geoscience and Remote Sensing, vol. 38, no. 1, pp. 357–365, 2000.

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Lightning electromagnetics: Volume 2 T. J. Cui and W. C. Chew, “Accurate model of arbitrary wire antennas in free space, above or inside ground,” IEEE Transactions on Antennas and Propagation, vol. 48, no. 4, pp. 482–493, 2000. T. Sarkar, “Analysis of radiation by arrays of parallel vertical wire antennas over imperfect ground,” IEEE Transactions on Antennas and Propagation, vol. 23, no. 5, pp. 749–749, 1975, doi: 10.1109/TAP.1975.1141153. P. R. Bannister, New Formulas that Extend Norton’s Farfield Elementary Dipole Equations to the Quasi-Nearfield Range, Naval Underwater Systems Center New London CT, 1984. K. A. Michalski and J. R. Mosig, “Multilayered media Green’s functions in integral equation formulations,” IEEE Transactions on Antennas and Propagation, vol. 45, no. 3, pp. 508–519, 1997, doi: 10.1109/8.558666. K. A. Michalski, “Extrapolation methods for Sommerfeld integral tails,” IEEE Transactions on Antennas and Propagation, vol. 46, no. 10, pp. 1405– 1418, 1998, doi: 10.1109/8.725271. C. E. Baumann and E. E. Sampaio, “Electric field of a horizontal antenna above a homogeneous half-space: implications for GPR,” Geophysics, vol. 65, no. 3, pp. 823–835, 2000. J. R. Wait, “Influence of finite ground conductivity on the fields of a vertical traveling wave of current,” IEEE Transactions on Electromagnetic Compatibility, vol. 41, no. 1, p. 78, 1999, doi: 10.1109/15.748141. G. J. Burke, A. J. Poggio, J. C. Logan, and J. W. Rockway, “Numerical electromagnetic code (NEC),” in 1979 IEEE International Symposium on Electromagnetic Compatibility, 1979, pp. 1–3. R. L. Gardner, “Effect of the propagation path on lightning-induced transient fields,” Radio Science, vol. 16, no. 03, pp. 377–384, 1981. J. R. Wait and D. A. Hill, “Ground wave of an idealized lightning return stroke,” IEEE Transactions on Antennas and Propagation, vol. 48, no. 9, pp. 1349–1353, 2000, doi: 10.1109/8.898767. R. K. Pokharel, M. Ishii, and Y. Baba, “Numerical electromagnetic analysis of lightning-induced voltage over ground of finite conductivity,” IEEE Transactions on Electromagnetic Compatibility, vol. 45, no. 4, pp. 651–656, 2003, doi: 10.1109/TEMC.2003.819065. H. K. Hoidalen, J. Sletbak, and T. Henriksen, “Ground effects on induced voltages from nearby lightning,” IEEE Transactions on Electromagnetic Compatibility, vol. 39, no. 4, pp. 269–278, 1997, doi: 10.1109/15.649810. C. Yang and Bihua Zhou, “Calculation methods of electromagnetic fields very close to lightning,” IEEE Transactions on Electromagnetic Compatibility, vol. 46, no. 1, pp. 133–141, 2004, doi: 10.1109/ TEMC.2004.823626. A. Shoory, R. Moini, S. H. H. Sadeghi, and V. A. Rakov, “Analysis of lightning-radiated electromagnetic fields in the vicinity of lossy ground,” IEEE Transactions on Electromagnetic Compatibility, vol. 47, no. 1, pp. 131–145, 2005, doi: 10.1109/TEMC.2004.842104.

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[31] Y. Baba and V. A. Rakov, “Voltages induced on an overhead wire by lightning strikes to a nearby tall grounded object,” IEEE Transactions on Electromagnetic Compatibility, vol. 48, no. 1, pp. 212–224, 2006. [32] V. Cooray, “Horizontal fields generated by return strokes,” Radio Science, vol. 27, no. 04, pp. 529–537, 1992. [33] M. Rubinstein, “An approximate formula for the calculation of the horizontal electric field from lightning at close, intermediate, and long range,” IEEE Transactions on Electromagnetic Compatibility, vol. 38, no. 3, pp. 531–535, 1996, doi: 10.1109/15.536087. [34] V. Cooray, “Some considerations on the ‘Cooray-Rubinstein’ formulation used in deriving the horizontal electric field of lightning return strokes over finitely conducting ground,” IEEE Transactions on Electromagnetic Compatibility, vol. 44, no. 4, pp. 560–566, 2002. [35] A. Borghetti, J. A. Gutierrez, C. A. Nucci, M. Paolone, E. Petrache, and F. Rachidi, “Lightning-induced voltages on complex distribution systems: models, advanced software tools and experimental validation,” Journal of Electrostatics, vol. 60, no. 2–4, pp. 163–174, 2004. [36] A. Zeddam and P. Degauque, “Current and voltage induced on telecommunication cables by a lightning stroke,” Electromagnetics, vol. 7, nos. 3–4, pp. 541–564, 1987, doi: 10.1080/02726348708908197. [37] V. Cooray, “On the accuracy of several approximate theories used in quantifying the propagation effects on lightning generated electromagnetic fields,” IEEE Transactions on Antennas and Propagation, vol. 56, no. 7, pp. 1960–1967, 2005. [38] M. Darveniza, “A practical extension of Rusck’s formula for maximum lightning-induced voltages that accounts for ground resistivity,” IEEE Transactions on Power Delivery, vol. 22, no. 1, pp. 605–612, 2006. [39] C. A. Balanis, Advanced Engineering Electromagnetics. New York, NY: John Wiley & Sons, 2012. [40] E. Zauderer, Partial Differential Equations of Applied Mathematics. New York, NY: John Wiley & Sons, 2011. [41] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. London: Academic press, 2014. [42] F. Delfino, R. Procopio, M. Rossi, and L. Verolino, “Lightning current identification over a conducting ground plane,” Radio Science, vol. 38, no. 3, pp. 15–1, 2003. [43] V. A. Rakov and M. A. Uman, “Review and evaluation of lightning return stroke models including some aspects of their application,” IEEE Transactions on Electromagnetic Compatibility, vol. 40, no. 4, pp. 403–426, 1998, doi: 10.1109/15.736202. [44] C. A. Nucci and F. Rachidi, “Experimental validation of a modification to the transmission line model for LEMP calculation,” in 8th Symposium and Technical Exhibition on Electromagnetic Compatibility, 1989, no. CONF. [45] W. Rudin, Real and Complex Analysis. New York, NY:Tata McGraw-Hill Education, 2006.

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Lightning electromagnetics: Volume 2 W. L. Anderson, “Numerical integration of related Hankel transforms of orders 0 and 1 by adaptive digital filtering,” Geophysics, vol. 44, no. 7, pp. 1287–1305, 1979. P. J. Davis and P. Rabinowitz, Methods of Numerical Integration. Chelmsford:Courier Corporation, 2007. W. T. Vetterling and W. H. Press, Numerical Recipes in FORTRAN: The Art of Scientific Computing. Cambridge: Cambridge University Press, 1992. A. R. Krommer and C. W. Ueberhuber, Numerical Integration: On Advanced Computer Systems, vol. 848. New York, NY: Springer Science & Business Media, 1994. R. L. Smith-Rose, “The electrical properties of soil at frequencies up to 100 megacycles per second; with a note on the resistivity of ground in the United Kingdom,” Proc. Phys. Soc., vol. 47, no. 5, p. 923, 1935, doi: 10.1088/0959-5309/47/5/318. B. D. Fuller and S. H. Ward, “Linear system description of the electrical parameters of rocks,” IEEE Transactions on Geoscience Electronics, vol. 8, no. 1, pp. 7–18, 1970. S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics. New York, NY: John Wiley & Sons, 1994. G. P. De Loor, “The dielectric properties of wet materials,” IEEE Transactions on Geoscience and Remote Sensing, no. 3, pp. 364–369, 1983. V. A. Rinaldi and F. M. Francisca, “Impedance analysis of soil dielectric dispersion (1 MHz–1 GHz),” Journal of Geotechnical and Geoenvironmental Engineering, vol. 125, no. 2, pp. 111–121, 1999, doi: 10.1061/(ASCE)1090-0241(1999)125:2(111). C. L. Longmire and K. S. Smith, A Universal Impedance for Soils, Santa Barbara, CA: Mission Research Corp, Oct. 1975. Accessed: Oct. 23, 2020. Available: https://apps.dtic.mil/sti/citations/ADA025759. Scott, J., “Electrical and magnetic properties of Impedance analysis of soil dielectric dispersion rock and soil,” U.S. Geological Survey. C. Portela, “Measurement and modeling of soil electromagnetic behavior,” in 1999 IEEE International Symposium on Electromagnetic Compatability. Symposium Record (Cat. No. 99CH36261), 1999, vol. 2, pp. 1004–1009. S. Visacro and R. Alipio, “Frequency dependence of soil parameters: experimental results, predicting formula and influence on the lightning response of grounding electrodes,” IEEE Transactions on Power Delivery, vol. 27, no. 2, pp. 927–935, 2012. F. Delfino, R. Procopio, M. Rossi, and F. Rachidi, “Influence of frequencydependent soil electrical parameters on the evaluation of lightning electromagnetic fields in air and underground,” Journal of Geophysical Research: Atmospheres, vol. 114, no. D11, 2009. J. R. Wait, Electromagnetic Waves in Stratified Media: Revised Edition Including Supplemented Material. New York, NY: Elsevier, 2013. A. Shoory, A. Mimouni, F. Rachidi, V. Cooray, R. Moini, and S. H. Sadeghi, “Validity of simplified approaches for the evaluation of lightning

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electromagnetic fields above a horizontally stratified ground,” IEEE Transactions on Electromagnetic Compatibility, vol. 52, no. 3, pp. 657–663, 2010, doi: 10.1109/TEMC.2010.2045229. [62] F. Delfino, R. Procopio, and M. Rossi, “Lightning return stroke current radiation in presence of a conducting ground: 1. Theory and numerical evaluation of the electromagnetic fields,” Journal of Geophysical Research: Atmospheres, vol. 113, no. D5, 2008.

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Chapter 7

Lightning electromagnetic field propagation: a survey on the available approximate expressions Daniele Mestriner1, Renato Procopio1, Massimo Brignone1 and Federico Delfino1

In this chapter, a review of the main approximate expressions available in literature to evaluate lightning electromagnetic fields that propagate over and under a lossy ground is presented together with their validation against exact expressions or full Maxwell approaches. First the case of homogeneous ground is analyzed presenting the two main approximate expressions for the horizontal electric field above the ground (Cooray Rubinstein formula, Section 7.1) and below the ground (Cooray’s formula, Section 7.2). Then, the most popular expressions for electromagnetic fields propagating over a horizontally (Section 7.3) and vertically (Section 7.4) stratified ground are reported and validated.

7.1 Lightning electromagnetic fields over a homogeneous soil The propagation of the lightning electromagnetic fields over and under a lossy homogeneous ground has been intensively studied in the last decades. Researchers have focused their attention mainly on the horizontal component of the electric field for two reasons: 1.

The finite ground conductivity mostly affects such component while the vertical component of the electric field and the azimuthal component of the magnetic one can be evaluated supposing perfectly conducting ground without compromising the accuracy of the final result [1]. This assumption is generally valid but some particular cases should be taken into account as presented in Section 7.1.2.

1 Department of Electrical, Electronics and Telecommunication Engineering and Naval Architecture Department (DITEN), University of Genoa, Italy

244 2.

Lightning electromagnetics: Volume 2 When dealing with the effect of lightning on power and telecommunication lines, the horizontal component of the electric field is the source term of the field-to-line coupling differential equations and so it must be evaluated in all the line points. The vertical component of the electric field, on the other hand, only appears in the boundary conditions describing the line terminations.

7.1.1

Horizontal electric field – Cooray–Rubinstein (CR) formula

The exact evaluation of such component involves the Sommerfeld integrals which appear prohibitive from a computational point of view [2], especially when the fields have to be computed in many different points of the line and, most of all, when a statistical analysis of the effect of lightning on power lines has to be carried out (e.g. lightning performance studies [3]). For this reason, approximate expressions have been developed to obtain a good compromise between accuracy and computational performance. The first contribution was provided by Cooray [4] who developed an approximate frequency domain expression for the surface impedance able to evaluate the horizontal electric field at the ground surface from the incident azimuthal magnetic field. Such formula was then improved by Rubinstein [5] adding a term that allows the calculation of the horizontal electric field at any height above ground. The resulting expression became known as the Cooray–Rubinstein (CR) formula, which reads: Er ðw; z; rÞ ¼ Hji ðw; 0; rÞZ ðwÞ þ Eri ðw; z; rÞ in which Z is the surface impedance defined as: pffiffiffiffiffi m0 Z ðwÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffi s e þ jw

(7.1)

(7.2)

and where Er ðw; z; rÞ is the horizontal component of the electric field at angular frequency w, height z and horizontal distance r from the lighting channel, while Hji ðw; 0; rÞ is the azimuthal component of the magnetic field at the same frequency and horizontal distance and at the ground level and Eri ðw; z; rÞ is the horizontal component of the electric field both assuming perfectly conducting ground. Moreover, m0, e, and s are the magnetic permittivity the dielectric constant and the conductivity of the soil respectively. It should be noted that in (7.1) neglecting propagation effects on the azimuthal magnetic field is reasonable for medium-high values of the soil conductivity (greater than 0.001 S/m); it remains a valid assumption only for distances lower than 200 m for conductivities of about 0.001 S/m and lower than 100 m for conductivities of about 0.0001 S/m [6]. However, when the CR formula has to be inserted in time domain methods, an IFFT procedure has to be adopted to evaluate the lightning horizontal field. As well known, all the FFT-IFFT-based methods have two inherent drawbacks to calculate the quantities of interest in time domain. One is the requirement of huge computational resources, and the other is the error due to the truncation of the

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lightning current frequency spectrum. So, the direct evaluation of Er in time domain has been studied in recent years. The first expression of the CR formula in time domain was given in 2006 by Caligaris et al. [7]; its derivation is here briefly summarized. Under the assumption that the return-stroke wavefront starts traveling up at time zero, we are in presence of quantities that are identically zero for t < 0. This implies that the existence of the Fourier transform =ðf Þ of a function f ensures the existence of its Laplace transform Lðf Þ [21]. So, for further manipulations, it is convenient to express (7.1) in the Laplace domain. For the sake of simplicity, we denote with the same name both Fourier and Laplace transform, due to the fact that the relation between the two is given by s ¼ jw: from here on we will assume that if the argument is s then we are working in the Laplace domain, if w in the Fourier one. Then we get: Er ðs; z; rÞ ¼ Hji ðs; 0; rÞZ ðsÞ þ Eri ðs; z; rÞ rffiffiffiffiffi Where pffiffiffiffiffi m0 m0 1 1 pffiffiffiffiffiffiffiffiffiffiffi ¼ h pffiffiffiffiffiffiffiffiffiffiffis : Z ðsÞ ¼ pffiffiffiffiffiffiffiffiffiffis ¼ e 1 þ ses eþs 1 þ se pffiffiffiffiffiffiffiffiffi Observing that lims!1 Z ðsÞ ¼ m0 =e ¼ h, it is convenient to write: pffiffiffiffiffi m0 g ðsÞ ¼ Z ðsÞ  h ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  h e þ s=s The inverse Laplace transform of g is known analytically, namely  s i  1  s s h s  L ðgÞ ðtÞ ¼ h e2et I1 t  I0 t 2e 2e 2e

(7.3)

(7.4)

(7.5)

In being the modified Bessel function of first type and order n. Moreover, denoting with d the Dirac function, one has that: fLðdÞgðsÞ ¼ 1; so, combining (7.6) and (7.5), it readily follows that: ns s h s   s i o  1  L ðZ Þ ðt Þ ¼ h e2et I1 t  I0 t þ dðtÞ : 2e 2e 2e

(7.6)

(7.7)

With the aid of the convolution integral properties, one gets the time domain CR formula for the horizontal electric field er at height z, distance r and time t: ðt (7.8) er ðt; z; rÞ ¼ h hji ðt; 0; rÞ½K ðt  tÞ þ dðt  tÞdt þ eri ðt; z; rÞ 0

being

s  s i s  th  s  t  I1 t ¼ aeat ½I0 ðatÞ  I1 ðatÞ K ðtÞ ¼ e 2e I0 2e 2e 2e s a¼ 2e

(7.9)

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and where eri is the horizontal electric field at height z, distance r and time t, calculated as if the ground was a perfect conductor and hfi the azimuthal magnetic field at height z, distance r and time t, calculated as before in the case of perfectly conducting ground. The terms eri and hfi are, respectively, the inverse Laplace transform of Eri and Hfi appearing in (7.3) and can be evaluated directly in the time domain, as presented in the previous chapter. Finally, recalling the properties of the Dirac function, one has: er ðt; z; rÞ ¼ h

ðt

hji ðt; 0; rÞK ðt  tÞdt þ eri ðt; z; rÞ  hhji ðt; 0; rÞ

(7.10)

0

The implementation of (7.10) presented in [7] is a classic example of linear convolution, which is reported here to show its weaknesses that have originated subsequent research. Let us consider np time samples and assume hji ðt; 0; rÞ as constant in each interval between two samples, that is to say, with tn ¼ Dtðn  1Þ, n ¼ 1::np , and hn ¼ hji ðtn ; 0; rÞ: hji ðt; 0; rÞ  hn

8 t 2 ½tn ; tnþ1 Þ

(7.11)

With this assumption, one can compute the integral in (7.10) at each time sample tn between t1 = 0 and tnp as: CIðtn Þ ¼ CIn ¼

ð tn

Kðt  tÞhji ðt; 0; rÞdt 

t1

¼

n1 X m¼1

Kðtn  tÞhm dt

m¼1 tm

ð tmþ1 hm

n1 ð tmþ1 X

Kðtn  tÞdt

(7.12)

tm

(for n > 1; CI1 = 0). The integral appearing in (7.12) can be rewritten as follows: ð tmþ1

Kðtn  tÞdt ¼

tm

ð ðnmÞDt ðnm1ÞDt

D

Kðt0 Þdt0 ¼ Knm

(7.13)

So (7.12) becomes 8 CI ¼ 0 > < 1 n1 X > ¼ hm Knm ; n ¼ 2::np ; CI : n

(7.14)

m¼1

The integral in (7.13) was calculated using a Gaussian quadrature method, opening two problems: if one has to evaluate the late time response of the horizontal fields, with typical values of s and e, the values assumed by the argument at of the modified Bessel functions can generate problems of overflow. This can be

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explained simply by recalling the asymptotic behavior of the modified Bessel function, namely: 1 ex X ð1Þk Gðn þ k þ 1=2Þ þ f ðx Þ (7.15) In ðxÞ  pffiffiffiffiffiffiffiffi 2px k¼0 ð2xÞk k!Gðn  k þ 1=2Þ with f a suitable function such that limx!þ1 f ðxÞ ¼ 0 and where G is the incomplete Gamma function. It is apparent that the term ex is the responsible for the overflow. The problem can be circumvented observing that in (7.5) the Bessel functions In(x) are multiplied by the term ex. So, if one sets up a routine which directly computes In(x)ex, he can avoid to calculate the exponential ex. At time sample tn, one has to perform one more integral (7.13) and n1 products (and n1 sums); in other words, the number of computations to be performed at each iteration grows linearly with the number of iterations, i.e. the algorithm has a time complexity order n2. For these reasons, the authors of [8] proposed the following improvement. The surface impedance can be expressed as: rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sm0 m0 s stG (7.16) ¼h ¼ ZðsÞ ¼ se þ s e s þ s=e stG þ 1 with tG ¼ e=s. If we can find a rational approximation (RA) for the function: rffiffiffiffiffiffiffiffiffiffiffiffi NRA X s0 rk 1þ zðs0 Þ ¼ 0 0a s þ1 s k k¼1

(7.17)

where NRA is the number of poles used in the expansion, (7.3) can be approximated as: ECR ðs; z; rÞ ¼ Hji ð0; r; sÞZ ðsÞ ¼ hHji ðs; 0; rÞ  h

NRA X rk Hji ðs; 0; rÞ k¼1

stG  ak

(7.18) The RA in (7.17) can be effectively computed by means of the vector fitting (VF) technique [9]. It should be noticed that (7.17) and thus the poles ak and residues rk do not depend on tG , i.e., do not depend on the values of ground permittivity and conductivity. Therefore, they have to be computed only once. The computations illustrated as numerical examples in the last section were carried out using an expansion with NRA=12, obtained by fitting the values of z sampled on 105 “frequencies” w0 [i.e., replacing s0 with jw0 in (7.17)], ranging from 10-4 to 100. The fitting accuracy can be appreciated in Figures 7.1 and 7.2. All poles are real. Table 7.1 shows the obtained values for poles and residues. In order to re-state the problem in time domain, we rewrite (7.18) in the classical state-space equation form. The method adopted here presents the advantage of directly making use of the poles ak and residues rk from the vector fitting. Furthermore, it allows a straightforward transformation from continuous time to discrete time, and it is very simple to code (Table 7.1).

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Lightning electromagnetics: Volume 2

[z]

1

0.5

0

0

10

20

30

40

50 ω'

60

70

80

90

100

0

10

20

30

40

50 ω'

60

70

80

90

100

N =12 exact

0.4

[z]

0.3 0.2 0.1 0

Figure 7.1 Rational approximation of z. Adapted from [8]

0.8 [z]

0.6 0.4 0.2 0 0

0.1

0.2

0.3

0.4

0.5 ω'

0.6

0.7

0.8

0.9

1

0.1

0.2

0.3

0.4

0.5 ω'

0.6

0.7

0.8

0.9

1

N =12 exact

0.4

[z]

0.3 0.2 0.1 0 0

Figure 7.2 Rational approximation of z, zoom. Adapted from [8] The aforementioned classical state space form is: 8 Hji ðs; 0; rÞ > > ; k ¼ 1::NRA X ðsÞ ¼ > > < k stG  ak NRA X > > > > : ECR ðs; z; rÞ ¼ hHji ðs; 0; rÞ  h rk Xk ðsÞ k¼1

(7.19)

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Table 7.1 Poles and residues ak

rk

9.5767718047658079E01 6.9826012409778959E01 4.1509910805114025E01 2.2286140573279206E01 1.1343452528471004E01 5.5577444158951196E02 2.6246991903765934E02 1.1888088239587109E02 5.1013756832941235E03 2.0025909738681470E03 6.2900517731833501E04 1.0993593611408876E04

2.4807255056699237E01 1.4853965557307094E01 6.4924454704342230E02 2.4809658479110788E02 8.9808401793780475E03 3.1389835104789613E03 1.0559668692280658E03 3.3870559700515702E04 1.0243347132959180E04 2.8760866085666423E05 6.9477373464759209E06 7.8176216694433083E07

or: 8 ak 1 > sX ðsÞ ¼ Xk ðsÞ þ Hji ðs; 0; rÞ; > > < k t t G

G

k ¼ 1::NRA

NRA X > > > E ðs; z; rÞ ¼ hH ð s; 0; r Þ  h rk Xk ðsÞ : CR ji

(7.20)

k¼1

where Xk ðsÞ are the NRA state variables of the system. System (7.20) can be reformulated in time domain as follows: 8 ak 1 > > > x_ k ðtÞ ¼ t xk ðtÞ þ t hji ðs; 0; rÞ; k ¼ 1::NRA < G

G

NRA X > > > : eCR ðt; z; rÞ ¼ heji ðt; 0; rÞ  h rk xk ðtÞ

(7.21)

k¼1

(we assume xk ð0Þ ¼ 0; k ¼ 1::NRA , as initial conditions). Now this continuous time system has to be approximated with a discrete time system, in order to implement it in a computer code. This discretization is achieved by considering the magnetic field as constant at each time interval: hji ðt; 0; rÞ  hn 8 t 2 ½tn ; tnþ1 Þ

(7.22)

where tn are the time samples tn=Dt(n1), n=1..np. Now, solving the first NRA of (7.21) between tn and tn+1: 8 < x_ ðtÞ ¼ ak xk ðtÞ þ 1 hn k tG tG ; k ¼ 1::NRA ; t 2 ½tn ; tnþ1  (7.23) : xk ðtn Þ ¼ xk;n

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Lightning electromagnetics: Volume 2

we obtain: xk ðtÞ ¼ xk;n eðak =tG Þðttn Þ þ

 1  ðak =tG Þðttn Þ e  1 hn ak

(7.24)

The discrete counterpart of (7.21) can be found simply by evaluating (7.24) and the last of (7.21) itself at t = tn+1: 8  1  ðak =tG ÞDt > ðak =tG ÞDt > ¼ x e þ e  1 hn ; x > k;nþ1 k;n > < ak NRA X > > > e ¼ hh  h rk xk;nþ1 > nþ1 : CR;nþ1

k ¼ 1::NRA (7.25)

k¼1

Note that the coefficients eðak =tG ÞDt and ðeðak =tG ÞDt  1Þ=ak can be evaluated only once and hold for all the iterations. Differently from the previous approach, the algorithm corresponding to system (7.25) requires about 3NRA products (and 2NRA sums), with NRA of the order of ten; the number of computations at each iteration is constant, so the algorithm has a time complexity order n. As observed in [10], the possible issue of this approach is that the accuracy of the horizontal electric field is dependent on the performance of the vector-fitting technique. Meanwhile, a frequency truncation is necessary when using the vector-fitting technique, which could be an uncertain factor to influence the final result. So, from 2011 on, researchers have mainly focused their attention on 2 main issues: improve the computational efficiency of the convolution integral (7.10) and analyze properties of the integral kernel (7.5) in order to find suitable approximations that do not involve special functions. Concerning the first research line, the first contribution was given by Zou et al. in 2012 [10]. As summarized in [11], the method proposed in [10] proposed a piecewise quadratic convolution and quadrature routine of a truncated integral based on K to evaluate the convolution term in (7.10). More in details, given a function f : D ! h þ r; 4pe0 jz
hj3

Ez
¼ Ez


(11.3)

where e0 is the electric constant/dielectric constant of free space. Since the ground is a conductor, the induced charges of the opposite sign will appear on the ground surface. The electric field of these induced charges in the atmosphere coincides with the field of the mirror electric image of the ball located at depth z ¼
h (Figure 11.5). The left side of Figure 11.5 shows a model of electric charges in a cloud and their electrical images in the ground. The net electric field taken along

432

Lightning electromagnetics: Volume 2 90 z 80 70

2

60 z, km

3 50

4

40 1 30 20

E z=h

10

E

0 –3

–2

z = –h

–1

1 0 log ‌ Ez ‌ , kV/m

2

3

4

Figure 11.5 Model calculations of thunderstorm QE field preceding sprite discharge. Absolute value of the vertical electric field is shown with line 1, which corresponds to the thunderstorm charge q ¼
150 C. Altitude dependences of the breakdown electric field that correspond to different air breakdown mechanisms are shown with dotted lines 2
4 the z-axis is given by Ez ¼ Ez
þ Ezþ , where Ez
is defined by (11.3), and the field of induced charges has the form: Ezþ ¼


q 4pe0 ðz þ hÞ2

;

ðz > 0Þ:

(11.4)

where q < 0 is the charge of thundercloud. Note that at high altitudes under requirement z  h the (11.3) and (11.4) are simplified. In the first approximation, for the small parameter h=z we obtain: Ez ¼ Ez
þ Ezþ 

qh : pe0 z3

(11.5)

This dependence on height is not surprising, since the cloud charges and their electric image in the ground form an electric dipole with a dipole moment 2qh. The direction of the vector of the total field Ez under and above the charged ball is indicated in Figure 11.5 by vertical arrows. The dependence of the absolute value of the vertical electric field on the height z, calculated by using (11.3) and (11.4), is shown in Figure 11.5 with line 1. In making this plot the following numerical values of the parameters have been used q ¼
150 C, h ¼ 10 km, and r ¼ 2 km. Such a thundercloud charge could arise immediately after the +CG discharge, which results in the CMC of the order of 1,500 Ckm.

Lightning effects in the mesosphere

433 80 km

70 km

60 km

50 km

t = 5.29 ms

t = 5.71 ms t = 5.99 ms t = 6.41 ms

t = 6.69 ms

t = 6.96 ms

t = 7.38 ms

t = 10.29 ms

Figure 11.6 High-speed images of the sprite development from small dim inhomogeneities as observed at Yucca Ridge Field Station on August 13, 2005, at 03:43:09.4 UT. The nucleation point of the left sprite element is marked with a white arrow. Adapted from Cummer et al. [42]

QE field decreases with altitude according to the power law, approximately inversely proportional to z3 . The conventional breakdown threshold in (11.2) decreases exponentially with height, i.e. more rapidly than the QE field does. Air breakdown can occur at those altitudes where the total QE field of the thundercloud charges and the charges induced in the ground exceeds the breakdown threshold Ec . As is seen from Figure 11.5, this breakdown condition is valid for z > 82 km. However, the sprite nucleation can also occur at lower altitudes near some atmospheric inhomogeneities, where the local electric field is greater than Ec . The origin of the initial sprite streamers can occur in the altitude range of 60
80 km, where QE field (line 1 in Figure 11.5) begins to exceed the threshold values required for the propagation of streamers (lines 3 and 4 in Figure 11.5).

11.2.3 Sprite development High-speed video observations have shown that a carrot-shaped sprite can be initiated several ms after the causative lightning spontaneously from a bright spot in the mesosphere. For example, Figure 11.6 displays a series of high-speed images illustrating sprite development and structure [42]. The time in these pictures is counted since the moment of the causative lightning stroke. This sprite consists of two distinct sprite elements, marked on the first and second pictures with the symbols A and B. These elements start to grow from two small and dim streaks located between 70 and 75 km altitude. Consider, for example, the left sprite element shown with a white arrow in the first picture. The downward streamer propagation is accompanied by branching into multiple channels and by enhancement of luminescence. Taking into account that the QE field is directed downwards, one can assume that this is a positive streamer. The upper portion of the sprite starts to grow up and develops into a bright column which is larger in size and significantly brighter than the initial downward streamers. In the fifth and subsequent pictures, this sprite portion branches and terminates in diffuse glows. Pre-existing kilometer-scale plasma irregularities arising before the streamer initiation in halo emissions have been observed near the altitude  of 75 km [43].

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Lightning electromagnetics: Volume 2

These spatial irregularities descend rapidly along with the sprite halo and then slow down producing spots of stationary glow. Finally, the streamer suddenly starts from this formation. Figure 11.7 shows this scenario for the development of the sprite. This event illustrates the sprite development from brightening inhomogeneities at the bottom of a halo [42]. The bright inhomogeneities arise at the lower edge of the originally homogeneous halo 2.2 ms after the return stroke. The downward propagating sprite streamers begin to develop from these spots and then branch in the same way as the previous example. The upward streamers propagate at velocity ð0:5
2Þ  107 m/s and terminate in diffusive emissions. The expanding bright columns in the upper portion of the sprites are wider and more diffuse than that visible in the sprites shown in Figure 11.6. The last images in Figure 11.7 illustrate the gradual fading of the glow and the disintegration of the sprite. Early observations have shown that the initial streamer’s velocities are greater than 107 m/s [44]. Using a multi-anode photometer, McHarg et al. [45] have observed the downward and upward propagating streamer at the velocities on the order of 107
108 m/s. On the basis of high-speed imagining with 0:1 ms resolution, McHarg et al. [46] has shown that the highest part of the upward sprite streamers can accelerate on the order of 1010 m/s2 at the initial stage of streamer development. Plots of the streamer head velocity versus time and velocity versus altitude are shown in Figure 11.8. As is seen from this Figure, the lowest region of downward streamers accelerates initially to a maximum velocity ð1
3Þ  107 m/s and then immediately decelerates at approximately constant value 1010 m/s2 which is saved until the sprite is stopped [47].

80 km 70 km 60 km 50 km

t = 1.66 ms

t = 2.06 ms

t = 2.26 ms

t = 2.46 ms

t = 2.66 ms

t = 3.06 ms

40 km 80 km 70 km 60 km 50 km

t = 3.46 ms

t = 3.66 ms

t = 4.06 ms

t = 4.66 ms

t = 6.26 ms

t = 7.86 ms

40 km

Figure 11.7 High-speed images of the sprite development from the sprite halo as observed at Yucca Ridge Field Station on August 13, 2005, at 03:12:32.0 UT. Adapted from Cummer et al. [42]

Lightning effects in the mesosphere × 107

3

Center

1×

1

10 10

Left

V, m/s

V, m/s

2

m/ 2 s

0 0 0.5 × 107 2

1.0 1.5 Time, ms

2.0

V, m/s

V, m/s

Left

1

Center

1×

Left

10 10

m/s 2

1

3

Center

× 107

2

0

2.5

435

0 0.4 × 107

0.8 1.2 Time, ms

1.6

2.0

Center Left

2 1

0 75

70

65 60 A‫׀‬t, km

55

50

0 75

70

65

60 55 A‫׀‬t, km

50

45

40

Figure 11.8 The downward streamer head velocity versus time (upper row), and the streamer velocity versus altitude (lower row). In making the first and second columns the high-speed images shown in Figures 11.6 and 11.7 were used. Adapted from [47] Flashes of extremely strong sprite are occasionally accompanied by appearance of red spots arising after the flash down in the lowest tendrils near the cloud tops [48]. These phenomena which have been termed Trolls (Transient Red Optical Luminous Lineaments) consist of a rapid series of events. The video observations with high time resolution have shown that each individual event starts with the formation of a red spot with faint red tails like a sprite tendril. Then this spot “flows” downward. Each following event starts higher than the previous one. As a whole, this series of events looks like a blur propagating upward from near cloud top to 40–50 km altitude at a velocity of about 1:5  105 m/s.

11.2.4 Sprite models To gain a better insight into the mechanism of sprite formation and its properties, consider a few theoretical models describing the sprite evolution. Free electrons make a significant contribution to the electrical conductivity of air at mesospheric altitudes since the mobility of electrons is much greater than that of ions. In strong electric fields, the free electron production is predominately due to the impact ionization of oxygen and nitrogen molecules according to the following reactions (e.g., [49,50]) e
þ O2 ! 2e
þ Oþ 2;

e
þ N2 ! 2e
þ Nþ 2:

(11.6)

In addition, free electrons can appear due to photo-ionization near the charged streamer head, where the charge density and electric field are more significant.

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Lightning electromagnetics: Volume 2

Under an appreciable electric field, a major role is played by the competing processes of dissociative two-body or three-body attachment of electrons to either O2 or N2 molecules e
þ O2 ! O þ O
;

e
þ O2 þ A ! O
2 þ A;

(11.7)

where A is another neutral molecule. It is generally accepted that the dissociative two-body attachment is more important than the three-body attachment at mesospheric and lower ionospheric altitudes [14]. For simplicity, we consider a three-component plasma consisting of electrons and single charged positive and negative ions. Let ne be the electron number density while nþ and n
be the number densities of positive and negative ions, respectively. To treat the electromagnetic phenomena associated with sprite development including its streamer structure and relationship with intra-cloud (IC) process, a set of transport kinetic and electrodynamic equations are required. The equations describing the variations in the number densities of electrons and ions must take into account not only the ionization and electron attachment to neutrals but also the electron-ion recombination, electron drift and diffusion, etc. Let ni and na be the ionization and attachment rates, while me and De be the electron mobility and diffusion coefficient, respectively. Then the basic kinetic equations are as follows: @ t ne ¼ Ic þ Iph þ ðni
na
bÞne
bd ne nþ
r  ðne me EÞ þ r  ðDe rne Þ; (11.8) @ t n
¼ ðna þ bÞne
bi n
nþ ; @ t nþ ¼ Ic þ Iph þ ni ne
bd ne nþ
bi n
nþ :

(11.9) (11.10)

Here the symbol @ t denotes partial time derivative, Ic stands for the rate of primary ionization due to cosmic rays, solar radiation, electron precipitation and etc.; Iph is the source of nonlocal photo-ionization which can play a significant role in the vicinity of the streamer head, bd is the effective coefficient of dissociative recombination, bi is the effective coefficient of ion-ion recombination, and b is the effective electron attachment rate in the absence of the electric field E. In the strong electric field ni and na are much greater than b. The functions me , and De , both depend on the neutral molecule number density N , which falls off approximately exponentially with altitude z according to (11.1). The electric field in the mesosphere is caused by both thundercloud charges and charges resulting from the electric breakdown of air. To complete the set of (11.8)–(11.10) a proper model of thundercloud QE field and the following Maxwell equation are needed e0 r  E ¼ eðnþ
n

ne Þ;

(11.11)

where e is the elementary charge. If the transient magnetic field can be neglected, then the electric field can be expressed through the potential j via E ¼
rj. The ionization and attachment rates, ni and na , are strongly dependent on both the applied electric field through electron temperature and on the neutral molecule

Lightning effects in the mesosphere

437

number density. For model calculations, an approximation of these functions similar to the Townsend approximation can be used [51]:

 E i ðN Þ E a ðN Þ ; na ¼ me aa Eexp
: (11.12) ni ¼ me ai Eexp
E E Here me is the electron mobility while ai and aa are the inverse of the electron mean free paths between ionization or attachment events. The products of me ai and me aa do not depend on the air number density, since ai and aa are proportional to the N whereas me / N
1 . Therefore, me ai ¼ me0 ai0 and me aa ¼ me0 aa0 , where the subscript 0 means the functions taken at sea level. Critical fields Ei and Ea responsible for the ionization and electron attachment rates, are proportional to N , and thus exponentially depend on altitude; that is, Ei ¼ Ei0 N =N0 and Ea ¼ Ea0 N =N0 . The difference between the ionization rate and electron attachment rate given by  ¼ E=N . (11.12) is shown in Figure 11.9 as a function of the reduced electric field E In making this plot we have used altitude 70 km and the following numerical values of the parameters me0 ¼ 3:8  10
2 m2 V–1s–1, ai0 ¼ 4:3  105 m–1, aa0 ¼ 2  103 m–1, Ei0 ¼ 2  107 V m–1, and Ea0 ¼ 3  106 V m–1 [51]. As is seen from this Figure, the inequality ni < na is valid at a low electric field. If an excess of electron density is formed at such fields, then it will fall off quickly due to the dissociative attachment of electrons to electronegative species, essentially to molecules of O2. In the inverse case, that is, ni > na , the electron density may increase exponentially with time thereby producing electron avalanches. The critical value of electric field known as

vi 2

– va,

103 s–1

1

0

–1

0.2

0.4

1.2 1.0 0.8 0.6 Emin E2 E* E1 E0

E N,

10–19 V m2 τ –1

–2 –3

Figure 11.9 The difference between the ionization rate and electron attachment  ¼ E=N as calculated rate as a function of reduced electric field E from (11.12) at an altitude of 70 km. The attachment instability can  min where the graph has a minimum. develop to left from the point E Adapted from Surkov and Hayakawa [52]

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Lightning electromagnetics: Volume 2

the conventional breakdown threshold Ec is derivable from the condition ni ¼ na and (11.12). From this, we arrive at (11.2) for Ec with the parameter Ec0 ¼ ðEi0
Ea0 Þ=lnðai =aa Þ  32 kV=cm. In the analysis that follows, we start with some analytical results. First of all, let us consider the so-called avalanche-to-streamer transition. Evidently that the sprite development starts with the formation of electron avalanches at the presence of ambient electric field exceeding the breakdown threshold Ec of air at mesospheric altitudes The number of electrons in avalanche increases in time: Ne ¼ expðgi me Ec tÞ, where gi is the ionization coefficient [39]. The growth of an individual electron avalanches is accompanied by an accumulation of considerable space charges at the avalanche heads followed by the generation of the proper electric field E0 of these charges. The field of these charges can strengthen the ambient field that results in the avalanche-to-streamer transition. To study this transition in a little more detail, the head of the electron avalanche can be approximated by a ball withradius Ra [39]. The electric field of a uniformly charged ball is given by E0 ¼ eNe = 4pe0 R2a . With increasing the avalanche size and the electron number, the electrostatic repulsion of the electrons becomes more and more significant. The electron drift due to their repulsion in the field E0 results in the ball expansion at the rate dRa =dt  me E0 . Taking into account the above expressions for Ne and E0 we arrive at the following equation dRa eme expðgi me Ec tÞ ¼ : dt 4pe0 R2a

(11.13)

The solution of this differential equation under requirement Ra ð0Þ ¼ 0 is given by 3e 3eNe : (11.14) R3a ¼ fexpðgi me Ec tÞ
1g  4pe0 gi Ec 4pe0 gi Ec Rearranging this equation we obtain that Ra ¼ 3E0 =ðgi Ec Þ. Actually, the increase in the head radius in accordance with (11.14) is limited since Ra must be less than the ionization length g
1 i . The increase in the avalanche head slows down, or even ceases 0 completely if Ra reaches the value about g
1 i [39]. At this point, the electric field E of 0 the head charges reaches a critical value Eс  Ec =3. The total electric field on outer side of the avalanche head increases up to the value of E  Ec þ Eс0 , while on the side of the head facing the inside of the avalanche, the field is approximately equal to E  Ec
Eс0 . The weakening of the total field behind the avalanche head promotes the formation of a quasi-neutral plasma in the tail of the avalanche and to the rebuilding of the avalanche structure to a streamer form. In fact, the criterion for the avalanche-to-streamer transition depends not only on ionization and electron repulsion but also on electron diffusion and attachment rate. In the model by Qin et al. [53], the avalanche head is also approximated as a charged ball, but they took into account not only the electrostatic repulsion of electrons but also the diffusive spreading of the electron avalanche. It was assumed that the electron drift due to their repulsion in the field E0 and diffusion gives rise to the ball expansion at the rate dRa 2De ¼ me E 0 þ : (11.15) dt Ra

Lightning effects in the mesosphere

439

As the first term on the right-hand side of (11.15) is neglected, the solution of this equation under zero initial condition has a form Ra ¼ ð2De tÞ1=2 , which is typical for diffusion processes. In the model by Qin et al. [53], it is assumed that ionization and dissociative electron attachment in the thundercloud QE field play a major role in the excitation of the sprite streamer. In this case, only the ionization and attachment rates can be retained in (11.8), neglecting the other terms on the right-hand side of the equation dne ¼ ðni
na Þne : dt

(11.16)

Just after + CG lightning, the uncompensated negative electric charges appear in a thundercloud for a short time. Equations (11.15) and (11.16) were used to approximate the inception of sprite streamers from sprite halo in the electric field of these charges. In this model, the condition Eс0  Ec =3 was applied as a criterion for the avalanche-streamer transition. The numerical modeling has shown that the sprite streamer initiation depends strongly on the CMC of causative + CG lightning and the ambient electron density profile. In semi-analytical models by Surkov and Hayakawa [52], a sprite is approximated through an expanding plasma ball that develops from a plasma inhomogeneity situated at altitudes of 70–80 km above the thundercloud. To simplify the problem, the streamer structure of the sprite leaves out of the account, and the plasma ball is assumed to be homogeneous and conductive. At the mesospheric altitudes, the mobility of the electrons is much greater than that of the ions, so only the electron conductivity is taken into account inside the ball, that is, s ¼ ene me . Combining this relationship and (11.16) and taking into account the continuity equation for electric current, yields r  j ¼ 0;

ds ¼ ðni
na Þs; dt

(11.17)

where j ¼ sE þ e0 @ t E is the total electric current density. Just after causative lightning, the uncompensated charges arise in the thundercloud. The plasma ball is polarized in the electric field in the thundercloud electric field E0 ðtÞ and polarization charges appear on its surface thereby producing the time-dependent dipole-type field outside the ball. In this model, the total electric field is given by   pðtÞ  r ; (11.18) Eðr; tÞ ¼ E0 ðtÞ
r 4pe0 r3 where pðtÞ is the dipole moment of the polarization charges while r denotes the position vector drawn from the ball center which is located at the height h. The total electric field (11.18) at the lowest point of the ball is assumed to be equal to kEc , where k  1 is the dimensionless factor. The polarization charges weaken the electric field inside the ball. But, if this field exceeds the air breakdown threshold, then the ionization of the air and the production of free electrons begin to prevail

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Lightning electromagnetics: Volume 2

over their attachment to the neutral molecules followed by the increase in the ball plasma conductivity in (11.17). In this model, the rates of ionization and electron attachment to molecules are governed by the electric field (11.18) in accordance with (11.12). That is, given some electric field E0 ðtÞ, the problem is reduced to a set of ordinary nonlinear equations for the functions pðtÞ, sðtÞ, and RðtÞ. These equations should be supplemented by the proper boundary conditions on the ball surface, that is, the continuity of electric potential and normal component of the total current density. The ball radius begins to increase as soon as the electric field of the thundercloud exceeds the conventional breakdown threshold of air Ec ðhÞ. The analysis of the approximate analytical solution of this problem showed that the initial acceleration a0 ¼ d 2 R=dt2 is practically independent of the rate of change of the thundercloud electric field and is given by [54]
    me0 kEc ðhÞsa ðhÞHa Ei0 Ea0
aa0 exp
: (11.19) ai0 exp
a0  3e0 kEc0 kEc0 For night conditions, the conductivity of the ambient air sa ðhÞ at altitude h ¼ 70 km is about 10
9 S/m while Ec ðhÞ  0:51 kV/m. Substituting the above numerical values of the parameters as well as Ha ¼ 8 km and k ¼ 1:2 in (11.19), we obtain that a0  1010 m/s2. The numerical solution of this model problem is illustrated in Figure 11.10 as the thundercloud QE field is approximated via E0 ðtÞ / ðt=tr Þexpð
t=tr Þ, where tr is

4.5

× 107 2 ms 5 ms 9 ms

4 3.5

V, m/s

3 2.5 2 1.5 1 0.5 0

0

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Figure 11.10 Model calculation of the expansion velocity of the plasma ball immersed in thundercloud QE field. Taken from Surkov and Hayakawa [54]

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the relaxation time. In this figure, the maximum expansion velocity of the plasma ball varies within ð3:2
4:4Þ  107 m/s depending on tr . Despite this model being far from perfect, the above estimates of the ball expansion velocity and acceleration are close in magnitude to the observed values of the sprite streamer velocity and acceleration shown in Figure 11.8. To proceed analytically, it is necessary at this point to construct a more complete model of the sprite which takes into account the structure of a developed sprite consisting of individual moving streamers. Even in this simplified form, the set of (11.8)–(11.12) describing the plasma phenomena associated with sprite evolution is rather complicated. Thus, the most portion of the theoretical studies of sprites are based on numerical simulations of the (11.8)–(11.12) or similar equations (e.g., [55–57]). The reader is referred to that works for details about sprite numerical simulations.

11.2.5 Inner structure and color of sprites Figure 11.1 displays a typical structure of the developed carrot-shaped sprite. It is usually the case that upper diffuse region in red color gradually changes to lower tendril-like filamentary region in blue color below approximately 50 km. The predominance of red and blue colors in the optical sprite emission is due to the excitation of molecules of N2 by electron impact followed by the optical emission of excited molecules of N2. At altitudes above 50 km the first positive band system of N2 emission (N21P) makes the main contribution to the red region (600
760 nm) of the optical spectrum of the sprite whereas below 50 km the strong collisioninduced quenching of the electron-excited state B3 Pg gives rise to the suppression of this emission [5,58]. In the altitude range below 50 km, the blue color begins to prevail in the sprite optical emission due to the excitation of the second positive band of N2 (N22P).

Figure 11.11 A sprite telescopic image at wide (a) and narrow (b) field of view (FOV) as observed over northwestern Mexico on July 13, 1998, at 06:00:00 UT. The small rectangle on panel (a) is shown on panel (b) at a large scale. The images have been false-colored and saturated at 685 kiloRayleigh (kR) in order to better represent intensity (see color bar). Taken from Gerken et al. [59]

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Figure 11.11 shows sprite telescopic images taken over northwestern Mexico on July 13, 1998, at low (a) and high (b) resolutions [59]. The small white rectangle on panel (a) is shown on panel (b) at a large scale in order to highlight the fine structure of the sprite. It is evident from this image that the tendril-like region of the sprite consists of densely packed branching streamers. These high-resolution measurements have shown that the mobile bright compact balls, shown in this image with red color, are the highly ionized streamer heads. In the altitude range of 60
64 km, the transverse scale of filamentary structures ranges from 60 to 145 m whereas the size of streamer heads varies from 10 to 100 m and the streamer speed lies in the range of 106
107 m/s [46,59]. The diameter of the CG lightning streamer is smaller than 1 mm and its length amounts to tens of meters, whereas the size of sprite streamers is much greater. This fact can be explained based on the similarity theory for streamers that establishes similarity relations for different streamer parameters and the air number density ([40,41,60]). First of all, we consider the scaling in size of a glow discharge. According to the similarity theory the typical streamer length Ls , the streamer head radius Rs , and the charge qs in the head vary inversely proportional to the neutral gas density N . For example, this means that Rs ðzÞ ¼ Rs0 N0 =N ðzÞ, where Rs0 denotes the streamer head diameter at sea level and z is the streamer altitude. Substituting (11.1) for N ðzÞ into the above relationship and using the parameters z ¼ 62 km, Rs0 ¼ 1 mm, and Ha ¼ 7 km, we obtain that Rs  7 m. Kanmae et al. [61] have shown that the sprite streamer diameters have to be larger than that predicted by this simple similarity law possibly due to the effects of the photoionization and an expansion of the streamer head along its propagation over a long distance. In this notation, the above estimate is compatible with the streamer head size observed by Gerken et al. [59]. The dynamics of the individual sprite streamer and its internal structure are rather complicated. After the passage of the streamer heads, the ionized channels arise in their wake. These channels exhibit intricate luminous patterns of alternating bright and dark spots, conventionally called beads and glows. The sprite beads appear as persistent, localized spots of light emission that often punctuate a streamer channel [62]. Their lifetime is much longer than the duration of the streamer head propagation. One conceivable reason for the existence of beads and glows is assumed to be the attachment instability of electric discharges developing in the streamer channel [63]. The attachment instability builds up as a result of the fluctuation of the electron number density ne at certain electric field in the streamer channel (e.g., [39]). To explain this kind of instability, consider again (11.16) describing the variation of electron number density due to the ionization and the dissociative attachment of electron to neutral molecules. The difference between the ionization rate and electron attachment rate versus the reduced electric field is shown in Figure 11.9. Let I ¼ ene me SE be the time-dependent electric current in a streamer channel, where S is the cross-section of the streamer channel. For simplicity, this current is assumed to be uniform and constant along a streamer channel. This implies that all

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 min the parameters in the expression for I are taken in an average sense. Let E denotes the reduced electric field which corresponds to minimum of the graph  min in some  0 is less than E shown in Figure 11.9. Suppose that the electric field E cross-section of the channel. Since the electric current is constant along the channel, the small negative fluctuation of the electron number per unit of the channel length, ne S, in this section will result in a small increase of the local electric field to  > E  0 . This change in the electric field, which is schematically shown the value E in Figure 11.9 with a blue arrow, leads to a decrease in the difference ni
na . From (11.16), it is clear that this change in ni
na will cause an additional decrease of ne ,  and so on thereby exciting the attachment which in turn results in the increase in E instability. Following Luque et al. [63], we now examine the mechanism of beads generation in the presence of exponentially decreasing electric current I ¼ I0 expð
t=tÞ in the streamer channel. Here t  1 ms is the current relaxation time. Assuming for the moment that S is constant, then the equation for the current in the streamer channel can
1 be rewritten to the view I
1 ðdI=dtÞ ¼ n
1 e ðdne =dt Þ þ E ðdE=dt Þ. Substituting the expression for electric current and (11.16) for dne =dt into the above relationship yields   dE=dt ¼ E na
ni
t
1 : (11.20) 1 There are three stationary points in (11.20); that is, E ¼ 0 and two points E
1  and E 2 which are the roots of the equation na
ni ¼ t . These roots are defined by the intersection points of the graph na
ni with a red horizontal line t
1 shown in Figure 11.9. It follows from (11.20) that the electric field can enhance in the 2  1 to E streamer channel areas if the electric field lies within the interval from E shown in Figure 11.9. Additionally, the reduction of the free electrons due to their attachment to neutral molecules gives rise to the resistance of these areas. This results in the Joule dissipation and enhancement of the heat release, which can cause the transformation of these areas into quasi-stationary bright beads. The model based on attachment instability claims only a qualitative explanation of the observed beads and glows in the streamer channel. Actually, the beads appear in the streamer wake at different times after the streamer head passage. In this notation, it seems likely that the streamer’s current variations and the streamer interaction can play a significant role in bad and glow excitation.

11.2.6 ELF/VLF electromagnetic fields produced by sprites The time delays between causative CG strokes and the sprites vary widely from less than 1 ms to a few hundred depending on the sprite type, meteorological conditions, etc. [32,33]. The carrot sprite occurrence can be accompanied by a burst of VHF electromagnetic radiation starting 25–75 ms before the causative CG stroke whereas the column sprites exhibit little VHF activity [64]. These effects are indicative of a correlation between intra-cloud processes and sprite generation mechanisms. The simultaneous optical and ground-based observations of the spriteproducing lightning events have shown that the ELF/VLF electromagnetic field

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data contained not only the signal radiated by the lightning current but also the peak which coincides in time with the appearance of sprite luminosity [65]. This fact was established by using the ISUAL instruments on board the FORMOSAT-2 satellite. Figure 11.12 displays an example of ELF/VLF variations detected by the groundbased magnetometer located approximately 2780 km from lightning discharge [66]. The radiation caused by the sprite current manifests itself as 1
2 ms pulses that follow a lightning return stroke by a few milliseconds to a few hundred milliseconds. It should be emphasized that these pulses are caused by the currents generated inside the sprite itself and they are not associated with the M-component of continuing current of causative +CG discharge. In contrast to return stroke, the sprite spectrum almost entirely belongs to the ELF region. In theory, one of the main characteristics of the ELF field at large distances from a lightning discharge is the so-called lightning current moment, which is approximately equal to the product of the discharge current I ðtÞ and the length LðtÞ of the lightning channel. The current moment waveform of the observed signals can be extracted from the ELF data by using the model approximation of the lightning and sprite currents. To extract this information, the solution to the problem of the ELF electromagnetic field propagation in the Earth-ionosphere waveguide is used [66]. The time dependence of the lightning and sprite current moments is selected in such a way that the calculated ELF field pulse would be close to the observed ELF magnetic field variations. On the basis of several case studies, the sprite current moments have been estimated as much as 200
300 kAkm [66]. A network of ground-based stations equipped with different sensors has been operative during Japanese sprite campaigns in winters 2004/2005 and 2006/2007 [33]. This network included optical instruments at two-spaced observatories, VLF and ELF electromagnetic field antennas, and magnetometers. High-sensitive cameras have been used to capture optical emissions of causative lightning and sprite images. The coordinates and onset time of lightning flashes were measured by using three VHF antennas of the SAFIR interferometric lightning system. The

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Figure 11.12 Azimuthal component of the ELF/VLF magnetic field variation recorded at Duke University, North Carolina on October 3, 2004 at 0426:55.6 UT. The peaks on the graphs are presumably due to the current pulses generated by the causative +CG lighting and sprite. Adapted from Cummer et al. [66]

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technique of determining the current moment of sprite-producing lightning discharges used in these experiments differs from the technique described above. The current moment has been estimated by comparing the measured ELF lightning spectra with the spectra obtained by solving the model problem [28]. As an example, Figure 11.13 illustrates the sprite-producing lightning event recorded in Hokuriku area, Japan on February 3, 2007 [33,67]. North-South component of ELF magnetic field shown in this figure has a peak marked by the vertical red line which coincides with the onset time of the sprite. Other ELF transients; that is, East-West magnetic component and vertical electric field have the same waveform including the sprite-producing peak. The power spectral density of these ELF magnetic field variations is shown in Figure 11.14. The spectrum resonance structure below 7 Hz is apparently due to excitation of the ionospheric Alfve´n resonator (IAR) (e.g., [68]), but several peaks are in the region 7
25 Hz. An interesting property of the spectrum is that the envelope of the spectrum, shown with the green line, is subject to oscillations with a “period” of about 15–20 Hz.

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Figure 11.13 North-south component of ELF magnetic field caused by a spriteproducing lightning event recorded at Hokuriku area, Japan on February 3, 2007 [33,67]. The peak marked by a vertical red line coincides in time with the appearance of sprite luminosity. Adapted from Surkov et al. [67], with permission, Elsevier

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Figure 11.14 Power spectral density of the ELF magnetic field variations which were shown in Figure 11.13. Adapted from Surkov et al. [67], with permission, Elsevier

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To explain this property one should take into account that this power spectrum contains the contribution of the magnetic fields of both causative lightning discharge, Bc , and the delayed sprite, Bs . Let t be the lag of time between the causative lightning and sprite occurrences, while rc and rs be their position vectors with respect to the observation site. The total spectrum is then given by [67]: BðwÞ ¼ Bc ðrc ; wÞ þ Bs ðrs ; wÞexpðiwtÞ:

(11.20)

The power spectrum is proportional to jBðwÞj2 which contains the oscillatory factors cos ðwtÞ and sin ðwtÞ whence it follows that the power spectrum amplitude has to be modulated with “period” t
1  15
20 Hz. So, the ULF/ELF power spectra permit us to obtain information not only about the lightning and sprite current moments but also about the sprite delay time.

11.2.7 Effects of sprites on the ionosphere The intense lightning discharges often result in observable changes in the amplitude and phase of VLF electromagnetic waves propagating in the Earth–ionosphere waveguide and passing over a thunderstorm region [4,69–71]. One possible cause for these changes is the so-called lightning-induced electron precipitation (LEP) event (e.g., [70]). The basic mechanism for the LEP effect is due to the fact that, firstly, a portion of lightning-radiated wave energy is transferred into magnetospheric whistler mode waves, which interact in gyroresonance with relativistic radiation belt electrons and, secondly, the electron scattering into the loss cone and precipitation in the upper atmosphere followed by the secondary air ionization [72]. The VLF amplitude and phase changes occur 1 c after the return stroke. This time lag is due to the delay of both the VLF whistler mode propagating to the equatorial region of the radiation belts and the electrons traveling from the equatorial region to the lower ionosphere. It was later discovered that in some events the observed VLF wave perturbations follow within only a few milliseconds of the lightning discharge in contrast to the LEP effect. Most of them had positive amplitude changes whereas LEP events most commonly result in negative amplitude changes. This kind of VLF perturbation has been termed “early” or “early/fast” events [73]. The lightning impact on the ionosphere is assumed to be due to heating and ionization of the D region of the ionosphere caused by lightning electromagnetic radiation or QE field of the thundercloud. This leads to local changes in conductivity and reflection coefficient in the D region, which in turn results in the amplitude and phase perturbations of the subionospheric VLF wave passing over a thunderstorm region (e.g., [74]). Similar early/fast effects associated simultaneously with sprite appearance have been observed by Inan et al. [73]. Analysis of VLF data gathered during the 2003 EuroSprite campaign has shown that the sprite occurrences are accompanied by early/fast VLF perturbations in a one-to-one correspondence [75]. Figure 11.15 shows an example of the sprite-produced early/fast event observed during this campaign [76]. The onset time of this event has a short

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VLF amplitude (dB)

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Figure 11.15 A typical example of an “early/fast” VLF event associated simultaneously with a sprite at Crete during the 2003 EuroSprite campaign. Adapted from Haldoupis et al. [77]

duration smaller than 20 ms and coincides with the sprite onset time. This data counts in favor of the suggestion that these sprite-related early/fast events are caused by the interaction of VLF waves with the conductivity perturbation in the upper D region. The typical recovery time of the early/fast effects varies from 10 to 300 s and is comparable to the LEP recovery time [4,77]). It is generally believed that this recovery time is basically determined by recombination processes in the ionospheric plasma since the dissociative recombination of electrons and single positive ions prevails over electron attachment at the altitudes of the lower ionosphere [78]. In consideration of this approximation, (11.8) describing the perturbations of electron and ion number densities is simplified to dne ¼
bd ne nþ : dt

(11.21)

Let ne0 be the electron number density arising just after the short-term stage of interaction of the sprite-radiated electromagnetic wave with the lower ionosphere. Taking into account that ne  nþ and performing integration of (11.21) under the requirement ne ð0Þ ¼ ne0 , we come to the usual law for binary plasmas ne ¼

ne0 : 1 þ bd ne0 t

(11.22)

Substituting ne ¼ ne0 =3 into (11.22) we obtain the rough estimate of the recovery time: tr  2=ðbd ne0 Þ. Using the typical values of the parameters bd ¼ ð1
3Þ  10
7 cm3/s [78] and ne0 ¼ 6  104 cm–3 [77] gives the value of tr  ð1
3Þ  102 s, which is compatible with the observed recovery time.

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Figure 11.16 A typical example of an “early/slow” event associated with a spriteproducing lightning was observed at Crete during the 2003 EuroSprite campaign. The onset time of the sprite is shown with the vertical dashed line. Adapted from Neubert et al. [4] Another type of the sprite effect on phase and amplitude of VLF waves propagating in the Earth–ionosphere waveguide is referred to as “early/slow” event [4]. Figure 11.16 shows an example of such event observed in Crete at a distance of about 2,000 km from a convective storm in central France. The sprite appears at the time marked by the vertical dashed line in Figure 11.16. The “early/slow” events have a long onset duration of up to 2.5 s, which is much greater than the “early/fast” onset duration shown in Figure 11.15. The ground-based VLF measurements have shown that the growth phase of the early/slow events is accompanied by bursts of spherics which are commonly related to intra-cloud (IC) lightning discharges whereas spherics were not detected during onsets of early/fast events [74]. This implies that the long duration of the “early/slow” onset time can be associated with the enhancement of IC electric activity before and after the sprite appearance. Horizontal IC discharges can have the greatest impact on the ionosphere since the horizontal currents radiate most of the electromagnetic energy in the directions perpendicular to the current line. A series of electromagnetic pulses radiated upward from horizontal IC discharges can accelerate sprite-produced electrons followed by secondary ionization buildup in the upper D region of the ionosphere. This leads to a gradual buildup of conductivity changes below the nighttime VLF wave reflection heights. One may assume such an effect could be responsible for the long onset durations of the observed early/slow events. The VLF measurements thus provide us with important information about the role played by IC processes in the perturbation of the lower ionosphere above the sprite occurrence.

11.3 Blue jet, blue starter, and gigantic jet 11.3.1 Basic properties and morphology of blue and gigantic jets A class of upward propagating beams of blue light that start at the top of thundercloud and terminate in the upper stratosphere have been discovered during the Sprites94 aircraft campaign [6,7]. These beams of luminosity have been termed blue jet (BJ). They originate from the thundercloud top and move upwards in a narrow cone of about 15
20 with the mean velocity of the order of 100 km/s to terminal altitudes of about 25
40 km. A picture of a typical BJ which was

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observed over a large evening thunderstorm in the Indian Ocean [8] is shown in Figure 11.3. Figure 11.17 demonstrates the inverted black and white image of the BJ shown in Figure 11.3 [8]. At the base of BJ, its diameter is about 400 m. At 30 km altitude, the jet diameter broadens to about 2 km. It is evident that the main cause of the optical emission of BJs is the upwarddirected large-scale electric discharge in the stratosphere that cause the ionization of air and the excitation of air molecules including N2 by electron impact. The red N2 emissions are strongly quenched at these altitudes. So, the blue color in the optical emission of BJs is caused by the predominance of the emission of the excited energy levels of the second positive band of N2 (N22P) (e.g., [79]). As compared to the sprites, BJ luminosity is brighter than the sprite ones while the BJ tip moves slower than the sprite streamer head. Unlike sprites, the BJs do not seem to require the occurrence of lightning before the BJ discharge. Blue starters (BSs) are a kind of BJs, which propagate upward from cloud tops at 17
18 km altitudes and terminate abruptly at altitudes below 25 km. Basically, they differ from BJs by a lower terminal latitude [9]. Analysis of data gathered by

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Figure 11.17 The inverted black-and-white image of this blue jet which was shown in Figure 11.3. Adapted from Wescott et al. [8]

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the ISUAL payload on board the FORMOSAT-2 satellite has shown that BSs have a length about 8 3 km with a width of  ð2
4Þ km [80]. As compared to BJs and BSs, the GJs are more intensive discharges propagating upwards from the thundercloud top through the stratosphere and mesosphere to lower ionosphere. When the GJ terminates at altitude of about 85
90 km, it produces the electrical connection between the thundercloud top and the conducting D- and E-layers of the ionosphere [10,81,82]. The first images of GJ over an oceanic thunderstorm were captured at the Arecibo Observatory in Puerto Rico [10]. Figure 11.18 demonstrates an example of GJ observed over the Pilbara region in the north of Western Australia on March 28, 2017. The bright blue/purple stem below the streamer corona is thought to be a leader channel. The top edge of the bright stem is usually located at 30
50 km altitude [83,84]. The upper streamer zone at mesospheric altitudes exhibits red color, in analogy to the sprites, mainly due to the excitation of the first positive band system of N2 emission. A majority of GJs are of negative polarity and can transfer more than 100 C of negative charge from the cloud top to the ionosphere (e.g., [85]). The ISUAL experiment observed global rates of 0.50, and 0.01 events per minute for sprites and gigantic jets, respectively [86]. Although the instruments with a larger sensitivity and coverage will detect many more such events, it is evident that the GJ occurrence is a much rarer event compared to a sprite occurrence. The GJs are more frequent over intense tall tropical thunderstorms of 14
18 km altitude with convective surges or even overshooting [87–89] although Yang et al. [90] have reported observations of GJs over a mesoscale convective system in the middle latitude region in eastern China. The GJ over a maritime thunderstorm not only with 6.5 km tall but also with overshooting has been observed by van der Velde et al. [84].

Figure 11.18 A gigantic jet over Pilbara, Australia, on March 28, 2017, captured by Jeff Miles. Taken from https://watchers.news/2017/03/31/gigan tic-jets-over-pilbara-australia-on-march-28-2017

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According to morphological characteristics, GJs can be divided into several categories such as the tree-shaped GJs like that shown in Figure 11.4 or carrot-like GJs with beads and patches at the top of a massive central area similar to those seen in carrot sprites [91]. One more type of GJs, which have two clearly separated main branches that can develop separately; that is, not simultaneously is shown in Figure 11.18.

11.3.2 Development of gigantic jet The video sequence of the images of upward propagating GJs can be used in order to estimate the dependence of the GJ top altitude on time. Triangles and circles shown in Figure 11.19 indicate the altitudes of the GJ top depending on time as observed by a Pasko et al. [10] and b Soula et al. [89]. The initial and final stages of this dependence can be approximated by two dashed straight lines which correspond to approximately constant velocities of the GJ top [49]. At the initial stage, the apparent velocity of GJ upward propagation is of the order of ð5:7
6:3Þ  104 m/s that is close to a typical leader velocity observed during usual CG lightning discharge. Some GJs may initially develop as ordinary IC lightning followed by the origination of the leader which begins to propagate upward from the thundercloud top [83]. At the final stage, as is seen in Figure 11.19, the GJ top velocity increases up to ð1:2
2:3Þ  106 m/s. Based on recent measurements, the GJ evolution can be divided into several main stages: the leading jet (LJ), fully developed jet (FDJ), and trailing jet (TJ) (e.g., [81]). Figure 11.20 illustrates these stages of the GJ development with an example of the image sequences captured from the north coast of Colombia in 2018 [91]. The LJ stage numbered by 1 in Figure 11.20 manifests itself as a weak luminescence above the cloud top starting  150 ms before the FDJ stage. The area marked with a dotted line in the left upper corner of this figure depicts three faintly 90

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Figure 11.19 Illustration of GJ evolution as observed by a Pasko et al. [10] and b Soula et al. [89]. Triangles and circles indicate the heights of the GJ top depending on time. The initial and final stages of GJ evolution are approximated with dashed straight lines. Taken from da Silva and Pasko [49]

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glowing segments, which become visible at an altitude of 32.6 km to 34.4 km only 5.0 ms before the start of the FDJ stage (numbered by 2–4). These close-up images are shown with 0.2 ms resolution. The segments exhibit the brief downward extension of a streamer starting at the same time as the onset of upwardpropagating streamers at the beginning of the FDJ stage. The time-averaged speed of the streamer upward extension during the LJ stage is about ð0:5
2Þ  104 m/s as observed with a slow camera while the LJ current is estimated at 100 A. In the late LJ stage, sometimes there occurred a stepping process with step sizes of 2
5 km, 5
10 ms intervals between them, and 0.5 ms step duration [91]. On account of the step duration 0:5 ms, one can estimate the upward step speed of the order of ð0:4
1Þ  107 m/s, which is about two orders of magnitude greater than the time-averaged speed at this stage. Stepping is typical for negative leaders in lightning flashes. So, these observations count in favor of the hypothesis that the negative lightning leader reaching  40 km altitude can be generated in GJ followed by the formation of the streamer zone, which in turn can make the jump to the ionosphere [49]. In contrast to the typical negative leader in the CG lightning, however, the LJ stepping process exhibits a much longer duration, no optical pulses upon connection, and less brightness. Although the absence of optical emission can be due to strong absorption in the blue/ultraviolet part of the GJ spectrum in the atmosphere at large distances. The FDJ stage starts with the final growth of the GJ to the ionosphere followed by a sudden increase of the luminosity which is much brighter than that at the LJ stage. The lower panel in Figure 11.20 illustrates the FDJ stage in an enlarged time scale with a time resolution of 0.2 ms. The origin of the FDJ is located at the

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GJ 12 19 November 2018 02:13:39

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1 image = 0.2 ms

Figure 11.20 An example of the image sequences of the GJ development captured from the north coast of Colombia, Cartagena on November 19, 2018. The upper panel shows the selected images recorded by the high-speed camera with a time resolution of 1 ms. A dotted line in the left upper corner indicates the brief downward extension of a streamer starting at the same time as the final upward jump. The lower panel demonstrates a zoom in time of the FDJ stage of the GJ with a temporal resolution of 0.2 ms. The background in this figure was removed and the contrast was improved. Taken from van der Velde et al. [91]

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altitude range of 35
50 km. From this point, the GJ develops in a bidirectional fashion in such a way that upper portion of the GJ propagates toward the ionosphere without interruption forming the diverging branched structure of the negative streamers whereas the power portion of the GJ transforms into downward extending filaments/positive streamers, which gradually fade over time. Van der Velde et al. [91] have reported that the upward negative streamers propagate at mean speeds of ð1
2:5Þ  107 m/s, accelerating above 70 km up to the speed 7:5  107 m/s; that is, one order magnitude greater than that reported by Pasko et al. [10] and Su et al. [81]. Notice that no downward-propagating return strokes were detected upon connection to the ionosphere in contrast to CG lightning. Simultaneous optical and ground-based ELF observations have shown that GJs result in the generation of low-frequency magnetic fields [83,85,92]. For example, the ascending streamers during the final jump and decay of the FDJ are accompanied by the appearance of a sharp peak in the magnetic field perturbation with a duration of about 5 ms. This peak is assumed to be due to the increase in the FDJ current up to 350 A during the final jump [91]. The TJ stage (numbered by 5–7) begins after the GJ reaches the ionosphere and then lasts for about 200 ms. The upper part of the TJ luminescence attenuates gradually during this stage although the middle part of the jet starts to form a brighter section after a few milliseconds [89,91]. This bright section/trailing jet reaches a maximum luminosity 60
80 ms after the time when the FDJ stage is completed. Beads and new patches appearing above the TJ indicate that an electric field may exist in this region. The upward velocity of the TJ top slows down from 9  105 m/s to 2  104 m/s after reaching maximum luminosity and then the TJ terminates at 55
62 km altitudes. The highest current moment during the TJ stage is estimated to be 37.5 kA km, which corresponds to the current of about 850 A while the total CMC by the end of the TJ stage is 2,300 C km.

11.3.3 Models of gigantic jet It is generally believed that the conditions under which a BJ or GJ can occur largely depend on the distribution of electric charges in the cloud. Classical, normally electrified thunderstorms have a stratiform electrical structure close to a typical dipolar structure. Most part of negative charges predominantly accumulates in the middle region of the thundercloud, whereas the majority of positive charges tend to pile up at the thundercloud top [29,93,94]. These main charges are supplemented by a small positive charge located at the bottom and negative screening charge at the upper cloud boundary. An example of the electric field measured by a balloon in the typical thundercloud is shown in Figure 11.21, which demonstrates strong variations of the electric field with altitude [95]. The arrow L indicates the height at which the electric field is close to the conventional breakdown threshold. In order to explain a normal and nonstandard electrical structure of the cloud, several models of the charge distribution in the cloud have been proposed. The models consisting of four charges areas located at different altitudes are illustrated in Figure 11.22 [94]. These space charges are assumed to have a Gaussian spatial distribution. The

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Figure 11.21 Examples of altitude dependence of balloon-measured vertical electric field in a thundercloud. The arrows L indicate the heights at which a CG lightning leader may start to develop. Adapted from Marshall et al. [95] negative and positive charge areas are shown with blue and red contours while the total charges of these areas are indicated to the right from contours (in C). Figure 11.22a, c displays the model charge distribution at which the IC lightning (a) can occur between the two most highly charged regions, and the model that corresponds to the low-altitude IC lightning (c). Negative CG discharge can arise if the storm accumulates lower positive charge as shown in Figure 11.22b. Observations of the so-called “bolt-from-the-blue” (BFB) lightning discharges can be explained in terms of the model shown in Figure 11.22e. The negative BFB discharge begins as regular upward-directed IC discharge which does not terminate in the upper positive charge. Instead, it continues horizontally out the upper side of the storm and turns downward to ground [94]. It is generally believed that the positive BJ discharge may occur under the requirement that large amount of positive charge piles up near the top of cloud. This charge can be covered from above by a negatively-charged screening layer as shown in Figure 11.22d. It seems likely that the BJ discharges once triggered would propagate upward through the negative-charged layer and escape the cloud top into the stratosphere. Since the GJ usually bears a large amount of negative charge, in contrast to the BJ discharges, the GJ events can be associated with the accumulation of great deal of negative charges at middle level of the thundercloud and opposite charges at upper level. This charge distribution provides an alternative way of neutralization the mid-level negative charge, by discharging it to the upper atmosphere due to GJ rather than to ground due to CG lightning. Figure 11.22f shows the model distribution of space charges in the cloud, which may precede the occurrence of the

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(km) 21 18

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Figure 11.22 Simplified models of electric charge distribution in a thundercloud. Blue and red contours indicate negative and positive charge areas. The numbers in the columns on the right indicate the charges of the corresponding areas in coulombs. Taken from Krehbiel et al. [94] CG discharge [94]. This model consists of four space charges: 25,
120, 82.5, and
3 C distributed around the altitudes 4.3, 8.0, 13.2, and 15.4 km, respectively. To study the QE field associated with the GJ events in a little more detail, the simplest model was used, which assumes that these charges form four spherically symmetric regions arranged one above the other along the vertical z-axis. The spatial charges are uniformly distributed in these regions. The results of numerical

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3

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Ɀ

, km

14 12 10 8

2

6

1

4 2 0 –300 –200

–100

0 EⱿ

100 200 , kV/m

300

400

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Figure 11.23 Numerical simulation of the altitude dependence of the thunderstorm QE field, which may precede the occurrence of the CG discharge. Vertical component of the QE field versus altitude is shown with solid line. A runaway breakdown field and the electric fields required for propagation of positive and negative streamers in the air are shown with dashed lines 1, 2, and 3, respectively. Adapted from Surkov and Hayakawa [60] modeling of the vertical electrical field taken along z-axis versus altitude z are shown in Figure 11.23 with solid line [60]. The dashed lines 1
3 indicate the altitude dependence of the runaway breakdown threshold and minimum fields required for propagation of positive and negative streamers in the air. The occurrence of GJ discharge is possible at those altitudes where the thundercloud QE field exceeds one of these three threshold fields. Initiation of the GJ discharge due to the runaway breakdown mechanism at low altitudes is very questionable because of the lack of seed relativistic electrons and great length required for a relativistic avalanche multiplication process (e.g., [79]). As is seen from Figure 11.23 the negative GJ discharge can be initiated within the altitude range of 10–12 km where the thunderstorm QE field is close to electric fields required for propagation of negative streamers in the air. The arrow indicates the most probable altitude for the GJ initiation. The negative GJ once triggered can penetrate through the above narrow layer with positive electric field (10
15 km), and then escape the thundercloud toward the ionosphere. So, the GJ once triggered starts to propagate as a normal IC discharge between the main mid-level negative charge and a screening positive upper-level charge, that continues to propagate upward out of the top of the thundercloud [94]. The upward-propagating BJ and GJ discharges and normal CG lightning have a few properties in common despite the BJ and GJ have the inverted structure and other spatiotemporal scales. The BJ can be considered as an upward-propagating

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positive leader with a streamer corona on the top in close analogy to usual CG lightning [96] whereas the most of GJ manifest themselves during the LJ stage as upward-propagating negative leaders (e.g., [91]). A great number of the short-lived cold streamers produce a streamer corona of BJ and GJ discharges. In contrast to CG discharge, the length and radius of individual streamers emitted from the hot leader head and a typical size of the streamer corona increase with altitude because of exponential decreasing air density. In the model by Raizer et al. [97–99], the BJ/GJ originated from a bidirectional leader starting at the height where the electric field is maximal. It is assumed that the ascending leaders prevail over the descending ones due to the exponential decrease in atmospheric pressure with altitude. The minimum field required for the streamer propagation is given by equation similar to (11.2); that is, Es ¼ Es0 expð
z=Ha Þ, where Es0 is the critical field at sea level. Let hl be the altitude of the leader tip, which plays a role in the streamer source. The individual streamer born at this altitude can grow up to “infinity” if the leader tip has the potential ð1 Es dz ¼ Es0 Ha expð
hl =Ha Þ: (11.23) U¼ hl

From here one may obtain an order-of-magnitude estimate of the escape altitude hl . Such an upward-propagating streamer can reach to the lower ionosphere if it starts from the altitude [98] hl ¼ Ha ln

Es0 Ha : U

(11.24)

Using empirical data by van der Velde et al. [84] and by Neubert et al. [100], Milikh et al. [101] have estimated the parameters in (11.24) and found that hl  42 km. This rough estimate is compatible with the GJ observations since the streamer corona of the GJ start to grow up at the altitude range of 40–50 km (e.g., see Figure 11.20). The streamer corona of the BJ discharge arises at lower altitudes and therefore it terminates in the stratosphere before reaching the altitudes of the ionosphere. One of the challenges of the BJ and GJ research is to know enough about the streamer-to-leader transition and initiation of lightning leader. Laboratory tests with long sparks and numerical simulations have shown that the initial lightning leader can result from the contraction of the streamer’s currents into a small radius channel [39,102]. This contraction has been assumed to be due to the development of an ionization-thermal instability that is well-known in the theory of a glow discharge [40,103]. The numerical simulations of the initiation of lightning leader and streamer-toleader transition in the air have shown that the transition from a uniform discharge to the contracted state occurs as the electric current in the discharge channel exceeds the critical value which increases with the pressure decreasing [101]. This means that the critical current required for the BJ/GJ discharge contradiction increases with altitude. The BJ/GJ leader thus terminates at such an altitude where the critical current becomes as high as the leader current. Taking into account that

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the observed altitude of GJ leader termination is about 48 km, and comparing this altitude with that derived from their numerical calculation, Milikh et al. [101] have estimated the critical current as 3.3 kA.

11.4 Elves Divergent rings of brief optical emissions at the bottom of the ionosphere sometimes occurring immediately after intense return strokes of lightning discharges were first observed from the Space Shuttle in 1991 [15]. These phenomena are referred to as Elves that is an abbreviation for emission of light and VLF perturbations due to EMP sources [16]. Generally, the Elves first arise at  90 km altitude and then expand over 300
700 km laterally and 10
20 km in thickness for an extremely short time of less than 0.1 ms [15,16,104,105]. Figure 11.24 illustrates an example of Elves above a powerful thunderstorm in the Czech Republic taken from the ground by a low-light video camera by Martin Popek. It is generally accepted that the Elves are the visible manifestation of the ionospheric response to a strong electromagnetic pulse (EMP) radiated by the CG discharge current of either polarity (e.g., [104–106]). Elves are produced as a result of heating and secondary ionization of the lower ionosphere by the EMP from a lightning discharge [105,107]. The Elves predominantly radiate in red color basically due to the nitrogen fluorescence resulting from the molecules excitation by

Figure 11.24 Elves above a powerful thunderstorm in the Czech Republic captured by Martin Popek at night on April 2, 2017. This image was taken from spaceweathergallery.com

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the EMP. The short duration of this effect is basically due to a short lifetime of excited states of molecules in the ionosphere. The electromagnetic radiation pattern of a vertical lightning discharge is partly similar to that of a vertical dipole antenna over a conducting ground. Due to that, the lightning radiation intensity increases with increasing zenith angle. This property of the radiation pattern can explain the fact that most observed elves have a “doughnut” shape. Elves are much more common than other types of TLEs. Analysis of data gathered by the ISUAL instrument aboard the FORMOSAT-2 satellite has shown that the global occurrence rate of the Elves can be estimated as 3.23 events per minute while about 90% of elves happened over sea [86,108]. The data of the JEM-GLIMS mission (Global LIghtning and sprite MeaSurements at Japanese Experimental Module of ISS) gathered for the three-year observation period have shown that about 6.1% of lightning events are accompanied by Elves appearance [109]. The unusual Elve, which exhibits a distinct striped structure unlike the vast majority of symmetric, torus-shaped “classic” Elves were first reported by Yue and Lyons [110]. Figure 11.25 shows the event called a “tiger Elve”, which illustrates the striations in the Elves luminosity observed using a high-speed camera system near Fort Collins, Colorado on June 12, 2013 [110]. Simultaneous and independent observations by a co-located color near infrared camera revealed a pattern of internal gravity wave (IGW) in the airglow at the OH layer located at height of 85 km. Since the banded structure in the Elve and the IGW roughly coincided in space, this suggested that the

Figure 11.25 A distinctly striated Elve was observed using an intensified high-speed camera system at the Yucca Ridge Field Station near Fort Collins, Colorado on June 12, 2013. Taken from Yue and Lyons [110]

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“tiger Elve” structure is due to the Elve interaction with a strong IGW propagating at lower ionosphere altitudes. This effect can be explained by the fact that the ionization rate in the D-region ionosphere is inversely proportional to the air density, which is modulated by the IGWs. Numerical simulation of the lightning EMP interaction with the ionospheric plasma has shown that the observable Elves striations at altitudes near 85 km can be generated by the IGW with a plasma density perturbation of as low as 5% [111]. So, a strong lightning discharge followed by Elves can make it visible the IGW in the ionosphere. About 4% of Elves events produced by strong lightning EMP are accompanied by the so-called Elves doublets; that is, a pair of Elves occurred with a very small time delay on the order of 100 ms [112,113]. The first Elve in the doublet is due to EMP reflection from the ionosphere, whereas the second one is caused by triple reflection; that is twice from the ionosphere and once from the Earth. This conclusion is supported by the facts that the time lag in the appearance of the second Elves depends on the altitude of the discharge, the altitude of the ionospheric reflection, and the distance to the camera.

11.5 Other transient atmospheric phenomena possibly related to lightning activity 11.5.1 Gnomes and Pixies In this section, we make a small excursion into the recently discovered optical phenomena, whose direct connection with lightning discharges has not yet been established, although these phenomena are most commonly observed over thunderclouds and active thunderstorms. So-called Gnomes manifest themselves as brief lightning-like channels, presumably white in color, propagating upward from the top of a large thundercloud’s anvil [18]. Typical Gnome’s lifetime is about tens milliseconds, their lateral size is about 150
200 m and they do not grow more than 1 km above the cloud top at the speed 104 m/s. Gnomes are partly similar to BSs although they are brighter and much more compact in shape than the BSs. A very small, brief spot of light sometimes appearing at the overshoot dome similar to those produced the Gnomes have been referred to as Pixies [18]. These mini flashes of white color have only 100 m in size and last for about 10 ms. It also appears that the Gnomes and Pixies are neither temporally nor spatially associated with specific CG and IC lightning flashes and occur independently of the lightning appearance.

11.5.2 Transient atmospheric events One more type of faint optical flash in the atmosphere near thundercloud, which is presumably different from known TLEs, has been recently observed onboard loworbit satellites [114]. The optical measurements of this transient atmospheric event (TAEs) were performed in the UV (300
400 nm), red, and infrared (>610 nm) ranges. The majority of these flashes were detected in the equatorial region in

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cloudy areas although the most powerful flashes occurred more frequently over the oceans at higher latitudes. The total optical energy emitted by TAEs is estimated to be about 25
50 kJ [115], which is much less than the average optical emission energy of the sprites (300 kJ, [116]) and lighting (1 MJ, [29]). Typical TAE may contain one burst of light or a series of multiple bursts. The individual burst of light has a duration of about 0.1 ms or even shorter while the multiple bursts can last for tens milliseconds. Conventionally, this kind of faint flashes can be split into to two types: “luminous” and “dim” transients depending on the number, Nph , of emitted photons [117]. The “dim” transients is a portion of the short-term TAE with Nph < 1021 photons while the “luminous” transients are more intense event with Nph > 1023 photons and longer duration. It is generally believed that the optical emission spectra of TAEs are due to the fluorescence of the atmospheric nitrogen molecules, analogously to the emission spectra of sprites and other mesospheric flashes, which are mainly composed of the emission spectral lines of nitrogen molecules. Although the mechanism for the fluorescence excitation seems to be distinct from that of TLEs. Analysis of the “luminous” transients has shown that the photons number emitted in the UV range was approximately three times greater than that of the red-IR range [114]. This property is typical for the emission spectra of nitrogen molecules located at high altitudes greater than 50 km [118]. Thus, it can be expected that the sources of the “luminous” TAEs are located in the mesosphere. It was hypothesized that the molecule fluorescence can be excited by electron flows with energies on the order of tens of eV [19]. However, the reason for the appearance of such electron flows in the atmosphere is a puzzle. Another model explaining the TAE’s origin is based on assumption that large-scale areas with a low space charge density and a low electric breakdown threshold can occur in the mesosphere thereby producing electric discharges between the charged areas [115]. One possible cause for the existence of such charged areas at 50
70 km altitudes is the abnormally low air conductivity due to the presence of suspended dust particles [119]. The decrease in air conductivity is caused by free electron and ion attachment to both the dust particles and positive ion clusters, which build up from nitrogen ions N þ due to their hydration with water molecules. In the model, the TAEs parameters are best suited to the onboard observations as the charged areas are situated at altitudes of 60
70 km. The best-fit parameters are so that the charged areas of 10
15 km in size have to contain a total charge of about 1 C. In such a case the theory predicts that the optical energy radiated during a single electric discharge between the charged areas can be on the order of tens kJ while the discharge duration is about several milliseconds [115]. Despite these theoretical estimates being compatible with the observations much remains to be done in this area.

11.5.3 Terrestrial gamma-ray flashes Another interesting phenomenon is the brief bursts of upward-directed X-ray and gamma-ray radiation detected on board low-orbit satellites near active thunderstorm. This phenomenon termed terrestrial gamma-ray flashes (TGFs) was first observed in 1994 by the Burst and Transient Source Experiment (BATSE) on board

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the Compton Gamma-Ray Observatory [120]. These studies were continued by other space missions, for example, Reuven Ramaty High-Energy Solar Spectroscopy Imager (RHESSI) spacecraft designed to detect X-rays and gammarays [121] and Atmosphere-Space Interactions Monitor (ASIM) mounted at the International Space Station, which is orbiting at about 400 km altitude [122–124]. The ASIM payload contains instruments for TGF and optical measurements of lightning flashes; that is, the Modular X- and Gamma ray Sensor (MXGS) and the Modular Multi-spectral Imaging Array (MMIA). Typically the TGFs consist of several pulses of the gamma-ray radiation lasting of about 10
100 ms. However, this is much shorter than duration of typical gamma-ray burst incident on the Earth from the space. The photon energy spectrum of the TGF extends from 25 keV to 30
40 MeV [121,125,126]. The total energy of the radiated photons is estimated to be 1
102 kJ. The mean fluence of an individual TGF flash is about 0:1
1 photon/cm2 which corresponds to the total photon number 1017
1019 photons radiated per flash [127,128]. The presence of high-energy photons implies that the main source of the TGF photons is the bremsstrahlung of relativistic electrons in the atmosphere. It is now commonly accepted that the production and acceleration of the energetic electrons in the atmosphere at time scales of several microseconds is due to the development of relativistic runaway electron avalanche (RREA) in a strong electric field [129,130]. To activate the RREA process, the seed population of relativistic electrons is required. The energetic particles including the high-energy electrons arise from the interaction of a cosmic ray shower with air molecules. These electrons can then be accelerated to relativistic energies in a strong large-scale QE field produced by electric storm charges [131–133]. It has been also proposed that the seed electrons can reach the relativistic energy in the strong inhomogeneous electric field in the vicinity of the lightning leader channel and its tip ([134–142]). To gain a better understanding of the physical mechanism of TGF generation, it is necessary to establish the TGF source location in the atmosphere. It is generally believed that the most likely TGF source is a powerful IC lightning discharge or, more exactly, an upward-propagating IC lightning leader, which carries a negative charge [143–147]. Cummer et al. [148] and Lyu et al. [149] have reported a few TGF events caused by strong current pulses with a current amplitude greater than 200 kA. The altitudes of the IC lightning leader pulses associated with the TGFs can be determined from VLF/LF electric field measurements. Using TGF events detected by the instruments of Gamma-ray Burst Monitor (GBM) on the Fermi satellite, Cummer et al. [143] showed that some of TGFs were produced a few milliseconds after the onset of radio signals produced by the ascending IC leader. These TGFs occurred after the leader had extended 1:5
2 km from its initiation point at an altitude of about 8
11 km. The simultaneous TGF and optical observations of lightning flashes by ASIM satellite favor the assumption that the TGFs can occur during the initial phase of IC lightning. Analysis of these data has shown that the onset time of TGFs frequently preceded the onset of the lightning optical emission by 250
400 ms [123]. This implies that the majority of TGFs start to develop during upward propagation of the

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leader just before a large current pulse heats up the leader channel and emits a main optical pulse. In some instances, the TGF-produced lightning current had such a high amplitude, a prerequisite for Elves, that both TGF and Elves were observed simultaneously [150]. Recently Belz et al. [151] have reported the first high-resolution ground-based observations of downward-directed TGFs, which were detected by the large-area Telescope Array cosmic ray observatory in a proximity (3
4 km) to the lightning flash. The TGFs consisting 5
10 ms duration bursts of gamma rays were observed during strong initial breakdown pulses in the first few milliseconds of negative cloud-to-ground and low-altitude IC flashes. The number of gamma photons produced by such TGF was estimated as high as 1012–1014; that is, several orders of magnitude less than satellite-detected photon number [152]. We cannot come into detail about this interesting phenomenon since the researches are still continuing. The interested reader is referred to a review by Kumar and Pooja [153] and references herein for details.

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Skeltved, A. B., N. Østgaard, N. Lehtinen, A. Mezentsev, and B. Carlson (2017), Constraints to do realistic modeling of the electric field ahead of the tip of a lightning leader, J. Geophys. Res. Atmos. 122, 8120–8134. https:// doi.org/10.1002/2016JD026206. Xu, W., S. Celestin, and V. P. Pasko (2012), Source altitudes of terrestrial gamma-ray flashes produced by lightning leaders, Geophys. Res. Lett. 39, L08801. https://doi.org/10.1029/2012GL051351. Cummer, S. A., F. Lyu, M. S. Briggs, G. Fitzpatrick, O. J. Roberts, and J. R. Dwyer (2015), The lightning leader altitude progression in terrestrial gamma-ray flashes, Geophys. Res. Lett. 42, 7792–7798. https://doi.org/ 10.1002/2015GL065228. Lu, G., R. J. Blakeslee, J. Li, et al. (2010), Lightning mapping observation of a terrestrial gamma ray flash, Geophys. Res. Lett. 37, L11806. doi:10.1029/2010GL043494. Mailyan, B. G., M. S. Briggs, E. S. Cramer, et al. (2016), The spectroscopy of individual terrestrial gamma-ray flashes: Constraining the source properties. J. Geophys. Res. Space Phys. 121(11), 11346–11363. doi:10.1002/ 2016JA022702. Shao, X., T. Hamlin, and D. Smith (2010), A closer examination of terrestrial gamma ray ash related lightning processes, J. Geophys. Res. Space Phys. 115, A00E30. doi:10.1029/2009JA014835. Stanley, M. A., X. M. Shao, D. M. Smith, et al. (2006), A link between terrestrial gamma-ray flashes and intracloud lightning discharges, Geophys. Res. Lett. 33, L06803. https://doi.org/10.1029/2005GL025537. Cummer, S. A., M. S. Briggs, J. R. Dwyer, et al. (2014). The source altitude, electric current, and intrinsic brightness of terrestrial gamma-ray flashes, Geophys. Res. Lett. 41(23), 8586–8593. doi:10.1002/2014GL062196. Lyu, F. S., S. A. Cummer, and L. McTague (2015), Insights into high peak current in cloud lightning events during thunderstorms, Geophys. Res. Lett. 42, 6836–6843. doi:10.1002/2015GL065047. Neubert, T., N. Østgaard, V. Reglero, et al. (2019b), A terrestrial gammaray flash and ionospheric ultraviolet emissions powered by lightning, Science, 367(6474), 183–186. doi:10.1126/science.aax3872. Belz, J. W., P. R. Krehbiel, J. Remington, et al. (2020), Observation of the origin of downward terrestrial gamma-ray flashes, J. Geophys. Res. Atmos. 125(23), e2019JD031940. doi:10.1029/2019jd031940. Surkov, V. V. and V. A. Pilipenko, Estimate of the source parameters of terrestrial gamma-ray flashes observed at low-Earth-orbit satellites, J. Atmos. Solar-Terr. Phys., 237, 105920, 2022, doi:10.1016/j.jastp.2022.105920. Kumar, P. M. and R. Pooja (2021), Terrestrial gamma-ray flashes and other energy atmospheric phenomena: An overview, Disaster Adv. 14(4), 107–125.

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Chapter 12

The effects of lightning on the ionosphere/ magnetosphere: whistlers and ionospheric Alfve´n resonator M. Hayakawa1,2 and Y. Hobara3,4

This chapter will present possible effects of atmospheric lightning on the upper atmosphere such as the ionosphere and magnetosphere composed of ionized plasmas. Though there exist several phenomena on the effect of lightning discharges onto the ionosphere/magnetosphere, we introduce only two major attractive topics: (1) lightning-induced whistlers in the ionosphere/magnetosphere, and (2) ionospheric Alfve´n resonator (IAR) in an altitude region between the lowest ionosphere and lower magnetosphere, where one likely candidate of its source is lightning discharges. The former is quite a well-known phenomenon, and whistlers are bursts of ELF/VLF waves produced by lightning discharges. Some part of VLF/ELF lightning energy penetrates through the ionosphere, propagates along the magnetic field line in the magnetosphere, and penetrates again through the ionosphere in the opposite hemisphere, followed by reception on the ground as a whistler. We show initially the phenomena of ground-based whistlers, their brief theoretical explanation, and their use in the diagnostics of ionospheric/magnetospheric electron density. Also, earlier satellite observations of nonducted whistlers are presented. Further, we will describe recent satellite observations of short-fractional hop whistlers and VLF/ELF electromagnetic waves, with special reference to their use in the study of global lightning activity. On the other hand, the latter phenomenon, IAR in the ULF/ELF band is a rather new subject as compared with whistler studies, and so we pay more emphasis on IAR in this chapter. IARs exhibit an interesting feature of fingerprint resonance structures on the dynamic spectra. This IAR is apparently considered to be a kind of resonance of Alfve´n waves in a region between the lowest ionosphere and lower magnetosphere, whose resonance frequencies (f = 1-10 Hz) are lower than the well-known Schumann resonances. We 1 Advanced Wireless and Communications Research Center, The University of Electro-Communications (UEC), Japan 2 Hayakawa Institute of Seismo Electromagnetics, Co., Ltd. (Hi-SEM), Japan 3 Graduate School of Informatics and Network Engineering, The University of Electro-Communications (UEC), Japan 4 UEC, Center for Space Science and Radio Engineering, Japan

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present our own statistical results on morphological characteristics of IARs (spectral resonance structures) at middle latitudes as your basis to understand the resonance structure of IARs. Then we will review the physical mechanisms, in other words, the energy source of IAR signatures seems to depend on latitude. Though there are considerable uncertainties in the physical modeling, we will suggest a plausible hypothesis as the origin of IARs at middle and low latitudes; the link to nearby lightning discharges. Lastly, a summary will follow.

12.1 Introduction The physics and effects of lightning discharges taking place in the atmosphere are extensively discussed in most chapters of this monograph, and hence this chapter will deal with the consequence of atmospheric lightning onto the upper atmosphere such as the ionospheric/magnetospheric plasma. There are various possible effects of lightning on the ionosphere/magnetosphere, but we restrict our attention to the following two interesting phenomena: (1) lightning-induced whistlers in Section 12.2 and (2) Section 12.3 concerned with ionospheric Alfve´n resonator (IAR) where one possible source is either distant or nearby lightning discharges. In Section 12.2, we will discuss the first part of whistlers. Section 12.2.1 provides you with the history and general description of whistlers. In early times whistlers are generally observed on ground-based stations, so they are defined by bursts of ELF (extremely low frequency, f < 3 kHz)/VLF (very low frequency, 3 kHz < f < 30 kHz) waves, originated in lightning discharges in the opposite hemisphere of the receiver. While the bulk of the energy from the causative lightning discharges propagates in the Earth-ionosphere waveguide as atmospherics (or sferics) [1–5], some part of its energy is known to penetrate through the upper ionosphere, propagate in the magnetosphere, probably along the magnetic field line (ducted propagation), and penetrate again through the ionosphere in the opposite hemisphere, to be received as a whistler. On the other hand, most whistlers observed on board rockets and satellites (either deep in the magnetosphere or in the ionosphere) are considered to be propagating in the nonducted mode, so these nonducted whistlers have never been detected on the ground. There have already been published several excellent books and reviews on the topic of whistlers [6–11]. Section 12.2.2 deals with the general theoretical background of plasma waves including whistler mode and also Alfve´n mode (this will be the topic of IAR in Section 12.3). Particularly in Section 12.2.2.4, we will explain the whistler dispersion (D) as the most useful parameter of ground-based whistlers. Section 12.2.3 is aimed at the use of whistlers as a diagnostic tool. Section 12.2.3.1 deals with the diagnosis of magnetospheric electron density profile with the use of ground-based whistler dispersions at different latitudes, and Section 12.2.3.2 will be concerned with early satellite observations of various kinds of nonducted whistlers. Section 12.2.3.3 will present recent LEO (low-Earth-orbit) satellite observations of short-fractional hop whistlers and VLF/ELF electromagnetic waves (both waveform and power spectra data) to be utilized to study the global lightning activity and lightning characteristics.

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In Section 12.3, we will consider IARs. As compared with whistler studies, this phenomenon of IAR in the ULF (ultra low frequency, f 0) and mass m which moves with velocity v. Only the motion of electrons can be considered because the mass of electrons is significantly lighter than that of ions (e.g. mass of protons (the lightest ion) is about 2,000 times heavier than that of electrons). The electrons are influenced by the wave electric and magnetic fields as well as the DC Earth’s magnetic field (B0) during the propagation of electromagnetic waves and the currents generated by the electrons re-generate electromagnetic waves. For electrons, the equation of motion in the Cartesian coordinate is given by m

dv ¼
eðE þ v  ðB0 þ BÞÞ dt

(12.1)

The electromagnetic electric and magnetic field vectors are E and B, and the ambient Earth’s magnetic field is B0 = B0az (az is the unit vector directed towards positive z direction) as in Figure 12.5. Here collisions between electrons and heavy neutral particles are ignored. We are also justified in neglecting the force on the electron from the time-varying magnetic field (v  B). The force due to the timevarying magnetic field is generally negligible in comparison with the force due to the electric field since the ratio of these forces is roughly v/c and v/c « 1 where c is the speed of light. By assuming the time dependence of e jwt ( j, w, and t are imaginary unit, angular frequency (w = 2p f, f frequency), and time respectively). The equation of motion of electrons will be written as follows. 2 3 2 3 2 3 uy Ec uc e eB0 4 (12.2)
uc 5 jw4 uy 5 ¼
4 Ey 5
m m uz Ez 0

z B0 k, vp

θ y

x

Figure 12.5 Coordinate system for the wave propagation. The Earth’s magnetic field B0 is in the z direction, and the wave propagation (k) is in the xz plane and makes an angle q with B0.

Whistlers and ionospheric Alfve´n resonator

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Then we obtain the three components of electron velocities driven by the wave electric field and ambient Earth’s magnetic field are, nc ¼


e jw Ec
wH Ey m wH 2
w2

ny ¼


e wH Ec þ jw Ey m wH 2
w2

nz ¼


e Ez m jw

(12.3)

where wH is the electron angular gyrofrequency given by wH (= 2pfH) = eB0/m, which is easily calculated by the simplified form fH(Hz) = 28B0(nT). This gyrofrequency means the angular frequency at which the electron makes circular orbits in a plane perpendicular to the Earth’s magnetic field. The current density is given by, J ¼
Nev

(12.4)

where N is the electron density. Each component of this current is shown by using (12.4). Jx ¼ ðNe2 =mÞðjw E x
wH Ey Þ=ðwH 2
w2 Þ Jy ¼ ðNe2 =mÞðwH Ex þjw Ey Þ=ðwH 2
w2 Þ  2  Jz ¼ Ne =m Ez =jw

(12.5)

Then the conductivity tensor is given by, sxx ¼ ðNe2 =mÞjw =ðwH 2
w2 Þ ¼ syy sxy ¼
ðNe2 =mÞwH =ðwH 2
w2 Þ ¼
syx

(12.6)

szz ¼ ðNe2 =mÞ=jw sxz ¼ szx ¼ syz ¼ szy ¼ 0 and the dielectric tensor is also described by using ¼ e0 hIi
j=w; ! w2p exx ¼ eyy ¼ e0 1 þ wH 2
w2 exy ¼
eyx ¼ je0  ezz ¼ e0 1


wp 2 wH wðwH 2
w2 Þ  2

wp w2

exz ¼ ezx ¼ eyz ¼ ezy ¼ 0

(12.7)

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where hIi is a unit tensor and the electron plasma frequency is defined by w2p ¼ ð2pf p Þ2 Þ ¼ N e2 =me0 (e0, dielectric constant of free space) ðf p ðkHzÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 N ðcm
3 ÞÞ: We consider now Maxwell’s equations for the case of time-harmonic electromagnetic fields that vary in space as, exp½jðwt
k  rÞ

(12.9)

Here we have used the time dependence assumed previously and now include a spatial variation specified by the term of –j k  r, where k is the propagation vector (or wave normal direction) (its direction is the direction of wave propagation and its magnitude k equals 2p divided by the wavelength) and r is the position vector. Figure 12.5 is the coordinate system of our propagation study, in which the static Earth’s magnetic field B0 is already assumed to be in the z-axis and the wave propagation (k) is assumed to be in the x-z plane and to make an angle (q) with B0. For an electromagnetic wave with temporal and spatial variation given by (12.9), we can write Maxwell’s equations as (m0, magnetic permeability of free space)
jk  E ¼
jwmH

(12.10)


jk  H ¼ jwD

(12.11)

Combining the above two equations, yields to, k  ðk  EÞ þ w2 m0 < e > E ¼ 0

(12.12)

This equation is the so-called wave equation of electromagnetic waves in the anisotropic plasma with the presence of B0. Equation (12.12) is decomposed into (x, y, z) components as follows. 2 2 2 32 3 Ec
k 2 sin q cos q k cos q
w2 m0 exx
w2 m0 exy 2 2 4 w2 m exy 5 4 Ey 5 ¼ 0 k
w m e 0 0 0 xx Ez 0 k 2 sin2 q
w2 m0 ezz
k 2 sin q cos q (12.13) The solution to any equation of this form (that is, in order to have a non-trial solution (E)) is found by setting to zero the determinant of the matrix in (12.13). This result is inserted in (12.13), which can be solved for the wave electric field. Once both the electric field and the propagation direction are known, the wave magnetic field can be calculated from (12.10). The phase velocity vp and group velocity vg can be expressed by a vector representation as, w w ¼ k k k2 @w vg ¼ @k

vp ¼

(12.14)

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The phase velocity vp is defined as the direction normal to the wave front, so that it is easily understood as being parallel to k. On the other hand, the group velocity vg is defined as the direction of electromagnetic wave energy (ray direction) and the most interesting peculiarity of the anisotropic plasma is that vg is not parallel to vp for the general oblique propagation (consult Helliwell [9], Park [10], and Hayakawa [8] for further details). Only when q = 0, the direction of vg is parallel to vp and the value of vg is easily given by vg ¼ @ðnwÞ=@w (n: refractive index).

12.2.2.3 Special cases of longitudinal and perpendicular propagation and general oblique propagation By using (12.13), we will study first the special cases of propagation; that is, (1) longitudinal propagation (q = 0) and (2) transverse propagation (q = p/2) to facilitate better understanding of wave propagation in an anisotropic plasma. (a) Longitudinal propagation The longitudinal propagation means that the propagation vector (k) is parallel to the Earth’s magnetic field (B0); that is, q = 0. Substituting q = 0 in (12.13) yields the following wave equation.  2   Ec k
w2 m0 exx
w2 m0 exy ¼0 (12.15) w2 m0 exy k 2
w2 m0 exx Ey ðezz ÞEz ¼ 0

(12.16)

Since (12.16) represents only the plasma oscillation (w = wp), we will not discuss this oscillation further. The presence of non-trivial solution of Ex and Ey in (12.15) requires the determinant of the coefficients in (12.15) equal to zero, which leads to the following. 2



(12.17) k
w2 m0 ðexx
jexy Þ k 2
w2 m0 ðexx þ jexy Þ ¼ 0 Equation (12.17) is found to yield the following two characteristic mode waves. (i) Right-hand polarized wave Taking the first term on the left-hand side of (12.17) being zero, the following dispersion relation is obtained. wp 2 k 2 c2 ¼R (12.18) n2 ¼ 2 ¼ 1 þ w wðwH
wÞ where n is called refractive index, and this wave is right-handed circularly polarized as shown below. The rotation of wave electric field is exactly the same as the gyration of electrons in the Earth’s magnetic field. j

Ex ¼
1 Ey

(12.19)

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(ii)

This wave mode is so-called “whistler” mode when the wave frequency is below wH. Left-handed polarized wave

Similarly taking the second term on the left-hand side of (12.17) being zero, the dispersion relation for this wave is, n2 ¼

wp 2 k 2 c2 ¼L ¼1þ 2 w wðwH þ wÞ

(12.20)

and this wave is found to be left-handed circularly polarized as is evidenced by the following relation. j

Ex ¼1 Ey

(12.21)

(b) Transverse propagation We now consider the transverse propagation characterized by q = p/2; that is, propagation direction (k) is perpendicular to B0. In this case, the wave equation (12.14) becomes,    Ec
w2 m0 exx
w2 m0 exy ¼0 (12.22) w2 m0 exy k 2
w2 m0 exx Ey ðk 2
w2 m0 ezz ÞEz ¼ 0

(12.23)

That is, there exist two mode waves as follows. (i) Ordinary wave (O wave) (k?B0 and EkB0) The wave mode obtained by (12.23) is characterized by its electric field being parallel to B0 (Ez 6¼ 0) with Ex = Ey = 0. By setting the first term of the lefthand side of (12.23) to zero, we obtain the following dispersion relation for the ordinary mode. n2 ¼ 1


wp 2 ¼P w2

(12.24)

This wave is linearly polarized and (12.24) is identical to the dispersion relation of the plasma oscillation without any effect of the Earth’s magnetic field (12.16). This is the reason why we use the terminology of “ordinary” wave. (ii) Extraordinary wave (X wave) (k ?B0 and E ?B0 ) In order to have non-trivial solutions (Ex 6¼ 0, Ey 6¼ 0) in (12.22), the determinant of the matrix on the left-hand side of (12.22) is set to zero, which leads to the following dispersion relation. n2 ¼

exx 2 þ exy 2 2RL ¼ exx RþL

(12.25)

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487

The wave polarization of this extraordinary wave is estimated from (12.22) as, exy wp 2 wH Ex ¼
¼
j Ey exx wðwp 2 þ wH 2
w2 Þ

(12.26)

so that the wave electric field is elliptically polarized in the xy plane. (c) Cutoffs and resonances Cutoff is defined as a frequency at which the value k2 goes to zero (or n2 ! 0), so these cutoff frequencies can be obtained by setting the refractive index n2 ! 0 (or k2 ! 0). Hence we expect that the wavelength and phase velocity becomes infinity. When the wave propagates through the plasma with a spatial gradient of refractive index including n2 = 0, the wave is reflected at the point of cutoff. The cutoff frequencies can be easily estimated by setting n2 ! 0 in the dispersion relation for each mode. The cutoff frequencies for the longitudinal propagation can be obtained from (12.18) and (12.20): rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wH wH 2 þ wp 2 þ (12.27) wR ¼ 2 4 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wH wH 2 þ wp 2 þ (12.28) wL ¼
2 4 The cutoff frequencies for transverse propagation are obtained from (12.24) and (12.25). For the extraordinary wave, the cutoff frequencies occur when n2! 0 (or R = 0 or L = 0), so that they are identical to those for the longitudinal propagation, wR and wL. For the ordinary wave, the cutoff frequency is expected from n2 = P = 0; that is, wp. Next, we move on to the explanation of resonances. Resonance is defined at a frequency at which n2 = ? (k2 ! ?). When the wave approaches a resonance condition, k2 goes to infinity so that the phase velocity becomes zero, then the wave is mainly dissipated due to the significant interaction with plasma. The resonance frequencies for the longitudinal propagation are obtained when the denominator of (12.18) and (12.20) vanishes. That is, the righthanded polarized (R) mode exhibits a resonance at w = wH, and at this frequency, the wave electric field continuously accelerates electrons because the wave field is in phase with the gyro-motion of electrons. This resonance is called electron cyclotron resonance. Also, the left-hand polarized (L) mode is found to have a resonance at w = wHi (ion gyrofrequency). Though in our former equations we have only considered the motion of electrons, the motion of ions becomes important when the frequency becomes smaller than wHi and it is very easy for us to take into account the effect of ions in the dispersion relation. The resonance for the transverse propagation can be obtained exactly in the same way as above. The ordinary wave has no resonance because the denominator of

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Lightning electromagnetics: Volume 2 (12.24) may not vanish except w = 0. On the other hand, the extraordinary wave (12.25) exhibits a resonance at a particular frequency of wUHR (upper hybrid resonance frequency), so that the wave electromagnetic energy is converted to the upper hybrid oscillation. A similar hybrid resonance is observed as well at wLHR (lower hybrid resonance frequency).

(d) Oblique propagation We have discussed two representative cases of propagation, that is, longitudinal (q = 0) and transverse (q = p/2), which gives us a lot of useful ideas on the propagation characteristics of electromagnetic waves in the anisotropic plasma. Here, the dispersion relation of the waves propagating obliquely to B0 is obtained by solving the following equation by setting the determinant of the coefficients of E in (12.13) to zero. An4
Bn2 þ C ¼ 0

(12.30)

A ¼ exx sin q þ ezz cos q 2

2

B ¼ ðexx 2 þ exy 2 Þsin2 q þ exx ezz ð1 þ cos2 qÞ

(12.31)

2

C ¼ ezz ðexx þ ezy 2 Þ A ¼ Ssin2 q þ Pcos2 q B ¼ RLsin2 q þ PSð1 þ cos2 qÞ C ¼ PRL 1 1 S ¼ ðR þ LÞ; D ¼ ðR
LÞ 2 2

(12.32)

(12.33)

Equation (12.30) gives us the presence of two characteristic modes in the plasma at any frequency as, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B  B2
4AC 2 n ¼ (12.34) 2A The dispersion relations (w–k diagram) of different modes in the plasma are illustrated in Figure 12.6 for different q values. In the figure, the longitudinal propagation (q = 0) is indicated by k, while the transverse propagation (q = p/2), is by ?. Shaded regions in Figure 12.6 correspond to the oblique propagation. The plasma condition of wp > wH (satisfied nearly almost in all regions in the magnetosphere) is assumed in Figure 12.6, and Figure 12.6 indicates the presence of three frequency branches as follows. (i)

Free space waves (w > wp, wH) This plasma wave branch is called quasi-free-space mode. There are the extraordinary mode (R–X and Z mode) and ordinary mode (L–O), but this branch is not so interesting in our chapter, so that we do not go into the details of this branch. Please look at the books by Parks [30] and Kivelson and Russell [24] for further details and the corresponding wave phenomena in the magnetosphere including Z mode in the magnetosphere.

Whistlers and ionospheric Alfve´n resonator ω

489

n2 = R

║

n 2 = 2RL/ (R + L) n2 = L

ωR ωPe

┴ n2 = P ┴

n 2 = 2RL/ (R + L) ║

ωUHR

┴

┴

║ ωL

whistler–mode ωHe n2 = R ωLHR ┴ n 2 = 2RL/ (R + L)

║

ωHi

║

n2 = L k

Figure 12.6 Dispersion curve in the form of w-k diagram for an electron-ion plasma for the case of wp = 2wH. k and ? mean the longitudinal (q = 0) and transverse propagation (q = p/2), and the dispersion curve for oblique propagation must lay in the shaded area. wR and wL are the cutoff frequencies of right-and left-handed polarized mode waves. While wUHR and wLHR are the lower and upper hybrid resonance frequencies. n2 = R and n2 = L for q = 0 indicate the right- and left-handed circularly polarized waves, respectively. (ii)

(iii)

Whistler-mode branch (wHi < w < wH) This is the so-called “whistler-mode” branch, which is our main interest for this chapter. In this frequency range we have only this whistler mode because another L wave is evanescent (i.e., non-propagating) (n2 = L < 0 in this frequency). The whistler mode wave has right-handed polarization and tends to propagate along B0. This will be discussed later in some more detail. Alfve´n wave branch (w < wHi) This frequency range is the Alfve´n wave branch, and there exist two characteristic modes (i.e., Alfve´n wave and modified Alfve´n wave). Alfve´n wave is considered to be the counterpart of whistler-mode in the sense that it is lefthanded polarized and its ray direction tends to be very parallel to B0. Modified Alfve´n wave is considered to be the lower frequency part of the whistler branch. The Alfve´n wave is responsible for IARs to be discussed in the next section.

12.2.2.4 Whistler propagation and dispersion Many important properties of whistlers can be explained by using the theory outlined in Sections 12.2.2.2. and 12.2.2.3. To understand how the electromagnetic

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waves from lightning propagate into the ionosphere, we first examine the application of Snell’s law to the boundary between the free space and ionosphere, sin qi ¼ ni  sin qt

(12.34)

where ni is the refractive index of the ionosphere and that of free space is taken as unity. And qi is the incidence angle measured from the vertical downward, and qt is the transmitted angle in the ionosphere measured from the vertical upward. We will show that ni »1 and hence qt in the ionosphere must be nearly zero; that is, essentially vertical for any incidence angle (qi) of the wave in free space impinging on the boundary. Using (12.18), we can have the whistler-mode refractive index to good approximation as, ni 2 ¼

wp 2 wðwH
wÞ

(12.35)

This refractive index exceeds much that of free space. For example, at the ionospheric F region where the electron density is near 106 cm
3, ni is close to 100 at frequencies near 5 kHz. After the whistler enters the magnetosphere, propagates along a field-aligned duct, and reaches the ionosphere in the conjugate hemisphere, there happens the reverse interface situation. Snell’s law can also be involved to show that if the wave’s normal direction is inside a relatively narrow angle around the vertical (this is called transmission cone) [8,9], then it can penetrate through the ionosphere and hence be detected on the ground. When the wave is ducted and also we consider the high to middle latitudes where the Earth’s magnetic field is close to the vertical, this condition is easily satisfied. However, if not (e.g., at low latitudes), then the wave will be reflected (totally reflected) back into the magnetosphere, sometimes leading to the generation of echo train whistlers. On the assumption that the whistler wave is propagating along a magnetic field line, the time of propagation T(w) from the source to the receiver, can be computed from the knowledge of group velocity of the wave. ð ds (12.36) T ðwÞ ¼ path vg In the case of q = 0, vg is easily estimated by vg ¼ @ ðnwÞ=ð@wÞ and when w « wH (so-called low-frequency approximation in Figure 12.2(a)), (12.36) becomes, ð wp 1 T ðwÞ ¼ (12.37) pffiffiffiffiffiffiffiffiffiffi ds 2c path wwH We can define the dispersion D as, ð wp 1 D ¼ T ðf Þf 1=2 ¼ pffiffiffiffiffiffi pffiffiffiffiffiffiffi ds wH 2c 2p

(12.38)

which is independent of the wave frequency f and which can be determined only by the plasma conditions of the magnetosphere. Equation (12.38) suggests that the

Whistlers and ionospheric Alfve´n resonator

491

greatest contribution comes from the region around the apex of the propagation path because of the smallest wH there. This equation is known as the Eckersley’s dispersion law and is valid at frequencies far below the nose frequency. The fact that the dispersion D is independent of frequency is illustrated in Figure 12.3(c). By utilizing the more general (12.18) for a given path, the integral in (12.36) enables us to deduce a nose frequency fn for which the propagation time is a minimum. For a homogeneous plasma, the nose frequency fn = fH/4. As seen from (12.38) the ionosphere does not contribute much to the observed time delays; the most important contribution is expected at the highest part of the path (equatorial plane) of the magnetosphere.

12.2.3 Use of whistlers as a diagnostic tool of the ionosphere/magnetosphere 12.2.3.1 Diagnosis of magnetospheric electron density profile The ground-based observation of lightning-generated whistlers at different latitudes enables us to obtain the latitudinal distribution of D, together with the extensive use of nose whistlers at higher latitudes. Then, we can estimate the general electron density profile in the equatorial plane of the magnetosphere as in Figure 12.7. The full line refers to the profile obtained with ground-based whistlers, in which we can find the presence of plasmapause with a sharp density drop [10,33,34]. Some white circles in Figure 12.7 refer to the observation by satellites, to be compared with

60˚

neq (el – cm–3)

104

56˚

(56˚) (40˚)

59˚ (28˚)

61˚

63˚

(23˚) (20˚)

103

102 Gringauz et al. 101

1

3 2 4 Geocentric distance (earth radii)

5

Figure 12.7 Electron density profile in the equatorial plane of the magnetosphere (full line) with the use of whistlers. White circles indicate the electron density observed by satellites. After Darrouzet et al. [36]

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ground results. Furthermore, we can monitor the dynamic behavior (spatial and temporal variations) of the magnetospheric electron density profile including the plasmapause structure [25,33,35,36].

12.2.3.2

Satellite observations of nonducted whistlers in the magnetosphere

While ground-based whistlers are known to be trapped in field-aligned ducts (ducted propagation), most of the whistlers observed aboard spacecraft are propagating in the nonducted mode. Various kinds of those nonducted whistlers on board satellites have been observed in early times and please refer to former reviews for further details [8,10,19]. Here we will list only a few examples and we will make a brief description of each. ●

●

Magnetospherically reflected (MR) whistlers This MR whistler is observed deep inside the magnetosphere, which is composed of multiple discrete components (whistlers), all originated from a single lightning discharge [37]. A lightning discharge illuminates the ionosphere over a wide latitude range with the reasonable assumption of a nearly vertical wave normal angle of each component as the wave enters the ionosphere. At a given frequency there are usually a number of nonducted paths (as in Figure 12.1(d)) from the lightning discharge to the satellite. We have to emphasize here that the effect of ions is playing the main role in this nonducted propagation [38], which makes it possible for the ray to propagate across the magnetic field line. The spectrograms of those MR whistlers are often utilized to deduce the electron density profile of the magnetosphere and also the presence of fieldaligned ducts. Ion cyclotron whistlers Another typical effect of ions appears in the form of ion cyclotron whistlers [39]. This whistler appears for a short-fractional whistler, or 0+ whistler in the view concept of Figure 12.1(b), and propagation of the ion cyclotron mode is possible only if “more than two species” are present. When there are many species of ions, the dispersion relation (Figure 12.6) as in Section 12.2.2.3 (d) becomes rather complicated, so that we direct the reader to consult the original paper by Gurnett et al. [39] or a review by Hayakawa [8] for further detailed exposition on the generation mechanism of ion whistlers. These ion-cyclotron whistlers are used to study the presence of different ion species in the ionosphere and their dynamic behavior.

12.2.3.3

Satellite observations of short-fractional hop (0+) whistlers and their link to global distribution of lightning activity and lightning characteristics

Prior to going to the LEO satellite observations, we will review the earlier whistler studies, primarily based on ground-based observations. Many problems remained to be resolved at that time: (1) overall propagation feature of whistlers from the

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source to the receiver, and (2) duct characteristics and duct formation mechanism. The first topic was to discuss the one-to-one correspondence between a causative lightning discharge and its whistler and then the detailed propagation mechanism in the magnetosphere. This one-to-one correspondence was attempted by many scientists, especially at low latitudes [40–42]. In these studies, they made full use of the direction finding of locating causative sferics [43] and also the direction finding to pinpoint the ionospheric exit regions of short whistlers [44–47]. As for the second topic, characteristics of field-aligned ducts (spatial scale, enhancement factor, life time, etc.) are of essential interest in whistler propagation, and also the physical mechanism of duct formation has yet to be established, though there have been proposed a few hypotheses (e.g., [48,49]). The second topic is still poorly understood even now, but the first topic has been recently studied extensively because of the abundance of LEO satellite observations, which will be presented below. We first present a few earlier LEO satellite observations related to whistler mode waves (mainly 0+ whistlers [50]) in the ionosphere [51,52]. Hayakawa [51] measured the wave intensities at a few ELF and VLF point frequencies on board the Ariel 4 satellite. High values of the observed mean/minimum and peak/mean intensity ratios in each sampling interval allowed him to infer that the measured ELF/VLF radio noises at low latitudes are impulsive and so were considered to be due to lightning discharges. His world distribution suggested that VLF noises are localized around three lightning chimneys of the world, and also a clear day/night asymmetry was compatible with the day/night difference of transionospheric absorption with full-wave computations. We move on to recent satellite observations. Chum et al. [53] investigated the correspondence between 0+ whistlers on the DEMETER and Magion-5 satellites and lightning discharges detected by the ground-based European Lightning Detection Network (EUCLID). And they demonstrated that the maximum whistler amplitude at the satellite depends primarily on the proximity to the source lightning relative to the magnetic footpoint of the satellite, and the area in the ionosphere through which the electromagnetic energy induced by a lightning discharge enters the ionosphere, is up to several thousands of kilometers wide. Then, using the same satellite and EUCLID, Fiser et al. [54] found pairs of causative lightning and corresponding whistlers, and processing data from 200 paths over Europe it is found that the mean amplitude decreases monotonically with the horizontal distance up to 1,000 km from the lightning discharge, and nighttime amplitude is about three times the daytime value. Using a similar detection technique to Fiser et al. [54] adopted to the burst mode data on DEMETER, Compston [55] studied the global distribution of lightning activity. Then, using VLF wideband signals on board the DEMETER, Ferencz et al. [56] have found a specific signal structure of numerous 0+ whistlers and they termed them as “Spiky” whistlers as shown in Figure 12.8. These signals appear to be composed of a conventional 0+ whistler combined with the compound mode patterns of tweek sferics [57–60]. This finding is not so surprising, because they have detected just the leakage of tweek sferics into the ionosphere, propagating in whistler mode up to the satellite. A more general effect fundamentally based on the

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frequency (kHz)

15

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Figure 12.8 An example of “spiky” whistlers observed in burst mode VLF recording on board DEMETER (satellite position 34.7 N, 5.0 E, altitude 702 km, L = 1.49) (after Ferencz et al. [56]) same principle, had already been observed on the ground-based short whistlers [40,42,61]. Shimakura et al. [61] found, for the first time, such tweek mode patterns superimposed on low-latitude short (1 hop) whistlers as observed simultaneously at two stations of Sakushima and Kagoshima in Japan. The simultaneous observations of whistlers and causative lightning discharges enabled them to conclude that the causative sferics are located exactly just below the duct entrance in the southern hemisphere and then additional tweek traces might be due to the subionospheric propagation from the far ionospheric exit point (about 3,000 km east of the stations) in the northern hemisphere to the VLF stations. Next Parrot et al. [62] have studied the occurrence rate of 0+ whistlers with the automatic detection with the use of neural network on board the DEMETER [63], and Figure 12.9 illustrates an example of their world distribution of the occurrence rate of 0+ whistlers during nighttime. They have shown that the whistler rate calculated as a function of longitude varies between 1 and 6 s
1 during nighttime and 0.5–0.7 s
1 at the day. Also, whistler rate is found to be anti-correlated with the F10.7 cm solar flux, being consistent with earlier ground-based results at low latitudes [64,65], though there have been some publications on the positive correlation between the lightning activity and solar activity [66]. The decreased lightning activity at the solar minimum is largely counterbalanced by the increase in whistler rates in the ionosphere due to the decrease in ionospheric absorption. Finally, the measurements of electromagnetic wave intensity unlike the use of wideband spectra before, have recently been extensively utilized to study lightning activity. Using DEMETER data, Nemec et al. [67] studied the relationship between median intensities of electromagnetic emissions in the VLF range and lightning. The analysis of 3.5-year data demonstrated that electromagnetic emissions may be due to lightning activity changes. The effect of lightning activity is most

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Figure 12.9 Global distribution of 0+ whistlers with small dispersion (D less than 3.2 s1/2) detected on DEMETER during the period of 2005 and 2010 for nighttime condition regardless of geomagnetic activity (Kp value). After Parrot et al. [62]

pronounced at f 2 kHz, forming a continuous band and being strongest at night. Colman and Starks [68] constructed a climatology of VLF wave intensity from lightning in the plasmasphere. Using OTD/LIS (optical transient detector/lightning imaging sensor) lightning data (1995–2005) and assuming a linear relationship between optical flash rate and VLF power flux and that VLF amplitude drops as 1/distance, the proximity for VLF power is calculated. These values are mapped along the magnetic field line in order to compare them with the electric field spectral densities observed on the DEMETER (2005–2009). Good overall agreement was found with previous observations. Using data from DEMETER (period of 2004–2010) and Van Allen probes (2012–2016), Zahlava et al. [69] have studied the longitudinal dependence of the intensity of whistler mode waves. A significant longitudinal dependence is observed in the nighttime in the frequency range from 40 Hz to 2 kHz, but almost no dependence at the day. The observed results are compared with the previous OTD/LIS data by Christian et al. [70], and it is found that lightning-generated electromagnetic waves may be responsible for the observed electromagnetic effect. Electric and magnetic field amplitudes of VLF lightning-generated waves have been studied by Ripoll et al. [71] on the basis of 11.5 years of observation by the Van Allen probes. Their satellite data have been compared with those by the ground-based World Wide Lightning Location Network (WWLLN). Mean amplitudes are found to be low compared with whistler mode waves (1  1.6 pT, 19  59 mV/m). They have found excellent correlations between WWLLN-based power and wave amplitudes in space at various longitudes and strong dayside ionospheric damping.

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Finally, Jacobson et al. [72] have suggested a new direction of direction finding and polarization measurement with the data from C/NOFS (Communications/Navigation outage Forecasting System) satellite observation of vector electric field in the band of 0–16 kHz during a period of 2008–2014. Based on the statistics of these burst-mode recordings, of which 6,890 records meet stringent quality-control criteria, it is found that the wave normal makes an angle of 40–90 degrees with the magnetic field, indicating nonducted propagation, which seems to be consistent with the common view. Work will continue in the future on the use of satellite observations to study the global distribution of lightning activity (longitudinal dependence, diurnal, seasonal, and solar cycle variations, etc.) and the characteristics of lightning discharges. The information on ELF/VLF wave intensities would be of essential importance in the study of wave-particle interactions (e.g., [73,74]), particle precipitation (e.g., [75]), mechanisms of ELF/VLF emissions (e.g., [76–78]) as the background wave intensity.

12.3 Ionospheric Alfve´n resonator (IAR) 12.3.1 Brief history and general introduction of IAR Various kinds of resonance phenomena are known to be present in the near-Earth environment. From the higher frequency, there is a transverse resonance in the Earth-ionosphere waveguide of lightning discharges in the ELF and VLF range (known as cutoff of tweek sferics) (e.g., [4,60,58,79]). When we go to the lower frequency down to the ELF range, we know the presence of longitudinal resonance in the Earth-ionosphere waveguide due to global lightning discharges, which is known as Schumann resonance [12,79–81]. Its fundamental frequencies are 8, 14, and 20 Hz, etc., and recently the observation of the Schumann resonance intensity is found to serve as a kind of global thermometer because of the close relationship of Schumann resonance intensity to the equatorial surface temperature (e.g. [79,81,82]). At the frequency below the Schumann resonance region, there is an additional resonance phenomenon called “ionospheric Alfve´n resonator” (IAR), which is the topic of this section (see a book by Surkov and Hayakawa [12]). Polyakov [83] and Polyakov and Rapoport [84] predicted theoretically the existence of such Alfve´n quasi-resonances in the ionosphere, and the IAR plays an important role in the understanding of the physical phenomena in the coupled magnetosphere-ionosphere system (e.g., [11,85–95]). The fundamental idea of this IAR is that there is the presence of IAR as a resonance in the region between the two regions of sharp boundary where the Alfve´n velocity vA ¼ B20 =m0 ni mi changes abruptly (B0 is the Earth’s magnetic field intensity, and mi and ni are the ion mass and density (this ni is different from the previous ni in Section 12.2.2.4)). One is the lowest ionosphere (D/E layer) and the other is a height of 500–1,000 km. Based on these spatial scales the resonance frequencies are found to range from 1 to 5 Hz in the ULF/ELF range, and IAR appears as a unique “fingerprint” structure. However, the generation mechanism of

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this IAR is not yet agreed. As you imagine, there are two probable possibilities depending on latitude, that is (1) magnetospheric origin; any noise in the upper magnetosphere or (2) lightning origin; that is, the effect of lightning discharges in the atmosphere. No matter whether the origin of IAR is either magnetospheric or atmospheric lightning effect, the mode responsible for this IAR is left-handed polarized Alfve´n mode as discussed in Section 12.2.2.3. The findings of IARs have been obtained from ground-based observations, and we will cite the papers mainly based on “long-term” observations. IAR was initially discovered at a middle latitude station (Nizhni Novgorod, Russia; L = 2.65 (McIlwain L is defined by the geocentric distance where the magnetic field crosses the magnetic equator in the unit of Earth’s radius), geomagnetic latitude = 52.12) and at a high latitude station (Kilpisjarvi, Finland; L  6, geomagnetic latitude = 66) by Belyaev et al. [96–98]. Further experimental evidence for the existence of IAR at high latitudes was later confirmed by Demekhov et al. [91] with the data at Kilpisjarvi observatory. Yahnin et al. [99] studied diurnal and seasonal variations of SRS occurrence rate based on continuous observations for more than 4 years at a high latitude station, Sodankyla¨ (L = 5.2, geomagnetic latitude of 64 ). They found a clear tendency of decrease in both the resonant frequencies and difference in resonance frequencies DF from the minimum to maximum solar activity. The high-resolution measurements of IAR signatures were also made at low-latitude stations such as Crete (L = 1.3, geomagnetic latitude of 28 ) by Bo¨singer et al. [100] during half a year and Muroto, Japan (L = 1.206, geomagnetic latitude of 24.4 ) with the 2.5-year data by Nose et al. [101]. Very recently Beggan and Musur [102] have reported on the characteristics of IAR in middle latitude from their long-term (5 years) observation at Eskdalemuir, UK (L = 3.46, geomagnetic latitude = 55.3), which are consistent with early works by Molchanov et al. (2014) to be presented later. Potapov et al. [95] have attempted the simultaneous IAR observation at two mid-latitude stations, Ulaanbaatar (L = 1.9) and Mondy (L = 2.2) and one high-latitude station, Istok (L = 6.1), but unfortunately not continuous. Their results during 4 days are found to be consistent with early results. While there are so many ground-based measurements of IARs, there have been recently some attempts to make in-situ observations of IARs from data by LEO satellites. Simoes et al. [103] have found few cases, with the data obtained by the C/NOFS satellite, of detecting a distinct picture of IAR (and Schumann resonances) during the minimum solar activity during the cycle 23/24 in the low-latitude ionosphere (most evident during nighttime). Further observational evidence of IAR signatures was obtained in low latitudes by the Ukrainian satellite, Chibis-M [104] and by the Freja satellite in auroral latitudes [105]. Dudkin et al. [104] have shown that their satellite observations have not revealed any long-term IAR signatures unlike the ground-based observations (as will be shown later). Also, in contrast to the dominant view (as will be shown later), IAR has been found to be effectively excited on the dayside as well. In Section 12.3.2, we initially present our statistical properties of IARs observed at middle latitude on the basis of ground-based observations for more than 2 years at a mid-latitude observatory at Karimshino (Kamchatka, Russia, L = 2.1, geomagnetic latitude of 46 ) [106,107] in order for the reader to understand what

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IAR looks like. Next in Section 12.3.4, we will review the generation mechanisms of IAR signatures. It is definitely certain that this phenomenon is a resonance effect of Alfve´n mode waves between the lowest ionosphere and lower magnetosphere, but the energy source (or initial agent) of those IAR signatures is very controversial, seemingly strongly dependent on latitude of our observation. And, in Section 12.3.5, we have suggested one of the most promising candidate hypothesis or excitation of IAR at middle and low latitudes by nearby thunderstorms, which might be very useful for the lightning community, that is, readers of this book.

12.3.2 Ground-based observations of IARs at middle latitude 12.3.2.1

Observations and data processing

The continuous registration of the ULF/ELF magnetic field variations at the geophysical observatory, Karimshino started in June 2000 and is still going on. The observatory is located in Karimshino (geographic coordinates: 52.94 N, 158.25 E, L = 2.1) at a distance of about 50 km from Petropavlovsk-Kamchatsky (see the details in Uyeda et al. [108] and Gladychev et al. [109]). Three axial induction magnetometers are used to measure geomagnetic field variations in the p frequency ffiffiffiffiffiffi range of 0.003–40 Hz. The sensitivity threshold is better than 20 pT/ Hz at a pffiffiffiffiffiffi frequency of 0.01 Hz, and it corresponds to 0.02 pT/ Hz at frequencies above 10 Hz. The sampling rate per channel is 150 Hz, and the sampling resolution is 24 bits. The accuracy of absolute and relative (between channels) timing of digital data is 5 ms and better than 10 ms, respectively. See Schekotov and Hayakawa [110] for further details of the equipment and measurement. Because of relatively high crust conductivity in the region of observations the vertical magnetic signal amplitude is low in comparison with the horizontal ones. The amplitude of the vertical component in the frequency range of 1–3 Hz is comparable with the sensor sensitivity, and thus the resonant structures cannot be detected for the vertical component with the equipment used. The technique and results of the signal analysis for two horizontal components are given below. The signal spectra were evaluated with Welch’s method in the frequency range 0.1–5 Hz with resolution of 0.05 Hz in the time window of 30 min and many parameters of resonance structures were estimated. An example of the evolution of nighttime dynamic frequency spectra is shown in Figure 12.10 for about one week from September 12 to 18, 2000. The bottom three panels illustrate the dynamic spectra for the three magnetic field components (H, D, and Z), respectively. The second panel indicates the coherency between the two magnetic components H and D and the third panel indicates the wave polarization by using the two horizontal magnetic field components. The top panel indicates the temporal evolution of the Kp index (geomagnetic activity) for comparison. Figure 12.11 is another type of presentation of Figure 12.10 in the form of spectral power density versus wave frequency for the two horizontal components D (in thick line) and H (in thin line). The local time is LT = 19 h through LT = 06 h on a particular day, September 13 and 14, 2000. We can recognize from this figure the IAR spectra with typical resonance (fingerprint) structures, especially in the local time interval from LT = 21 h to 01h, but weak (not so conspicuous) resonance structures are also seen at other LTs.

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Figure 12.10 An example of the temporal evolution of SRS of IARs from LT = 19 to 06 h on September 13–14, 2000. SRS is most clearly seen at LT = 22 h (DF  0.5 Hz). (LT = UT + 12 h). The top panel indicates the Kp index, the second one, the coherency between the horizontal magnetic field components, the third, wave polarization by means of the horizontal magnetic field components, and the last three panels, the dynamic spectra for H, D, and Z components. The values are indicated in color. After Molchanov et al. [107]

We can notice from Figures 12.10 and 12.11 very clear resonance structures on the horizontal magnetic field components (H and D). However, when we use the “polarization spectrum,” it is much easier for us to identify such resonance structures (SRS), which is clearly recognized in the second top panel of Figure 12.10, when we compare it with the corresponding amplitude dynamic spectra (e.g., the 2nd panel from the bottom in Figure 12.10). The observed wave is found to be lefthanded polarized (or Alfve´n mode wave). The algorithm of automatic SRS detection and calculation of its main parameters have been described below. Various parameters obtained from the analysis of the spectra are as follows: ● ● ●

averaged separation frequencies of resonance DF, intensity of the resonance signal, “quality” parameter Q of the resonance structure.

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The algorithm of data analysis is schematically illustrated in Figure 12.12. The SRS with the frequency difference DF is given in the top panel (a). Its fifth-order polynomial approximation is shown with a dashed line as a trend. The difference between the raw spectrum and its approximation (trend) (hereafter, we call it spectral KARIMSHINO, from 19:00 LT /13/09/2000 to 06:00 LT /14/09/2000 19 h

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Figure 12.12 The method of SRS identification and determination of the SRS parameters. (a) SRS (in solid line) and its trend (in dashed line), (b) SRS difference, and (c) Spectrum of the SRS difference. Its maximum location is the averaged DF in the SRS. SRS exists if the distribution is narrow banded, that is, quality Q1. After Molchanov et al. [107]

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density variation) is given in the middle panel (b). The total power can be estimated as a sum of the resonant power and the background power approximately corresponding to the curve going through the minima of spectra in Figure 12.12(a). The intensity of the resonant signal is numerically estimated as the mean of the absolute value of spectral density variation, but the relative accuracy of this approximation is low. Both spectra of the total signal and its resonant part have several approximately equidistant maxima. The average distance between the maxima is estimated with a help of the Fourier transform of the spectrum of the resonant signal, and this spectrum is depicted in Figure 12.12(c) as a function of frequency. Its maximum corresponds to the averaged frequency difference between the SRS maxima. We define the “quality” Q of the resonant structure as the ratio of this maximal frequency to the half-width of the maximum df using the formal similarity with the parameters of damping oscillations. A resonance structure by definition exists if at least two maxima are found in the spectrum, that is, Q > 1. This allows us to exclude a possibility of false IAR detection caused by some other effects like Pc1 geomagnetic pulsations.

12.3.2.2

Seasonal and diurnal variations of the SRS parameters

The IAR occurrence rate (occurrence probability) is defined as the ratio of number of nighttime (21h–03h LT) intervals with IARs to the total number of intervals, and this is plotted in Figure 12.13(a). The average frequency difference between adjacent spectral maxima is shown in Figure 12.13(b). The top of each grey rectangle means the average DF for a certain month, and the range of the dark rectangle indicates the error bar of estimation. When only one event was registered during a month, the frequency difference is not shown because of a big error in estimation. The seasonal variation averaged over all periods of observations of the IAR occurrence rate, frequency difference DF and intensity are shown in Figure 12.14 from the top to the bottom. It is seen from the figures that during all the periods of observations the probability of IAR occurrence is maximal in autumn-winter and it vanishes in springearly summer. A clear maximum of DF is also observed in winter in the second panel of Figure 12.14. The bottom panel of Figure 12.14 exhibits a broad maximum, so that we can suggest that the IAR power density depends relatively weakly on the season. So, the seasonal variations of the two quantities of IAR power and SRS IAR occurrence rate seem to be in anti-correlation. The diurnal variations of IAR occurrence rate for the four seasons are summarized in Figure 12.15 (winter, spring, summer, and autumn, from top to the bottom). In winter and autumn when the occurrence rate is maximal there is a clear pre-midnight maximum. Besides, in winter a secondary weaker maximum is found in early morning (04h–06h LT). Only few events were registered in spring, so that no estimates can be made for them. The IAR occurrence rate in summer seems to be shifted to post-midnight hours.

12.3.2.3

Dependence on geomagnetic activity

SRS IAR occurrence rate both at middle and high latitudes [99] is found to increase with a decrease in geomagnetic activity. This observational result is confirmed by

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Figure 12.13 The temporal evolution of monthly IAR occurrence rate (probability) during all the observation periods from July 2000 to December 2002 (top panel). The bottom is the corresponding temporal variation of DF during the same period. The top of each grey rectangle indicates DF, and the range of the dark rectangle means the error bar. After Molchanov et al. [107]

our statistical analysis. Correlation coefficients of SRS occurrence rate with the Kp index of global magnetic activity for different 1-year intervals are shown in Figure 12.16 (days with data gaps and the last three months, April–June 2002 (see seasonal dependence) were excluded from the analysis). The stable and reliable negative correlation of IAR with Kp is noticed from the figure and it may be possible that SRSs of IAR are masked under disturbed geomagnetic conditions by high background activity.

12.3.2.4 Summary of morphological characteristics of IARs at middle latitudes We characterize SRS by the fundamental frequency and averaged DF. The parameter Q characterizing the “quality” of the SRS has been introduced, which is controlled by the IAR quality and the ratio of the amplitude of resonant signal to the background noise. If the resonance “quality” parameter exceeds a threshold value, we can detect any SRS of IAR. We have found that the seasonal variation of

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Figure 12.16 Correlation coefficients of the IAR occurrence rate with the Kp index for different 1-year periods (shown above). All the correlation coefficients are negative for time delay of 0–1 day. After Molchanov et al. [107] The main results of this observation can be summarized as follows: 1.

2. 3. 4.

5. 6.

During the observation for 2.5 years, IAR structures occur during approximately one-quarter of the observation period (250 nights). There is an evident seasonal variation in the occurrence rate with a maximum in the autumn–winter period and almost complete absence of IAR structures at the spring–early summer time. The occurrence maximum in the diurnal variation is found at the LT of 21 h– 23 h, and almost all the IAR structures are observed at local nighttime. The averaged DF is about 0.2–0.5 Hz in the summer–autumn period, but it increases up to 0.5–0.7 Hz in the winter time. IARs are mostly polarized along the azimuthal direction (D-component), and the diurnal variations in the two horizontal components are sometimes not identical. IAR is found to be left-handed polarized (that is, Alfve´n mode waves). There is an anti-correlation between the IAR occurrence rate and Kp index of the global geomagnetic activity.

We briefly compare the present experimental findings at our middle latitude (L = 2.1) station with the previous and recent observations at high latitude (L = 5.2) [99] and at low latitudes (L = 1.3, Bo¨singer et al. [100] and L = 1.206, Nose´ et al. [101]). Yahnin et al. [99] utilized an extremely long-term observation of more than

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4 years, Bo¨singer et al. [100] used the data only for half a year, and Nose´ et al. [101] used the data for about 2.5 years. When we think about the generation mechanism of IAR, the latitudinal dependence of different morphologies of IAR would be of essential importance (this will be described in the next subsection). Several results have been reported on the basis of a shorter database [95–97] even at middle latitudes, but this section has provided the first results of middle latitude IAR characteristics by using sufficiently long-term observation. Point (1) in the above summary suggests that IAR structures occur for approximately one-quarter of the observation periods (250 nights). This means that the IAR phenomena are not so rare, and are rather a regular phenomenon, which seems to be consistent with the conclusion by Bo¨singer et al. [100,111] and Nose´ et al. [101] at low latitudes. Point (2) on LT dependence is likely to be consistent with the nighttime maximum in previous or recent publications at high and low latitudes ([99,101]; Bo¨singer et al. [100]. Next, the seasonal variation in Point (1) seems to be consistent with Yahnin et al. [99] at high latitudes, but Nose´ et al. [101] at low latitudes have found a slightly higher occurrence in May–September, which is considerably different from our seasonal dependence at middle latitude. Finally, as far as Kp dependence is concerned, we have found a clear anti-correlation at middle latitudes (Point (6)) being consistent with high-latitude results [99], but Nose´ et al. [101] have noticed no significant correlation with Kp index at low latitudes. So these existing consistencies and discrepancies in morphological characteristics of IARs at different latitudes will be of great potential in studying the energy source of IARs at different latitudes. These findings with ground-based measurements have been confirmed by the LEO satellites [103,104] but few examples. Further, Dudkin et al. [104] have found the daytime occurrence as well, which suggests that the absence of IAR during the day on the ground may be due to the stronger daytime absorption during the ionospheric transmission.

12.3.3 Generation mechanisms of IAR Though the fundamental idea of IAR is a resonance of Alfve´n waves in the altitude region between the lowest ionosphere and lower magnetosphere, the physical mechanism, in other words, the origin (or source) of IAR has not yet been established, leading to proposals of different hypotheses. Also, it is highly likely to be dependent on latitude of our observation. At high latitudes, Trakhtengertz and Feldstein [85,112–114] proposed a mechanism of the IAR excitation due to an instability of magnetospheric convective flows that cause the formation of a turbulent Alfve´n boundary layer near the upper wall of the IAR. The “feedback” instability due to the ionospheric modification by energetic particle precipitation can also stimulate the IAR [86–88,115–117]. Evidently, these proposed plausible mechanisms seem to work only at high and auroral latitudes, but no observational evidence has been obtained yet. As a possible source of the IAR excitation at middle and low latitudes, the generation of the ionospheric turbulence and currents by the neutral wind

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fluctuations in the conductive lower ionosphere has been theoretically suggested by Molchanov et al. [107], Surkov et al. [118] and Surkov and Hayakawa [12]. According to this model, the IAR excitation might be expected to take place above the regions with intense atmospheric turbulence such as hurricanes, typhoons, atmospheric fronts, etc. However, this expectation has not yet been validated experimentally or observationally. The other plausible energy source for the IAR excitation is related to atmospheric lightning discharges. The world thunderstorm centers in the tropical regions were suggested to be the primary source of IAR excitation [12,96,100]. However, their theoretical estimates have shown that the contribution of distant global thunderstorm centers with typical charge movements of negative cloudto-ground (–CG) flashes to middle latitude electromagnetic field in the IAR band is about two orders of magnitude lower than that observed actually [12,119]. To resolve this issue, Shalimov and Bo¨singer [120] proposed that relatively infrequent (10%) but more intense (stronger by about an order of magnitude) +CG flashes can provide a necessary ULF background at large distances. Even in this case, however, the estimated spectral power would be inconsistent with observations. Thus, the possible role of distant thunderstorms in the IAR excitation is still very controversial. As shown in Section 12.3.3, IAR signatures seem to be a persistent feature of the upper nighttime ionosphere and should be seen by LEO satellites. Dudkin et al. [104] have found the daytime IAR as well, which will provide an interpretation that ground-based IARs cannot be observed during day because of the enhanced absorption during the ionospheric penetration. As an alternative mechanism, it was suggested that local thunderstorms are able to generate signals in the IAR range with sufficient intensities [121]. Schekotov et al. [122] investigated a ULF magnetic response to the regional lightning activity, and showed that the mechanism of the SRS is not related to the oscillatory response of the upper ionosphere, as has been commonly assumed, but is caused by the specific multi-pulse structure of geomagnetic disturbances produced by a lightning discharge. As mentioned above, the generation mechanism of IARs has not yet been established, but in Section 12.3.5, we will introduce the latest idea of the role of nearby lightning discharges in the IAR excitation by Surkov et al. [121], which might be of great interest to the lightning community, the readers of this book. Lastly, we have indicated some further theoretical studies on the modeling of the IAR excitation in the upper atmosphere by lightning discharges, sometimes with taking into account the excitation of the horizontal ionospheric waveguide by fast magnetosonic wave (i.e., modified Alfve´n mode wave in Figure 12.6) [93,123–126].

12.3.4 Excitation of IAR by nearby thunderstorms It is a fact that the low frequencies determine the near field of the lightning discharge, while the high frequencies make a main contribution to the far-field spectrum. On the other hand, the quasi-static field falls off faster with distance than the

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wave field, which means that in the ELF frequency range the discharge spectrum from nearby lightning is more intense than that of more distant discharges [121]. This is the basic idea of trying to correlate IARs to nearby lightning. As it is seen from the examples in Figure 12.10, there are many impulses that can be associated with thunderstorm activity, and only some of these impulses are accompanied by sharp impulses in the frequency range of 0.25–4 Hz. The former is thus assumed to be a result of nearby lightning discharges. In order to estimate the number of lightning discharges per unit of time, we choose the signal discrimination level as 5 pT. Total number of the impulses, DN, with amplitude which is greater than this level, increases with time as shown in Figure 12.17. Lines 1 and 2 correspond to the frequency filters 6–20 Hz and 0.25–4 Hz, respectively. Averaging over an interval of 1 hour results in a mean occurrence rate of the impulses of about n1 = 0.144 s
1 and n2 = 0.023 s
1, which is typical for nighttime conditions at Karimshino station. Note that about 2,000 thunderstorms operate simultaneously in the whole world producing a total current of about 1,800–2,000 A [79]. Taking into account that a lightning discharge usually brings the charge 20–30 C, one can find that the mean occurrence rate of the lightning discharges v = 60–100 s
1. Hence a thunderstorm typically produces a rate number of about 0.03– 0.05 s
1, which is 600

500 1

∆N

400

3

300

4 200 2

100

0 0

10

20

30 t, min

40

50

60

Figure 12.17 Temporal variations of the sum of magnetic impulses. The threshold level for the impulse amplitude is 5 pT. Lines 1 and 2 correspond to the frequency channels 6–20 and 0.25–4 Hz, accordingly. The impulses of solely H component (6–20 Hz) are shown with dash lines 3, while the D component is shown with dash line 4. After Surkov et al. [121]

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close to n2. From here we may assume that one nearby thunderstorm and 3–6 remote ones make a major contribution to the rate number shown in Figure 12.17. As for the signals of remote thunderstorms, it should be noted that the intensity of H component is larger than that of D component. The impulses of H component (6–20 Hz) that are displayed in Figure 12.17 with dashed line 3 occur more frequently than in the D component shown with dashed line 4. On the other hand, in the frequency range of 0.25–4 Hz the occurrence rate numbers of both components are very close to each other. Most of the intense signals which can be associated with a nearby thunderstorm have a bipolar structure. It appears that the first impulse in the signals is due to the primary wave radiated by the return stroke. The interval between positive and negative impulses is typically 2 s. One may assume that such a shape of the signal results from Alfve´n wave reflection from the gradient in Alfve´n velocity at the upper boundary of the resonance cavity. If the typical size of the resonance cavity is 500–1,000 km, the arrival time of the reflected Alfve´n wave is estimated as 2 s, which is close to the signal duration. The signal occasionally contains three distinct impulses at least. This implies a possibility for multiple wave reflections from the IAR upper boundary. Let N be the number of nearby thunderstorm centers simultaneously operating around the ground-based recording station. A local coordinate system has the x-axis eastward, the y-axis to the north, and z-axis vertically upward. Since we are interested in solely nearby thunderstorms, a plane-stratified model of the medium is used. Let rm and jm be the polar coordinates of the thunderstorm epicenter, where m = 1, 2, . . . N as shown in Figure 12.18. A typical size of the thunderstorm is assumed to be smaller than the distance from the recording station. We ignore the lightning discharge distribution inside a thunderstorm area, and this implies that all

y'

y N-S E-W

x'

z'

rµ z Br

φµ

x

BI

Figure 12.18 Schematic picture of reference and local coordinate systems. There, z and z0 axes are “out of paper.” Br and Bf are the component of magnetic variations due to the thunderstorm located at r = rm. After Surkov et al. [121]

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the lightning discharges related to the same thunderstorm must have the same coordinates; that is, the coordinates of a given thunderstorm. Now we also introduce a reference frame x0 , y0 , and z0 fixed to the thunderstorm with the number m. Let B (rm, t–tnm) be the magnetic field due to the lightning discharge that happened at the accidental moment tnm, where nm= 1, 2, . . . is the number of the lightning discharge. It is usually the case that the lightning discharges are vertical and transfer a negative electric charge to the ground [6,127,128]. If we use a cylindrical coordinate system in which the lightning discharge is in the direction of the polar z0 -axis, the magnetic field B is independent of azimuthal angle j. According to Belyaev et al. [97], only radial component Br among the two horizontal ones contains the resonance factor, which dominates the IAR resonance properties. In the Cartesian reference frame fixed to the ground-recording station, the horizontal magnetic field can be expressed through the radial and azimuthal components as follows: Bx ¼ Bf sin jm
Br cos jm By ¼
Bf cos jm
Br sin jm

(12.39)

On the ground z = z0 = 0 the components Br and Bf are random values, which depend on rm and t–tnm. Far from the lightning discharge, the electromagnetic field of the lightning discharge can be characterized by the current moment of the discharge m(t) = I(t)l(t), where I(t) is the current produced by the median return strokes of lightning and l(t) denotes the lightning channel length. According to Surkov et al. [118], we approximate the actual current moments with the function m(t) = MF(t), where the magnitude M of the current moment is assumed to be a random value, whereas F(t) is a universal function of time. Hence the horizontal magnetic field of a single lightning discharge can be written as Br ;f ðrm ; t
tnm Þ ¼ Mnm Gr ;f ðrm ; t
tnm Þ

(12.40)

where the functions Gr,f is supposed to be equal to zero as t < tnm. These functions are derived from Maxwell’s equations, which should be supplemented by proper boundary conditions at the ground and the atmosphere–ionosphere boundary. The net magnetic perturbation at a ground-recording station is a random quantity Bran which equals the sum of the magnetic perturbations caused by separate lightning discharges. The net horizontal field of all the thunderstorms located around the station is then Bran ðtÞ ¼

N X m¼1

Bm ðtÞ; Bm ðtÞ ¼

X

b Gn ; Mn m A m m

(12.41)

nm

where Bm is the horizontal magnetic field due to the thunderstorm with number m, the matrix Aˆm and the vector Gnm are given by   sinjm b m ¼
cosjm ; (12.42) A
sinjm
cosjm

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 Gr ðrm ; t
tnm Þ ; Gj ðrm ; t
tnm Þ

(12.43)

We now turn to the modeling of a single lightning discharge. A typical lightning discharge consists of several return strokes. According to Jones [129], Uman and Krider [128], Uman [127], MacGorman and Rust [130], and Rakov and Uman [6], the current produced by the median return stroke of lightning is modeled as, IðtÞ ¼

4 X

Im expð
wm tÞ

(12.44)

m¼1

where wm are inverse time constants and the amplitudes Im of individual currents 4 P Im ¼ 0. The vertical current channel grows term must satisfy the condition m¼1

upward with the velocity dl/dt = V0 exp (
Wt), where V0 = 8 107 m/s is the current wave velocity at the ground level and W is the relaxation time parameter whence it follows that the final channel length is l1 = V0/W. The current moment of the single return stroke m1(t) = I(t)l(t) can be written as (see, e.g., [79,131,132]) m1 ðtÞ ¼ l1 ½1
expð
WtÞ

4 X

Im expð
wm tÞ

(12.45)

m¼1

It is usually the case that the lightning discharge contains n = 2–6 return strokes. More frequently, there are three return strokes with a characteristic duration of about 100 ms. The mean interval between them is of the order of t0 = 40 ms. Following Jones [129], we assume that the final length of the return stroke increases with its number n as ln = l1 + (n–1) Dl, where l1 = 4 km and Dl = 1 km. The current relaxation time parameter Wn is related to the channel length ln as follows:
1

Wn ¼


1 ln Dl ¼ W1 þ ðn
1Þ; V0 V0

(12.46)

where W1 = V0/l1 = 2  104 s–1. The net magnetic moment of multiple discharges can be written as m(t) = MF(t), where M = l1|I1| is the “magnitude” of magnetic moment. while the dimensionless function F(t) describes the shape of multiple discharge FðtÞ ¼

4 

 X  Im  h t0n 1
exp
Wn t0n exp
wm t0n l jI j n¼1 1 m¼1 1

n0 X ln

(12.47)

where t0n ¼ t
ðn
1Þt0 and h(x) denotes the step-function, that is h = 1 if x 0 and h = 0 if x < 0. In this model, all the current impulses have the same shape while the increase in lightning channel length results in a gradual enhancement of the

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electric current moment. To simplify the problem, we assume now that the lightning discharge contains n0 = 3 return strokes. Taking the typical parameters for the models of the return strokes (see, e.g., [79,131–134]) I1–4 =
28.45, 23.0, 5.0, 0.45 (in kA) and w1–4 = 6.0  105, 3.0  104, 2.0  103, 147.0 (in s
1), one can estimate the typical magnitude of magnetic moment as M = 170 kA  km. The spectrum of the function F(t) is F ðw Þ ¼

n0 X 4 W1 X Im exp½jwt0 ðn
1Þ 2p n¼1 m¼1 jI1 j ðwm
jwÞðWn þ wm
jwÞ

(12.48)

Here we will evaluate the electromagnetic field spectrum of the lightning discharge in the upper atmosphere, and we approximate the actual conductivity distribution with the plane-stratified model [118] which is a reasonable approximation to the variation of the conductivity with altitude. Figure 12.19 illustrates a schematic medium model. The ground z 0 is considered to be a uniform conductor with constant conductivity sg. The vertical lightning discharge has appeared in the atmosphere at the altitude z = h above the ground at the moment t = 0. When considering far distances, the actual lightning discharge can be replaced with the vertical lumped electric current moment m(t) located at the z-axis at the altitude z = h. Since a lightning discharge transfers the negative electric charge to the ground, the vector of current moment is therefore vertically upward. The atmospheric slab 0 < z < d is supposed to be an insulator. The ionospheric plasma z > d

z B0

z

z=d+1+L Magnetosphere

z

F layer

L

VA

z=d+l l

E layer z=d

VAI

σP

σH

VAM x

y Atmosphere

d

σg

z=0 Earth

σ

Figure 12.19 Schematic illustration of a stratified medium model. The plots of the Alfve´n velocity and the ionosphere/ground conductivities are shown in the right panel. After Surkov et al. [121]

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is immersed in the constant Earth’s magnetic field B0. The actual profile of the Hall and Pedersen conductivities strongly varies with altitude. The plasma conductivity peak lies inside the ionospheric E layer, which is considered in the thinslab approximation. This condition is not so burdensome on account of smallness of the typical wavenumber k allowing for the condition kl « 1, where l denotes the characteristic width of the E layer. In the model, the region above the E layer (i.e., F layer and magnetosphere) is supposed to be the area consisting solely of cold collisionless plasma. The IAR formation in this region is due to the plasma density fall off with altitude, which makes the Alfve´n velocity, vA, rapidly increase upward. The ionospheric resonance cavity is bounded from below by the conductive E layer and from above by the region where the gradient of the Alfve´n velocity reaches a peak, which makes it possible to have reflection of the Alfve´n wave from the upper space. The typical vertical scale of the resonance cavity is L  103 km. It should be noted that the same resonance cavity can serve as a waveguide for the magnetosonic/compressional mode (modified Alfve´n wave in the previous section) (e.g., [135]). According to Pokhotelov et al. [87], we use a suitably idealized model of the resonance cavity which describes the Alfve´n velocity in terms of a piece-wise function so that vA = vAI within the resonance cavity (d < z < d + l +L) and vA = vAM in the outer magnetosphere (z > d + l +L), where vAI and vAM (vAM » vAI) are constant quantities referring to the ionosphere (I) and magnetosphere (M), respectively. The geomagnetic field B0 is supposed to be vertically upward. A more accurate model which takes into account the dip angle of the local geomagnetic field gives rise to very complicated equations [136–138]. For simplicity, we adopt the model of the vertical geomagnetic field in order to avoid the complexities connected with magnetic field inclination. On account of the axial symmetry of the field produced by the vertical current moment, the cylindrical coordinates z, r, and f are used. In this case, all the quantities are free of f so that @=@f ¼ 0. In the atmosphere, the primary field of the current moment contains only three components, Er, Ez, and, Bf termed as TM mode. When the electromagnetic wave of TM mode penetrates into the ionosphere, it produces excitation of the TE mode, that is, Ef , Br, and Bz due to the mode coupling via Hall conductivity in the ionosphere. The shear and compressional Alfve´n waves can get trapped in the F layer of the ionosphere thereby exciting the IAR. Low-frequency resonant oscillations are leaking back into the atmosphere so that all the components of the electromagnetic field, which are TM and TE modes, can be detected on the ground. In the framework of the model, we note that the spectra of the electromagnetic perturbations due to a solitary lightning discharge have been obtained by [139]. The reader is referred to those works for details about the derivation of the formulae. Let SW ¼ 1=ðm0 vAI Þ be the Alfve´n parallel conductance, ap ¼ Sp =Sw and ap ¼ SH =Sw stand for the dimensionless height-integrated Pedersen and Hall conductivities, respectively, and x0 = wL/vAI be dimensionless frequency. The Fourier transforms of the magnetic perturbation dBr and dBf on the ground level z = 0 can be written as follows,

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dBr ðr; wÞ ¼ Mgr ðr; wÞ

(12.49)

dBf ðr; wÞ ¼ Mgf ðr; wÞ

(12.50)

The mathematical derivations of gr ðr; wÞ and gf ðr; wÞ are rather complicated, so that we want to avoid these. Please look at our original paper by Surkov et al. [121] for further details of electromagnetic fields in each region and the application of boundary conditions to each boundary. A model calculation of the lightning-generated spectra of the resonance component dBr for the typical nighttime parameters of the mid-latitude ionosphere is given in Figure 12.20. The discharge contains three CG return strokes. The numerical parameters of the lightning discharge used in making this plot are given above. The numerical values for the various magnetospheric, ionospheric, and other parameters are vAI = 500 km/s, vAM = 5  103 km/s, L = 500 km, d = 100 km, l = 40 km, z = 0, sg = 210
3 S/m, SP = 0.2 Ohm
1, and SH = 0.3 Ohm
1 (nighttime conditions). In this figure, lines 1–4 correspond to the distances r = 100, 300, 1,000, and 10,000 km, respectively. It is obvious from Figure 12.20 that the spectra exhibit distinct resonance structures in such a way that the resonance frequencies are close to the IAR eigenfrequencies as observed. Owing to the

0.05 1

2

| δ Br | . pT . Hz –1

0.04 2' 0.03 3 0.02

0.01 4 0

0.5

1

1.5

2 f, Hz

2.5

3

3.5

4

Figure 12.20 Model calculation of the nighttime IAR spectra excited by a solitary cloud-to-ground lightning discharge. The radial/resonant component Br on the ground is shown with lines 1–4, which correspond to the distances r = 100, 300, 1,000, and 10,000 km, respectively. The approximate analytical solution at distance r = 300 km is shown with dotted line 2’. After Surkov et al. [121]

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symmetry of the problem the radial component of the magnetic perturbation must tend to zero when r ! 0. The calculations have shown that the spectrum magnitude reaches a peak at a distance of about 300 km. Surkov et al. [121] have further considered the random magnetic variations produced by the CG lightning discharges treated as a stochastic process. The obtained frequency spectra are found to be close to the observations. The theoretical consideration by Surkov et al. [121] can be summarized as follows: 1. 2.

3.

The model computations of the power spectra are in favor of nearby thunderstorms as a possible cause for the IAR excitation at middle latitudes. The solitary CG lightning discharges in the neighborhood of the ground station may result in the impulse IAR excitation, which is capable of producing an observable SRS signature on the ground. The random lightning discharges in the range of 1–2 thousands km make a main contribution to the mid-latitude IAR power spectrum since the predicted resonant frequencies and peak intensities are practically consistent with those observed.

Finally, we have to conclude that even though there have been proposed several hypotheses as the origin of IARs at different latitudes, any mechanism is far from being accepted. So appreciable work will be highly required observationally in order to validate even the most promising hypothesis of nearby lightning discharges in the excitation at middle/low latitudes in Section 12.3.4.

12.4 Summary of lightning effects on the ionosphere/ magnetosphere In this chapter, we have discussed two major effects of atmospheric lightning discharges on the upper atmosphere of the ionospheric/magnetospheric plasma. The first is an extremely well-known phenomenon; whistlers. This whistler signal originated in the VLF/ELF part of the causative lightning discharge in the opposite hemisphere, it propagates in the ionosphere/magnetosphere, and is received by a ground-based receiver. General characteristics and the general theory of groundbased whistlers are presented, and we have presented the use of this lightninginduced ground-based whistler as the diagnosis of magnetospheric plasma density and also earlier satellite observations of nonducted whistlers. Furthermore, we have introduced abundant recent LEO satellite observations of 0+ whistlers and ELF/ VLF electromagnetic waves (either waveform or power spectral data) extensively utilized to study the global lightning activity and lightning characteristics. The second item of this chapter is IAR in the ULF/ELF band, which has a relatively short history. This IAR is known to take place in the frequency range below the Schumann resonance and the eigenfrequencies are in the ULF/ELF range from 1 to 5 Hz. The frequency spectra of IARs are just like fingerprint structure (SRS). Theoretically, this IAR was predicted around 1980, but it is only recently that many workers have paid extensive attention to this IAR. The characteristic

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IAR eigenfrequencies can be roughly estimated as frnnvA/(2L) where n = 1,2, . . . . (mode number) and L is the length of field line piece within the resonance cavity. Two possible hypotheses depending on the latitude of observation have been proposed so far as the excitation mechanism of this IAR: (1) lightning in the atmosphere, and (2) magnetospheric effect. After presenting our own long-term observational results at middle latitudes as a basis for the readers, we have compared our mid-latitude results with those at high and low latitudes. Finally, we have presented one likely mechanism by nearby lightning discharges for the excitation of middle and low-latitude IARs. However, further appreciable efforts will be highly required to obtain deeper understanding of the mechanisms of IARs. Finally, we would like to add one more phenomenon of the lightning effect on the upper atmosphere including the mesosphere and ionosphere. During the last three decades there has been an enormous progress in the study of transient luminous events such as sprites, blue jets, elves, etc., in possible association with lightning discharges, and the readers interested in this topic are directed to our latest review [140].

References [1] Wait, J. R., Electromagnetic Waves in Stratified Media, Pergamon Press, Oxford, 1970. [2] Budden, K. G., The Waveguide Mode Theory of Wave Propagation, Logos Press, London, 1961. [3] Davies, K., Ionospheric Radio, Peter Peregrinus, London, 580p., 1990. [4] Al’pert, Ya. L., Propagation of Radio Waves in the Ionosphere, Plenum Press, New York, NY, 1974. [5] Galejs, J., Terrestrial Propagation of Long Electromagnetic Waves, Pergamon Press, Oxford, New York, 376p, 1972. [6] Rakov, V. A., and M. A. Uman, Lightning: Physics and Effects, Cambridge University Press, Cambridge, 2003. [7] Walker, A. D. M., The theory of whistler propagation, Rev. Geophys. Space Phys., 30, 411–421, 1976. [8] Hayakawa, M., Whistlers, in H. Volland (Ed.), Handbook of Atmospheric Electrodynamics, Vol. 2, CRC Press, Boca Raton, FL, pp. 155–193, 1995. [9] Helliwell, R. A., Whistlers and Related Ionospheric Phenomena, Stanford University Press, Stanford, CA, 1965. [10] Park, C. G., Whistlers, In Handbook of Atmospherics, Vol. 2, Ed. by H. Volland (eds.) , CRC Press, Boca Raton Florida, pp. 21–79, 1982. [11] Pilipenko, V.A., Impulsive coupling between the atmosphere and ionosphere/ magnetosphere, Space Sci. Rev. 168, 1–4, 2011, doi: 10.1007/s11214-0119859-8. [12] Surkov, V. and M. Hayakawa, Ultra and Extremely Low Frequency Electromagnetic Fields, Springer, Tokyo, 486p, 2014. [13] Barkhausen, H., Zwei mit Hilfe der neuen Versta¨rker entdeckte Erscheinungen, Physik Z., 20, 401–403, 1919.

518 [14] [15] [16] [17] [18] [19] [20] [21]

[22] [23] [24] [25]

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Chapter 13

On the NOx generation in corona, streamer and low-pressure electrical discharges* Vernon Cooray1, Marley Becerra1 and Mahbubur Rahman2

13.1 Introduction An assessment of the global distribution of nitrogen oxides (NOx) is required for a satisfactory description of tropospheric chemistry and in the evaluation of the global impact of increasing anthropogenic emissions of NOx [1]. In the mathematical models utilized for this purpose, it is necessary to have the natural as well as man-made sources of NOx in the atmosphere as inputs. Thunderstorms are a main natural source of NOx in the atmosphere and it may be the dominant source of NOx in the troposphere in equatorial and tropical South Pacific [2]. In quantifying the production of NOx by thunderstorms, scientists have until recently concentrated on the lightning return strokes neglecting all other processes associated with thunderstorms [3]. This view is gradually changing as the theories and experimental data show that not only the return strokes in ground flashes but other discharge events in ground and cloud flashes, such as continuing currents, are also contributing significantly to the NOx emissions [4]. For example, the study conducted by Rahman et al. [4] shows that the contribution by continuing currents to the NOx production in lightning flashes is comparable, if not overwhelm, to the contribution by return strokes. However, the physics behind the process that makes continuing currents as efficient as return strokes in producing NOx is still unknown. Based on the ionization process that leads to the production of free electrons, electrical discharges taking place in the atmosphere can be divided into two types, namely, ‘cold’ and ‘hot’ electrical discharges. In cold electrical discharges, free electrons are produced solely by the collisions between energetic electrons and atoms. In these discharges, the electron temperature may reach several tens of thousands of degrees whereas the gas and ion temperature remains close to ambient temperature. Corona discharges, streamer discharges and Townsend-type electrical

*This

material was published previously by the same authors in Open Atmospheric Science Journal, vol. 2, pp. 176–180, 2008. It is reproduced here by permission from the journal. 1 Department of Electrical Engineering, Uppsala University, Sweden 2 School of Engineering, Royal Institute of Technology, KTH, Sweden

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discharges taking place at low pressure are several examples of cold discharges taking place in the atmosphere. In hot electrical discharges, the gas and ion temperature can also reach several tens of thousands of degrees and the main mechanism in the discharge that generates free electrons is the thermal ionization. In these discharges, ions and neutral atoms are heated to such high temperatures through a process called thermalization that facilitates the transfer of energy from free electrons to neutrals via ions [5]. This transfer of energy from electrons to neutrals causes the temperature of the neutrals to go up. This increase in the temperature of the neutrals leads to production of copious amount of electrons by the energetic collisions between neutral particles (i.e. thermal ionization). Several examples of hot discharge processes taking place in the atmosphere are return strokes, leaders, M components and continuing currents. In addition to hot discharges mentioned above, an active thundercloud also produces cold discharges in the form of corona and streamer discharges. However, in quantification of NOx produced by thunderstorms, scientists neglect the contribution of the cold discharge processes. Moreover, the recent discovery of thunderstorm-created ionization processes in the stratosphere and mesosphere, known as sprites, blue jets and elves, also poses a question as to the effects of these ionization processes on the chemistry of the upper atmosphere. These ionization processes can also be categorized under ‘cold’ discharges because the pressure at which these discharges take place does not support thermal ionization, making electron impacts the main source of ionization. Indeed, there is a need today to develop procedures to quantify the NOx production in cold discharges. The effects of solar proton events in the NOx production in the upper atmosphere and the effect of NOx on the chemical balance of the stratosphere were a major concern of the atmospheric scientists since the discovery of the importance of NOx in ozone production and destruction [6]. In quantifying the NOx production from these events, scientists utilized the connection between the ionizing events in the atmosphere and the number of resulting NOx molecules. To facilitate further discussion, let us denote the number of NOx molecules produced per ionizing event in the atmosphere by the parameter k. The theoretical work of Nicolet [7] set the value of k close to unity, whereas the investigations of Jackman et al. [8] predicted that at altitudes larger than about 80 km, k is about 1.5 and for low altitude it is about 1.2–1.3. These results have been used extensively to study the NOx production by cosmic rays and solar radiation impinging on the Earth’s atmosphere. Recently, Rahman et al. [9] investigated the validity of this theoretical calculation by studying the NOx production in air by alpha particles emitted by a radioactive source. The results of this study confirmed these theoretical predictions fixing the value of k to about 1.0. Since the main source of ionization during proton impacts are high energetic secondary electrons, it is reasonable to assume that the number of NOx molecules produced in discharge processes in which electrons are the main source of ionization is approximately equal to the number of ion pairs produced in the discharge. This hypothesis is tested in this chapter using the data obtained from corona discharges and then it is utilized to quantify the NOx production in lowpressure discharges and streamer discharges.

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13.2 Testing the theory using corona discharges The working hypothesis of this chapter is that in cold electrical discharges, the total number of NOx molecules generated by the discharge is k times the total number of ion pairs produced in the discharge. Let us test this hypothesis by using the experimental data on corona discharges. In a recent study, Rehbein and Cooray [10] conducted an experiment to quantify the NOx production in corona discharges. In the study, a corona discharge is maintained in a coaxial geometry and the discharge voltage and the current are measured simultaneously with the concentration of NOx produced in the discharge chamber. If one neglects the ionization loss processes (i.e. attachment and recombination), the steady state current gives the rate of production of electrons and hence the rate of occurrence of ionizing events in the discharge. Since the total NOx production in the discharge over a given time interval is known, the data can be used to quantify the number of NOx molecules produced per ion pair. The results of this calculation gave k=0.6 for negative corona and k=1.0 for positive corona. Even though the NOx generating efficiency per ion pair is lower in negative corona than in positive corona, this result confirms that the number of NOx molecules produced in the discharge is approximately equal to the number of ionization events.

13.3 NOx generation in electron avalanches and its relationship to energy dissipation Consider a single electron accelerating in a background electric field of strength E. The number of ionizations caused by the electron in moving a unit length per unit is equal to the Townsend’s primary ionization coefficient a(p, E), which is a function of pressure p and the electric field E. Assuming that one ionization event corresponds to k number of NOx molecules, the total number of NOx molecules Nul produced by the ionization processes as the electron moves a unit length is given by Nul ¼ kaðp; EÞ

(13.1)

Now, the energy dissipated as the electron moves a unit length in the background electric field is given by eE, where e is the electronic charge. Thus, the number of NOx molecules produced per unit energy, Nue, is given by Nue ¼

kaðp; EÞ eE

(13.2)

Since a/p is a function of the reduced electric field E/p alone, the above equation predicts that Nue is only a function of E/p. Figure 13.1 shows how the value of Nue varies as a function of E/p. First note that the NOx production efficiency increases with increasing E/p. This also shows that for a given pressure, the NOx production per unit energy is not unique but depends also on the applied electric field. Consequently, in any experiment that is designed to obtain the NOx production efficiency of cold

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NOx molecules/J

6 × 1016

4 × 1016

2 × 1016

0 × 100 0 × 100

1 × 104 2 × 104 E/p (V/m/Torr)

3 × 104

Figure 13.1 NOx production per unit energy Nue as a function of E/p in electron avalanches electrical discharges, it is necessary to utilize voltage impulses in which the reduced electric field E/p remains the same.

13.4 NOx production in streamer discharges As an electron avalanche propagates towards the anode of a discharge gap, low mobile positive space charge accumulates at the avalanche head. When the avalanche reaches the anode, the electrons will be absorbed into it leaving behind the net positive space charge. Due to the recombination of positive ions and electrons, the avalanche head is a strong source of high energetic photons. These photons will create other avalanches in the vicinity of the positive space charge. If the number of positive ions in the avalanche head is larger than a critical value, the electric field created by the space charge becomes comparable to the background electric field and the secondary avalanches created by the photons will be attracted towards the positive space charge. The electrons in the secondary avalanches will be neutralized by the positive space charge of the primary avalanche, leaving behind a new positive space charge, little bit closer to the cathode. The process repeats itself and the positive space charge head travels towards the cathode as a consequence [11]. This discharge that travels towards the cathode from the anode is called a positive streamer. At atmospheric pressure, the total number of ions in the streamer head, w, is about 108 and the radius of the streamer head, Rs, is about 100 mm [12]. The streamer needs a background field of about 5–10 kV/cm, depending on polarity for continuous propagation. Consider a streamer moving in a uniform electric field of strength E. In order for the streamer to move a unit length in this electric field,

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the total number of ionizing events taking place in the vicinity of the streamer head is about w/2Rs and the total number of NOx molecules created by the streamer in moving a unit length, Ns, is Ns ¼

kw 2Rs

(13.3)

On the other hand, the amount of energy dissipated by the streamer channel in moving the unit distance is Eew. Thus, the production efficiency of NOx in streamer discharges, PNOx , in molecules/J is PNOx ¼

k 2eERS

(13.4)

Substituting k = 1, Rs = 10-4 m and E = 5  105 V/m (for positive streamers), we find that the streamer will make about 6  1016 NOx molecules/J. In a recent study, Cooray and Rahman [13] conducted an experiment in coaxial geometry to measure the NOx production efficiency of streamer discharges. Let us consider the cylindrical coaxial geometry. Assume that the peak voltage applied to the central conductor of the coaxial system is V. This voltage will create an electric field in the cylinder, which has its highest value at the surface of the conductor and then decreases inversely with increasing radial distance. Consider a streamer discharge initiated close to the inner electrode and moving in this electric field. Assume that the streamer discharge will propagate to a distance where the background electric field reaches Es, the critical electric field necessary for streamer propagation. If the applied voltage is V, then the charge that will be induced on a unit length of the inner conductor of the coaxial arrangement is given by 2pe0 V lnðb=aÞ

Q¼

(13.5)

where a and b are the radii of the inner and outer conductors. The electric field at a radial distance r from the inner conductor is then given by Er ¼

V r lnðb=aÞ

(13.6)

Since the streamers propagate to a distance where the background electric field is Es, the length of the streamers, ls, in the coaxial arrangement is given by ls ¼

V a lnðb=aÞEs

(13.7)

Then the number of ionizing events, Ns, generated during the propagation of the streamer is given by Ns ¼

wls 2Rs

(13.8)

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On the other hand, the energy released by the movement of the streamer head in the gap, Us, is given by Us ¼ ew

V lnðls =aÞ lnðb=aÞ

(13.9)

Thus, the number of NOx molecules produced per unit energy, Nue, is given by Nue ¼

k 2Es Rs lnðls =aÞ

(13.10)

For the experimental conditions reported in [13], i.e. a = 0.001 m, b = 0.15 m and V = 83  103 V, the calculated production efficiency is about 1.6  1016 molecules/J. The measurements produced 1–2  1016 molecules/J.

13.5 Discussion and conclusions Studies on the NOx production in the atmosphere by proton impacts show that the number of NOx molecules created is almost equal to the number of ion pairs created during the impact. By comparing theory with available experimental data it is shown in this chapter that this result is also valid for electrical discharges in which electron impacts are the main source of ionization. Based on this observation, the NOx production in electrical discharges where the electron production depends solely on the impact of electrons with neutral atoms is evaluated. The types of electrical discharges considered in this chapter are the corona discharges, electrical discharges at low pressure and streamer discharges. In a corona discharge associated with current amplitude I, the rate of occurrence of ionizing events (neglecting attachment and recombination) is I/e, where e is the electronic charge. This is equal to 6.25  1018I. Thus, the number of NOx molecules produced per second in the discharge is given by 6.25  1018kI in which the value of k depends on polarity (i.e. k  1 for positive polarity and k  0:6 for negative). It is of interest to note that this equation can be utilized to estimate the global production of NOx by ground corona associated with thunderstorms. Various studies estimate that the global corona current associated with ground corona is about 1–2 kA [14]. Substituting this value in the above equation and using k = 1.0 for positive corona, the number of NOx molecules produced globally by ground corona per second is estimated to be about (6–12)  1021. This is equivalent to an annual production of 0.01 Tg (N). During thunderstorms, it is not only at ground level that corona discharges are initiated. The thunderstorm itself is a large source of corona discharges. For example, in a thundercloud, charges may disperse from regions of high concentration to low concentration through corona discharges and the same could be the vehicle that transports the charges induced in conducting channels during neutralization events in to the bulk of the cloud. The theory as presented in this chapter could be applied to obtain the NOx production from these processes once the magnitude of the currents associated with these processes are known.

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Let us consider the results presented in Figure 13.1. These results are obtained using several assumptions. First, it is assumed that the electron generating mechanism in the discharge is the impact ionization due to energetic electrons. Second, it is assumed that the background electric field is uniform in the region in which the discharge is taking place. However, the calculation procedure can be modified rather easily to take into account the effect of non-uniform electric fields. Third, it is assumed that the space charge accumulated in space during the discharge will not distort the electric field locally. Due to these assumptions, the results as given in Figure 13.1 are valid for ‘Townsend’s like’ low-pressure electrical discharges where the space charge effects can be neglected. On the other hand, the data given in Figure 13.1 can be utilized to study the NOx production in any cold discharge with space charge distortions provided that the quantity E is replaced by the effective electric field in which the electron avalanches are generated. An example of a cold discharge in which the space charge effects cannot be neglected is a streamer discharge. In streamers the electron avalanches are initiated in the electric field of the streamer head that is much stronger than the background electric field in which streamers are propagating. In evaluating the NOx production of streamer discharges in this chapter, instead of evaluating the NOx production in each individual avalanche taking place at the streamer head, a simpler but an equivalent procedure based on the known mechanism of the streamer is utilized. In a streamer discharge, electron avalanches are generated in a specially varying electric field. The maximum value of the field, which is about 2  107 to 4  107 V/m, occurs at the head of the streamer (assuming 108 ions located within a radius of about 50–100 mm) and it decreases to about 3  106 V/m at a distance of about 200 mm from the head. Recall that our analysis gave about 2  1016 NOx molecules/J for streamer discharges moving in a background electric field of about 5  105 V/m. Comparison of this NOx production efficiency with the data given in Figure 13.1 indicates that in streamer discharges the electron avalanches are generated in an effective electric field of about 5106 V/m. Streamer discharges occurring in the atmosphere are associated mainly with the lightning leaders in ground flashes and cloud flashes. Actually, the propagation of the leader is mediated by streamer bursts emanating from the tip of the leader. They may also originate during the neutralization of these leaders by return strokes. Cooray et al. [15] utilized the theory as developed in this chapter to study the NOx production by streamers in lightning leaders. The experimental observations indicate that the space charge effects cannot be neglected in upper atmospheric discharges known as sprites even though these discharges are taking place at considerably low atmospheric pressures. For example, observations indicate that sprites give rise to streamer-like structures. Therefore, the electric field in which the avalanches are growing during the development of sprites may be considerably higher than the background electric field generated by the thunderclouds. However, if the dimension and the ion concentration in these streamer heads are known, an analysis similar to that utilized in this chapter to study the NOx production in streamer discharges at atmospheric

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pressure could be used to evaluate the NOx production in streamers in sprites. On the other hand, the elves are not associated with streamers and the results presented in Figure 13.1 could be applied directly if the background electric field that drives them is estimated. The reason for the development of streamers in sprites, even though they are taking place at considerably low atmospheric pressure, is the large dimensions involved with the region of ionization associated with these electrical discharges. This allows the accumulation of space charge over large volumes affecting the local electric field. On the other hand, laboratory discharges generated at the same pressures would not have streamer-type discharges because the special dimensions are not large enough in laboratory discharges to support them. The results presented in Figure 13.1 would be applicable directly in low-pressure laboratory discharges. In a study conducted recently by Peterson et al. [16], NOx generated by laboratory discharges at different pressures were measured, and based on the results an estimation was made on how the NOx production efficiency in molecules/J varies as a function of pressure. Based on the results the authors predicted that the NOx production efficiency increases with decreasing pressure. Unfortunately, in that study the applied voltage is not given and therefore it is difficult to find out whether the voltage applied at different pressures is such that the E/p ratio is the same at different pressures. As shown in the present chapter, the NOx production efficiency in these discharges depends on the ratio E/p and any change in this ratio when one moves from one pressure to another will also affect the NOx production.

References [1] Crutzen, P. J., The influence of nitrogen oxides on the atmospheric ozone content, Quart. J. R. Met. Soc., vol. 96, 320–325, 1970. [2] Gallardo, L. and H. Rodhe, Oxidized nitrogen in the remote pacific: the role of electrical discharge over oceans, J. Atmos. Chem., vol. 26, 147–168, 1997. [3] Chameides, W. L., The role of lightning in the chemistry of the atmosphere, in The Earth’s Electrical Environment, National Academy Press, Washington D.C, 1986. [4] Rahman, M., V. Cooray, V. A. Rakov, et al., Measurements of NOx produced by rocket-triggered lightning, Geophys. Res. Lett., vol.34, L03816, 2007, doi:10.1029/2006GL027956. [5] Orville, R. E., Lightning spectroscopy, in R. H. Golde (ed.), Lightning, Volume 1, Physics of Lightning, Academic Press, London, 1977. [6] Crutzen, P. J., Atmospheric interactions-homogeneous gas reactions of C, N and S containing compounds, in B. Bolin and R. B. Cook, (eds.), The Major Bio-geochemical Cycles and Their Interactions, SCOPE, Paris, 1983. [7] Nicolet, M., On the production of nitric oxide by cosmic rays in the mesosphere and stratosphere, Planet. Space Sci., vol. 23, 637–649, 1975. [8] Jackman, C. H., H. S. Porter and J. E. Frederick, Upper limits on production rate of NO per ion pair, Nature, vol. 280, 170, 1979.

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[9] Rahman, M., V. Cooray, G. Possnert and J. Nyberg, An experimental quantification of the NOx production efficiency of energetic particles in air, J. Atmos. Solar-Terr. Phys., vol. 68(11), 1215–1218, 2006. [10] Rehbein, N. and V. Cooray, NOx production in spark and corona discharges, J. Electrostat., vol. 51–52, 333–339, 2001. [11] Cooray, V., Mechanism of electrical discharges, in V. Cooray (ed.), The Lightning Flash, The Institution of Electrical Engineers, London, 2003. [12] Van Veldhuizen, E. M. and W. R. Rutgers, Pulsed positive corona streamer propagation and branching, J. Phys. D: Appl. Phys., vol. 35, 2169–2179, 2002 [13] Cooray, V. and M. Rahman, Efficiencies for production of NOx and O3 by streamer discharges in air at atmospheric pressure, J. Electrostatics, vol. 63, 977–983, 2005. [14] Roble, R. G. and I. Tzur, The global atmospheric-electrical circuit, in The Earth’s Electrical Environment, National Academy Press, Washington, D.C., 1986. [15] Cooray, V., M. Rahman and V. Rakov, NOx production in lightning flashes, Proceedings of International Conference on Atmospheric Electricity, Beijing, 2007. [16] Peterson, H., M. Bailey, J. Hallett and W. Beasley, NOx production in laboratory simulated blue jets and sprite discharges, 12th Conference on Cloud Physics, P2.13, Madison, WI, 2006.

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Chapter 14

On the NOx production by laboratory electrical discharges and lightning* Vernon Cooray1, Mahbubur Rahman1 and Vladimir Rakov2

14.1 Introduction An assessment of the global distribution of nitrogen oxides is required for an adequate description of tropospheric chemistry and in the evaluation of the global impact of increasing anthropogenic emissions of NOx [1]. In the mathematical models utilized for this purpose, one needs to specify as inputs the natural as well as man-made sources of nitrogen oxides in the atmosphere. Lightning is one of the main natural sources of nitrogen oxides in the atmosphere, and it may be the dominant source of nitrogen oxides in the troposphere in equatorial and tropical South Pacific regions [2]. Thus, an accurate quantification of nitrogen oxide production by thunderstorms is necessary for further development of the chemical models of the troposphere and in the evaluation of the effects of the man-made nitrogen emissions in the terrestrial atmosphere. Because of the difficulty of making direct measurements of NOx produced by natural lightning flashes, researchers have employed indirect methods to quantify the global production of NOx [3–12]. Because of a large number of uncertainties involved in these methods, the estimates of global NOx production by lightning flashes available in the literature vary by two orders of magnitude, from 1 to 100 Tg (N) per year. In estimating lightning produced NOx by indirect methods scientists have usually utilized the following two procedures: (1) a laboratory measurement of the number of NOx molecules per unit energy for a laboratory spark is made and the result is extrapolated to lightning by multiplying this measured value by estimated energy of lightning event. (2) A ground-based NOx measurement is made in the vicinity of a natural lightning flash and from it the source strength is estimated by making suitable assumptions concerning the fluid dynamics of the NOx flow from the source to the measurement point.

*This

material was published previously by the same authors in Journal of Atmospheric and SolarTerrestrial Physics, vol. 71, pp. 1877–1889, 2009. It is reproduced here by permission from the journal. 1 Department of Electrical Engineering, Uppsala University, Sweden 2 Department of Electrical and Computer Engineering, University of Florida, USA

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In studies of NOx production by lightning flashes, only return stroke is assumed to be the NOx source and the effects, if any, of leaders, continuing currents, M components, and K processes are neglected. Many studies have excluded cloud flashes from NOx estimates assuming their contribution to be insignificant. Recent direct measurements of NOx produced by triggered lightning flashes [13] show, however, that it is not only the return stroke in ground flashes but also other slow processes such as continuing currents are significantly contributing to the NOx production. This calls for a more thorough investigation of the problem including different processes in both ground and cloud flashes. In this chapter, we describe how this could be achieved. First, we quantify the NOx production by different discharge processes of lightning flashes such as return strokes, leaders, continuing currents, and M components and subsequently this information is utilized to quantify NOx production by a typical lightning flash containing all these elements. The results are used to obtain a global estimate of the lightning produced NOx. While performing this analysis, we will also attempt to provide answers to the following important questions related to the quantification of NOx emission by lightning flashes: (a) Is the energy of discharge the correct scaling factor to extrapolate NOx emission from laboratory discharges to lightning flashes? (b) Does the shape of discharge current waveform influence the NOx emission? (c) What are the relative contributions from leaders, return strokes, continuing currents, M components and K processes to the NOx production by lightning flashes? (d) Can one neglect the cloud flashes in evaluating the global NOx production?

14.2 NOx production by laboratory sparks In order to evaluate NOx production from electrical discharges or sparks it is necessary to estimate the amount of air heated to a given temperature in the discharge. This requires information concerning the dimension of the hot core of electrical discharges and how it will change for different current waveforms. Therefore, let us first investigate how the hot core of the electrical sparks varies as a function of its current.

14.2.1 Radius of spark channels According to Braginskii [14], the radius of spark channel at time t, r(t), as a function of current is given by rðtÞ ¼ kro1=6 i1=3 t1=2

(14.1)

where r(t) is in m, t is in microseconds, i is the instantaneous current in the spark channel in kA, k is a constant and ro is the air density at atmospheric pressure (1.29  103 g/cm3). In deriving this expression, Braginskii assumed that the current increases linearly with time. However, the current in sparks decays after reaching the peak value, and Braginskii noted that the value of constant k (originally set to 0.93  103) may have to be changed if the equation is to predict the time variation in

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channel radius of sparks. Cooray and Rahman [15] have made a comparison of the results predicted by (14.1) with the constant suggested by Braginskii [14] with the experimental data published by Flowers [16] and Higham and Meek [17] and found that it overestimates the radius of spark channels. Table 14.1 gives channel radii observed in the experiment and constants k in (14.1) that give the best fit to the experimental data. Based on this comparison, Cooray and Rahman [15] estimated that k = (0.328  0.05)  103. Recently, Perera et al. [18] analysed the diameter of spark channels of length 30 cm using a photographic technique and the maximum channel diameter is obtained as a function of spark peak current for both positive and negative polarities. Perera et al. [18] compared the measured maximum channel diameter with the one obtained from (14.1) using the current waveform measured in the discharge as input. The results of that study confirm that the Braginskii’s original constant overestimates the channel diameter, whereas the constant suggested by Cooray and Rahman [15] (i.e. 0.328  103) provides a reasonable fit to the data. Based on these experimental validations, the value of the constant suggested by Cooray and Rahman [15] is used in the analysis presented here. Lightning currents were directly measured on tall towers and at the triggered lightning channel base (e.g. Berger [19]; Fisher et al. [20]). Mathematical expressions to describe the waveform of typical first and subsequent return-stroke currents recorded by Berger [19] are found in CIGRE Study Committee 33 Report [21] and Nucci et al. [22]. First let us utilize the typical current waveforms found in the above references to study how the radii of the first and subsequent return strokes vary as a function of peak current. Figure 14.1 shows how the maximum radius of the channel given by (14.1) varies as a function of peak current for first and subsequent return strokes. Note that the radius of a typical first return stroke (with a peak current of 30 kA) is about 2 cm and that of a typical subsequent stroke (peak current 12 kA) is about 1 cm. These values are in agreement with the radii of return-stroke channels estimated in photographic studies as summarized by Orville [23]. Table 14.1 Measured spark radii and values of k in (14.1) giving the best fit to data Reference

Peak current (kA)

Rise time (ms)

Decay time (ms)

r (tm) (cm)

tm (ms)

Value of k giving the best fit (10–3)

Flowers (1943)

17 18 22 26 0.185

8 6 3.3 8

60 60 60 60 20

0.8 0.7 0.615 0.9 0.19

8 6 3.3 8 12

0.37 0.35 0.41 0.35 0.259

20 10 10 10

0.145 0.225 0.195 0.16

12 12 12 9

0.254 0.34 0.32 0.31

Higham and Meek (1950)

0.250 0.500 0.400 0.300

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Radius (m)

0.025 (i) 0.02 (ii)

0.015 0.01 0.005 0

20

40 60 Peak current (kA)

80

Figure 14.1 The maximum radius of the channel as given by (14.1) as a function of peak current for first (i) and subsequent (ii) return strokes. In these calculations, the typical current wave-shapes for first and subsequent return strokes found in the literature were used

14.2.2 The volume of air heated in a spark channel and its internal energy Detailed studies of the temporal variation in temperature and pressure of lightning discharges and long laboratory sparks show that the channel temperature reaches a peak of about 25,000–30,000 K and the channel pressure is of the order of 10 atm in a few microseconds after the commencement of current flow through the channel [24–26]. Then the channel plasma cools down mostly due to the channel expansion, loss of heat due to radiation and engulfing cold air from outer zones. This phase may last for about few tens of microseconds. At the end of this phase, the channel temperature reduces to a value of about 15,000 K and the pressure inside the channel attains the atmospheric pressure clamping down the pressure driven expansion of the channel. Experimental data that support this scenario are provided, both for laboratory sparks and for lightning flashes, by Orville [24,25,27] and Orville et al. [26]. Another experimental observation that supports this scenario is the following. The experimental data obtained by Orville [24] on the temperature of the lightning stepped leader channel show that at the formation of the step the channel temperature increases to about 25,000 to 30,000 K. Subsequently, the temperature decreases to about 15,000 K and remains at that level until the channel is retraced by the return stroke. The theoretical calculations of Paxton et al. [28], Hill [29] and Plooster [30] also indicate that by the time the pressure inside the spark channel reduces to the atmospheric pressure the average temperature in the channel is close to 20,000–15,000 K. After the pressure equilibrium is reached,

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the cooling of the channel takes place mainly due to the entrainment of cold air across the channel boundaries into the hot core of the channel effectively reducing the diameter of the hot core [31]. Similarly, (14.1) predicts that the spark channel expands initially with time and its maximum radius is attained in a few microseconds to a few tens of microseconds, depending on the peak and the duration of current. After this, the radius of the hot channel decreases, while the channel resistance starts to increase with time. Based on the experimental and theoretical data [24–26,28–30], we assume in our analysis that the pressure in the channel at the time of maximum radius is close to atmospheric pressure and that the average temperature of the hot air in the channel at this time is close to 15,000 K. In reality, the channel temperature is not uniform across the cross-section of the channel. But the theoretical calculations of Paxton et al. [28] and Hill [32] indicate that after a few tens of microseconds the radial temperature distribution is relatively flat with a sharp decrease to ambient temperature at the channel boundaries. This justifies the use of an average value to describe the channel temperature. Further, the temperature of 15,000 K estimated by Orville [27] when the channel was close to pressure equilibrium is the average temperature across the channel justifying our selection of 15,000 K as the average temperature. Thus, we assume that in a cylindrical spark channel the volume of air, V, heated to a temperature of about 15,000 K is given by 2 V ¼ lprmax

(14.2)

where l is the length of the spark and rmax is the maximum radius of the spark channel given by (14.1) with k = 0.328  103.

14.2.3 NOx production in spark channels It is believed that in an electrical discharge mainly NO is being produced through a series of high temperature reactions, which is confirmed by different laboratory experiments. Depending on the presence of excess O2 and O3 and the residence time, the experimentally found NO/NOx ratios vary significantly. However, the total number of molecules of NOx (NO + NO2) produced by a discharge is equal to the total number of NO molecules produced in the discharge, which is calculated in this section. As mentioned previously, a procedure outlined by Borucki and Chameides [33] has been adopted to quantify the number of NO molecules that will be ‘fixed’ as the discharge channel cools down to ambient temperature. The amount of NO produced by a discharge via high temperature reactions is determined by the freezeout temperature, Tf. This temperature is defined as follows: Let us denote by tNO(T) the time required by NO to reach thermodynamic equilibrium at a given temperature, T. The freeze-out temperature is defined in such a way that tNO(Tf) = tT(Tf) where tT(T) is the characteristic cooling time of the heated gas. When T > Tf then tNO (T) < tT (T) and the chemical reactions are fast enough to keep NO in thermodynamic equilibrium. If T < Tf then tNO (T) > tT(T) and chemical reactions are too slow to adjust to the rapidly decreasing temperature. In this case, the amount of

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NO at Tf in the mixture is frozen out. Thus, if N is the number of air molecules heated above Tf, the number of NO molecules generated by the process, MNO, is given by MNO ¼ Nf ðTf Þ

(14.3)

where f (Tf) is the fraction of NO molecules in the gas at temperature Tf [33,34]. The value of Tf depends on the cooling rate of the gas in the discharge. For lightning-like discharges, Tf and corresponding f (Tf) were estimated to be around 2,660 K and 0.029 respectively [33,34]. These values have been used in the calculations presented here. In order to evaluate MNO it is necessary to evaluate N, the number of molecules heated above Tf, which can be found from the following equation:

 Eh To  No  (14.4) N ¼V Ef Tf where V is the volume of hot air in the discharge channel given by (14.2), Eh and Ef are the internal energy of air per unit volume at the temperatures Th and Tf, respec tively, To is the standard temperature and No is the number of molecules per unit volume at standard temperature and pressure. In the calculations, we have assumed that Th = 15,000 K, Tf = 2660 K, To = 273 K and No = 2.69  1025 m3. In deriving (14.4) we have assumed that, as the channel cools after reaching the pressure equilibrium due to the entrainment of cold ambient air into the discharge channel, not much of the energy escapes the channel as radiation. Moreover, in the derivation we have neglected the production of NO, if any, by the shock wave generated by the discharge. Both these assumptions are supported by the calculations of Hill et al. [31]. The number of NO molecules produced by the discharge is then given by

 Eh T0  N0  (14.5) MNO ¼ f ðTf Þ  V  Ed Tf The values of Eh and Ef are calculated using the set of equations given by Plooster [30,35] describing the variation in internal energy of air as a function of temperature and pressure. Since the radii of spark channels and hence the amount of air heated to a given temperature depend not only on the peak current but also on the temporal variation of the current, the amount of NOx produced by a spark depends both on the peak current and the current waveshape. Thus, if the current waveform in the discharge channel is known, then (14.1) to (14.5) can be used to evaluate the number of NO molecules produced by the discharge.

14.2.4 Efficiency of NOx production in sparks with different current wave-shapes Let us now check the validity of (14.5) by comparing its predictions with the available experimental data. The efficiency of NOx production by electrical discharges is evaluated in the references [33,36–41]. Unfortunately, only in a few cases the current waveform associated with the electric discharges used in the experiment is given. These are the studies conducted by Wang et al. [40], Rehbein and Cooray [39] and Rahman et al. [41]. The current waveform associated with the

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sparks analysed by Wang et al. [40] has a rise time of 30 ms and a decay time (time taken by the current to decay to 1/2 of its peak value) of 400 ms. The current waveform associated with the sparks in the experiment conducted by Rehbein and Cooray ([39]; the current waveforms are given in Ref. [42]) was oscillatory with a frequency of 2.8 MHz and decay time of 2 ms. The current waveform in the spark experiments conducted by Rahman et al. [41] had a rise time of about 0.3 ms and a decay time of about 25 ms. The NOx production per unit length by these discharges is evaluated using the equations presented in the previous section and the results, together with the experimental data, are shown in Figure 14.2. First note that there is a reasonable agreement between the experimental data and the theory. Second, note how the NOx production depends on the wave-shape of current. For a given peak current, a current with a longer duration gives rise to more NOx than a current with a shorter duration. The reason for this is that the volume of the discharge channel increases with increasing the duration of current waveform. Since the volume of the discharge channel is a measure of the internal energy retained in the 1E+022 (a) (b) (c) (d)

NOx molecules/m

1E+021

1E+020 (e) 1E+019

1E+018

1E+017

100

1000 10000 Peak current (A)

100000

Figure 14.2 NOx production efficiency of laboratory sparks and lightning return strokes as a function of peak current. (a) Theoretical prediction based on the current waveform of the study conducted by Wang et al. [40], (b) prediction based on the typical first return-stroke current waveform, (c) prediction based on the typical subsequent return-stroke current waveform, (d) prediction based on the current waveform in the sparks studied by Rahman et al. [41] and (e) prediction based on the current waveform in the sparks studied by Rehbein and Cooray [39]). The experimental data corresponding to different studies (Rehbein and Cooray: hollow triangles; Wang et al.: hollow circles; Rahman et al.: solid circles) are also shown in the figure

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discharge channel, in long duration currents more energy goes into the internal energy of the discharge than in short-duration currents. As the internal energy of the discharge increases the mass of air that is being heated beyond the NOx, freeze-out temperature also increases leading to a higher NOx production. To the best of our knowledge, this is the first time that the dependence of NOx production on the shape of the current waveform flowing in the discharge channel is explicitly recognized.

14.2.5 NOx production in sparks as a function of energy In order to calculate the energy dissipated in the discharge channel, we will employ again the spark discharge channel model of Braginskii [14]. Two simplifying assumptions made in developing this model are as follows: (a) The conductivity of the channel is uniform across the channel cross-section. (b) The conductivity of the channel does not vary as a function of time. With these assumptions, the energy dissipated in the discharge channel is given by

ð1 U ¼i

i2 prðtÞ2 s

dt

(14.6)

0

where s is the effective conductivity of the spark channel, r(t) is the radius of the channel at time t (given by (14.1)) and l is the length of the channel. Braginskii [14] recommended the use of 104 S/m as the effective conductivity of the channel. In order to test the validity of this equation, the energy in the sparks studied by Rahman et al. [41] was evaluated by integrating the product of voltage and current waveforms. For 35 current waveforms, the total energy calculated from the above equation agrees within 15%, when the value of s is assumed to be 0.65  104 S/m. Paxton et al. [28] studied the development of lightning channel taking into account the detailed physics of the complex electrohydrodynamic and thermodynamic processes. The current waveform used by Paxton et al. had a linear rise to peak followed by an exponential decay. The peak value, rise time and decay time of the current waveform used by Paxton et al. were 20 kA, 5 ms, and 50 ms respectively. The calculated total energy dissipation in the discharge up to 50 ms was about 5 kJ/m. Equation (14.6) for the same current predicts the same energy dissipation when s = 104S/m. These comparisons suggest that (14.6) can give a reasonable value for the total energy dissipated in the discharge for values of s ranging from 0.65  104 to 104 S/m. In the calculations to follow, we will use s = 0.8  104 S/m. Figure 14.3 depicts the energy dissipation per unit length in electrical discharges having current signatures similar to those of typical first and subsequent strokes, as a function of peak current. According to Figure 14.3, typical first (30 kA) and subsequent (12 kA) return strokes will dissipate about 20 and 2.5 kJ/m, respectively, in channel sections close to ground. In Figure 14.4, the calculated yield of NOx as a function of the energy dissipated in the discharge is depicted for current waveforms corresponding to the experiments conducted by Wang et al. [40], Rehbein and Cooray [39] and Rahman et al. [41]. For

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80,000

Energy (J/m)

60,000

(i) 40,000

(ii) 20,000

0

0

20

40 60 Peak current (kA)

80

Figure 14.3 The energy dissipation per unit length in (i) first return strokes and (ii) subsequent return strokes as a function of peak current 1E+022 (a) 1E+021 NOx molecules/m

(b) 1E+020

(c)

1E+019

1E+018

1E+017 1

10

100 1,000 Energy (J/m)

10,000 1,00,000

Figure 14.4 NOx production efficiency of laboratory sparks and lightning return strokes as a function of the energy dissipated in the discharge. (a) Theoretical prediction based on the current waveform of the study conducted by Wang et al. [40], (b) prediction based on the current waveform in the sparks studied by Rahman et al. [41] and (c) prediction based on the current waveform in the sparks studied by Rehbein and Cooray [39]). The experimental data corresponding to different studies (Rehbein and Cooray: hollow triangles; Wang et al.: hollow circles; Rahman et al.: solid circles) are also shown in the figure

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comparison purposes the experimental data are also shown in the same diagram. It is important to note that the experimental evaluation of the energy dissipated in spark channels is not trivial. Errors may result from the measurement of voltage across the spark channel either due to inductances in the circuit or due to the response characteristics of high-voltage dividers. Moreover, for large currents the discharge channel transverse diameters may reach several centimetres and in the case of short gaps (about 3 cm), as in the case of Wang et al. [40] study, the electrode effects may influence the total energy measured. Nevertheless, as one can see in Figure 14.4, there is a reasonable agreement between the theory and the experiment. One important conclusion that can be made from the results of this analysis is that the energy dissipated in the discharge cannot be used as a scaling factor in extrapolating laboratory data to lightning. The reason for this is that for a given energy the NOx production efficiency of a spark depends on the waveform of the discharge current.

14.3 NOx production in discharges containing long-duration currents Currents having relatively long durations ranging from several milliseconds to hundreds of milliseconds are associated with different lightning processes. One such process is the stepped leader. A stepped leader may carry currents of tens to hundreds of amperes with durations of some tens of milliseconds. Long-duration currents initiated by return strokes and flowing along the channel to ground are known as continuing currents. Discharge processes taking place inside the cloud can also generate long-duration currents. The theory presented in Section 14.2.3 cannot be applied to calculation of NOx production in channels carrying long-duration currents. When the current duration is long, ample time is available for the mixing of cold air into the discharge channel while the current is still flowing in the channel. Thus, the energy dissipated in the channel is continuously being utilized to heat cold air coming into the channel. At the same time hot air leaving the channel as it cools down creates NOx in gas volumes adjacent to the discharge channel. In the case of long-duration current, this turbulent mixing of cold air into the channel and hot air leaving it has to be taken into account in the calculation of NOx production. This prevents us from using the procedure for calculating the NOx production in spark channels, described in Section 14.2.3. Recent measurements conducted by Rahman et al. [13] show that the NOx production by steady currents in rocket-triggered lightning is proportional to the charge transferred along the channel. According to their measurements, the NOx production efficiency of long-duration currents is equal to about 2  1020 molecules/m/C.

14.4 NOx production in streamer discharges The propagation of leaders in long laboratory sparks and lightning is facilitated by streamer discharges taking place at the forward moving leader tip. Streamer discharges may also be responsible for the leakage of charge from the hot leader

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channel core, which is at extremely high potential, into the corona sheath. The air temperature in a streamer is close to the ambient temperature, whereas the electron temperature can be several tens of thousands of degrees Kelvin. The collision between energetic electrons and neutral molecules leads to the dissociation of N2 and O2 in the streamer discharges and the resulting chemistry gives rise to both NOx and O3. However, the theory developed for the NOx generation in hot sparks cannot be utilized here, because the NOx production process is not controlled by temperature variation. Recently, Cooray et al. [43, see chapter 13] demonstrated that a theory developed for studying NOx production by solar proton events [1,44] could be utilized to calculate the NOx production from corona and streamer discharges. According to this theory, the NOx production rate is approximately equal to the rate of production of ion pairs during the proton impact. Since the bulk of ionization in such events is produced by secondary electron impacts, Cooray et al. [43] applied the same concept to study the NOx production in low pressure gas discharges, corona discharges and streamer discharges in which the source of ionization is the electron impacts. Let us assume that the radius of the streamer channel is Rs and the number of charge particles at the streamer head is Nhead. Since the number of ionizing events created by a streamer in moving a unit length is equal to Nhead/2Rs, according to Cooray et al. [43], the number of NOx molecules produced by a streamer in propagating a unit distance is given by kNhead/2Rs, where k is the number of NOx molecules generated per ionizing event. Using experimental data for corona Cooray et al. [43] demonstrated that k  1 for positive polarity and k  0.6 for negative polarity.

14.5 NOx production in ground lightning flashes Having outlined the procedure to evaluate the efficiency of NOx production in sparks, continuing currents and streamer discharges we are now in a position to incorporate them all into a single model that can be used for evaluating the NOx production by lightning flashes.

14.5.1 The model of a ground lightning flash As summarized by Cooray [45], a ground flash is initiated by an electrical break down process in the cloud that is called the preliminary breakdown. This process leads to the creation of a column of charge called the stepped leader that extends from cloud to ground in a stepped manner. On its way towards the ground a stepped leader may give rise to several branches. Once the connection of the stepped leader to ground is made, a nearly ground-potential wave and the associated luminosity wave travel along the leader channel towards the cloud at a speed comparable to that of light. This wave is called the return stroke. Although the current signature associated with the return stroke proper tends to have duration of a few hundred microseconds, the return-stroke current may not go to zero within this time, and a low-level current may continue to flow for tens to hundreds of milliseconds. Such long-duration currents are called continuing

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currents. Continuing current longer than 40 ms tends to follow subsequent (as opposed to first) strokes, described below. The arrival of the first return-stroke front at the cloud end of the return stroke channel leads to a change of potential in the vicinity of this point. This change in potential may initiate a positive discharge that travels away from the upper end of the return-stroke channel (the so-called J-process). When a fresh discharge is created in the previously ionized channel, the process that follows depends on the conditions along the channel. If the channel carries a continuing current, there will be a wave that travels towards the ground and produces a reflection there. This process is called the M component. If the channel carries essentially no current, the downward-moving wave may take the form of dart leader that travels towards the ground and produces a return stroke there. Such return stroke is called the subsequent return stroke. Processes similar to those occurring after the first return strokes may also take place after subsequent return strokes. Unsuccessful dart leaders and other transient processes involving in-cloud channels are referred to as K changes. In evaluating the NO x production by a ground flash, one has to consider all these processes. In the present study, an attempt is made to include the various lightning processes in the estimation of the global NOx production by ground flashes. In this evaluation, a ground flash is represented by the following model. The geometry of the lightning channel consists of a vertical section of height H and a horizontal section of length L. The horizontal channel that is located in the cloud consists of n branches of equal lengths. Each branch is created by a leader discharge and during its creation the leader current is confined to that particular branch, i.e. it does not flow along other branches. In lightning flashes giving rise to continuing currents, the source is confined to a single branch, i.e. the current passes through a branch and follows the vertical channel to ground. Processes to be taken into account are the leaders, return strokes, continuing currents and M components in the vertical channel section and leaders and K changes in the horizontal channel sections. A cloud flash is represented by two networks of horizontal channels, one in the positive charge region and the other in the negative, connected to each other by a vertical channel. The geometry of the horizontal channels is identical to the one assumed for ground flashes.

14.5.2 NOx production in different processes in ground flashes 14.5.2.1 Leaders Corona sheath A leader channel consists of a hot core surrounded by a corona sheath. The corona sheath is created through the action of streamer discharges, and the charge deposited in the corona sheath by the streamers is supplied by the current flowing in the hot core. Both these processes (i.e. streamer discharges and current flow along the core of the leader) have to be considered in evaluating the NOx production by leader discharges. In this section, we will concentrate on the former.

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According to bidirectional leader concept, the vertical channel of a negative ground flash is forged by negative stepped leaders and the channels in the cloud are created by positive leaders moving away from the point of origin of the flash. In the analysis to follow, we assume that the magnitude of the charge deposited per unit length of the leader channel is the same on both vertical and horizontal channels. Let us denote this by r. We also assume that most of this charge resides in the corona sheath and the transport of this charge into the corona sheath is mediated by streamers. Since most of the charge of the streamer is located at the head of the streamer [46], the number of streamers, Ns, per unit length of the leader channel is Ns ¼

r eNhead

(14.7)

where Nhead is the charge on the head of the streamer channel. Applying the Gauss law over a cylindrical surface encompassing the whole corona sheath, one obtains the radius of the corona sheath of a positive leader, Rc+, as Rcþ ¼

r ð2peo Esþ Þ

(14.8)

where Es+ is the critical background electric field necessary for the propagation of positive streamers. Note that in writing down the above equation, we assume that all the charge in the corona sheath is located inside the radius Rc+ and the electric field at this outer edge of the corona sheath is equal to Es+. If the electric field in the streamer region remains constant at this critical electric field, in order to satisfy the boundary conditions in coaxial geometry the volume charge density in the streamer region should decrease inversely with radius. Since most of the streamer charge is located at its head, this condition requires the number of streamers moving out from the central conductor to decrease linearly with radial distance. In other words the number of streamers having a given length l is two times the number of streamers of length 2l provided that 2l < Rc+. Thus, streamers travel, on average, a distance of Rc+/2 in creating the corona sheath. Using the expression for the number of NOx molecules generated by a single streamer in moving a unit length derived previously, we find the number of NOx molecules created by positive streamers per unit length of positive leader channel, hstr+, as hstrþ ¼

K þ r2 8pe0 eEs þ Rs

(14.9)

where k+ is the number of NOx molecules generated per ionizing event in positive discharges. Similarly, the number of NOx molecules generated by negative streamers per unit length of the negative leader channel is given by hsrt ¼

k  r2 8pe0 eEs Rs

(14.10)

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where the parameters have the same definition as in previous equation but corresponding to negative streamers. Thus, the total number of NOx molecules generated by corona sheath of the leaders in the whole ground flash is given by NOxleadersheath ¼ hstr H þ hstrþ L

(14.11)

The charge per unit length of lightning stepped leaders is expected to be in the range of 0.0005–0.001 C/m. In the calculations, we assumed that r = 0.0005 C/m. The values of Es+ and Es- are equal to 500 kV/m and 1 MV/m respectively [47]. We assume that N = 108 [46]. We also assumed that the values of k = 0.6 and k+ = 1.0 are independent of pressure. Recall that k- and k+ refer to the number of NOx molecules created by an ionizing event. The assumption is based on the study of Jackman et al. [48] whose theoretical calculations predict that NOx molecules per ionizing events does not change significantly with increasing altitude and hence with pressure. We have also assumed that the values of Es+ and Es- do not vary with pressure. In reality, they decrease linearly with pressure but that effect is somewhat compensated by the increase in the size of the streamer head with decreasing pressure. Substituting these values in (14.9), we obtain hstr+  2  1020 for r = 0.0005 C/m.

NOx production in the hot core of the leader

The average speed of propagation of lightning stepped leaders is about 2  105 m/s and to supply a charge per unit length equal to 0.0005 C/m the current flowing along the hot core should be about 100 A. As the leader progresses, this current will continue to flow in any given channel section as long as the conditions are suitable for the continuous propagation of the leader head. Let us represent the current flowing along the stepped leader channel by Ils. If the linear charge density along the leader channel is constant and equal to r then Ils = vr where v is the speed of propagation of the head of the leader channel. Let hlea be the number of NOx molecules generated per unit length per unit charge by the current flowing in the core of the leader channel. Thus, the number of NOx molecules generated by a unit length of the leader channel due to this current is hleaIls td, where td is the time over which a current of amplitude Ils flows along the core of the channel section. Consider the vertical channel of length H. In a channel element of length dz located at a height (H–z) above ground level the duration of this current is (H–z)/vs, where vs is the speed of progression of the leader head. Note that this time is equal to the time needed for the leader head to travel the distance from the channel element to the ground. The total number of NOx molecules generated in the channel element by the core current after correction for the pressure is hleaIls (H–z)e–(H–z)/lr dz/vs. The total contribution from the vertical channel section can be obtained by integration of this expression from 0 to H. In constructing the above equation, we have assumed that the atmospheric pressure decreases exponentially with height with a decay height constant lp and the efficiency of NOx production by hot discharges decreases linearly with pressure [49,50].

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The evaluation of the contribution to the NOx production from non-vertical incloud channels is more complicated. The channel system inside the cloud may consist of many branches and at a given time only a few of these branches may be developing [51–53], and hence the core current is active only along those branches at that time. Thus, one problem in evaluating the NOx production in the core of the non-vertical in-cloud channel is the difficulty of knowing the length of the channel sections in which the current is flowing at a given time. Consider that the nonvertical in-cloud channels comprise n identical branches connected to the top of the vertical channel. We presume that these channels in the cloud are also created by processes similar to that of lightning leaders observed in ground flashes, and therefore, their current and speed of development are also identical to those of these leaders. As mentioned in Section 14.5.1, we assume that the core current will flow in each branch only when that particular branch is being formed. In reality, core current may pass from an active branch to a previously formed branch thus increasing the total length of the channel sections supporting a core current at a given time. Consider the development of a horizontal channel section inside the cloud. Let us direct the coordinate x along the channel section. Consider an element dx on this channel located at a distance x from the origin of the section. In this element, the current flows for a duration of (l – x)/vs where vs is the speed of development of the channel and l is the length of the channel. Thus, the number of NOx molecules produced in this channel element is hleaIls–H/lp(l–x)dx/vs. The total NOx production in the channel section can be obtained by integrating the above expression from 0 to l. The result of this integration is hleaIlsI2e–H/lp/2vs. Since we have assumed that there are n identical branches and the total length of the horizontal channels is L the number of NOx molecules produced by the current flowing through the core of the channels inside the cloud is given by hleaIlsL2e–H/lp/2vsn. A similar procedure can be used to evaluate the NOx pro duction along the vertical section of the leader channel, but the mathematics is slightly more complicated due to the fact that the pressure varies along the channel. After applying the mathematics, one can show that the total number of NOx molecules produced by the core current in the stepped leader channel of the ground flash (including the branches in the cloud) is  h Ils  2 lp  lp eH=lp ½H þ lp  NOxslcore ¼ lea vs (14.12a) h Ils þ lea L2 eH=lp 2vs n Using the same procedure the number of NOx molecules produced in the dart leader channel core can be written as NOxdlcore ¼

hlea Ild 2 ðlp  lp eH=lp ½H þ lp Þ Vd

(14.12b)

where Idl is the current in the dart leader and vd the speed of dart leaders. To evaluate this equation, it is necessary to have values for hlea, n, vs, Ils, vd and Ild. As

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pointed out in Section 14.3, the efficiency of NOx production by long-duration currents is about 21020 molecules/m/C. We assume that this is also true for the currents flowing along the leaders channels, i.e. hlea = 2  1020 molecules/m/C. This assumption is justified because, like continuing currents, leaders also support currents with amplitudes on the order of hundred amperes or more for durations of many milliseconds depending on the time of travel. The channel structure inside the cloud, as revealed from interferometric studies, can be approximated by a few large channels connected to the main channel [51,52]. Thus, the value of n may lie in the range of, say, 3–10. Optical observations of the stepped leaders and the interferometric studies show that the speed of development of stepped leader channels in virgin air, vs, is about 2  105 m/s [51,54]. Measurements conducted with both natural and triggered lightning show that the speed of dart leaders, vd, is about 107 m/s [55–57]. The currents in either the stepped leaders or dart leaders cannot be measured directly. But, inferences based on electric field measurements show that stepped and dart leaders are associated with currents of the order of 100 A and 1 kA [58–60], respectively.

14.5.2.2

Return strokes

In assessing the NOx production by first and subsequent strokes, we make the following assumptions. (a) The shape of the current waveform at the channel base of the first and subsequent return strokes are similar to the typical waveforms constructed by CIGRE Study Committee 33 [21] and Nucci et al. [22]. (b) The return-stroke peak current decreases linearly along the vertical section of the channel and reducing to zero amplitude at the cloud end of the vertical channel. (c) The return-stroke channel is vertical from ground level to the height of the charge centre, H. With these assumptions, the NOx produced by the first return stroke after correcting for the decrease in pressure with height is NOxfr ¼ hfr lp ½1 

lp lp H=lp þ e  H H

(14.13)

where hfr is the number of NOx molecules produced by a unit length of the discharge at atmospheric pressure having a current identical to that of a typical first return stroke. In the calculation, we assume that the peak current of a typical first return stroke is 30 kA. Note that, as depicted by curve b in Figure 14.2, NOx production depends on the peak current of the first return stroke. From this figure we estimate that hfr=5.91020 molecules/m. Similarly, the number of NOx molecules generated by a subsequent return stroke is given by NOxsr ¼ hsr lp ½1 

lp lp H=lp þ e  H H

(14.14)

where hsr is the number of NOx molecules produced by a unit length of the discharge at atmospheric pressure having a current identical to that of a typical subsequent return stroke. In the calculation, we assume that the peak current of a typical subsequent return stroke is 12 kA. Note again that, as depicted by curve c in

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Figure 14.2, NOx production also depends on the peak current of the subsequent return stroke. From this figure, we estimate that hsr = 1.4  1020 molecules/m. The use of return-stroke current instead of the energy dissipated in return strokes as an input parameter in quantifying NOx production in return strokes, as done above, has at least one important advantage. The energy dissipated in lightning flashes cannot be measured directly but has to be inferred by indirect methods leading to large inaccuracies in the estimated NOx production. On the other hand, the current at the channel base of ground flashes can be measured and a large amount of data on this parameter is available in the literature. In deriving (14.13) and (14.14), it has been assumed that the return-stroke peak current decreases with height. The experimental observations show that the luminosity of both first and subsequent return strokes decreases with height indicating that the return-stroke current peak also decreases with height [54,61]. Of course, since the exact nature of how the return-stroke peak current decreases with height is not known, one has a freedom to select any other form of decay for the peak current than the linear decay assumed in the calculation. However, the expression describing a linear decay with current amplitude decreasing to zero at cloud level involves only one parameter, i.e. height of the vertical channel, and it also specifies the boundary conditions for the current at the cloud end of the channel. Moreover, the linear decay has been shown to produce electric fields at both far and close distances that are similar to those measured when used in returnstroke models [62,63].

14.5.2.3 M components and K processes As pointed out earlier, the development or extension of the lightning channels located inside the cloud is mediated by leaders. As these leaders develop, the changes in the potential at the extremities of the leader may cause K processes that travel along the channel reducing the potential differences. The interferometric observations indicate that it is usual to have a few K processes in the developing stage of a given channel section [51, 52]. Thus, if hk is the number of NOx molecules produced per unit length of the channel by a K process, then the total number of NOx molecules generated by K processes during the development of the channels inside the cloud after correction for the pressure is hknkLe–H/lp . In this expression, nk is the number of K changes taking place in the development of a particular branch. In writing down the above expression, we have also assumed that the current associated with a given K change occurring during the development of a given branch travels only along that branch. In the calculations to be conducted later, we assume that nk=3. Thus, the total number of K changes per flash is equal to nnk, where n is the number of branches in the channel. In a lightning flash with five branches in the cloud, the total number of K changes would be 15. If these K changes end up in a channel carrying a continuing current to ground, then the resulting current will propagate to ground as an M component. The number of NOx molecules produced in the vertical channel by the M components after correction for the pressure is given by nmhmlp(1–e–H/lp), where nm is the number of M components travelling along the vertical channel of a typical ground flash. It is also

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assumed that the NOx production efficiency of a typical K change is identical to that of a typical M component, i.e. hk = hm. This assumption is not unreasonable since, as mentioned above, they share a common origin and therefore they probably have similar currents. The total number of NOx molecules generated by K changes and M components after the pressure correction is NOxm ¼ ½nm hm lp ð1  eH=lp Þ þ nk hk LeH=lp 

(14.15)

Now, let us evaluate the magnitude of hm. A typical M component current has a more or less symmetric bell-shaped waveform with a rise time of about 400 ms and a peak of about 160 A [64]. Calculations done with such a current waveform show that hm = 2  1020. Thus, hk is also equal to 2  1020. M components travel along the vertical channel carrying continuing currents. In a typical ground flash having a continuing current, M component may outnumber the number of return strokes by 1 to 4. Therefore, a typical ground flash containing continuing currents may support about 16 M components along the vertical channel. In making the above statement, we have assumed that a typical ground flash contains four return strokes. On the other hand, the percentage of ground flashes containing long (longer than 40 ms) continuing currents is about 30–50% [65,66]. This fixes the value of nm to about 5.

14.5.2.4

Continuing currents

In the model under consideration, we assume that the continuing currents are flowing only through the vertical section of the ground flash. Of course, they also flow along horizontal channels, but our knowledge at present on how continuing current is distributed in channels in the cloud is rather meagre. In this case, the total number of NOx molecules generated by these continuing currents in a ground flash after pressure correction is given by NOxcv ¼ ½kc hcon Icon tc lp ð1  eH=lp Þ

(14.16)

where hcon is the number of NOx molecules produced per Coulomb by a unit length of the discharge channel carrying continuing current, Icon is the magnitude of continuing current, tc is the typical duration of continuing current and kc is the fraction of ground flashes that support continuing currents. In the calculations we assumed that hcon = 2  1020 molecules/m/C (see Section 14.3). According to experimental data, about 30–50% of the lightning flashes contain continuing currents and the amplitude and the duration of a typical continuing current are about 100 A and 100 ms respectively [65,66]. Thus, kc = 0.3 and tc = 0.1 s. The exact nature of the source that drives continuing current along the vertical channel is not known. Most likely the source is the development and the charge transfer along the upper channel sections. Since we have taken this activity into account in Section 14.5.2.1 under leaders, it is reasonable to consider only the vertical channel here.

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14.5.2.5 NOx production in a typical negative ground flash One can sum the contributions to the NOx production from different processes taking place in a ground flash as done in the following equation:  h Ils  2 NOxground ¼ hstr H þ hstr þ L þ lea lp  lp eH=lp ½H þ lp  vs   lp lp H=lp hlea Ils 2 H=lp Le þ hfr lp 1  þ e þ 2vs n H H   ns hlea Ild 2 H=lp lp  lp e ½H þ lp  þ vd   lp lp H=lp þ nm hm lp ð1  eH=lp Þ þ hsr hs lp 1  þ e H H þ nk hk LeH=lp þ kc hcon Icon tc lp ð1  eH=lp Þ (14:17) We made an attempt above to specify numerical values for the constants that appear in this equation. One has to admit, of course, that our knowledge on the numerical values of different parameters is not complete and more work has to be done before the above equation could be applied with confidence. However, this equation provides a foundation on which the procedure to estimate NOx production in lightning flashes could be built as one obtains more information concerning the parameters. In Appendix 1, we have summarized our current knowledge on each of the parameters that appear in (14.17) Note that since M components are discharges propagating in channels carrying continuing currents, one may wonder whether the contribution from M components to the NOx production is already taken into account in the production of NOx by continuing currents. The reason why we have included both contributions (i.e. continuing currents and M components) in the above equation is the following. The lightning channel becomes more luminous when M components are travelling through it. This shows that they cause additional atomic excitation and ionization because the channel becomes luminous while they propagate along it. Furthermore, the optical observations show that during the propagation of the M components the diameter of the channel through which a continuing current is already propagating gradually increases from about 0.5 cm to about 3 cm [67]. Moreover, M components were also observed to produce thunder [68]. Thus, it is reasonable to assume that the M component will enhance the NOx production beyond the NOx production level of the continuing current during its passage through the channel. This justifies adding the contribution of the M components to that of the continuing current. The same reasoning applies to the addition of the contribution from K processes to the NOx yield from the leader currents flowing through the hot core of the developing leaders. Now, let us illustrate the use of (14.17). In Figure 14.5, we have depicted the contributions from leaders (separated into contribution from streamers in the corona sheath and current flow along the core), return strokes, M components, K processes and continuing currents as a function of the horizontal channel length L. First, note that the contributions from return strokes and the continuing currents do

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Figure 14.5 The number of NOx molecules produced by different processes associated with a ground flash as a function of horizontal channel length. (a) Total, (b) streamers in corona sheaths, (c) core current in leaders, (d) M components and K changes, (e) continuing currents and (f) return strokes (four strokes). The vertical channel length is assumed to be 5 km and the number of main branches in the cloud is assumed to be 5. not vary with increasing length of the horizontal channels because these currents are assumed to propagate only along the vertical channel. Second observe that the return strokes produce the least contribution to the NOx production, whereas the largest contribution is made by leaders with the current flowing along the core being the main contributor. More than 90% of the contribution to the NOx production is coming from leaders, M components and K processes. Leaders alone contribute about 50% to the NOx production. Note that these observations are true also for horizontal channel lengths as short as 10 km. On the other hand, in the literature, the return stroke is often assumed to be the NOx source in ground flashes. Our study shows that this assumption is incorrect. Using VHF lightning channel mapping technique, Laroche et al. [69] observed that the mean total channel length in over 20,000 cloud and ground flashes is 45 km. Taking this length as a typical value the results presented in Figure 14.5 show that an average ground flash with four return strokes will generate about 4 x 1025 NOx molecules per flash.

14.6 NOx production by cloud flashes Cloud flashes normally occur between the main negative and upper positive charge regions of the cloud. Much of the information available today on the mechanism of the cloud flash is based on electric field measurements. Also Proctor [70–72],

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Shao et al. [51] and Shao and Krehbiel [52] made important discoveries utilizing VHF radio imaging techniques. Based on this information, Cooray [45] summarized the activities during a cloud flash as follows (see also [65]). The cloud flash commences with a movement of negative leader discharge from the negative charge region towards the positive one in a more or less vertical direction. The vertical channel develops within the first 10 to 20 ms from the beginning of the flash. This channel is a few kilometres in length and it developed with a speed of about 2.0  105 m/s. Even after the vertical channel was formed, one could detect an increase in the electrostatic field indicative of negative charge transfer to the upper levels along the vertical channel. The main activity after the development of the vertical channel is the horizontal extension of the channels in the upper level (i.e. the channels in the positive charge region). These horizontal extensions of the upper level channels are correlated to the brief breakdowns at the lower levels, followed by discharges propagating from the lower level to the upper level along the vertical channel. Thus, the upper level breakdown events are probably initiated by the electric field changes caused by the transfer of charge from the lower levels. For about 20 to 140 ms of the cloud flash, repeated breakdowns occur between the lower and upper levels along the vertical channel. These discharges transported negative charge to the upper levels. Breakdown events of this type can be categorized as K changes. In general, the vertical channels through which these discharges propagate do not generate any radiation in the VHF range, which indicates that they are conducting. This is so because, in general, conducting channels do not generate VHF radiation as discharges propagate along them. Occasionally, however, a discharge makes the vertical channel visible at VHF and then the speed of propagation can be observed to be about (5–70)  106 m/s, typical of K changes. This active stage of the discharge may continue to about 200 ms. In the latter part of this active stage (140–200 ms), significant extensions of the lower level channels (i.e. the channel in the negative charge region) take place, but they occur retrogressively. That is, successive discharges, or K changes, often start just beyond the outer extremities of the existing channels and then move into and along these channels, thereby extending them further. These K changes transport negative charge from successively longer distances to the origin of the flash, and sometimes even to the upper level of the cloud flash as inferred from VHF emissions from the vertical channel. Sometimes, these K changes give rise to discharges that start at the origin of the flash and move away from it towards the origin of the K changes. Such discharges can be interpreted as positive recoil events that transport positive charge away from the flash origin and towards the point of initiation of the K change. At the final part of the discharge, the vertical channel and the upper level channels were cut off from the lower level channels. This is probably caused by the decrease in the conductivity of the vertical channel. The above description shows that a cloud flash can be described as an electrical activity that collects the charge from the main negative charge centre and redistribute it in the positive charge centre after transporting it along a more or less vertical channel. The recent observations based on three-dimensional interferometry also confirm

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the basic features of cloud flashes described above [73–76]. It is important to mention here that, since positive discharges do not radiate efficiently in HF and VHF, the channels created by positive discharges could be detected only when negative recoil discharges travelling along them. Thus, the channel structure of lightning flashes inside the cloud available today may not be complete. Let us assume that the total length of the channels in the negative charge centre is equal to that in the positive charge centres. Denote this length by L. Assume further that these channels are oriented in a horizontal direction. The electrical activity taking place in the negative and positive charge centres during a cloud flash are not very different from the electrical activity taking place in the negative charge centre in the case of a ground flash (i.e. creation and extension of channels by leaders and intermittent occurrence of K changes). Taking into account the fact that the atmospheric pressure is different at the heights where negative and positive charges are located in a cloud, we can describe the NOx production by a cloud flash by the following equation. NOxcloud

h h ¼ hstrþ L þ hstr L þ lea Ilea L2 eHn =lp þ lea Ilea L2 eHp =lp 2vn 2vn  kc hcon Icon tc lp eHn =lp  eHp =lp þ nk hk LeHn =lp þ nk hk LeHp =lp  þnm hm lp eHn =lp eHp =lp (14.18)

where Hn is the height of the negative charge centre and Hp is the height of the positive charge centre. We assume that Hp = 10 km. In writing down the above equation, it was assumed that the percentage of cloud flashes supporting continuing currents and the number of M components in a cloud flash are identical to those of ground flashes. The results obtained for different values of L are shown in Figure 14.6. In many studies dealing with the global production of NOx the assumption is made that the cloud flashes do not contribute significantly to the NOx production in thunderstorms. This assumption was challenged previously by Gallardo and Cooray [77]. There are also field measurements showing the importance of cloud flashes in NOx production. For example, airborne measurements of Dye et al. [78] show that NOx from a storm that produced exclusively cloud discharges was comparable to other observations, where both cloud and ground discharges are occurring. Further, the modelling of airborne NOx measurements by DeCaria et al. [9,79] show that intracloud lightning (or the intracloud part of the ground flashes) was the dominant source of NOx for the thunderstorms investigated in the study. The results presented in Figure 14.6 show that for a given channel length both the ground flash and the cloud flash generate more or less equal number of NOx molecules.

14.7 Global production of NOx by lightning flashes The results presented above can be used to evaluate the global production of NOx by lightning flashes if the flash rate of the lightning flashes is known. In the analysis, we have not treated positive flashes separately but indirectly assumed that the

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Figure 14.6 The number of NOx molecules produced by different processes associated with a cloud flash as a function of horizontal channel length. (a) Total, (b) streamers in corona sheaths, (c) core current in leaders, (d) K changes, (e) continuing currents and (f) M components. The vertical channel length is assumed to be 5 km long and the number of main branches in the cloud is assumed to be 5 NOx production from a typical positive flash is similar to that of a typical negative flash. In the literature, the flash rate is assumed to lie in the range of 40–300 flashes/s [80,81]. In the results to be presented, we assumed a global lightning flash rate of 100/s. There is no reason to separate the flash rate into ground and cloud flashes because both types of flashes produced more or less the same amounts of NOx. In Figure 14.7, we have depicted the annual NOx production by lightning flashes as a function of the horizontal channel length. If one assumes an average total channel length of 45 km for a lightning flash, the global NOx production by lightning flashes will be about 4 Tg(N)/year. One has to understand that the global lightning flash frequency is not a constant and it may vary from one year to another. Moreover the number 100 flashes/s is based on thunderstorm observations and satellite data suggest values in the range of 40–60 flashes/s. But, of course, this estimation too depends on the detection threshold level of the satellite and the possible screening of light by cloud cover. The important point however is that the NOx production rate is proportional to the global lightning flash frequency. The number 4 Tg(N)/year is based on 100 flashes/s and if it varies between 40 and 300 flashes/s, the global NOx production rate will also vary between 2 and 12 Tg(N)/ year. Lee et al. [82] studied the various sources and sinks of NOx in the atmosphere and concluded that the contribution from lightning should be in the range of 4–8 Tg (N)/year. Our results agree with this prediction. The present global estimates of NOx based on theoretical and laboratory studies vary between 1 and about 100 Tg(N)/year. The two orders of magnitude variation in this estimate are due to

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Figure 14.7 Annual production of NOx by lightning flashes as a function of horizontal channel length. The flash rate is assumed to be 100/s, the vertical channel length in ground flashes is assumed to be 5 km and the number of main branches in the cloud is assumed to be 5 the different results obtained for the NOx production efficiency of laboratory discharges and the different values assumed for the energy dissipation in lightning flashes. As pointed out previously, the variation in the efficiency of NOx production in laboratory discharges is probably due to the differences in the current waveforms associated with these discharges. The energy dissipation in lightning flashes is a parameter that cannot be measured directly and therefore is not a good scaling quantity in NOx studies. Our estimate is free from both these drawbacks. Another interesting point of our study is the observation that most of the NOx production in lightning flashes is due to cloud flashes or the cloud portion of ground flashes. Thus, the injection of NOx by thunderstorms into the atmosphere takes place primarily at a height of 5–10 km. The theoretical studies conducted by Gallardo and Rodhe [2] show that in order to account for the nitrate deposition in the remote marine regions, the strength of the NOx source due to lightning should be about 5 Tg(N)/year and the source should be located at the cloud height. Our study confirms this inference.

14.8 Conclusions The results presented in this chapter show that the NOx production efficiency of electrical discharges depends not only on the energy dissipated in the discharge but also on the shape of the current waveform. This provides an explanation for the different values of NOx molecules/J obtained by different researchers in different experiments. Thus, energy dissipated in a discharge is not suitable as the scaling

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quantity for extrapolating the laboratory data to lightning flashes. In this chapter, we present a theory that can be used to evaluate the NOx production in electrical discharges, if the discharge current is known. The results obtained are compared with the available experimental data and a good agreement is found between theory and experiment. The study shows that the primary contribution to NOx from thunderstorms is coming from the electrical activity inside the cloud, with only a small fraction being contributed by return strokes. Using the proposed theory, we estimated the global NOx production by lightning flashes taking into account different lightning processes such as leaders, return strokes, M components, K changes and continuing currents. The results show that the efficiency of NOx production in ground flashes and cloud flashes are similar and for an average total channel length of 45 km the global production of NOx by lightning flashes, based on lightning flash frequency of 100 flashes/s, is about 4 Tg(N)/year.

Appendix 1 hstr – The number of NOx molecules generated in the corona sheath during the creation of a leader channel. The current estimate is 2  1020 molecules/m. hcon – The number of NOx molecules generated per unit length per unit charge by a continuing current. The best estimate is 2.0  1020 molecules/m/C. hlea – The number of NOx molecules generated per unit length per unit charge by the leader current flowing through the channel core. The best estimate is 2.0  1020 molecules/ m/C. lP – The decay height constant for the atmospheric pressure. This is equal to 8500 m. H – The height of the negative charge centre. The value used in the calculations is 5000 m. Hp – The height of the positive charge centre. The value used in the calculation is 10,000 m. L – The total length of the horizontal sections in the cloud. Current estimates place it somewhere between 30 and 50 km. hfr – The number of NOx molecules generated per unit length in a discharge channel carrying a current waveform similar to that of a typical first return stroke. The best estimate is 5.9  1020 molecules/m. hsr – The number of NOx molecules generated per unit length in a discharge channel carrying a current waveform similar to that of a typical subsequent return stroke. The best estimate is 1.4  1020 molecules/m. ns – The number of subsequent return strokes in a typical ground flash. The best estimate is 3. hm – The number of NOx molecules generated per unit length in a discharge channel carrying a current waveform similar to that of M component. The best estimate is 2.0  1020 molecules/m. hk – The number of NOx molecules generated per unit length by a K change. It is assumd that hk = hm. nk – The average number of K changes taking place during the development of a given channel branch in the cloud. It is assumed to be 3. nm – The average number of M components in a typical ground flash. The best estimate is 5. This number is based on the fact that a ground flash with a continuing current can support about 16 M components and about 30% of the ground flashes contain continuing currents.

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(Continued) kc tc Icon Ils Ild vs vd n

– The fraction of ground flashes containing continuing currents. The best estimate is 0.3. – The average duration of continuing current. The best estimate is 100 ms. This figure is actually valid for long continuing currents. – Magnitude of typical continuing current. The best estimate is 100 A. – Magnitude of typical stepped leader current. The best estimate is 100 A. – Magnitude of typical dart leader current. The best estimate is 1 kA. – The average speed of development of lightning leader channels in virgin air inside the cloud. The best estimate is 2105 m/s. – The average speed of dart leaders. The best estimate is 107 m/s. – The number of major branches inside the channel. The available VHF interferometric images show that it may vary from about 3 to 10. It is assumed to be 5.

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Lightning electromagnetics: Volume 2 Jordan, D. M. and M. A. Uman, Variation in lightning intensity with height and time from subsequent lightning return strokes, J. Geophys. Res., vol. 88, pp. 6555–6562, 1983. Rakov, V. A. and A. A. Dulzon, Calculated electromagnetic fields of lightning return stroke, Tech Elektrodinam., vol. 1, pp. 87–89, 1987. Thottappillil, R., V. A. Rakov and M. A. Uman, Distribution of charge along the lightning channel: Relation to remote electric and magnetic fields and to return-stroke models, J. Geophys. Res., vol. 102, pp. 6987–7006, 1997. Thottappillil, R., J. D. Goldberg, V. A. Rakov, M. A. Uman, R. J. Fisher and G. H. Schnetzer, Properties of M components from currents measured at triggered lightning channel base, J. Geophys. Res., vol. 100, pp. 25711– 25720, 1995. Rakov, V. and M. Uman, Lightning Physics and Effects, Cambridge University Press, Cambridge, UK, 2003. Saba, M. M. F., O. Pinto Jr. and M. G. Ballarotti, Relation between lightning return stroke peak current and following continuing current, Geophys. Res. Lett., vol. 33, pp. l23807, doi:10.1029/2006gl027455, 2006. Idone, V. P., The luminous development of Florida triggered lightning, Res. Lett. Atmos. Electr., vol. 12, pp. 23–28, 1992. Rakov, V. A., M. A. Uman, K. J. Rambo, G. H. Schnetzer and M. Miki, Triggered-Lightning Experiments Conducted in 2000 at Camp Blanding, Florida, (Abstract), Eos Trans. Suppl., AGU, vol. 81, no. 48, Nov. 28, p. F90, 2000. Laroche, P., E. Defer, P. Blanchet and C. Thery, Evaluation of NOx produced by storms based on 3D VHF lightning mapping, ICAE 99 – 11th Int. Conf. On Atmospheric Electricity, Huntsville, AL(USA), June 07–11, 1999. Proctor, D. E., Lightning flash with high origins, J. Geophys. Res., vol. 102, pp. 1693–1706, 1997. Proctor, D. E., VHF radio pictures of cloud flashes, J. Geophys. Res., vol. 86, pp. 4041–4071, 1981. Proctor, D. E., Regions where lightning flashes began, J. Geophys. Res., vol. 96, pp. 5099–5112, 1991. Lojou, J. Y. and K. L. Cummins, On the representation of two- and three dimensional total lightning information. Conf. on Meteorological Applications of Lightning Data, San Diego, Cal., Amer. Meteor. Soc., paper 2.4 (2005). Lojou, J. Y., M. J. Murphy, R. L. Holle and N. W. S. Dimetriades, Nowcasting of thunderstorms using VHF measurements, in Lightning: Principles, Instruments and Applications, eds. D. Betz, U. Schumann and P. Laroche, Springer, the Netherlands, 2008. Coleman, L. M., T. C. Marshall, M. Stolzenburg, et al., Effects of charge and electrostatic potential on lightning propagation. J. Geophys. Res., vol. 108, p. 4298, doi:10.1029/ 2002JD002718, 2003. Morimoto, T., Z. Kawasaki and T. Ushio, Lightning observations and consideration of positive charge distribution inside thunderclouds using VHF

On the NOx production by laboratory electrical discharges and lightning

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broadband digital interferometry, Atmos. Res., vol. 76, no. 1–4, pp. 445–454, 2005. Gallardo, L. and V. Cooray, Could cloud-to-cloud discharges be as effective as cloud-to-ground discharges in producing NOx?, Tellus, Ser B, vol. 48, pp. 641–651, 1996. Dye, J. E., B. A. Ridley, W. Skamarock, et al., An overview of the Stratospheric – Tropospheric Experiment: Radiation, Aerosols and Ozone (STERAO) – Deep convection experiment with results for the July 10, 1996 storm, J. Geophys. Res., vol. 105, pp. 10023–10045, 2000. DeCaria, A. J., E. Pickering, G. L. Stenchikov, et al., A cloud scale model study of lightning- generated NOx in an individual thunderstorm during STERAO-A, J. Geo phys. Res., vol. 105, pp. 11601–11616, 2000. Turman, B. N. and B. C. Edgar, Global lightning distribution at dawn and dusk, J. Geophys. Res., vol. 87, pp. 1191–1206, 1982. Christian, H. J, R. J. Blakeslee, D. J. Boccippio, et al., Global frequency and distribution of lightning as observed from space by the Optical Transient Detector, J. Geophys. Res., vol. 108, no. D1, p. 4005, doi:10.1029/ 2002JD002347, 2003. Lee, D. S., I. Ko¨hler, E. Grobler, et al., Estimation of global NOx emissions and their uncertainities, Atmos. Environ., vol. 31, pp. 1735–1749, 1997.

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Chapter 15

Lightning and climate change Earle R. Williams1, Joan Montanya2, Joydeb Saha3 and Anirban Guha3

Lightning is a widely recognized source of damage and disruption to electrical power systems worldwide. The climate is changing, with both natural and anthropogenic origins. This chapter is concerned with the response of lightning to changes in temperature and aerosol loading of the atmosphere that is expected to accompany climate change. In the present climate, lightning is shown to increase with both temperature and with the boundary layer populations of cloud condensation nuclei (CCN). In a future climate characterized by the continued consumption of fossil fuels, the threat from lightning is expected to increase.

15.1 Introduction Lightning is a natural phenomenon originating in the high voltage differences encountered in thunderstorms (up to one billion volts) and exhibiting currents as large as hundreds of kiloamperes. The lightning threat to worldwide energy infrastructure is widely recognized [1]. Lightning dominates the damage to electrical/ electronic equipment in homes, commercial installations, and industrial facilities. The total cost is dependent on both the total exposure and the worldwide lightning activity. Both these contributions to cost are increasing with time as a result of a growing infrastructure worldwide. Much attention is given today to extreme events in a warmer climate (e.g., [2]). This attention serves to place lightning at center stage, to the extent that lightning is a manifestation of the extreme form of moist convection—largest clouds, strongest updrafts, and most hazardous precipitation. One can expect volatile behavior in the tail of any distribution, and for this reason alone, the recent selection of lightning as a climate variable [3] is most appropriate.

1 Parsons Laboratory, Department of Civil and Environmental Engineering, Massachusetts Institute of Technology (MIT), USA 2 Department of Electrical Engineering, Universitat Polite`cnica de Catalunya, Spain 3 Department of Physics, Tripura University, India

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This chapter is concerned with an assessment of how global lightning may respond to global climate change. This turns out to be a difficult problem. Some understanding of this difficulty is derived from the limits of our current ability to understand the general behavior and global distribution of lightning in the present climate. One particular challenge is that both temperature and aerosol play important roles in lightning activity in the present climate. Accordingly, this aspect shall be the point of departure in this chapter. A global climatology for lightning measured from optical sensors on satellites in space is shown in Figure 15.1(a). This integration is based on nearly two decades of observation. The most conspicuous feature of the global distribution is the strong preference of lightning for land, with a 10- to 20-fold contrast between land and ocean (see also [4]). The leading factor in this contrast is the number of thunderstorms, with a secondary contribution from a greater flash rate per storm in the continental case [6]. Since the majority of the world’s population density and infrastructure is also over land, this land dominance aggravates the lightning threat. However, since the lightning is also strongly centered on equatorial regions (for reasons that are soon to be discussed) where population and infrastructure are reduced in comparison to higher latitudes in the northern hemisphere, the overall threat is ameliorated to some extent on a global basis. Three major continental zones straddling the equator—the Americas, Africa, and the Maritime Continent (southeast Asia, Indonesia, and northern Australia)—dominate the global lightning activity. The same three zones are also the major players in the Earth’s global electrical circuit [7,8]. From a climate perspective, the three major continental zones have previously been ranked in their continentality [9], with Africa leading, followed by America, and with the Maritime Continent closest to oceanic behavior. Both the total lightning activity and the aerosol burden in these three regions follow the same order, whereas rainfall amounts follow the reverse order. The energy involved with global lightning activity is derived from the much larger latent heat released when water vapor condenses. Rainfall is also a product of the condensation process. The global distribution of rainfall (Figure 15.1(b)), more readily measured than condensation, provides some global measure of the distribution of latent heat release. In marked contrast with the lightning distribution, rainfall and latent heat release are as prevalent over the ocean as over the land, but as will be shown by the evidence in this chapter, the vertical profile of latent heat release is markedly different between land and ocean. Both thermodynamic and aerosol effects are at play in this difference, by virtue of their impact on thunderstorm updrafts, and both are important in considerations of how lightning will respond to climate change. The traditional explanation [10,11] for the contrast in lightning activity between land and ocean, aptly illustrated in Figure 15.1(a), is based on thermodynamics: land is hotter and more unstable to vertical motion. This greater instability over land pertains to both dry and moist convection. In recent years, a growing body of evidence [12–23] has shown that the atmospheric aerosol, and in particular the CCN that provides the embryos for cloud droplets, is also playing a key role in this contrast. The global aerosol population shown in Figure 15.1(c) also shows a prominent land– ocean contrast, with more polluted conditions over the land. The markedly less

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Figure 15.1 Global climatologies of (a) lightning flash density (lightning imaging sensor), (b) rainfall (NASA TRMM), and (c) aerosol concentration (as measured with satellite aerosol optical depth). Adapted from [5]

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distinct contrast in aerosol concentration at the land/ocean boundary is a suggestion that thermodynamics plays the leading role in the land/ocean lightning contrast. Since climate change will invariably involve a change in both thermodynamics and aerosol, the physical basis for both controls needs to be explored. The treatment of the thermodynamic contribution appears in Section 15.4 and for the aerosol part in Section 15.5. Ahead of these discussions, some attention is warranted on the workings of the thunderstorm in the next Section 15.2.

15.2 Basics of thunderstorm electrification and lightning The deepest and most vigorous convective clouds in the atmosphere are thunderstorms, and extend deeply into the cold portion (defined here at T < 0
C) of the atmosphere (Figure 15.2). Considerable evidence has accrued [24,25] that the mechanism for charging a thunderstorm and for the production of lightning flashes is based on the

0° C

Figure 15.2 The mechanism of the thunderstorm: storm cloud with colliding ice particles and positive electric dipole

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collisions of two kinds of particles: small ice crystals and larger graupel particles. Both ice particles are the product of mixed-phase conditions involving water substance in all three thermodynamic phases: vapor, liquid, and solid (ice). The liquid phase at “cold” temperature (T < 0
C) is referred to as supercooled water. The ice crystals form by diffusion of water vapor in the so-called Bergeron process based on the asymmetry in equilibrium vapor pressure between liquid and ice. The mass of ice crystals increases at the expense of the supercooled cloud water. The graupel particles grow by the accretion of supercooled cloud droplets, which freeze in contact with the graupel surface. The collisions between graupel particles and ice crystals result in the transfer of negative charge to graupel and positive charge to the crystals, by a mechanism at the molecular scale that has long eluded scientists [25,26], but very likely involving mobile protons as main agents of charge transfer. The descent of negative graupel with respect to positive ice crystals under gravity sets up the macroscopic positive dipole of the thunderstorm (Figure 15.2). This hydrometeor-based mechanism of differential charge separation (based on different fallspeeds of ice crystals and graupel) is immune to the effects of turbulence, which often shows a strong presence in thunderstorms.

15.3 Thermodynamic control on lightning activity A number of basic thermodynamic parameters as well as relationships from physical meteorology deserve discussion when one considers possible changes in lightning in a changing climate. These items are here addressed in turn.

15.3.1 Temperature The most commonly used thermodynamic parameter in global climate change is the temperature of surface air, typically measured at “screen level.” The formal meteorological quantity is “dry bulb temperature” to distinguish it from “wet bulb temperature” and “dew point temperature” both of which involve the water vapor content of the air. Traditional estimates of the global mean temperature [27,28] and global warming involve averages of 4,000–6,000 thermometer readings of dry bulb temperature over the Earth’s surface. This temperature parameter is also a key factor in other thermodynamic quantities of interest here: saturation water vapor concentration, convective available potential energy (CAPE) and cloud base height (CBH), described in greater detail below.

15.3.2 Dew point temperature The dew point temperature Td is a direct measure of the water vapor concentration in surface air and is typically measured by cooling a metal surface to a temperature at which condensation, or “dew,” appears. For water-saturated conditions (i.e., inside a cloud), the dew point temperature is equal to the dry bulb temperature T. In contrast, in a dry desert environment, the dew point temperature can be several tens of
C lower than the dry bulb temperature.

15.3.3 Water vapor and the Clausius–Clapeyron relationship The working substance of a thunderstorm is water vapor. Energy is released when water vapor rises and condenses to form cloud. The latent heat of

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condensation Lv is 2.5  106 J/kg of water, sufficient energy to raise the condensate 250 km against gravity if this transformation took place with perfect efficiency. The water vapor concentration in the atmosphere in a condition of thermodynamic equilibrium is controlled by temperature in an exponential dependence known as the Clausius–Clapeyron relation. In differential form: de  ðT Þ=dT ¼ Lv e  =Rv T 2

(15.1)

where T is the absolute temperature (K), e*(T) is the saturation vapor pressure of water vapor, and Rv is the gas constant for water vapor (461 J/kg/K). The integral form of this relationship (see e.g., [29]) in terms of the water vapor mixing ratio is shown graphically in Figure 15.3. As a rough rule of thumb, the equilibrium water vapor concentration e*(T) doubles for every 10
C of temperature increase. This result has much to say about the sparsity of thunderstorms in polar regions and their predominance in tropical latitudes. A change in temperature of 30
C between equatorial regions and high latitudes amounts to nearly an order of magnitude difference in available water vapor. A quantitative 100

Saturation Mixing Ratio (g/kg)

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0.0001 –80

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Figure 15.3 Equilibrium water vapor mixing ratio (g/kg) versus temperature: the Clausius–Clapeyron relationship

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consideration of the global lightning climatology shows that two of every three lightning flashes lie within 23
of the equator [30]. The slope de*/dT in the Clausius–Clapeyron relationship in (15.1) is a valuable benchmark for judging results on the response of lightning to temperature on various time scales (Section 15.4). This slope depends on temperature, but at the mean Earth surface temperature (14
C), the slope is 7% per 1
C. Cumulonimbus clouds are the primary agents for transporting water vapor from the planetary boundary layer in the lower troposphere to the upper troposphere. Consistent with this general picture, Price [31] has shown variations in upper tropospheric water vapor correlated with lightning variations over the African continent.

15.3.4 Convective available potential energy and its temperature dependence The maintenance of thunderstorm mixed-phase conditions and the associated “factory” for ice and electric charge requires an energy source for the updraft. That energy source is CAPE and is illustrated in Figure 15.4. CAPE is represented as the area on a thermodynamic diagram involving height (or pressure) and temperature. This area is bounded on the left by the temperature sounding in the storm environment and on the right by a “wet bulb adiabat” which is a theoretical prediction for the temperature of the air in an updraft parcel that is buoyant with respect to the storm’s environment. At any given altitude, the different between the wet bulb adiabat and the environment is a measure of the buoyancy force acting on the updraft parcel of air. The buoyant force per unit mass at an altitude is given simply as g(DT/T) where DT is the temperature contrast between the updraft and the environment and where g is the acceleration of gravity (9.8 m/s2). Since the wet bulb adiabat is determined by purely thermodynamic quantities of temperature and dew point temperature of the surface air ingested by the storm to form the updraft, it would seem that CAPE is also a purely thermodynamic quantity. A complication arises here, however, making CAPE dependent on both thermodynamic and aerosol characteristics, and adds to the challenge of disentangling thermodynamic from aerosol influences on lightning activity. The updraft parcel buoyancy depends not only on temperature but also on the mass of condensate within the parcel. For example, if the temperature contrast is 1
C, a typical value, the local cloud buoyancy force per unit mass is roughly 1/300 g = 0.03% of g. Since the density of surface air is 1.2 kg/m3, a mass of condensate as small as 4 g/m3 would completely negate the thermal buoyancy and strongly impact the dynamics of air parcels at that level. This point will be elaborated on below. Accurate estimates of condensate mass are lacking in real thunderstorms, and theoretical CAPE calculations typically resort to two extreme assumptions, neither of which is entirely satisfied. In the most common pseudo-adiabatic (or “irreversible”) approach, all the condensate is removed as the updraft parcel ascends (e.g., [13]). In this situation, only the temperature contrast (and a smaller contribution from the water vapor component) affects the parcel buoyancy. The implication is that the

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Figure 15.4 Temperature sounding, wet bulb adiabatic approximation for the updraft temperature, and the area representing convective available potential energy (CAPE) transformation from cloud water to precipitation is very efficient. In the context with small CCN concentration, typical of clean oceanic conditions. The observation of initial radar echoes in maritime convection at altitudes of only a few km (and in the “warm” portion of the troposphere) [32] is evidence of an efficient precipitation process in clean conditions. The other extreme assumption in the evaluation of CAPE is that all condensate is retained as the updraft parcel ascends. The process is reversible and the wet bulb adiabat has a different mathematical form [33]. The implication is that the transformation from cloud water to precipitation has zero efficiency because the cloud droplets remain too small to coalesce. This situation is typical of rich CCN concentrations, as in polluted continental conditions. In early investigations of tropical oceanic convection [34,35] the reversible process was favored in computing CAPE,

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but it is now recognized that for maritime convection drawing from clean boundary layer air, the pseudoadiabatic irreversible process was a more appropriate choice. Given the negative buoyancy contribution of condensate loading, reversible CAPE is expected to be systematically less than irreversible CAPE. This expectation is consistent with numerous published results [29,36]. In early considerations of the reversible and irreversible processes [37], it was concluded that the “differences between the products of condensation as falling out or being retained are so small as to be negligible in practice,” but today this difference is acknowledged as being all important in deep moist convection. Williams and Renno [36] also showed that CAPE in the current climate was well predicted by the wet bulb potential temperature of surface air, though different relationships were apparent for land and ocean. Global maps of wet bulb potential temperature show that maximum CAPE over land is greater than over ocean. Lucas et al. [38,39] claimed that CAPE over land was similar to values over ocean but they did not consider the diurnal variation of CAPE over land. This plays an important role in the transition of cumulus congestus clouds to thunderstorms over land, with important contributions from thermodynamics. A global climatology of CAPE has been prepared [40]. These results also show that CAPE is larger over continents than over oceans, though no consideration was given to aerosol-related effects in condensate loading. The land–ocean CAPE contrast is qualitatively consistent with the land–ocean lightning contrast, but on closer examination [11,38,41,42], the contrast is not sufficient to account for the order-of-magnitude contrast in lightning. In the latter work however, precipitation is considered at the surface rather than in the cold region aloft where electrical energy is generated by the relative descent of graupel particles with respect to ice crystals. This aspect will be revisited below in the context of CBH. More recently [39] have shown evidence for giant sea salt nuclei in suppressing lightning in oceanic convection. Given the primary role of CAPE in the charge separation and lightning activity of thunderstorms, the variation of CAPE with temperature on the long-time scale of global warming is of considerable interest. This problem is nontrivial because the entire temperature profile is involved, as well as the condensate-related ambiguities of the wet bulb adiabat. In early work [43] CAPE was postulated to be a climate invariant. However, many GCM results show CAPE to increase with global warming [44,45,46,47], and Del Genio et al. [48] have found increases in cumulonimbus velocity in climate models in a warmer world. Furthermore, still more recent theoretical works [42,49,50] support a scaling of CAPE with the Clausius–Clapeyron exponential temperature dependence. On this basis alone, one expects to have more lightning in a warmer climate. However, one recent model results [51] shows the opposite result for the tropics. This contrast in predictions is not well understood at present.

15.3.5 Cloud base height and its influence on cloud microphysics The contrast in physical characteristics between land and ocean surfaces exerts an important influence on the behavior of thermodynamic parameters of surface air. The contrast in heat capacity and mobility between land and ocean affect the

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surface air temperature, with ocean water and overlying air resisting temperature increase in response to solar heating, in comparison with dry land surfaces. The diurnal variation of ocean surface temperature is typically a fraction of 1
C. In contrast, the diurnal variation of land surface temperature invariably exceeds 1
C but the surface temperatures over deserts can vary by tens of
C. The contrast in available surface water between land and ocean affects the dew point temperature and relative humidity of surface air. Land surfaces are generally both hotter (larger T) and drier (smaller Td) than oceans, and as a consequence of both of these contributions, the dew point depression of surface air (T  Td), a purely thermodynamic quantity, is invariably larger over land than ocean. [Global maps of daytime dew point depression (and equivalently CBH) would show marked land–ocean contrasts as in Figure 15.1(a) and (c).] See [52]. The convenience of cloud physics is that the lifted condensation level (LCL) and CBH are both proportional to T  Td. If T = Td, the air is saturated (RH = 100%) and the cloud extends downward to the surface. Over oceans, typical CBHs

W WT W

0° C

Balance Level

WT

(a)

W

W

Balance Level

WT 0° C

WT

(b)

Figure 15.5 Illustration of typical continental and maritime convection at (a) the time of radar first echo and (b) at the cumulonimbus stage. The balance level heights in all cases are also indicated

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are 500 m (corresponding to typical relative humidity of 80%), but over land can vary from 1,000 to 5,000 m (with relative humidity in the range of 70%–20%). The contrast in CBH between continental and maritime environments is illustrated in Figure 15.5(a), for afternoon clouds at the first-radar-echo stage. The oceanic cloud achieves the first echo while still a warm cloud. In contrast, the continental cloud with systematically higher CBH is usually extended into the cold region (< 0
C) of the atmosphere at first-echo stage. The heights of the 0
C isotherm are similar for land and ocean, near 4,500 m MSL, but the CBHs differ markedly for thermodynamic reasons. Figure 15.5(b) shows deeper clouds for both land and ocean that include the mixed-phase region bounded by the 0
C and 40
C isotherms where active charge separation can occur under appropriate conditions of cloud vertical development. The CBHs remain the same and often coincide with the top of the planetary boundary layer. The cloud widths are different based on observations showing that continental clouds are broader than maritime ones [11,53,54]. The updrafts in clouds are fundamentally important in regulating cloud microphysics and electrification. Scaling analysis indicates a sensitive fourth power relationship of lightning flash rate on updraft speed [55]. Accordingly, modest changes in CAPE can have substantial effects on lightning flash rate. Figure 15.5 also includes vertical arrows to contrast the updraft speeds in different regions (including CBH in Figure 15.5(a)) of both shallow and deep convection. Regarding the subcloud region, Zheng and Rosenfeld [56] have found larger ascent speeds (by 50%–100%) in the continental boundary layer than the oceanic one, and larger speeds at CBH, consistent with predictions based on thermodynamics and the contrast in surface properties in Williams and Stanfill [11]. Puzzlingly, model results on deep clouds with greater CBH do not show evidence for larger updraft speeds [57]. It should be remembered however that the land/ ocean lightning contrast is more strongly controlled by numbers of thunderstorms than by lightning activity per storm [6]. Earlier studies [53,54] have shown systematically larger updraft speeds in deep moist continental convection over land than over ocean. The contrast in ascent speeds is unmistakably linked with a contrast in the ice phase microphysics and lightning activity between land and ocean, but the explanation for the contrast in ascent speeds remains a controversial issue [11,41,57]. When thunderstorms over land alone are examined, lightning flash rate and CBH are positively correlated in global comparisons with the lightning imaging sensor in space and with surface thermodynamic observations of dew point depression [58]. These measurements need to be controlled for CAPE and aerosol variations to narrow down the physical causality. The formation of precipitation within the updraft of convective clouds is important because the precipitation can load the updraft, and ultimately reduce the updraft speed. This issue was raised initially in the context of CAPE, but here one can be more quantitative by estimating the precipitation content that will offset the effect of thermal buoyancy. Simple considerations of Archimedean buoyancy show that the force per unit mass associated with a temperature perturbation DT is simply DT/Tg, where T is the ambient temperature. The negative buoyancy contribution (again force per unit total mass) from additional mass loading m in a parcel with mass of air M is simply

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m/(m + M)g. For a parcel at the 0
C isotherm with nominal air density of 0.6 kg/m3, the condensate loading needed to offset a thermal buoyancy of 1
C is 2.0 g/m3. Extensive documentation on the detection of first radar echo and the formation of precipitation in deep convection has come from radar studies (good summary in Ludlam [32], with additional examples in [59]). The D6 dependence of the radar cross-section of precipitation particles gives radar considerable sensitivity in detecting the formation of precipitation, given the runaway nature of coalescence of cloud droplets in the diameter range 25 mm [60] where the rate of droplet coalescence varies steeply with droplet size as D5. The radar studies have shown that cloud depths (cloud top height minus CBH) in the 4,000–5,000 m range over continents and as small as 2,000 m over oceans [32,59] are needed for first echo development. These radar-based estimates are broadly consistent with aircraft in situ measurements of cloud depths needed to achieve critical cloud droplet size [60]. The relevance of these results in the context of Figure 15.5 is that the warm cloud depth in the maritime case (4,000 m), a thermodynamic effect, is large in comparison with that needed for the formation of precipitation. In the continental case, this condition is not fulfilled. These comparisons are consistent with observations of radar first echoes that appear consistently in the “warm” part of the cloud over oceans, but more typically in the “cold” part of the cloud over land [32,59,61]. In a more global context, the results are also consistent with observations that warm precipitating clouds are prevalent over oceans and scarce over land [11,62]. Still, the conundrum remains between a thermodynamic effect and an aerosol effect (Section 15.5). The lower CBHs over ocean overlie cleaner air with more dilute CCN concentrations [63]. Accordingly, following the discussion in Section 15.5, the cloud droplets above CBH will be larger and more prone to form precipitation and radar first echoes at lower heights. Braga et al. [63] give emphasis to the aerosol effect and neglect the thermodynamic effect. The formation of precipitation in moist convection is important because it can then descend with respect to the air parcel in which it forms, and thereby load the updraft column beneath. (In barotropic conditions typical of tropical convection, the updraft is vertical.) This is sometimes called “super-adiabatic” loading because the precipitation condensate is added to the adiabatic condensate at lower levels. (In baroclinic conditions (Section 15.3.7) more typical of convection at higher latitudes, the updraft can be tilted and then the updraft can unload its precipitation, as is assumed in the irreversible calculation of CAPE.) Based on observations with a precipitation radar in space, warm rain clouds over oceans (where the warm cloud depths are greatest) are capable of achieving precipitation concentrations up to 2–3 g/m3 [62]. These mass loadings are commensurate with cloud buoyancy at the 1
C level as was shown earlier. These superadiabatic loadings represent reductions in the condensate that is delivered to the mixed-phase region by the updraft, and where the conversion of supercooled water to ice can invigorate the updraft by the latent heat of freezing. This process can strongly influence the nature of the vertical profiles of latent heat release and larger ice-phase hydrometeors and help explain the marked land–ocean contrasts in differences in lightning (Figure 15.1(a)) and rainfall (Figure 15.1(b)). In short, the warm rain cells over ocean may be substantially prevented from becoming thunderstorms by virtue of the raindrop loading

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they achieve. Further evidence for this suggestion is found in the next Section 15.3.6.

15.3.6 Balance level considerations in deep convection The unloading of the updraft laden with warm rain and the delivery of condensate to the mixed-phase region are both sensitive to the updraft profile and to the location of the balance level [64,65], as illustrated in Figure 15.5. Vertically pointing Doppler radar observations of moist convection show a zero-crossing of mean Doppler velocity, with upward motions above and downward motions below. In the maritime case this balance level is located initially in the warm rain region, whereas in the continental case [65], its location is often in the mixed phase above. The precipitation particle fallspeed is balanced by the updraft w at the zero-crossing. Since the fallspeed of raindrops varies as D1/2 at the balance level, the particle mass (D3) varies as W6, and the reflectivity contribution varies as D6 or as W12. Following these sensitive dependences, a reduction in W by 40% (H2) by adiabatic loading in the maritime case relative to the continental case can cause an 8-fold reduction in mass and a 64-fold (18 dB) reduction in radar reflectivity. Eventually, the supercooled raindrops will freeze and then one needs to consider the gravitational power contribution from the ice particles [66] which will scale as D7/2, leading to an 11-fold difference in gravitational power between the land and ocean case. Indirect evidence for the severe loading of an updraft by raindrops comes from experience in vertical mine shafts in South Africa [67–69]. Air that is nearly saturated with water at 30
C at the bottom of a mineshaft (with vertical extent of 1,500 m) is forced vertically by a powerful ventilator system. At a vertical air speed near 10 m/s, matched with the fallspeeds of the largest raindrops, the load on the ventilator system frequently exceeded its capacity and failed, allowing the suspended water to fall to the bottom of the shaft in a deluge. In a vertical shaft, no opportunity was afforded for unloading of the updraft, as in the irreversible thermodynamic process discussed in Section 15.3.4. In moist convection over the ocean for which the radar first echo appears below the 0
C isotherm [32,59], this mineshaft experiment is likely applicable and is reminiscent of suggestions by Zipser [70] that the updraft speeds in oceanic cumulonimbus clouds [54] would be limited to the fall speeds of the raindrops within them. The mineshaft experiment is also a reminder that vertical updrafts cannot unload their condensates. In continental convection in which the radar first echo is found typically above the 0
C isotherm [32,59], the balance level updraft loading by raindrops near 10 m/s is avoided, and larger ascent rates are possible. Now in the mixed-phase region, the main hydrometeors are graupel and so the new balance level there manifest in triple Doppler radar measurements [65] can be substantially higher. In severe storms, still larger ascent rates are possible, with a balance level accommodating the growth of hailstones that may reach softball size in updrafts approaching 100 m/s. In these situations, a BWER (bounded weak echo region) in radar observations is indicative of the balance level above and may be two to three times higher than the balance level in warm rain cells. But in this strongly baroclinic situation (see Section 15.3.6) the updraft is strongly tilted and the hailstones

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can fall out of the updraft. In a special class of supercells (called LP for “low precipitation” but storms invariably productive of hail) with high CBHs (occasionally with sub-0
C cloud base temperatures), no condensate is lost in a warm rain process and the adiabatic cloud water is available for the growth of the hail.

15.3.7 Baroclinicity The pole-to-equator temperature difference is an important consideration for both weather and climate [71]. The local latitudinal temperature gradient exerts a decided influence on the organization of thunderstorm activity, and may also affect the flash rate behavior of thunderstorms. This temperature contrast is set up by the latitudinal imbalance between the incoming shortwave radiation from the Sun and the outgoing longwave radiation to space. Instabilities in the prevailing westerly winds at mid-latitudes draw on the potential energy of the pole-to-equator temperature difference to produce synoptic scale weather disturbances there [72]. The condition of a latitudinal gradient in temperature is “baroclinicity” and the thermal wind equation in atmospheric dynamics [71,73] links a baroclinic atmosphere with a vertical shear in the horizontal wind. This vertical wind shear can tilt the updrafts of thunderstorms (embedded in these large-scale disturbances) from the vertical, and thereby strongly influence the unloading of condensates from the updraft. In the near-equatorial zone of the tropics, the latitudinal temperature gradient nearly vanishes and the atmosphere is characterized as barotropic rather than baroclinic [73]. Without the organizing effects of vertical wind shear, air mass thunderstorms prevail, with an expectation for vertical rather than tilted updrafts. Yoshida et al. [74] have compared the flash rates of thunderstorm cells in continental and oceanic zones at different latitudes. In general, the mean flash rates are increasing with latitude away from the tropics. Bang and Zipser [75,76] have also found that oceanic convection is more likely to produce lightning in the presence of vertical wind shear than without it. Some of the most dramatic outbreaks of lightning over the open ocean occur in the presence of strong baroclinicity [77,78]. These findings may have explanation for baroclinicity in unloading of the updraft in the warm rain region of the storms so as to provide greater invigoration of the mixed phase. On this basis, and with all other factors the same, one might expect to have more lightning activity globally with a larger pole-to-equator temperature contrast. Supercells are rotating thunderstorms characteristic of strongly baroclinic environments in the springtime, and to a lesser extent in the fall, in both northern and southern hemispheres [79–81]. These storms are parent to a large portion of severe weather episodes, including strong winds, large hail, and tornadoes [80,81]. Supercell thunderstorms also produce exceptional total lightning flash rates [58,82–86].

15.4 Global lightning response to temperature on different time scales The global temperature is known to vary on a number of natural time scales: the diurnal, the semiannual, the annual, on the El Nino/La Nina time scale, and on the

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11-year time scale of the solar cycle. In understanding global lightning’s response to climate change on the long-time scale, it is valuable to look for consistent patterns of response on other time scales whose physical origin is better understood. It is also possible that systematic changes in aerosol may accompany these natural variations in temperature. Here again, we encounter the problem of disentangling aerosol and thermodynamic effects [87].

15.4.1 Diurnal variation On a planet covered with ocean, the variation of global temperature over the 24-h period of the Earth’s rotation in sunlight is expected to be nil. But based on the contrasting properties of land and ocean discussed in Section 15.3.5, with preferential heating of land relative to ocean in response to incoming shortwave radiation from the Sun, a consistent diurnal variation of global temperature in UT time is established. The local diurnal variation of surface air temperature typically shows a maximum value shortly after local noon, whereas the lightning activity peaks later in the afternoon, near 4 pm [7,8,88,89]. The globally integrated effect is a consistent global variation of temperature in universal time [90] that would appear to be the physical basis for the Carnegie curve of atmospheric electricity [7,8,91–93]. The regions that are sequentially heated by the Sun are the three “chimneys” of global lightning activity—the Maritime Continent, Africa, and the Americas, prominently manifest in the climatology of global lightning activity, as shown in Figure 15.6. In the global surface skin temperature [90] found a peak-topeak diurnal variation of 3.0
C for the globe. For a variation in the global lightning activity of 60% [88,94,95], this amounts to a sensitivity of 20% per
C [9,44,96]. By using surface temperature at airports with hourly variation during a period of active measurement of ionospheric potential (Vi) of the DC global electrical circuit, Markson [97,98] established positive correlations between Vi and tropical continental air temperature on the UT diurnal time scale, with a sensitivity of 7% per
C. The variation of aerosol and CCN concentration on the diurnal time scale has been investigated (e.g., [99]) but observations are insufficient to compile a climatology of UT variation. A climatological variation in storm flash rate in local time [89] shows a maximum between the time of maximum temperature and the time of maximum number of storms. This finding is inconclusive in distinguishing thermodynamic and aerosol contributions to lightning activity.

15.4.2 Semiannual variation A consistent repetition of weather twice per year is a foreign concept to midlatitude observers but is a very consistent feature of the near-equatorial region over land [100]. The sensible semiannual variation then in a number of meteorological quantities is a direct result of the Sun’s traversal of the equator twice per year at equinoxes [101]. The variation in solar insolation for the global tropics (23
latitude) is 7% (max–min/mean). The corresponding peak-to-peak amplitude variation in global tropical surface air temperature is about 1
C.

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Figure 15.6 The Carnegie curve of atmospheric electricity over 24 h of universal time and a global map showing three continental lightning chimneys. Global lightning observations are taken from references [95] and [102].

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Clear evidence for a semiannual variation in global lightning with the same phase as the temperature variation has been provided by optical observations of lightning from space [102]. This signal is most conspicuous when analysis is concentrated in the near-equatorial zone. Here the sensitivity of lightning to temperature is 20%–30% per 1
C. Rainfall observations are also consistent with double rainy seasons in northern South America and in near-equatorial Africa. The discharge record from the extensive Congo River basin, straddling the equator in central Africa, also shows a distinct semiannual variation. Several independent observations of the global electrical circuit (both DC and AC) show evidence for semiannual signals [92,101,103–105] that are plausibly linked with the semiannual variation of temperature. The aerosol contribution to the semiannual variation in global lightning activity has not been studied. We are presently unaware of long-term investigations of CCN concentration in the near-equatorial zone over continents that would shed further light here.

15.4.3 Annual variation The global distribution of land/ocean area is decidedly asymmetrical in the extratropics [101], with a fivefold greater land/ocean area ratio in the northern than southern hemispheres. The smaller heat capacity of land relative to ocean assures that the temperature of the surface air is greater in northern hemisphere summer than in northern hemisphere winter. This asymmetry is in large part responsible for the annual variation in mean global temperature, showing a maximum in August and with peak-to-peak amplitude variation of about 4
C. Many observations of global lightning have shown a seasonal variation in phase with this variation in surface air temperature, both in optical observations of lightning from space [88,94,102] and in surface-based observations of the intensity of the Earth’s Schumann resonances (e.g., [106,107]). The variation of global flash rate is nearly a factor of two, and the computed sensitivity of flash rate to temperature is 11% per
C. It has also been established that the major contribution to global flash rate on the annual cycle is number of thunderstorms rather than mean flash rate per storm. If temperature is the controlling thermodynamic variable, this would imply that higher temperature is favorable for more frequent release of conditional instability rather than in increasing CAPE. It is remarkable to have a globally integrated quantity that nearly doubles on the annual time scale. This finding is testament to the volatile nature of lightning in its apparent response to temperature. Similar to the discussion on the semiannual time scale (Section 15.4.2), the contribution of aerosol to lightning activity on the annual time scale has not been evaluated, for lack of comprehensive observations of the global aerosol climatology. It stands to reason that substantially larger CCN concentration will be available for storm ingestion in northern hemisphere summer than in winter, for the same reasons pertaining to land/ocean asymmetry, and the dramatic contrast in aerosol optical depth and CCN between land and ocean (Figure 15.1(c)). New

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methods now under development for the observation of CCN at CBH from satellite [14] will be particularly valuable in this context. Ambiguity about the annual phase of the DC global electrical circuit began when Lord Kelvin observed [108] in the United Kingdom that the Earth’s electric field was greatest in winter, contrary to the general picture based on global lightning activity and temperature. Whipple [7] identified this apparent contradiction but did not resolve it. Adlerman and Williams [109] pointed out the local effect of wintertime aerosol (in both hemispheres) in decreasing the electrical conductivity of the atmosphere and thereby increasing the electric field. A more reliable groundbased measurement of the DC global circuit is the air-earth current which represents the product of the electric field and conductivity. For example, long-term measurements of the air-earth current in Athens [110] have been shown to peak in August. Seasonal averages of the ionospheric potential [98], the preferred measure of the DC global circuit, also show a maximum in northern hemisphere summer.

15.4.4 ENSO Early interest in a strong relationship between the phase of the El Nino-Southern Oscillation and lightning developed out of analysis of a single magnetic coil recording of the first resonance mode of Schumann resonances [30], for a single ENSO event (in 1973–1974). A remarkably sensitive response of lightning to temperature was inferred from this analysis. Though the sign of this response has been corroborated in more recent work (e.g., [111,143]), the magnitude of the response in the data obtained by C. Polk is unprecedented, and may not be valid. Based on the global temperature analysis of Hansen and Lebedeff [27] and the classical analysis of rainfall variations [112] showing less rainfall over land in the warm phase, the general picture that developed was one in which all tropical land regions warmed in the El Nino warm phase, with the major upwelling in the central/eastern Pacific region causing large scale subsidence over tropical continental regions. This picture is in keeping with reductions in total river discharge in large drainage basins (Amazon and Congo), serving as continental-scale rain gauges, straddling the equatorial region [113] in the warm phase. This investigation was later extended to the Nile and the Ganges [114] River basins, with similar ENSO phasing. This picture of regional rainfall variability is also consistent with the majority of multiple ENSO events pictorialized in Allan et al. [115]. The regional behavior of lightning over the ENSO cycle shows somewhat less consistency overall, but a definite tendency in behavior is evident. This is most apparent in the Maritime Continent (including southeast Asia, Indonesia, and tropical Australia), where the ENSO studies are most numerous and where lightning is more prevalent in the warm El Nino phase [116–122]. Generally speaking, in this part of the world, the warm ENSO phase is also the drier (lower relative humidity and higher CBH) phase and the more polluted phase. As noted above, the wet cool phase (La Nina) is more abundant in rainfall. So here again we have the entanglement issue of thermodynamics and aerosol. The opposite tendencies for rainfall and lightning on the ENSO time scale are at first peculiar, but highly variable lightning/rainfall relationships, temporarily and regionally, are now widely recognized [123,124] and are linked with differences in the vertical development of the

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precipitation in the cold part of the cloud. This situation of opposite ENSO phase relationships for rainfall and lightning has also led to speculation that the two global electrical circuits (DC and AC) may have opposite tendencies over ENSO cycles [125,126], though coordinated synchronous measurements are lacking to check on this prediction. In this context, it seems likely that the contribution of electrified shower clouds [127–130] will be more potent over land during the cold La Nina phase, when continental rainfall is greater [115]. In South America, both Chronis et al. [131] and [111] using lightning observations from space found greater amounts of lightning in northeast Brazil during the cold La Nina phase. In contrast, Pinto [132] using thunder day observations over many ENSO cycles found a conspicuous tendency for greater numbers in the warm El Nino phase. An extreme El Nino event in 1926 has been documented by Richey et al. [133] and by Williams et al. [134] in the Amazon basin, with exceptional hot and dry characteristics but no information on lightning activity is available. Chronis et al. [131] and Sa´tori et al. [111] agree in finding more lightning in the warm phase in southern Brazil and eastern Argentina, where the discharge of the Parana River is also maximum [113]. This region appears to be the southern component of the north– south rainfall dipole anomaly identified by Grimm and Natori [135]. This distinctly extra-tropical tendency for greater lightning in the warm phase was found earlier in Argentina [136] and in the opposite hemisphere by Goodman et al. [137] in the Gulf of Mexico region of the United States, in a similar range of latitude. Among three tropical lightning “chimneys” [9], Africa appears to show the weakest lightning variation on the ENSO time scale. Chronis et al. [131] found some enhancement in the La Nina phase, whereas Sa´tori et al. [111] reported a modest lightning increase in the warm phase. Given Africa’s status as the most distant lightning chimney from the Pacific Ocean source of convective upwelling, it seems likely that global scale subsidence would have the least effect on Africa, while leaving conspicuous effects on adjacent chimneys Maritime Continent and America. Dowdy [138] has considered the effect of season on the lightning response to temperature on the ENSO time scale and this may impact the generality of a simple positive response in the warm phase. This seasonal aspect has also been discussed by LaVigne et al. [125]. Extreme El Nino events, such as the drought of the century in South America, can lead to so much warming and drying as to prevent both moist convection and lightning. Evidence for this situation may be found in the 1926 drought on the Amazon basin [133,134,139]. Despite the great sparsity of oceanic lightning (recall Figure 15.1(a)), easily discernible regional variations are discernible on the ENSO time scale. The general tendency for oceans is opposite to that for land: greater lightning over ocean in the cold La Nina phase [111,140,141]. This has been interpreted on a basis consistent with that for the warm phase: greater regional subsidence signifies less overall cloudiness, and hence greater surface heating and greater instability to drive moist convection. This situation contributes to the heat uptake of the tropical ocean during the La Nina phase. On a worldwide basis, Satori et al. [111] documented greater lightning in the ENSO warm phase than in the cold phase. Harrison et al. [142,143] have found evidence for inferred increases in the global electrical circuit in the warm phase. As

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this book chapter was going to press, evidence for greater global lightning activity during the 2019 El Nino was evident, followed by reductions in global lightning in the La Nina years of 2020-2023 (See also [52]). Recent improvements in understanding lightning variations on the ENSO time scale Williams et al. [144] have been achieved by considering superlative events in the Super El Nino category, of the same kind considered earlier by Williams [30]. In contrast with earlier studies that compare lightning totals in the warm (El Nino) and cold (La Nina) phase, the full evolution of lightning activity in individual ENSO events has been considered, including the important transition phase from cold to warm periods. Although more lightning is present in the warm phase than the cold phase, as, in earlier work, the peak lightning activity is found in the transition phase, when the vertical temperature profile is most out of equilibrium with the surface. In another ENSO study Guha et al. [145], peak lightning activity was also found in the transition phase, but in that case in the transition from warm to cold periods.

15.4.5 Decadal time scale Modest changes (0.1%) in the integrated energy output of the Sun have long been recognized over the 11-year solar cycle, and the corresponding changes in global temperature have been investigated. The peak-to-peak variation of temperature from this analysis is about 0.1
C [146–148]. Attempts to see these changes in global records of thunder day data have been mostly unsuccessful [149,150]. No solar cycle signal has been found in the Lightning Imaging Sensor/OTD satellite record of global lightning. It should be noted however that solar cycle variations have been detected in the analysis of thunder day records at selected stations in Brazil [151] on the basis of wavelet analysis. Solar cycle variations in the intensity of Schumann resonances at high latitude are not plausibly explained by temperature-related variations in lightning activity [152]. Koshak et al. [153] studied variations in lightning incidence over the continental United States (CONUS) in the decadal period 2003–2012. The trend in cloud-toground (CG) lightning was negative over this period, with a 12% decrease from the interval 2003–2007 compared to the interval 2008–2012. The trend in wet bulb temperature over the CONUS was also negative for the same period but the dry bulb temperature showed an increase. The total lightning activity measured by the lightning imaging sensor in space showed no significant trend over the same period. In retrospect, the decadal period examined in this study lay within the period now frequently referred to as the hiatus in global warming (1998–2013). It is also interesting to speculate about a possible decadal increase in CBH, as the average dry bulb temperature increased and a moisture variable decreased, together entailing an increase in the dew point depression which is proportional to the CBH. Boccippio et al. [154] had found a large enhancement in the IC/CG ratio in a region of the CONUS (extending from eastern Colorado northward through the Dakotas) with elevated CBH [58].

15.4.6 Multi-decadal time scale For studies on lightning variability on time scales longer than the typical lifetimes of lightning detection networks, researchers have resorted to the use of thunder day

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data (and model calculations: [155]). Thunder day observations have been underway at meteorological stations and airports worldwide for more than a century. One interesting recent application here has focused on an upward lightning trend in the Sea of Japan [156]. A rough doubling in thunder day counts since 1930 has been linked with observed increases in sea surface temperature of 1.2
C–2.2
C. The tendency for the current global warming to predominate at high northern latitudes is widely recognized [28]. At Fairbanks, Alaska (latitude 64.8
N) both the temperature and the thunder day counts have been increasing conspicuously [44]. Figure 15.7 includes plots of both quantities with least squares fits for trend. The number of thunder days has more than doubled in 50 years. Anecdotal reports indicate that Canadian meteorological stations at the highest latitudes have noted thunder days for the first time on record. The long-term record of mean global temperature [27,28,157] based on averaging of surface thermometers shows an increase of the order of 1
C on the 100year time scale but is interrupted by shorter intervals when the temperature is flat or declining with time. The two most notable intervals are the so-called “Big Hiatus” from 1940 to 1975, and the more recent (and more controversial) hiatus in global warming from 1998 to 2013. Both these intervals have been addressed recently by Williams et al. [158]. Appeal was made to previously published thunder day observations to address the Big Hiatus and separate analyses for both North America and Siberia show flat or declining counts of thunder days. A 15% decline in mean annual thunder days is evident from 1940 to 1970. For the more recent hiatus in global warming, satellite optical observations are available for nearly the entire interval from the lightning imaging sensor in space. Several global temperature data sets were examined and it was shown that both the

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Figure 15.7 Upward trends in (a) temperature and (b) thunder days for Fairbanks, Alaska. Linear least squares fits are also included to illustrate these trends

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temperature trend and the trend in global lightning flash rate were statistically flat during the hiatus period [158]. This period ended with a pronounced El Nino event in 2015 and a strong resumption in the increase in global temperature [157] that had been underway between the time of the Big Hiatus and the recent hiatus. Unfortunately, the Lightning Imaging Sensor is no longer operational in space to check the lightning behavior in this later period. Overall, the available results are not inconsistent with the hypothesis that global lightning is responsive to global temperature.

15.4.7 Hiatus in global warming and “warming hole” In the study by Williams et al. [158], the regional reductions in temperature and thunder day activity were interpreted as being part of the “Big Hiatus” in global warming (1940–1975). However, in light of the evidence for a “warming hole” (a region not exhibiting a warming trend) over a large portion of the central United States [159], attributable to changes in agricultural activity, it is possible that the reductions reported in Williams et al. [158] were more local than global. China also exhibited a “warming hole” over the multi-decadal period (1910–1949 versus 1970–2009) [159], but the regional behavior of temperature and thunder day activity there are currently unknown and deserve to be examined.

15.5 Aerosol influence on moist convection and lightning activity 15.5.1 Basic concepts When condensation of water vapor occurs during the ascent of air parcels in the real atmosphere, every cloud droplet that forms is dependent on some nucleus to initiate its formation. The process is known as heterogeneous nucleation. The subset of the atmospheric aerosol population that serves this role is called cloud condensation nuclei (CCN) [160]. Were it not for the ubiquitous presence of CCN throughout the atmosphere, large departures in water vapor concentrations from the equilibrium predictions of the Clausius–Clapeyron relation (Section 15.3.4) would develop in a thunderstorm updraft, and these departures are not generally observed (but see recent findings in [20]). As an air parcel ascends in a thunderstorm updraft, the adiabatic cloud water content that appears (enforced by Clausius–Clapeyron) is shared roughly equally among all the available nucleation sites. This means that the cloud droplet concentration is matched with the CCN population at CBH, and when the CCN population is large (polluted conditions) the cloud droplets are smaller than they are in clean conditions. Since the tendency of cloud droplets to coalesce and form precipitation particles is strongly dependent on their size [60], the CCN concentration can be influential on the development of convection [12,13,15,161,162]. The contrast in conditions for convection growing over clean and polluted boundary layers is illustrated in Figure 15.8. Three broad ranges of CCN population can be considered: (1) clean conditions with CCN concentrations typical of maritime air (10–100 per cc),

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Maritime Regime

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Figure 15.8 Illustration of the aerosol effect on convection and lightning in clean and polluted situations

(2) more polluted conditions (a few hundred to 1,000 per cc) typical for continents, and (3) ultra-polluted conditions with concentration exceeding several 1,000 per cc. These regimes have been treated in cloud models [162]. Condition (1) may favor the rapid formation of precipitation and the production of rain, which may ultimately contribute to the superadiabatic loading of the air parcel (Section 15.3.5). Condition (2) may enable the retention of condensate in the updraft (consistent with the assumptions of reversible ascent ([13] and Section 15.3.5)) until the mixed-phase region is attained. The most polluted condition (3) may lead to cloud droplets so small that the formation of graupel particles in the mixed-phase region is prevented, thereby enabling the cloud water to rise as high as the 40
C isotherm, where homogenous nucleation of the cloud water may occur.

15.5.2 Observational support A major shortcoming in the evaluation of aerosol effects on convective vigor and lightning activity, and making detailed comparisons with competing

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thermodynamic contributions, has been the general absence of information on the CCN concentrations involved in specific situations. The innovative development of satellite methods, both for observing the cloud droplet sizes from space [163] and for obtaining the CCN concentration at cloud base height (CBH) [14], is having a dramatic impact at the time of this writing on the understanding of the influence of aerosol on cloud microphysics and lightning. Orville et al. [164] documented an approximate doubling of CG lightning flash density in the vicinity of oil refineries in the vicinity of Houston, TX, that was attributed to an aerosol effect. A more elaborate investigation of both thermodynamic and aerosol effects in the same general area [21] shows evidence that the lightning enhancement there is more likely due to an effect of CCN than to a heat island effect. Even in the absence of this direct measurement of CCN, several studies have appeared that make use of global proxies for aerosol to compare with lightning activity obtained with other platforms. Stolz et al. [19,165] have used aerosol estimates from GEOS-chem (www.geos-chem.org) to compare with NASA TRMM Lightning Imaging Sensor observations to show that aerosol contributions to lightning activity are comparable with thermodynamic ones. Altaratz et al. [166] have examined the influence of aerosol (estimated with satellite measurements of aerosol optical depth) and CAPE on lightning recorded with the World Wide Lightning Location Network (WWLLN) on a regional basis. They found that statistically significant increases in lightning activity were associated with more polluted conditions. Published examples of perturbations in lightning activity when aerosol is introduced in maritime convection have produced the most convincing evidence for aerosol control. In the first case [167] volcanic aerosol documented by satellite was ingested by oceanic cumulonimbus clouds whose exceptional lightning activity was documented from space, by and specific controls were placed on thermodynamic influence. In a more recent study by Thornton et al. [22], a rough doubling of lightning activity along sharply defined oceanic shipping lanes in Southeast Asia has been documented with the WWLLN. In this case, it can be shown that the warming of the sea by engine cooling exhaust water by the sea-going vessels is of negligible consequence, and diesel exhaust is rich in fine aerosol [168]. The introduction of rich aerosol concentrations in the pristine environment of the ocean represents a dramatic change in cloud microphysical conditions, with a maximum likelihood of manifestation on lightning when none is present in the pristine state. The collection of different kinds of lightning studies by Yuan et al. [164], Altaratz et al. [163], Stolz et al. [18], and Thornton et al. [22], all over oceans, may reflect this more sensitive response of lightning to aerosol in that regime. One may contrast the aerosol sensitivity in the maritime regime with the situation with high CBH (and shallow warm cloud depth) over land [58,169,170]. Li et al. [169] concluded that in this regime, there was little change in cloud microphysical behavior in response to aerosol variations. One reason for this is that the cloud droplet sizes in the mixed-phase region are already small by virtue of the proximity to cloud base [60] and the absence of a deep warm rain process to

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mediate with aerosol. Without the depletion by warm rain, the cloud water contents in the mixed-phase region are expected to be large in such continental storms, promoting the growth of hail and also conditions conducive to thunderstorms with inverted electrical polarity [58,84,170,171]. Lyons et al. [172] had earlier found that storms ingesting smoke from fire were exhibiting greater numbers of positive ground flashes. Additional evidence that aerosol has a pervasive influence not only on lightning but other aspects of meteorology are studies that show reduced activity on weekends when the anthropogenic contribution to aerosol in industrialized regions has been shown to be reduced. Weekend effects on rainfall [173], lightning [174], hail [175], and even tornadoes [175] have been documented in recent years, with statistically verified results.

15.5.3 Lightning response to the COVID-19 pandemic Unprecedented reductions in global aerosol occurred in conjunction with the COVID-19 pandemic (e.g., Sanap, 2021) [176]) linked with reductions in fossil fuel consumption during a lockdown phase beginning in the month of April 2020 in many sectors of the global economy. Preliminary investigation of the lightning response to this aerosol reduction, using global VLF networks GLD360 and WWLLN, as well as ELF (Schumann resonance) methods, have shown lightning reductions of global scale of the order of 10% in comparisons with adjacent years. The transition from El Nino to La Nina from 2019 to 2020 makes ambiguous the interpretation of the lightning reduction however, and here we have entanglement of aerosol and thermodynamic effects [52].

15.5.4 Work of Wang et al. (2018) on the global aerosollightning relationship Wang et al. [177] have made valuable use of independent satellite measurements of aerosol and lightning to investigate the dependence of lightning flash rate on aerosol conditions for both biomass-burning smoke and for mineral dust aerosol. At modest values of aerosol optical depth (AOD) (0–0.25), the observed flash rate increases quasi-linearly with AOD, in both cases. It was not possible however in this study to completely disentangle thermodynamic from aerosol effects.

15.6 Lightning as a climate variable Lightning frequency is changing as the climate changes. For example, lightning’s close relationship to thunderstorms and precipitation makes it a valuable indicator of storminess, making lightning an instrumental means for monitoring a variable climate [30,102,178,179]. What’s more, lightning is not only an indicator of climate change; it also affects the global climate directly. Lightning produces nitrogen oxides, which are strong greenhouse gases. In recent years, lightning measurements have become more extensive, and new satellite instruments have further enhanced measurement coverage [180]. However, lightning monitoring for climate science

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and services is still limited globally and at an early stage of development. To overcome these shortcomings and to explore the opportunities and challenges of lightning observations for climate, the scientists involved with the Global Climate Observing System (GCOS) sought to ensure that data necessary for climate studies is available to the public. The Commission for Climatology (CCl) of the World Meteorological Organization (WMO) established a Task Team for Lightning Observations for Climate Applications (TTLOCA) in October 2017. Lightning has been added to the Global Climate Observing System’s (GCOS) list of Essential Climate Variables (ECVs) to improve on climate monitoring [3,179].

15.7 Lightning activity at high latitude Despite the general understanding that lightning activity is greatly diminished at high latitude relative to the tropics, based on considerations of surface air temperature and water vapor (Section 15.3.3), lightning can and does occur in the colder polar regions. For example, consultation of the NOAA GSOD (Global Summary of Day) dataset on thunder day observations [181] for recent decades shows that 390 surface stations north of the Arctic circle show at least one thunder day in their records. Following the Franklin Lecture on “Lightning and Climate” at the American Geophysical Union meeting in 2012, one Canadian meteorologist reported that thunder days were being recorded for the first time at the highest latitude stations in recent years. Some improvement in the characterization of lightning trends in high latitude regions has been achieved quite recently with regional lightning networks, for both the Arctic as a whole using a subset of the World Wide Lightning Location Network [182] and for the State of Alaska [183] using the Alaska Lightning Detection Network (ALDN). In both cases, the assessment of decadal trends is compromised by the changes in the networks over time. The quantitative results from these studies are here discussed in turn below. Interest in lightning at high latitudes also derives from the role of lightning in initiating fire in relatively dry forests there [183–186]), and the possibly positive feedback linked with aerosol and lightning [186].

15.7.1 The Arctic No region on Earth is warming more rapidly than the Arctic [187]. Recent reports show a warming rate in surface air temperature that is four times greater than the rest of the Earth. The leading explanation for this behavior is a positive feedback effect involving ice albedo, with greater surface absorption of solar radiation as the ice retreats [188]. The dramatic changes here have prompted recent investigation of lightning variations with the World Wide Lightning Location Network (WWLLN) by Holzworth et al. [182]. Since the number of stations in the WWLLN network has nearly tripled in the last 20 years, Holzworth et al. [182] give more emphasis to changes in the Arctic lightning as a fraction of the global total, rather than the actual stroke counts. (The latter counts increase markedly in the period of steepest increase in the total number of WWLLN receiving stations.) If one takes the Arctic

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stroke counts at face value to obtain a lightning sensitivity to temperature over the past 10 years, over which Ballinger et al. (2021) [187] report a 1.5 C mean increase in surface air temperature, that amounts to a 300% increase in lightning for 1.5 C, or 200% per C. Clearly, this value is out of line with other estimates (Section 15.4), and also points to the problem with time-dependent detection efficiency.

15.7.2 Alaska Evidence for upward trends in thunder days, surface air temperature, and lightning activity in recent decades in Alaska can be found in Section 15.4.6 (Figure 15.7), Bieniek et al. [189] and in Bieniek et al. [183], respectively. The magnitudes of the temperature trends show the largest values above the Arctic circle, consistent with the evidence for the pronounced warming of the Arctic noted above. As noted by Bieniek et al. [189], “the low frequency variability confounds the identification of trends in the time series of temperature.” Given the network changes over time, the lightning data are also caveated [189] by the following: “These changes in detection efficiency and accuracy through the record make the data challenging to use for assessing the variability and trends of lightning over the historical period.” For best comparison with the 30-year lightning record in Bieniek et al. [183], our selection of representative temperature increase (0.8 C) is based on the 30-year period of summertime (June, July, and August) temperature in Bieniek et al. [189] coinciding with the months of greatest lightning activity. Given their estimate of a 17% increase in lightning over the same period [183, p. 1148] we derive a rough sensitivity of 17%/0.8 C = 21% per C. This sensitivity is broadly consistent with the estimates obtained in Section 15.4. Despite the evidence for unparalleled increases in lightning activity at high latitude in recent years, the total contribution to global lightning from this remote part of the world is inconsequential. To show this, one can compare the annual lightning totals recorded by Holzworth et al. [182] and by Bieniek et al. [183] with the totals in the tropical continental “chimneys” discussed in Section 15.1. The total flash rates in the tropical chimneys are of the order of 10 flashes per second. In contrast, the largest annual-averaged WWLLN stroke rate amounts to 8  103 strokes/sec. For Alaska, the peak annual stroke rate is 1  105 strokes/yr, or 3  103 strokes/s. This comparative situation gives the global VLF networks some superiority over ELF methods in monitoring the changes in high latitude lightning into the future, so long as the networks are stable with time.

15.8 Winter-type thunderstorms and lightning Lightning occurrence in winter thunderstorms has been a concern due to their high damage potential (e.g., [190]). In the winter cold period (e.g., winter in middle latitudes), diurnal heating is weak and moisture content is much reduced over land compared to the warm season. Low-pressure systems are more vigorous and bring cold air masses from polar and arctic regions southward to mid-latitudes, creating strong vertical temperature gradients as cold continental air flows over relatively warm seas and ocean currents. Air parcels near the surface then experience no

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inhibiting warm layers on their way to the equilibrium level, often the tropopause, and occur over large regions over the sea behind cold fronts. The tropopause is found between 10 and 15 km at mid-latitudes in summer but can descend to 5–10 km in winter, limiting the vertical extent of convection. Just as in summer, low-level convergent winds organize the triggering of storms, but over the sea, these are often found near and upwind of coastlines, where enhanced friction and sloping terrain create a relative stagnation and ascending flow. The importance of the moisture source from seas and oceans in winter thunderstorms deserves a look in the context of the ocean gyres. Every major ocean basin (North Atlantic, North Pacific, South Atlantic, South Pacific, etc.) contains a basinscale closed rotating current: a gyre with clockwise (counterclockwise) rotation in the northern (southern) hemisphere. The primary drive for these gyres is the zonal wind stress from prevailing easterly winds in the tropics otherwise known as the trade winds, and from prevailing westerly winds at mid-latitude. In this near-equatorial portion of the gyre, the ocean surface is warmed substantially by sunlight and at the western limit of this equatorial transit, this warm oceanic flow is diverted northward and southward, depending on hemisphere. Since the surface air over continents is increasingly colder away from the equator and tends to be moving eastward off the continents at midlatitude, one has a consistent situation in all gyres that warm ocean water in this poleward current is found beneath colder air away from the equator. This configuration is inherently unstable and can produce vigorous atmospheric convection and thunderstorm activity. The Gulf Stream along the North American coast and the Kuroshio Current along the eastern coast of Asia (China, Japan, Korea, and Russia) are prime examples in which lightning activity over warm ocean water is prevalent during winter. In contrast, the return current on the eastern boundaries of oceanic gyres, moving equatorward, is colder than the air overlying it. This situation is stable against convection and accordingly lightning is absent. A prime example is the Eastern Pacific Ocean. Similar behavior is present in the oceanic gyres of the southern hemisphere.

15.8.1 Effects of global warming on winter thunderstorms Evidence has accrued that global warming is shifting large-scale extratropical atmospheric circulation poleward [191–193]. The effect on large-scale ocean gyres is still not clear. Recently, based on observations of sea surface height and temperature it has been found that the major ocean gyres are also shifting poleward [194]. Supported by climate model simulations, it seems evident that the observed shift is likely a response to global warming [195]. The polar shift of the major ocean gyres and the temperature warming of mid-latitude and polar regions also suggest a shift in the occurrence of winter thunderstorms. But, on the other hand, the influence of polar vortex weakening and the meandering of the jet stream allowing cold air to penetrate to lower latitudes and to produce extreme cold weather has also been documented. Winter thunderstorms can be common in these episodes. The suggested reason for this behavior is the shift of the polar vortex from North America to Eurasia. Studies indicate that in a few decades this asymmetry will increase and cause more frequent extreme cold episodes [196].

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15.9 Storms at the mesoscale The mesoscale involves a size an order of magnitude larger than the scale of an individual thunderstorm. The expectation for larger and more frequent storms at the mesoscale in a warmer world has been given much emphasis in the popular literature (e.g., [197]). This expectation is based on the general influence of the Clausius-Clapeyron relation involving the exponential dependence of equilibrium water vapor on temperature discussed in Section 15.3.3. The largest storms in the tropics are the tropical cyclones (hurricanes and typhoons), to be discussed in the next section. The largest storms on planet Earth are found in the extratropical land areas as Mesoscale Convective Systems (MCSs), and the largest MCSs are Mesoscale Convective Complexes [198]. These storms will be discussed in the context of lightning megaflashes in Section 15.12. Seen from a satellite, an MCC is a roughly circular cloud shield with large size (200–300 km in at least one direction). An MCC may include squall lines, bow echoes, and/or isolated convective cells, each of which has its own life cycle. More than 70% of MCSs and all MCCs evolve from the merger of multiple convective clusters, which resulted in larger systems than those that developed from a single cluster [199].

15.10 Tropical cyclones These vast spinning masses of air and moisture have different names in different parts of the world: “hurricanes” in the Atlantic Ocean, “typhoons” in the western Pacific Ocean, and “storms” in the southern Pacific and Indian oceans. The tropical cyclones with the strongest vertical development, often taking the form of Category 4 and 5 storms on the Saffir–Simpson scale, are also the most frequent producers of lightning in the inner storm core. Accordingly, as we consider the response of tropical cyclones to climate change, a lightning component to that change is also expected. The physical impacts of tropical cyclones are often profound. Wind damage can be extreme, especially in Category 5 storms with sustained wind in excess of 70 m/s (250 km/hr). Both storm surge effects (with the strong surface wind stress piling up water) and the abundant rainfall over large coastal areas [200,201] can lead to extensive flooding. The lightning impact can be considered less serious because these large storms require the warm ocean presence for their energy supply, and few persons and ships are in harm’s way there. Figure 15.9 shows the vertical cross-section (with a 210-mile horizontal extent) of a Category 5 hurricane (Emily, July 17, 2005, south of Cuba), and serves to illustrate the three main regions [202]: the eye, the eyewall, and the outer rainbands, all of which have physical connections with lightning production in these storms. The eye of the hurricane is cloud-free air descending over the full vertical depth of the storm, and so neither condensate nor lightning can occur there. In marked contrast, immediately adjacent to, and outside the eye, is the eyewall, where in strong hurricanes like this one the strongest vertical air motion is found,

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INTENSE EYEWALL THUNDERSTORM heavy

Rain Intensity

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Figure 15.9 A vertical slice through the center of Hurricane Emily (July 17, 2005, near Cuba) shows the rain structure across 210 miles (340 km) of the storm. Note the lack of precipitation inside the eye of the storm compared to the intense eyewall convection just outside of it. The areas of heaviest rainfall are shown in red, and the lightest are in blue. The prominent horizontal feature in red is the radar bright band near 0C and often characterizes the outer rainbands. The rainfall structure shown here was measured by NASA’s ER-2 Doppler Radar

much like the updraft of an exceptional thunderstorm. The vigor of the eyewall convection and attendant lightning is often maximum during cyclone intensification and falling central pressure (i.e., “deepening”) [203–206]. Further out in radius from the eye, the ascent speed of the air is much reduced, as evidenced by the red horizontal feature near 5 km altitude: the radar “bright band”, close to the 0 C isotherm. Ice phase condensate is in evidence above the bright band to the top of the storm, an important source for charge separation and lightning in this region. The total vertical extent of the eyewall convection, using the bright band height as a “yardstick” is 15 km. Here the local lightning flash densities are much reduced in comparison to the eyewall lightning, but because of the substantially greater convective area in these outer hurricane rainbands, the total lightning production generally exceeds that in the eyewall [205,207]. Theoretical considerations of tropical cyclones that treat these storms as giant heat engines (transporting heat from the warm ocean surface to the cooler upper troposphere) predict increases in cyclone intensity with global warming [208]. Numerical models also show globally-averaged increases in maximum wind speeds by 2–11% by the year 2100 [209–211]. Using numerical global climate modeling, Lynn et al. [212] were among the first to project increases in hurricane intensity by the end of the century based on simulations of Category 5 Hurricane Katrina under climate change scenarios. However, difficulties with establishing an upward trend

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in the incidence of tropical cyclones in actual observations are similar to that in establishing an upward trend in global or regional lightning: the natural variability is great and the data records are relatively short. Despite considerable variability in tropical cyclone occurrence on multi-decadal time scales (e.g., [213]), the establishment of a century-scale trend with anthropogenic origin (now well established in the case of the global record of surface air temperature), has proven challenging [211,214]. Other studies on these shorter time scales by Holland and Bruye`re [215] have shown that the proportion of Categories 4 and 5 hurricanes has increased at a rate of 25–30% per
C of global warming after accounting for analysis and observing system changes. This has been balanced by a similar decrease in Categories 1 and 2 hurricane proportions, leading to the development of a distinctly bimodal intensity distribution, with the secondary maximum at Category 4 hurricanes. This global signal is reproduced in all ocean basins. The observed increase in Categories 4–5 hurricanes may not continue at the same rate with future global warming. The analysis suggests that following an initial climate increase in intense hurricane proportions a saturation level will be reached beyond which any further global warming will have little effect [215]. These results raise the interest in possible increases in tropical cyclone lightning activity, given the evidence that the high flash rate scenarios in hurricanes are generally found in the eyewall region of Categories 4 and 5 storms [205,207,216,217]). Indeed, the cores of Categories 4 and 5 storms are often so electrically active [216,217] that they resemble continental supercells. Normally speaking, oceanic convection is lightning-sparse compared to continental convection (as discussed earlier in Section 15.1). Both thermodynamic and aerosol-related explanations for this prodigious electrical activity in eyewall convection have been proposed. Given the evidence that both the mean rainband flash density and flash rates in Categories 2 and 3 storms are greater than in categories 4 and 5 [205,207] in the present climate, the expected temperature-dependent incidence of cyclone category [215] may lead to rather substantial increases in total tropical cyclonerelated lightning.

15.11 Cloud-to-ocean lightning The measured peak current in negative polarity cloud-to-ground lightning has been shown to be greater over ocean than over continents [172,218]. The physical explanation for this contrast, and why a similar contrast is less apparent for lightning with positive polarity, is presently in debate. Predictions for increases in peak lightning current (also known as lightning “intensity”) to the ocean surface in response to global warming have appeared [219]. These predictions are based on expected changes in the chemical and electrical properties of seawater in a warmer climate [219,220]. For example, increased ocean acidification may result in increases in hydrogen ions in seawater, thereby modifying the seawater conductivity. As these predictions for increases in peak current are based on

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laboratory-scale experiments [219], and boundary layer experiments with suspended vertical wires [221], it is essential that the equivalent circuit implemented in these experiments be compared with that of real lightning to place the effect of the variations in the oceanic electrical medium in proper physical context. Other more recent studies [222] show evidence that lightning peak current is insensitive to variations in the conductivity of the ocean. Alternative explanations for larger peak current in ocean strokes are based on considerations of the differences in cloud potential in storms over land and over ocean (see [223]), and not on the changes in medium properties. In any case, the information presently available suggests that the monitoring of the lightning superbolt population [224] may provide another useful diagnostic for climate change.

15.12 Lightning superbolts and megaflashes In searching for evidence in lightning documentation for indicators of climate change, a sound strategy is to examine the extreme categories of lightning (e.g., [9]). Two lightning extremes are superbolts and megaflashes. Superbolts are lightning events with superlative peak current (and/or stroke energy), with corresponding exceptional brightness. Megaflashes are events with exceptional horizontal extent, >100 km by present definition [225], and so require large storms to contain them. Superbolts were first identified optically with the VELA satellite by [226] over the Pacific Ocean as being 100–1,000 times brighter than ordinary lightning. Much more recent use has been made of the WWLLN to make a global map of superbolt locations, where in this case the superbolt is defined in terms of a threshold in VLF radiation energy (> 1 megajoule) [224]. The majority of these events are located over the oceans, with a great abundance over the Mediterranean Sea and north Atlantic Ocean in winter. With regard to megaflashes, lightning is sometimes casually described as “a huge spark.” According to the Glossary of Meteorology (American Meteorological Society 2015), a lightning flash is a transient, high-current electric discharge with path length measured in kilometers. So how “big” can a lightning flash get? The length in the vertical dimension is generally limited by the altitude of the main charge centers in the cloud (typically 6–15 km) and certainly by the cloud top, rarely more than 20 km high. However, the horizontal extent of a flash within the cloud can be much longer, reaching “mesoscale” dimensions in large storm systems [225, 227]). Besides being intrinsically interesting as extreme events, exceptionally long lightning discharges propagating through the stratiform precipitation region of a mesoscale convective system (MCS) sometimes produce exceptionally powerful positive cloud-to-ground flashes (+CGs), which induce several unusual phenomena, including sprites in the mesosphere [228]. Such exceptional CGs result from extending lightning channels tapping into vast reservoirs of positive charge present within the MCS stratiform region [229–234]. A frequent scenario involves a lightning discharge originating near the top of the convective cells in the leading line (8–10 km altitude) and then traveling rearward and downward, following the trajectory of descending positively charged ice crystals, often to near the melting

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layer [232,235–237]. Some space-based lightning mapping can measure flash extent and duration continuously over broad geospatial domains. These new satellite instruments, including the Geostationary Lightning Mappers (GLMs) on the R-series Geostationary Operational Environmental Satellites (GOES-16 and 17) and their orbiting counterparts from Europe (the Meteosat Third Generation (MTG) Lightning Imager) and China (FY-4 Lightning Mapping Imager), have enabled the establishment of new lightning records. The advent of the 3D Lightning Mapping Array (LMA) systems [238] confirmed that lightning discharges exceeding 100 km in length and spawning multiple CGs, of both polarities, often separated by considerable distances, were not uncommon [233]. On the basis of analysis with the Geostationary Lightning Mapper in space, Peterson et al. [239] identified and validated two lightning megaflash events accepted as records by the World Meteorological Organization (horizontal mesoscale lightning discharges of >100 km in length). Since that publication, the World Meteorological Organization (WMO) has announced a new world record for a megaflash. The longest single flash covered a horizontal distance of 768  8 km (477.2  5 miles) across parts of the southern United States on April 29, 2020. A plan view of this discharge is shown in Figure 15.10, showing that it extends over GOES16 ABI/GLM 98˚ W 96˚ W

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Figure 15.10 Satellite image of record extent of a lightning flash observed with the Geostationary Lightning Mapper over the southern United States on April 29, 2020 covered a horizontal distance of 768  8 km (477.2  5 mi). The horizontal structure (white line segments) and maximum extent (gold  symbols) of this megaflash are overlaid. As of this writing, this is the record-setting megaflash for scale

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four separate States (Texas, Louisiana, Alabama, and Florida) and the Gulf of Mexico. Since this distance is more than twice the threshold diameter [198] of a Mesoscale Convective Complex, it is likely that many of the largest megaflashes will require these superlative storms for their existence. The association of the largest lightning flashes with the largest storms suggests that the monitoring of megaflashes in a warmer climate may be a useful diagnostic for climate change.

15.13 Nocturnal thunderstorms The great majority of thunderstorms worldwide follow the diurnal variation in local time that has been discussed in Section 15.4.1, with a typical 4 pm maximum in activity when thermodynamic and dynamic conditions (e.g., outflow boundaries) are most favorable. This exclusive afternoon prevalence was assumed in the early analysis of global thunderstorms [240] that underlies the analysis of the global electrical circuit [7,8,130]. However, the most spectacular displays of lightning occur in the contrast of night, in nocturnal thunderstorms. Satellite-based studies at nighttime [241,242] show that the large land/ocean contrast in lightning prevalence apparent in daytime is reduced by about a factor of two at nighttime [242]. Thunderstorms that occur outside the usual afternoon diurnal cycle are of special interest in the lightning and climate context for two reasons: (1) their existence and the nocturnal land–ocean contrast provides an additional perspective on the disentanglement of thermodynamics and aerosol, and (2) the global warming signal has been dominated by nighttime temperatures over certain decadal intervals [243,244], most prominent in the period 1965 to 1985 for reasons that are not well understood. It is appropriate first to summarize special configurations of land and water that are favorable for nocturnal thunderstorms. Nighttime lightning activity may occur in low-lying areas, both lakes and valleys, adjacent to more mountainous terrain. Examples from the literature include the large tropical lakes and inland seas: Lake Victoria in Uganda, Africa [245], and Lake Maracaibo [246,247] in Venezuela, South America. Another example is when mountainous terrain lies adjacent to inland seas, as in the Mediterranean Sea [248]. Other examples of mountains adjacent to valleys productive of nocturnal thunderstorms are Phoenix, Arizona [249], and Albuquerque, New Mexico (personal observations, 1994) as well as the mountainous terrain of the Andes in Colombia [88] in South America. In all of these cases, the favored convective development in the afternoon is found in the elevated terrain where strong daytime heating of the surface is prevalent. The cold downdrafts of the ensuing thunderstorms then descend into the valleys and over lake surfaces to lift the air there and initiate deep convection where only subsidence from the adjacent mountain convection was present earlier in the day. The most violent nocturnal convection in the Earth’s atmosphere [250] occurs in two separate but conjugate regions in the Americas: in the Great Plains of North America east of the Rocky Mountains, and in Argentina, South America east of the Andes Mountains [88]. The peculiar phase of the diurnal cycle in Argentinian thunderstorms was identified in the report on worldwide thunder occurrence in

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Brooks [240], and Wallace [251] has systematized the local diurnal cycle in the Great Plains of North America. Prodigious nocturnal lightning activity in both areas may ultimately lead to some important reconsideration of the behavior of the global electrical circuit in universal time [7,8]. In both areas, air in the prevailing westerly flow is warmed in the afternoon in traversing the respective heated mountain ranges, and then progresses further east over low-level flow laden with moisture. The warm air aloft creates a temperature inversion in the local temperature profiles, otherwise known as a capping inversion, which serves to suppress the afternoon convection while allowing the buildup of temperature and moisture below the cap throughout the daytime that contributes to a large CAPE [29]. When the capping inversion weakens and special triggering processes occur at night, the moist convection develops with exceptional vigor [252,253]. The prominence of thunderstorms with large hail in the nighttime activity [252] is evidence for exceptional updrafts in nocturnal convection. In all of the scenarios described above, the nocturnal thunderstorms described occur only when the usual afternoon thunderstorms are absent at the same location. This situation has special implications for the thermodynamic boundary conditions for the nighttime storms. One implication is that the nighttime air temperatures are substantially larger on the evenings with earlier afternoon storms [249,253]. There are two important reasons for this observation. One is that the customary cold downdraft air was not present. Second, the blanket of boundary layer water vapor that accumulates continuously in the daytime, and which is not reduced by its transformation to rainfall in afternoon thunderstorms, serves to reduce the outgoing longwave radiation that would otherwise serve to cool the Earth’s surface. It is important to note that these conditions contrast starkly with the usual nighttime conditions (without thunderstorms) in the tropics when the relative humidity reaches 100%, the dew point depression vanishes, and dew appears at the surface (e.g., [53,254]). In such circumstances, the contrast in CBH between land and ocean would not serve as a viable thermodynamic contributions to the land– ocean contrast in lightning activity, as discussed in Lucas et al. [38], Williams and Stanfill [11], and Zipser [41,70]. But an appeal to the literature shows that nocturnal thunderstorms do not occur in these circumstances. As one example study in the Great Plains [253], the development of nocturnal convection is characterized by the presence of unsaturated air at the surface and an elevated CBH (3.4–3.5 km MSL) linked with most unstable air that is elevated from the surface. CAPE values are linked with the most unstable values of wet bulb potential temperature and are also large (>3,000 J/kg) in the case of nighttime storms. The evidence that nocturnal thunderstorms are a subclass of “elevated thunderstorms” that feed on a source of most unstable air removed from the surface was established earlier by Colman [255]. Regarding the contribution of aerosol to lightning activity in the context of nocturnal thunderstorms, generally speaking, one would not expect major changes in CCN from daytime to nighttime. Accordingly, the reduced land/ocean lightning at nighttime [242] is likely of thermodynamic origin. Further studies on this issue are warranted.

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Interest in nocturnal thunderstorms is also of interest in the climate context because of the earlier idea that most of the global warming was found in nighttime temperatures [98,243,256] in many regions worldwide. For reasons linked with the sensitivity of lightning to temperature (Section 15.4), greater lightning activity might be expected in nocturnal thunderstorms in those periods, though no analysis specific to nighttime has been undertaken to the author’s knowledge. More recent global analysis of the DTR [244] shows that the most conspicuous decrease occurred in the period 1950–1980, roughly coinciding with the so-called “Big Hiatus” in global warming. If the global temperature trend is flat the decline of DTR with a flat mean temperature guarantees a decline in the nighttime temperatures.

15.14 Meteorological control on lightning type The greater danger to both mankind and infrastructure posed by CG lightning in comparison with intracloud (IC) lightning motivates some discussion on the meteorological conditions favoring the former over the latter. A related issue pertains to how a changing climate may influence the relative numbers of CG and IC lightning. The abundant evidence that CG lightning does not occur until late in the lightning life cycle of thunderstorms [61] is strongly suggestive that clouds with substantial width of the main negative charge region are needed to provide a discharge from cloud to ground, and that multistroke CG flashes are more likely, the greater that width [257]. Narrow storms with extraordinary vertical development tend to be dominated by IC lightning in the author’s personal experience. But the meteorological controls on thunderstorm width are still not well tied down. Conspicuous variations in the IC/CG ratio over the continental United States have been shown by Boccippio et al. [154]. When the map of this ratio is compared with the climatology of summertime CBH, it is apparent that the largest IC/CG ratios are found in a region (eastern Colorado and extending northward into the Dakotas) with elevated CBH, and also a region where the climatology of CG lightning shows a diminished incidence of CG flash density (e.g., [153]). It has also been suggested [258] that CBH may be influencing the latitudinal variation of the IC/CG lightning ratio. Given the evidence that the surface relative humidity is quasi-invariant with climate change [259–264], it may also be true that the relative incidence of CG lightning may not change appreciably in a warmer climate. This general area of research is in need of further attention.

15.15 The global circuits as monitors for destructive lightning and climate change The conductive Earth and the conductive ionosphere sandwich the more electrically insulating atmosphere to form two global electrical circuits. In the classical DC global circuit, a quasi-steady DC voltage of 250 kV known as the

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“ionospheric potential” [98,265,266] is maintained between the Earth and ionosphere by electrical source currents from thunderstorms and electrified shower clouds [130,267–269] that together provide about 1000 amperes globally. For the AC global circuit, otherwise known as Schumann resonances [107,269–275], the insulating space between two spherical conductors serves as a giant waveguide that supports resonant electromagnetic waves maintained continuously by the vertical charge transfers enacted by global lightning activity. The naturally occurring waveguide signals are manifest in two ways: as the “background” Schumann

Classical DC Global Circuit

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Figure 15.11 Illustration of the two global electrical circuits: (a) the classical DC global circuit, and (b) the AC global circuit, otherwise known as Schumann resonances

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resonances consisting of the overlapping waveforms of ordinary lightning produced at rates of order 100 per second globally, and as the transient resonances (or “Q-bursts”) produced by exceptional mesoscale lightning with global rates of only a few per minute, but which singlehandedly ring the global waveguide to amplitude levels that dominate all the other lightning contributions combined. Simple illustrations of the two global circuits are shown in Figure 15.11. The simultaneous behavior of the two global circuits has been considered recently by Williams et al. [276]. The two global circuits provide natural frameworks for global monitoring. In the case of the DC global circuit [98,263,264,277], the ionospheric potential is a measure of all the electrified weather underway at any time, including thunderstorms and electrified shower clouds that provide current to the ionosphere without producing lightning. For the AC circuit, the background resonances can be monitored at multiple stations (of the order of ten stations) to produce chimney-resolved measures of lightning activity in units of coul2km2/sec [275,278–280]. The AC global circuit can also be used to locate and monitor the special population of mesoscale lightning flashes worldwide that are most damaging to mankind and infrastructure, by virtue of their exceptional charge transfers and their long continuing currents [281–288]. Conventional lightning detection networks operating in the LF and VLF range do not have sufficient low-frequency bandwidth to identify particularly hazardous ground flashes with long continuing current and large charge moment change. Early attempts to evaluate the DC and AC global circuits as diagnostics for global temperature were made by Williams [30] for the AC global circuit and by Markson and Price [289] and Markson [97,98] for the DC global circuit. A substantial challenge remaining is to provide for the continuous monitoring of both global circuits over the long time scales that are relevant to climate change (One approach for the continuous monitoring of air-earth current over large areas would be the use of long bare-wire transmission lines that are out of regular service). When this problem is overcome, comprehensive diagnostics of global weather will be available in the electromagnetic field.

15.16 Expectations for the future The foregoing discussion has given emphasis to the response of lightning to climate variables on a number of time scales as well as the growing body of evidence supporting an important role of aerosol in cloud electrification and lightning. On the basis of this body of evidence, one may speculate about changes in lightning and its effect on mankind and infrastructure in the future. As global warming proceeds, uncertainties remain about what quantities are invariant and what quantities are changing. Inferences about changing quantities are often based on their behavior in the current climate, but still uncertainties remain. If total water vapor in the Earth’s atmosphere follows the Clausius–Clapeyron relationship, one expects greater total water and more condensate in a warmer world, and given the need for condensates for lightning, greater lightning is expected. Global climate models [262,263,290] do show increased precipitation in a warmer world,

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though this change is regionally dependent. It must also be recognized that the nature of the precipitation increase is important and that an increase in warm rain alone is unlikely to be accompanied by an increase in lightning. Given the evidence for CAPE as a driver for lightning in the present climate, more lightning is expected in a warmer climate if CAPE increases. Early speculation showed CAPE to be a climate invariant [43]. Romps et al. [291] have predicted increases over the CONUS in a warmer climate on the basis of both increases in CAPE and increases in precipitation. More recent global climate models show larger CAPE in warmer climates [44]. Theoretical calculations in equilibrium atmospheres [292] show CAPE scaling with the Clausius–Clapeyron relationship. Estimations of CAPE changes in the western United States based on the advection of dry desert area over the moist boundary layer with origin in the Gulf of Mexico also lead to predictions that CAPE should scale with Clausius–Clapeyron [49]. For all of these reasons, one expects greater lightning in a warmer world. A frequent assumption in the climate community is that surface relative humidity is a climate invariant [259,264]. This assumption is based in part on the empirical evidence in the present climate for a quasi-fixed relative humidity near 80% over large areas of the tropical oceans, with an associated CBH for moist convection near 500 m. If the ocean surface temperature were to increase globally, the mean dew point temperature would also increase so as to keep the dew point depression (T – Td) and the associated CBH (above local terrain) both constant. But the height of the 0
C isotherm would increase following a presumed lifting along a dry adiabat. Accordingly, the warm cloud depth (distance between the 0
C isotherm and the CBH) would increase. Based on findings in the current climate that the lightning flash rate is decreasing with increased warm cloud depth, in this scenario, one might expect less lightning in the warmer world. However, the CAPE change in this scenario also deserves consideration. The expected increases in both T and Td both contribute to increases in the wet bulb potential temperature of the boundary layer, a result that certainly favors greater CAPE in the current climate. But the ultimate CAPE change depends also on the change in the overall temperature profile in a warmer climate, and this aspect is indeterminate in the context of the foregoing assumptions. One recent study [51] predicts less tropical lightning in a warmer world. The prediction is model-based and with the finding that less ice phase is reaching the upper portion of the troposphere where active charge separation is known to occur. More information about this model, and in the context of expected changes in warm cloud depth, is needed to understand this result. Other recent observations on extremes in tropical rainfall in a warmer climate showing enhancements exceeding the simple predictions based on Clausius–Clapeyron [293] would seem to contradict the model predictions of Finney et al. [51]. Further discussion on this issue and predictions for the behavior of lightning in a warmer climate can be found in Yair [294]. Given the recent evidence for an important role of CCN in increasing lightning activity in the present climate [23], one can surely speculate about changes in lightning in a more polluted world on the basis of this effect. China for example continues to undergo major industrialization and with a major reliance on coal

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[295] and so one can expect an increasing aerosol production from that region alone. From what is known at present, the lightning enhancements are expected over a finite range of CCN concentrations, from typical oceanic concentrations of a few per cc to up to about 1,000 per cc [21,23]. Above that level (typical of continental conditions described in Section 15.5), the lightning activity is expected to flatten and then decrease, for reasons that are evident in model calculations [162]. An assessment of where the world stands relative to this important threshold will be much aided by a new satellite method for the estimation of CCN concentration at CBH [14].

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Hales, J.E., Jr., On the relationship of convective cooling to nocturnal thunderstorms at Phoenix, Mon. Wea. Rev., 105, 1609–1613, 1977. Zipser, E.J., C. Liu, D.J. Cecil, S.W. Nesbitt and D.P. Yorty, Where are the most intense thunderstorms on Earth?, Bull. Am. Meteorol. Soc., 87(8), 1057–1071, 2006, doi:10.1175/BAMS-87-8-1057. Wallace, J.M., Diurnal variations in precipitation and thunderstorm frequency over the conterminous United States, Mon. Wea. Rev., 103, 406– 419, 1975. Reif, D.W. and H.N. Bluestein, A 20-year climatology of nocturnal convection initiation over the central and southern Great Plains during the warm season, Mon. Wea. Rev., 145, 1615–1639, 2017. Wilson, J.W., S.B. Trier, D.W. Reif, R.D. Roberts and T.M. Weckwerth, Nocturnal elevated convection initiation of the PECAN 4 July hailstorm, Mon. Wea. Rev., 146, 243–262, 2018. Dessens, J., Severe convective weather in the context of a nighttime global warming, Geophys. Res. Lett., 22, 1241–1244, 1995. Colman, B.R., Thunderstorms above frontal surfaces in environments with positive CAPE. Part I: a climatology, Mon. Wea. Rev., 118, 1103, 1990. Vose, R.S., D.R. Easterling and B. Gleason, Maximum and minimum temperature trends for the globe: An update through 2004, Geophys. Res. Lett., 32, L23822, 2005, doi:10.1029/2005GL024379. Williams, E.R., E.V. Mattos and L.T. Machado, Stroke multiplicity and horizontal scale of negative charge regions in thunderclouds, Geophys. Res. Lett., 43, 5460–5466, 2016, doi:10.1002/2016GL068924. Mushtak, V.C., E.R. Williams and D.J. Boccippio, Latitudinal variations of cloud base height and lightning parameters in the tropics, Atmos. Res., 76, 222–230, 2005. Dai, A., Recent climatology, variability, and trends in global surface humidity, J. Clim., 19, 3589–3606, 2006. Held, I.M. and B.J. Soden, Water vapor feedback and global warming, Annual Rev. Energy Environ., 25, 441–475, 2000. O’Gorman, P.A. and C.J. Muller, How closely do changes in surface and column water vapor follow Clausius-Clapeyron scaling in climate change simulations? Env. Res. Lett., 5, 025207, 2010, doi:10.1088/1748-9326/5/2/ 025207. Soden, B.J. and I.M. Held, An assessment of climate feedbacks in coupled ocean–atmosphere models, J. Clim., 19, 3354–3360, 2006a. Soden, B.J. and I.M. Held, Robust responses of the hydrological cycle to global warming. J. Clim., 19, 5686–5699, 2006b. Byrne, M.P. and P.A. O’Gorman, Trends in continental temperature and humidity directly linked to ocean warming, Proc. Nat’l Acad. Sci., 115, 4863–4868, https://doi.org/10.1073/pnas.1722312115, 2018. Markson, R., L.H. Ruhnke and E.R. Williams, Global scale comparison of simultaneous ionospheric potential measurements, Atmos. Res., 51, 315–321, 1999.

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Mu¨hleisen, R., The global circuit and its parameters, 467–476, in H. Dolezalek and R. Reiter (eds.), Electrical Processes in Atmospheres, Steinkopff, Darmstadt, 1977. Bering, E.A., A.A. Few and J.R. Benbrook, The global electric circuit, Physics Today, 51, 24–30, 1998. Chalmers, J.A., Atmospheric Electricity, 2nd ed., Pergamon Press, London, 1967. Williams, E.R., The global electrical circuit: a review, Atmos. Res., 91, 140–152, 2009. Heckman, S., E. Williams and R. Boldi, Total global lightning inferred from Schumann resonance measurements, J. Geophys. Res., 103, 31775– 31779, 1998. Madden, T.R. and W.B. Thompson, Low-frequency electromagnetic oscillations of the Earth-ionosphere cavity, Rev. Geophys., 3, 211, 1965. Nickolaenko, A.P. and M. Hayakawa, Resonances in the Earth-ionosphere Cavity, Kluwer Academic Publishers, London, 2002. Polk C., Schumann resonances, in H. Volland (ed.), CRC Handbook of Atmospherics, Vol. I, CRC Press, Boca Raton, FL., pp. 55–82, 1982. Sentman, D.D., Schumann resonances, in H. Volland (ed.), Handbook of Atmospheric Electrodynamics, Vol. I, CRC Press, 408 pp., 1995 (Chapter 11). Williams, E.R. and E.A. Mareev, Recent progress on the global electrical circuit, Atmos. Res., 135–136, 208–227, 2014. Williams, E., R. Boldi, R. Markson and M. Peterson, Comparative behavior of the DC and AC global circuits, in XVI International Conference on Atmospheric Electricity, Nara city, Nara, Japan, 17–22 June, 2018. Mu¨hleisen, R.P., New determination of the air-earth current over the ocean and measurement of ionospheric potential, PAGEOPH, 84, 112–115, 1971. Dyrda, M.A., M. Kulak, M. Mlynarczyk, et al., Application of the Schumann resonance decomposition in characterizing the main African thunderstorm center, J. Geophys. Res., Atmospheres, 119, 13338–13349, 2014. Williams, E., V. Mushtak, A. Guha, et al., Inversion of multi-station Schumann resonance background records for global lightning activity in absolute units, Fall Meeting, American Geophysical Union, San Francisco, CA, December, 2014. Pra´cser, E., T. Bozo´ki, G. Sa´tori, E. Williams, A. Guha and H. Yu, Reconstruction of global lightning activity based on Schumann Resonance measurements: Model description and synthetic tests, Radio Science, 54, 254–267, 2019. Guha, A., E. Williams, R. Boldi, et al., Aliasing of the Schumann resonance background signal by sprite-associated Q-bursts, J. Atmos. Sol. Terr. Phys., 165–166, 25–37, 2017. Hobara, Y., E. Williams, M. Hayakawa, R. Boldi and E. Downes, Location and electrical properties of sprite-producing lightning from a single ELF

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Index

absolute peak amplitude 181 active region 8 adaptive mesh refinement (AMR) 111 advection mixing 55 advective forms 41 aerosol effects 570 aerosol optical depth (AOD) 593 aggregation 27 Agrawal, Price, and Gurbaxani model 295–6 Alaska 595 Alaska Lightning Detection Network (ALDN) 594 Alfve´n wave branch 489 ambient temperature 547 Ampe`re’s law 381, 383 analytical expression 103 analytical solutions 305 annual variation 585–6 antenna-mode 293 antenna-theory (AT) models 157, 174, 292 Arctic 594–5 Atmosphere-Space Interactions Monitor (ASIM) 416, 462 attachment coefficient 2 avalanche-to-streamer transition 133, 438 average length of grid mesh 63 axial symmetry 94 axisymmetric model 25 azimuthal magnetic field 256–9, 333

backflashover 325, 346 back-of-the-envelope calculation 153 Barbosa kernel 254 baroclinicity 582 Barthe–Molinie´ scheme 63 beads 442 Bergeron process 573 Berger’s data 177–8 Bessel function 245, 247, 251 bi-directional discharge 6 bi-directional probes 342 bi-Gaussian function 384 Big Hiatus 589, 604 bin microphysics 27, 29 BLT (Baum, Liu, Tesche) equations 303 blue jets (BJ) 426, 448–58, 528 blue starters (BSs) 427, 448 basic properties and morphology of 448–51 bolt-from-the-blue (BFB) 454 Boltzmann’s constant 56, 80, 83 Boltzmann’s equation 82, 86, 90 Boolean operations 88 branching algorithm 63 branch points 209–18 breakdown streamers 9 breakeven electron 59 buildings 338–40 bulk microphysics 27–8 Burst and Transient Source Experiment (BATSE) 461

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Lightning electromagnetics: Volume 2

capping inversion 603 carbon dioxide 1 carrot sprite 429 Cartesian coordinates 25, 99, 482 cascade models 160 channel-base current 157 channel-base lightning current statistics 157 charge continuity 40–2 charge density 431 charge mixing ratio 40 charge moment change (CMC) 430 charge simulation method 15 charging processes in clouds applications 64 charge structure and lightning type 66–70 ion and inductive mechanisms 64–5 non-inductive graupel–ice sensitivity 65 electrification modelling 33–7 lightning parameterizations 58 pseudo-fractal lightning 62–3 stochastic lightning model 60–2 model descriptors 24 basic terminology 24–6 cloud models 31–2 electrification mechanisms 30–1 terms related to microphysics 26–30 parameterization of electrical processes 37 charge continuity 40–2 inductive charging mechanism 51–4 non-inductive graupel–ice collision mechanism 42–51 small ion processes 54–7 circuit models 160

circuit theory-based simulation 347 circuit theory models 158, 164 classical state-space equation 247 classical telegrapher’s equations 295 Clausius–Clapeyron relationship 573–5, 590 climate change 570 global circuits as monitors for 604–6 climate variable 593–4 cloud base height (CBH) 573 cloud condensation nuclei (CCN) 590 cloud droplets 27, 52 cloud flashes 548, 556–8 cloud models 23, 31–2 cloud-to-ground (CG) lightning 36, 375, 429, 588 cloud-to-ocean lightning 599–600 CN Tower Lightning Studies Group (CNTLSG) 182 coalescence 27 cold cloud models 27 cold electrical discharges 527 collisional ice–ice mechanism 30 collision integral 82 collision kernel 49 column of charge 133 Commission for Climatology (CCl) 594 Communications/Navigation outage Forecasting System (C/NOFS) 496 compact intracloud discharges (CIDs) 396 Compton Gamma-Ray Observatory 462 computational methods 157 Comsol Multiphysics 86–7 connecting leader 134 orientation of 150

Index conservation of charge 40 constant non-zero reflection coefficients 158 continuing current (CC) 429, 546–8, 554 continuity equations 26, 40 anelastic form 31 convection–diffusion equations 92 convective available potential energy (CAPE) 573, 575–7 convective charging hypothesis 36 convective mechanism 64 conventional breakdown threshold 430–1, 438 Cooray formula 172–3, 243, 260–2, 285 Cooray–Rubinstein approximation 172 Cooray–Rubinstein (CR) formula 172–3, 243–56, 264, 285 corona current 108 corona discharges 9–10, 86, 112, 527, 529 corona electrode 100 corona model 87–99 corona region 99 cosmic rays 1, 55 Coulomb forces 10 Courant criterion 119 Courant–Friedrichs-Lewy condition 117 COVID-19 pandemic 593 critical impulse flashover voltage (CFO) 325 critical radius and critical streamer length concepts 137–8 Cumulonimbus clouds 575 curl operator 328 cutoff frequencies 487–8 cutoff of tweek sferics 496

629

cylindrical coordinates 167 cylindrical plasma channel 19 dancing sprites 429 dark discharge 81 dart leader 548 data processing 498–502 decadal time scale 588 dedicated algorithms 171–2 delta function 167 diffusivity 56 digital simulations 326 dip angle 392 Dirac function 245, 246, 260 Dirichlet conditions 62 Dirichlet type 113 discharge plasma conductivity 120 discharge processes 546 corona discharges 9–10 electron avalanche 2–4 leader discharges 11–13 leader inception based on thermalization of discharge channel 18–19 low-pressure electrical discharges 11 mathematical modelling of positive leader discharges 13–18 streamer discharges 4–8 thermalization 10–11 dispersion relation 480–5 distributed-circuit models 291 distribution function 83 diurnal variation 583 domain transformation technique 114 Doppler radars 25, 32 Doppler velocity 581 Doppler wind fields 32 down-coming negative stepped leader 150

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Lightning electromagnetics: Volume 2

downward propagating waves 159 D-region ionosphere 460 D-region ionospheric profiles 389 of different slopes and ionospheric reflection heights 392 effect of Earth’s magnetic field 392–6 typical daytime/nighttime ionospheric profiles 389–92 drift-diffusion equations, solving 111 drift-diffusion model 85 drift velocity 4 of secondary electrons 111 drop–droplet separation probability 64 ducts 478 dynamic mesh 111 Earth-ionosphere waveguide (EIWG) 375, 384, 444, 476 narrow bipolar events 396–9 VLF/LF signal of lightning EM fields propagation 384 different D-region ionospheric profiles 388–96 Earth’s curvature 384–6 ground conductivity 386–8 Earth’s atmosphere 1 Earth’s curvature 375, 384–6 Earth’s magnetic field 378, 381 Earth’s Schumann resonances 585 Eckersley’s dispersion law 491 effective ionization coefficient 2 Einstein coefficient 408 Einstein relation 56 electrical discharge 1 electrical gas discharges 77 electrical processes parameterization of 37 calculating the electric field 37–40

charge continuity 40–2 inductive charging mechanism 51–4 non-inductive graupel–ice collision mechanism 42–51 small ion processes 54–7 electric field 2–3 integral equation 164 radial component of 216–17 vertical component of 213–16 electric potential 62 electrification mechanisms 30–1 electrification modelling 33–7 electrification processes inductive 30 non-inductive 30 electrodynamic equations 436 electrogeometrical model (EGM) 136, 145–8 electromagnetic field computation 165 expressions for perfectly conducting ground 166 comparison between different engineering models 168–71 effect of the tower 171 turn-on term 167–8 for finitely conducting ground 171 dedicated algorithms 171–2 numerical methods 173–4 simplified approaches 172–3 electromagnetic field interaction 291 with buried cables 311 field-to-buried cables coupling equations 311–14 frequency-domain solutions 314–15 lightning-induced disturbances 315–16 time-domain solutions 315

Index Agrawal, Price, and Gurbaxani model 295–6 analytical solutions 305 application to lightning-induced voltages 306–11 coupling to complex networks 302 frequency-domain solutions 302–3 inclusion of losses 298–9 multiconductor lines 299–301 Rachidi model 296–7 Rusck model 297–8 single-wire line above a perfectly conducting ground 294 Taylor, Satterwhite, and Harrison model 294–5 time-domain solutions 303–5 transmission line theory 292–3 electromagnetic (EM) fields 376 electromagnetic field theory 342 electromagnetic models 164, 174, 291 electromagnetic pulses (EMPs) 375, 417, 458 electromagnetic transient program (EMTP) 164, 308, 326, 347 electron avalanches 2–4, 80–2, 109, 133 electron cyclotron resonance 487 electron mobility 378 electron multiplication 8 electron-neutral collision frequency 378 electron streamer 80–2 electrosphere 39 electrostatic energy 123 electrostatics equation 88 elevated strike object 160–1 elliptic equations 31, 39 Elves 428, 458–60, 528

631

Elves doublets 416, 460 energetic in-cloud pulses (EIPs) 416 engineering models 157–8, 291 current distribution along the channel as predicted by 161–3 on distributed source representation 158–60 of elevated strike object 160–1 lightning current data and associated electromagnetic fields 174 Berger’s data 177–8 data from short towers 176–7 data from tall towers 181–90 data obtained using short towers 178–81 experimental data 175–6 on lumped series voltage source 160 reflection coefficients at top and bottom of strike object 163–4 engineering return-stroke models 157–8 ENSO 586–8 equations of electron motion 480–5 equivalent charges 113 Essential Climate Variables (ECVs) 594 Eulerian models 32 European Cooperation for Lightning Detection (EUCLID) 187 experimental data 151, 175–6 Extended Rusck Model (ERM) 348, 364 extrapolation method 164 FD-FCT technique 112 field-to-buried cables coupling equations 311–14 field-to-transmission line coupling equations 299, 303

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Lightning electromagnetics: Volume 2

field-to-transmission line coupling models 166 final jump condition 134 finite-differences (FD) methods 109, 111, 113 finite-difference time-domain (FDTD) technique 173–4, 285, 292, 304, 375 finite element method (FEM) 111, 339, 375 fluid model 77 flux-corrected transport (FCT) 111 flux-correction 111 flux forms 41 flux-limiting 111 FORMOSAT-2 satellite 444, 450, 459 FORTRAN routines 112 Fourier transform 245 fractal formula 63 Frank-Condon factor 408 Franklin rod 15 free electrons 80, 435 free space waves 488–9 frequency-dependent soil parameters 267–8, 273–7 frequency-domain solutions 174, 302–3, 314–15 Fukui tower in Japan 187 fulchronograph 175 full-scale power lines 326 full simulation model 24, 31 full-wave ray theory 384 fully developed jet (FDJ) 451 Gaisberg Tower in Austria 181 Gamma function 247 Gamma-ray Burst Monitor (GBM) 462 gantry model 346 Gardner’s magnetic link sensor 175

gas charged species in 78–9 discharge plasma 82–6 dynamic models 291 losses of charged species in 79–80 gas discharge problems, solving 86 computer implementation of corona model 87–99 positive corona between coaxial cylinders 99–104 positive corona in rod-plane electrode system 104–9 simulations of corona in air 86–7 Gaussian spatial distribution 453 Gauss integral 252 Gauss method 251 Gauss numerical method 252 Gauss’s law 37 general oblique propagation 485–9 geomagnetic field 392 geometry function 115 Geostationary Lightning Mappers (GLMs) 601 gigantic jets (GJs) 427, 448 development of 451–3 models of 453–8 global aerosol-lightning relationship 593 Global Climate Observing System (GCOS) 594 global digital elevation model version 2 (GDEM V2) 400 global warming on winter thunderstorms 596 glow corona 9 glows 442 Gnomes 428, 460 Godunov’s method 111 gradient operators 82

Index Green’s functions 202, 302, 314 for electromagnetic field 228–32 lightning return stroke over-ground electromagnetic field 208–9 for overground electromagnetic field 204–8 Sommerfeld’s integrals 209 theory 203–4 GREMPY code (GRanada ElectroMagnetic PYthon simulator) 417 Grenet–Vonnegut mechanism 64 groudwave 386 ground admittance 299, 301 ground-based European Lightning Detection Network (EUCLID) 493 ground-based measurements 425 ground conductivity 386–8 ground-Cooray formula 259–62 grounded structure 133 ground lightning flashes 547 continuing currents 554 leaders 548–52 M components and K processes 553–4 model of 547–8 return strokes 552–3 in typical negative ground flash 555–6 ground losses on induced overvoltages 308–11 on overvoltages due to a direct strike 306–8 ground resistivity 333 gyrofrequency vector 392 Heaviside function 250, 260 Heaviside unit-step function 159 Heidler’s functions 255

633

Helmholtz equation 206, 228 Hermitian glow 9 Hermitian sheath 9 heterogeneous nucleation 590 Hiatus in global warming 590 homogeneous wave equation 259 hot electrical discharges 527 hybrid electromagnetic/circuit theory 158 hybrid electromagnetic model (HEM) 158, 164 hydrometeors 30 ice multiplication 27 ideal current 159 ideal fields 232 Imager of Sprites and Upper Atmospheric Lightning (ISUAL) 430 image theory 165 inclusion of losses 298–9 indirect strokes 347 analysis of complex situations 357–63 theoretical models 347–57 inductive charging mechanism 51–4 initial continuous current (ICC) 175–6 integral tail 214 intermediate distances 279–83 internal gravity wave (IGW) 459 International Center for Lightning Research and Testing (ICLRT) 315, 326 International Reference Ionosphere (IRI) model 378, 389, 408 International Space Station (ISS) 416, 428 interpolation function 92 intra-cloud (IC) lightning 375, 416, 448

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Lightning electromagnetics: Volume 2

intra-cloud (IC) process 436 inverse exponential (IE) distribution 28 inverse Fourier transform algorithm 213, 304 inverse Laplace transform 245, 246 ion and inductive mechanisms 64–5 ion cyclotron whistlers 492 ionization coefficient 125 ionization potential 78 ionization processes 2, 4 ionosphere/magnetosphere 476 lightning effects on 516–17 lightning-induced whistlers in 477 characterization of 479–80 diagnosis of magnetospheric electron density profile 491–2 dispersion relation 480–5 equations of electron motion 480–5 general description of 477–9 general oblique propagation 485–9 longitudinal and perpendicular propagation 485–9 Maxwell’s equations 480–5 satellite observations of nonducted whistlers in 492 satellite observations of shortfractional hop (0+) whistlers 492–6 whistler propagation and dispersion 489–91 ionospheric Alfve´n resonator (IAR) 445, 476, 496 brief history and general introduction of 496–8 excitation, by nearly thunderstorms 508–16 generation mechanisms of 507–8

ground-based observations of IARs at middle latitude 498 dependence on geomagnetic activity 502–3 morphological characteristics of 503–7 observations and data processing 498–502 seasonal and diurnal variations of SRS parameters 502 ionospheric potential 605 Jianghuai Area Sferic Array (JASA) 396 Joule heating 77 J-process 548 kinematic model 24, 32 kinetic approach 82 K processes 553–4 Lagrangian models 32 Lagrangian reference 40 Lalande’s stabilization field equation 138–9 Langevin equation 376 Laplace transform 245 Lax–Wendroff algorithm 304 leader inception model of Becerra and Cooray (SLIM) 139 leader inception models 137 critical radius and critical streamer length concepts 137–8 Lalande’s stabilization field equation 138–9 leader inception model of Becerra and Cooray 139 Rizk’s generalized leader inception equation 138 leaders 12, 548 attachment models 139–42

Index charge distribution 153 corona sheath 548–50 discharges 11–13 NOx production in hot core of 550–2 potential 153 progression models 139 propagation models 150 leader tip potential function 144 leading jet (LJ) 451 Les Renardie´res Group 153 lifted condensation level (LCL) 578 lightning 569 activity at high latitude 594 Alaska 595 Arctic 594–5 aerosol influence on moist convection 590 basic concepts 590–1 global aerosol-lightning relationship 593 lightning response to COVID-19 pandemic 593 observational support 591–3 as climate variable 593–4 cloud-to-ocean lightning 599–600 global circuits as monitors for destructive lightning 604–6 global lightning response to temperature on different time scales 582 annual variation 585–6 decadal time scale 588 diurnal variation 583 ENSO 586–8 Hiatus in global warming 590 multi-decadal time scale 588–90 semiannual variation 583–5 warming hole 590 meteorological control on lightning type 604

635

nocturnal thunderstorms 602–4 storms at the mesoscale 597 superbolts and megaflashes 600–2 thermodynamic control on lightning activity 573 balance level considerations in deep convection 581–2 baroclinicity 582 cloud base height and its influence on cloud microphysics 577–80 convective available potential energy and temperature dependence 575–7 dew point temperature 573 temperature 573 water vapor and Clausius– Clapeyron relationship 573–5 thunderstorm electrification and 572–3 tropical cyclones 597–9 winter-type thunderstorms 595 lightning activity 460 Gnomes and Pixies 460 terrestrial gamma-ray flashes 461–3 transient atmospheric events 460–1 lightning bipole pattern 70 lightning channel model 331–3 lightning current moment 444, 539 lightning electromagnetic field homogeneous soil 243 azimuthal magnetic field 256–9 Cooray–Rubinstein (CR) formula 244–56 ground-Cooray formula 259–62 vertical electric field 256–9 horizontally stratified ground 262 far distances 268–77 near and intermediate distances 265–8

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Lightning electromagnetics: Volume 2

for two-layer horizontally stratified ground 263–5 vertically stratified ground 277 far distances 283–4 intermediate distances 279–83 near distances 278–9 for two-layer vertically stratified ground 277–8 lightning electromagnetic field calculation 201 lossy and horizontally stratified ground 228 Green’s functions for electromagnetic field 228–32 lightning electromagnetic field 232–3 reflection coefficient R 233–6 lossy ground with constant electrical parameters 202 branch points 209–18 over-ground electromagnetic field 203–9 underground electromagnetic field 218–21 lossy ground with frequencydependent electrical parameters 221 numerical simulation of overground electromagnetic field 226–7 soil conductivity and permittivity on frequency 222–6 underground lightning electromagnetic field 226–7 lightning electromagnetic pulse (LEMP) 292, 339 lightning flash comparison of EGM against SLIM 145–8 connection between leader potential and return stroke current 150–2

downward lightning flash 133 inclination of leader channel 152 leader inception models 137 critical radius and critical streamer length concepts 137–8 Lalande’s stabilization field equation 138–9 leader inception model of Becerra and Cooray 139 Rizk’s generalized leader inception equation 138 leader progression and attachment models 139–42 main assumptions of SLIM 152–3 orientation of connecting leader 150 orientation of stepped leader 148–50 potential of stepped leader channel and striking distance 142 Armstrong and Whitehead 142 leader potential extracted from charge neutralized by return stroke 142–4 striking distance based on leader tip potential 144–5 striking distance 135–7 upward lightning flash 133 lightning-induced electron precipitation (LEP) 417, 446 lightning intensity 599 lightning interaction with ionosphere D-region ionosphere 376 2D symmetric polar model 381–3 3D spherical model 378–81 lightning EM field propagation over a mountainous terrain 400–3 optical emissions of lightninginduced transient luminous events 403–16 parameterization of lower D-region ionosphere 376–8

Index earth-ionosphere waveguide 375 narrow bipolar events 396–9 VLF/LF signal of lightning EM fields propagation 384 lightning parameterizations 58 pseudo-fractal lightning 62–3 stochastic lightning model 60–2 lightning return stroke channel 291 lightning return stroke models 157, 291 lightning return stroke parameters 272 lightning striking model 134, 153 linear system solver 96 LIOV-EMTP code 339, 349, 364 logarithmic function 299 longitudinal propagation 485–9 left-handed polarized wave 486 right-hand polarized wave 485–6 lossless transmission line 157 lossy soil 201 lower D-region ionosphere 378, 417 low-frequency approximation 490 low precipitation (LP) 582 low-pressure electrical discharges 11 low-voltage (LV) terminals 335 lumped series voltage source 160 MacCormack’s method 111 magnetic field 217–18 magnetic link 174 magnetospherically reflected (MR) whistlers 492 Marshall–Palmer distribution 28 mathematical modelling of positive leader discharges 13–18 Maxwell approaches 243 Maxwell–Bolzmann statistics 78 Maxwell’s curl equations 268, 376 Maxwell’s equations 157, 171, 291–2, 327, 436, 480–5, 511

637

M components 553–4 medium voltage (MV) networks 335 megaflashes 600–2 Meso-NH model 36 mesoscale convective systems (MCSs) 70, 597, 600 mesosphere 436 meteorological control on lightning type 604 Meteosat Third Generation (MTG) 601 method of moments (MoM) 157, 174, 292 microphysics 24–5 mid gap streamer 6 mixed-phase models 27 model-dependent attenuation function 159 modeling lightning strikes to tall structures 157 electromagnetic field computation 165 electromagnetic models 164 engineering models 158–64 hybrid electromagnetic model 164 Model Navigator window 87 modified transmission-line model with exponential current decay (MTLE) 163, 255, 384, 401 Modular Multi-spectral Imaging Array (MMIA) 462 Modular X- and Gamma ray Sensor (MXGS) 462 Mount San Salvatore in Lugano 176 MTLL models 163 multi-avalanche mechanism 81 multiconductor lines 299–301 multi-decadal time scale 588–90 multigrid method (MG) 116

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Lightning electromagnetics: Volume 2

narrow bipolar events (NBEs) 376, 386, 396, 416–17 narrow bipolar pulses (NBPs) 396 Neudorf E-field measurement station 401 Neumann type 113 nitrogen oxides (NOx) 37, 527 in discharges containing longduration currents 546–7 in electron avalanches 529–30 in ground lightning flashes 547 continuing currents 554 leaders 548–52 M components and K processes 553–4 model of 547–8 return strokes 552–3 in typical negative ground flash 555–6 production by cloud flashes 556–8 production by laboratory sparks 538 radius of spark channels 538–40 in spark channels 541–2 in sparks as function of energy 544–6 in sparks with different current wave-shapes 542–4 volume of air heated in spark channel and internal energy 540–1 production by lightning flashes 558–60 production in streamer discharges 530–2 using corona discharges 529 nocturnal thunderstorms 602–4 non-inductive charging processes 30 non-inductive graupel–ice collision mechanism 42 general formulation 49–51 parameterized laboratory results 42–9

non-inductive graupel-ice sensitivity 65 nonlinear D-region ionosphere 376 Norton’s approximation 202 nose whistler 478 nuclear electromagnetic pulse (NEMP) 302 nucleation 27 numerical methods 173 finite difference time domain technique 173–4 method of moments 174 numerical modelling 23, 70 numerical Poisson solvers 39 numerical simulations of non-thermal electrical discharges electro-physical processes in gaseous medium under electric fields 77 charged species in gas 78–9 dynamics of densities of charge carriers in discharge plasma 80 electron avalanche and streamer 80–2 losses of charged species in gas 79–80 hydrodynamic description of gas discharge plasma 82–6 solving gas discharge problems 86 computer implementation of corona model 87–99 corona in air 86–7 positive corona between coaxial cylinders 99–104 positive corona in rod-plane electrode system 104–9 streamer discharges in air 109 negative streamer in weak homogeneous background fields 123–6 positive streamer in weak homogeneous background field 117–23

Index oblique propagation 488 Alfve´n wave branch 489 free space waves 488–9 whistler-mode branch 489 observational support 591–3 Ohm equation 376 one-and-one-half dimensional model 35 one-dimensional continuity equation 111 one-dimensional FDTD method 173 one-dimensional model 25–6 one-dimensional (1-D) radial system 18 1.5D model 113 onset streamers 9 optical-electrical converter 348 optical sensors 570 Ostankino tower 181–2 over-ground electromagnetic field 203 overhead lines 333–4 parameterizations 23 partial differential equations (PDE) 84 particles distribution function 82 particle tracing 32 Paulino kernel 254 Peek’s formula 103 Peissenberg tower in Germany 162, 169 phase velocity 485 photoionization 19 piecewise cubic interpolation method 92 pilot system 13–14 Pixies 428, 460 plasma ball 439 plasmasphere 480 plasma spot 118 plasma waves 479

639

Pockels on Mount Cimone in Italy 174 Poisson’s equation 39, 62, 85, 92, 112–13, 117 polarization spectrum 499 positive streamer 6, 530 potential–distance diagram 152 potential of stepped leader channel and striking distance 142 Armstrong and Whitehead 142 leader potential extracted from charge neutralized by return stroke 142–4 striking distance based on leader tip potential 144–5 preliminary breakdown 547 primary avalanche 81 propagation model 145 pseudo-fractal lightning 62–3 pulse voltages 331 quadratic finite elements 87 quasi-electrostatic (QE) field 431 quasi-free-space mode 488 quasi-image method 173 quasi-neutral layer 118 quasi transverse electromagnetic (quasi-TEM) 293 Rachidi model 296–7 radial electric field 201 radiated electromagnetic fields 157 radiation transfer theory 110 radio active gases 1 radius of spark channels 538–40 rainfall 570 random number 61 rational approximation (RA) 247 red sprite/sprite 425 reduced electric field 403

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Lightning electromagnetics: Volume 2

reference current 160 Reference Line 350 reflected sky waves 386 reflection coefficient R 233–6 at top and bottom of strike object 163–4 refractive index 485 relative permittivity 201 relativistic runaway electron avalanche (RREA) 462 remote electromagnetic fields 157 return stroke channel 2, 331, 547 return-stroke models 157, 291, 552–3 return-stroke speed 157 Reuven Ramaty High-Energy Solar Spectroscopy Imager (RHESSI) 462 Riemann sheet 265 riming 27 Rizk’s generalized leader inception equation 138 rocket-triggered lightning technique 326 Romberg method 213 runaway electron 59 Rusck model 297–8, 348 S91 simulations 66 Sa¨ntis telecommunication tower in Northeastern Switzerland 188 Sato’s equation 85 scale modeling 327–9 electromagnetic environment 329 buildings 338–40 ground resistivity 333 lightning channel 331–3 overhead lines 333–4 surge arresters 335–8 transformers 334–5 transmission line towers 340–3

lightning surges in power lines 343 with direct strokes 343–7 with indirect strokes 347–63 scale model technique 326 scaling analysis 579 scattered current 297 Schumann resonance region 496, 605 secondary electrons 8 self-consistency 158 self-consistent leader inception 145 self-consistent model 85, 134 semi-analytical models 439 semiannual variation 583–5 short-circuit current 160 single-moment schemes 28 single-phase distribution line 334 slab-symmetric models 25–6 SLIM 145–8, 152–3 small ion processes 54–7 Snell’s law 259, 490 soil conductivity and permittivity on frequency 222–6 Sommerfeld’s integrals 171, 201, 209–10, 244, 255, 263, 285 SP98 simulation 66 space charge density 114 space leader 13–14 spark channels 541–2 spark discharge channel model 544 spatial distribution 157 spatial–temporal distribution 158 spatial variations 26 spectral resonance structures (SRSs) 477, 499 splintering 27 split explicit method 31 sprite discharge 425 sprite halos 428 sprites 429, 528, 533

Index basic properties and morphology of 429–30 development 433–5 ELF/VLF electromagnetic fields produced by 443–6 inner structure and color of 441–3 on ionosphere 446–8 mechanism of sprite nucleation 430–3 models 435–41 stability field 117 stepped leader 133, 547 approaches 133 orientation of 148–50 step-wise ionization 85 stochastic lightning model 60–2 streamer discharges 4–8 streamer inception criterion 139 streamer stem 11, 14 streamer to leader transition 134 striking distance 135–7 strong very high frequency (VHF) 396 structured rectangular grids 113 subdomain settings 92 super-adiabatic loading 580 superbolts 600–2 surge arrester model 335–8 Task Team for Lightning Observations for Climate Applications (TTLOCA) 594 Taylor, Satterwhite, and Harrison model 294–5 TCS models 163 Telegrapher’s equations 295 temporal distribution 157 temporal variations 26 terrestrial gamma-ray flashes (TGFs) 416, 461–3

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Test Line 351 theoretical models 347–57 thermal equilibrium 10 thermalization 10–11, 528 thermodynamic effects 583 thermodynamic equation 32 thermodynamic fields 24 thermodynamic model 134 thermodynamic processes 544 thermo-hydrodynamic model 18, 138 three-dimensional model 25 three-dimensional (3D) spherical FDTD model 378–81 3-D FDTD method 363 3D Lightning Mapping Array (LMA) systems 601 thunderstorm charge distribution 25 thunderstorm-created ionization processes 528 thunderstorms 537 tiger Elve 459–60 time-domain solutions 303–5, 315 Toronto CN Tower in Canada 160 Townsend approximation 437 Townsend’s first ionization coefficient 2 Townsend’s ionization coefficient 78 Townsend’s or dark discharge 81 trade winds 596 trailing jet (TJ) 451 transformers 334–5 transient atmospheric events 460–1 transient ground resistance matrix 304 transient luminous events (TLEs) 403, 425 translational energy 10 transmission cone 490 transmission line (TL) model 163, 293, 331, 348

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Lightning electromagnetics: Volume 2

transmission line (TL) theory 292–3, 316 transmission line towers 340–3 transverse propagation 486 extraordinary wave 486–7 ordinary wave 486 Trichel pulses 10 Trolls 428, 435 tropical cyclones 597–9 turbulent mixing 55 2D cylindrical coordinates 114 two-dimensional model 25 two-dimensional (2D) symmetric polar FDTD model 378–81 undisturbed current 159 upward lightning flashes 133–4 upward propagating waves 159 Van Leer’s approach 111 vector fitting (VF) technique 247, 250 velocity space 82 vertical electric field 201, 256–9 vibrational energy 10 Wait’s formula 278–9, 282 warm cloud models 27

warming hole 590 wave equation of electromagnetic waves 484 waveguide mode theory 375 Weibull distributions 226 Welch’s method 498 wet bulb adiabat 575 whistler-mode branch 478, 486, 489 wildfire technique 59 Wilson’s selective ion capture mechanism 33 wind fields 24 wind turbine generator systems 326 winter-type thunderstorms 595 World Magnetic Model (WMM) 392 World Meteorological Organization (WMO) 594, 601 world wide lightning location network (WWLLN) 495, 592, 594 Yee discretization scheme 268 zero charge/symmetry 94 zero-crossing 169–70 zero-dimensional cloud models 25 zero order diffusion 111 zeroth moment 28