zotero-db/storage/7ZMLIE7U/.zotero-ft-cache

arXiv:1910.14022v4 [astro-ph.SR] 27 Mar 2024

PRINCIPLES OF HELIOPHYSICS: a textbook on the universal processes
behind planetary habitability
by Karel Schrijver, Fran Bagenal, Tim Bastian, Ju¨rg Beer, Mario Bisi, Tom Bogdan, Steve Bougher, David Boteler, Dave Brain, Guy Brasseur,
Don Brownlee, Paul Charbonneau, Ofer Cohen, Uli Christensen, Tom Crowley, Debrah Fischer, Terry Forbes, Tim Fuller-Rowell, Marina Galand, Joe Giacalone, George Gloeckler, Jack Gosling, Janet Green, Nick Gross, Steve Guetersloh, Viggo Hansteen, Lee Hartmann, Mihaly Horanyi, Hugh Hudson, Norbert Jakowski, Randy Jokipii,
Margaret Kivelson, Dietmar Krauss-Varban, Norbert Krupp, Judith Lean, Jeff Linsky, Dana Longcope, Daniel Marsh, Mark Miesch, Mark Moldwin, Luke Moore, Sten Odenwald, Merav Opher, Rachel Osten,
Matthias Rempel, Hauke Schmidt, George Siscoe, Dave Siskind, Chuck Smith, Stan Solomon, Tom Stallard, Sabine Stanley, Jan Sojka,
Kent Tobiska, Frank Toffoletto, Alan Tribble, Vytenis Vasyliunas, Richard Walterscheid, Ji Wang, Brian Wood, Tom Woods, and Neal Zapp
Principles of heliophysics, V 2.1

2
Copyright ©2024 The authors ISBN-13: 9798847272711 (paperback) ISBN-13: 9798353812982 (hardcover) https://arxiv.org/abs/1910.14022 Cover design for the Amazon/KDP version by Karel Schrijver Printed in the United States of America

Contents

Preface (do read it!)

vii

1 Stars, planetary systems, and the local cosmos

1

1.1 Preparing for the future . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Considering planetary habitability . . . . . . . . . . . . . . . . . . . . 2

1.3 Heliophysics: unification, coupling, exploration . . . . . . . . . . . . . 3

1.4 The language of heliophysics . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 A timeline of exploration of planetary systems . . . . . . . . . . . . . . 10

2 Neutrals, ions, and photons

13

2.1 Conditions in the local cosmos . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Gravitationally stratified atmospheres and stellar winds . . . . . . . . 16

2.3 Photons, collisions, ionization, and differentiation . . . . . . . . . . . . 25

2.4 On collisions and currents, and on neutrals and pickup ions . . . . . . 31

2.5 Sources of plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 MHD, field lines, and reconnection

39

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 (Magneto-)Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2.1 MHD equations, individual terms, and special cases . . . . . . 46

3.2.2 The induction equation . . . . . . . . . . . . . . . . . . . . . . 49

3.3 Waves in magnetized plasmas . . . . . . . . . . . . . . . . . . . . . . . 52

3.4 MHD, magnetic field lines and reconnection . . . . . . . . . . . . . . . 54

3.5 A few notes about conditions . . . . . . . . . . . . . . . . . . . . . . . 61

3.5.1 Solar atmosphere vs. terrestrial magnetosphere . . . . . . . . . 61

3.5.2 Heliosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4 Dynamos of Sun-like stars and Earth-like planets

63

4.1 Dynamo settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.1.1 Earth and other terrestrial planets . . . . . . . . . . . . . . . . 67

4.1.2 The Sun and other stars . . . . . . . . . . . . . . . . . . . . . . 69

4.2 Dynamo principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3 Essentials of fluid motions in dynamos . . . . . . . . . . . . . . . . . . 72

i

ii

CONTENTS

4.4 Insights from approximate stellar dynamo models . . . . . . . . . . . . 75 4.5 Mean-field dynamo models . . . . . . . . . . . . . . . . . . . . . . . . 77 4.6 Dynamos in other stars . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.7 Dynamos in terrestrial planets . . . . . . . . . . . . . . . . . . . . . . 90

5 Flows, shocks, obstacles, and currents

95

5.1 Introductory overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.2 Low-velocity interactions versus shocks . . . . . . . . . . . . . . . . . . 98

5.3 Elementals of shocks and other discontinuities . . . . . . . . . . . . . . 100

5.4 The magnetized solar wind and the Parker spiral . . . . . . . . . . . . 105

5.5 Flow-based interactions in heliophysics . . . . . . . . . . . . . . . . . . 107

5.5.1 Solar-wind stream interactions . . . . . . . . . . . . . . . . . . 107

5.5.2 A non-conducting body without atmosphere . . . . . . . . . . . 112

5.5.3 Flow around a conducting body . . . . . . . . . . . . . . . . . . 114

5.5.4 Plasma flow around a permanently magnetized body . . . . . . 116

5.5.5 A closed magnetosphere . . . . . . . . . . . . . . . . . . . . . . 116

5.5.6 The open magnetosphere . . . . . . . . . . . . . . . . . . . . . 118

5.5.7 Solar wind-magnetosphere-ionosphere interaction . . . . . . . . 122

5.5.8 A large-scale flow impinging on a fast outflow . . . . . . . . . . 127

6 Magnetic (in-)stability and energy pathways

129

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.1.1 Introducing solar flares and coronal mass ejections . . . . . . . 131

6.1.2 Introducing geospace (sub-)storms . . . . . . . . . . . . . . . . 132

6.2 Terrestrial magnetospheric disturbances . . . . . . . . . . . . . . . . . 135

6.2.1 Energy pathways and reservoirs . . . . . . . . . . . . . . . . . . 135

6.2.2 What leads to explosive energy releases? . . . . . . . . . . . . . 137

6.2.3 Terrestrial magnetospheric substorms . . . . . . . . . . . . . . 140

6.2.4 Terrestrial magnetic storms . . . . . . . . . . . . . . . . . . . . 141

6.3 Models of solar impulsive events . . . . . . . . . . . . . . . . . . . . . 142

6.3.1 The magnetic reservoir . . . . . . . . . . . . . . . . . . . . . . . 142

6.3.2 Two-dimensional force-free models . . . . . . . . . . . . . . . . 143

6.3.3 Three-dimensional force-free models . . . . . . . . . . . . . . . 146

6.3.4 Formation of the pre-eruption field . . . . . . . . . . . . . . . . 148

6.3.5 Observed signatures of flares and CMEs . . . . . . . . . . . . . 150

6.4 Magnetic instabilities and reconnection . . . . . . . . . . . . . . . . . . 154

7 Torques and tides

159

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

7.2 Magnetic torques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

7.2.1 Stellar winds and magnetic braking . . . . . . . . . . . . . . . . 161

7.2.2 Planetary magnetospheric torque . . . . . . . . . . . . . . . . . 162

7.2.3 Magneto-rotational coupling . . . . . . . . . . . . . . . . . . . . 165

7.2.4 Disk winds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

CONTENTS

iii

7.3 Gravitational tides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 7.3.1 Spin-orbit interactions . . . . . . . . . . . . . . . . . . . . . . . 168 7.3.2 Orbital interaction . . . . . . . . . . . . . . . . . . . . . . . . . 171
7.4 Planetary atmospheric tides . . . . . . . . . . . . . . . . . . . . . . . . 172

8 Particle orbits, transport, and acceleration

175

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

8.2 Single particle motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

8.3 Phase space density and Liouville’s theorem . . . . . . . . . . . . . . . 181

8.4 The collisionless Boltzmann equation . . . . . . . . . . . . . . . . . . . 182

8.5 Particle scattering and transport . . . . . . . . . . . . . . . . . . . . . 185

8.5.1 Solar energetic particles . . . . . . . . . . . . . . . . . . . . . . 188

8.5.2 Galactic cosmic rays . . . . . . . . . . . . . . . . . . . . . . . . 190

8.6 Particle acceleration in shocks . . . . . . . . . . . . . . . . . . . . . . . 193

8.7 Relativistic particles in planetary radiation belts . . . . . . . . . . . . 201

8.7.1 Electron acceleration mechanisms . . . . . . . . . . . . . . . . . 201

8.7.2 Proton acceleration in the radiation belt . . . . . . . . . . . . . 205

8.7.3 Radiation belt losses at Earth . . . . . . . . . . . . . . . . . . . 205

9 Convection, heating, conduction, and radiation

207

9.1 Convective and radiative energy transport . . . . . . . . . . . . . . . . 207

9.2 Heating and cooling of the solar outer atmosphere . . . . . . . . . . . 211

9.3 Magnetic activity and atmospheric radiation . . . . . . . . . . . . . . . 214

10 Evolution of stars, activity, and asterospheres

217

10.1 Evolution of stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

10.2 Stellar activity and its evolution . . . . . . . . . . . . . . . . . . . . . 221

10.2.1 Overall activity level . . . . . . . . . . . . . . . . . . . . . . . . 221

10.2.2 Flares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

10.2.3 Rotation rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

10.2.4 Stellar infancy: birth to the zero-age main sequence . . . . . . 227

10.2.5 Stellar teenage years: ZAMS - 1 Gyr . . . . . . . . . . . . . . . 227

10.2.6 Stellar adulthood: 1-5 Gyr . . . . . . . . . . . . . . . . . . . . . 228

10.3 Evolution of astrospheres . . . . . . . . . . . . . . . . . . . . . . . . . 229

10.3.1 Effects of a variable ISM on heliospheric structure . . . . . . . 229

10.3.2 Long-term evolution of stellar winds . . . . . . . . . . . . . . . 232

10.3.3 Astrospheric field patterns in time . . . . . . . . . . . . . . . . 236

11 Formation of stars and planets

239

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

11.2 (Exo-)Planets and (exo-)planetary systems . . . . . . . . . . . . . . . . 241

11.2.1 Exoplanet formation . . . . . . . . . . . . . . . . . . . . . . . . 243

11.2.2 Exoplanet migration . . . . . . . . . . . . . . . . . . . . . . . . 245

11.2.3 Exoplanet geology . . . . . . . . . . . . . . . . . . . . . . . . . 246

iv

CONTENTS

11.2.4 Exoplanets and binary star systems . . . . . . . . . . . . . . . 247 11.3 Formation and early evolution of stars and disks . . . . . . . . . . . . 247
11.3.1 Observations of star-forming processes . . . . . . . . . . . . . . 247 11.3.2 Properties of young stars . . . . . . . . . . . . . . . . . . . . . 251 11.3.3 The rotation rate of very young stars . . . . . . . . . . . . . . . 253 11.3.4 Protoplanetary disks and gravity . . . . . . . . . . . . . . . . . 255 11.3.5 Dust-disk evolution . . . . . . . . . . . . . . . . . . . . . . . . . 257 11.3.6 Disk evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . 259

12 Irradiance, atmospheres, and habitability

263

12.1 Evolving planetary habitability . . . . . . . . . . . . . . . . . . . . . . 263

12.1.1 Earth’s formative phase . . . . . . . . . . . . . . . . . . . . . . 264

12.1.2 The habitable zone . . . . . . . . . . . . . . . . . . . . . . . . . 265

12.1.3 Oxygen, methane, and carbon dioxide over time . . . . . . . . 267

12.1.4 Water over time . . . . . . . . . . . . . . . . . . . . . . . . . . 268

12.2 Atmospheres and climates of Venus, Earth, and Mars . . . . . . . . . . 269

12.3 Irradiance, orbits, spin, and climate . . . . . . . . . . . . . . . . . . . 271

12.3.1 Atmospheric effects and albedo . . . . . . . . . . . . . . . . . . 271

12.3.2 Orbital changes . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

12.4 Planetary atmospheres, geological activity, and stellar winds . . . . . . 280

12.4.1 On time scales beyond millions of years . . . . . . . . . . . . . 280

12.4.2 On time scales of up to several millennia . . . . . . . . . . . . . 286

13 Upper atmospheres and magnetospheres

289

13.1 Upper-atmospheric chemistry and insolation . . . . . . . . . . . . . . . 289

13.2 Maintaining ionospheres . . . . . . . . . . . . . . . . . . . . . . . . . . 293

13.2.1 Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

13.2.2 Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

13.2.3 Venus and Mars . . . . . . . . . . . . . . . . . . . . . . . . . . 297

13.3 Setting geospace climate . . . . . . . . . . . . . . . . . . . . . . . . . . 298

13.3.1 Geospace climate response to solar photon irradiation . . . . . 298

13.3.2 Geospace climate at earlier terrestrial ages . . . . . . . . . . . . 303

13.3.3 Geospace climate and Earth’s magnetic field . . . . . . . . . . 305

13.3.4 Geospace climate dependence on the solar wind . . . . . . . . . 308

14 Cosmic rays and magnetic fields over time

311

14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

14.2 Long-term energetic-particle exposure of Earth . . . . . . . . . . . . . 312

14.2.1 Generation of cosmogenic radionuclides . . . . . . . . . . . . . 312

14.2.2 Transport and deposition of cosmogenic radionuclides . . . . . 315

14.3 Radionuclides as proxies of magnetic variability . . . . . . . . . . . . . 317

14.3.1 Geomagnetic field . . . . . . . . . . . . . . . . . . . . . . . . . 318

14.3.2 Solar variability . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

14.3.3 Very-long time scale variability in cosmic-ray exposure . . . . . 320

CONTENTS

v

14.4 Exposure to supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . 321

15 Applied heliophysics, mutatis mutandis

323

15.1 Activities to take you beyond the present-day solar system . . . . . . . 325

Activities

16 Activities

327

Chapter 1. Stars, planetary systems, and the local cosmos . . . . . . 327

Chapter 2. Neutrals, ions, and photons . . . . . . . . . . . . . . . . . . 328

Chapter 3. MHD, field lines, and reconnection . . . . . . . . . . . . . 332

Chapter 4. Dynamos of Sun-like stars and Earth-like planets . . . . 334

Chapter 5. Flows, shocks, obstacles, and currents . . . . . . . . . . . 339

Chapter 6. Magnetic (in-)stability and energy pathways . . . . . . . 342

Chapter 7. Torques and tides . . . . . . . . . . . . . . . . . . . . . . . . 344

Chapter 8. Particle orbits, transport, and acceleration . . . . . . . . 345

Chapter 9. Convection, heating, conduction, and radiation . . . . . . . . . . 346

Chapter 10. Evolution of stars, activity, and asterospheres . . . . . . 349

Chapter 11. Formation of stars and planets . . . . . . . . . . . . . . . 351

Chapter 12. Irradiance, atmospheres, and habitability . . . . . . . . . 354

Chapter 13. Upper atmospheres and magnetospheres . . . . . . . . . 356

Chapter 14. Cosmic rays and magnetic fields over time . . . . . . . . 358

Chapter 15. Applied heliophysics, mutatis mutandis . . . . . . . . . . 360

17 Solutions and supplemental text for selected activities

363

17.1 Activity 13: Solar wind travel time . . . . . . . . . . . . . . . . . . . . 363

17.2 Activity 15: Parker solar wind; basics . . . . . . . . . . . . . . . . . . . 363

17.3 Activity 16: Mean free paths and MHD . . . . . . . . . . . . . . . . . . 365

17.4 Activity 21: Penetration depth of energetic particles . . . . . . . . . . 367

17.5 Activity 22: Collisions and conductivities . . . . . . . . . . . . . . . . . 367

17.6 Activity 28: cgs to SI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

17.7 Activity 35: Plasma β in the Parker solar wind . . . . . . . . . . . . . 368

17.8 Activity 37: Thin flux tube and hydrostatic equilibrium . . . . . . . . 369

17.9 Activity 51: Mean-field induction equation . . . . . . . . . . . . . . . . 370

17.10Activity 53: Babcock-Leighton surface flux dispersal . . . . . . . . . . 370

17.11Activity 64: Solar surface to heliosphere: PFSS . . . . . . . . . . . . . 371

17.12Activity 66: Solar wind behind a non-conducting body . . . . . . . . . 373

17.13Activity 69: Temperature at the wind stagnation point . . . . . . . . . 374

17.14Activity 70: Magnetopause distances . . . . . . . . . . . . . . . . . . . 375

17.15Activity 85: Coronal loop cooling time scales . . . . . . . . . . . . . . . 375

17.16Activity 92: Magnetic braking of the Sun . . . . . . . . . . . . . . . . . 376

17.17Activity 94: Tidal breakup . . . . . . . . . . . . . . . . . . . . . . . . . 377

17.18Activity 105: Drift velocity . . . . . . . . . . . . . . . . . . . . . . . . . 377

17.19Activity 107: SEP paths and scattering . . . . . . . . . . . . . . . . . . 377

vi

CONTENTS

17.20Activity 112: Setting the scale of granulation . . . . . . . . . . . . . . 378 17.21Activity 113: Acoustic cutoff . . . . . . . . . . . . . . . . . . . . . . . . 379 17.22Activity 115: Optical depth and field strength . . . . . . . . . . . . . . 380 17.23Activity 121: Properties of coronal loops . . . . . . . . . . . . . . . . . 380 17.24Activity 126: Least-massive post-main-sequence star . . . . . . . . . . 381 17.25Activity 138: Earth masses in the Sun-forming cloud . . . . . . . . . . 381 17.26Activity 140: The Jeans Mass . . . . . . . . . . . . . . . . . . . . . . . 382 17.27Activity 174: Molecular diffusion coefficients . . . . . . . . . . . . . . . 382 17.28Activity 186: Earth’s magnetopause distance over time . . . . . . . . . 382 17.29Activity 189: 10Be production on a hypothetical Mars . . . . . . . . . 383 17.30Activity 193: Exposure to supernovae . . . . . . . . . . . . . . . . . . . 383 17.31Activity 200: Reaching Earth’s climate from scratch . . . . . . . . . . 383

Appendices

Version history

385

List of Figures

390

List of Tables

391

Bibliography

401

Subject index

403

Physical constants, plasma quantities, and vector identities

411

Preface (do read it!)
Heliophysics is the system science of the physical connections between the Sun and the solar system. As the physics of the local cosmos, it embraces space weather and planetary habitability. The wider view of comparative heliophysics forms a template for conditions in exoplanetary systems and provides a view over time of the aging Sun and its magnetic activity, of the heliosphere in different settings of the interstellar medium and subject to stellar impacts, of the space physics over evolving planetary dynamos, and of the long-term influence on planetary atmospheres by stellar radiation and wind.
Based on a series of NASA-funded Summer Schools for early-career researchers, this
Heliophysics
helio-, pref., on the Sun and environs, from the Greek helios.
physics, n., the science of matter and energy and their interactions.
Heliophysics is the
• comprehensive new term for the science of the Sun - Solar System Connection. • exploration, discovery, and understanding of our space environment. • system science that unites all of the linked phenomena in the region of the cosmos
influenced by a star like our Sun. Heliophysics concentrates on the Sun and its effects on Earth, the other planets of the solar system, and the changing conditions in space. Heliophysics studies the magnetosphere, ionosphere, thermosphere, mesosphere, and upper atmosphere of the Earth and other planets. Heliophysics combines the science of the Sun, corona, heliosphere and geospace. Heliophysics encompasses cosmic rays and particle acceleration, space weather and radiation, dust and magnetic reconnection, solar activity and stellar cycles, aeronomy and space plasmas, magnetic fields and global change, and the interactions of the solar system with our galaxy.
From NASA’s “Heliophysics. The New Science of the Sun - Solar System Connection: Recommended Roadmap for Science and Technology 2005 - 2035.”
Table 1: Heliophysics: definition.
vii

viii

Preface

Table 2: Chapters and their authors in the Heliophysics book series sorted by theme (continued on the next page), not showing introductory chapters.
Universal and fundamental processes, diagnostics, and methods I.2. Introduction to heliophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T. Bogdan I.3. Creation and destruction of magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . M. Rempel I.4. Magnetic field topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Longcope I.5. Magnetic reconnection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T. Forbes I.6. Structures of the magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Moldwin et al. II.3 In-situ detection of energetic particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Gloeckler II.4 Radiative signatures of energetic particles . . . . . . . . . . . . . . . . . . . . . . . . . . T. Bastian II.7 Shocks in heliophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Opher II.8 Particle acceleration in shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Krauss-Varban II.9 Energetic particle transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Giacalone II.11 Energization of trapped particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Green IV.11 Dusty plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Hor´anyi IV.12 Energetic-particle environments in the solar system . . . . . . . . . . . . . . . . N. Krupp IV.13 Heliophysics with radio scintillation and occultation . . . . . . . . . . . . . . . . . M. Bisi
Stars, their planetary systems, planetary habitability, and climates III.3 Formation and early evol. of stars and proto-planetary disks . . . . L. Hartmann III.4 Planetary habitability on astronomical time scales . . . . . . . . . . . . . . . D. Brownlee III.11 Astrophysical influences on planetary climate systems . . . . . . . . . . . . . . . . J. Beer III.12 Assessing the Sun-climate relationship in paleoclimate records . . . T. Crowley III.14 Long-term evolution of the geospace climate . . . . . . . . . . . . . . . . . . . . . . . . J. Sojka III.15 Waves and transport processes in atmosph. and oceans . . . . . . R. Walterscheid IV.5 Characteristics of planetary systems . . . . . . . . . . . . . . . . . . . D. Fischer & J. Wang IV.7 Climates of terrestrial planets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Brain The Sun, its dynamo and its magnetic activity; past, present and future I.8. The solar atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Hansteen II.5 Observations of solar and stellar eruptions, flares, and jets . . . . . . . . . . H. Hudson II.6 Models of coronal mass ejections and flares . . . . . . . . . . . . . . . . . . . . . . . . . . T. Forbes III.2 Long-term evolution of magnetic activity of Sun-like stars . . . . . . . . C. Schrijver III.5 Solar internal flows and dynamo action . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Miesch III.6 Modeling solar and stellar dynamos . . . . . . . . . . . . . . . . . . . . . . . . . . P. Charbonneau III.10 Solar irradiance: measurements and models . . . . . . . . . . . . . J. Lean & T. Woods IV.2 Solar explosive activity throughout the evol. of the solar system . . . . . R. Osten

textbook is intended for students in physical sciences in later years of their university training and for beginning graduate students in fields of solar, stellar, (exo-)planetary, and planetary-system sciences. The lecturers at the Summer Schools developed a series of five volumes on Heliophysics (four published in printed form by Cambridge University Press, and one online at the Heliophysics Summer School website) contain in total 1919 pages of text and figures, in 56 topical chapters (see Table 2): Vol. I:

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Table 2: (Continued from the previous page) Chapters and their authors in the Heliophysics book series sorted by theme, not showing introductory chapters.
Astro-/heliospheres, interstellar environment, and galactic cosmic rays I.7. Turbulence in space plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Smith I.9. Stellar winds and magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Hansteen III.8 The structure and evolution of the 3D solar wind . . . . . . . . . . . . . . . . . . J. Gosling III.9 The heliosphere and cosmic rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Jokipii IV.3 Astrospheres, stellar winds, and the interst. medium . . . . B. Wood & J. Linsky IV.4 Effects of stellar eruptions throughout astrospheres . . . . . . . . . . . . . . . . . O. Cohen
Dynamos and environments of planets, moons, asteroids, and comets I.10. Fundamentals of planetary magnetospheres . . . . . . . . . . . . . . . . . . . . V. Vasyliu¯nas I.11. Solar-wind magnetosphere coupling . . . . . . . . . . . . . . . . . F. Toffoletto & G. Siscoe I.13. Comparative planetary environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. Bagenal II.10 Energy conversion in planetary magnetospheres . . . . . . . . . . . . . . . . V. Vasyliu¯nas III.7 Planetary fields and dynamos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U. Christensen IV.6 Planetary dynamos: updates and new frontiers . . . . . . . . . . . . . . . . . . . . . S. Stanley IV.10 Moons, asteroids, and comets interact. with their surround. . . . . . M. Kivelson
Planetary upper atmospheres I.12. On the ionosphere and chromosphere . . . . . . . . . T. Fuller-Rowell & C. Schrijver II.12 Flares, CMEs, and atmospheric responses . . . . . T. Fuller-Rowell & S. Solomon III.13 Ionospheres of the terrestrial planets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Solomon III.16 Solar variability, climate, and atmosph. photochemistry . . . G. Brasseur et al. IV.8 Upper atmospheres of the giant planets . . . . . . . . . . . . . . . . . . . . . . . L. Moore et al. IV.9 Aeronomy of terrestrial upper atmospheres . . . . . . . . . . D. Siskind & S. Bougher
Technological and societal impacts of space weather phenomena II.2 Introduction to space storms and radiation . . . . . . . . . . . . . . . . . . . . . . . S. Odenwald II.13 Energetic particles and manned spaceflight . . . . . . . . . . S. Guetersloh & N. Zapp II.14 Energetic particles and technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Tribble V.2 Space weather: impacts, mitigation, forecasting . . . . . . . . . . . . . . . . . . . S. Odenwald V.3 Commercial space weather in response to societal needs . . . . . . . . . . . . W. Tobiska V.4 The impact of space weather on the electric power grid . . . . . . . . . . . . . D. Boteler V.5 Radio waves for communication and ionospheric probing . . . . . . . . . . N. Jakowski

Schrijver and Siscoe (2011); Vol. II: Schrijver and Siscoe (2012b); Vol. III: Schrijver and Siscoe (2012a); Vol. IV: Schrijver et al. (2016); and Vol. V: Schrijver and Siscoe (2015). The present volume presents a selection of these texts, while adding new text as connecting or summarizing material, with an overall text length that is about one-fifth of the original textbooks.
The topics in this volume are organized to emphasize universal processes from a perspective that draws attention to what provides Earth (and similar (exo-)planets) with a relatively stable setting in which life as we know it can thrive. This text aims to serve as a textbook-style volume for which the original Heliophysics books are the

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extended ‘readers’ with much more detail, and domain-specific topical chapters. Note that references from the original texts were omitted here (see the original volumes for those); references for new texts can be found in the Bibliography, where also source references to figures are provided as needed.
This volume is intended for students in physical sciences in later years of their university training and for beginning graduate students in fields of solar, stellar, (exo-)planetary, and planetary-system sciences. This contrasts with the intended audiences for the Heliophysics volumes which included the community of mid-toadvanced graduate students, the cohort of early postdoctoral researchers, and those professional researchers looking for review-like introductions into fields of heliophysics adjacent to their own. In targeting the audience of advanced undergraduate and beginning graduate students, many of the deeply technical details discussed in the original volumes were omitted, introductions were broadened, and the emphasis was placed on processes rather than on details of equations, states, or numerical experiments.
Throughout this work the original text from the Heliophysics volumes is directly quoted, following a volume and chapter reference, where between double quotation marks, but with equations, units (here cgs-Gaussian throughout with a few exceptions [1]), and symbols modified where needed for homogeneity throughout this work, with edits (and some corrections) shown between brackets, with many parenthetical notes removed, and with citations of the professional literature left out (and those to other sections in the books modified as appropriate).
The source texts in the series of Heliophysics books are referenced in the margins as #[roman]:#[arabic].#[arabic]. For example, Vol. I, Section 9 in Chapter 2 would be referred to as “I:2.9” in the margin or in captions. The original sources of all of the figures can be found in the figure captions of the Heliophysics books, but for many here a reference to the original publication is included for figures not made by the Heliophysics authors but whose original authors have given permission to have their artwork used in this volume. A few figures were replaced by color versions or by alternative figures.
Activities for the reader
New here compared to the printed volumes is the inclusion of 200 ’activities’ (starting in version 1.3; see the chapter on ’Version history’ for a description of changes) in the form of problems, exercises, explorations, literature readings, and ’what if’ challenges. Many contain additional information complementing the main text, so I suggest you read them as you go along, if not on first reading, then at least on review. {Ⓢ0} [2] Some were developed by the teachers for the Heliophysics Summer School but
1 A good resource for unit conversions (and many other things related to plasma physics) is the online NRL plasma formulary.
2 Exercises are flagged as {A#} or {ⓈA#}, also in the margin, with continuous numbering throughout (this numbering is Version dependent!). Activities can be found in Ch. 16. For a selection of of these Activities, solutions and/or supplemental reading are provided in Ch. 17 – these selected Activities are marked Ⓢ in the margin and in the compilation in Ch. 16.

I:2.9

{Ⓢ0}

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’Space weather’ is the term used to describe an ensemble of changing conditions in the vicinity of Earth and, by extension, any other body in a planetary system, typically occurring on time scales up to a few days. Often, the term is implicitly taken to refer also to the conditions from the solar dynamo outward to the furthest reaches of the heliosphere that are involved in space weather around Earth. Much of what is described in this volume therefore concerns space weather: heliophysics contains the science of space weather. However, where the science of space weather focuses on phenomena that can impact society through short-term variability, this text takes the long view by putting the spotlight on evolutionary changes in the states of star-planet systems. As such, this text does describe the foundational processes of space weather, but is not concerned with the impacts of space weather on technological infrastructure, does not address the challenges of forecasting space weather, and skips coupling mechanisms such as ground-induced currents (GICs) associated with geomagnetic disturbances and ground-level enhancements (GLEs) of energetic particles. This choice of focus is motivated by my desire to introduce the reader to the science of heliophysics from the perspective of habitability on time scales on which stellar and planetary atmospheres change, and indeed up to time scales on which stars and their planets evolve, and to do that in a relatively compact form. As you go through this text, you should realize that many of the processes described here have consequences for society, ranging from system design choices to potentially substantial failures in one or more of the infrastructures that we have come to rely on, including continuous and reliable electric power, positional information, and means of communication. Interruptions in quality or availability of any of these can have substantial consequences that may be costly or life-threatening on scales that may involve single individuals or populations of millions. Descriptions of the impacts of space weather can be found in the Heliophysics books in Chapters II:2, II:13, II:14, H-V:2, H-V:3, H-V:4, and H-V:5; another resource is a ’roadmap’ document (Schrijver et al., 2015) that reviews the state of our knowledge of space weather and its technological and societal impacts, and what is needed to advance our abilities to forecast space weather.
Table 3: Heliophysics and space weather

most are newly created specifically for this volume. They are meant to let the reader look up a definition, to introduce a moment of reflection on an equation or figure, to see connections to similarities elsewhere, to get a feel for the magnitude of things or the relative importance of processes, or to consider what would happen under conditions other than those encountered in our Solar System; they are not meant to particularly exercise mathematical skills. There are five classes of activities as indicated at their start: “Look up” to familiarize you with processes, numbers, and definitions, “Consider” to make you think about processes, “Show” for applications of equations or numerical estimates, “Background” for further information, and “Advanced/Group” for larger activities to explore beyond the textbook and/or to undertake with a group of students in a class. At the end of the book, in Activity 200, the reader is asked to reflect back

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on all the processes that are involved in the habitability of Earth and, by analogy, of exoplanets elsewhere in the Universe.
Terminology
As you go through this volume, you will encounter words that have somewhat different meanings in different communities. For example, ’convection’ is often used in the magnetospheric community to describe movement that in astrophysics would be referred to as ’advection’, while ’convection’ in that discipline is reserved for overturning plasma motions involved in the transport of thermal energy. Another example is that of the word ’dynamo’ which in astrophysics and planetary sciences is used to describe the ensemble of processes maintaining a magnetic field against decay, often with an alternating temporal character. In ionospheric physics, it is often used for processes where differential motions of (neutral plus ionized) gas and magnetic field exchange energy through work.
You will also note that terms may describe locations or something in a location, or a property of what is in that location. For example, ’ionosphere’ may sound like a location descriptor but actually refers to only the ionized medium in an atmosphere (with ’thermosphere’ used for the overlapping neutral environment). The term ’chromosphere’, which describes a stellar environment in some respects not dissimilar to an ionospherethermosphere, encompasses both the ionized and neutral components; it is often used as an indicator of a volume above a stellar surface in a certain thermal range, but is defined formally (as you will see later) by the properties of the radiative transfer of the medium.
Finally, there are words like ’late-type star’ that have nothing to do with a temporal attribute, but which survived an older era where the nature of stars was not yet understood and where cooler was erroneously interpreted as older.
I hope that all terms are properly defined where first used. Here, I want to raise your awareness that as you talk to colleagues in other disciplines they may not only be puzzled by processes that you study, but that communication may be hampered by misinterpretation of the terms that you use: language can be a very precise tool, but only if the user is aware of how the listener/reader may interpret the words that are being used.
Online resources associated with the Summer School
• If you are looking at a paperback or hardcover version of this book from Amazon/KDP then look for a free e-version at https://arxiv.org/abs/1910.14022 which uses hyperrefs for easy navigation. It also has many figures in color, as does the hardcover edition, that are shown in gray-scale in the paperback.
• The Summer School’s home is at https://heliophysics.ucar.edu. • Many of the Heliophysics lectures can be found on YouTube by searching for
’Heliophysics Summer School’.

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A few notes on other resources
The figures published in the Heliophysics book series are available on-line at the website of the Heliophysics Summer School (https://heliophysics.ucar.edu), where you can also find labs (with instruction manuals) and many recorded lectures sorted by theme (in part hosted on youtube).
There is a subject index in this volume but note that the online version of this book can be searched with the tools of web browsers and pdf viewers to provide an effective and entirely comprehensive alternative way to find topics.
This volume focuses on processes, not on their measurements. For an introduction to some of the aspects of remote and in-situ sensing within the Heliophysics series I refer to the following chapters that focus on that aspect in particular: Chs. II:3, II:4, IV:5, and IV:13.
You can find a list of English-language textbooks, popular texts, topical monographs and book series, related to the science of space weather and published since 1990 in Knipp and Cade (2020).
Explore NASA’s Scientific Visualization Studio at https://svs.gsfc.nasa.gov for a variety of images and movies — real and simulated.
Navigating the pdf version at arXiv:2001.01093
References to sections, figures, tables, and equations in this book are shown in red, pointers to the bibliography are shown in green, and references to web pages are shown in blue. Clicking on any of these jumps to that location or web page. How you get back to reading where you left off depends on how you are viewing this file and on what type of device. For example, this web site shows a list of keyboard shortcuts to move around the pdf version of this book with Acrobat Reader. Using that on a Mac, you can return to the page you came from by pressing [command + left-arrow] after clicking on a link to a figure, section, or activity. When using Mac OS Preview, look under the ’Go’ menu for navigation shortcuts (where you will see that the equivalent of the above is [command + left bracket]).
Corrections and updates
This version of the textbook is subject to corrections and updates. I welcome input from students, teachers, and colleagues: if you see a typo or an explanation that you think is in error, or if you believe a serious update is in order, please email me: heliophysicsnutshell@gmail.com! Be as specific as you can about where the text is that you think should be changed, what to change it to, and why it needs such a change. Your input will help improve this text for all users.
Acknowledgements
I thank all the Heliophysics authors and other teachers in the Heliophysics Summer School for the skill with which they taught me as one of the participants in the School as well as for their patience with me as one of the editors of the book series. This

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volume was supported by the Johannes Geiss Fellowship of the International Space Science Institute. I am indebted to Ju¨rg Beer, Paul Charbonneau, Terry Forbes, Marina Galand, Dana Longcope, Sten Odenwald, and Matthias Rempel for their insightful comments on an earlier version of the manuscript, and to Nick Gross for working through most of the Activities, noting errors, adding tasks, and developing new Activities. Special thanks go to Tom Bogdan who worked through the entire draft volume.
Karel Schrijver, March 27, 2024 karelschrijver.com
“A physical understanding is a completely unmathematical, imprecise, and inexact thing, but absolutely necessary for a physicist.”, R.P. Feynman, in The Feynman Lectures on Physics, Vol. II.

Chapter 1
Stars, planetary systems, and the local cosmos
Chapter topics:
• The rationale for this book • Heliophysics: the system science of the physical connections between the Sun
and the solar system • Magnetohydrodynamics as the basic language of heliophysics • Basic glossary for heliophysics • A timeline of exploration of (exo-)planetary systems
1.1 Preparing for the future
By the time you reach the end of this book, you will have the basic set of tools of scientific imagination involved in understanding what couples stars and planets. What you will learn is universal, literally: it does not matter which stars and planets we speak of: whether of those few nearby or of the many distant ones. Nor does it matter whether they are those few that we are long familiar with or the many that we know about, so far, only in a statistical sense. It does not matter either whether your particular interests lie within the Solar System or beyond it: the same principles apply in our local cosmos as in the most distant planetary system we shall ever have access to.
But looking forward to your science-based career, whether as a researcher or as a teacher, as a journalist or as a politician, you need to be familiar with what is known. That is particularly true in order to discover something new. And to appreciate the value of a discovery, you need to know how to apply what you know to what is not (yet) known. You will need to imagine things no one has ever seen, but not arbitrarily: science demands that you come up with what appears most probable, not merely with things that are possible. Richard Feynman (in The meaning of it all ) said it this way: ’It is surprising that people do not believe that there is imagination in science. It is a very interesting kind of imagination, unlike that of the artist. The great difficulty is in
1

2

1 – Stars, planetary systems, and the local cosmos

trying to imagine something that you have never seen, that is consistent in every detail with what has already been seen, and that is different from what has been thought of.’
The pace at which exoplanets are being discovered is simply amazing. What we can learn from them, and from our Solar System, offers so many opportunities to learn yet more. I realize that going through the first nine chapters will be hard, because they have to build your foundation, because they cover so many different branches of science, and because they look at things so different from everyday life. But these first nine chapters look for commonalities, for ’universal processes’ that help create in your mind a virtual laboratory: in the astronomy we cannot turn dials to explore things under different conditions, but we can compare environments and look for what they have in common and for what sets them apart.
The final six chapters require prior digestion of the first nine. These final six invite you to imagine, scientifically, Earth in the distant past and future, Earth-like planets in a variety of orbits around Sun-like stars, and the space environments and climates of tropospheres of exo-worlds. Future discoveries have their beginnings in lessons from the past:
1.2 Considering planetary habitability
Planetary system are, statistically speaking, about as common as stars. We have learned a lot about stars over the century that followed the realization that they are huge nuclear fusion reactors and that most, like the Sun, function also as giant dynamos. In contrast, firm evidence that planetary systems are common companions to stars was only obtained within the past two decades. It is therefore no surprise that much still needs to be learned about how planets form, how planetary systems evolve, and what the conditions are near planetary surfaces (if indeed a solid or liquid surface exists). The combination of exoplanetary science and the study of the local cosmos is enlightening us as much about the history and future of our Solar System as about the growing number of planetary systems that have been observed in some detail. Whether life exists anywhere beyond Earth remains to be established, but scientists are making rapid headway in knowing about the conditions that life on Earth has been subjected to since its genesis and also the conditions that any life on any other planet would be subjected to depending on the properties of their central star and companion planets.
Heliophysics deals with all of the aspects of ’living with a star’ on time scales from fractions of a second to billions of years. The series of Heliophysics books offers an introduction to a large cross section of that vast scientific field. In the present volume, we focus on the universal processes that tie together the branches of heliophysics with particular emphasis on those processes that are relevant to what one might describe as ’planetary habitability’. With life having been found on only a single astrophysical body we do not have a particularly well-considered concept of what ’planetary habitability’ might mean, of course. But we have an intuitive feel for it: a long-lived planet orbiting a long-lived star, with a fairly substantial planetary atmosphere that is neither too hot nor too cool to allow chemistry to be complex (and, in many minds, restricting that to chemistry that involves liquid water), shielded well enough (but not necessarily

Heliophysics: unification, coupling, exploration

3

I:2.9

I:2.1

perfectly) from energetic radiation (both electromagnetic and particulate) by that atmosphere and by a planetary magnetic field. The star’s irradiance onto a ’habitable planet’ should not vary too much, comet and asteroid impacts should be limited, atmospheric erosion slow, . . . If that sounds like we are describing the Sun-Earth system then that is no surprise: we know it has made the Earth habitable to a diversity of life that is on one hand astoundingly diverse and on the other – at the molecular level – remarkably homogeneous.
As the number of known exoplanetary systems is bound to keep growing rapidly, and as our instrumentation and methods are bringing exoplanetary atmospheric science within our grasp, it is clear that our understanding of the Solar System and its central star provide crucial guidance to the study of ’planetary habitability’ and – some day rather soon, one should anticipate – the study of extraterrestrial life. That expectation has guided the selection of topics covered in this volume.
[3]“If we gaze upon the uncountable array of stars strewn across the vault of the heavens, one may know that the remarkable things one will come to know about heliophysics in the pages that follow are presently unfolding around those very stars and planetary systems that give light to the night sky. Heliophysics is truly a universal science.”
1.3 Heliophysics: unification, coupling, exploration
“Walk along an island beach on a clear, breezy, cloudless night, or stand on the spine of a barren mountain ridge after sunset, and behold the firmament of stars {A1}
glittering against the coal-black sky above. They fill the sky with their timeless, brilliant flickering [(mostly caused by the terrestrial atmosphere)]. With binoculars or even a small telescope one finds that even the lacy dark matrix between the vast sea of stars is populated with still more stars that are simply too faint to be seen with the naked eye. Within the Milky Way galaxy, that stretches from horizon to horizon, the density of stars against the background sky is even greater.
Each twinkling point of light is a star not too unlike our own Sun. The Sun is an ordinary star that features so prominently in our lives and on the pages of [the Heliophysics book series, as in this volume,] because of its proximity. The next closest star, α Centauri (which is a triple system in which Proxima Centauri is currently the closest to Earth), is almost a million times farther away (at 4.22 light years), and the others are farther still. We may now say with some confidence that many of the stars are surrounded by planets of various sizes. {A2} Some of these orbital companions are so immense that they are stars in their own right: double-star systems are quite common. {A3}
With the same measure of confidence we may assert that most of these stars possess magnetic fields; that these magnetic fields create hot outer atmospheres, or coronae, that drive magnetized winds from their stars; and that these variable plasma winds
3 Throughout this work the original text from the Heliophysics volumes is directly quoted (with edits between brackets) with references shown in the margin like this: #[roman]:#[arabic].#[arabic]. So, for example, Vol. I, Section 9 in Chapter 2 would be referred to as I:2.9.

{A1}

{A2} {A3}

4

1 – Stars, planetary systems, and the local cosmos

Table 1.1: Basic glossary for domains and phenomena in heliophysics (continued on the next page).
• active region: a bipolar area of relatively strong magnetic flux, mostly consisting of magnetic plage (underlying the chromospheric plage) and, by definition, containing one or more sunspots at some point in its evolution (cf. Fig. 4.4). Collectively they form two active-region belts located on opposite hemisphere.
• ast(e)rosphere: equivalent of a heliosphere around another star (Sect. 10.3) • aurora: A visual phenomenon associated with geomagnetic activity visible mainly in
the high-latitude night sky, resulting from collisions between atmospheric gases and precipitating charged particles (mostly electrons) guided by the geomagnetic field from the magnetotail. • chromosphere: domain above the Sun’s visible ’surface’, with temperatures around 10,000–20,000K(see Table 2.3) • corona: the hottest domain of the Sun’s atmosphere, at ≥ 1 MK (Table 2.3) • coronal hole: formally a coronal region that is dark in X-rays and EUV; generally identified with a region where the Sun’s magnetic field is ’open’, i.e., reaches into the heliosphere (e.g., Sect. 2.2) • coronal loop: a high-temperature atmosphere within the Sun’s corona, constrained to the volume of a magnetic ’flux tube’ (e.g., Sect. 3.4) • coronal mass ejection: impulsive expulsion of magnetized material from a star into an astrosphere (e.g., Fig. 5.1, Sect. 3.1) • current sheet: defined in Table 3.1 • exosphere: outermost domain of an atmosphere in which collisions are rare and ballistic trajectories dominate for constituent particles (e.g., Sect. 2.3) • facula and bright point: a small flux tube in near-photospheric layers viewed towards the solar limb or disk center, respectively (e.g., Sect. 9.1) • flare: impulsive conversion of magnetic energy in a stellar atmosphere into thermal and non-thermal particles and bulk plasma motion, and appearing as a brightening over much of the stellar spectrum, although not significantly in total stellar brightness except for the most energetic events (e.g., Sect. 3.1) • filament/prominence: A volume of gas at chromospheric temperatures suspended within the corona by magnetic forces, seen as dark ribbons threaded over the solar disk. A filament beyond the edge of the solar disk seen in emission against the dark sky is called a prominence • flux tube/rope: defined in Table 3.1

blow past the orbiting planets, distorting their individual magnetospheres, and push outward against the surrounding interstellar medium. Where the ram pressure of the stellar wind becomes comparable to the surrounding pressure (gas, magnetic, and cosmic ray) of the interstellar medium, a bow shock forms. This serves to mark the farthest extent of the mechanical impact of the star on its surrounding environment: a sphere of influence, so to speak. [4] {A4}
4 See Tables 1.1 and 1.2 for definitions of the most common descriptors used for domains in, or phenomena related to, heliophysics. For a more extensive glossary of terms used in this volume, see, for example

{A4}

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5

Table 1.2: Basic glossary for domains and phenomena in heliophysics (continued from preceding page).
• (super-)granulation: granulation is the pattern of convective cells visible in the solar photosphere with a typical scale length of 1 Mm (see Fig. 9.1); supergranulation is a much larger cellular pattern with a scale length of 20–30 Mm that manifests primarily in velocity maps and in its ordering of the magnetic network (see Fig. 4.4)
• geomagnetic (sub-)storm: Storm: A worldwide disturbance of the Earth’s magnetic field, distinct from regular diurnal variations. Substorm: A geomagnetic perturbation lasting 1 to 2 hours, which tends to occur during local post-midnight nighttime. The magnitude of the substorm is largest in the auroral zone. (e.g., Sects. 6.2 and 6.1.2)
• heliosphere: the extended region where the solar wind dominates over the interstellar medium (e.g., Fig. 5.1)
• ionosphere: the ionized component of a planetary atmosphere, largely overlapping with the thermosphere (e.g., Sect. 2.3)
• magnetosphere: a magnetic environment, generally of a planet, in which the intrinsic or induced magnetic field of the central body dominates over external fields or flows (e.g., Fig. 5.1)
• mesosphere: layer between stratosphere and thermosphere (Sect. 2.2) • photosphere: ’surface’ of a star, at the rapid transition from opaque to transparent • stratosphere: at Earth, the domain between troposphere and mesosphere where
temperature rises with height and convection is rare (e.g., Sect. 2.2) • solar (or sunspot or activity) cycle: quasi-cyclic variation in the number of sunspots
seen on the solar surface when averaged over time scales of months (e.g., Fig. 4.5) • (spectral, total) solar irradiance: solar input into a planetary atmospheric system in the
form of photons (e.g., Sect. 2.2) • sunspot: a ’flux tube’ in the near-surface layers, with suppressed internal convection,
and large enough that lateral influx of radiation cannot prevent the interior from cooling relative to the surrounding photosphere and thereby appearing relatively dark (e.g., Sect. 4.1.2) • thermosphere: outer layers of a planetary atmosphere in which the temperature increases with height (e.g., Sect. 2.2), specifically the neutral particles • (solar) transition region: a domain between chromosphere and corona with a very strong temperature gradient dominated by conduction (see Sect. 9.2) • troposphere: the lower layers of a planetary atmosphere (e.g., Sect. 2.2)

Our Sun has and does all of these things, and we refer to the sphere of influence carved out by the solar wind as our heliosphere. It is not really spherical and it varies in extent with solar activity. But in broad terms we may safely think of it as extending about 100 times further from the Sun than the Earth’s orbit. We have yet to agree on the name for such spheres of influence around the other stars (for which astrospheres has been proposed), but there can be little doubt that such environments are as commonplace as the many points of light we see strewn across the sky on a dark
https://hesperia.gsfc.nasa.gov/sftheory/glossary.htm; for a glossary of terms related to space weather, see https://www.swpc.noaa.gov/content/space-weather-glossary.

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I:2.2

and cloudless night.”
“Heliophysics encompasses the study of the various physical processes that take place within the sphere of influence of the Sun (i.e., the heliosphere), and by analogy, those environments surrounding most other typical stars. But heliophysics also defines a specific method of study. This method embraces a holistic connected-system approach. It emphasizes a comparative context in which to understand a process by the many facets it presents in its various incarnations throughout the heliosphere. Taken together, each diverse facet serves to fill out a complete and physically satisfying picture of a given process or phenomenon.
The physical processes and phenomena that we will encounter in [this volume] are themselves especially diverse. They include the rapid and efficient energization of thermal particles to suprathermal energies, the generation and annihilation of magnetic field, stellar variability and activity cycles, space weather, turbulent transport of energy and momentum, [the coupling between ionized and neutral atmospheres, and atmospheric chemistry,] to name just a few. Heliophysics fills a critical need to establish a unified science that connects these seemingly unrelated concepts in a manner that emphasizes complementarity over individuality, function over form, and generality over specificity.
Along with unification, coupling provides the second principal pillar upon which heliophysics rests. The heliosphere is a collection of coupled systems. It is fortunate that many of the linkages essentially operate only in one direction. That is to say, system A impacts system B, but B has little influence on A. Under these circumstances it is expedient to treat system A independent of the behavior of system B. This provides a certain economy of effort and scale, and it often reduces the (apparent!) complexity of a problem. For example, complex geomagnetic activity has no impact on solar flares, and the solar wind does not influence the Sun’s cyclic variability.
Linkages, especially when several are present and working at cross purposes, can lead to confusion and spirited debate over what is a root cause and what is simply a resulting effect. The cause and effect relationship between solar flares and coronal mass ejections is a good case in point. Consider, for example, what the purported cause and effect relationship might be between a sore throat and a fever. Because a sore throat often starts before a fever develops one might be tempted to assign the effect to the fever and take the sore throat to be the cause. Fortunately, medical research informs us that both are effects and the root cause is the influenza virus. Heliophysics is needed to play this very same role in sorting out the appropriate relationships (or lack thereof) between any variety of physical effects that often occur contemporaneously throughout the heliosphere.
Solar variability does influence our climate here on Earth. This fact is certainly not negotiable in a purely scientific context and is arguably one of the most important linkages between the Sun and the Earth. Satellites have confirmed that the solar irradiance is variable on time scales from minutes to decades. The fluctuations are greatest on the shortest time scales. Day-to-day irradiance changes are on the order of a percent versus tenths of a percent over a solar cycle. The magnitude and sense of

The language of heliophysics

7

irradiance trends over centuries and millennia are currently difficult to determine with any measure of certainty. Slow but steady progress on this question is being made through the studies of paleoclimate records. Over much longer time scales, stellar evolution theory provides assurances that significant changes in solar irradiance have taken, and will take, place with dramatic impacts on our climate and way of life.
What is debatable, however, is precisely what the direct relationship is between solar variability and climate change over any particular time scale, or epoch, of interest. For example, various opinions have been advanced that span the entire gamut from wholly inconsequential to complete solar responsibility for the gradual warming of the planet that has been observed since the middle of the 20th Century. Yet, it may not even make sense to speak of direct relationships between drivers and the behavior of systems which are as nonlocal, nonlinear and plagued by various hystereses as is our climate here on Earth.
The third and final pillar upon which heliophysics rests is the exploration of Earth’s neighborhood in space. As a space-faring civilization we have visited all the planets, [several asteroids and] comets and numerous planetary satellites. We have ventured to the boundaries of the heliosphere and have flown through various parts of our magnetosphere. We have a spacecraft [that passed] the Pluto/Charon system [and after that flew by Kuiper-belt object 486958 Arrokoth, provisionally known as 2014 MU69 and nicknamed Ultima Thule]. Heliophysics enables our exploration to be successful and at the same time gains in knowledge and understanding from our exploration initiatives.
In summary, heliophysics is the systems-mediated study of the physical processes that take place within the Sun’s sphere of influence. It is based upon the three pillars of unification of physical processes and phenomena, coupling of distinct physical systems, and the exploration of our neighborhood in space. And it is broadly applicable to the environments around most ordinary stars.”
1.4 The language of heliophysics
“The language of heliophysics is mathematics. And the body of literature from which heliophysics draws its substance and in turn records its accomplishments is the physics of magnetized plasmas. With only a rudimentary knowledge of a language, a literature is incomprehensible, except, perhaps in translation. And even in translation so much of the original meaning and the nuance the author wished to convey are inevitably lost, or worse, misinterpreted by even the most conscientious translator.
The most precise, and intellectually demanding, literary prose of heliophysics assigns a phase-space distribution function to each individual species of particle. By a species one may simply mean free electrons, protons, or oxygen molecules, or even photons. In some applications it might be necessary to distinguish between oxygen molecules in different excited (vibrational, rotational and electronic) states, or between iron atoms at different stages of ionization, or between different senses of photon polarization. In any case, the evolution of each distribution function is obtained by setting the total time derivative equal to the net production/loss of an individual species by various

I:2.3

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{A5}

collisional or radiative processes. Such evolution equations are commonly referred to as Boltzmann, or Vlasov equations. When there is no net gain or loss, then Liouville’s Theorem asserts that the vanishing of the total time derivative of the distribution function conserves the phase space density for each species [(see Sect. 8.4)].
In specifying the total time derivative it is necessary to determine the forces acting upon a given particle species. For uncharged particles, gravitational attraction is the only important consideration. Accordingly, to the system of equations for the individual distribution functions one must add Poisson’s equation in order to specify the gravitational field based on the mass distribution provided by those particles with mass. Charged particles are also subject to electromagnetic interactions. Thus we must also include Maxwell’s equations to deduce the electric and magnetic fields based on the distribution of charges and currents provided by the charged particle species.
{A5}
In principle, this suffices to provide a complete description of the grammar and syntax of heliophysics at a very elegant, learned and precise level. In practice the task of following through with this program (a) is prohibitively difficult with or without the assistance of the computer, (b) is subject to the problem that the initial conditions are not known with any degree of certainty, (c) is complicated by the fact that many of the collisional and radiative transition probabilities are not even approximately known, and (d) requires that certain conditions be fulfilled so that electromagnetic interactions can be separated into large-scale fields and small-scale collisions. Finally, this comprehensive description usually provides far more information than is usually necessary for comparing with observations or understanding the predictions of a theory over specified temporal and spatial scales.
At the opposite extreme from the scholarly literary prose is the common vernacular. For heliophysics, if high literary prose centers on Poisson, Maxwell, Boltzmann and Vlasov, then the vernacular is single-fluid, ideal, magnetohydrodynamics, or MHD for short (see Ch. 3). MHD is a continuum fluid description that does not distinguish between particle species, averages (in some sense) over particle collisions, ignores radiative effects altogether, and is based on velocity moments of the underlying distribution functions. It retains Poisson without modification, but takes certain liberties with Maxwell. Boltzmann and Vlasov drop out of the picture entirely.
MHD can be rigorously derived from Poisson/Maxwell/Boltzmann/Vlasov under various conditions that are not altogether unreasonable for very many heliophysical applications. Usually this involves following the behavior of a physical process or phenomenon over course-grained spatial and temporal scales. In other words, it is a useful, and indeed often very accurate, description of the ’big picture’. Because of its relative simplicity, ideal MHD provides a useful context in which to interpret and understand the behavior of magnetized plasmas at a basic and often extremely intuitive level. On the other hand, ideal MHD is often applied to processes or phenomena to which it does not actually apply. Generally speaking, if collisional and radiative relaxation times are short compared to the coarse-grained time scale of interest then ideal MHD is likely to be a reasonable option. But ’gotchas’ are always present.

The language of heliophysics

9

The successful derivation of the MHD equations requires a closure prescription, which may be regarded as a consequence of the familiar, ’no free lunch’ maxim. Closure entails specifying a tractable procedure to determine the pressure tensor (second-order velocity moment) in terms of the fluid density (zeroth-order velocity moment), the bulk fluid velocity (first-order velocity moment), and the magnetic field. The so-called polytropic approximation — in which the pressure is a scalar proportional to the particle number density raised to a specified power — is the simplest option. A power law index of unity corresponds to an isothermal process (constant temperature). A power law index equal to the ratio of specific heats describes an isentropic (constant specific entropy) process that also manages to conserve energy. More complicated options are possible and are often tailored to accommodate specific situations. A successful and accurate closure scheme is inevitably based on some additional a priori knowledge of the behavior of the particle trajectories, or the general nature of the particle distribution functions.
In contrast to the Poisson-Maxwell-Boltzmann-Vlasov description, ideal MHD is a system of nine partial differential equations for nine dependent variables [(shown in Table 3.3 and discussed in Ch. 3)]: the gravitational potential, the fluid density and pressure, the fluid velocity (3 components) and the magnetic field (3 components). These equations are (a) the Poisson equation to describe gravity, (b) the continuity equation expressing the conservation of mass, (c) the closure relation to specify the pressure tensor, (d) the equation for the conservation of momentum, or the force-balance equation (3 components), and (e) the magnetic induction equation (3 components).
Of course, between ideal MHD and the Poisson-Maxwell-Boltzmann-Vlasov description lies a vast real estate filled with a plethora of compromise or hybrid descriptions. The number of such schemes is limited only by the imagination and ingenuity of the investigators. Multi-fluid treatments allow for individual densities, velocities and pressures associated with different particle species or groupings of particle species, but retain a single gravitational potential and magnetic field applicable to every fluid. This formulation is useful when the time scales of interest are short compared to characteristic inter-species collisional relaxation times, but long compared to the analogous intra-species times.
Another intermediate scheme employs high-order moment closures. These schemes are necessary when the species distribution functions deviate significantly from the fully-relaxed Maxwellian. Often this situation occurs when significant spatial gradients are imposed on the system. Additional partial differential equations are then used to describe the time-evolution of the components of the pressure tensor. The closure is postponed to the next higher level of the heat flux tensor (third-order velocity moment), or in extreme circumstances to even higher-order moments.
Hybrid schemes treat some species as fluids and retain a Boltzmann-Vlasov – or kinetic – description for others. Indeed even a single species of particle may be partitioned in such a fashion that some of the particles are treated kinetically (generally the high energy suprathermal tail of the distribution function) while the remainder are described as a fluid (the thermal core of the distribution). Such schemes are

10

1 – Stars, planetary systems, and the local cosmos

particularly useful in describing the energization of charged particles; [we see this in action in Ch. 8].
In summary, there is a bewildering array of schemes that are presently invoked to describe the behavior of magnetized plasmas in the heliosphere. They encompass an extremely wide range of complexity. Each is specifically tailored to a given physical process and phenomenon. They are not simply interchangeable, but have their own individual strengths and weaknesses. One should always choose the simplest description that will suffice for understanding the problem in hand. Use all the information and knowledge you have at your disposal about the nature and behavior of a physical system in selecting a scheme. If the heliophysics concepts can be adequately framed in the common vernacular, then eschew the sophisticated flowery prose unless nothing less will do.”
1.5 A timeline of exploration of planetary systems
NASA’s Heliophysics Division within the Science Mission Directorate was previously known as the Sun-Earth Connections Division. That earlier name reflected that much of its research focused on how solar activity impacts our home planet. As probes explored ever more of the solar system, researchers realized that learning about the science of terrestrial space weather and of the evolution of Earth’s climate system was boosted by the incorporation of discoveries from around the solar system; the name change of the Division reflected the shift to a broader perspective that was already taking place in the research community. As exoplanets were found to be more common than stars, the application of the science of heliophysics to the exploration and understanding of processes in exoplanetary systems, and in particular to exoplanetary habitability, presents a natural development of the discipline. The multi-disciplinary science arena that looks into star-planet couplings has accelerated rapidly alongside astronomical exploration.
In 1969, half a century ago, astronauts first landed on Earth’s sole moon. The first successful robotic landers touched down on the much more distant Venus and Mars in 1970 and 1976, respectively, and in the same decade spacecraft flybys provided the first, fleeting close-ups of Jupiter and Saturn. It was not until two decades later, however, that missions that explicitly targeted these giant planets revealed how fundamentally distinct these worlds are from our own.
The Galileo satellite started exploring the Jupiter system in late 1995, swinging by moon after moon. The Cassini-Huygens mission reached Saturn in 2004, exploring the giant planet, its rings and satellites, and even sending a lander onto Titan, the only moon in the solar system with a substantial atmosphere. These spacecraft uncovered a fascinating diversity of environments on dozens of moons: many are cold worlds enrobed in miles-thick ice; some with volcanoes spewing molten rock but others whose volcanoes somehow gush liquid water or nitrogen; and then there is Titan with its seas of liquid methane and ethane. Their pictures are as stunning and diverse as the scientific discoveries enabled by these spacecraft. The far reaches of the Solar System continue to offer surprises: dwarf planets Haumea and Makemake, objects in

A timeline of exploration of planetary systems

11

the distant Kuiper belt, were not discovered until 2004 and 2005, respectively. As the close-up exploration of the largest planets in the solar system got underway,
a revolution was about to befall astronomers looking much further out. It started in 1995 with the announcement of the first exoplanet, now known as 51 Pegasi b, orbiting a star like our own Sun. There are now well over 5,000 exoplanets on the books [5] (almost half of which were found with NASA’s Kepler satellite), but the number expected to exist is vastly larger: by carefully quantifying what our available methods can and cannot observe, scientists estimate that there are over a hundred billion planetary systems in our Milky Way galaxy alone, with perhaps of order ten billion planets with some similarity to Earth.
Apart from its very existence, 51 Pegasi b had another surprise in store: at 150 Earth masses and orbiting its star almost 20 times closer than Earth does the Sun, this “hot Jupiter” should not have existed by theories of the time. These and many subsequent observations have changed our ideas on how planetary systems form and evolve: we now realize that orbits can change so that planets may be discovered well away from where they formed; planets can engage in gravitational fights that can cause losers to be ejected as lone ’nomads’ into interstellar space; planets exist that have two stars to cast twin shadows on their surfaces; . . . Many planets orbit their stars at distances where water, if there is any, may exist in liquid form on their surface for billions of years, as on Earth where it enabled the development of life.
These discoveries have intensified the astronomers’ hunt for extraterrestrial life in which also solar-system scientists participate. Organic molecules cause the haze in the icy-cold atmosphere of Saturn’s Titan and are vented in cryo-volcanic plumes rising from the ice-locked deep ocean of nearby Enceladus. There are many sizable moons and dwarf planets in the solar system that are rich in water, although much of it is frozen solid. The combination of liquids and organics in many places around our solar system fuels theories of life and plans for space missions designed to look for it near to home.
But exoplanet astronomers have the advantage of the vast number of systems. Their challenge is that even the largest telescopes can image exoplanets no better than as an unresolved blur the size of the instrumental point spread function, if indeed they can separate the reflected light from the exoplanets from the light of the stars that they orbit. In fact, most of what we learn about exoplanets comes from analyzing how their star’s light is modified in brightness or color by the exoplanets, either by adding some reflected starlight or by taking away some light should they move in front of their star during their orbit. Careful study of these effects as observed with the most powerful telescopes can reveal which gases contribute to the changes. This is receiving a big boost from NASA’s James Webb Space Telescope that started operations in 2022 and more from future telescopes. So much was discovered in the most recent few decades; what will the next several decades bring?

5 See https://exoplanetarchive.ipac.caltech.edu and http://exoplanet.eu.

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Chapter 2
Neutrals, ions, and photons
Chapter topics:
• Conditions in the local cosmos • Gravitational stratification • Cycle-driven variability of the solar spectral irradiance • Penetration depth of sunlight and its impact on the terrestrial atmosphere
Key concepts:
• Pressure scale height and differential stratification by atomic/molecular mass • The role of electron heat conduction in powering the solar wind • Collisional mean-free path • Optical depth
2.1 Conditions in the local cosmos
The local cosmos discussed in this book exhibits an enormous diversity of conditions. Figure 2.1 is one perspective of this in its comparison of number densities and temperatures: densities range over more than 28 orders of magnitude (more than the contrast between solid rock and the ’vacuum’ of low-Earth orbit) and temperatures over 5 orders of magnitude. The magnetic field, another crucial parameter that is explored starting in Ch. 3, provides another dimension and adds its own physical processes. All together, these physical parameters cover a wide range of states that include solids, liquids, gases, and ionized and magnetized particle ensembles called plasmas.
Matter in most of the domain of heliophysics is electrically conducting, being generally at least partially or even fully ionized as will be abundantly clear from the chapters in this volume. Ionization can be a consequence of high-speed collisions between particles in a hot medium and/or of high energies in the thermal radiation associated with high temperatures. A hot medium can result from the transport and conversion of different forms of energy where a balance of thermal sources and sinks may only be reached at high temperatures. Examples of such settings are the interior and the atmospheric domains of the Sun. In these environments, internal collisional
13

14

2 – Neutrals, ions, and photons

Figure 2.1: Temperature versus mass density for a variety of conditions within the local cosmos. Some typical ranges are indicated, and labeled with magnetic field strengths (in Gauss) found in that domain, followed by estimated ranges of the plasma β, i.e., the ratio of energy density in plasma over that in the magnetic field (Eq. 3.24), in this scaling for a fully ionized hydrogen-dominated plasma. [Dash-dotted lines show where the plasma β equals unity for the field strengths shown near the top of the diagram, making the same assumptions about the plasma. Fig. I:1.1]

Conditions in the local cosmos

15

Table 2.1: Present characteristics and climates of the terrestrial planets. [Modified after Table III:7.1, with added surface gravity, escape velocity, and escape energies Eesc for protons and atomic oxygen.]

Venus

Earth

Mars

Radius Orbital radius

6050 km 0.72 AU∗

6400 km 1 AU

3400 km 1.52 AU

Rotation period Surface gravity

243 days 8.9 m/s2

24 hours 9.8 m/s2

24.6 hours 3.7 m/s2

Escape velocity Eesc for H+, O Surface temp.

10 km/s 0.5, 9 eV
740 K

11 km/s 0.6, 10 eV
288 K

5 km/s 0.1, 2 eV
210 K

Surface pressure

92 bar

1 bar

7 mbar

Composition
H2O content Precipitation

96% CO2 3.5% N2 20 ppm None at surface

78% N2 21% O2 10,000 ppm Rain, frost, snow

95% CO2 2.7% N2 210 ppm
Frost

Circulation

1 cell/hemisph.; 3 cells/hemisphere; 1 cell/hemisphere

quiet at surface local and regional

or patchy

but very active

storms

circulation; global

aloft

dust storms

Maximum

surface wind

∼3 m/s

>100 m/s

∼ 30 m/s

Seasons

None

Comparable northern Southern summer

and southern seasons more extreme

∗ An AU, or Astronomical Unit, is the average distance between Sun and Earth.

{A6}

ionization and recombination, as well as excitation and de-excitation processes dominate in balancing ionization and recombination rates. Alternatively, ionization can be the result of impacts of externally-generated high-energy particles (such as solar energetic particles or particles accelerated in a planetary magnetosphere) or be caused by irradiation by solar photons of sufficiently high energy (typically X-ray and [extreme] ultraviolet) such as occurs in planetary ionospheres and cometary tails. {A6}
Much of what is described in this volume deals with the physics of magnetized plasmas, and much of that physics is approximated by a description known as magnetohydrodynamics, or MHD, as introduced in Ch. 3. In the present chapter, however, we first look at the more familiar situation of neutral gases, also because many of the phenomena discussed in this volume occur in the layers of planetary atmospheres for which the concept of hydrodynamics – in which magnetic field is ignored – gives us a good starting point. Later in this chapter, we focus on where ionization becomes important. For now disregarding the effects of magnetic fields, the limits of pure (’non-magneto-’) hydrodynamics are reached in the high tenuous layers of planetary atmospheres where collisions are infrequent and other processes enter into our discussion, such as chemical differentiation subject to gravity or even outflows from the body in question.

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Figure 2.2: Average vertical temperature profile through Earth’s atmosphere. The general shape of the temperature profile is reasonably consistent to the point where it can be used to define the four main neutral atmosphere ’layers’, from the troposphere to the thermosphere. The temperature of the uppermost layer, the thermosphere, increases steeply with altitude due to absorption of solar extreme ultraviolet (EUV) and far-ultraviolet (FUV) radiation. The thermosphere and upper mesosphere are partially ionized by the same EUV radiation, which varies by a factor of three over the solar cycle, and by auroral particle precipitation. [The effect of absorption by ozone is specifically highlighted. Fig. I:12.1] For a color version of this figure, see arXiv:2001.01093.
A great variety of phenomena in the local cosmos have their foundation in the electrical conductivity of the media within which they occur. This may be in the generation and maintenance of magnetic field deep inside the Sun and in most of the planets, in the many phenomena driven by the interaction of the magnetized flow of the solar wind with Solar-System bodies, or even in the processes in the ionized domains of atmospheres of many of these bodies. In most situations discussed in this volume, that conductivity has its origin not in the metallic behavior of the medium as it does deep inside Earth but rather in the ionization of matter: whereas in metals ions are relatively immobile and share some of their electrons, in a plasma both the ions and the electrons are entirely unbound on microscopic scales. This chapter introduces electrical conductivity in a magnetized medium, here looking at plasma with a low degree of ionization; fully ionized plasmas are discussed in Ch. 3.
2.2 Gravitationally stratified atmospheres and stellar winds
Among the planetary atmospheres in the Solar System, those of Venus and Mars are most similar to those of Earth. The abundances of their primary constituents – mostly CO2 and N2, and, on Earth, N2 and O2 – are compared in Table 2.1. Note that the order of the most abundant components as well as the absolute base pressures differ markedly.
A sketch of the Earth’s atmospheric vertical thermal structure is shown in Fig. 2.2. The temperature gradually drops from the surface – where the bulk of the conversion of solar irradiance into heat occurs – through the troposphere due to adiabatic expansion.

Gravitationally stratified atmospheres and stellar winds

17

Table 2.2: Extent and important species for upper atmospheric regions of the terrestrial planets. [Table IV:7.3; added planetary radii Rp (km). More detailed information is provided in the text and figures of Ch. 13.]

Thermosphere Ionosphere Exosphere

Venus
Rp,♀ = 6052 ∼120-250 km
CO2, CO, O, N2 ∼150-300 km O+2 , O+, H+ ∼250-8,000 km
H

Earth Rp,⊕ = 6378 ∼85-500 km O2, He, N2 ∼75-1,000 km NO+, O+, H+ ∼500-10,000 km H, (He, CO2, O)

Mars
Rp,♂ = 3396 ∼80-200 km
CO2, N2, CO ∼80-450 km O+2 , O+, H+ ∼200-30,000 km
H, (O)

At greater altitudes the absorption of short-wavelength sunlight by tenuous gas that is less efficient in cooling through radiation leads to increased temperatures in the stratosphere (mainly by photons between about 2,000 ˚A and 3,000 ˚A) and in the thermosphere (for wavelengths mostly short-ward of 2,000 ˚A). Energy leaves the Earth’s atmospheric domains mainly by infrared radiation from the lower regions, which also leads to a decrease in temperature above the stratosphere by radiation from the mesosphere. The densities in the thermosphere are so low, and the dominant chemical constituents such inefficient radiators, that downward thermal conduction exceeds radiative losses above about 100 km (see Ch. IV:9). Table 2.2 compares the properties of the upper atmospheres of the three terrestrial planets (with significant atmospheres), i.e., the thermospheres, the ionized constituents referred to as the ionospheres that largely overlap with the thermospheres, and the exospheres beyond that; the reasons for the apparent chemical mismatch between the neutral molecular and the ionized components are discussed in Ch. 13. [6]
For the Sun’s atmosphere, there is a comparable pattern of temperature with height: moving upward, the temperature drops throughout the lower atmosphere (the ’photosphere’ from which the bulk of the solar irradiance is emitted; also referred to as the ’solar surface’ by astronomers, despite the fact that the Sun is entirely gaseous throughout), but then increases again in the chromosphere (extending a few thousand km above the photosphere) and then shoots up to form an extended, hot corona. Some of the physical properties of these domains (along with a rough definition of the terms) are summarized in Table 2.3. The reasons behind this similarity in pattern are partly the same, partly completely different. A similarity is that energy is most efficiently radiated from the low, dense atmospheric layers, and poorly from high, tenuous layers where conductive redistribution plays an important role. But the heat input differentiates the two: the solar chromosphere and corona are not heated by absorption of photons from the solar surface (which is thermodynamically impossible because the atmospheric temperature is higher than the surface temperature) but by
6 This volume focuses on terrestrial planets; we refer to Ch. IV:8 for an introduction to the upper atmospheres of the giant planets.

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Table 2.3: Basic parameters for, and definitions of, domains in the solar atmosphere. Note that all regions of the solar atmosphere are very inhomogeneous and that these values are only meant to give a rough idea of their magnitudes. [Table I:8.1, here converted to cgs-Gaussian units, and with solar properties added. ne and nH are the densities of electron and neutral hydrogen; the plasma β is defined in Eq. (3.24)]

Region

n ne/nH

T

B

β

[cm−3]

[K]

[Gauss]

Photosphere1

1017

Chromosphere2

1013

Transition region3 109

Corona4

108

10−4 10−3
1 1

6 103 2 104 − 104 104 − 106
106

1 − 1500 10 − 100
1 − 10 1 − 10

> 10
10 − 0.1 10−2
10−2 − 1

Sun: radius R⊙ = 7 105 km; surface gravity g⊙ = 274 m/s2; bolometric luminosity Lbol = 4 1033 erg/s; effective temperature Teff = 5772 K, defined such that Lbol ≡ σTe4ff 4πR⊙2

Definitions: 1 the photosphere is the layer from which the bulk of the electromagnetic radiation leaves the Sun (this layer has an optical thickness τν <∼ 1 in the near-UV, visible, and near-IR spectral continua, but it is optically thick in all but the weakest spectral lines); 2 the chromosphere
is optically thin in the near-UV, visible, and near-IR continua, but optically thick in strong spectral lines - it is often associated with temperatures around 10, 000−20, 000 K; 3 the transition
region is a thermal domain between chromosphere and corona in which thermal conduction leads to a steep temperature gradient; 4 the corona is optically very thin over the entire electromagnetic
spectrum except at radio wavelengths and in a few spectral lines - the term is often used to
describe the solar outer atmosphere out to a few solar radii with temperatures exceeding ∼ 1 MK.

dissipation of electrical currents and a variety of waves running through the plasma (both generated by the convective flows below the solar surface, and coupled into the outer atmosphere via the Sun’s magnetic field; see Sect. 9.3). The amount of energy converted in the solar outer atmosphere from chromosphere to corona and solar wind is a function of the instantaneous magnetic activity. This activity exhibits an 11-year quasi-cyclic pattern that is often referred to as ’the sunspot cycle’ because it was discovered from multi-decade records of sunspot counts.
The Sun’s radiative input into the Earth’s atmosphere (known as the spectral irradiance, S(λ)) exhibits a significant variability depending on solar magnetic activity (Figure 2.3). The overall emission from the solar photosphere varies little with magnetic activity, that from the warm chromosphere mildly, and that from the hot corona strongly. {A7} As a result, the relative variability in S(λ) through the solar cycle increases markedly short-ward of about 3,000 ˚A: (Smax − Smin)/Smin climbs from below one part in 1,000 long-ward of 3,000 ˚A to near unity short-ward of 1,000 ˚A. The absorption of the most variable segment of the spectral irradiance in Earth’s atmosphere occurs primarily above about 50 − 100 km (Fig. 2.4), causing the high atmosphere to evolve strongly in temperature and density in response to the solar sunspot cycle (see Fig. 2.6),

{A7}

Gravitationally stratified atmospheres and stellar winds

19

Figure 2.3: Comparison of the solar spectrum and the black body spectrum for radiation at 5770 K (the approximate temperature of the Sun’s visible surface). Also shown is an estimate of the variability of the solar spectrum during the 11-y solar cycle, inferred from measurements (at wavelength below 4000 ˚A) and models (at longer wavelengths) and, for reference (dashed line), the solar cycle 0.1% change in the total solar irradiance. [Fig. III:10.1]
Figure 2.4: [Altitude of penetration of the solar radiation as a function of wavelength [from Xrays through 3600 ˚A]. The color range shows the amount of energy deposited in the different layers of the atmosphere for the different parts of the solar spectrum (on a logarithmic scale, in units of mW/m3/nm [or 10−3 erg/s/cm3/˚A]). [Fig. III:13.3] For a color version of this figure, see arXiv:2001.01093.

20

2 – Neutrals, ions, and photons

I:12.2.1

further modulated as Earth goes through its weakly elliptical orbit around the Sun and its rotation about a tilted axis, and with variable contributions from geomagnetic activity (see Chs. 12 and 13).
Wavelengths short-ward of about 2400 ˚A and about 1250 ˚A can dissociate O2 and N2, respectively, and short-ward of about 900 ˚A can ionize, e.g., O atoms. Consequently, the atomic and ionic components in the Earth’s atmosphere do not show up significantly below around 100 km in altitude because all ionizing and dissociating wavelengths have been absorbed by that depth into the atmosphere (see Ch. 13); above that altitude, the abundances of the ionic and atomic components all reflect solar, orbital, and diurnal cycles.
The density stratification in much of the lower atmosphere of the terrestrial planets (defined as below about 100 km for Earth) can be understood to first order by looking at the behavior of a stationary gas subject only to gravity. “The frequent collisions of molecules in a gas close to thermal equilibrium enable the Maxwellian [velocity distribution (with corresponding exponential energy distribution)] of the individual particles to be characterized by the basic fluid properties of pressure, p, temperature, T , number density, n, and mass density, ρ, that are related by the perfect gas law:

p = nkT = (ρ/m)kT = ρRT /µ,

(2.1)

where k [(1.4 × 10−16 erg/K)] and R [(8.31 × 107 erg/K/mol)] are the Boltzmann and universal gas constants, respectively, and m is the mean molecular mass [while µ the mean molecular mass in atomic units]. {A8} {A9} The fluid concept of pressure in the atmosphere represents the weight of the column of gas above.
The neutral gas under the influence of the planet’s gravitational force gives rise to the concept of hydrostatic balance, which states that the change in pressure with height, dp, is closely balanced by the weight of the fluid, nmgdh (where m is the mean molecular mass in [grams] and h is the height), under the action of the planet’s gravitational acceleration, g. The concept is expressed mathematically as:

dp dh = −ρg = −p/Hp.

(2.2)

This basic equation describes the exponential decrease in gas density with altitude, and results in the concept of the pressure scale height,

Hp = kT /mg,

(2.3)

which represents the [height difference] through which the gas pressure [in an isothermal atmosphere] will decrease by a factor of e = exp (1). Earth’s upper atmosphere extends for about a dozen scale heights above 100 km altitude, with scale heights changing from about 5 km to 50 km with increasing altitude, as the temperature increases from about 180 K to over 1000 K (see Fig. 2.2). {A10}
[The] quasi-equilibrium implied by hydrostatic balance does not exclude the possibility of vertical winds. The assumption simply demands that the rate of [flow] is such that the atmosphere adjusts at a comparable rate. The term quasi-hydrostatic

{A8} {A9}

{A10}

Gravitationally stratified atmospheres and stellar winds

21

{A11} {A12}

Figure 2.5: Composite image of solar coronal loops over an active region (observed in the SDO/AIA 193 ˚A channel, most sensitive to coronal temperatures around 1.5 MK). The foreground image that masks the on-disk corona is an SDO/HMI intensity image showing a large sunspot group approaching the limb from which most of the coronal loops in this composite image emanate. The dark band between the solar limb in visible light and the overlying corona is the chromosphere which is itself EUV-dark and opaque to coronal emissions, thus blocking the EUV emission from the corona behind it, showing up partially masked by foreground coronal emission. The images were taken on 2022/09/04 around 20 UT, and were rotated by 115◦ counterclockwise from solar north.

balance is the more correct expression in the case of accommodating vertical winds in

the system. [. . . ] Vertical winds in Earth’s upper atmosphere of the order of 100 m/s

can be accommodated within the quasi-hydrostatic assumption.” {A11}

{A12}

The quasi-hydrostatic description applies not only to planetary atmospheres but is also used for the interior of the gas giants, for the interior and lower-atmosphere of the Sun, and – as we shall see later – even inside magnetic ’containers’ in the solar atmosphere that are known as flux tubes (see Table 3.1 for a definition), one incarnation of which are ’coronal loops’ – which is the general term describing the emitting structures seen in EUV and X-ray images of the Sun’s hot outer atmosphere. Table 2.3 summarizes characteristic physical parameters for the domains within the solar atmosphere from the solar ’surface’ (photosphere) up into the corona (see also Fig. 2.5 for an example image showing coronal loops above a sunspot region). These numbers should be seen as characteristic values only: all these domains span a few orders of magnitude in density and all are very dynamic at any given location, while moreover the solar magnetic field plays a key role in them as it structures multitudes of adjacent distinct atmospheres along magnetic field bundles (Sect. 3.4). The solar corona is visible at X-ray and EUV wavelengths up to several hundred thousand kilometers. The coronal plasma is mostly contained in magnetic structures relatively low down, but increasingly with height the gas pressure forces the magnetic field to ’open’ into the heliosphere. The plasma on ’open field’ streams out to form the solar wind, resulting in a low-density and consequently dark lower coronal region known as

22

2 – Neutrals, ions, and photons

Table 2.4: Basic parameters of the fast and slow solar wind [near Earth; modified after Table I:9.1. Notes: (1) subscripts ’e’, ’p’, and ’i’ are used to denote electrons, protons, and other ions, respectively; (2) ’FIP’ stands for ’first ionization potential’; ’low-FIP’ is a group of elements with first-ionization potentials below 10 eV.]

Property (1 AU) Speed Ion density Flux Magnetic field Temperatures
Anisotropies Structure Composition
Minor species
Associated with

Slow wind
430 ± 100 km/s ≃ 10 cm−3
(3.5 ± 2.5) × 108 cm−2s−1
60 ± 30 µG Tp = (4 ± 2) × 104 K Te = (1.3 ± 0.5) × 105 K > Tp
Tp isotropic filamentary, highly variable
He/H≃ 1 − 30%
low-FIP enhanced
ni/np variable Ti ≃ Tp vi ≃ vp
streamers, transiently
open field

Fast wind
700 − 900 km/s ≃ 3 cm−3
(2 ± 0.5) × 108 cm−2s−1
60 ± 30 µG Tp = (2.4 ± 0.6) × 105 K Te = (1 ± 0.2) × 105 K < Tp
Tp⊥ > Tp∥ uniform, slow changes
He/H≃ 5%
near-photospheric
ni/np constant Ti ≃ (mi/mp)Tp
vi ≃ vp + vA coronal holes

a ’coronal hole’. The quasi-hydrostatic description even forms a useful, albeit very crude, approxi-
mation for that part of extended atmosphere of the Sun that is the inner-heliospheric domain of the solar wind: whereas there is in fact an outflow, this ’vertical wind’ leaves the stratification nearly hydrostatic for many solar radii above the solar surface, as we shall see shortly.
Table 2.4 summarizes a few characteristics of the solar wind near Earth orbit. Outside of dynamic coronal mass ejections (Ch. 6), the solar wind is predominantly in one of two states, referred to as the ’fast wind’ and the ’slow wind’. These states originate from distinct environments on the Sun, and because the Sun rotates underneath the radially outflowing wind, slow and fast streams unavoidably interact – see Sect. 5.5.1. For what follows here, we focus on domains where only one of these types of wind prevail for several days, which is the time it takes to flow from Sun to Earth (the geometry of the magnetic field that it carries is discussed in Sect. 5.4). {ⓈA13}
The medium of the heliosphere is fundamentally distinct from that of the lower 100 km of the terrestrial atmosphere: the solar wind is primarily made up of hydrogen with a lesser amount of helium, is hot and therefore almost fully ionized, and is threaded by a magnetic field. The dynamics of the solar wind and the ways in which it interacts with planetary magnetospheres is modulated by that magnetic field but, as first noted by Parker (1958), the basic stratification and flow of the solar wind can be understood

{ⓈA13}

Gravitationally stratified atmospheres and stellar winds

23

I:9

from its high temperature: because it is hot and ionized, the electrons in the solar wind are very efficient at conducting heat, and that is all it takes to understand how it can lead unavoidably to a fast wind that can escape solar gravity. It is not simply an ’evaporation’ off the Sun; after all, even at some millions of degrees, “the sound speed cs — essentially the mean ion speed — is much smaller than the [escape speed vesc which can be derived by equating a particle’s kinetic energy with its gravitational potential energy at the surface:

vesc = 2GM/r.

(2.4)

For the solar corona, the sound and escape speeds are]

cs ≈ kT /m ≈ 100 km/s ≪ vesc = 2GM⊙/R⊙ = 618 km/s,

(2.5)

where k is Boltzmann’s constant, T the coronal temperature, m the mean particle mass, G the universal gravitational constant, and M⊙ and R⊙ the solar mass and radius, respectively.
Mass and momentum balance radially away from the Sun [in an assumed uniform, strictly radial flow] at heliocentric distance r can be written

d (ρv4πr2) = 0 dr

dv ρv
dr

=

dp −
dr

−

ρ

GM⊙ r2

,

(2.6) (2.7)

with ρ the mass density, v the flow speed. Then p = 2nkT is the gas pressure in an electron–proton plasma with n representing the electron or proton number density, and ρ = mn where m is the mean particle mass which is given by m ≈ mp/2 for an electron–proton plasma. {A14}
[The] consequence of the thermal conduction in a million degree corona is to extend the corona; i.e., the temperature falls off slowly with distance from the Sun. Thus, in a hypothetical static atmosphere, we find a pressure at infinity given by

dp dr

=

−nm

GM⊙ r2

,

p(r)

=

p0 exp

− mGM⊙ 2k

r dr R⊙ r2T (r)

.

(2.8) (2.9)

Thus, if the temperature falls less rapidly than 1/r, we find that limr→∞ p(r) > 0, we expect a non-vanishing pressure at infinity when the corona is extended. In particular, we find that for reasonable temperatures and densities n0, T0 at the ’coronal base’ this pressure is much larger than any conceivable interstellar pressure.
[The observed slow decrease of temperature with distance from the Sun, caused by the efficient thermal conduction that is mostly carried by electrons, implies that the solar wind must expand supersonically into interstellar space. For a spherically symmetric, single-fluid, isothermal outflow,] the equations of mass and momentum conservation (Eqs. 2.6, 2.7) can be rewritten to give {ⓈA15}

{A14}

{ⓈA15}

24

2 – Neutrals, ions, and photons

{ⓈA16} {A17}

1 dv v2 − 2kT

v dr

mp

=

4kT mpr

−

GM⊙ r2

(2.10)

[The solar wind starts slow, but is supersonic further out in the heliosphere; such a] transonic wind passes through a critical point at

rc

=

mpGM⊙ 4kT

where

vc =

2kT mp

(2.11)

[(note the dependence on stellar mass). {ⓈA16}

{A17} Formally, the equations

allow such a flow] to match any pressure as r → ∞ [although in reality the reach of

the flow is limited by the existence of an interstellar medium (Sec. 5.5.8)].

Let us examine this transonic wind solution in somewhat greater detail. If we

integrate the force balance, Eq. (2.7), from the coronal base to the critical point rc we

find a density ρc at the critical point given by

ρc = ρ(rc) = ρ0 exp

− mpGM⊙ + 3 2kT R⊙ 2

.

(2.12)

Note that this density is almost exactly the same as if there had been no solar wind flow: The subsonic corona in the solar wind is essentially stratified as a static atmosphere.
We can also find the resultant mass flux for the wind by examining the density and the velocity at the critical point:

(nv)r

=

ncvc

rc2 r2

∝

ρ0T −3/2 exp

C −
T

(2.13)

where ρ0 is the density at the coronal base [and C a constant]. The mass flux is proportional to the density at the coronal base and depends exponentially on the coronal temperature.” The actual solar wind is not only driven by thermal conduction from the coronal environment (which supplies energy for the work of driving the wind against gravity), but also by magnetic waves, known as Alfv´en waves, whose fluctuations act as an additional pressure term, and whose dissipation aids in heating far above the solar surface, all of which is particularly important for the fast wind streams; more on that in Sect. I:9.5. Another note on more detail is found in Sect. I:9.6, which begins to explain why for a more realistic solar wind description that also allows for helium, the exponential dependence of the solar mass loss on temperature is much weakened into a power-law dependence of temperature.
Note that it is not only the efficient thermal conduction per se that leads to a significant solar wind, but also the high temperature and low particle mass, and that that is the reason for the contrast with Earth’s atmosphere. In Eq. (2.2) gravity is approximated by a constant, leading to a formal solution for the pressure stratification of the terrestrial atmosphere that tends to zero exponentially even for an isothermal atmosphere; this is not a bad approximation for an atmosphere in which the pressure scale height (at most some 50 km) is well below 1% of the planet’s radius, so gravity

Photons, collisions, ionization, and differentiation

25

III:3.1

changes little even over many scale heights above the surface. But in the hot corona, the pressure scale height for the hydrogen-dominated gas at ≈ 2 MK is about 0.15R⊙, so gravity diminishes noticeably in the first few pressure scale heights, hence its distance dependence needs to be reflected in Eq. (2.7). The relatively weaker gravity (and the correspondingly reduced escape energy) at large heights leads to a transonic wind at coronal temperatures.
On a side note (to which we return in Ch. 11), the same equation Eq. (2.7 also informs us about an accelerating inflow (for which vdv/dr > 0 as both v < 0 and dv/dr < 0) enabling the formation of stars and planetary systems: gravity can win out over a pressure difference on very large scales in the Galaxy on which stars form, because now gravity in fact is built up by the infalling matter itself so that M⊙ needs to be replaced to read

dv dp G

ρv dr

=

− dr

− ρr2

r
ρ4πr2dr.
0

(2.14)

“To make a star of a given mass M from a gas with temperature T , gravity must

overcome the pressure support. [One way to estimate the required properties of a

cloud involved in the initiation of star formation is to look at Eq. (2.14) and see

when conditions cannot remain in a stationary balance, i.e., when v = 0 cannot be

maintained. That occurs when] the radius R of the protostellar cloud exceeds the

critical radius]

Rc(M, T )

>∼

GM c2s

=

GM µmp , kT

(2.15)

where cs is the sound speed and mp is the mass of the hydrogen atom. Taking a mean molecular weight µ = 2.3, appropriate for molecular hydrogen plus helium, and a

typical cold molecular cloud temperature of T = 10 K, Eq. (2.15) implies that a solar mass star must collapse from a cloud of radius R ∼ 2 × 104 astronomical units ([i.e.,

Sun-Earth distances; shorthand] AU).”

You will see the logic used in these examples applied throughout this book, and

indeed astrophysics in general: approximations in functional forms, simplifications

about geometries, and order of magnitude estimates are used throughout to aid in the

basic understanding what is going on. With these tools, analytical and – far more

commonly – numerical solutions become interpretable in terms of the basic, common

processes. How much can be simplified to show the basics, however, depends on the

environment: heliophysics, as is physics in general, is about simplifying as much as is

allowed, but no more.

2.3 Photons, collisions, ionization, and differentiation
In our everyday lives we can get away with taking it for granted that the atmosphere around us is the same no matter where we are. Moreover, we may take it to be true that this atmosphere is a mixture of mostly N2 and O2. And that this atmosphere is a very poor electrical conductor and that its winds are unaffected by the planetary magnetic field. As it turns out, none of these properties that we take for granted apply outside

26

2 – Neutrals, ions, and photons

of the domain where we live: the chemical mixture depends on height in planetary atmospheres and is affected by the variable spectral irradiance from the Sun’s outer atmosphere, ions and thus electrical conductivity are important in most of the local cosmos, and magnetic fields influence flows and vice versa almost everywhere in space. In this section, we focus on the processes that make the atmospheric composition dependent on location, primarily altitude. In the next section, Section2.4, we start looking at the role of ions in electrical conductivity and flows, although the role of magnetic fields in that is the focus of Ch. 3.
The scale height for different atmospheric constituents depends on the molecular or atomic mass, and is thus in principle different for different chemicals. But as long as the mixing by winds and (turbulent) convection is fast enough compared to the time scale by which the chemical separation can occur by diffusive settling, the atmospheric composition will remain uniform, and all major species will share the same scale height. When collisions become relatively infrequent above the homopause (at about 100 km for Earth), and diffusive settling exceeds mixing by flows, separation of chemicals by molecular mass occurs; see Ch. 13. The rate of separation depends on the diffusion coefficients, which themselves depend on chemical species and density, and on the chemical reactions that couple species (and, in the ionosphere, also through ion-neutral interactions), relative to turbulent mixing efficiency; see the discussion in Ch. IV:9.
Still higher in the atmosphere, where collision frequencies become so low that the mean free path approaches or exceeds the formal pressure scale height, the description of the medium as an ideal gas fails. That environment, where particles essentially move ballistically over long distances subject only to gravity (still disregarding any effects of electric and magnetic fields), is known as the exosphere. The exospheric base height can be estimated by looking at collision frequencies.
The characteristic frequency at which a particle in a non-magnetized plasma or a non-ionized gas of identical particles, all characterized by a temperature T and at particle density n, collides with other such particles is given by

kT 1/2

ν = σccvreln = σcc m

n,

(2.16)

where σcc is the mutual collision cross section and vrel is the velocity of one particle relative to another. In computing the mean free path, the velocity cancels out, leaving only the density as a variable:

λmfp

=

vrel ν

=

1 .
σccn

(2.17)

By way of example, let us look at neutral atoms with a collisional cross section of order, say, 3 × 10−16 cm2 (as for hydrogen atoms). For these, a density of 3 × 108 cm−2 (reached at roughly 500 km in Earth’s atmosphere, depending on solar activity) would correspond to λmfp ≈ 100 km. This order-of-magnitude estimate shows that this density in the Earth’s atmosphere roughly forms the point at which a vertically moving atom could jump over a scale height, or essentially through the bulk of overlying matter,

Photons, collisions, ionization, and differentiation

27

so where the assumption that we can work with the medium as a gas of electrically neutral particles fails; this is about the point where the Earth’s atmosphere transitions into an exosphere where neutral atoms move essentially ballistically.
On the Sun, in contrast, the neutral hydrogen population could still be described by hydrodynamics at that density because of the much larger scales involved, if matter were largely neutral there; however, that density is reached only in the corona where high temperatures cause hydrogen and helium to be fully ionized (see Table 2.3), and collisions occur via long-range electromagnetic forces between charged particles (see Table 3.4 for mean-free path estimates in an ionized medium, which, with Eq. (2.17), shows the larger effective collision cross section for Coulomb collisions). Lower down in the solar atmosphere where neutrals do dominate, the mean free path lengths are significantly smaller: the plasma throughout the Sun up to the inner corona behaves like a gas in which (often turbulent) flows counter gravitational separation. There are fractionation effects deep inside the Sun where mixing by flows is negligible on solar evolutionary time scales. Chemical differentiation is also seen in the atmosphere in the minority species, specifically determined by the energy required for first ionization of the atom (see Fig. I:9.2); this differentiation, not by diffusive settling but likely related to MHD waves and by EUV and X-ray irradiation of the chromosphere from the higher atmosphere, is still inadequately understood and not further discussed here.
Below the Earth’s exosphere and above the mesosphere, in a domain ranging from roughly 110 km to around 500 km in altitude, i.e., throughout much of the bulk of the thermosphere, lies a domain where collisions are frequent enough that the gas approximation is largely valid but not frequent enough to maintain uniform mixing of the chemicals that make up the terrestrial atmosphere up to that height: the atmosphere up to heights of about 110 km “is known as the homosphere and is constantly being mixed by turbulent wave eddies. It is only at altitudes above about 110 km that turbulent mixing gives way to molecular mixing processes, where each species begins to be distributed vertically under its own pressure scale height or hydrostatic balance, see Eq. (2.2). A heavy species, such as carbon dioxide, will decrease in concentration with height more rapidly than a lighter species, such as atomic oxygen (see Fig. 2.6). Each species, i, will have its own characteristic scale height Hpi, where Hpi = kT /mig, which is the vertical distance a species will decrease in partial pressure and number density by a [factor of e]. The upper atmosphere differs from the lower atmosphere in this respect such that the mean mass of the fluid will change with altitude, as well as other gas parameters such as the specific heat, cp. [. . . ]
The vertical distribution of species also has a global seasonal/latitudinal structure from large scale [. . . ] inter-hemispheric circulation from summer to winter. Closure of this circulation drives an upwelling of material across surfaces of constant pressure in the summer hemisphere and a downwelling in the winter hemisphere. The upwelling causes the heavier molecular rich gas, which had diffusively separated at lower altitudes, to be transported upwards to increase the mean molecular mass in summer. In winter the downwelling reduces the mean mass.”

I:12.3

28

2 – Neutrals, ions, and photons

Figure 2.6: Comparison of the global mean vertical profiles of the major species in the neutral upper atmospheres of a) Venus, b) Earth, and c) Mars for low and high solar activity. SMIN and SMAX indicate solar minimum and maximum conditions. Note that the turbopause heights (where turbulent mixing and diffusive separation are comparable) are 135, 110, and 125 km for Venus, Earth, and Mars, respectively. [Note: the International Space Station orbits at an altitude of ∼400 km. Figs. I:12.2, IV:9.1; source: Bougher and Roble (1991).]
Figure 2.7: Three planet global mean temperature profiles for solar minimum (SMIN) and maximum (SMAX) conditions. [Note the differences in horizontal and vertical scales. Fig. IV:9.3; source: Bougher and Roble (1991).]
The seasonal changes in insolation and the resulting circulations subject to Coriolis forces on the rotating planet are modulated by the effects of space weather. These effects include the X-ray and (E)UV part of the solar spectral irradiation, dissipation of electrical currents, and energetic particles precipitating from the magnetosphere. All of these (and others discussed in Chs. 5, 6 and 14) lead to heating, ionization, and dissociation of the high atmosphere. “Early investigation of the terrestrial ionosphere through its effect on radio waves resulted in description by means of layers, principally the D, E, and F layers, the latter subdivided into F1 and F2 [(sketched in Fig. 2.8)]. This terminology continues to influence our current concept of the nature of energy

III:13.1

Photons, collisions, ionization, and differentiation

29

III:13.1

Figure 2.8: Sketch (not to scale) of the ’regions’ in the terrestrial ionosphere. Source: NASA/SVS. For a color version of this figure, see arXiv:2001.01093.
deposition in atmospheres, although the misleading term ’layer’ has given way to ’region’. The term ’layer’ arose from the observation of systematic variation in the height at which the critical frequency of reflection occurs in ionospheric radio sounding; this method cannot detect ionization above the peak of a region, which explains the appearance of layers. Radar and spacecraft measurements now give a more complete picture of peaks and valleys and reveal the complex morphology of the ionosphere. [. . . ] An overview of the altitude dependence and variability of Earth’s ionosphere is given in Figure 2.9, showing the diurnal and solar-cycle changes and the locations of the named regions.”
“An additional historical artifact in terminology is the word ionosphere itself. Because the atmospheric ionization was discovered before the neutral thermosphere in which it is contained, anything above the stratosphere is often referred to as the ionosphere, resulting in a common misconception that this region of the atmosphere is mostly ionized. In fact, it is mostly neutral, ranging from less than a part in a million ionized during the day at 100 km altitude to about 1% ionized at the exobase (∼600 km, depending on solar activity; compare Fig. 2.12). Even at 1000 km, there is only of the order of 10% ionization. At several thousand km, where ions (mostly protons) finally become dominant, the region is defined as the plasmasphere. [. . . In the bulk of the terrestrial ionosphere] O+ is the most important ion, particularly in the extensive F2 region above ∼ 200 km. The F1 region from ∼150 to ∼200 km appears as a mere plateau in the profile, but is distinguished by a transition to molecular ions, particularly NO+. The low levels of N+2 , given the dominance of N2 at these altitudes, is noteworthy. {A18} The E region from ∼100 to ∼150 km exhibits a small peak, dominated by O+2 and NO+.”
“Much of the external sources of heating, ionization, and dissociation of a planetary atmosphere comes from the absorption of photons or particles impinging on the neutral atmosphere. The physics defining the altitude profile of the three processes is the same. For example, the rate of ionization, q [(cm−3 s−1)], by solar radiation intensity, I(h)

{A18} I:12.4.1

30

2 – Neutrals, ions, and photons

Figure 2.9: [left:] Overview of the altitude distribution of Earth’s ionosphere for daytime and nighttime conditions, at high and low solar activity. [Fig. III:13.1] [right: One of multiple different conventions between planetary scientists and astrophysicists is that the height coordinate is usually displayed vertically for planetary scientists and horizontally for stellar scientists. This flipped and rotated version of the figure conveys the difference in appearance.]

[(erg cm−2 s−1)], at some height in the atmosphere of number density, n(h), can be expressed as a product of four terms:

q = σaI(h)n(h)ηi,

(2.18)

where σa [(cm2) is the atomic] absorption cross section [for a wavelength interval matching that of I,] and ηi [(erg−1)] is the ionizing efficiency; ηi could equally be the heating or dissociation efficiency. The intensity of the radiation gradually decreases along the path through the atmosphere starting from an initial intensity of I(h = ∞). The altitude deposition profile depends on the absorption coefficient and on the atmospheric number density, which varies exponentially with height. Clearly the product of the intensity of the radiation, I, that decreases as the source penetrates the atmosphere, and on the atmospheric number density, n(h), that increases with increasing depth into the atmosphere, must reach a maximum at some altitude or, more correctly, at some pressure level [(except, of course, for visible wavelengths for which the atmosphere is largely transparent, in which case the surface absorption and reflection need to be taken into account)]. The level of penetration is referred to as the opticaldepth, τ , which is expressed mathematically as

τ

=

σan(h)

Hp(h) cos(χ)

,

(2.19)

On collisions and currents, and on neutrals and pickup ions

31

{A19}

Figure 2.10: The vertical profile of the classical Chapman profile appropriate for heating, ionization, or dissociation in a stratified hydrostatic atmosphere irradiated from above, relative to a reference height, shown for different slant angles χ. [Fig. I:12.4]

where the product of the number density n(h) at height h with the scale height Hp(h) at that level represents the integrated content of a column of gas above that point, and χ is the angle from the zenith at which the radiation penetrates a planar atmosphere. [The above expression is valid as long as the curvature of the atmosphere can be neglected, so for angles χ <∼ 75◦.] {A19}
The profile of the rate of heating, ionization, or dissociation from these processes takes the form of the classical Chapman profile, as depicted in Fig. 2.10, and is given mathematically by

q(h) = I∞ exp

−σa

n(h)

Hp(h) cos(χ)

ηiσan(h).

(2.20)

{A20} The peak of the profile is at unit optical depth, which depends on the mass of atmosphere above traversed by the energetic photon or particle. This corresponds to a fixed pressure level for a given angle of incidence. The depth of penetration into the atmosphere of a photon or particle in pressure coordinates therefore does not change with the gas temperature or the degree of thermal expansion. Even with the changing heating over the solar cycle or during a [magnetospheric] storm that might cause a thermal expansion of the atmospheric gas, that same radiation will still penetrate and produce heating or ionization at the same pressure level. The altitude associated with that pressure and the local number density would, of course, be different since they depend explicitly on gas temperature.” {ⓈA21}

2.4 On collisions and currents, and on neutrals and pickup ions
The terrestrial upper atmosphere is coupled to the Earth’s magnetic field through the ionized component of the atmosphere (referred to as the ionosphere) that is in turn collisionally coupled to the neutral molecular and atomic medium within which

{A20}

{ⓈA21}

32

2 – Neutrals, ions, and photons

it is embedded. The dynamics of these couplings in the overall system of solar wind,

magnetosphere, and ionosphere are discussed mostly in later Chs. 5, 6, and 13. Here,

we look at the consequence of the ionized medium threaded by a dynamic magnetic field

and embedded in moving neutral gas: electrical currents. In the terrestrial atmosphere,

the effects depend sensitively on the magnetic latitude because of the orientation of

the magnetic field: at high latitudes, where the field is predominantly vertical, the

connection with the magnetosphere dominates and the dissipated power can lead to

substantial heating. At mid and low latitudes, where the field is mostly horizontal,

internal processes dominate that provide less dissipative power than at higher magnetic

latitudes, but that do contribute to transport of plasma.

A moving electrical charge subject to a magnetic field experiences a Lorentz force

perpendicular to its velocity and to the magnetic field, in a direction that depends on

the sign of the charge. Also allowing for an electrical field to be present, the total force

equals:

dv

q

FL

=m dt

= qE +

v × B. c

(2.21)

In case E = 0 and in the absence of collisions, electrons and ions thus would spiral about the magnetic field line in opposite directions [(much more on that in Sect. 8.2)]. Their gyration radii and frequencies are very different because of their difference in mass and thermal velocity (see Table 18.2). Where the gyration radii are well below the gradients in the magnetic field, these opposite circular motions do not lead to a net current in the absence of collisions. However, when field gradients are substantial within the gyration radii of the particles (most readily for the ions, in particular the more energetic ones) the particles drift perpendicular to the field in directions opposite for opposite charges, thus leading to a current; one important heliophysical setting in which this occurs is in the Earth’s inner magnetosphere, where the gradient drift of primarily the energetic ions leads to the ’ring current’ (see Sect. 8.2).
In the variety of settings in heliophysics, collisions may occur among the electron and ion populations (see Ch. 3 for that), or with neutral particles (the focus here). In ionospheres, the neutral particles are atoms and molecules of a body’s atmosphere. In, say, the environments of comets, planetary rings, or in the outer heliospheric solar wind the neutral particles, in contrast, may be either dust particles, escaping atmospheric gas, or inflowing neutral interstellar atoms.

Let us start with a collision in which no charge-transfer occurs in a setting where the charged particle senses both a magnetic and electric field. In each such collision of an electron or ion with a neutral particle, the gyro-motion of the electron or ion involved is modified. Because of the opposite charges of the electron and ion populations, they attempt to gyrate about the magnetic field in opposite directions as they are accelerated by the electric field; consequently, they exhibit a net drift perpendicular to the magnetic field with ions and electrons moving in the same direction and at the same rate. There is no net current (see the left-hand side of Fig. 2.11), but if

On collisions and currents, and on neutrals and pickup ions

33

Figure 2.11: Schematic of interactions of plasma with neutrals. Left: Initial motion of pickup ions and electrons. The gray circle represents a neutral composed of a positively charged ion and a negatively charged electron. The directions of plasma flow velocity, u, of the magnetic field, B, and of the electric field, E, are indicated. In the image, following dissociation, the ion path starts upward and the electron path starts downward. Although initial motion is along E for the ion, the Lorentz force causes the path to twist, resulting in motion around B at the ion cyclotron period, leading to a net drift at a velocity of E × B/B2. The electron initially moves in the −E direction. Its motion also rotates around B, but at the electron cyclotron frequency. The net effect is a transient current in the direction of E. Right: Schematic of the effect of collisions with neutrals for a case with the collision frequency of order the ion cyclotron frequency. Triangles represent neutrals. The effect of collisions is to slow the motion in the E × B direction of the ions but not of the electrons [(which have other collision and gyrofrequencies)] and to displace the ions in the direction of E. A net current arises, with one component along −E × B (a Hall current) and one component along E (a Pedersen current). [Fig. IV:10.3]
there are collisions roughly at the same frequency as the gyrofrequency (different for particles of different masses), the situation changes fundamentally: collisions interrupt the gyromotion, and this results in a net separation of the charges. A graphic example, discussed in more detail below, is given in Fig. 2.11(right).
If collisions are very infrequent, or to be precise if the electrons or ions can gyrate about the magnetic field many times between collisions, the electrical conductivity across the magnetic field is very low. If collisions are very frequent for both ions and electrons, hardly any charge separation can occur between collisions, and the electrical conductivity perpendicular to the magnetic field is also very low. Peak perpendicular conductivity depends on the direction relative to the electric field and is reached depending on the ratios of collision and gyro-frequencies, as shown below.
Collisions between the populations of charged and neutral particles in the presence of a magnetic field while allowing for bulk flows is described through multiple equations. One of these captures the transfer of momentum that affects the force balance (touched upon towards the end of this section) almost entirely by looking at ions because they carry the bulk of the mass. Another accommodates electrical currents that arise from the differential behavior of the ions and electrons subject to the magnetic field. A third describes the energy transfer through the collisional effects formulated as Ohmic dissipation in the energy balance.
How collision frequencies influence currents in the ionosphere/thermosphere, where the neutral component is the most common, can be approximated as follows (collisions between charged particles are ignored here because collisions with the abundant neutrals

34

2 – Neutrals, ions, and photons

I:12.6

are far more common in the bulk of the terrestrial ionosphere). “If we take the magnetic

field to be aligned with the z axis, then the generalized Ohm’s law [(the derivation of

which is shown for a fully ionized plasma in Ch. 3)], j = Σe · E0 (where E0 is the total

electric

field:

E0

=

E+

1 c

v

×

B),

contains

the

conductivity

tensor

 σP σH 0  Σe =  −σH σP 0  ,
0 0 σ∥

(2.22)

where the Pedersen (⊥ B, ∥ E⊥), Hall (⊥ B, ⊥ E⊥), and parallel (∥ B) conductivities are given [by:

σP

=

neec B

Me 1 + Me2

+

e qi

Mi 1 + Mi2

;

σH

=

neec B

Me2 1 + Me2

−

e qi

Mi2 1 + Mi2

;

σ∥

=

neec B

Me +

e qi

Mi

(2.23) (2.24) (2.25)

(where the equations from Sect. I:12.6 were rewritten to the above by using the expression for ωe,i below). For characteristic values of these conductivities in the terrestrial ionosphere, see Figure I:12.5. Here, Me,i = ωe,i/νe,i are the electron and ion magnetizations, with ωe,i = |qe,i|B/me,ic the electron and ion (with charge qi) gyro-frequencies around the field of strength B, me,i are the electron and ion masses, νen and νin the electron-neutral and ion-neutral collision frequencies.] The effect of the collisions is to rotate the net current from the direction of E at high altitudes towards the negative E × B direction at low altitudes. In the terrestrial ionosphere, the] current and dissipation reach a peak at the altitude where the Pedersen and Hall conductivities are equal, around 125 km. For high-frequency currents, like those that may occur in the solar chromosphere, the dissipation may increase markedly (see Sect. I:12.8). Note that σP is generally dominated by the ion term.” {ⓈA22}
The collisional coupling between ions and neutrals causes momentum exchange (through the drag force that works to reduce the velocity difference between these two populations) and energy dissipation (in the form of Joule heating). “The electrodynamic properties can be conveniently separated into a high [magnetic] latitude region, where the current flow in the ionosphere is connected to the magnetospheric current system, and a mid and low latitude region, where the majority of the current flow and polarization electric fields are controlled internally by the thermosphere-ionosphere conductivity and dynamics.” “In the ionosphere, currents flowing perpendicular to the magnetic field are produced by electric fields and neutral winds. Although collisions between ions and the neutral gas are relatively infrequent [in Earth’s upper atmosphere] above ∼160 km, they are sufficient to accelerate the neutral component, i.e., the thermosphere, at high latitudes to many hundreds of m/s over periods of tens of minutes or more [to speeds well in excess of those associated with solar heating]. [. . . ]

{ⓈA22} I:12.6

I:12.6.1

On collisions and currents, and on neutrals and pickup ions

35

Solar chromosphere
Terrestrial ionosphere

Figure 2.12: Comparison of densities, n (cm−3), and ionization fractions, fion, for a characteristic dayside ionosphere (dashed) and mean chromosphere (solid). The diamonds mark the mean values for the ionospheric D and F2 regions, centered on about 80 km and 300 km, respectively. The triangles denote the base of the chromosphere (defined here as at a continuum optical depth of τ5 = 0.004) and the top of the chromosphere (where the temperature exceeds 30 000 K). [Fig. I:12.13]
At low altitude, ∼100 km, the ions are forced to move with the neutral gas, whether stationary or moving. The large-scale wind system at this altitude is driven by the tidal and planetary waves propagating from the lower-atmospheric terrestrial weather system, and the mass of the atmosphere is such that ion drag has little or no impact on the neutral dynamics. The altitude range between 100 and 160 km altitude is the narrow altitude range that is responsible for most of the dissipation of electromagnetic energy from the magnetosphere. The neutral dynamics and conductivity in this boundary region between space and atmospheric plasma are critical.” “At mid and low [magnetic] latitudes the electric fields [in Earth’s ionosphere] arise largely from internal dynamo processes driven by the conversion of neutral wind kinetic energy to electromagnetic energy, and are typically an order of magnitude smaller (a few mV/m) than high-latitude fields. The energy involved is also much smaller. The importance of the small electric fields at low latitudes is no longer the Joule heating and momentum dissipation, but rather their role in the redistribution of plasma.” Some of these effects are touched upon generically in Sect. 5.5.7, with a more comprehensive discussion for Earth’s ionosphere in Sect. I:12.6.
In much of the discussion of magnetized plasma in the Sun’s interior and atmosphere in subsequent chapters, the Hall and Pedersen conductivities are often assumed to be negligible. A similar approximation is often seen in the study of the heliosphere and

I:12.6.2

36

2 – Neutrals, ions, and photons

I:12.8.3

planetary magnetospheres. The Sun’s chromosphere, however, is an environment with a strong neutral population and with collision frequencies not so high that Pedersen and Hall conductivities are effectively ignorably small. The chromosphere is located immediately above the photosphere (which itself has a thickness of roughly a single scale height of about 100 km), and extends over a height range of some 2,500 km, spanning roughly a dozen pressure scale heights in a highly dynamic setting that is strongly patterned by the magnetic field, before the transition region is reached in which the temperature rapidly rises to coronal values.
“The Earth’s ionosphere has a range of degrees of ionization, starting from the essentially neutral troposphere below, reaching an ionization fraction of about 10−4 − 10−3 around 200 km in height, and exceeding a few percent by 1 000 km. In the case of the chromosphere, the ionization fraction starts at about 10−4 around photospheric heights, drops through 10−5 through the classical ’temperature minimum’ around 500 km in height, and then increases through a few percent around 1 500 km in height, continuing to near-complete ionization in the solar corona. Figure 2.12 compares the densities and ionization fractions for mean states characteristic of the ionosphere and chromosphere. Note that the neutral densities in the D–F2 ionospheric region are comparable to those in the chromosphere, but the ion densities are at least 1 000 times lower at any given neutral density, resulting in a much weaker ion-neutral coupling in the ionosphere than in the chromosphere.
Let us look back at Eqs. (2.23)-(2.25) and assess their meaning for both chromosphere and ionosphere. In the limit of a weak magnetic field or a high collision frequency, the ion and electron magnetizations Me,i = ωe,i/νe,i → 0, σP → σ∥, σH → 0; hence, currents are more readily aligned with the electric field, as expected. As the collision frequencies with the neutral population decrease, the above expressions would have current and magnetic field aligned (as both σP,H → 0) [. . . ]
In the chromosphere of a solar [sunspot] region, Me(500 < h < 2000 km) = O(100), decreasing rapidly towards the photosphere to Me(h = 0) = O(0.01) at the solar surface. Some studies find the proton magnetization to remain below unity throughout the chromosphere, up to the transition into the corona (these findings depend on the atmospheric domain, of course, and on the models used [. . . ]). Consequently, the bulk of the active-region chromosphere has an anisotropic conductivity of at least a factor of 10 difference between the field-aligned and transverse components. Conduction in the corona is almost exclusively field aligned (and thus essentially free of Lorentz forces), while photospheric conduction is nearly isotropic. [. . . ]” {A23}
Now, let us look at different environments, and illustrate not only currents but also the effects of momentum transfer. “At comets and in the vicinity of moons, such as Io and Enceladus, that are significant sources of neutral gas, various processes that convert neutral atoms or molecules into ions are important to consider. Neutrals can be ionized by photons (photoionization) or by collisions with other particles, typically electrons (impact ionization). An additional process that affects the interaction region is charge exchange in which a neutral gives up a charge to an ion. The original ion, now neutral, carries off its incident momentum while the original neutral becomes an

{A23} IV:10.3.2

On collisions and currents, and on neutrals and pickup ions

37

{A24}

ion at rest in the frame of the neutral gas. The ions introduced into the plasma by ionization of neutrals modify the bulk
properties of the plasma. Consider a situation in which the neutrals are at rest relative to [a location] towards which the plasma flows at (bulk) velocity u. Photoionization and impact ionization add mass to the plasma whereas charge exchange between the ionized or neutral form of the same element does not change the mass density. All three processes slow the bulk flow because the new ions must be accelerated so that their average motion matches that of the bulk plasma and the process extracts momentum from the incident plasma. These processes also change the thermal energy of the plasma and may modify the plasma composition. The complex effects associated with pickup can significantly modify the interaction region surrounding a moon or a comet.
The relation between pickup and currents is shown schematically in Fig. 2.11a. The newly ionized ion senses the electric field of the flowing plasma and begins to move in the direction of this electric field. The electron that has separated from the ion is initially accelerated in the opposite direction. After one gyroperiod, the average separation of the gyrocenters of the two charges is close to one ion gyroradius

rgi = mionvionc/qB

(2.26)

where mion is the ion mass, vion is its thermal velocity, and q is its charge (see Sect. 8.2 for details on single-particle motions). {A24} The result of the separation of charges is to produce a transient current density in the direction of the electric field. If the pickup is occurring at a rate n˙ , where n˙ is the number of ionizations per unit volume and time, then the pickup current [density] is

jpickup = qn˙ rgi.

(2.27)

Because pickup current flows across the background field, a cloud of pickup ions acts much like a solid conducting obstacle in the flow and imposes the same types of perturbations, i.e., it slows and diverts the incident flow” in a way outlined in Ch. 5.
In this volume, we do not go into the behavior of dusty plasmas. The interested reader is referred to Ch. IV:11, which introduces the subject as follows: “The study of dusty plasmas bridges a number of traditionally separate subjects, for example, celestial mechanics, mechanics of granular materials, and plasma physics. Dust particles, typically micron and submicron sized solid objects, immersed in plasmas and UV radiation collect electrostatic charges and respond to electromagnetic forces in addition to all the other forces acting on uncharged grains. Simultaneously, dust can alter its plasma environment by acting as a possible sink and/or source of electrons and ions. Dust particles in plasmas are unusual charge carriers. They are many orders of magnitude heavier than any other plasma particles, and they can have many orders of magnitude larger (negative or positive) time-dependent charges. Dust particles can communicate non-electromagnetic effects, including gravity, neutral gas and plasma drag, and radiation pressure to the plasma electrons and ions. Their presence can influence the collective plasma behavior by altering the traditional plasma wave modes and by triggering new types of waves and instabilities. Dusty plasmas represent the

IV:11.1

38

2 – Neutrals, ions, and photons

{A25}

most general form of space, laboratory, and industrial plasmas. Interplanetary space, comets, planetary rings, asteroids, the Moon, and aerosols in the atmosphere, are all examples where electrons, ions, and dust particles coexist.” {A25}
2.5 Sources of plasma
There are many sources of plasma around the heliosphere: all it takes is some neutral medium subjected to sufficient energy to ionize particles. The bulk source medium can be the largely neutral gas in the Sun’s surface layers, but dissipation of (magnetic) waves and currents, as well as the acceleration of particles in electric fields result in heating and ionization of the Sun’s outer atmosphere. The larger planets have neutral atmospheres of which the top layers are ionized by solar radiation and by suprathermal particle precipitation. Moons may be large enough to have their own atmosphere (as is the case for Titan at Saturn), while others may still have some matter around their surfaces because these are subjected to sputtering by the solar wind or, for moons within planetary magnetospheres, by magnetospheric particles, or matter may be supplied by geysers (as on Enceladus at Saturn) or volcanoes (as on Io at Jupiter) that contribute molecules (including SO2, SO, S2, H2S, . . . ) as well as atoms. Comets have a coma of gas that sublimates off the nucleus, along with dust. And dusty material is around in the rings of all the giant planets. Whereas the magnetized and ionized components of the interstellar medium cannot penetrate each other (as discussed in Chs. 3 and 5), neutral interstellar-medium particles can make it deep into the heliosphere, following free-fall trajectories in the collisionless environment until they are subjected to a charge-exchange collision with solar-wind ions.

Chapter 3
MHD, field lines, and reconnection
Chapter topics:
• The fundamental difference between gravity and magnetism • MHD as a low-order description of single-fluid plasma dynamics • Alfv´en, fast-mode, and slow-mode waves • Processes and scales of reconnection
Key concepts:
• General, force-free, and potential magnetic fields • Magnetic pressure and tension • Magnetic structures: current sheets and flux tubes • Reconnection: failure of ideal MHD and of frozen-in field lines
3.1 Introduction
“Absent the magnetic field, neither solar activity nor magnetic storms – the solar and terrestrial sources of [variable conditions referred to as space weather [7] – would exist. . . . ] Although in principle fossil magnetic fields could have remained from the creation of the Solar System, this appears not to be the case. Witness the 22-year magnetic cycle of the Sun and the reversals of the Earth’s magnetic field. On shorter time scales, the magnetic topography of the solar surface changes so rapidly that it must be monitored constantly as input for space weather forecasts. {A26}
[The contrast between magnetic variability and gravitational persistence has its origin in the sources of the two fields: the magnetic field, B, has its origin in a variable source, namely the relative motion of differently charged particles, while the gravitational field, g, springs forth from a conserved (positive definite) source.] The
7 For introductions to the impacts of space weather on society and its technological infrastructure we refer to Chs. II:2, II:12, and H-V:1-5.
39

II:1

{A26}

40

3 – MHD, field lines, and reconnection

conserved source of the gravitational field is mass, as can be seen in the [non-relativistic] field equations that apply to the gravitational field:

∇ · g = −4πGρ, ∇ × g = 0,

(3.1)

where G is the gravitational constant and ρ is the mass density. Thus, gravity is determined by the amount of mass present and its distribution. Because mass is conserved and the gravitational force causes matter to collapse into systems in which the gravitational force is almost perfectly balanced by thermal pressure or inertial forces, gravitationally organized matter tends to be stable over eons [. . . ] In contrast, the pertinent field equations for the magnetic field are

4π ∇ · B = 0, ∇ × B = j
c

(3.2)

[(the second expression holds if all velocities involved are well below light speed).]

The source term for the magnetic field in these equations is electrical current, j,

which, unlike mass, is not a conserved quantity [(although ∇ · j = 0) and which

can point in any direction]. Thus we see that B is a product of dynamo or other

magnetohydrodynamic (MHD) processes that generate current in real time. The crucial

distinction is that unlike the gravitational field, which is in effect a byproduct of a

conserved, definite quantity of mass and so is inherently persistent, the magnetic field

is generated by a variety of plasma motions in the Sun, in the solar wind, and in

planetary magnetospheres on time scales shorter than what would be needed to reach

an equilibrated state. Hence, the local cosmos is constantly adjusting and attempting

to relax, but it never gets to such a quasi-stationary state. The consequence of this is

what we call weather, including [. . . ] space weather.”

“There is an important difference regarding the types of volumes that the gravi-

tational and magnetic tension forces organize. The gravitational field has no shield-

ing currents (∇ × g = 0) [because its source is the positive-definite mass density

(∇ · g = −4πGρ); consequently, gravity] has no discontinuities because that would

require an infinite mass density. Hence, the gravitational field is relatively homogeneous;

it varies smoothly and continuously in space. On the other hand, [owing to the fact

that electrical charges can be of either sign, a magnetic field can contain] shielding

currents

(∇

×

B

=

4π c

j)

which

spontaneously

form

discontinuities,”

that

are

commonly

referred to as current sheets [(see Table 3.1 for a definition)] despite the fact that

their geometry is generally quite complex in the local cosmos. The combination of

the distinct behaviors of gravitational and magnetic forces yields a rich diversity of

phenomena in the local cosmos and beyond that emerge from the ‘universal processes’

captured in the MHD description of magnetized plasma.

Among the universal concepts in heliophysics one pair stands out in particular,

namely that of magnetic lines of force – or commonly magnetic field lines – and of

their reconnection. {A27} Field lines are abstractions; they are 1-dimensional

virtual devices that are used to outline the geometry of magnetic structures in the

local cosmos, in a way in which the tangent of the field line anywhere along it has the

II:1

{A27}

Introduction

41

Table 3.1: Structures in the magnetic field.
Current sheet: Examples on large scales: heliospheric current sheet; magnetospheric current sheet. E.g., Fig. 5.4. [I:6.2] “Our focus here is mainly on current sheets in the form of tangential discontinuities or rotational discontinuities that evolve into tangential-like discontinuities. Tangential discontinuities are non-propagating surfaces across which no magnetic flux passes as the magnetic field changes direction or strength or both, while total (magnetic plus thermal) pressure is continuous. [. . . ] current sheets (tangential discontinuities) inevitably form in naturally occurring turbulent plasmas; [. . . they] form in the corona through the expansion of magnetic flux tubes that poke out of the photosphere [and] expand until at some altitude they press against each other forming a beehive pattern of flux tubes separated by current sheets [unless and until the currents dissipate and the field becomes potential]. [. . . ] Interplanetary space is a honeycomb of outwardly advecting current sheets. [. . . ] In the magnetospheric case, the solar wind snags magnetic field lines from the planet’s two poles on the sunward side and stretches them anti-sunward to form the characteristic two-lobe magnetospheric tail across which the magnetosphere’s analog of the heliospheric current sheet separates the two lobes.” Flux tube, flux rope: Examples: compact sunspots, pores, ’bright points’; as an entity, they are bounded by a current sheet. E.g., Fig. 9.3(top) and 9.1. [I:6.4] “A flux tube is the volume enclosed by a set of field lines that intersect a simple closed curve. The frozen-in flux condition of ideal MHD describes a parcel of plasma threaded by magnetic field lines as a conserved entity whose motion can be followed.” In the solar photosphere, flux tubes may emerge as preformed entities, or may form from by ’convective collapse’. A flux rope is a flux tube twisted about itself (and thus carrying an internal net current); many magnetic configurations emerge into the solar photosphere as flux ropes; many form in the corona by the dynamics of the reconnecting field; coronal mass ejections inject ropes into the heliosphere (there known as ’magnetic clouds’) while others form by reconnection across current sheets; at magnetopauses, flux ropes (’flux transfer events’) form by reconnection; and flux ropes (’plasmoids’) form by reconnection across the magnetospheric current sheet. Cell: Examples: planetary magnetosphere; heliosphere. E.g., Fig. 5.1. [I:6.7] “Magnetic fields tied to gravitating bodies will expand to fill all space unless prevented from doing so. [. . . ] the magnetic field’s expansionist ambition is checked by some other magnetic field-bearing plasma expanding from somewhere else. Each magnetic field is therefore encased within a definable volume, which we refer to as a cell. In the Sun’s case, the cell is the heliosphere. In the other cases mentioned, the cells are planetary magnetospheres. [. . . The cellular structure] is like a Russian nesting doll in which one cell is encased within another. [. . . ] within the heliosphere, the scale sizes of the objects already mentioned cover seven orders of magnitude”

same direction as the field there while the local field line density is equivalent to the magnetic field strength.
In a vacuum, magnetic field lines have no intrinsic temporal continuity. For example, consider the field between and surrounding magnets or electrical wires at time t0 and again at time t1 after having moved the magnets or wires into new positions. The lines of force used to visualize the field at times t0 and t1 are completely independent, the result only of the magnetic fields at the two instances in combination with two sets of points, one for t0 and one for t1, selected by a researcher from which to compute the lines of force. In a plasma, however, field lines can be thought of as structures

42

3 – MHD, field lines, and reconnection

whose continuity in time derives from the ionized matter that is contained in the flux bundle or tube that is centered on the field line. In our thinking, we should map these ‘lines of force’ to their 3-dimensional equivalent, the ‘flux tube’: as long as the ions and electrons once contained never move out of the flux tube, the field line has some temporal continuity. Whenever matter does migrate out of the flux tube, the attribute of continuity for the field line fails. However, if the locations where this occurs are compact compared to the field line’s length, one can think of field lines – that can never end in the divergence-free magnetic field except close onto other field lines – as being cut and connected onto another field line. Where that happens, the concept of ‘magnetic reconnection’ is then introduced to salvage that of the ‘field line’ as something that has an identity over time, at least while matter remains constrained to within the flux tube.
Field lines and their reconnection are but two of the concepts related to a variety of processes that occur in ionized gases (‘plasmas’) that are threaded by magnetic field. We come across such processes in the vastly different environments of the solar interior and of the far reaches of the heliosphere, and in the depths of planets as well as in the most tenuous parts of their outer atmospheres. Temperatures and densities (and, as we shall discuss later in this chapter, magnetic field strengths) differ by many orders of magnitude; a summary of some of the conditions encountered in the local cosmos that surrounds us is visually represented in Figure 2.1.
In everyday life we tend to ignore the Earth’s magnetic field, but we can do so only because of the low temperatures in which we live (which renders most material, except metals, non-conducting) combined with the high densities; together, as we shall see in more detail later in this chapter, these conditions make the forces exerted by the terrestrial magnetic field utterly negligible in our day-to-day affairs, except where we take special care to uncover them, such as in magnetic compasses. Conditions are markedly different, however, in the layers underneath the atmosphere of the Sun, throughout the extended solar atmosphere, and in the outermost reaches of atmospheres of all bodies in the Solar System: there, magnetism is effectively coupled to matter while the inertia of that matter is in much of the domain significantly lower in comparison to magnetic forces than in our daily settings. There, the magnetic field is an important player that adds a significant force to compete with pressure, gravity, and inertia. It provides a medium for a variety of waves (which this text merely touches upon), and changes the transport of thermal energy and energetic particles. Add to that the fact that the magnetic field is evolving on a range of spatio-temporal scales, and you have a source of continual change in conditions throughout the local cosmos.
The mathematical formulation of what happens in a magnetized plasma is often simplified through an ensemble approximation that is equivalent to the hydrodynamics used in the description of gases, but here including the magnetic field in what is called magnetohydrodynamics, or MHD for short. MHD is a description of the multitude of constituent particles in the local cosmos that relies on statistical averaging carried out by the medium itself, namely through interactions that lead to essentially Maxwellian velocity distributions, often assumed to be isotropic (but in some formulations distinct

Introduction

43

Table 3.2: MHD approximation and the concept of ’closure’.
Philosophy of magnetohydrodynamics: The fundamental assumption underlying the MHD equations as shown in Table 3.3, and the principal criterion to judge the applicability of that MHD approximation under given circumstances, is that the medium can be suitably described as a continuum. This presents us with a statistical criterion: MHD can be applied beyond a fiducial length, say L, such that there are sufficient particles in a volume L3 such that statistical means – like density, mean velocity, pressure and so forth – have small variances or fluctuations about them. Within that volume, collisions (or wave-particle interactions) result in average properties of the medium that transform the need to describe each particle separately in its interaction with all others into an enormously truncated set of descriptions of statistical averages. This truncation is known as ’closure’: the continuum description requires a closure relation at some level that relates an unknown high-order moment of the full particle distribution function, such as pressure, to lower-order moments (see Sect. 8.4 for more on that). An equation of state, as in Eq. (3.8), is predicated on there being ample collisions to isotropize the random motions and achieve a thermodynamic equilibrium, with its characteristic Maxwellian velocity distribution (or more than one if a multi-fluid description is used). The MHD equations as in Table. 3.3 describe a 5-moment continuum closure scheme using mass density, temperature, pressure, energy density, and velocity. As collisions become less frequent one is required to enforce closure at higher levels, examples of which lead to, e.g., Eqs. (3.11) and (3.27). More generally dielectric and magnetization properties of the material enter in the definitions of D and H. Therefore, if by some other means (e.g., by observation) you know how to close the moments (like in Sect. 2.2 for the solar wind by using the observationally motivated approximation that the temperature is constant throughout the heliosphere and the pressure is an isotropic scalar) then you can use the continuum fluid description to answer some questions even about a medium where collisions are a rare thing.

for directions along and perpendicular to the magnetic field) and for velocity equilibrium between electrons and ions. To this, a few other assumptions are made about local conditions: processes described by MHD assume that ion and electron interactions as well as their gyrations about the field occur on scales that are small compared to the gradients in the magnetic field while at the same time large compared to a distance (known as the Debye length) over which electrical charges can exist unshielded by other particles, with velocities well below relativistic, and only allowing for wave-like phenomena that are slow enough that electrical neutrality is achieved well within any time scale of interest and that are slow compared to the plasma frequency and electron/ion gyro-frequencies. However, interactions between particles should be infrequent enough that the medium should allow the electron and ion populations to move differentially with relative ease, i.e., conditions should allow the medium to conduct electrical currents rather effectively.

44

3 – MHD, field lines, and reconnection

MHD treats the ionized medium as a fluid by working with ensemble properties. In hydrodynamics this is generally allowed because of a high frequency of molecular collisions relative to the time scales of the processes on macroscopic scales. In many environments in heliophysics, however, collisions can be so rare that distances between collisions can be comparable to the scale of the system under consideration, while the solar wind is entirely collisionless beyond a few dozen radii from the Sun . . . and yet MHD has been shown to be a useful approximation. The key factor in making MHD useful is that the medium should not be able to maintain a significant electric field in its own reference frame. Even if collisions are rare in such a medium, long-range flights of the particles are impeded: the gyration of particles about the magnetic field reduces the scale of flight perpendicular to the field, while wave-particle interactions have a similar effect along the field. Consequently, the movement of individual charged particles in a plasma is coupled to the collective of its environment, resulting in a fluid-like behavior even if collisions are rare.
However, where binary interactions are important in the MHD description, the anisotropy imposed by the magnetic field does affect what approximations can be made. Most importantly, these effects are seen on gas pressure and viscosity. In a collisional plasma, these terms are generally essentially isotropic and thus described by scalars. But in a collisionless plasma, pressure and viscosity are anisotropic, and thus are approximated by tensors. In this volume, we generally use a scalar for pressure, and capture anisotropy in conductivity in Hall and Pedersen terms (see below and Ch. 2).
3.2 (Magneto-)Hydrodynamics
The equations of magnetohydrodynamics, or MHD, are based on the assumption that the plasma can be described as a continuum; see Table 3.2 for a very concise description of what that entails. The approximations used here lead to six equations that describe magnetized plasma subject to gravity, as shown in Table 3.3 (note that processes involving radiative transfer are largely omitted from this volume). {ⓈA28} Five of these are essentially equations of hydrodynamics, namely continuity, momentum, energy, gravity, and the equation of state (EOS), with two important modifications: the
magnetic, or Lorentz, force (1/c)j × B ○6 is added in the momentum description, and there are additional terms ○10 in the energy equation. We return to these terms and
equations below, and discuss the additional equation, namely the induction equation Eq. (3.3), which couples the magnetic field to macroscopic flows and microscopic collisions, in some detail in Section 3.2.2. {A29}
In order to assess the validity of the assumption made to derive the MHD equations for the vastly different conditions with which heliophysics concerns itself we can look at a variety of dimensional and dimensionless numbers. Table 18.2 lists frequently used length and time scales, as well as some commonly used ratios, some of which have been given a name. Some of these are pertinent to microscopic, particle-level conditions and some are pertinent to macroscopic, system-level conditions. We introduce them here only briefly – most will be looked at explicitly later on – in order to give you an

{ⓈA28}

{A29}

(Magneto-)Hydrodynamics

45

Table 3.3: Equations of magnetohydrodynamics for a fully-ionized plasma, ignoring radiative energy transport and radiation pressure, to be complemented by initial and boundary conditions to specify the solution.

Single-fluid non-relativistic magnetohydrodynamics (MHD):

Induction Continuity Momentum
Internal energy Gravity

○1

○2

∂B ∂t

=

∇×(v

×

B)

−

∇×(η ∇×B)

○3

○a

∂ρ ∂t

+

(v

·

∇)ρ

=

−ρ∇

·

v

+

(S

−

L)

○4 ○5

○6

ρ

∂v ∂t

+ ρ(v

· ∇)v

=

+ ρg

− ∇p +

1 4π

(∇×B)

×

B

○7

○b

○c

+∇ · τ − v(S − L) + (Sp − Lp)

○8

○9

○10

ρ

∂e ∂t

+

ρ(v

·

∇)e

=

−p∇

·

v

+

∇

·

(κ∇T

)

+

(Qν

+

Qη )

∇ · g = ∇2Φ = −4πGρ

EOS

p = (γ − 1) ρ e

Complemented by initial and boundary conditions

(3.3) (3.4)
(3.5) (3.6) (3.7) (3.8)

Online resources:

Plasma physics:

online NRL Plasma Formulary

Vector calculus:

Wikipedia page

Introduction to MHD ’Essential magnetohydrodynamics for astrophysics’

(Spruit, 2013)

Symbols: B magnetic field; v fluid velocity; e = CV T specific internal energy (e.g., energy

per

unit

mass;

3 2

kT

/µ

for

an

ideal

gas

with

µ

the

average

mass

per

particle);

p

gas

pressure;

ρ mass density; Φ the gravitational potential and G Newton’s gravitational constant; g

gravity, τik

= 2ρν

Λik

−

1 3

δik

∇

·

v

the viscous stress tensor with the deformation tensor

Λik

=

1 2

+ ∂vi

∂vk

∂xk ∂xi

; Qν viscous heating; and Qη = η|j|2 = η (c/4π)2 |∇×B|2 the resistive

(Ohmic) dissipation; ν, ηe and κ represent the viscosity, magnetic diffusivity, and the thermal

conductivity tensor (which is highly anisotropic, with heat most effectively conducted by

electrons moving along the magnetic field); γ = Cp/CV is the adiabatic index, the ratio of

specific heats for constant pressure and constant volume. In an ideal, mono-atomic gas with

3 degrees of freedom γ = 5/3. S, L, Sp and Lp are source and loss terms for mass and momentum by introduction or loss of ions from a non-ionized reservoir.

impression of which types of processes or relative scales are important. For example, we can look at the length scale on which ions gyrate around the local magnetic field relative to the gradients in the field to assess whether the ions sense the magnetic field

46

3 – MHD, field lines, and reconnection

in an ensemble sense such as required of a fluid or whether higher-order descriptions are needed. Or one can ask whether length scales involved are large enough that the plasma can be viewed as not having significant charge separation; the length scale on which electrostatic potential of any particle is effectively shielded by the surrounding plasma is known as the Debye length. Or one can look at the ratio of the average time between collisions and the time needed to complete one gyration around the magnetic field in order to assess whether the magnetic field can effectively be followed by the charged particles and whether the Hall current needs to be considered.
3.2.1 MHD equations, individual terms, and special cases
First, let us briefly review what the MHD equations express, the role of the individual terms, and some special cases:
• v, p: The velocity v reflects the mass-weighted bulk velocity (the first-order moment of the velocity distribution) of the electron-ion plasma. For a fullyionized hydrogen plasma this equals (mivi + mive)/(mi + me), which can be taken as the velocity of the center of mass of the ion-electron pairs that comprise the ’particles’ of a ’single-fluid plasma.’ The pressure p is the sum of the pressures of the electron and ion populations.
• Eq. (3.3): The induction equation (a combination of Faraday’s law with Ohm’s law, see Sect. 3.2.2) states that any local change in the magnetic field is associated with a ‘curl’, or ’circulation’, in the component of plasma flows working perpendicular to the magnetic field and/or to the slippage of plasma relative to the magnetic field through finite diffusivity. Note that this form of the induction equation is linear in B so that if B(t = 0) = 0 then no field can arise at a later time. Sect. 3.2.2 touches on the fact that some terms were ignored to arrive at this form, some of which can act as a source term for magnetic field; this is not further discussed in this volume as the clouds out of which stars and planetary systems form initially are threaded by a galactic seed field from the outset (interested readers could look for ’battery effects’, including the ’Biermann battery’).
• Eq. (3.4): Continuity requires that the local plasma density changes only because of flow through a volume and by compression or dilation in doing so, unless there are sources or sinks within the volume.
• Eq. (3.5): The momentum (or force) equation (Newton’s second law in volumetric form) summarizes how the plasma velocity is affected, as in hydrodynamics, by gravity, pressure gradients, and viscosity, but here also by the Lorentz force associated with currents flowing across the magnetic field.
• Eq. (3.6): The local energy density (here shown in a per-mass formulation of the first law of thermodynamics) is affected by flows, including compression or dilation, thermal conduction, and by viscous and resistive heating.

(Magneto-)Hydrodynamics

47

{A30}

• Eq. (3.7): As mass is a positive definite quantity, it can only strengthen gravity, which can be represented by the gradient of a potential.

• Eq. (3.8): The equation of state couples pressure, density, and internal energy.

• Eqs. (3.3, 3.5): The induction and momentum equations are derived from the (mass-weighted) difference (see Sect. 3.2.2) and the sum of the equations of motions for the electrons and for the ions, each of which includes a term for their collisional coupling.

• ○1 : In case the term ○2 is negligible, Eq. (3.3) describes what is known
as ‘ideal MHD’. In this case (see Sect. 3.4) the plasma and magnetic field must move with each other for velocity components perpendicular to the magnetic field, whereas plasma movement along the field is not affected by that field. In this condition, the field is said to be ‘frozen in’ the plasma. In such a state, the lines of force (‘field lines’) are advected with the flow while unable to break their connectivity between any plasma elements along their length; in non-ideal, or resistive, MHD such connections can be broken through a process known as ‘reconnection’. The concepts of field lines and reconnection are described in Section 3.4. {A30}

• ○2 : This term quantifies the effects of resistivity on the magnetic field

by the dissipation and diffusion of the electrical current j magnetic diffusivity η is constant throughout the medium,

=the4cπn∇te×rmB.○2If

the can

be rewritten as η∇×(∇×B) = −η∇2B (because ∇ · B = 0), which shows that
it causes the magnetic field to decay diffusively; in the absence of ○1 , such as

in a stationary plasma, this makes Eq. (3.3) a diffusion equation for decaying

magnetic field.

• ○3 : For an incompressible fluid, ρ is constant as material flows throughout
the volume under study, which consequently means that ∇ · v = 0, i.e., that the
velocity field is divergence free, and – unless there are terms like ○a to consider – Eq. (3.4) vanishes from the set. That also removes term ○8 from Eq. (3.6), so
that the energy density of the medium can only change by thermal conduction
○9 and by viscous and resistive dissipation ○10 (disregarding here, as we do
throughout this chapter, the effects of radiation). {A31}

• ○5 : As formulated here, the isotropic part of the pressure tensor is expressed
as a scalar, while the other terms are captured in the stress tensor. If only this scalar term is carried, then particle microscopic velocity distributions are taken to be isotropic.

• ○6 : Term ○6 measures the interaction of the Lorentz force and the plasma
flow. The vector product (∇×B) × B can be reformulated (see Eq. 3.23) into the sum of a pressure-like term (that works to expand unless countered) and a

{A31}

48

3 – MHD, field lines, and reconnection

term that is equivalent to a tension (which works to straighten unless countered), showing that the magnetic field in a plasma behaves as if it were both like a gas and like a flexing rod or taut string. There is a special class of magnetic fields in which currents run parallel to the magnetic field; in that case (∇×B) × B = 0, i.e., there is no Lorentz force, and these are consequently referred to as ’force-free fields’, of which the potential field is a special (and lowest-energy) state. As the field is parallel to the current, there is a scalar field α such that ∇×B = αB. If α is a uniform constant, this field is called ‘linear force free’ (which is mathematically easier to work with, but does not develop in general astrophysical settings); if not, the corresponding field is a ‘non-linear force-free’ field (to which we return in Sect. 6.3).
• ○9 : As for term ○2 with uniform magnetic diffusivity η, here a uniform thermal conductivity κ would allow rewriting of term ○9 to be proportional to
∇2T , quantifying diffusion of thermal energy.
• ○a ,○b ,○c : These terms reflect source and loss terms for mass and momen-
tum density per unit volume through, e.g., (de-)ionization of neutrals (including charge exchange) that are important, for example, where comets add gas and dust or around geysers on low-gravity moons (Sec. 2.4), or to the inflow of neutral matter into the solar wind from outside the heliosphere.

A few special cases:

• B = 0: A field-free state (or a non-conducting, and thus current-free, gas in which the field does not apply force to the gas; see also under ’Potential’ below) transforms Eqs. (3.3)–(3.8) into regular hydrodynamic equations.
• v = 0: A static plasma is described by Eqs. (3.3)–(3.8) without terms ○1 , ○3 , ○7 , ○8 , and Qν in ○10 . Moreover, with no flows, no change can occur that
involves bulk flows, so that, for example, the left-hand side of the momentum Eq. (3.5) has to equal 0. This yields an equation for magnetohydrostatic balance in which gravity, pressure gradient, and Lorentz force sum to zero.

•

∂ ∂t

=

0:

Stationary

situation

in

which

none

of

the

variables

can

change.

In

particular,

∂v ∂t

=

0

is

a

situation

with

stationary

flows,

which

can

be

maintained

only for limited times.

• Potential: In the case of a potential field, there are no currents in the
system, i.e., ∇×B = 0. Consequently, term ○6 vanishes because there is no Lorentz force. Term ○2 also vanishes, leaving only term ○1 in the righthand
side of induction Eq. (3.3) (equivalent to the infinitely conducting case of ideal
MHD with frozen-in field, or in which the field is maintained from outside of a
current-free volume). To see to full consequence of this state, however, we need
to realize that ∇× B = 0 means that there is a magnetic potential Φm such that B = −∇Φm from ∇2Φm = 0. Such a Laplace equation, once the boundary

(Magneto-)Hydrodynamics

49

I:3.2

condition is specified, has a unique solution. And for a current-free system with fixed boundary conditions that, in turn, means that B cannot change in
time, such that term ○1 then implies that there is a scalar field Ψ such that
v × B = ∇Ψ, of which one particular case has v ∥ B.
• Force free: See at ○6 above in this listing.

3.2.2 The induction equation
“The induction equation, Eq. (3.3), arises from Ohm’s law combined with the non-relativistic approximation of the Maxwell equations. In its most general form Ohm’s law is a relation between electric current, electric field, magnetic field, plasma motions and electron pressure gradients. Ohm’s law is derived from an equation of motion for electrons in which the interaction with ions (defining the bulk motion of the plasma with velocity v [because the ions, here taken to be dominated by singly-ionized species, by far outweigh the electrons]) is described through a collisional drag term related to the differential motion:

ne

me

dve dt

=

−ne e(E

+

1 c ve

× B) − ∇pe

v + ne me

− ve τei

.

(3.9)

Here ve denotes the electron velocity, τei the collision time between electrons and ions,

e the electron charge, me the electron mass, ne the electron density, and pe the electron

pressure” (omitting gravity). E and B are the electric and magnetic vector fields.

By noting that the differential velocity between ions and electrons is proportional

to the current,

j = ne e (v − ve)

(3.10)

we can reformulate Eq. (3.9), when combined with the analogous version for the ions, to yield a formulation of Ohm’s law (here ignoring electron inertia and assuming pressure to be a scalar, i.e., isotropic; compare with Section 3.4):

j

=

τeinee2 (E me

+

1 v
c

×

B)

−

τeHiealljte×rmB mec

+

τeie me

∇pe

.

(3.11)

{A32} Note that when the electric field expressed through the electron pressure gradient is ignored, this equation can be rewritten to an equivalent Ohm’s law discussed for the ionosphere in Ch. 2 that has a conductivity tensor with components as in Eqs. (2.23)-(2.25), from which terms with Mi disappear for the fully-ionized plasma because there are only ion-electron collisions, and in which the electron magnetization subject to collisions with neutrals, Me, is replaced by Mei = eB/(mcνei) for νei = 1/τei. In other words, the Hall term in Eq. (3.11) takes care of the anisotropic part of the conductivity in the fully ionized plasma. The Pedersen current, directed along E is part of the first term on the right-hand side. {A33}
Specifically, “the second term on the right-hand side describes the Hall current, which becomes important if the collision time is longer than the electron gyration time, i.e., when τei ωL > 1, where ωL = eB/me denotes the Larmor

{A32}

{A33} I:3.2

50

3 – MHD, field lines, and reconnection

(or [electron] gyro-)frequency. The Hall term leads to anisotropic plasma conductivity with respect to the magnetic field direction and is typically important in low-density plasmas in which τei can be very large”. In many settings in heliophysics, the last two terms in Eq. (3.11) are ignored “(unless high-frequency plasma oscillations are considered), leading to the simplified Ohm’s law

1

j

=

σe(E

+

v c

×

B)

(3.12)

with the plasma conductivity

σe

=

τeinee2 me

.

(3.13)

Using

Amp`ere’s

law,

∇×B

=

4π c

j

,

yields

for

the

electric

field

in

the

laboratory

frame

1

c

E = − v × B + ∇×B

c

4πσe

(3.14)

leading to the induction equation through one of the Maxwell equations:

∂B = −c∇×E = ∇× (v × B − η ∇×B)
∂t

(3.15)

with the magnetic diffusivity

c2

η=

.

4πσe

(3.16)

In MHD, the equations are typically expressed in terms of the magnetic field B

and flows v, with electric fields and currents eliminated from the system. This is done

primarily out of mathematical convenience, since formulating the problem in terms

of currents leads to intractable equations involving integrals of the currents over the

entire volume under study.”

Whether a formulation in terms of the magnetic field or electrical currents is more

convenient also depends on the inhomogeneity and anisotropy of the conductivity.

The most extreme example is electrical engineering where cables give full control over

the current, and thus a current-based description is clearly the method of choice. A

formulation in terms of currents can be easier to work with also when currents can only

flow along the field or are restricted to relatively thin layers with high conductivity, such

as is the case in the ionosphere. In most MHD problems with highly conducting fluids,

however, there is no a priori control over where currents flow, so that dealing with the

magnetic field is typically the better choice. Because of this, solar and heliospheric

physicists generally use arguments primarily based on the magnetic field; in space

physics, however, and in particular in ionospheric physics, currents are often discussed.

Of interest to the induction equation Eq. (3.3) is the relative importance of the

advection and diffusion-like terms on the right-hand side. One way to assess that is

to reformulate it into characteristic scales and a frequently occurring dimensionless

number: “Let Lt be a typical length-scale and vt a characteristic velocity of the problem. Expressing the time in units of Lt/vt and the spatial derivatives in the

I:3.2.3

(Magneto-)Hydrodynamics

51

induction equation Eq. (3.3) in units of Lt leads to the dimensionless form of the

induction equation

∂B

1

= ∇× v × B − ∇×B

∂t

Rm

(3.17)

with the magnetic Reynolds number

Rm

≡

vt Lt η

.

(3.18)

The limit Rm ≪ 1 is referred to as diffusion dominated regime, in which the (dimensional) induction equation reduces to a diffusion equation of the form

∂B = η∇2B . ∂t

(3.19)

Here we made the additional simplifying assumption of a constant magnetic diffusivity

η. Assuming that the magnetic field has a typical length scale Lt, we can estimate [its

decay time scale:]

τd

∼

L2t η

.

(3.20)

The limit Rm ≫ 1 is referred to as the advection-dominated regime, in which the induction equation reduces to the equation of ideal MHD (except for possible boundary

layers where diffusivity could be still important)

∂B = ∇× (v × B) .
∂t

(3.21)

Expanding the expression of the right-hand side of this ideal induction equation leads

to

∂B

= −(v · ∇)B + (B · ∇)v − B (∇ · v) .

(3.22)

∂t

While the first term on the right-hand side describes the advection of magnetic field,

the last two terms describe the amplification by shear (second term) and compression

(third term).” Of interest to the momentum equation Eq (3.5) is that a vector identity allows us
to reformulate the Lorentz force “to equal:

1

B2 1

(∇×B) × B = −∇ + (B · ∇)B,

4π

8π 4π

(3.23)

which shows that the Lorentz force is a sum of an isotropic pressure-like force and

a tension force related to the curvature of the field” (note that both of these are

insensitive to a reversal of the direction of the magnetic field). Because the pressure and tension terms, as does therefore the full Lorentz force, scale as O(Bt2/Lt) (where the subscript ’t’ denotes a typical value of the quantity) they can be compared in

magnitude to the pressure gradient force O(pt/Lt); the ratio of magnetic and gas pressure terms in Eq. (3.5) yields an often-used dimensionless number in heliophysics,

the plasma β: {A34}

{ⓈA35}

8πp β ≡ B2 .

(3.24)

I:3.2

{A34} {ⓈA35}

52

3 – MHD, field lines, and reconnection

IV:10.2.1

3.3 Waves in magnetized plasmas
Before we proceed with a discussion of field lines and reconnection, we look into an important aspect of a magnetized plasma, namely how it carries waves. Waves are important, among other things, in communicating information about changes in the field’s structure or in boundary conditions or the effects of obstacles embedded in flows, while moreover they transport energy. “The waves that carry information through a magnetized plasma differ from the sound waves of a neutral gas, partly because of the anisotropy imposed on the fluid by a magnetic field and partly because the waves must be capable of carrying currents that modify the properties of both matter and magnetic field. The properties of such waves can be derived from the MHD [equations] by analyzing the evolution of small perturbations.
Consider a uniform plasma with constant pressure and density (p and ρ) whose center of mass is at rest (v = 0). Assume that a constant background field (B) is present and that neither sources nor losses need be considered. Small departures from this background state are taken to vary with space (x) and time (t) as ei(k·x−ωt). Here, k is the wave vector and ω is the angular frequency of the wave. Perturbations occur in density dρ, velocity dv, pressure dp, current j, and field b. Terms linear in small quantities in Eqs. (3.4) and 3.5) satisfy

−ωdρ + ρk · dv = 0

(3.25)

1

1

−ωρdv = −kdp + b(k · B) − k(b · B).

4π

4π

(3.26)

[If we assume an isentropic (i.e., adiabatic and reversible) process, then Eq. 3.8 becomes pρ−5/3 = constant, so that] the pressure perturbation in terms of the density

perturbation is

dp/p = γdρ/ρ

(3.27)

and [the ideal induction equation Eq. (3.3 (with η ≡ 0)])

ωb = dv(k · B) − B(k · dv).

(3.28)

The solutions to Eqs. (3.25) to (3.28) are the roots of the equation

(ω2 − vA2 k2 cos2 θ)[ω4 − ω2k2(c2s + vA2 ) + k4vA2 c2s cos2 θ] = 0,

(3.29)

where θ is the angle between k and B, and the Alfv´en speed (vA) and the sound speed (cs) have been introduced. These quantities characterize the speed of propagation of waves in a magnetized plasma and are defined by

vA2 = B2/4πρ,

(3.30)

c2s = γp/ρ.

(3.31)

The sound speed has the form familiar for a neutral gas. The Alfv´en speed is a second natural wave speed characteristic of a magnetized plasma. Just as [one can work with

MHD, magnetic field lines and reconnection

53

the dimensionless] sonic Mach number as the ratio of the flow speed to the sound speed, it is useful to define a dimensionless Mach number, the Alfv´enic Mach number

MA = v/vA,

(3.32)

related to the Alfv´en speed. As mentioned [at Eq. (3.23)], the quantity B2/8π is the pressure exerted by the
magnetic field, so both of the basic wave speeds are proportional to the square root of a pressure divided by a density. [. . . When β = 8πp/B2 ≪ 1,] magnetic effects dominate the effects of the thermal plasma, but in a high-β plasma, the plasma effects dominate.
Equation (3.29) is of sixth order in ω/k with three pairs of roots. One pair results from setting the first factor in Eq. (3.29) to zero; the resulting dispersion relation is

(ω2 − vA2 k2 cos2 θ) = 0.

(3.33)

This solution describes waves referred to as Alfv´en waves. For this dispersion relation to apply, the magnetic perturbation must be perpendicular to both B and k (see Fig. IV:10.1a). This orientation implies that to first order in small quantities, the Alfv´en wave does not change the field magnitude [(B + b)2 = B2 + 2(B · b)2 + b2 ≈ B2]. The wave phase speed is vph = ω/k and vph = ±vA cos θ. [Wave packets] carry information at the group velocity, vg = ∇kω, where the subscript on the gradient indicates that the derivatives are taken in k space; the solution is vg = ±Bˆ vA where Bˆ is a unit vector along the background field. The remarkable property of these waves is that they carry information only along the background field, and they bend the field without changing its magnitude. These properties are of considerable importance in interpreting the interaction of a flowing plasma with the solid bodies of the Solar System [(discussed in Ch. 5)].
Eq. (3.29) has two more pairs of roots, the zeroes of the fourth order polynomial in square brackets in Eq. (3.29), i.e., the solutions

vp2h

=

ω2/k2

=

1 2

c2s + vA2 ± [(c2s + vA2 )2 − 4vA2 c2s cos2 θ]1/2

.

(3.34)

The solutions (two pairs, one positive and one negative, of roots) correspond to what
are unimaginatively referred to as fast-mode (or magnetosonic) and slow-mode waves.
The wave perturbations of both modes may have magnetic perturbations along and
across B (see Fig. IV:10.1b). Perturbations along B change the field magnitude and the
thermal pressure. The fast mode changes of thermal and magnetic pressure are in phase
with each other; this implies that the total pressure fluctuates. The slow mode changes
of thermal and magnetic pressure are in antiphase, and the total pressure fluctuations
are very small. For waves propagating along the background field (cos θ = ±1), the solutions to Eq. (3.34) are c2s and vA2 , with the larger of the two applying to the fast mode. For waves propagating at right angles to the background field (cos θ = 0), the [solutions] are c2s + vA2 and 0, indicating that only fast-mode waves propagate across the field.” {A36}

{A36}

54

3 – MHD, field lines, and reconnection

I:4.1

3.4 MHD, magnetic field lines and reconnection

“One of the most idiosyncratic aspects of space physics is the central role assigned

to magnetic field lines. Particularly in studies of the Sun, the heliosphere and the

magnetosphere, magnetic field lines are treated as full-fledged physical objects with

their own dynamics. The electrical current, when needed, is derived from the magnetic

field lines. These practices appear at odds to the basic approach, followed in elementary

electrodynamics, of deriving the magnetic field from a current distribution and treating

magnetic field lines at best as fictitious curiosities. However, physical laws such as

Amp`ere’s law (without displacement current [because velocities are assumed to be well

below relativistic]),

4π ∇ × B = j,
c

(3.35)

do not attribute a causative nature to either side of the equality; they simply state the

equality of two quantities. So either approach to satisfying Eq. (3.35), beginning with

either j or B, is a valid one.

[The central role of B in space physics has been furthered tremendously by the

introduction of the concept of the field line.] A magnetic field line, sometimes called a

line of force, is a space-curve r(ℓ) which is everywhere tangent to the local magnetic

field vector, B(x). This description can be cast as the differential equation

dr B[r(ℓ)]

=

,

dℓ |B[r(ℓ)]|

(3.36)

whose solution, starting from some initial point r(0), is a magnetic field line. [. . . ] A field line is a curve, and therefore has zero volume. A flux tube may be constructed by bundling together a group of field lines. The net flux, Φ, of the tube is the integral
B · da over any surface pierced by the entire tube. Because ∇ · B = 0, the tube must have the same flux at every cross section.
The only way, in general, to find a field line is to integrate the differential Eq. (3.36). A solution to the field line equation, Eq. (3.36), can in principle be found for a magnetic field at any instant. What is not immediately evident is why such a curve should be physically significant, even if one concedes that the magnetic field itself is significant. There is, in fact, no single reason that field lines will be significant under general circumstances — this is why students are often warned not to attribute undue importance to them. There are, however, numerous circumstances arising in space physics whereby a magnetic field line can achieve a degree of [utility]. The following is a brief list of the most common, applicable to a wide variety of plasma regimes from general 1, to the fluid regime 2, to MHD 3, to ideal MHD 4.

1. General: single particle motion. Subject to no other forces than a relatively stationary magnetic field, [the guiding center of a charged particle will remain tied to a single field line while the particle gyrates about that] according to its mass and charge [(discussed in detail in Sect. 8.2)]. Drifts will displace the particle’s guiding center by several gyro-radii after it has traversed a length

MHD, magnetic field lines and reconnection

55

comparable to the field’s curvature radius or gradient scale. Global scales of space plasmas are typically much, much greater than the gyro-radii of their electrons, and to a lesser extent of their heavier ions (the Earth’s geomagnetic ring current is a counterexample to this[; see Activity 24]). Waves in the field may scatter particles (important in, e.g., the Earth’s radiation belts, [and for solar and galactic cosmic rays propagating through the heliosphere, see Ch. 8]), but this too is generally unimportant. Field lines therefore serve as excellent approximations of the electron orbits. [. . . ]
2. Fluid regime: thermal conductivity and solar coronal loops: In a diffuse, hightemperature plasma, thermal energy is conducted principally by electrons. When electrons are strongly magnetized (i.e., the cyclotron frequency is much greater than the collision frequency) their orbits will follow field lines over long distances between collisions at which point they scatter a perpendicular distance no greater than a single gyro-radius. The huge disparity between parallel and perpendicular scattering distances makes thermal conductivity highly anisotropic. Consequently, heat is conducted parallel to the magnetic field far more readily than perpendicular to the field.
Due to this anisotropic conductivity, heat deposited somewhere in a plasma is rapidly and efficiently conducted to all points on the same field line, at least while collision frequencies remain relatively low. [In the coronal setting, for example, the] plasma β is also generally low, so plasma flows are mechanically confined by the field. This means that a bundle of field lines will behave as a one-dimensional autonomous atmosphere, at least as long as reconnection is relatively unimportant. [. . . ]
3. MHD: Alfv´en wave propagation: Low-frequency waves in a magnetized plasma [(see Section 3.3)] comprise three branches: slow magnetosonic, fast magnetosonic and shear Alfv´en waves. The group velocity of the shear Alfv´en wave is exactly parallel to the local magnetic field. In the limit of very short wavelengths, any small localized disturbance will therefore propagate along a path following a magnetic field line. This means that a given field line will ’learn’ of perturbations anywhere along itself at the Alfv´en speed. In this sense the magnetic field line has a dynamical integrity similar to that of a piece of string. Indeed, it is common to derive the Alfv´en speed intuitively using the analogy to a string under tension. [. . . ]
4. Ideal MHD: frozen-in field lines: [. . . ] At its simplest, the frozen-in-field-line theorem states that if two fluid elements lie on a common field line at one time, then they lie on a common field line at all times past and future. This follows directly from the ideal induction equation, ([Eq. (3.3) with η ≡ 0]), and from the fact that fluid elements move at the same velocity v that appears in it.”
The mathematics of ideal MHD is such “that differentiation along a field line is interchangeable with differentiation along a flow trajectory. From this it follows

I:4.1

56

3 – MHD, field lines, and reconnection

I:2.5

{ⓈA37}

that a field line linking two fluid elements can be traced either before or after following the flow of those elements. That is a restatement of the [frozen-in-fieldline] theorem introduced above. One can thereafter imagine ’labeling’ all the fluid elements along a given field line and then following those fluid elements as they move at their own velocities, v. These material elements, which are manifestly real, will trace out a single field line at all times, so that [the field line is a useful concept in thinking about plasma motions. Wherever η ̸= 0 in Eq. (3.3) field lines lose their nature as coherent entities; more on this below where we discuss reconnection.]”

Field lines and flux tubes have taken on a remarkable degree of utility in the thinking of many working in the various branches of heliophysics. “The motion of plasma along the magnetic field does not stress the field and incurs no dynamic back-reaction on the plasma through the action of the Lorentz force. Magnetic field lines therefore serve as conduits for moving energy, mass, momentum and energetic particles from point to point in the heliosphere. Heliophysics accordingly focuses on the magnetic connectivity of the Earth to the Sun, of the magnetotail to the polar caps, of the Io plasma torus to the Jovian magnetic field, and so forth. Magnetic field lines are truly the interstate highway system, the Autobahn network, the autostrada web of the heliosphere.” {ⓈA37}
Field lines as true, persistent entities have their greatest utility in ideal (nonresistive) MHD. But ideal MHD, in which field lines always connect the same parcels of plasma, fails when magnetic diffusivity becomes important in the MHD approximation, or when the basic assumptions of MHD itself fail on the smallest time or length scales. Then field is no longer ’frozen in’ wherever that happens, and the very concept of continuity of field lines in space and time loses validity. Failure of the field-line concept as it is discussed above is captured by the term ’reconnection.’ This term, widely used, turns out to be very loosely defined. “It can be used to refer to the changing connectivity in a vacuum potential field as much as to the decoupling of particle motions from the background magnetic field by any number of concepts, ranging from inertia to wave-particle interactions, or from resistivity to infinitesimal current sheets. It is thus as much a culturally accepted term for something that we really do not understand, as a descriptor of a well-understood consequence: we can say that reconnection occurs whenever the approximation of frozen-in flux fails.”
Non-ideal MHD sees reconnection as a consequence of resistivity. “To determine a realistic resistivity for a collisionless plasma requires consideration of the generalized Ohm’s law. For a fully ionized plasma it can be written as

○i ○ii

○iii

○iv

○v

E

1 =− v×B+
c

j σe

+

me ne2

∂j + ∇ · (vj + jv)

+ j × B − ∇ · pe ,

∂t

nec

ne

(3.37)

where vj and electron stress

jv are tensor.

dyadic tensors [(with components vnjm
Term ○i on the right-hand side of this

and

vmjn)]

and

p
e

is

the

equation is the convective

electric field, while the term ○ii is the field associated with Ohmic dissipation caused by

II:1

I:5.2.2

MHD, magnetic field lines and reconnection

57

electron-ion collisions. The conductivity, σe, is the inverse of the electrical resistivity, η.
The next group of terms ○iii describes the effects of electron inertia [(which is ignored

in Eq. (3.11) as another approximation of Ohm’s law, while the latter describes also
simplifies pressure by assuming it to be isotropic). The next term, ○iv ,] is the Hall

effect. Ion inertia is negligible because the large mass of the ions means that they do
not contribute significantly to a change in the current density. Finally, the term ○v

includes the electron gyro-viscosity, which is considered by many to be important at

[any point where the magnetic field vanishes, i.e., at a] magnetic null. For a partially

ionized plasma, collisions between charged particles and neutrals lead to additional

terms associated with ambipolar diffusion.

Although all of the terms on the right-hand side of the generalized Ohm’s law,

other than the first, allow field lines to slip through the plasma, they do not all produce
dissipation. For example, the inertial terms in ○iii do not cause the entropy of the

plasma to increase. Thus, even though one may speak of inertial effects as creating an

effective resistivity, this resistivity does not necessarily lead to dissipation.

Which terms are important in a particular situation depends not only upon the

plasma parameters, but also upon the length and time scales for variations of these

parameters. For magnetic reconnection, we normally want to know which non-ideal

terms are likely to be significant within the current sheet where the frozen-flux condition

is violated. Because each non-ideal (i.e., diffusion) term in the generalized Ohm’s

law contains either a spatial or temporal gradient, we can estimate the significance of

any particular term by computing the gradient scale-length, Lt, required to make the

term

as

large

as

the

value

of

the

convective

electric

field,

1 c

v

× B,

outside

the

diffusion

region.

Consider, for example, the three inertial components of term ○iii . If we assume that

∇ ≈ 1/Lt, |j| ≈ (c/4π)Bt/Lt and ∂/∂t ≈ vt/Lt, say, where Lt is a typical length-scale
and vt a typical velocity, then these three components of ○iii will be of the same order

as the convective electric field if

cme vtBt 4πne2 L2t

≈

VtBt . c

(3.38)

In other words, in order for the inertial terms to be important in a current sheet, its thickness ℓinertia should be

ℓinertia ≈

c2me 4πne2

1/2
≈ λe,

(3.39)

where

c λe = ωpe =

c2me 4πne2

1/2
= 5.3 × 105 n−1/2

(3.40)

is the electron-inertial length or skin-depth [(which characterizes the depth in a plasma into which electromagnetic radiation can penetrate)], c is the speed of light and

ωpe = (4πne2/me)1/2 = 5.6 104n1e/2

(3.41)

58

3 – MHD, field lines, and reconnection

is the electron plasma frequency.
Similarly, for the Hall term ○iv

Bt2 ≈ VtBt 4πneLt c

(3.42)

or

c ℓHall ≈ MA

µmp 4πne2

1/2
≈

λi ,

MA

(3.43)

where

c λi = ωpi =

µc2mp 4πne2

1/2
= 2.3 × 107

µ 1/2 n

(3.44)

is the ion-inertial length or skin-depth [(below which ions decouple from electrons, and

the magnetic field may no longer be frozen into the plasma overall but instead into the

electron fluid), and µ = mi/mp]. The Alfv´en Mach number equals [MA = Vt/vA,] and ωpi = (4πne2/mi)1/2 the ion plasma frequency, and vA the Alfv´en speed [(see Eq. 3.30
and Table 18.2)].
For the electron-stress term ○v we can write

nkTt ≈ VtBt

ne Lt

c

(3.45)

if we assume |pe| ≈ nkTe and Te ≈ Ti ≈ T . Solving for Lt leads to

β1/2 ℓstress ≈ MA rgi,

(3.46)

where [the plasma β is given by Eq. (3.24) and the ion-gyro radius for the average thermal velocity (vT i) equals rgi = (kT mi)1/2c/eB; see also Table 18.2.]
Finally, for the collision term ○ii , j/σe,

cBt ≈ MAvABt .

4πσeLt

c

(3.47)

[where σe−1 is] also the magnetic diffusivity, η. Using Spitzer’s formula for the collisional resistivity, η, of a plasma (see ) we obtain

η

=

(kmeTe)1/2 ne2λmfp

,

(3.48)

where

λmfp

=

3 4π1/2

(kTe)2 ne4 lnΛ

=

1.1

×

105 Te2 n lnΛ

(3.49)

is the mean-free path for electron-ion collisions. Combining these expressions with

those for the electron and ion inertial lengths we obtain [an estimate for the length

scale below which effects of collisions become important to field diffusion:]

ℓcollisions

≈

β1/2 MA

λeλi . λmfp

(3.50)

I:3

MHD, magnetic field lines and reconnection

59

Table 3.4: Comparison of order-of-magnitude plasma parameters in different environments (cgs-Gaussian units – i.e., length scales in cm, n in cm−3, T in K, B in Gauss, electric fields in statV cm−1). [Modified after I:5, merging two tables in SI units]

Parameter
region scale Ls density nt temperature Tt field strength Bt Debye length λD ion gyro radius rgi ion inertial length λi Coulomb logarithm ln (Λ) coll. mean-free path λmfp 5 ℓinertia(λe) ℓHall(λi) ℓstress ℓcollision plasma β Lundquist no. Lu(≈ Rm) Dreicer field ED EA(= vAB/√c) ESP(= EA/ Rm)

Laboratory experiment1
10 1014 105 103 10−4 10−1 1 11 1 10−2 1 10−1 10−2 10−2 103 10−1 1 10−2

Terrestrial
magnetosphere2 109 10−1 107 10−4 105 107 108
33 1018 106 108 107 10−5 10−1 1014 10−17 10−6 10−13

Solar corona3
1010 109 106 102 10−1
10 103
19 106
10 103 10−1 10−5 10−4 1014 10−6 101 10−7

Solar interior4
109 1023 106 105 10−8 10−2 10−4
3 10−7 10−6 10−4
1 10−1 104 1010 107
1 10−6

1 The Magnetic Reconnection eXperiment (MRX) at Princeton Plasma Physics Laboratory; 2 plasma sheet; 3 above a solar active region; 4 at the base of the solar convection envelope [at a depth of about 200,000 km around which many consider primary dynamo mechanisms to operate; 5 note that this is a purely collisional mean-free path, ignoring other couplings that may occur through the magnetic field].

Note that the length-scale, ℓcollision, of the spatial variations required to achieve significant field-line diffusion is inversely proportional to the mean-free path, λmfp. As λmfp increases, the diffusion caused by collisions becomes less effective, and increasingly sharper gradients are required to maintain the size of the dissipation term, j/σe.
[Table 3.4 lists] various plasma parameters along with the characteristic scale-lengths for four different regions where reconnection is thought to occur. The parameter Ls is the global (system-level) scale-size of the region, and the fundamental quantities from which all other parameters are derived are the density n, temperature T , and magnetic field B. For convenience, we assume that the Alfv´en Mach number MA is unity and that the electron and ion temperatures are roughly equal. The most extreme plasma environments listed in Table 3.4 occur in the magnetosphere, which is completely

60

3 – MHD, field lines, and reconnection

collisionless, and in the solar interior which is highly collisional.
In addition to the parameters discussed above, Table 3.4 also lists the value of the Debye length [whose expression is shown in Table 18.2.] The number of particles within a Debye sphere (i.e., 4πnλ3D/3) must be larger than unity in order for the generalized Ohm’s law to hold. Otherwise, the collective behavior which characterizes a plasma breaks down. The number of particles in a Debye sphere for the environments shown in Table 3.4 ranges from 1014 for the magnetosphere to only about four for the solar interior at the base of the convection zone. Also shown in the table is the Lundquist number, Lu, which is the same as the magnetic Reynolds number, Rm [introduced in Eq. (3.18)], when the flow and Alfv´en speeds are the same. For a collisional plasma the Lundquist number based on Ls can be expressed as

Lu

=

vA vd

=

LsvA η

= 2. × 108 LsTe3/2Bt (µn)1/2ln(Λ)

(3.51)

[. . . ] In the expression on the right, η has been replaced by Spitzer’s formula for the electrical resistivity of collisional plasma.
The characteristic scale-lengths in Table 3.4 provide an indication of which terms in the generalized Ohm’s law of Eq. (3.37) are likely to be important for reconnecting current sheets. As with MHD shocks and turbulence, the large-scale dynamics of the flow cause the current sheet to thin until it reaches a length-scale where field-line diffusion is effective. Thus, in principle, the term with the largest characteristic lengthscale in Table 3.4 is the one that will be most important. Because the Hall term has the largest length in every environment except the solar interior, one might conclude that it is generally the most important. However, this conclusion does not take into consideration the fact that the Hall term tends to zero in the region of a magnetic null point or sheet. The Hall term on its own does not contribute directly to reconnection, since it freezes the magnetic field to the electron flow. [. . . ] An excessively small scale does indicate that any process associated with that term is unlikely to be important. Therefore, on this basis, we can conclude that collisional diffusion is not important in the terrestrial magnetosphere or the solar corona, and that the electron-inertial terms and the Hall term are not important in the solar interior. [. . . On the other hand, if a term is not associated with an obviously ’excessively small’ scale, it is difficult to know whether a particular term is really as important as suggested by its relative length scale; evaluating such cases] requires a complete analysis of the kinetic dynamics, which is a rather formidable task.
Although the collision length-scale, ℓcollision, is equally small in both the magnetosphere and the corona, the general importance of collisions for these two regions is quite different. In the magnetosphere the collision-mean-free path, λmfp, is nine orders of magnitude larger than the global scale-size, Ls, but in the corona it is four orders of magnitude smaller than the global scale. Thus, we can be confident that collisional transport theory applies to large-scale structures in the corona even though it is not applicable within thin current sheets or dissipation layers. By contrast, in the magnetosphere, collisions are so few that collisional transport theory does not apply at

A few notes about conditions

61

any scale. Another important issue concerning the applicability of collisional theory is the
strength of the electric field in a frame moving with the plasma. If this field exceeds the Dreicer electric field defined by

ED

=

e ln(Λ) λ2D

=

4πe3 k

ln(Λ) n Te

=

10−11 n ln(Λ) , Te

(3.52)

runaway acceleration of electrons will occur. The most likely location for the production of runaway electrons in a reconnection process is in a thin current sheet that forms at the null point. This field could be as large as the convective electric field based on the Alfv´en speed, that is

EA

=

1 c vABt

=

7.2

Bt2 (µn)1/2

,

(3.53)

or as low as the Sweet-Parker electric field

ESP

=

EA , Rm1/2

(3.54)

where Rm is the magnetic Reynolds number based on the inflow Alfv´en speed (i.e., the inflow Lundquist number). As shown in Table 3.4, the Dreicer field in the magnetosphere is much smaller than EA or ESP, so runaway electrons will always be generated by reconnection there. On the other hand, in the solar interior the Dreicer field is so large that runaway electrons never occur. In the intermediate regimes of the laboratory and the solar corona, the Dreicer field lines between EA and ESP, so perhaps runaway electrons are only produced when very fast reconnection occurs.
Even in completely collisionless environments like the Earth’s magnetosphere, it is still sometimes possible to express the relation between electric field and current density in terms of an anomalous resistivity. For example, [. . . ] the electron inertial
terms ○iii in the generalized Ohm’s law of Eq. (3.37) lead to an anomalous resistivity

1 σe∗

=

πB⊥ , 2ne

(3.55)

where B⊥ is the field normal to the current sheet. This resistivity is derived solely from a consideration of the particle orbits, and in the magnetotail current sheet it may be larger than any anomalous resistivity due to wave-particle interactions. A typical example of the latter is the anomalous resistivity due to ion-acoustic waves”.
For some discussion of reconnection in two and three dimensions in the Heliophysics books, see I:5.3 and I:5.4. More on the effects of reconnection follows in Chs. 6 and 8.

3.5 A few notes about conditions
3.5.1 Solar atmosphere vs. terrestrial magnetosphere
The scale lengths estimated for the importance of terms in Ohm’s law in Table 3.4 are very much smaller than the scale of the corona itself and even compared to any

62

3 – MHD, field lines, and reconnection

I:11.3

active region, but importantly also very much smaller than the angular resolution achievable by imaging instruments (currently about 1 arcsec or ∼ 700 km for spacebased EUV imagers). Consequently, the scale on which reconnection occurs in the corona is not observed, while the effects of such reconnection become apparent in the magnetic geometry and plasma atmospheres on scales well above the reconnection itself.
In contrast, in the terrestrial magnetosphere all but the length scale, ℓcollision, on which collisional effects could contribute significantly as a term in the generalized Ohm’s law are large enough that spacecraft can scan reconnection regions as they fly through, while constellations of spacecraft can probe reconnection in multiple dimensions.
Another significant distinction is that in the terrestrial magnetosphere the ion-gyro radius (particularly for relatively heavy and energetic particles) is not small compared to the scale on which these particles probe the magnetic field. This is an important cause behind what is known as the ring current (see Sect. 8.2) that is largely carried precisely by such particles. For solar conditions, in contrast, such effects of particle gyration are not directly evident on any observable scale.
3.5.2 Heliosphere
“Adopting typical solar wind values near Earth of nt = 5 particles/cm3 for density, vt = 400 km/s solar wind speed and Bt = 50 µG magnetic field strength (values consistent with Table [2.4]) we can evaluate the expected energy density of the solar wind, which can be broken down into three components: flow, magnetic and thermal. [. . . ] The flow energy density is estimated to be

ev,⊙ ≡

1 ρv2 2

sw

≈

1 2

mpnvs2w

≈

7

×

10−9

n(cm−3) 5

The energy density of the solar wind’s magnetic field is

v(km/s)

2
erg/cm3. (3.56)

400

eB,⊙ ≡

B2 8π

≈ 1. × 10−10

B(µG) 50

2
≈ 0.015ev,⊙ erg/cm3,

(3.57)

while the thermal energy density using values from Table [2.4] is

3 eT,⊙ ≡ 2 nk(Te + Ti)

≈ 2.5 × 10−10 n(cm−3) 6
≈ 0.03ev,⊙ erg/cm3

Te (K ) 1.2 × 105

+

Ti (K ) 1.4 × 105

(3.58)

where Ti,e are the solar wind ion and electron temperatures. Taking the [values from Table 2.4], the above [These] estimates show that the bulk of the energy in the solar wind at Earth is in the flow:” ev,⊙ ∼ 30eB,⊙ ∼ 70eT,⊙ {A38}

{A38}

Chapter 4
Dynamos of Sun-like stars and Earth-like planets
Chapter topics:
• Basic stellar structure (colors, masses, radii), and (cyclic) magnetism • Basic structure of the terrestrial planets and the contrast of the rotation/convection
time scales (in the Reynolds number) with stars • Principle of dynamos: the conversion of mechanical energy into electromagnetic
energy through induction • Energy sources and fluid flows that enable stellar and planetary dynamos
Key concepts:
• Rotation (through Coriolis forces) introduces a source term for magnetic field in the induction equation
• The mean-field dynamo approximation relies on a separation of scales not strictly supported by the convective spectrum
• Back reaction of the magnetic field on the flow limits dynamo strengths • An ’interface dynamo’ relies on storage of field below the convective envelope,
and a ’flux-transport dynamo’ involves global meridional circulation
4.1 Dynamo settings
Stellar and planetary dynamos thrive wherever sufficiently vigorous flows of a conducting medium transport substantial thermal energy in an adequately spinning body. The energy transported has to come from a reservoir that may date back to the formation of the body (in planets or very young stars) or may have its origin in nuclear fusion (in stars) or in solidification – the latter often accompanied by chemical separation – or nuclear fission (in terrestrial planets). The flows that transport the energy may be dominated by Coriolis forces (in planets where flows are slow compared to spin rates) or by stratification (including chemical gradients in planets, while in stars pressure gradients of the compressible medium limit how far matter can efficiently
63

64

4 – Dynamos of Sun-like stars and Earth-like planets

I:3.3

III:5.1

rise before overturning). The amount of energy transported is regulated by the source in the deep interior as well as by the sink at the top of the dynamo region. In Sun-like stars that sink is the stellar surface, and the properties of radiative transfer through these surface layers are important in determining the internal structure of the entire star as it balances the energy produced by nuclear fusion with its luminosity. In a planet like Earth, the energy transport in the dynamo region of the core is determined to a large extent by the convective motions in the enveloping mantle that transport heat to where it is ultimately lost through the surface.
“The formal difference between the type of dynamos that we are interested in here and the self-excited dynamos in power plants is the homogeneous distribution of conductivity (that would lead to a short-circuit situation) that does not put any constraints on electric currents (electric wires could be considered as special cases of a highly inhomogeneous conductivity distribution). For this reason these dynamos are also called homogeneous fluid dynamos.”
In stars, “[t]hermonuclear fusion in their cores converts matter into thermal energy and electromagnetic radiation which, in the Sun, is transported outward via the diffusion of photons. In the solar envelope, the plasma becomes more opaque as the temperature drops, which inhibits radiative diffusion and steepens the temperature gradient relative to the adiabatic temperature gradient. The stratification soon becomes superadiabatic [(i.e., has a temperature gradient steeper than for adiabatic conditions in which no energy enters or leaves a volume of gas)] and thermal convection [gradually] takes over as the primary mechanism for transporting energy to the solar photosphere where it is radiated into space. {A39} [All stars with a mass of somewhat above that of the Sun or less than that have such a convective envelope during their ’main-sequence’ (equilibrated hydrogen-fusing) phase (see Figs. 4.1 and 4.2); the least massive stars are fully convective. All of these stars power a dynamo during the longest-lived mature phase, and all stars do during their initial birth phases and in the last phases of their lives, both of which are short compared to the mature phase (Ch. 10). Stars cool enough to have a convective envelope reaching into their surface layers are known as ’cool stars’. [8] {A40}
The solar convection zone occupies approximately the outer 30% of the Sun by radius. It is here where [a small fraction of the] internal energy of the plasma is converted to kinetic energy and then [a small fraction of that] to magnetic energy, aided by radiation and gravity. Radiative heating [of the bottom of the] convection
8 Astronomers characterize the properties of stars based on their spectrum. The overall shape gives an indication of the surface temperature, while details of spectral lines (generally in absorption, but some in emission) provide finer detail used in classification schemes. One such scheme frequently used is that of ’spectral type’ in the Morgan-Keenan (MK) scheme: only after the classes were introduced was a monotonic mapping to temperature established, going from hot to cooler: O, B, A, F , G, K, M , L, and T (with the last two fairly recent additions for very cool, very faint stars, with T reaching the domain of ’brown dwarfs’). The letter is followed by a subclass from 0 to 9, and commonly an indicator of ’luminosity class’: a roman numeral indicative of the size of the star: I, II, III, IV, and V for supergiants, bright giants, giants, subgiants, and main-sequence or dwarf stars. The term ’main sequence’ refers to a band in brightness-color diagrams, such as Fig. 4.2, within which stars spend most of their lives, as long as they are steadily fusing hydrogen into helium.

{A39}

{A40}

Dynamo settings

65

Figure 4.1: Schematic representation of the radiative (light grey) and convective (dark grey) internal structure of main-sequence stars. The thickness of the outer convection zone for the A-star is here greatly exaggerated; drawn to scale it would be thinner than the black circle delineating the stellar surface on this drawing. Relative stellar sizes are also not to scale: a B0 V star has a radius of ∼ 7.5 R⊙, and and M0 V star has a radius of ∼ 0.6 R⊙, i.e., 12 times smaller. [Fig. III:2.10]
zone and radiative cooling in the photosphere maintain a superadiabatic temperature gradient that sustains convective motions by means of buoyancy. In a rotating star, convection transports momentum as well as energy, establishing shearing flows and global circulations. These mean flows work together with turbulent convection to amplify and organize magnetic fields through hydromagnetic dynamo action, giving rise to the rich display of magnetic activity so striking in modern solar observations.”
The Sun’s large scale magnetic field exhibits a quasi-periodic modulation on a roughly 11-year basis during which the level of magnetic activity waxes and wanes as a pattern of activity migrates from mid to low latitudes, then to pick up again at higher latitudes, with some temporal overlap in the early and late phases of these cycles. For stars like the Sun, the mean level of activity as expressed by the surface-averaged absolute magnetic flux density ranges over more than three orders of magnitude, depending on the stellar rotation rate, age, and internal structure (more on that in Sect. 9.3; see also III:2). “[T]he existence of solar and stellar magnetic fields is in itself not really surprising; any large-scale fossil field present at the time of stellar formation would still be there today at almost its initial strength, because the Ohmic dissipation timescale is extremely large for most astrophysical objects [(Eq. 3.20)]. The challenge is instead to reproduce the various observed spatiotemporal patterns [. . . ], most notably the cyclic polarity reversals on decadal timescales.”
As to planetary dynamos, “[s]pace missions revealed that most planets in the Solar System have internal magnetic fields (see Ch. I:13), but there are exceptions (Venus, Mars). Some planets seem to have had a field that is now extinguished (e.g., Mars). In many cases with an active dynamo the axial dipole dominates the field at the planetary surface, but Uranus and Neptune are exceptions. Saturn is special because its field is extremely symmetric with respect to the planet’s rotation axis. The field strengths at the planetary surfaces differ by a factor of 1000 between Mercury and Jupiter [(cf. Table 5.3)]. A full understanding of this diversity in the morphology and strength

III:6.1

III:7.1

66

4 – Dynamos of Sun-like stars and Earth-like planets

For main-sequence stars

Sp. M/M⊙ Teff BC type

A0 3,2

9600 -0.25

F 0 1.7

7300 0.02

G0 1.1

5900 -0.07

K0 0.78 5000 -0.19

K5 0.69 4400 -0.62

M 0 0.60 3900 -1.17

Absolute bolometric magnitude Mbol,∗ (a logarithm of the brightness of a star normalized to a standard distance), absolute visual magnitude MV and stellar luminosity L∗ expressed in solar units (L⊙) are related through:
L∗/L⊙ = 10 , 0.4(Mbol,⊙−Mbol,∗)

while Mbol = MV +BC, where BC is the ’bolometric correction’ that corrects the brightness of the star in the V (’visual’ filter) bandpass to the bolometric brightness.

Figure 4.2: A Hertzsprung-Russell diagram showing stars with substantial magnetic activity in shaded or hatched domains, which are distinguished in groups of solar-likeness as indicated in the legend. The main sequence where stars spend most of their lifetime fusing hydrogen into helium in their cores is indicated by a solid curve; well above that lies the domain of the supergiant stars, with the giant star domain in between. Also indicated is the region where massive winds occur and where hot coronal plasma appears to be absent. Some frequently studied stars (both magnetically active and nonactive) are identified by name. The axes above the main panel show the spectral types (see footnote 8) for supergiant, giant, and main-sequence stars for the corresponding spectral color index B − V or corresponding V − R index. [Fig. III:2.8, with an added information panel on the right; figure source: Linsky (1985).]

of planetary magnetic fields is still lacking, but a number of promising ideas have been suggested and backed up by dynamo simulations. Some of the differences can be explained by a systematic dependence of the dynamo behavior on parameters such as rotation rate or energy flux, whereas others seem to require qualitative differences in the structure and dynamics of the planetary dynamos.”

III:7.4.1

Dynamo settings

67

90

95

100

105

110

115

120

60

65

70

75

80

85

90

30

35

40

45

50

55

60

0

5

10

15

20

25

30

Time B.P. [Ma]

Figure 4.3: Polarity of the geomagnetic field for the past 120 million years, with time running backward from left to right in each row (before present - B.P., i.e., 1950 - in units of millions of years). Dark regions indicate times when the dipole polarity was the same as today, in white regions it has been opposite. [Fig. III:7.4]

“Earth serves as the prototype for the terrestrial planets. [. . . ] There is a core with radius Rcore ≈ 0.55Rplanet, [the outer part of which is liquid]. The small inner core, with a radius 0.35Rcore, is [solid]. The core appears to consist predominantly of iron. [. . . ]
The total internal heat flow at the Earth’s surface is 4.6 1020 erg/s (although a large number, it is only 0.03% of the total power coming into the Earth’s atmosphere by insolation). Roughly one half of it is balanced by the heat generated by the decay of uranium, thorium and the potassium isotope 40K inside the Earth. The remainder of the heat flow is due to the cooling of the Earth. The loss of gravitational potential energy associated with the contraction of the Earth contributes a modest amount, but is much less important than it is in young stars or in gas planets. How much of the Earth’s heat flow comes from the core is rather uncertain. Recent estimates that are based on different lines of evidence mostly fall into the range (0.5 − 1.5) 1020 erg/s, although values as low as (0.3 − 0.4) 1020 erg/s have also been discussed. Most of the radioactive elements reside in the silicate crust and mantle. Some amount of potassium may be present in the core, but the majority of the core heat must be due to cooling. It is important to note that the heat loss from the core is regulated by the slow solid-state convection in the mantle. The core, which convects vigorously in comparison to the mantle and which is thermally well-mixed, delivers as much heat as the mantle is able to carry away.”
4.1.1 Earth and other terrestrial planets
Solar System bodies that have a present-day active dynamo include Mercury and Earth among the terrestrial planets, the jovian moon Ganymede, and all the giant planets; see Table 5.3 for their global properties.
“The surface magnetic field of the Earth has a strength of about 0.5 G with mainly dipolar character. [The dipole axis is tilted by a variable amount over time with respect to the axis of rotation, such that the magnetic north pole has wandered from as far

I:3.1.1

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III:7.4.1

south as about 70 degrees in geographic latitude to within a few degrees from the geographic north pole over the past two centuries]. From studies of rock magnetism (when rocks cool below the Curie point they preserve the magnetic field that was present in them at that time) it is known that the Earth had a magnetic field over the past 3.5 × 109 years and that the strength and orientation of the field varied significantly on time scales of 103 to 104 years. A given polarity typically dominates for about 200 000 years with quick reversals on a time scale of a few thousand years in between [(see Fig. 4.3)]. While the orientation of the axis of the dipole changes significantly with time, the dipole moment is aligned with the axis of rotation when averaged over ∼ 104 years.”
In contrast to the case of cool stars, “[r]adiative heat transfer is not an issue in planetary cores, but liquid metal is a good thermal conductor. The heat flux that can be transported by conduction along an adiabatic temperature gradient, (dT /dr)ad = T /HT , is sometimes called the ‘adiabatic heat flow’ (T is absolute temperature, HT = cp/(ζg) is the temperature scale height with cp the heat capacity, ζ the thermal expansivity and g the local gravitational acceleration). In terrestrial planets, the adiabatic heat flow can be a large fraction of the actual heat flow, or it may exceed the actual heat flow, in which case at least the top layers of the core would be thermally stable. Near the top of Earth’s core approximately (0.3 − 0.4) 1020 erg/s can be conducted along the adiabat, i.e., close to the minimum estimates for the entire core heat flow. But even if all the heat flux near the core-mantle boundary were carried by conduction, a convective dynamo can exist thanks to the inner core. At the inner core boundary, the adiabatic temperature profile of the convecting outer core crosses the melting point of iron. The latter increases with pressure more steeply than the adiabatic gradient, which is the reason why the Earth’s core freezes from the center rather than from above. As the core cools, the inner core grows with time by freezing iron onto its outer boundary. This has two important implications for driving the dynamo. The latent heat that is released upon solidification is an effective heat source, which contributes to the heat budget approximately the same amount as the bulk cooling of the core. [. . . ] A second, perhaps more important effect is that the light elements in the outer core are preferentially rejected when iron freezes onto the inner core. Hence, they become concentrated in the residual fluid near the inner core boundary. This layering is gravitationally unstable because of the reduced density, which leads to compositional convection that homogenizes the light elements in the bulk of the fluid core. Compositional convection contributes as much as, or more than, thermal convection to the driving of the geodynamo in recent geological times.
Most predictions for the inner core growth rate imply that the inner core did not exist for most of the history of the Earth. Rather, it would have nucleated between 0.5 and 2 billion years ago. In the absence of an inner core, only thermal convection by secular cooling of the fluid core (and perhaps radioactive heating) can drive a dynamo, which is less efficient than the present-day setting. A change in the geomagnetic field properties might be expected upon the nucleation of the inner core, but no clear indication for such an event has been found in the paleomagnetic record.”

III:7.4.2

Dynamo principles

69

I:3.1.2

“No direct evidence on the existence or non-existence of a solid inner core is available for any planet other than Earth. But the possible absence of an inner core could explain why Venus and Mars do not have an active dynamo. On Earth, mantle convection reaches the surface in the form of plate tectonics, which is a fairly efficient mode of removing heat from the interior. None of the other terrestrial planets have plate tectonics. In their cases, mantle convection is confined to the region below the lithosphere, a rigid lid of some 100 − 300 km thickness through which heat must be transported by conduction. Without plate tectonics, the heat flow is expected to be significantly lower not only at the surface, but also at the top of the core, where it is very probably subadiabatic. If no inner core exists to provide latent heat, it is then subadiabatic throughout the core. Furthermore, compositional convection is also unavailable to drive a dynamo. The slower cooling of the planetary interior in the absence of plate tectonics concurs with the idea that an inner core has not (yet) nucleated in the cases of Mars and Venus. Early in the planets’ history the cooling rate was probably much higher and the associated core heat flow large enough for thermal convection. The demise of the dynamo must have occurred when the declining heat flow dropped below the conductive threshold.”
For discussion of dynamos in non-terrestrial planets, see Ch. III:7.
4.1.2 The Sun and other stars
“The Sun shows magnetic field on all observable scales [(Fig. 4.4)] with a significant range in field strength, from individual sunspots with magnetic field strengths of 2 500 to 3 000 G to the average field strength of the global field of only a few Gauss.
The most prominent feature of solar magnetism is the 11-year sunspot cycle (if one considers the field reversals the full period is 22 years), which is reflected in the changing number of sunspots appearing on the surface of the Sun. In the beginning of a cycle spots appear at latitudes of about 35◦, while close to the end they appear almost at the equator. This property is commonly summarized in the so-called solar butterfly diagram [(Fig. 4.5)]. During the epoch of minimum, the large-scale field of the Sun is most dipolar; the reversal of the poles takes place during solar maximum. On a longer time scale the magnetic activity changes significantly in amplitude and is interrupted by epochs of 100 − 200 years in duration where sunspots are [infrequent or] completely absent [(such as during the Maunder Minimum, about 1645–1715). . . . ] Observations of the stellar luminosity or of chromospheric (UV/optical) and coronal (X-ray) emission show that a majority of solar-like stars are magnetically active and around a third to a half show cyclic activity with periods in the range from 3 to 30 years.” {A41}
{A42}
4.2 Dynamo principles
“Dynamo action refers to the conversion of mechanical energy into electromagnetic energy through induction. In [stars and in planets alike], the mechanical energy is supplied by fluid motions in electrically conducting regions inside [these bodies] and the electromagnetic energy produces the observed [. . . ] magnetic fields. A dynamo is referred to as self-sustaining if it does not require any external magnetic field

{A41} {A42} IV:6.1

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4 – Dynamos of Sun-like stars and Earth-like planets

104

102

N (per day)

100

10-2

10-4

10-6

ER AR

100 101 102 103 104 105

Flux (1018 Mx)

Figure 4.4: Left: First solar magnetic map (magnetogram) of the current millennium, taken by SOHO’s MDI on 2001/01/01 00:03 UT. The magnetogram (with white/black for negative/positive line of sight polarity) shows a variety of active regions, embedded in patches of largely unipolar enhanced supergranular network, mixed-polarity quiet Sun regions, and low-flux polar caps (weak at this near-maximum phase of the cycle, and weakened further in the line-of-sight flux map because of projection effects on the near-vertical magnetic field). Right: Distribution function of emerging magnetic bipolar regions on the Sun, showing the emergence frequency per day per flux interval of 1018 Mx, estimated for the entire solar surface. The shaded region on the right envelopes the range of the active-region spectrum for solar cycle 22 (for half-year intervals around sunspot minimum and maximum). The histograms on the left are for the ephemeral regions; the shaded band shows where observations are least affected by spatial (lower cutoff ) and temporal (upper cutoff ) biases. The spectrum for regions below ∼ 1019 Mx has yet to be determined; the cutoff here is caused by the limited resolution of the SOHO/MDI magnetograph. [Fig. III:2.1]

contributions for regeneration (except initially for a starting seed field).
The fundamental equation governing this induction process is known as the Magnetic Induction Equation [Eq. (3.3) in Table 3.3; its derivation and its limitations are described in Sect. 3.2.2. That equation is complemented by the requirement that the currents and the driving flows that are associated with the magnetic field are entirely contained within the body, and that the transition to outside the body for the field is smooth (compare I:3.3). . . . ]
By inspecting the two terms on the right-hand side of Eq. (3.3) we see that magnetic
field can grow or decay in time through two processes. The first term ○1 involves
interactions of the velocity and magnetic fields through electromagnetic induction
and acts as a source/sink term for field generation. The second term ○2 represents
diffusion due to Ohmic dissipation. To ensure magnetic field does not decay away in time, field must be generated as fast as or faster than its diffusion. A necessary

Dynamo principles

71

Figure 4.5: ‘Butterfly diagram’ showing (top) sunspot latitudes (also activity belts) and (bottom) total fractional area coverage as a function time. [updated with data through 2018] For a color version of this figure, see arXiv:2001.01093.
condition for self-sustained dynamo action is therefore that the induction term ○1 be larger than the diffusion term ○2 in Eq. (3.3). By using characteristic scales for the
variables in the Magnetic Induction Equation (i.e., Bt for the magnetic field scale, vt for the velocity scale and Lt for a length scale) we derive a common measure of the ratio of field generation to field diffusion known as the magnetic Reynolds Number: Rm ≡ vt Lt/η, see Eq. (3.18).]
Upon first glance, it seems reasonable that the magnetic Reynolds number must be larger than unity for dynamo action to be possible. However, more rigorous theoretical analyses suggest that the lower bound for Rm is instead closer to π2 and planetary numerical dynamo simulations typically find Rm must be larger than ∼ 20 − 50 for self-sustained dynamo action to occur. These higher values are due to the complexities in the velocity field morphologies that cannot be captured in the simple estimate given in Eq. (3.18):” after all, it is a big leap from small-scale field generated on the scale of the flow (such as sketched in Fig. 4.6) to a large-scale field. In cool stars, Rm typically far exceeds critical values for dynamo action because of the large scales and relatively fast flows involved (see Sect. III:5.3.2).
A perspective of what actually supplies the energy to power the dynamo is provided by integrating the induction equation Eq. (3.3) over the object’s volume to establish

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4 – Dynamos of Sun-like stars and Earth-like planets

Figure 4.6: Illustration of two possible flux-rope dynamos. In both cases the field amplification takes place during the stretch operation. The twist-fold (top) and reconnect-repack (bottom) steps are required to remap the amplified flux-rope into the original volume element so that the process can be repeated. Magnetic diffusivity is essential to allow for the topology change required to close the cycle. Each cycle increases the field strength by a factor of 2. [Fig. H-1:3.3]

the total energy in the system:

d

B2 dV = −

S · nˆdS − η j2dV − v · (j × B)dV.

dt V 4π

∂V

V

V

(4.1)

The first term on the right is the Poynting flux S = (1/4π)B × (v × B), which is the energy via the electromagnetic field through a surface into or out of the system across the closed boundary surface ∂V (ignorable if the stellar wind does not take too much power away compared to the total). The second term is the dissipative loss (assuming here that η is uniform). The final term shows that the magnetic energy in the system can be maintained against the dissipative losses only if there are sufficient flows working against – i.e., have an antiparallel component relative to – the Lorentz force F = (1/c)j × B. {A43}

4.3 Essentials of fluid motions in dynamos
In essence, to drive a large-scale stellar or planetary dynamo, the magnetic field must be subjected to a combination of flow components of a different nature that have their origin in convection and rotation. “Fig. 4.6 illustrates the basic ingredients required to amplify a closed magnetic field loop. After a full cycle, the magnetic field strength and the flux have doubled (two loops, each with the original magnetic flux) and the process can be repeated. This very simple illustration points out already a few fundamental properties of a dynamo process. To be able to remap the magnetic field configuration into the original volume element, three-dimensional motions are required. Amplification through stretching is possible in a strictly two-dimensional domain, but there is no way to move the resulting field to return to the right-hand side of the image. The two examples also point out the crucial role of diffusivity in changing the topology of the field. The ’stretch-twist-fold’ mechanism (excluding diffusive steps) leads to loops of increased complexity, while the ’stretch-reconnect-repack’ process explicitly involves magnetic diffusivity and ends up with two flux ropes [(see Table 3.1 for a definition)] of similar topology. A reconnection step at the end of the ’stretch-twist-fold’

{A43}

1:3.3.4

Essentials of fluid motions in dynamos

73

Figure 4.7: Panel a: Power spectrum of the convective velocity field in the solar photosphere obtained from Doppler measurements, plotted as a function of spherical harmonic degree ℓ. Mean flows and p-modes are filtered out. The falloff beyond ℓ ∼ 1500 reflects the resolution limit of the Michelson Doppler Imager (MDI) instrument onboard the SOHO spacecraft from which these data were obtained and is therefore artificial. Shaded areas indicate the approximate size ranges of supergranulation (SG), mesogranulation (MG) and granulation (G). Note that the expected granulation spectral peak at ℓ ∼ 4400, corresponding to L ∼ 1 Mm, is not resolved. Panel b: The solar internal rotation profile inferred from helioseismic inversions. Angular velocity Ω/2π is shown as a function of fractional radius r/R⊙ for several latitudes as indicated. Symbols and dashed lines denote different inversion methods, known as subtractive optimally localized averages (SOLA) and regularized least squares (RLS) respectively. Vertical 1-σ error bars (SOLA) and bands (RLS) are indicated and horizontal bars reflect the resolution of the inversion kernels. The vertical dashed line indicates the base of the convection zone. [Fig. III:5.1; panel a is based on data from this source: Hathaway et al. (2000); source panel b: Thompson et al. (2003).]
process leads to a similar result. In the case of the ’stretch-twist-fold’ dynamo the sign of the twist does not matter.”
The driving flow of dynamos in stars and planets is energy-transporting convection. “Thermal convection is familiar to most of us from our daily experience; warm air rises and cooler air sinks. When a fluid is heated from below it overturns, provided the temperature gradient is large enough, which here means that it must not only be greater than the adiabatic temperature gradient (the Schwarzschild criterion) but it must also overcome stabilizing influences such as thermal and viscous diffusion, rotation, compositional gradients (the Ledoux criterion), and magnetic flux. An intuitive way to think about convection (and to derive the Schwarzschild and Ledoux criteria) is to consider a small isolated volume, or parcel, of fluid that will buoyantly rise like a hot air balloon if its density is less than that of its surroundings or sink like a stone if its density is greater (the parcel is assumed to be in pressure equilibrium with its surroundings so density and temperature are anticorrelated). [For a compressible medium, t]his is the conceptual framework behind mixing length theory which goes on to say that the parcel will lose its identity, dispersing into the background, after traveling a vertical distance of order a pressure scale height Hp. [. . . ]

III:5.2

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4 – Dynamos of Sun-like stars and Earth-like planets

Figure 4.8: Columnar convection in a rotating spherical shell near onset. The inner core tangent cylinder is shown by broken lines. Under Earth’s core conditions the columns would be much thinner and very numerous. [Fig. III:7.6]

With this intuitive picture in mind, we may expect that the vertical scale of solar convection should vary tremendously from the deep convection zone where the stratification is relatively gentle (Hp ∼ 35 Mm) to the solar surface layers where the density and pressure drop precipitously (Hp ∼ 36 km) as [radiation escapes freely into space]. The associated drop in temperature near the surface triggers the recombination of hydrogen and other ions, which modifies the opacity, decreases the particle number density, and releases latent heat, altering the thermodynamics (in particular the equation of state and the specific heats) and contributing to the convective enthalpy transport. Add in radiative energy transfer and the result is what we call solar granulation; the continually shifting pattern (lifetime ∼ 5 min) of small-scale convection cells (with a horizontal extent ∼ 1 Mm) that blankets the solar surface and accounts for the dappled appearance of the solar photosphere (Fig. I:8.3).”

{A44}

Also the global-scale flows are important in the solar dynamo. The solar surface

exhibits a differential rotation: the equator rotates faster than the poles, with a smooth

latitudinal gradient between these. {A45} “Helioseismology now reveals that this

monotonic decrease in angular velocity with increasing latitude persists throughout the

convection zone, with an abrupt transition to nearly uniform rotation in the radiative

interior (Fig. 4.7b). The transition region near the base of the convection zone is known

as the solar tachocline [. . . ]. There is also a less dramatic but no less significant

near-surface shear layer in which the rotation rate systematically decreases by about

10-20 nHz from r = 0.96R⊙ to the photosphere. This is most apparent at low latitudes

but may also occur at higher latitudes. [. . . ] {A46}

{A47}

The striking difference in the rotation profile of the convective envelope and that

of the radiative interior implicates convection as the primary source of the differential

{A44}

{A45} III:5.2.3

{A46} {A47}

Insights from approximate stellar dynamo models

75

{A48}

{A49}

III:5.5.4

rotation. Furthermore, it tells us that giant cells are large enough and slow enough to be influenced by the rotation of the star. The magnitude of nonlinear advection [(v · ∇v)] relative to the Coriolis force [(Ω × v)] is quantified by the [Rossby number:

NR

=

vt , ΩLt

(4.2)

where vt and Lt are characteristic velocity and length scale, respectively.] In the deep solar convection zone it is of order unity or less whereas it is much greater than unity in the solar surface layers. {A48} Coriolis-induced velocity correlations in the convection redistribute angular momentum via the Reynolds stress, generating a substantial rotational shear: ∆Ω/Ω ∼ 30% where Ω(r, θ) is the angular velocity and ∆Ω is the angular velocity difference between equator and pole. {A49} Furthermore, the nature of the redistribution is such that the angular velocity increases away from the rotation axis, ∂Ω/∂dθ > 0 where dθ = r sin(θ) is [the distance to the axis of rotation]. This is in stark contrast to the behavior one would expect from isotropic turbulent diffusion (if ∆Ω/Ω ≪ 1) or from fluid parcels that tend to locally conserve their angular momentum as they move (∂Ω/∂dθ < 0), [which would behave as sketched in Fig. 4.8]. Giant cells must be a global phenomenon distinct from supergranulation.”
These solar flow patterns are in striking contrast to what fluid motions in the planets are thought to look like: “the latter often tend to be quasi-two-dimensional. This is largely a consequence of rapid rotation. Planets are smaller than stars and generally spin faster (with the exception of compact remnants such as pulsars). In the fluid cores and mantles of terrestrial planets and the extended atmospheres of many gas giant planets, the convective time scales are much longer than the rotation period, implying very low Rossby numbers [. . . T]his gives rise to elongated, quasi-2D convective structures such that the flow is relatively invariant in the direction parallel to the rotation axis (Fig. 4.8). In the atmospheres and oceans of terrestrial planets, on the other hand, quasi-2D dynamics arises simply by virtue of the geometry; global-scale horizontal motions are confined to thin spherical shells.”

4.4 Insights from approximate stellar dynamo models
Astrophysical dynamos have been studied for many decades, and whereas the fundamental ingredients may be known, there is no proper theory of dynamo action in stars and planets: there is no validated dynamo model that matches all stellar observations or that has been demonstrated to successfully forecast the Sun’s magnetism over multiple sunspot cycles, nor do planetary dynamo models successfully reproduce, for example, the quasi-irregular reversals in the terrestrial magnetic field. Nonetheless, dynamo concepts do guide our thinking as to the important ingredient processes as well as the possible internal structure and dynamics of both the magnetic field and the plasma/magma flows involved. The remainder of this chapter is an exploration of some of these to create a sense of how dynamos in stars and planets are thought to function.
“All solar and stellar dynamo models to be considered in this chapter operate within a sphere of electrically conducting fluid embedded in vacuum. We restrict ourselves

III:6.1

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4 – Dynamos of Sun-like stars and Earth-like planets

here to axisymmetric mean-field-like models, in the sense that we will be setting and solving evolutionary equations for the large-scale magnetic field, and subsume the effects of small-scale fluid motions and magnetic fields into coefficients of these partial differential equations. Working in spherical polar coordinates (r, θ, ϕ), we begin by writing:

v(r, θ) = vp(r, θ) + dθΩ(r, θ)eˆϕ , B(r, θ, t) = ∇ × (A(r, θ, t)eˆϕ) + B(r, θ, t)eˆϕ ,

(4.3) (4.4)

where dθ = r sin(θ), vp is a notational shortcut for the component of the large scale flow in meridional planes, and Ω is the angular velocity of rotation, which in the solar interior varies with both depth and latitude, and is now well-constrained by helioseismology. Note that in this prescription neither of these large-scale flow components is time dependent. This kinematic approximation is an assumption that is tolerably well-supported observationally. Substituting these expressions in the MHD induction equation in Eq. (3.3) allows separation into two coupled 2D partial differential equations for the scalar functions A and B defining respectively the poloidal and toroidal components of the magnetic field:

∂A ∂t

=

η

∇2

−

1 d2θ

A

−

vp dθ

·

∇(dθ A)

,

∂B ∂t

=

η

∇2

−

1 d2θ

B + 1 ∂(dθB) ∂η − dθ ∂r ∂r

B dθ∇ · dθ vp + dθ(∇ × (Aeˆϕ)) · ∇Ω ,

(4.5) (4.6)

where we retain the possibility that η varies with depth. The shearing term (∝ ∇Ω) on the right-hand side of Eq. (4.6) acts as a source of toroidal field. However, no such source term appears in Eq. (4.5). This is the essence of Cowling’s theorem which in fact guarantees that an axisymmetric flow of the general form given by Eq. (4.3) cannot act as a dynamo for an axisymmetric magnetic field as described by Eq. (4.4). The construction of solar and stellar dynamo models, therefore, hinges critically on the addition of an extraneous source term in Eq. (4.5). The physical origin of this source term is what fundamentally distinguishes the various classes of solar and stellar dynamo models described [below].
Shearing of the poloidal magnetic field into a strong toroidal component by differential rotation [(as illustrated in Fig. 4.9)] is an essential ingredient of all solar cycle models discussed below. The growing magnetic energy of the toroidal field is supplied by the kinetic energy of the rotational shearing motion, which makes for an attractive field amplification mechanism, because in the Sun and stars the available supply of rotational kinetic energy is immense (unless the dynamo were entirely confined to a very thin layer, for example the tachocline, [the shear layer just below the Sun’s convective envelope into which convection overshoots]). Moreover, a strong, axisymmetric and temporally quasi-steady internal differential rotation is likely responsible for the

Mean-field dynamo models

77

Figure 4.9: Left and center: Visualization of the effects of differential rotation and equatorto-pole meridional flow for Sun-like conditions: lines of equal longitude (with markers) are distorted into a spiral pattern. The center panel shows the distorted lines after 3 months. Right: Simulated magnetogram for a star like the Sun, [simulated with a flux-transport model with parameters as observed for the Sun,] but with an active-region emergence rate 30 times larger. The simulated star is shown from a latitude of 40◦ to better show the polar-cap field structure. [Fig. III:2.3]
observed high degree of axisymmetry observed in the Sun’s magnetic field on spatial scales comparable to its radius. This situation is very different from that encountered in planetary core dynamos, where differential rotation is believed to be much weaker, and energetics pose a much stronger constraint on dynamo action. Lacking the large-scale organization provided by differential rotation, planetary core dynamos also tend to produce non-axisymmetric large-scale fields. The one outstanding exception appears to be Saturn, and indeed in this case the high axisymmetry of the observed surface field may well reflect the symmetrizing action of differential rotation in the envelope overlying the metallic-hydrogen core. The important point remains that in the solar dynamo context, the assumption of an axisymmetric large-scale magnetic field is consistent with the observed and helioseismically-inferred axisymmetry and quasi-steadiness of internal differential rotation.”
4.5 Mean-field dynamo models
“Turbulence at a high magnetic Reynolds number Rm [(Eq. 3.18)] is known to be quite effective at producing a lot of small-scale magnetic fields, where ’small-scale’ is roughly Rm−1/2 times the length scale of the flow. In addition, under certain conditions, solar/stellar convective turbulence can also produce magnetic fields with a mean component building up on large spatial scales. These mean-field dynamo models remain arguably the most ’popular’ descriptive models for dynamo action in the Sun and stars, but also in planetary metallic cores, stellar accretion disks, and even galactic disks.
Under the assumption that a good separation of scales exists between the large-scale

III:6.2.1

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{A50}

I:3.4

’laminar’ magnetic field B and the flow v, and the small-scale turbulent field B′ and flow v′, it becomes possible to express the inductive and diffusive action of the turbulence on B in terms of the statistical properties of the small-scale flow and field. {A50}
The corresponding theory of mean-field electrodynamics is discussed in detail in Ch. I:3. The turbulent flow introduces on the right-hand side of the induction equation Eq. (3.3) a term of the form ∇ × E, where E is a mean-electromotive force.”
For a quick introduction to the origin of that term we can see what happens when “we decompose the magnetic field into a large-scale ’mean’ field and the small-scale components through an averaging procedure. We assume in the following that the averaging procedure obeys the Reynolds rules: For any function f and g decomposed as f = f + f ′ and g = g + g′, where the bar indicates the averaged and the prime the fluctuating quantity, we require that

f = f −→ f ′ = 0 f +g = f +g
f g = f g −→ f ′g = 0 ∂f /∂xi = ∂f /∂xi
∂f /∂t = ∂f /∂t .

(4.7) (4.8) (4.9) (4.10) (4.11)

The averaging procedures that are of interest in the context of mean-field theory are the ensemble average (meaning a chaotic system is averaged over several representations of the chaotic system) and the longitudinal average, in which B reflects the axisymmetric component of the large-scale magnetic field (multipole series with m = 0).” “In order to derive an equation for the time evolution of the mean field we apply the averaging procedure to the induction equation Eq. (3.3) which leads to

∂B = ∇× v′ × B′ + v × B − η∇×B . ∂t

(4.12)

The new term which enters this equation compared to the original induction equation is the second order correlation electromotive force (EMF)

E ≡ v′ × B′ .

(4.13)

While the fluctuating velocity component v′ is assumed to be known (kinematic approach), B′ has to be computed from the induction equation. An equation for B′ can be derived by subtracting the mean-field induction equation Eq. (4.12) from the microscopic induction equation Eq. (3.3), which leads to

∂B′ = ∇× v′ × B + v × B′ − η∇×B′ + v′ × B′ − v′ × B′ . ∂t

(4.14)

It is in general only possible to solve this equation by making strong assumptions, primarily because of the terms that are quadratic in the fluctuating quantities (closure problem).” For a more detailed description, see Sect. I:3.4.3. Here, we proceed with one

I:3.4.1

Mean-field dynamo models

79

{ⓈA51} III:6.2.1

particular such assumption that leads to the conclusion that for “mildly inhomogeneous and near-isotropic turbulence, E can be expressed in terms of the large-scale field B as:

E = αB − β∇ × B ,

(4.15)

with

α

=

−

1 3

τcorrv′

·

(∇

×

v′)

[cm s−1] ,

β

=

1 3

τcorrv′2

[cm2s−1] ,

(4.16)

where τcorr is the correlation time for the turbulent flow. Note that the α-term is proportional to the (negative) kinetic helicity [(v′ · (∇ × v′))] of the turbulence, which requires a break of reflectional symmetry. In stellar interiors and planetary metallic cores alike, this anisotropy is provided by the Coriolis force. Small-scale turbulence thus impacts the induction equation for the mean-field in two ways: it introduces a field-aligned electromotive force (the α-term), which acts as a source term and is called the ’α-effect’, and an enhanced ’turbulent diffusion’ (the β-term), associated with the folding action of the turbulent flow. In principle, the α and β coefficients can be calculated from the lowest-order statistics of the turbulent flow. In practice, more often than not they are chosen a priori, although with care taken to embody in these choices what can be learned from mean-field theory. {ⓈA51}
Under mean-field dynamo theory, Eqs. (4.5)—(4.6) are now taken to apply to an axisymmetric large-scale mean magnetic field. With the inclusion of the mean-field α-effect and turbulent diffusivity, scaling all lengths in terms of the radius R of star or planet, and time in terms of the diffusion time

τd = R2/η

(4.17)

based on the (turbulent) diffusivity in the convective envelope, these expressions become

∂A ∂t

=

η

∇2

−

1 d2θ

A

−

Rm dθ

vp

·

∇(dθ A)

+

CααB

,

∂B ∂t

=

η

∇2

−

1 d2θ

B

+

1 dθ

∂ (dθ B) ∂r

∂η ∂r

−

Rm dθ ∇

·

B dθ vp

+

CΩdθ(∇ × (Aeˆϕ)) · (∇Ω) + Cαeˆϕ · ∇ × [α∇ × (Aeˆϕ)] .

(4.18) (4.19)

We continue to use the symbol η for the total diffusivity, with the understanding that within the convective envelope this now includes the (dominant) contribution from the β-term of mean-field theory. Three non-dimensional numbers have materialized:

Cα

=

αtR η

,

CΩ =

ΩtR2 η

,

Rm

=

utR η

,

(4.20)

with αt, ut, and Ωt as reference values for the α-effect, meridional flow and envelope rotation, respectively. The quantities Cα and CΩ are dynamo numbers, measuring the importance of inductive versus diffusive effects on the right-hand side of Eqs. (4.18)– (4.19). The magnetic Reynolds number Rm here measures the relative importance of

80

4 – Dynamos of Sun-like stars and Earth-like planets

{A52} III:6.2.1.1

advection versus diffusion in the transport of A and B in meridional planes. {A52} Structurally, Eqs. (4.18)–(4.19) only differ from Eqs. (4.5)—(4.6) by the presence of two new source terms on the right-hand side, both associated with the α-effect. The appearance of this term in Eq. (4.18) is crucial for evading Cowling’s theorem.”
In what follows in this section, we first look at a simplified, linear mean-field dynamo model to illustrate the geometry and temporal evolution. Later, we look at non-linearities that lead to amplification and saturation of the field, and to the modulation of the magnetic cycles. First, the linear model: “In constructing mean-field dynamos for the Sun, it has been a common procedure to neglect meridional circulation, because it is a very weak flow. It is also customary to drop the α-effect term on the right-hand side of Eq. (4.19) on the grounds that with R ≃ 7 × 1010 cm, Ωt ∼ 10−6 rad s−1, and αt ∼ 102 cm s−1, one finds Cα/CΩ ∼ 10−3, independently of the assumed (and poorly constrained) value for η. Equations (4.18)—(4.19) then reduce to the so-called αΩ dynamo equations. In the spirit of producing a model that is solar-like we use a fixed value CΩ = 2.5 × 104, obtained by assuming [an equatorial angular velocity of] ΩEq ≃ 10−6 rad s−1 and η = 50 km2s−1, which leads to a diffusion time τd = R2/η ≃ 300 yr.
For the total magnetic diffusivity, we use a steep but smooth variation of η from a high value (ηCZ) in the convection zone to a low value (ηcore) in the underlying core [. . . ] A typical profile is shown in Fig. 4.10A (dash-dotted line). In practice, the core-to-envelope diffusivity ratio ∆η ≡ ηcore/ηCZ is treated as a model parameter, with of course ∆η ≪ 1, because we associate ηcore with the microscopic magnetic diffusivity, and ηCZ with the presumably much larger mean-field turbulent diffusivity. Taking at face values estimates from mean-field theory, one should have ∆η ∼ 10−9 to 10−6. The solutions discussed below have ∆η = 10−3 to 10−1, which is still small enough to illustrate important effects of radial gradients in total magnetic diffusivity.
All solar dynamo models discussed in this chapter utilize the helioseismicallycalibrated solar-like parametrization of solar differential rotation [. . . ]. The corresponding angular velocity contour levels are plotted in Fig. 4.10B. Such a solar-like differential rotation profile is quite complex from the point of view of dynamo modelling, in that it is characterized by multiple partially overlapping shear regions: a rotational shear layer, straddling the core-envelope interface, known as the tachocline, with a strong positive radial shear in its equatorial regions and an even stronger negative radial shear in its polar regions, as well as a significant latitudinal shear throughout the convective envelope and extending partway into the tachocline; for a tachocline of half-thickness w/R⊙ = 0.05, the mid-latitude latitudinal shear at r/R⊙ = 0.7 is comparable in magnitude to the equatorial radial shear, and its potential contribution to toroidal field production cannot be casually dismissed.
For the dimensionless function α(r, θ) we use an expression [. . . that] concentrates the α-effect in the bottom half of the envelope, and lets it vanish smoothly below, just as the net magnetic diffusivity does (see Fig. 4.10A). Various lines of argument point to an α-effect peaking in the bottom half of the convective envelope, because there the convective turnover time is commensurate with the solar rotation period, a

Mean-field dynamo models

81

Figure 4.10: Various ’ingredients’ for the dynamo models constructed in this chapter. Part (A) shows radial profiles of the total magnetic diffusivity η and poloidal source [terms: α(r) for the αΩ dynamo and for the Babcock-Leighton (B–L) dynamo]. Part (B) shows contour levels of the rotation rate Ω(r, θ) normalized to its surface equatorial value. The dotted line is the core-envelope interface at r/R⊙ = 0.7. Part (C) shows streamlines of meridional circulation, included in some of the dynamo models discussed below. [Helioseismic studies suggest that the meridional flow in the Sun is more complex than a single ’roll’ of the flow, but that there may be (at least) two stacked on top of each other. A key point for a flux-transport dynamo is that the meridional flow at the base of the convective envelope is equatorward. Fig. III:6.1]

most favorable setup for the type of toroidal field twisting at the root of the α-effect (see Fig. I:3.5). [The choice made here for α(r, θ) scales with latitude as cos θ, which] reflects the hemispheric dependence of the Coriolis force, which also suggests that the α-effect should be positive in the Northern hemisphere. The dimensionless number Cα, which measures the strength of the α-effect, is treated as a free parameter of the model. [. . . ]
In such linear αΩ models the onset of dynamo activity turns out to be controlled by the product of Cα and CΩ:

D ≡ Cα × CΩ

=

αtΩtR3 ηC2 Z

.

(4.21)

with positive growth rates materializing above a threshold value known as the critical

82

4 – Dynamos of Sun-like stars and Earth-like planets

Figure 4.11: Four snapshots in meridional planes of our minimal linear αΩ dynamo solution with defining parameters CΩ = 25000, ∆η = 0.1, and ηCZ = 50 km2/s. With Cα = +5, this is a mildly supercritical solution, with oscillation frequency ω ≃ 300 τd−1 (see Eq. 4.17). The toroidal field is plotted as filled contours (gray to black for negative B, gray to white for positive B, normalized to the peak strength and with increments ∆B = 0.2), on which poloidal field lines are superimposed (solid for clockwise-oriented field lines, dashed for counter-clockwise orientation). The long-dashed line is the core-envelope interface at r/R⊙ = 0.7. [Fig. III:6.2]
dynamo number. [. . . ] Figure 4.11 shows half a cycle of the dynamo solution, in the form of snapshots
of the toroidal (gray scale) and poloidal eigenfunctions (field lines) in a meridional plane, with the symmetry axis defined by the stellar rotation oriented vertically. The four frames are separated by a phase interval φ = π/3, so that panel (D) is identical to panel (A) except for reversed magnetic polarities in both magnetic components [halfway through the cycle with period Pcycle = 2π/ω]. Such linear eigensolutions leave the absolute magnitude of the magnetic field undetermined, but the relative magnitude of the poloidal to toroidal components is found to scale approximately as |Cα/CΩ|.
The [model’s magnetic field] is concentrated in the vicinity of the core-envelope interface, and has very little amplitude in the underlying, low-diffusivity radiative core. This is due to the oscillatory nature of the solution, which restricts penetration into

Mean-field dynamo models

83

III:6.2.1.2

the core to a distance of the order of the electromagnetic skin depth ℓskin = 2ηcore/ω. Having assumed ηCZ = 50 km2s−1, with ∆η = 0.1, a dimensionless dynamo frequency ω ≃ 300 corresponds to 3 × 10−8 s−1, so that ℓskin/R ≃ 0.026, quite small indeed.
Careful examination of Fig. 4.11(A)→(D) also reveals that the toroidal-poloidal flux systems present in the shear layer first show up at high latitudes, and then migrate equatorward to finally disappear at mid-latitudes in the course of the half-cycle. These dynamo waves travel in a direction given by α∇Ω × eˆϕ, i.e., along contours of equal angular velocity, a result known as the Parker-Yoshimura sign rule. Here with a negative ∂Ω/∂r in the high-latitude region of the tachocline, a positive α-effect results in an equatorward propagation of the dynamo wave, in qualitative agreement with the observed equatorward drift of the latitudes of sunspot emergences as the solar cycle unfolds (see Fig. 4.5).”
“Obviously, the exponential growth characterizing supercritical linear solutions must stop once the Lorentz force associated with the growing magnetic field becomes dynamically significant for the inductive flow. Because the solar surface and internal differential rotation show little variation with the phase of the solar cycle, it is usually assumed that magnetic back-reaction occurs at the level of the α-effect. In the meanfield spirit of not solving dynamical equations for the small-scales, it has become common practice to introduce an ad hoc algebraic nonlinear quenching of α directly on the mean-toroidal field B by writing:

α

→

α(B) =

αt 1 + (B/Beq)2

.

(4.22)

where Beq = (4πρu2t )1/2 is the equipartition field strength, of order 104 G at the base of the solar convective envelope. Needless to say, this simple α-quenching formula is an extreme oversimplification of the complex interaction between flow and field that is known to characterize MHD turbulence, but its wide usage in solar dynamo modeling makes it the nonlinearity of choice for the illustrative purpose of this [chapter: with this description, the only MHD equation that needs solving to experiment with dynamo action – as we do here – is the induction equation Eq. (3.3) that is now subjected to a parameterized coupling between the small-scale flow and field that may or may not be an appropriate approximation of reality. Note that α can, and in many models now is, time dependent, leading to what is called ’dynamical α-quenching’.]
Introducing α-quenching in our model renders the αΩ dynamo equations nonlinear, so that solutions are now obtained as initial-value problems starting from an arbitrary seed field of very low amplitude, in the sense that B ≪ Beq everywhere in the domain. [. . . ] At early times, B ≪ Beq and the equations are effectively linear, leading to exponential growth [. . . ]. Eventually, however, B becomes comparable to Beq in the region where the α-effect operates, leading to a break in exponential growth, and eventual saturation.
The saturation energy level increases with increasing Cα, an intuitively satisfying behavior because solutions with larger Cα have a more vigorous poloidal source term. The cycle frequency for these solutions is very nearly independent of the dynamo

84

4 – Dynamos of Sun-like stars and Earth-like planets

III:6.2.1.3

number, and is slightly smaller than the frequency of the linear critical mode (here by some 10 − 15%), a behavior that is typical of kinematic α-quenched mean-field dynamo models. Yet the overall form of the dynamo solutions very closely resembles that of the linear eigenfunctions plotted in Fig. 4.11.”
“The α-quenching expression in Eq. (4.22) implies that dynamo action saturates once the mean, dynamo-generated large-scale magnetic field reaches an energy density comparable to that of the driving small-scale turbulent fluid motions. However, various calculations and numerical simulations have indicated that long before the mean toroidal field B reaches this strength, the helical turbulence reaches equipartition with the small-scale turbulent component of the magnetic field. Such calculations also suggest that the ratio between the small-scale and mean magnetic components should itself scale as Rm1/2, where Rm = vtLt/η is a magnetic Reynolds number based on the turbulent speed but microscopic magnetic diffusivity. This then leads to the alternative quenching expression

α → α(B) =

αt 1 + Rm(B/Beq)2

,

(4.23)

known in the literature as strong α-quenching or catastrophic quenching (see Ch. I:3 in Vol. I). Because Rm ∼ 108 in the solar convection zone, this leads to quenching of the α-effect for very low amplitudes of the mean magnetic field, of order 0.1 G. Even though significant field amplification is likely in the formation of a toroidal flux rope from the dynamo-generated magnetic field, we are now a very long way from the 104 − 105 G demanded by simulations [needed for buoyantly rising flux ropes to survive emergence and to eventually lead to] sunspot formation.
[One] way out of this difficulty exists in the form of interface dynamos. The idea is beautifully simple: to produce and store the toroidal field away from where the α-effect is operating. [. . . ] in a situation where a radial shear and α-effect are segregated on either side of a discontinuity in magnetic diffusivity taken to coincide with the core-envelope interface, the constant coefficient, cartesian form of the αΩ dynamo equations support solutions in the form of traveling surface waves localized on the discontinuity in diffusivity. For supercritical dynamo waves, the ratio of peak toroidal field strength on either side of the discontinuity surface is found to scale as (ηCZ/ηcore)−1/2. With the core diffusivity ηcore equal to the microscopic value, and if the envelope diffusivity is of turbulent origin so that ηCZ ∼ Ltvt, then the toroidal field strength ratio scales as ∼ (vtLt/ηcore)1/2 ≡ Rm1/2. This is precisely the factor needed to bypass strong α-quenching, at least as embodied in Eq. (4.23).”
So far, this discussion has ignored the large-scale flow system known as meridional circulation. Such a flow “is unavoidable in turbulent, compressible rotating convective shells. The ∼ 15 m s−1 poleward flow observed at the surface has been detected helioseismically, down to r/R⊙ ≃ 0.85 without significant departure from the poleward direction, except locally and very close to the surface, in the vicinity of active region belts. Mass conservation requires an equatorward flow deeper down [(helioseismic measurements suggest that there may be two meridional overturning cells stacked

III:6.2.1.4