SOLAR MAGNETO HYDRODYNAMICS

Transcription

1 SOLAR MAGNETO HYDRODYNAMICS

2 GEOPHYSICS AND ASTROPHYSICS MONOGRAPHS Editor B. M. McCoIlMAc, Lockheed Palo Alto Research Loboratory, Palo Alto, Calif., U.S.A. Editorial Board R. GRANT ATHAY, High Altitude Observatory, Boulder, Colo., U.S.A. W. S. BR.OECKEll, Lamont-Doherty Geological Observatory, Palisades, New York, U.S.A. P. J. CoLEMAN, JR.., University of California, Los Angeles, Calif., U.S.A. G. T. CsANADY, Woods Hole Oceanographic Institution, Woods Hole, Mass., U.S.A. D. M. HUNTEN, University of Arizona, Tucson, Ariz., U.S.A. C. DE JAGER., The Astronomical Institute, Utrecht, The Netherlands J. KLECZEK, Czechoslovak Academy of Sciences, Ondrejov, Czechoslovakia R. LiisT, President Max-Planck Gesellschaftfiir Forderung der Wissenschaften, MUnchen, F.R.G. R. E. MUNN, University of Toronto, Toronto, Ont., Canada Z. SVESTKA, The Astronomical Institute, Utrecht, The Netherlands G. WEILL, Service d'aeronomie, Verrieres-le-Buisson, France VOLUME 21

3 SOLAR MAGNETO HYDRODYNAMICS ERIC R. PRIEST St. A.1Ulrewl UniYerlity, ScotlDnd D. REIDEL PUBLISHING COMPANY A MEMBER OF THE KLUWER ACADEMIC PUBUSHERS GROUP DORDRECHT / BOSTON / LANCASTER

4 library of Congress Cataloging in Publication Data Priest, Eric Ronald, Solar magneto-hydrodynamics. Bibliography: p. Includes index. 1. Solar magnetic field. 2. Magnetohydrodynamics. 1. Title. QB539.M23P '2 ISBN-13: DOl: / e-isbn-13: Published by D. Reidel Publishing Company, P.O. Box 17,3300 AA Dordrecht, Holland. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers. 101 Philip Drive, Norwell, MA 02061, U.S.A. In an other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, HoHand. Reprinted with corrections in 2000 All Rights Reserved 1982, 1984, 1987, 2000 by D. Reidel Publishing Company, Dordrecht, Holland No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner

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6 CONTENTS PREFACE ACKNOWLEDGEMENTS CHAPTER 1. A DESCRIPTION OF TIlE SUN 1.1. Brief History 1.2. Overall Properties Interior Outer Attnosphere 1.3. The Quiet Sun The Interior A. The Core B. A Model C. Convection Zone The Photosphere A. Motions B. Magnetic Field C. A Model The Chromosphere The Corona A. At Eclipses B. In X-rays C. Solar Wind 1.4. Transient Features Active Regions A. Development B. Structure C. Loops D. Internal Motions Sunspots A. Development B. Umbra C. Penumbra D. Motion E. Solar Cycle Prominences A. Introduction B. Properties xv xix vii

7 viii CONTENTS C. Development D. Structure E. Eruption F. Coronal Transients Solar Flares A. Basic Description B. Ground-Based Observations C. Space Observations CHAPTER 2. THE BASIC EQUATIONS OF MAGNETO HYDRODYNAMICS 2.1. Electromagnetic Equations Maxwell's Equations Ohm's Law Generalised Ohm's Law Induction Equation Electrical Conductivity 2.2. Plasma Equations Mass Continuity Equation of Motion Perfect Gas Law 2.3. Energy Equations Different Forms of Heat Equation Thermal Conduction Radiation Heating Energetics 2.4. Summary of Equations Assumptions Reduced Forms of the Equations 2.5. Dimensionless Parameters 2.6. Consequences of the Induction Equation Diffusive Limit Perfectly Conducting Limit 2.7. The Lorentz Force 2.8. Some Theorems Cowling's Antidynamo Theorem Taylor-Proudman Theorem Ferraro's Law of Isorotation The Virial Theorem 2.9. Summary of Magnetic Flux Tube Behaviour Definitions General Properties Flux Tubes in the Solar Atmosphere Summary of Current Sheet Behaviour Processes of Formation Properties

8 CHAPTER 3. MAGNETOHYDROSTATICS CONTENTS 3.1. Introduction Plasma Structure in a Prescribed Magnetic Field The Structure of Magnetic Flux Tubes (Cylindrically Symmetric) Purely Axial Field Purely Azimuthal Field Force-Free Fields 125 A. Linear Field 125 B. Nonlinear Fields 125 C. Effect of Twisting a Tube 126 D. Effect of Expanding a Tube 127 E. A Tube of Non-Uniform Radius Magnetostatic Fields Current-Free Fields Force-Free Fields General Theorems Simple Constant-a SC)lutions General Constant-a Solutions Non-Constant-a Solutions Diffusion Coronal Evolution Magnetohydrostatic Fields 149 CHAPTER 4. WAVES Introduction Fundamental Modes Basic Equations Sound Waves Magnetic Waves Shear Alfven Waves Compressional Alfven Waves Internal Gravity Waves Inertial Waves Magnetoacoustic Waves Acoustic-Gravity Waves Summary of Magnetoacoustic-Gravity Waves Five-Minute Oscillations Observations Models 177 A. Photospheric Ringing 177 B. Wave Trapping Wave Generation Strong Magnetic Field Regions The Future Waves in a Strongly Inhomogeneous Medium Surface Waves on a Magnetic Interface 183 ix II7

9 x CONTENTS A Twisted Magnetic Flux Tube A Stratified Atmosphere CHAPTER 5. SHOCK WAVES 5.1. Introduction Formation of a Hydrodynamic Shock Effects of a Magnetic Field 5.2. Hydrodynamic Shocks 5.3. Perpendicular Shocks 5.4. Oblique Shocks Jump Relations Slow and Fast Shocks Switch-Off and Switch-On Shocks The Intermediate Wave CHAPTER 6. HEATING OF THE UPPER ATMOSPHERE 6.1. Introduction 6.2. Models for Atmospheric Structure Basic Model Magnetic Field Effects Additional Effects 6.3. Acoustic Wave Heating Steepening Propagation and Dissipation 6.4. Magnetic Heating Propagation and Dissipation of Magnetic Waves Nonlinear Coupling of AIfven Waves Resonant Absorption of AIfven Waves Magnetic Field Dissipation A. Order of Magnitude B. Current Sheets C. Current Filaments 6.5. Coronal Loops Static Energy-Balance Models A. Uniform Pressure Loops B. Cool Cores C. Hydrostatic Equilibrium Flows in Coronal Loops CHAPTER 7. INSTABILITY 7.1. Introduction 7.2. Linearised Equations 7.3. Normal Mode Method Example : Rayleigh-Taylor Instability

10 A. Plasma Supported by a Magnetic Field B. Uniform Magnetic Field (*0"') = *0-) 7.4. Variational (or Energy) Method Example: Kink Instability Use of the Energy Method 7.5. Summary of Instabilities Interchange Instability Rayleigh-Taylor Instability Pinched Discharge Flow Instability Resistive Instability Convective Instability Radiatively-Driven Thermal Instability Other Instabilities CHAPTER 8. SUNSPOTS 8.1. Magnetoconvection Physical Effects Linear Stability Analysis Magnetic Flux Expulsion and Concentration 8.2. Magnetic Buoyancy Qualitative Effect Magnetic Buoyancy Instability The Rise of Flux Tubes in the Sun 8.3. Cooling of Sunspots 8.4. Equilibrium Struct1JR: of Sunspots Magnetohydrostatic Equilibrium Sunspot Stability 8.5. The Sunspot Penumbra 8,6. Evolution of a Sunspot Formation Decay 8.7. Intense Flux Tubes Equilibrium of a Slender Flux Tube Intense Magnetic Field Instability 8.7.3, Spicule Generation Tube Waves CHAPTER 9. DYNAMO THEORY 9.1. Introduction 9.2. Cowling's Theorem 9.3. Qualitative Dynamo Action Generation of Toroidal and Poloidal Fields Phenomenological Model 9.4. Kinematic Dynamos 253 2S n

11 xii CONTENTS Nearly-Symmetric Dynamo Turbulent Dynamo: Mean-Field Electrodynamics Simple Solution: Dynamo Waves Solar Cycle Models: The IX-W Dynamo 9.5. Magnetohydrodynamic Dynamos Modified Kinematic Dynamos Strange Attractors Convective Dynamos 9.6. Difficulties with Dynamo Theory CHAPTER 10. SOLAR FLARES Magnetic Reconnection Unidirectional Field Diffusion Region The Petschek Mechanism External Region Simple-Loop Flare Emerging (or Evolving) Flux Model Thermal Nonequilibrium Kink Instability Resistive Kink Instability Two-Ribbon Flare Existence and Multiplicity of Force-Free Equilibria Eruptive Instability The Main Phase: 'Post'-Flare Loops CHAPTER II. PROMINENCES Formation Formation in a Loop (Active-Region Prominences) Formation in a Coronal Arcade 1l.l.3. Formation in a Current Sheet A. Thermal Nonequilibrium B. Line-Tying Magnetohydrostatics of Support in a Simple Arcade Kippenhahn-Schliiter Model Generalised Kippenhahn-Schliiter Model The External Field Magnetohydrodynamic Stability Helical Structure Support in Configurations with Helical Fields Support in a Current Sheet Support in a Horizontal Field Coronal Transients Twisted Loop Models Untwisted Loop Models

12 Numerical Models Conclusion CONTENTS xiii CHAPTER 12. THE SOLAR WIND Introduction Parker's Solution Models for a Spheric.al Expansion Energy Equation Two-Fluid Model Magnetic Field Streamers an,d Coronal Holes Pneuman-Kopp Model A. Basic Model B. Angular Momentum Loss C. Current Sheet Coronal Hole Models Extra Effects APPENDIX I. Units APPENDIX II. Useful Values and Expressions APPENDIX III. Notation REFERENCES INDEX

13 PREFACE I have felt the need for a book on the theory of solar magnetic fields for some time now. Most books about the Sun are written by observers or by theorists from other branches of solar physics, whereas those on magnetohydrodynamics do not deal extensively with solar applications. I had thought of waiting a few decades before attempting to put pen to paper, but one summer Josip Kleczek encouraged an immediate start 'while your ideas are still fresh'. The book grew out of a postgraduate lecture course at St Andrews, and the resulting period of gestation or 'being with monograph' has lasted several years. The Sun is an amazing object, which has continued to reveal completely unexpected features when observed in greater detail or at new wavelengths. What riches would be in store for us if we could view other stars with as much precision! Stellar physics itself is benefiting greatly from solar discoveries, but, in tum, our understanding of many solar phenomena (such as sunspots, sunspot cycles, the corona and the solar wind) will undoubtedly increase in the future due to their observation under different conditions in other stars. In the 'old days' the solar atmosphere was regarded as a static, plane-parallel structure, heated by the dissipation of sound waves and with its upper layer expanding in a spherically symmetric manner as the solar wind. Outside of sunspots the magnetic field was thought to be unimportant with a weak uniform value of a few gauss. Recently, however, there has been a revolution in our basic understanding. Highresolution ground-based instruments have revealed a photosphere full of structure and with small-scale magnetic fields that are probably concentrated into intense kilogauss flux tubes. The chromosphere is now known to be made up of cool jets, and space experiments have shown the corona to be a dynamic, highly complex structure consisting of myriads of hot loops. At small scales in the corona, hundreds of X ray bright points are seen where new flux is emerging from below the solar surface and causing mini-flares. Also, coronal heating is now thought to be magnetic, either via various wave modes or by direct current dissipation, and the solar wind has been found to escape primarily from the localised regions known as coronal holes, where the magnetic field lines are open. Many of these new features are dominated by the magnetic field. Indeed, much of the detailed structure we now see owes its very existence to the field, and so solar MHD is at a most exciting stage as we attempt to explain and model the magnetic Sun. Magnetohydrodynamics (or MHD for short) is the study of the interaction between a plasma (or electrically conducting fluid) and a magnetic field. As the temperature of a material is raised, so it passes through the first three states of matter (solid, liquid and gas), and eventually it reaches the fourth state (plasma) when many electrons are no longer bound to the nuclei. A plasma, therefore. is an ionised xv

14 xvi PREFACE gas, which behaves quite differently from the other states. Most of the matter in the universe is in this plasma state, such as the gas in a glowing fluorescent light tube, the Earth's ionosphere or the atmosphere of the Sun. Indeed, we on the Earth represent a tiny enclave of solid, liquid and gas immersed in the outflow of solar wind plasma, like a pebble in a stream of water. A magnetic field affects a plasma in several ways. It exerts a force, which is able, for instance, to support material in a prominence against gravity or propel it away from the Sun at high speeds. It provides thermal insulation, and so allows cool plasma to exist alongside hotter material, as in prominences or cool loop cores. It also stores energy, which may be released violently as a solar flare. Solar MHO is an important tool for understanding many solar phenomena. It also plays a crucial role in explaining the behaviour of more general cosmical magnetic fields and plasmas, since the Sun provides a natural laboratory in which such behaviour may be studied. While terrestrial experiments are invaluable in demonstrating general plasma properties, conclusions from them cannot be applied uncritically to solar plasmas and have in the past given rise to misconceptions about solar magnetic field behaviour. Important differences between a laboratory plasma on Earth and the Sun include the nature of boundat:y conditions, the energy l)alance, the effect of gravity and the size of the magnetic Reynolds number (generally of order unity on the Earth and very much larger on the Sun). The importance of mathematical modelling in our subject must be stressed. The full nonlinear equations of MHO (including thermal and diffusive effects) are so complex that they often need to be approximated drastically by focusing on the dominant physical mechanisms in any particular phenomenon. One begins with a simple model for a coronal transient or a dynamo. forinstance, which perhaps admits analytical solutions to the equations. Then more and more effects may be added in an attempt to make the model more realistic. Thus the simple analytical model and the complex numerical computation can play the complementary roles of examining a new effect and attempting a more realistic simulation. The overall structure of the book is as follows. It begins with two introductory chapters on solar observations (Chapter 1) and the MHO equations (Chapter 2). Then the fundamentals of MHO are developed in chapters on magnetostatics (Chapter 3), waves (Chapter 4), shocks (Chapter 5), and instabilities (Chapter 7). Finally, the theory is applied to the solar phenomena of atmospheric heating (Chapter 6), sunspots (Chapter 8), dynamos (Chapter 9), flares (Chapter 10), prominences (Chapter 11) and the solar wind (Chapter 12). The chapter on heating was placed after chapters 4 and 5 because of its close relationship to them. Appendices discuss the question of units and notation and list some expressions and constants which may be found useful. The building blocks of the solar magnetic field are flux tubes and current sheets. Their properties are developed at numerous points throughout the book and are summarised in Sections 2.9 and Some of the chapter headings possess rather broad meanings. For instance, the chapter on sunspots includes sections on magnetoconvection, magnetic buoyancy and intense flux tubes, while the prominence chapter includes a discussion of coronal transients. The notation that has been adopted for cylindrical polar coordinates is (R, cp, z),

15 PREFACE whereas that for spherical polars is (r, e, cp). All quantities are measured in rationalised mks units, with the magnetic field in tesla (T) as far as most formulae are concerned (I T = 10 4 G). In the text, however, magnetic field strengths are commonly quoted in G. Lengths in formulae are usually measured in metres, although in the text they are often quoted in megametres (1 Mm = )06 m). I am extremely grateful to many people for their advice and help, notably colleagues in St Andrews (B. Roberts, M. Wragg, P. Browning, P. Cargill, T. Forbes, M. Gibbons, A. Hood), friends in Boulder (G. Athay, J. Christensen-Dalsgaard, P. Gilman, T. Holzer, A. Hundhausen, G. Pneuman, E. Zweibel), as well as T. G. Cowling, J. Heyvaerts, J. Hollweg, H. Spruit. P. Ulmschneider, N. Weiss and those who have so willingly supplied the figures. There are many others who have taught me what is written down here, although any errors or misconceptions are in no way their fault. My only hope is that this book may help others to understand a little more about the intriguing beha viour of solar magnetic fields and the mathematical language for its expression. xvii Sf Andrews, May, 1981 E. R. PRIEST

16 ACKNOWLEDGEMENTS The author gratefully acknowledges permission to reproduce the following copyright figures: Fig (American Geophysical Union), Fig (Associated Book Publishers Ltd., London), Figs and (Astronomy and Astrophysics Journal), Fig (S. Habbal), Fig. 8.4 (D. Galloway), Fig (Gordon and Breach), Fig (J. Heyvaerts), Fig. lui (A. Milne), Fig (T. Mouschovias), Fig (R. H. Munro), Figs. 6.6, 1.26, 8.11 (E. N. Parker), Fig. 2.2 (R. Rosner), Figs. 1.30, 1.37, 9.4, 10.6 (Royal Society of London), Figs. 1.3a-c, 1.8, 1.13, 1.37, (Association of Universities for Research in Astronomy, Inc., Sacramento Peak Observatory). The author is also most grateful to those who have supplied the many figures that are Reidel copyright, as indicated in the captions. xix