13
QUANTITATIVE ASPECTS OF MAGNETOSPHERIC PHYSICS
QUANTITATIVE ASPECTS OF MAGNETOSPHERIC PHYSICS
GEOPHYSICS AND
ASTROPHYSICS MONOGRAPHS
Editor
B. M McCORMAC, Lockheed Palo Alto Research Laboratory, Palo Alto, Calif, U.S.A.
Editorial Board
R GRANT ATIIA Y, High Altitude Observatory, Boulder, Colo., U. S.A.
W. S. BROECKER, Lamont-Doherty Geological Observatory, Palisades, New York, U.S.A.
P. J. COLEMAN, JR., University ofCalifomia, 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 Science, Ond1ejov, Czechoslovakia
R LUST, President Max-Planck Gesellschaft {iir F6rderung der Wissenschaften, Miinchen, 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, Ve"ieres-Ie-Buisson, France
QU ANTIT A TIVE ASPECTS OF
MAGNETOSPHERIC PHYSICS
L. R. LYONS
Space Environment Laboratory, National Oceanic and Atmospheric Administration, Boulder, Colorado. U.S.A.
and
D. J. WILLIAMS
Applied Physics Laboratory. Johns Hopkins University, Laurel. Maryland, U.S.A.
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
tt
Library of Congress Cataloging in Publication Data
Lyons, L. R. Quantitative aspects of magnetospheric physics.
(Geophysics and astrophysics monographs) Includes bibliographical references and index. 1. Magnetosphere. I. Williams, D. J., 1933-
II. Title. III. Series. QC809.M35L96 1984 538'.766 83-26886 ISBN 978-90-481-8391-3 ISBN 978-94-017-2819-5 (eBook) DOI 10.1007/978-94-017-2819-5
All Rights Reserved © 1984 by Springer Science+Business Media Dordrecht
Originally published by D. Reidel Publishing Company, Dordrecht, Holland in 1984
and copyright holders as speciiled on appropriate pages within. 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 infonnation storage and
retrieval system, without written permission from the copyright owner.
To Robin Lyons and Priscilla Williams, for their patience, understanding, and, most of all,
for being our wives.
TABLE OF CONTENTS
PREFACE ix
LIST OF SYMBOLS xi
CHAPTER 1: INTRODUCTION
CHAPTER 2: CHARGED-PARTICLE MOTION IN MAGNETIC AND ELECTRIC FIELDS 6
2.1. Guiding Center 2.2. Dipole Magnetic Field 2.3. Gyration 2.4. Bounce 2.5. Drift
2.5.1. Electric Fields 2.5.2. Magnetic Field Gradient 2.5.3. Field Line Curvature 2.5.4. 2.5.5.
Inertial Forces Total Drift
6 7 8
11 14 15 16 16 17 18
2.6. Particle Distribution Functions 20 2.6.1. Phase Space Density and Differential Flux 20 2.6.2. Omnidirectional Flux 22
2.7. Summary 22
CHAPTER 3: TRAPPING REGION AND CURRENTS DUE TO TRAPPED PARTICLES 28
3.1. Verification of Geomagnetic Trapping Coordinate Systems 28 3.2. Trapping Regions 30 3.3. Trapped Particle Currents 40
3.3.1. Particle Distributions and Currents 40 3.3.2. The Ring Current 44 3.3.3. Ring Current Generation 48 3.3.4. Ring Current Decay 50
CHAPTER 4: ELECTRIC FIELDS 56 4.1. Introduction 56 4.2 The Convection Electric Field 56
4.2.1. Proposed Large-Scale Convection Electric Field 56 4.2.2. Mapping of the Convection Electric Field to the Ionosphere
and Resulting Ionosphere Currents 59
viii
4.3.
4.4.
TABLE OF CONTENTS
4.2.3. Evidence for an Open, Polar Cap Magnetic Field 4.2.4'. Cold Plasma Convection and the Plasmapause 4.2.5. Energetic Plasma Convection Current Sheet Energization in the Tail 4.3.1. Estimate of Total Particle Energization Rate 4.3.2. Particle Motion in a Current Sheet 4.3.3. Effects of Current Sheet Energization of Ions in the Geomag
netic Tail Auroras and Parallel Electric Fields 4.4.1. Evidence for the Acceleration of Auroral Electrons by Parallel
Electric Fields 4.4.2. Association of Auroras with Field-Aligned Currents 4.4.3. Latitudinal Variations over Auroras 4.4.4. The Current-Voltage Relation along Auroral Field Lines 4.4.5. Generation of Large-Scale Inverted- V Precipitation Regions 4.4.6. Smaller Scale Discrete Auroral Structure
64 75 79 86 86 87
91 99
99 104 104 109 112 120
CHAPTER 5 : WAVE-PARTICLE INTERACTIONS 133 5.1. General Relations for Wave Growth and Particle Diffusion 133 5.2. General Results from Cold Plasma Theory 142 5.3. 1~12 in Terms of the Measurable Wave Intensity 143 5.4. Auroral Kilometric Radiation (AKR) 145 5.5. Whistler-Mode Waves and the Radiation Belts 156
5.5.1. Generation of Plasmaspheric Hiss 157 5.5.2. General Concepts on Pitch-Angle Diffusion in the Radiation
Belts 163 5.5.3. Pitch-Angle Diffusion of Radiation Belt Electrons within the
Plasmasphere 168 5.6. Loss of Ring Current Ions by Ion-Cyclotron Waves 180 5.7. Electrostatic Waves Outside the Plasma pause 190 5.8. Balance Between Radial Diffusion and Radiation Belt Particle Losses 195
5.8.1. Equilibrium Structure of Radiation Belt Electrons within the Plasmasphere 199
5.8.2. Equilibrium Structure of Radiation Belt Ions within the Plasmasphere 206
INDEX 229
PREFACE
The discovery of the earth's radiation belts in 1957 marked the beginning of what is now known as magnetospheric physics. The field has evolved normally from an early discovery phase through a period of exploration and into an era of quantitative studies of the dynamics of magnetized plasmas as they occur in nature. Such environments are common throughout the universe and have been studied in varying detail at the sun, the planets, pulsars, and certain radio galaxies.
The purpose of this book is to describe basic quantitative aspects of magnetospheric physics. We use selected examples from the earth's magnetosphere to show how theory and data together form a quantitative framework for magnetospheric research. We have tried to organize the material along the philosophy of starting simply and adding complexity only as necessary. We have avoided controversial and relatively new research topics and have tried to use as examples physical processes generally accepted as important within the earth's magnetospheric system. However, even in some of our examples, the question of whether the physical process applied to a particular problem is the dominant process, has yet to be answered.
MKS units are used in Chapters 1 through 4. Because of historical precedent and a desire to allow results in this book to be compared with the published literature, Gaussian units are used in Chapter 5, Wave-Particle Interactions. Chapter 1 presents a brief description of the earth's magnetosphere. Chapter 2 describes the motion of charged particles in magnetic and electric fields and introduces the useful approximation of guiding center particle motion in these fields. The earth's trapping regions and the currents established by particles trapped in these regions are presented in Chapter 3. Chapter 4 introduces the magnetospheric electric field and describes the major influence it has on magnetospheric dynamics. Chapter 5 describes wave-particle interactions and their role within the magnetospheric system.
Thanks are due to many people who have helped us in this effort. Specifically, we wish to thank Viola Hill and Gayle Snyder for typing, Jim Adams for drafting, and Lindsay Murdock for editing.
March 1983 L. R. LYONS D. J. WILLIAMS
ix
A A a B B(w)
Be B; Bk Bo BD BM Bs BVII BJ.I Bwave Bwave(w,6)
c c D D(w*,k) DLL DE
D~ Doe Dom DOl. 01.
DOI.v DvOl. Dvv q} d
LIST OF SYMBOLS
area vector potential semi-minor axis of ellipse magnetic field wave magnetic field as a function of w, normalized so that B~ave = JB2(W) dw equatorial magnetic field strength at earth's surface ionospheric magnetic field strength magnetic field of a wave mode as a function of k geomagnetic field strength in equatorial plane magnetic field at r = 0 due to a particle's azimuthal drift magnetic field at mirror point magnetic field at earth's surface magnetic field strength at top of a potential variation along field lines magnetic field at r = 0 due to particle's spiral motion about B total wave magnetic field as a function of position and time wave magnetic field strength as a function of wand 6 at a particular position and time speed of light as subscript, refers to cold plasma population quasi-linear diffusion matrix in velocity space D(w*, k) = 0 is the electrostatic wave dispersion relation radial diffusion coefficient radial diffusion coefficient from electric potential field fluctuations radial diffusion coefficient from magnetic field fluctuations DE for L = I and /J. = 0 D},f for L = I pit~-angle diffusion coefficient, element of D mixed diffusion coefficient, element of D mixed diffusion coefficient, element of D speed diffusion coefficient, element of D cold plasma parameter as defined by Equation (5.13) half-width of current sheet, also used for width of open magnetic field line region as mapped into the interplanetary medium electric field convection electric field in the equatorial plane electric field of wave mode as a function of k right-hand polarized component of the perpendicular wave electric field left-hand polarized component of the perpendicular wave electric field
xi
xii
ER Ewave e (£2)
F(t)
F(w!D.a F I
Ie
10 Ip
I(v, a)
G(L)
G g(ao) gw (8) [H] fl hll I 10 Ip It III
LIST OF SYMBOLS
corotation electric field total wave electric field as a function of position and time electronic charge, also, as subscript, refers to electrons mean square strength of convection electric field fluctuations time-varying part of equatorial particle distribution function when it is separated into two parts, 10 = F(t)g( ao) the function (QdW)2 (l - w!D.i)3 force particle distribution function in velocity space, normalized so that f I(x, v) d3 v =N(x) critical value of I for net growth of whistler-mode waves in the radiation belts, based on Kennel and Petschek (1966) (see Chapter 5) equatorial particle distribution function particle distribution function in momentum space, normalized so that fl(x, p) d3 p =N(x) number of particles having velocities between v and v + dv and pitch angles between a and a + da . weighting function for energy loss from Coulomb collisions, defined by Equation (5.57) operator defined by ([w/kll] - vII) (ajaVl) + Vl(ajaVII) time independent shape for the equatorial particle distribution function distribution of wave energy with wave normal angle at a particular w neutral hydrogen density Planck's constant divided by 27T ckll/nQ integral invariant for particle bounce motion between mirror points Bessel function of order zero and imaginary argument height-integrated ionospheric Pedersen current current per unit length total field-aligned current per unit distance as subscript, refers to ions, initial value, or ionospheric value current current due to gyration effects current due to magnetic field gradient and curvature drifts 2nd adiabatic invariant Bessel function of the first kind omnidirectional particle flux total electron flux per unit area precipitating into atmosphere current density differential particle flux, i.e., particles/unit area-s-ster-unit energy as subscript, plasma species ionospheric Hall current density ionospheric Pedersen current density magnetic field-aligned current density kinetic energy Hermitian part of the dielectric tensor
Kth K~
KII,min
K II , res k k
kll, res L Lpp
2 Q
Qc Q[
QM m
"!p n M N N(w)
Nb n pE pM
p~
PII P P fJJ Q(cxo)
q Q'(cxo) , '0 's Rc Re Rg [J1
S s(cxo) SD Sq
LIST OF SYMBOLS
thermal energy particle kinetic energy perpendicular to magnetic field minimum parallel kinetic energy [(1/2) mvn for cyclotron resonance parallel kinetic energy for resonance with a wave wave vector constant relating €p to VII value of kll for resonance with a wave McIlwain's L-parameter L-value of plasma pause cold plasma parameter as defined by Equation (5.13) distance along magnetic field lines distance along a single wave characteristic
xiii
length of open magnetic field region as mapped into the interplanetary medium distance of mirror point from magnetic equator along field line particle rest mass proton rest mass unit vector along outward normal to a magnetic field line earth's dipole moment plasma density normalization factor given by Equation (5.22) in transformation from Bwave (w, 8) to Bk bound electron density harmonic resonance number power spectrum of electric field fluctuations power spectrum of magnetic field fluctuations particle pressure normal to B particle pressure parallel to B integer> I momentum cold plasma parameter as defined by Equation (5.13) variation of bounce-averaged angular drift velocity with CXo multiplied by s(cxo); -(0.35 + 0.15 sin cxo)s(cxo) particle charge variation of D-DE with CXo multiplied by s(cxo) radial distance from origin equatorial crossing distance of a magnetic field line ,at earth's surface == IRe'" 6380 km radius of curvature of a magnetic field line one earth radius guiding center location cold plasma parameter as defined by Equation (5.13) surface separating closed and open geomagnetic field lines variation ofTb with pitch angle, -1.38 -0.32 (sin CXo + sinl/2 cxo) disturbed polar ionospheric current system quiet-time ionospheric current system
xiv
s~ y T Tc t UM
VII VII, 0
VD VDB VDe VDE VDl VDM Ve Vsw v Vg
vR
vII, min VII, 0
VII, res y W Wk x x xi
Xw Z a aLe ao ~ rw(ao) 'Y
'Yr g gp €
~ 11
LIST OF SYMBOLS
quiet-time polar ionospheric current system cold plasma parameter as defined by Equation (5.13) exponential decay time temperature of cold plasma population time total energy of earth's dipole field above earth's surface total magnetic field-aligned electric potential difference maximum value of VII drift velocity of particle guiding center total guiding center drift velocity due to magnetic field guiding center drift velocity due to magnetic field line curvature electric field drift velocity guiding center drift velocity due to inertial forces guiding center drift velocity due to magnetic field gradient volume of the earth, 1.08(10)21 m3
solar wind velocity velocity group velocity corotation velocity minimum parallel particle velocity for cyclotron resonance center of relativistic resonance ellipse in (Vi VII )-plane parallel particle velocity for resonance with waves volume sum of kinetic and potential energy total wave energy as a function of k, electromagnetic and kinetic position in space horizontal distance horizontal distance in the ionosphere half-width of large-scale auroral precipitation regions charge number of ions pitch angle equatorial pitch angle of loss cone equatorial pitch angle ratio of particle speed to speed of light, vic pitch-angle dependence of DiJ!L wave growth rate relativistic mass correction factor (1 - v2 Ic 2 )-1/2
electron energy flux per unit energy total electron energy flux per unit area precipitating into atmosphere sign of charge angle between the tangent to a single-wave characteristic and the Vi-axis fraction of total number of particles on a flux tube that are within the loss cone, given by Equation (5.36) wave normal angle (the angle between Band k) weighting factor in linear and quaSi-linear wave theory given by Equation (5.4)
A Xe AM J1. J1. J1.B J1.0 ~ P ~p a
aH ap TB Tc
TeE
TDB
TDE
Tm
TO
Tp Ts(O:o) cP CPj CPpp <P <PM <Ppc q, n nB nDB
nDE W
w* wLH
Wp
wUH
Wx
II 1
LIST OF SYMBOLS
geomagnetic latitude magnetic invariant latitude of a magnetic field line at the earth's surface geomagnetic latitude of mirror point first adiabatic invariant (magnetic moment = J1.h,) refractive index magnetic moment magnetic permeability of free space, 41T(10)7 henry m-1
angle between lobe magnetic field and midplane of current sheet gyroradius height-integrated ionospheric Pedersen conductivity cross section for charge exchange with neutral hydrogen ionospheric Hall conductivity ionospheric Pedersen conductivity particle bounce period gyroperiod ion lifetime from charge exchange
xv
particle drift period around magnetic field axis due to magnetic field gradients and curvature drift particle drift period around magnetic field axis due to electric field drift minimum lifetime for trapped particles under strong pitch-angle diffusion 4LRe/v lifetime of trapped particles from precipitation into the atmosphere particle source time scale as a function of 0:0
electric potential ionospheric potential electric potential of plasmapause magnetic flux through particle's gyro-orbit 3rd adiabatic invariant potential differences across polar cap longitude angle measured eastward with respect to 2400 LT particle gyro frequency frequency of particle bounce between mirror points frequency for particle drift around the earth due to magnetic field gradient and field line curvature frequency for particle drift around the earth due to electric fields real part of wave frequency complex wave frequency w* = w + ir lower hybrid frequency plasma frequency upper hybrid frequency cutoff frequency above ne as subscript, component of vector quantity parallel to B as subscript, component of vector quantity perpendicular to B
