THE EFFECT OF BOUNCE-RESONANCE ACCELERATION ON
FORMATION OF ELECTRON TEMPERATURE ANISOTROPY IN THE
LOW-LATITUDE BOUNDARY LAYER
A thesis
Submitted to the Faculty
in partial fulfillment of the requirements for the
degree of
MASTER OF SCIENCE
by
ELIZABETH PACKMAYER
Thayer School of Engineering
Dartmouth College
Hanover, New Hampshire
OCTOBER 2000
Examining Committee:
__________________________________
William Lotko, Chairperson
__________________________________
Bengt Sonnerup, Member
__________________________________
Anatoly Streltsov, Member
__________________________________
Dean of Graduate Studies
© 1999 Trustees of Dartmouth College
__________________________________
Elizabeth Packmayer, Author
Thayer School of Engineering
Dartmouth College
“The Effect of Bounce-Resonance Acceleration on Formation of Electron Temperature
Anisotropy in the Low-Latitude Boundary Layer”
Elizabeth Packmayer
Master of Science
Committee
William Lotko, Chairperson
Bengt Sonnerup, Member
Anatoly Streltsov, Member
ABSTRACT
A robust feature and reliable diagnostic of the low-latitude boundary layer (LLBL) on closed
field lines is the existence of an electron temperature anisotropy such that T/T>1. This
temperature ratio is observed as parallel heating of a magnetosheath population in the layer
and persists for conditions of both low and high magnetic shear at the magnetopause. A
mechanism for the formation of the anisotropy is described in this thesis based on results from
test-particle calculations of electron parallel heating by field line resonance (FLR) of a
dispersive shear Alfvén wave standing along closed LLBL field lines between the northern
and southern ionosphere. Such resonances are expected to form by coupling to MHD surface
waves in the LLBL. Satellite measurements provide some evidence for these resonant Alfvén
waves in the LLBL, although the measurements are usually complicated by difficulties in
deconvolving spatiotemporal behavior in a typically fast moving boundary region. In the
calculations of Strelstov et al. [1998], the resonance is produced by a numerical simulation
based on equations of two-fluid, finite ion Larmor radius MHD. In their model, Alfvén wave
dispersion sustains a parallel electric field. When the period of the wave electric field is about
one-half the bounce period of an electron, the particle is efficiently accelerated along the
magnetic field.
The work described in this thesis makes use of precomputed FLR fields by Strelstov et al.
[1998]. These fields are used to calculate the motion of 600,000 test electrons representing a
nominal distribution of magnetosheath electrons. The evolved distribution of test electrons is
then used to construct ensemble averages and to estimate the parallel heating of
magnetosheath electrons as they interact with a model resonant Alfvén wave. This heating is
consistent with the observed temperature anisotropy, but the evolved distribution of test
electrons appears as counterstreaming beams rather than the anisotropic thermal distributions
representative of the observations.
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TABLE OF CONTENTS
Table of Contents ...................................................................................................iii
List of Figures.......................................................................................................... v
Acknowledgments ................................................................................................. vii
Chapter 1. Introduction ........................................................................................... 1
1.1 Boundary Layer Phenomena ....................................................................... 6
1.2 Field Line Resonance .................................................................................. 7
1.3 Test-Particle Calculations ............................................................................ 8
Chapter 2. Guiding Center Equations ................................................................... 10
2.1 Basic Equations of Motion ........................................................................ 11
2.2 Guiding Center Approximation ................................................................. 12
2.3 Particle Drift Velocity ............................................................................... 13
2.4 Parallel Particle Acceleration .................................................................... 16
Chapter 3. Bounce-Resonance Test-Particle Model ............................................. 17
3.1 Dipolar Coordinate System ....................................................................... 17
3.2 Model Constraints ..................................................................................... 19
3.3 Bounce-Resonance Acceleration ............................................................... 21
3.4 Dipole Equations of Motion ...................................................................... 24
Chapter 4. Numerical Methods............................................................................. 26
4.1 Computational Domain ............................................................................. 26
4.2 Coordinate Transformation ....................................................................... 28
4.3 Numerical Integration ................................................................................ 28
4.3.1 Fourth Order Runga-Kutta Method .................................................. 29
4.3.2 Adams Fourth-Order Predictor-Corrector Method ........................... 30
4.4 Auxiliary Procedures ................................................................................ 30
4.4.1 Field Initialization and Interpolation ................................................ 31
4.4.2 Normalization Parameters ................................................................ 32
Chapter 5. Algorithm Confidence ........................................................................ 33
5.1 Bounce Acceleration ................................................................................. 34
5.2 Azimuth Drift Motion................................................................................ 36
5.3 Fixed Potential Effects .............................................................................. 38
5.3.1 Electric Potential .............................................................................. 39
5.3.2 Electron Energy Balance .................................................................. 40
5.3.3 Potential field accuracy .................................................................... 41
5.4 Error analysis ............................................................................................. 42
Chapter 6. Simulation Results and Discussion ..................................................... 45
6.1 Simulation Procedures ............................................................................... 48
6.2 Particle Distributions ................................................................................. 51
6.3 Electron Temperature Anisotropy ............................................................. 55
6.4 Discussion.................................................................................................. 59
6.5 Recommendations for Future Work .......................................................... 61
Appendix A ........................................................................................................... 63
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Appendix B ............................................................................................................ 68
B.1 Perpendicular Drift Motion ....................................................................... 68
B.2 Parallel Acceleration ................................................................................. 69
Appendix C ............................................................................................................ 71
C.1 Polarization Drift ...................................................................................... 71
C.2 Gradient Drift ............................................................................................ 72
C.3 Centrifugal Drift ....................................................................................... 73
C.4 Miscellaneous Drift Terms ....................................................................... 74
C.5 Parallel Acceleration Truncation .............................................................. 77
Appendix D ........................................................................................................... 78
References ............................................................................................................. 95
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LIST OF FIGURES
Number
Page
Figure 1-1. The superposed epoch analysis from Paschmann et al. [1993]
is constructed with AMPTE/IRM satellite data for low-shear
MP crossings. ..................................................................................... 4
Figure 1-2. This electron velocity distribution was taken at the flank of the
MP by the WIND spacecraft and provided by T. Phan. ..................... 5
Figure 2-1. A charged particle gyrating about its guiding center along the
magnetic field B is the definition of gyro motion of particles. ........ 12
Figure 3-1. Dipolar computational domain defined by Strelstov and Lotko
[1997]. .............................................................................................. 19
Figure 3-2. Illustration of wave-particle interaction due to bounceresonance in the presence of a FLR parallel potential and
dipole magnetic field. ....................................................................... 22
Figure 4-1. Dipolar computational domain, shown in box grid orientation,
illustrates the discretization along the L and  coordinate axes. ...... 27
Figure 5-1. Illustration from Baumjohann and Treumann [1997] depicting
the modes of particle motion within the Earth’s magnetic field. ..... 33
Figure 5-2. Top panel is a comparison of numerical solutions and
analytical solutions from Schultz and Lanzerotti [page 19], of
electron bounce periods varying with the mirror point
colatitude. ......................................................................................... 35
Figure 5-3. The numeric solutions of the azimuth drift motion for a static
transverse electric field of various electrons with energies from
10eV to 500eV are compared. ......................................................... 38
Figure 5-4. The electric potential function is directly compared to the
numerical approximation. ................................................................. 41
Figure 5-5. The bottom panel demonstrates the constancy of the total
energy, Wtotal  W  W||  e and the top panel depicts the
oscillatory behavior of the ratio, W|| W , due the bounce motion...... 42
Figure 5-6. Step size compared to local truncation error of the integrator
algorithm. Numerical bounce period solutions of a 100 eV
electron and 30˚ mirroring colatitude were calculated. .................... 43
Figure 6-1. The physical picture of the first electron transport case is
illustrated in this figure. The electrons are injected at a single
localized region upstream in the layer. ............................................. 47
Figure 6-2. The plasma transport injects electrons along the entire MP and
the layer is populated by a continual admixture of previously
energized electrons and recently injected low energy electrons....... 47
Figure 6-3. The flow chart details the particle integrator code. The methods
presented here can be used with different wave fields and with
different particles and motions. ........................................................ 50
v
Figure 6-4. The velocity distributions of the initial population and the first
eight equatorial crossings of the first transport case are
displayed. The distributions are each normalized to the initial
population then constrained to a maximum intensity of
3
3
100 s km . ....................................................................................... 53
Figure 6-5. Electron velocity distributions of the initial and the first eight
equatorial crossings of the second transport mode are
normalized to their total population and a maximum density of
3
3
100 s km for each frame. ............................................................... 54
Figure 6-6. Evolution of electron temperature (T||, T) an anisotropy (Ae)
for statistical model 1 (upper two panels) and statistical model
2 (lower panels) corresponding to the LLBL formation
processes depicted in Figures 6-1 and 6-2 respectively. .................. 58
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ACKNOWLEDGMENTS
I would like to thank my committee members for being so patient, especially, my
advisor, Bill Lotko, who never let me settle for “good enough”.
A special thank you to Tai Phan for sharing the electron velocity distributions taken by
the WIND spacecraft.
To my family who continually encouraged me to get done, thank you all for believing
that I could complete this milestone.
A heartfelt thanks to my dear friends Aimeé, Jennifer and Craig at Dartmouth. Thank
you for always taking me in. You were always there for me and kept me motivated.
Thank you Keith, for giving more than was necessary. Thank you for providing me a
safe haven to work and a reason to finish.
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