Chapter 6
SIMULATION RESULTS AND DISCUSSION
This chapter describes results from the test-particle bounce-resonance calculations
utilizing the third harmonic FLR of Streltsov et al. [1998] as the wave electromagnetic
field. The initial LLBL magnetosheath population has a thermal energy of 100 eV.
The results presented are the culmination of the test-particle model that has been
developed and tested in the previous chapters. Section 6.1 describes the simulation
procedure utilized for the calculations. The resulting velocity distributions of the
electrons are shown in Section 6.2. The third section evaluates the electron parallel
and perpendicular temperatures and the associated anisotropy for comparison with the
epoch analysis of Paschmann et al. [1993]. A qualitative discussion of the possible
outcomes of varied electron population and/or FLR parameters is contained in Section
6.4. The final section contains recommendations for extending the work in this thesis.
Several proposed mechanisms of plasma transport from the magnetosheath, across the
magnetopause, and into the LLBL were evaluated by Lotko and Sonnerup [1995].
Although precise LLBL formation is still uncertain, most modes described by Lotko
and Sonnerup [1995] appear to involve either an influx of plasma into the layer at a
localized region or plasma entry along its outer edge. These two modes of plasma
transport considered in this chapter are depicted in Figure 6-1 and Figure 6-2. Separate
statistical models are developed to represent each transport mode. Velocity
45
46
distributions and temperature calculations resulting from the model are constructed for
both cases. The overall effects of these statistical models on the results are discussed
in the Sections 6.2 and 6.3.
The first case, shown in Figure 6-1, injects electrons into the boundary layer at a single
localized region (depicted by the small arrows) near or at the magnetosheath
stagnation line. The electrons interact with the FLR fields while continuing to bounce
between mirror points, drift transversely within the flux tube and drift antisunward
along the boundary layer as illustrated by the thick black arrow of Figure 6-1. The
downstream velocity distributions are produced by a local injection of magnetosheath
plasma conditions near the subsolar point. It is assumed that the plasma density
decreases as the width of the layer increases with downstream distance such that the
relation N p w remains constant, where Np is the number density of particles and w
is the width change of the layer.
The second mode of transport illustrated in Figure 6-2 depicts the injection of
electrons at all points along the magnetopause. As with the first case, the electrons in
the boundary layer interact with the FLR wave fields while continuing their previous
bounce and drift motions. The uniform injection of low energy electrons into the
boundary layer attempts to create a more diffusive condition through a continual
admixture of magnetosheath and FLR energized electrons which is shown in Sections
6.2 and 6.3 to smooth the parallel energization processes.
47
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, shown by the small arrows.
The thick black arrow represents the drift motion of the electrons, antisunward, along the boundary
layer.
Figure 6-2. The plasma transport shown here injects electrons along the entire MP, shown by the
small arrows. Therefore, a continual admixture of previously energized electrons and recently
injected low energy electrons populates the layer.
48
6.1 SIMULATION PROCEDURE
The flow chart in Figure 6.3 illustrates how the programs, subroutines and functions
described Chapter 4 work together to estimate particle positions using the
precomputed FLR fields from the work of Streltsov et al. [1998]. The procedures
described are repeated for each of the 600,000 particles in the initial distribution used
for these calculations.
The precomputed FLR field data is initially read into the algorithm and placed into
three-dimensional matrices indexed by L, μ and time. The data is then normalized
using the parameters in Section 4.4.2 evaluated for a 100eV electron. Initial electron
velocity and pitch angle are chosen from a Maxwellian distribution using a random
number generator. The initial phase of the FLR wave is also assigned by a random
number generator.
The Runga-Kutta Fourth-Order integration described in Section 4.3.1 estimates the
first four iterations of particle position. The FLR field values at each of these
estimated positions are determined using the linear interpolation scheme discussed in
Section 4.4.1. The background magnetic field is evaluated functionally with spherical
coordinates transformed from the dipole coordinates at each estimated position (see
Section 4.2 and 4.4.1). These field values are used in evaluating the differential
functions shown in equations (3.5) and (3.6) for the next iteration of the numerical
integration. Having estimated the first four positions using the Runga-Kutta method,
49
the computationally faster Adams-Bashforth Fourth-Order Predictor-Corrector
integration method can now be used.
The algorithm checks for particle precipitation into the ionosphere. If the estimated
particle position is beyond the limits of the computational domain set in Section 4.1,
the particle is said to have precipitated. Its velocity and time at precipitation are
recorded. The procedure is repeated for the next particle with initial velocity, pitch
angle and wave phase chosen by random number generators. If the particle is still
within the bounds of the computational domain, the FLR and background magnetic
field values are again updated for the current position.
All particles are initially positioned at the equator and their energy (velocity) is
recorded at subsequent equatorial crossings up to the eleventh and final crossing. If the
estimated particle position is not crossing the equatorial plane, then the next particle
position is estimated with the Adams Fourth-Order Predictor-Corrector integration and
the process is repeated. If the particle is crossing the equator, its velocity, time of
crossing, and number of crossing are recorded. If it is not the final crossing, the
procedure is repeated starting with the estimation of the next position by the AdamsBashforth Fourth-Order Predictor-Corrector integration. If the particle is crossing the
equator for the final time, the velocity, time of crossing, and number of crossing are
again recorded and the next particle is initialized with a random velocity, pitch angle
and wave phase.
50
Particle parameters
initialized using Random
Number Generator
Data placed into 3D
matrices and their values
normalized
Raw FLR wave
field data read
Particle position
estimated by AdamsBashforth Fourth Order
Predictor-Corrector
Integrator
Check for
particle
precipitation
Field values updated for next
iteration:
Interpolate field values,
Coordinate transformation,
B0 function calculation,
Particle positions
estimated by RungaKutta Fourth Order
Integrator
Yes, record data and end particle loop
Start next
particle
simulation
No
Field values updated for next
iteration:
Interpolate field values
Coordinate transformation
B0 function calculation
Check for
equatorial
crossing
Yes, record data
No
Check for
final
crossing
Yes, end particle loop
No
Start next iteration
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.
51
6.2 PARTICLE DISTRIBUTIONS
The construction of velocity distributions of FLR interactions in the LLBL begins with
a distribution of 600,000 electrons initialized with velocity and pitch angle chosen
randomly from a gyrotropic Maxwellian velocity distribution. The electrons are
released at the equator and are found to interact immediately with the FLR. Velocity
distributions are recorded at each of the subsequent equatorial crossings for each of ten
such crossings.
The distributions shown in Figures 6-4 and 6-5 are constructed with the statistical
models representing the two modes of plasma transport described at the beginning of
this chapter. Figure 6-4 is representative of the first transport mode where electrons
are injected at one localized region. The dimensionless numerical values of phase
space density are constructed by dividing the phase space into v||-v bins, with each bin
of size v|| = 0.08 and v= 0.08 (47.2 km/s). The total number of electrons in each bin
is then counted. If this value is N v|| , v   , its normalized value N v|| , v  is
N v , v  N  100 s
||

MAX
3
km3 where NMAX is the maximum bin value in both phase
space and time.
A strong bi-directional coherence is evident in the distribution at the first equatorial
crossing in Figure 6-4. This bipolarization is observed in the subsequent distributions.
The distributions also appear to settle into a quasi-steady state by the eighth equatorial
crossing.
52
The distributions in Figure 6-5 represent the second mode of plasma transport into the
LLBL along the entire magnetopause. The statistical model for this case is assembled
with each subsequent distribution overlaid onto the previous one. The procedure for
constructing the “evolving” distributions is also to create bins in phase space. Again,
the electrons in each bin are counted. The total number of particles in each
distribution, NTOT, increases with each crossing so each bin value is first multiplied by
N0
NTOT  , where N0 is the number of particles in the initial distribution (i.e., 600,000
for these calculations). For this case, if the bin value is N v|| , v   , its normalized value
N v|| , v  is N v|| , v   N MAX  100 s 3 km3 where NMAX is the maximum scaled bin
value in both phase space and time. The panels of Figure 6-4 depict a similar approach
to a “steady state” condition as those found in Figure 6-5.
The absence of low energy particles (< 50 eV), which make up the bulk of the initial
distribution, is evident by the final crossing of both figures. The sample distribution
from the WIND spacecraft shown in Figure 1-2 does not correspond in detail to the
calculated distributions in Figure 6-4 and Figure 6-5. The calculated distributions also
differ from the observations reported by Paschmann et al. [1993] and Phan et al.
[1997]. The separation of the model electron distribution into two parallel
counterstreaming “beam” populations is most likely an artifact of the non-selfconsistent test-particle approach of the model. The observed distributions appear more
uniformly energized and their temperature anisotropy is on average less than what this
model predicts as shown in Section 6.3.
53
Figure 6-4. The velocity distributions of the initial population and the first eight equatorial
crossings (from right to left) for the first transport case are displayed here. The distributions are each
3
3
normalized to the initial population then constrained to a maximum intensity of 100 s km .
54
Figure 6-5. The electron velocity distributions here, for the initial and the first eight (right to left)
equatorial crossings for the second transport mode are each normalized to their population and
3
3
maximum density of 100 s km for each frame.
55
In this model, electrons are energized by the parallel electric field without an opposing
force to limit the immediate parallel acceleration response. If two-stream
microinstabilities were allowed to proceed self-consistently in the model, the initial
Maxwellian distribution might evolve more diffusely, possibly forming an elliptical
distribution similar to that of Figure 1-2. This diffusive evolution may also produce a
smaller temperature anisotropy than shown here.
6.3 ELECTRON TEMPERATURE ANISOTROPY
Observations indicate that the temperature anisotropy of electrons in the LLBL is a
robust diagnostic of the layer’s presence and magnetopause crossing. The qualitative
picture of bounce-resonance depicted in Figure 3-2 combined with the results of the
calculations depicted in Figure 6-5 demonstrate that when the particle has a bounce
period that is twice the wave period of a standing dispersive Alfvén wave, enhanced
parallel energization creates an anisotropy between the perpendicular and parallel
kinetic temperatures.
The concept of temperature within a plasma is not necessarily the same as in the
thermodynamic sense but rather it is a measure of the average energy of random
motion in a monatomic particle distribution. Temperature is a scalar quantity that is
calculated for a distribution when it is at or close to equilibrium. The kinetic
temperature of a plasma distribution is quantified as the second moment of the
velocity distribution defined in the form of a volume integral in Baumjohann and
Treumann [1997],
56
T
me
v  v b   v  v b  f v  d 3 v .
3kn 
Here, k is the Boltzmann constant, 1.380710-23 J / K. The velocity distribution f v 
found in Section 6.2 is used in the electron temperature and anisotropy calculations.
The field-aligned and transverse electron temperatures were approximated by
summations over the discretely binned velocity distribution, adapted from the triple
integral expressions by the trapezoidal rule,
T|| 
T 
2 me
3kn
v
2
 j || i
v


f v|| i , v  j  vb  2 ,
i, j


me
v 3 j f v|| i , v  j .

3kn i , j
Here, the velocity distribution is represented as a function of v|| and v. The average
bulk flow velocity, vb, is included in the expression for parallel temperature only
because the plasma displacement is along the parallel velocity axis while the
transverse flow remains unchanged. The particle density and bulk flow velocity are
also calculated by summation from integrals found in Baumjohann and Treumann
[1997],
n   v  j f v|| i , v  j 
i, j
 vb  
1
 v j v|| i f v|| i , v j  .
n i, j
The temperature anisotropy is defined by the ratio of electron parallel temperature to
perpendicular temperature minus one, Ae  T||e Te  1 . This form is used by
Paschmann et al. [1993], as shown in Figure 1-1. The values of temperature anisotropy
57
shown in Figure 6-6 are evaluated with the temperature calculated from the
expressions above.
The evolution of the electron temperature and anisotropy for the two statistical models
representing the boundary layer formation processes depicted in Figures 6-1 and 6-2
are shown in Figure 6-6. Because the electrons are energized primarily by the FLR
parallel electric field, both cases exhibit an increase in the parallel electron
temperature after interaction with the wave field while the perpendicular temperature
remains essentially unchanged. The result is consistent with the observed electron
temperatures and anisotropy reported by Paschmann et al. [1993] and Phan et al.
[1997].
The average electron temperature anisotropy of the superposed epoch analysis from
Paschmann et al. [1993] in Figure 1-1 is not quantitatively consistent with the results
shown in Figure 6-6. The values settle to an asymptote of approximately 0.4 by the
tenth equatorial crossing, while the average electron anisotropy reported by
Paschmann et al. [1993] is 0.2. The presence of an asymptotic value in both of the
formation processes serves as further evidence that the distributions have reached
some type of steady state.
58
Electron Temperature
Electron Temperature
Anisotropy
Figure 6-6. Evolution of electron temperature (T ||, T) and 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. The bottom horizontal axis in each panel
represents the number of equatorial crossings, where the distribution data was recorded. The top
axis is the mean time corresponding to the equatorial crossing numbers.
59
6.4 DISCUSSION
The test-particle calculations have shown that bounce-resonance acceleration of
electrons by low-harmonic, dispersive field line resonance in the Pc3/4 range produces
a temperature ratio T|| T  1 . This holds true wherever the source population is near
the optimal energy for resonant interaction with the standing Alfvén wave, i.e. bounceresonance, occurs when the equatorial electron velocity is essentially the Alfvén speed
in the LLBL. Magnetosheath electron thermal speeds are comparable to the Alfvén
speed in the LLBL indicating that the source population found in the LLBL is optimal
for efficient bounce-resonance interaction with low harmonic Pc3/4 standing Alfvén
waves. For the model calculations, this resonant acceleration produces a temperature
anisotropy of approximately 1.4 for a 100eV source population of electrons bouncing
in a dipole flux tube located at L=7.5.
A 50eV electron population, interacting with the same FLR, is likely to result in a less
dense, wider-set counterstreaming distribution. The lower energy particles in the testparticle calculations have a higher percent gain of energization and are expected have
a higher precipitation rate and subsequently a lower density than the 100eV
population. The implications of this case are touched upon in the discussion of
bounce-resonance presented in Chapter 3. An electron velocity distribution, with
thermal speed less than the Alfvén speed (i.e., the bounce period is greater than twice
the wave period) is initially accelerated nonresonantly by the slowly varying parallel
electric field until the average velocity becomes comparable to vA whereupon a
bounce-resonance interaction occurs. The bounce-resonance for the lower energy,
60
50eV distribution occurs when the equatorial electron velocity increases by about 50
percent, i.e. when it reaches the Alfvén speed in the LLBL. The additional energy gain
needed to reach resonance results in an increased parallel temperature differential and,
therefore, a greater electron temperature anisotropy for the 50eV electron distribution.
Similarly, changing the wave period will affect the distribution and its associated
anisotropy. Assuming the initial velocity distribution remains that of the test-particle
calculations (100eV electrons), a shorter wave period would result in a similar
situation to that described in the previous paragraph. The electrons would accelerate
with the wave parallel electric field until the distribution enters resonance. The
increased energy gain again results in a greater temperature differential and
anisotropy. Conversely, when the wave period is increased above the optimal value,
with respect to the velocity distribution, the thermal velocity of the distribution is
much greater than the Alfvén speed in the LLBL and the distribution is only mildly
perturbed.
The test-particle calculations described here involve wave-particle interaction with a
coherent, monochromatic 50 mHz Alfvén wave. In reality, a finite range of wave
periods (frequencies) can be expected in the LLBL. The expected particle response in
the presence of a finite bandwidth of frequencies would likely be a continuous electron
acceleration through the range of resonant frequencies. The electron bounce period
decreases as it is energized by the initial FLR; a shorter bounce period will likely
match another FLR frequency energizing the particle more. This process may occur
several times if the electron does not precipitate into the atmosphere first.
61
6.5 RECOMMENDATIONS FOR FUTURE WORK
A noise term added to the parallel equation of motion, representing fluctuations arising
from an expected electron-electron and/or electron-ion stream instability, would likely
inhibit the formation of counterstreaming electron beams that form in the test-particle
calculations. A model that includes this effect may provide a more realistic and
gradual approach of the asymptotic electron anisotropy.
The algorithms necessary to carry out detailed test-particle calculations of resonant
wave-particle interactions have been developed as part of this thesis project. An initial
study based on these algorithms demonstrates how bounce-resonant acceleration
creates an electron temperature anisotropy for a monochromatic wave in the LLBL. It
is unusual for the LLBL to be observed at L = 7.5, yet the bounce-resonance criterion
in equation (3.4) depends primarily on the Alfvén speed in the layer and is only
weakly dependent on the L-value. The results for bounce-resonance acceleration are
therefore not expected to change qualitatively from those discussed in Section 6.3 for
a FLR located, for example, at L = 10, corresponding to a more typical radial location
for the LLBL.
The parallel energy flux of electrons precipitating into the atmosphere, may provide
insights into micropulsations in auroral luminosity observed at the base of the LLBL
field lines [McHarg and Olson, 1992]. McHarg and Olson [1992] correlate ground
observations of magnetic pulsations with simultaneous measurements of fluctuations
in the intensity of auroral emissions. One might expect, based on the calculations
62
described in this thesis, that the precipitating electron flux will be modulated in the
Pc3/4 frequency range at the base of the LLBL field lines. An obvious extension of
this work might entail a closer examination of the parallel energy flux of precipitating
electrons and comparison of this flux to observed micropulsations in auroral
luminosity.
63