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Chapter 3
BOUNCE-RESONANCE TEST-PARTICLE MODEL
The guiding center equations developed in Chapter 2 are adapted in this chapter for
use in the test-particle simulation. Motion of particles in a model field line resonance
(FLR) are expressed using geomagnetic dipole coordinates [Streltsov and Lotko,
1997]. Dipolar coordinates are chosen to make use of the precomputed FLR fields
provided by Streltsov et al. [1998]. The dipole model in equation (3.1) is used to
represent a background geomagnetic field. It is noted that the actual background
magnetic field in the LLBL usually differs significantly from that of a simple dipole
model, so the model results presented in subsequent chapters may be compared with
observations only in a qualitative sense.
The first section of this chapter defines the dipolar coordinate system. The second
specifies the constraints placed on the model. The third section describes the
phenomena of bounce-resonance acceleration. The explicit differential equations
pertaining to this model, defined in dipole coordinates are derived in the final section.
3.1 DIPOLAR COORDINATE SYSTEM
Dipolar coordinates are a locally orthogonal system with respect to an ambient, dipolar
magnetic field. The precomputed FLR fields used in the calculations found in Chapter
6 are expressed in dipole coordinates. This coordinate system is derived from the
dipole representation of the geomagnetic field,
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

RE3
Br ,   31,000 3 rˆ 2 cos  θˆ sin  nT.
r
3.1
The geomagnetic field is expressed in terms of the spherical polar coordinates r, 
 
and unit vectors rˆ , θˆ . Here, r is the geocentric radial position; the mean radius of the
earth, RE, is 6371 km; and θ, measured from the north dipole axis, is sometimes
referred to as the magnetic colatitude.
Streltsov and Lotko [1997] define nondimensional dipole coordinates and their
respective metric factors, h, in terms of spherical polar coordinates as,
L
r
,
RE sin 2 θ
hL 
RE sin 3 θ
1  3 cos 2 θ
μ
φ,
,
h  r sin θ ,
RE2 cos θ
r2
h 
3.2
r3
RE2 1 3 cos 2 θ
.
3.3
The dipolar coordinate system is illustrated in Figure 3-1. The unit vector μ̂ , points
along the dipolar magnetic field lines; L̂ and φ̂ represent perpendicular outward
“radial-like” and azimuthal directions, respectively.
If ζ represents any one of the dipole coordinates, then the unit vectors ζ̂ along each
coordinate axis may be derived from the expression, ζˆ  ζ
 ζ . Using the
definitions for dipole coordinates given in 3.2 and 3.3, the dipolar unit vectors in terms
of spherical polar coordinates are:
ˆ
ˆ  rˆ sin   θ 2 cos ,
L
1  3 cos 2 
φ̂ ,
μˆ  
rˆ 2 cos  θˆ sin 
1  3 cos 2 
.
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Figure 3-1. Dipolar computational domain defined by Streltsov and Lotko [1997].
3.2 MODEL CONSTRAINTS
The particle motion is based on guiding center equations in the presence of an ambient
dipole magnetic field and precomputed dispersive Alfvén wave fields [Streltsov et al.,
1998]. The two-dimensional dispersive field line resonance model from Streltsov et al.
[1998] specifies the parallel electric field ( E  ), the meridonal perpendicular electric
field ( EL ) and the azimuthal magnetic field ( B ) as functions of (L,  , t). Because
the fields are periodic in time, it is sufficient to know, for the test-particle integration,
the spatiotemporal dependence of the fields for one wave period only.
The specific particles of interest here are electrons. In subsequent equations, the
particle charge is specified as the electron charge, q  e , and the particle mass is the
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electron mass, m  me . The magnetic moment, M, is conserved as mentioned in the
last chapter. The electrons here are non-relativistic, because the electron energies
calculated in this model are less than 1 keV. Collisions between particles are
neglected. Self-consistent contributions to the electromagnetic fields by the evolving
electron populations are also neglected.
To assess the effects of wave-particle interaction on electrons in a field line resonance,
the electron is initialized at the magnetic equator in the wave electromagnetic field
with a velocity chosen at random from a Maxwellian distribution with thermal energy
of 100 eV. The phase of the time-periodic wave field at the initial time is chosen
randomly with uniform weight over the wave period. The motion of 600,000 electrons,
which drift and accelerate according to equations (2.8) and (2.9), is recorded to
develop a statistical picture of the aggregate electron dynamics.
The electron population is initially placed in the boundary layer, presumed to be
located at L = 7.5. This location was chosen in order to make use of the precomputed
FLR specified by Streltsov et al. [1998], which straddles L = 7.5. Although it is
unusual for the LLBL to be observed at L = 7.5, the bounce-resonance criterion given
in the next section is shown to depend weakly on the L-value and, more significantly,
on the Alfvén speed in the layer. Therefore, the results on bounce-resonance
acceleration discussed later in Chapter 6 are not expected to change qualitatively from
those for a FLR located, for example, at L = 10, corresponding to a more typical radial
location for the LLBL.
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3.3 BOUNCE-RESONANCE ACCELERATION
The physical picture of bounce-resonance interaction in a standing dispersive Alfvén
wave is shown in Figure 3-2. Four half-cycles of the parallel electric wave-field, E||,
from Streltsov et al. [1998] are plotted along a magnetic field line. An electron is
initially near its northern hemispheric mirror point in the first frame and is allowed to
interact with the parallel electric field. The electric field points from positive to
negative and the electron accelerates in the opposite direction towards the equator. If
the wave E|| has changed its direction (one-half wave cycle later) after the electron has
reached the equator, it continues to see a negatively directed parallel electric field and
is then accelerated toward its southern hemispheric mirror point. Upon reaching its
southern hemispheric mirror point, the antisymmetric E|| has again evolved one-half
cycle; the electron now sees a positive E|| that accelerates it back towards the equator.
One-half wave cycle later when the electron again reaches the equator, the E|| remains
positive along the electron’s trajectory and continues to accelerate the electron back to
its northern hemispheric mirror point. The electrons are energized in this manner,
causing them to travel farther which lowers their mirror points towards the
atmosphere. Simultaneously, their bounce periods shorten, until they become
sufficiently short, causing the electrons to slip out of phase with the wave E||.
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e
t
t + 1/2  w
E|| (V/m)
t+w
t + 3/2  w
Distance along the field line (RE)
Figure 3-2. Illustrates the wave-particle interaction due to bounce-resonance in the presence of a
FLR parallel potential and dipole magnetic field. The top panel represents time t; the second frame
is one-half wave period later; the third panels depicts time at one wave period later; and the last
panel is at one full wave period and a half. The red dot represents an electron traveling along the
field line depicted as the horizontal axis in earth radii.
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The parallel energization of the electron by wave-particle interaction is optimal when
the phase of electron motion is synchronized with the wave phase as illustrated in
figure 3-2. The electron bounce period,  B , is twice the wave period,  w according to
the illustration. The bounce period in a dipole magnetic field without the perturbation
from the wave is [Baumjohann and Treumann, 1997],
τ B  LRE 3.7  1.6 sin  eq  W / m
1 / 2
.
Here, W represents the total energy of the particle and eq is the equatorial pitch angle.
The wave period, τ w , scales as the field line length divided by the Alfvén speed, vA, at
the equator. For large values of L the field line length is approximated by 2.4LRE , so
that, τ w  2.4LRE v A . Equating the expressions, τ B  2 τ w , results in the following,
 3.7  1.6 sin  eq
1
W  me ve2  me v A2 
2
4.8

ve  v A



2
3.7  1.6 sin  eq
3.4
3.4
The term within the absolute value brackets in equation (3.4) varies from 0.62 to 1.1
as the equatorial pitch angle varies from  2 to 0 radians, respectively. Therefore,
bounce resonance occurs when the equatorial electron velocity is essentially the
Alfvén speed in the LLBL. The motion of electrons in the LLBL with velocities
significantly exceeding vA will be only weakly perturbed by the presence of the wave
parallel electric field. Electrons with velocities much less than vA are initially
accelerated nonresonantly by the slowly varying parallel electric field until their
velocities become comparable to vA at which point a bounce resonance interaction
occurs. This interaction is confirmed by the numerical studies of Chapter 6.
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3.4 DIPOLE EQUATIONS OF MOTION
The guiding center equations (2.8) and (2.9) are rewritten in terms of the orthogonal
curvilinear dipole coordinate system described in Section 3.1. The FLR fields
computed by Streltsov et al. [1998] are used in the test-particle calculations to evaluate
the drift and acceleration motion of the electrons. This is facilitated using their dipole
forms in equations (2.8) and (2.9). The orthogonal dipolar coordinate system (L, , )
represents the general orthogonal curvilinear coordinates ( e1 , e2 , e3 ) defined in
Appendix A.
Evaluating the drift terms in equation (2.8) explicitly with the vector components
provided by the FLR model by Streltsov and Lotko [1997] and the ambient dipole
magnetic field gives:
vD 

E Lˆ  E μˆ  B φˆ  B μˆ 
L


B  B
2
0
2
0
μˆ E L B   Lˆ E  B   φˆ E L B0 
3.5
B2  B02
 v  μˆ  v L Lˆ  v φˆ .
where, v  EL B
B
2




 B02 , vL  E B B2  B02 ,


v  EL B0 B2  B02 . The
parallel acceleration in Chapter 2, equation (2.9), becomes,
v   
e
M 1 B0
E 
.
m
m h μ
3.6
The parallel gradient of the dipole magnetic field, B0, required to evaluate the mirror
force in equation (3.6) is calculated analytically in terms of the spherical polar
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coordinates r, θ  . Representing the parallel gradient as μ̂   and defining the μ̂ unit
vector in terms of spherical polar coordinates yields:
 2 cos θ

sin θ
 
 B0 .
μˆ  B0  


2
2

r

θ
r 1  3 cos θ
 1  3 cos θ

Evaluating the partial derivative using magnitude B0 from (3.1) then provides,
3BE cos θ
1 B0

4
h μ
RE r RE 

sin 2 θ 
 2 
 .
2
1

3
cos
θ


3.7
Finally inserting (3.7) into equation (3.6) gives the explicit form of the parallel
acceleration for use in the numerical calculations of Chapter 4,
v  
e
M 3BE cos θ
E 
m
m RE r RE 4

sin 2 θ 
 2 
.
1  3 cos 2 θ 

3.8
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