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Chapter 5
ALGORITHM CONFIDENCE
Confidence in the validity and accuracy of solutions obtained from the test-particle
algorithm is essential before one can draw meaningful conclusions when novel
electromagnetic fields are introduced into the system. In this chapter, the simulation is
tested against known analytical solutions of fundamental trapped particle behavior.
Figure 5-1 shows the reaction of charged particles in the presence of a dipole magnetic
field. Particles spiral around magnetic field lines with gyro motion, defined in Section
2.1; move between mirror points along the field lines with bounce motion; and revolve
about the Earth in drift motion.
Figure 5-1. Illustration from Baumjohann and Treumann [1997] depicting the modes of particle
motion within the Earth’s magnetic field. Electron and ion trajectories simultaneously spiral around
magnetic field lines; move along the field lines bouncing between the mirror points; and drift about
the Earth’s axis westward for ions and eastward for electrons.
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The first section, 5.1, describes a benchmark study involving pure electron bounce
acceleration in the presence of a dipole magnetic field. The study in Section 5.2
includes a transverse electric field component, which induces an azimuthal drift of the
particle about the Earth. Section 5.3 includes a static parallel potential drop; hence, an
electric field in  accelerates the electron and changes its mirror points relative to its
bounce motion in a pure dipole magnetic field. The overall reliability of the algorithm
is discussed in Section 5.4.
5.1 BOUNCE ACCELERATION
Charged particles in the presence of a dipole magnetic field, like the geomagnetic field
given by equation (3.1), may become trapped between mirror points in the northern
and southern hemispheres due to the mirror force in the parallel acceleration equation
from Chapter 2. The first benchmark study concerns one-dimensional parallel electron
motion in the presence of only the mirror force. The relevant equations of motion are:
v  
μ 
v
h
M 1 B
,
m h  μ
.
5.1
The bounce period of the motion is the time it takes for a particle to travel along the
field line to each mirror point and back to its original position. Baumjohann and
Treumann [1997] give the bounce period as,
τB 
LRE
W / m1 / 2
3.7  1.6 sin   ,
eq
5.2
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which is strongly dependent on the energy of the particle, the equatorial distance from
the Earth, LRE, and weakly dependent on the equatorial pitch angle. The dependence
of the bounce period on mirror point colatitude, which may be mapped uniquely onto
equatorial pitch angle [Baumjohann and Treumann, 1997], is shown in Figure 5-2.
Numerical bounce periods are calculated with the normalization used in Schultz and
Lanzerotti [1974],   4L  v , for a direct comparison to the analytical bounce
periods of a 100 eV electron as it varies with mirror point colatitude.
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. The
numerical values of bounce periods of a 100eV electron at L=7.5 are normalized to the coefficients
in Schultz and Lanzerotti,   4 L   v . The bottom panel illustrates the percent error between
the analytic and numeric solutions. Note that the analytical solution is itself an approximation. The
errors in this approximation are attributed to an irreducible integral are not included in the error of
the figure.
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The matrices representing the wave electromagnetic drift and acceleration terms of the
integrator algorithm were turned off (set to zero) for the test. Only the mirror force due
to the ambient magnetic field function contributes to particle motion.
To understand bounce motion in a physical sense, imagine releasing a particle in the
equatorial plane at a distance LRE from the Earth’s center; it travels down the dipole
field line. The particle parallel velocity decreases with increasing magnetic field
strength while the perpendicular velocity increases conserving the total energy of the
particle. The mirror point occurs when the parallel velocity is zero whereupon the
perpendicular velocity achieves its maximum value. At the mirror point, the particle
reverses its parallel motion and begins to accelerate back up the field line toward the
equator where the magnetic field strength is weaker. After crossing the equatorial
plane, the particle again experiences the parallel deceleration while approaching the
mirror point in the other geomagnetic hemisphere.
The bounce period solutions shown in Figure 5-2 exhibit an error of approximately
0.15 percent between the analytical and numerical models. Errors in the model are
introduced in the form of local truncation errors associated with integration and roundoff error due to multiple transformations between coordinate systems.
5.2 AZIMUTH DRIFT MOTION
Consider the drift motion about the earth shown in Figure 5-1, which occurs in the
presence of the dipole magnetic field. There is also an azimuthal component from the
electric drift velocity described in equation (3.4) when a transverse electric field, EL, is
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introduced. A static electric field transverse to the ambient magnetic field is defined
for this test, and the numerically calculated particle drift motion is proportional to the
value of this field.
The drift velocity expressed in terms of dipole coordinates is given in Section 3.1. The
relevant  component of the electric drift equation is,
 
vφ
h

E L B0
1
.
2
2
B  B0 r sin 
A static perpendicular electric field, proportional to the ambient magnetic field is
chosen to evaluate the azimuthal drift velocity:
EL  5B0 V/m.
5.3
This test exercises and verifies the accuracy of the field interpolation algorithm, as
well as, the particle integration for simple azimuthal drift motion combined with
dipole bounce motion. Mapping the electric field of equation (5.3) into the
computational domain, with respect to the spherical coordinates r, θ  , is completed
according to the generic matrix form established for the input fields in Section 4.3.
The velocity is constant; therefore, the associated error is dependent upon the accuracy
with which the electric field is mapped and interpolated in addition to the original
truncation and round-off errors associated with the integration algorithm already
discussed.
The algorithms described in Chapter 4 interpolate EL and predict a constant drift for
particles with energies varying from 10eV to 500eV, shown in Figure 5-3. The
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constant slope of the lines in Figure 5-3 represent the angular velocity of the particles
and is accurate to approximately 0.15 percent error.
Figure 5-3. The numerical solutions of azimuth drift motion for a transverse static electric field of
various electrons with energies from 10eV to 500eV are compared.
5.3 FIXED POTENTIAL EFFECTS
The third and final validation of the algorithm incorporates a specified parallel electric
potential distribution that changes the mirror point relative to that found in Section 5.1.
If an electric potential distribution is imposed along magnetic field lines between the
ionospheres, a static parallel electric field is introduced into the system:
E||   ||  .
5.4
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This electric field acts as the Coulomb force in equation (3.7), accelerating electrons in
the direction opposing the parallel electric field.
5.3.1 ELECTRIC POTENTIAL
For this benchmark, the electric potential is taken to be an explicit function of the
dipole magnetic field; therefore, it depends implicitly on its position along the field
line. The representation of the electric potential used for this benchmark is defined as,
   0 sin 1  B B p .
5.5
The variable  0 is the amplitude of the potential. The maximum potential occurs at the
equator and its minimum,   0 , occurs where B  B p , which is taken to be the dipole
field strength on a particular L-shell at an altitude of 100 kilometers above the Earth
where the particles are expected to precipitate.
The field-aligned gradient of the potential is not explicitly known because the electric
potential is an implicit function of position,   f B(r, ) . Applying the chain rule in
terms of dipole coordinates reveals that
μ̂   
1 B 
.
h  B
In Section 3.4 the dipole magnetic field gradient along the field was evaluated as
1 B 3BE cos

h  RE r RE 4

sin 2  
 2 
.
1  3 cos 2  

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Combining this expression and the partial derivative of the potential with respect to
the dipole magnetic field to form the parallel electric field yields,
E  
3BE cos 
B p RE r RE 
4


sin 2  
B 
.
 2 
 0 cos1 
2


B
1

3
cos

p 



5.6
Mapping the electric field above into the dipolar computational domain, with respect
to the spherical coordinates r,  , is completed according to the generic matrix form
established for the input fields in Section 4.3.
5.3.2 ELECTRON ENERGY BALANCE
The energy balance for an electron including the static electric potential is
W

 W||  e eq  W  e  m  W  W||  e i ,
5.7
where the subscript eq denotes values at the equator, the subscript m depicts values at
the mirror point and i represents the iterate values at any point during the simulation.
The value of the electric potential is calculated at every time step within the
integration algorithm. These numerical solutions are shown plotted with the analytic
function, equation (5.5), in Figure 5-4. The energy balance of equation (5.7) also
implies that the total energy of the particle must remain constant throughout the
simulation, illustrated in the bottom panel of Figure 5-5.
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Figure 5-4. The electric potential function is directly compared to the numerical approximation.
This approximation of the electric potential is calculated within the particle integrator every
iteration. The initial electron thermal energy, equatorial pitch angle and maximum amplitude of the
test case are displayed between the panels, respectively.
5.3.3 POTENTIAL FIELD ACCURACY
For the parallel potential problem, the continual coordinate transformations due to
dependence on particle position and magnetic field strength incurs additional roundoff errors compared to the other tests in this chapter. The complexity of evaluating
electric potential every iteration also contributes to the round-off error of the solutions.
These errors are in addition to the truncation error and interpolation round off inherent
in the algorithm.
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The analytic and numerical potential values in the top panel of Figure 5-4 are found to
differ on average by 0.001 percent which appears to be increasing with time at a rate
of 110-6 percent per second. The total numerically calculated energy of the particles
remained constant during the simulation illustrated in the bottom panel of Figure 5-5.
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. The
initial electron energy, equatorial pitch angle and maximum amplitude of the test case are displayed
between the panels.
5.4 ERROR ANALYSIS
The benchmarks presented in this chapter include an evaluated percent error between
numerical values and the corresponding known values. These errors arise from several
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sources and the estimates are the criteria for which this algorithm is considered
reliable. Round-off errors from continual coordinate transformations and linear
interpolation, along with local truncation errors associated with the numerical
integration, all contribute to the overall error. Any inherent errors associated with the
wave fields are also a factor.
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Figure 5-6. Step size is compared to the local truncation error of the integrator algorithm.
Numerical bounce period solutions for a 100 eV electron and 30˚ mirroring colatitude were
calculated. The percent error of these calculations compared to the analytical solution found in
Baumjohann and Treumann [1997] are plotted as a function of step size. The slope of this relation is
four indicating that the calculation is indeed fourth order (see Burden and Faires [1993]).
Local truncation error mentioned in Section 4.2 decreases with step size. The figure
below illustrates the relationship between error and step size (i.e., error increases with
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step size). This relationship normally holds until the increased resolution of the step
size creates instabilities in the code. The truncation error breaks down at this local
minimum and algorithm error increases as step size decreases. The amount of error
associated with a coarse step size, according to Figure 5-6, is acceptable and stable
while the algorithm operates within this range of resolution.
The anticipated errors discussed above appear to validate the assumptions made in
Chapter 3 and the numerical calculations described in Chapter 4. The functions and
subroutines developed for the solution have proven reliable and robust when subjected
to controlled phenomena. The direct comparisons give the range of precision of the
solutions generated by the aggregate algorithm.