Supporting Material
Code description:
The first versions of the Versatile Electron Radiation Belt (VERB) code included only radial
diffusion [e.g. Shprits et al., 2005; 2006a; Shprits et al., 2008] or performed 2D pitch angle and
energy simulations [Shprits et al., 2006b].
The VERB-3D code models the diffusion in the space of adiabatic invariants by solving the
modified 3D Fokker-Plank diffusion equation that incorporates energy diffusion, pitch-angle
diffusion (pitch-angle is the angle between the magnetic field and particle’s velocity), mixed
pitch-angle and energy diffusion, and radial diffusion for the drift and bounce-averaged particle
phase space density, f.


 DL* L* L* 2 f*
 

L  , J 
,J 


1 
f
f

p 2   D pp ( y, p ) 
  D py ( y , p ) 
2
p p y , L 
p y , L
y p , L 

1

f
f
T ( y ) y  Dyy ( y, p ) 
  Dyp ( y , p ) 

T ( y ) y y p , L
y p , L
p

2 
f
 L*
t
L*
(S1)
 f

 
y,L 
,
where y is the sine of the equatorial pitch angle, L* is defined as a radial distance in the
equatorial plane from the center of the Earth to the field line of interest if the field is
adiabatically changed to a symmetric dipole field, μ and J are the first and second adiabatic
invariants, p is the particle’s momentum, <Dpp>, <Dpy>, and <Dyy> are the bounce- and MLTaveraged components of the diffusion tensor, T(y)=1.3802-0.3198 (y+y1/2) is the function that
shows how bounce frequency depends on the pitch-angle, and  is the characteristic loss
timescale that is assumed to be infinite outside the loss cone and equal to a quarter bounce time
inside the loss cone.
VERB 3D was first presented in Shprits et al., [2008] and compared to observations by Shprits et
al. [2009]. Subbotin and Shprits [2009] presented a detailed description and numerical
verification of the two-grid numerical approach used in the VERB-3D code. Subbotin et al.
[2010] presented an updated version of the code that included mixed diffusion terms.
The long-term modeling results have a good agreement with observations for energetic and
relativistic electrons (100 keV - ~1 MeV). The simulation performed with the range of L* from 1
to 5.5 up to 7 [Subbotin et al., 2010; 2011a; Kim et al., 2012; Drozdov et al., 2015].
The locations of the inner and outer edge of the outer radiation belt, and the location of peak flux
in different models that include different scattering mechanisms were tested using the mean
absolute percentage error during periods of different geomagnetic activity [Kim et al., 2012]. The
VERB code was also used to model the dynamics of the specific event for ultra-relativistic
electrons [Shprits et al., 2013].
The simulations have been performed in 1-D, 2-D and 3-D modes with and without mixed terms.
The different types of waves may be included: lightning, VLF waves from anthropogenic
transmitters, hiss, chorus, and EMIC. Advective transport has been tested using initial condition
as a step function and sin function. The diffusion has been tested on the analytical solution of the
bell curve. A dipole field approximation was used in this study. Only day-side chorus, night-side
chorus, and hiss waves were used. Radial diffusion was parameterized following Brautigam and
Albert [2000].
At each time step, convective transport and local diffusion are calculated at each MLT and
radial distance, and radial diffusion is calculated at each MLT as a function of L. For future
applications in a non-dipole field, the same approach can be used by converting DL*L* to DLL by
using the Jacobian transformation as derived from the non-dipolar field geometry. In a future
version of the code, when the magnetic field changes, PSD will be transported radially for each
MLT on the computational grid in such a way that it will conserve all of the three adiabatic
invariants V, K, and L (and therefore conserves μ, J, and L). This remapping will account for
adiabatic changes.
We use parallel implementation of the Alternating Direction Implicit (ADI) method with
additional stability to calculate local diffusion at each MLT and radial distance [Shin and Kim,
2008]. This method has been compared with the Bulk method of Subbotin et al. [2010] and gives
similar results when the time step is relatively small. Radial diffusion is calculated implicitly
using tridiagonal numerical solutions [Press et al., 2002].
The VERB-4D code solves Eq. (2) as described in the main part of the manuscript. The grid in
, L, V, K (25;29;61;60) is uniform in all variables except for V. The grid in V is uniform in
logarithm to better resolve lower energies. The current version of the VERB-4D code uses a
Volland-Stern electric field model with parameterization of Maynard and Chen [1975].
Convection is calculated using a 9th order upwind numerical scheme, with a Universal Limiter
(the ULTIMATE method guarantees that monotonic profiles stay monotonic) and Distinguishing
algorithm that preserves peaks [Leonard, 1991; Leonard and Niknafs, 1991].
The boundary conditions used in the code are shown in the table below. A periodic boundary in
MLT ensures that flux values on both sides of midnight are the same and particles are neither
lost nor generated at the boundary. The outer boundary of fluxes may be set up from statistical
averages, as used in previous studies [e.g. Shprits et al., 2013], which was used to create Figure
3. We use exponential peace wise fits to the following table of energy and flux values following
Subbotin et al. [2011b].
E, MeV =
[10-5, 10-4, 10-3, 10-2, 10-1, 2 10-1, 1, 3, 10]
Flux, #/cm/s/sr/keV= [3 109, 2 107, 9 106, 3 105, 1 104, 2 103, 70 , 3 10-3, 1 10-3]
GOES observations were used for the boundary conditions for comparison with Van Allen
Probes in Figure 4 of the main part of the paper. The outer radial boundary condition serves as a
source or loss of particles. In this study the outer boundary condition is set up at L = 6.6, while at
the inner boundary we assume that there is an absence of electrons at the top of the atmosphere.
Zero PSD at the lower boundary for the radial diffusion operator does not affect the solution, as
particles are lost before they reach low L-shells. The lower energy boundary for the energy
diffusion is set up from the initial condition which is obtained by solving the steady state radial
diffusion equation. The upper boundary condition is set to zero at multi-MeV energies. Since this
boundary is far from the energies of interest in this study, this boundary does not create any
additional loss.
Table 1. Boundary Conditions for the VERB 4D simulations
Lower boundary
Range
Upper boundary
MLT, Φ
Periodic
0 MLT < Φ < 24
MLT
Periodic
Radial Distance, R
Zero PSD
1 RE < R < 6.6. RE
GOES observations
10 keV < E < 10
MeV
Zero PSD
0.3o < α < 89.7o
Zero PSD gradient
Energy, E
Pitch-angle, α
Upper boundary in
energy is held
constant at 0. Lower
boundary is data
driven, see text for
details.
Zero
Diffusion coefficients are computed using the Full Diffusion Code (FDC) [Shprits and
Ni, 2009], which accounts for Landau and multiple order resonances. Momentum diffusion is
responsible for the acceleration of electrons, pitch-angle scattering produces a loss of electrons to
the atmosphere, and radial diffusion redistributes relativistic electron phase space density by
accelerating electrons during inward transport and decelerating them during outward transport.
Equation S1 accounts for simultaneous acceleration by both Ultra Low Frequency (ULF) and
Very Low Frequency (VLF) waves. The time of calculation depends on the number of time
steps, the size of the grid, and the type of simulation. The presented simulations are much faster
than a wall clock and currently take approximately 12 hours on 16 CPU for the entire simulated
period presented in the manuscript. Execution time may significantly decrease if the code is run
on a large computer cluster.
The code can be run on a regular PC and can be compiled for Windows, MAC OS, and Linux. In
addition, the code has been executed on the UCLA Hoffman II computer cluster. Long-term
simulations may take several gigabytes of hard drive space, depending on the size of the grid and
time resolution required. The code does provide the option to output files only at a desired
interval to reduce the use of storage and to increase computation time.
The simulation speed can be increased using the multi-thread calculation option that is included
in the model. The parallelization can be implemented when solving convection, radial diffusion
and local diffusion on each step. The code is documented using the Doxygen system, and version
control is maintained by a dedicated GitLab server. The input errors are checked during code
operation, while the output errors are displayed as a warning and can be tracked with a generated
log file. The code is written on C++, and MATLAB scripts were written to support the code
input/output and visualization.
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