1
2
3
S1. Calculations of diffusion coefficients
The Kp, energy, and L shell dependent electric component (DLLE) and magnetic component
(DLLM) of the radial diffusion coefficient are expressed as [Brautigam and Albert, 2000]:
4
1 cE
T
E
DLL
 ( rms )2 (
)L6 ,
2
4 B0
1 ( DT / 2)
(S1)
5
DLLM  100.506Kp9.325 L10 ,
(S2)
6
respectively, where c is the light speed, B0 is the Earth’s equatorial magnetic field near the
7
surface (0.311 G), Erms is the root mean square (RMS) of the electric field amplitude, T is the
8
exponential decay timescale following the electric field fluctuations of the substorm activities
9
(2700 s), and ωD is the electron drift frequency. The total radial diffusion coefficient DLL is the
10
11
12
13
14
15
sum of DLLE and DLLM. When Kp > 1, ERMS is related to Kp as:
ERMS = 0.26(Kp-1) + 0.1 mV/m.
(S3)
The electron drift frequency for 90° pitch angle is:
D 
3c
,
 eL2 RE2
(S4)
where μ is the first adiabatic invariant, γ is the Lorentz factor, and e is the electron unit charge.
We used the UCLA Full Diffusion Code [Ni et al., 2008; Shprits and Ni, 2009] in a dipole
16
magnetic field to calculate the drift and bounce averaged quasi-linear diffusion coefficients of
17
electrons due to resonant interactions with plasmaspheric hiss. The hiss wave normal angles are
18
assumed to vary from field-aligned near the geomagnetic equator to highly oblique at higher
19
latitudes (45°), following the study of Ni et al. [2013]. The cyclotron harmonic resonances from -
20
10 to 10 and the Landau resonance are included in the calculation of diffusion coefficients.
21
Figure S1 presents the diffusion coefficients of 3.83 MeV electrons summed over different
22
ranges of harmonic numbers as a function of pitch angle at L = 4.5 and 5. At energies above ~3.5
23
MeV, although Ripoll and Mourenas [2013] have shown that higher order harmonics are needed
24
for a Gaussian wave frequency spectrum with a central frequency of 550 Hz, Figure S1 shows
25
that the summation of ten harmonics (black line) provides a reasonable estimate of the accurate
26
diffusion rates for the statistical wave frequency spectrum. The statistical hiss wave frequency
27
spectra during quiet periods are used for the 10-day period. The upper and lower cut-off
28
frequencies are 20 Hz and 4 kHz respectively based on the Van Allen probes observations. The
29
total plasma density ne is calculated using the empirical plasmaspheric density model of Sheeley
30
et al. [2001], which gives ne = 1390(3/L)4.83 cm-3. The pitch angle diffusion coefficients at L =
31
4.5 and L = 5.0 averaged over the 10-day period are shown in Figure 2b and Figure 2d,
32
respectively.
33
34
Figure S1: Diffusion coefficients for different ranges of harmonic number N due to
35
hiss wave scattering near noon at (a) L = 4.5 and (b) L = 5.0. Hiss wave amplitude is
36
10 pT; hiss wave power spectrum is adopted from the Van Allen probes survey
37
during quiet periods; hiss wave normal angle distribution is field-aligned near equator
38
and becomes oblique at higher latitudes, adopted from Ni et al. [2013].
39
The hiss diffusion coefficients are calculated using the statistical magnetic power spectrum
40
obtained from the recent Van Allen probes survey, which is different from the previously
41
commonly used Gaussian distribution with a central frequency of 550 Hz and a bandwidth of
42
300 Hz. Figure S2 presents the normalized wave power spectrum near noon at L = 4.5, the
43
resultant diffusion coefficients, and lifetimes of energetic electrons (MeV). The statistical power
44
spectrum (black line) peaks below 200 Hz, thus increasing the scattering of higher energy
45
electrons over that for the Gaussian spectrum results. At lower frequencies, the statistical wave
46
power spectrum falls faster than the Gaussian spectrum; therefore the diffusion coefficients drop
47
faster from 1 to 3 MeV. Figure S2 suggests that the accurate hiss power spectrum is necessary to
48
quantify the scattering of energetic electrons.
49
50
Figure S2: (a) Normalized statistical hiss wave frequency spectrum near noon at L = 4.5
51
and Guassian frequency spectrum with a central frequency of 550 Hz and a frequency
52
width of 300 Hz. (b) Diffusion coefficients near the loss cone. (c) Electron lifetimes for
53
MeVs energy range. Hiss wave amplitude is 10 pT; -10 ≤ N ≤ 10; hiss wave normal
54
distribution is adopted from Ni et al. [2013].
55
The diffusion coefficients due to the helium band EMIC waves are calculated from L = 4.0
56
to L = 5.5. The wave central frequency, frequency bandwidth, and lower and upper cutoff
57
frequencies are set as 3.6 fo+, 0.25 fo+, 3.35 fo+, and 3.85 fo+, respectively, where fo+ is the Oxygen
58
gyrofrequency. The wave normal is assumed to be field-aligned, and the waves have a maximum
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latitudinal distribution of 45°. The multi-species magnetospheric plasma is composed of 70% H+,
60
20% He+, and 10% O+, following Meredith et al. [2003] and Lee and Angelopoulos [2014]. The
61
EMIC wave intensity is Bw2 = 0.1 nT2 with an occurrence rate of 2% and a MLT coverage of
62
25% during modest periods when Kp ≥ 2, consistent with the recent study of Meredith et al.
63
[2014]. The pitch angle diffusion coefficients due to the EMIC waves at L = 4.5 and L = 5.0
64
when Kp ≥ 2 are shown in Figure 2c and Figure 2e, respectively.
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S2. The 3 dimensional radiation belt model
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The evolution of the radiation belt particle phase space density (PSD) due to the radial,
67
pitch angle, and energy diffusion processes can be described by the modified Fokker-Planck
68
equation [Schultz and Lanzerotti, 1974]:
f

f
 L *2
(DL*L* L *2
)
t
L *  ,J
L *  ,J

69



1

S( eq )sin  eq cos  eq  eq
1

S( eq )sin  eq cos  eq  eq
1 
p2  p
1 
p2  p
( p 2 D
L*, eq
eq p
( p 2 D pp
L*, eq
(S( eq )sin  eq cos  eq D
p,L*
(S( eq )sin  eq cos  eq D
p,L*
f
 eq
f
p
eq eq
)
p,L*
),
L*, eq
eq p
f
 eq
f
p
)
p,L*
)
L*, eq
(S5)
70
where f is the PSD and related to the differential flux j as f = j/p2, t is the time, αeq is the
71
equatorial pitch angle, p is the particle’s momentum, L* is the distance from the Earth’s center to
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the field line at the equator if the magnetic field was adiabatically relaxed to a dipole field, μ and
73
J are the first and second adiabatic invariants respectively, Dαeqαeq, Dpp and Dαeqp are the
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drift and bounce averaged pitch angle, momentum, and mixed pitch angle-momentum diffusion
75
coefficients, and S(αeq) is a function related to bounce period and in a dipole magnetic field it can
76
be simplified as S( eq )  1.38  0.32(sin  eq  sin 2  eq ) [Lenchek et al., 1961].
77
The Fokker-Planck equation (S5) is numerically solved in the radial distance from 2.5 RE to
78
5.5 RE with a step of 0.25 RE, and in the equatorial pitch angle range from 0° to 90° with a step of
79
1°. The energy range increases from the outer radial boundary to the inner radial boundary. To
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cover the essential energy range in which the Van Allen probes measured the gradual diffusion
81
processes, the energy range is set to be from 0.18 MeV to 5.8 MeV at L = 5.5, from 0.3 MeV to 8
82
MeV at L = 4.5, and from 1.1 MeV to 20 MeV at L = 2.5. We set the p grid as 90 logarithmically
83
spaced points from the lower to the higher energy boundaries.
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We solved the radial diffusion process along the constant μ and J lines in the radial grids.
85
The time step for radial diffusion is set as 60 s. The lower L shell boundary condition is set as f
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(L = 2.5) = 0, and the PSD at higher L shell boundary is set to be the particle data from the Van
87
Allen probes measurements. To simulate the radial diffusion process of PSD at each (L, α, p)
88
point, we calculated the PSD with the same μ and J values at different L shells by interpolation,
89
and used the fully implicit method to solve the radial diffusion equation. Therefore, the PSD
90
evolution at each (L, α, p) point is solved independently, and the results are directly used to
91
perform the local diffusion calculations.
92
We used the alternative direction implicit (ADI) method [Xiao et al., 2009] to simulate the
93
local diffusion processes due to the interactions between energetic electrons and the plasma
94
waves. The calculation at each L shell is performed independently. The PSD at the lower energy
95
boundary is set to be the particle data from measurements, and the higher energy boundary
96
condition is set as f (p = pmax) = 0 due to the absence of high energy electrons, where pmax is the
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momentum corresponding to the highest energy. The lower and higher pitch angle boundary
98
conditions are set as f (αeq  αeq,lc) = 0 and ∂f (αeq = 90°)/∂αeq = 0 respectively, where αeq,lc is the
99
equatorial pitch angle at the loss cone. The time resolution of the local diffusion process is 1 s.
100
The real-time wave amplitude distributions from observations are used in the simulation.
101
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102
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103
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