# Electron resonant interaction with whistler-mode waves around the Earth’s
bow shock II: the mapping technique

David S. Tonoian  Department of Physics, University of Texas at Dallas,
Richardson, TX, USA<EMAIL_ADDRESS>Faculty of Physics, National
Research University Higher School of Economics, Moscow, Russia, 105066.
Xiaofei Shi  Department of Earth, Planetary, and Space Sciences, University of
California, Los Angeles, USA<EMAIL_ADDRESS>Anton V. Artemyev
Department of Earth, Planetary, and Space Sciences, University of California,
Los Angeles, USA<EMAIL_ADDRESS>Space Research Institute, RAS, Moscow,
Russia  Xiao-Jia Zhang  Department of Physics, University of Texas at Dallas,
Richardson, TX, USA<EMAIL_ADDRESS>Department of Earth,
Planetary, and Space Sciences, University of California, Los Angeles, USA;
<EMAIL_ADDRESS>Vassilis Angelopoulos  Department of Earth, Planetary, and
Space Sciences, University of California, Los Angeles, USA<EMAIL_ADDRESS>
###### Abstract

Electron resonant scattering by high-frequency electromagnetic whistler-mode
waves has been proposed as a mechanism for solar wind electron scattering and
pre-acceleration to energies that enable them to participate in shock drift
acceleration around the Earth’s bow shock. However, observed whistler-mode
waves are often sufficiently intense to resonate with electrons nonlinearly,
which prohibits the application of quasi-linear diffusion theory. This is the
second of two accompanying papers devoted to developing a new theoretical
approach for quantifying the electron distribution evolution subject to
multiple resonant interactions with intense whistler-mode wave-packets. In the
first paper, we described a probabilistic approach, applicable to systems with
short wave-packets. For such systems, nonlinear resonant effects can be
treated by diffusion theory, but with diffusion rates different from those of
quasi-linear diffusion. In this paper we generalize this approach by merging
it with a mapping technique. This technique can be used to model the electron
distribution evolution in the presence of significantly non-diffusive resonant
scattering by intense long wave-packets. We verify our technique by comparing
its predictions with results from a numerical integration approach.

##  1 Introduction

The high-frequency electromagnetic whistler-mode wave is one of the most
intense wave modes that can resonate with solar wind electrons and provide
electron pitch-angle and energy scattering 72, 26, 25, 79. In particular,
intense whistler-mode waves are observed around interplanetary shocks 80, 27
and Earth’s bow shock 42, 41, 59, 65. Resonant interactions between electrons
and such intense waves may result in strong pitch-angle scattering 56, 55, 66,
which is important for electron participation in the stochastic-shock-drift
acceleration 7, 8. Moreover, in the presence of ambient magnetic field
gradients and shock-related electrostatic fields, whistler-mode waves may
directly accelerate solar wind electrons 47, 20. This important role of
whistler-mode waves for suprathermal electron dynamics drives many theoretical
investigations of wave-particle resonant interactions around shocks (see
discussions in Ref. 81, 7).

The basic theoretical approach for modeling wave-particle resonant
interactions, the quasi-linear theory 77, 28, can be used to estimate the
electron scattering rates provided by whistler-mode waves 78, 7. However, the
criteria for this theory (low wave intensity, broad wave spectrum; see Ref.
44, 64) can be violated around the shock, where the observed whistler-mode
waves are very intense and sufficiently narrow-band 41, 65, 20. Such waves may
resonate with electrons nonlinearly. Presently, models of nonlinear resonant
interactions are mostly developed for the Earth’s inner magnetosphere (see,
e.g., Refs. 68, 4, 40, 19 and references therein) and require significant
modifications to be suitable for the plasma and wave characteristics commonly
observed around the Earth’s bow shock. In this paper we seek to advance such
models for planetary shocks, and in particular Earth’s bow shock, their most
accessible representative.

Intense whistler-mode waves detected at the bow shock and in the foreshock
region are strongly modulated by low-frequency compressional waves 42, 41,
i.e., whistler-mode waves are propagating in short wave-packets. Examples of
such wave-packets are shown in Figure 1 of Ref. Electron resonant interaction
with whistler-mode waves around the Earth’s bow shock I: the probabilistic
approach, hereinafter referred to as Paper 1. Figure 1(left panel) shows the
probability distribution of whistler-mode waves in the space of wave amplitude
$B_{w}$, in $nT$, and wave-packet size $\beta$, in wave periods. Although
there are long and intense wave-packets with $B_{w}>100$pT and $\beta\geq 20$,
the majority of the observed intense waves propagate in short wave-packets,
characterized by $\beta<20$. Such a strong wave modulation (short packet size)
can significantly reduce the efficiency of nonlinear resonant interactions71,
6, 5, because electrons spend less time in resonant interactions with short
wave-packets and cannot gain substantial energy (see simulation results in
Refs. 49, 82, 9, 34).

Figure 1(right panel) illustrates how wave-packet size affects the electron
nonlinear resonant interactions. In the case of infinitely long wave-packets
($\beta\to\infty$), resonant electrons with the same initial energies and
pitch-angles are divided into two populations. The first population consists
of a small group (low probability) of phase-trapped electrons that experience
a significant energy gain ($\Delta E>0$). The second population comprises a
larger group (high probability) of phase-bunched electrons that undergo a
minor energy loss ($\Delta E<0$). For a finite wave-packet size ($\beta=50$),
the population of phase-trapped electrons increases due to a higher
probability of trapping. For comparison, we also show the $\Delta
E$-distribution for the quasi-linear scattering regime (small wave intensity)
with scaled $\Delta E\to\Delta E/B_{w}$. This distribution shows a spread of
$\Delta E$ within $[-\max\Delta E,\max\Delta E]$, where $\max\Delta E$ is
approximately the energy gain of the trapped population. The significant
disparity observed in the $\Delta E$-distributions between the quasi-linear
and the nonlinear resonant regime, as well as between the different values of
$\beta$ highlights the importance of incorporating realistic wave-packet
characteristics into the model of wave-particle resonant interactions around
the bow shock.

Figure 1: (left) The distribution of waves in $(B_{w},\beta)$ space for the
foreshock region (see details of wave statistics in Ref. 65). (right) Three
examples of probability distributions of energy change $\Delta E$ for a single
resonant interaction of electrons (having the same initial energy/pitch-
angle): (grey) low-intensity wave-packet, (blue) long intense wave-packet, and
(orange) short intense wave-packet. The wave magnetic field is normalized to
the minimum value of the ambient magnetic field (see details in the text), and
for the distribution with $B_{w}/B_{0}=10^{-5}$, we rescale $\Delta E\to\Delta
E\cdot 10^{3}$ to compare with the other two $\Delta E$-distributions
evaluated for $B_{w}/B_{0}=10^{-2}$.

For the two limits, long wave-packets ($\beta\geq 100$) and very short wave-
packets ($\beta\leq 20$), there are theoretical approaches modeling the
electron distribution evolution in the Earth’s radiation belts: systems with
$\beta\geq 100$ are described by a generalized kinetic equation (see Refs. 58,
40, 39, 17, 75), whereas systems with $\beta\leq 20$ are described by a
probabilistic approach (see Ref. 18). The probabilistic approach for
$\beta\leq 20$ has been adopted for the bow shock wave and plasma
characteristics in Paper 1. In this paper, we focus on a generalized approach
that can be applied to a wide range of $\beta$. We merge the probabilistic
approach and a mapping technique 16, 19 to describe wave-particle interactions
with the wave ensemble having arbitrary distributions in $(B_{w},\beta)$
space.

The paper is structured as follows: Section 2 describes the main equations and
concepts of nonlinear wave-particle resonant interactions, Section 3 provides
a theoretical model of the mapping technique for very long wave-packets,
Section 4 explains the effects of short wave-packets, Section 5 generalizes
the mapping technique for an arbitrary ensemble of wave-packets, Section 6
verifies the generalized mapping technique, and Section 7 discusses the
obtained results.

##  2 Hamiltonian equations for resonant systems

Following the same methodology as in the first paper, we employ two models:
one for the bow shock and the other for the foreshock region. Both models
describe electron (mass $m_{e}$, charge $-e$) motion in inhomogeneous magnetic
fields with field-aligned whistler mode waves 20:

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{{p_{\parallel}^{2}}}{{2m_{e}}}+\mu\Omega_{0}\left(s\right)-e\Phi\left(s\right)+U_{w}(s,\mu)\cos\left({\phi+\psi}\right)$
$\displaystyle U_{w}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{{2\mu\Omega_{0}\left(s\right)}}{{m_{e}c^{2}}}}\frac{{eB_{w}}}{k(s)}$
(1)

where $(s,p_{\parallel})$ are conjugate variables of parallel coordinate and
momentum, $(\psi,\mu)$ are conjugate variables of gyrophase and normalized
magnetic moment ($\mu=E\sin^{2}\alpha/\Omega_{0}$ where $E$ is the electron
energy and $\alpha$ is the electron pitch-angle), $\Omega_{0}(s)>0$ is the
electron gyrofrequency, $\Phi(s)$ is the electrostatic potential describing
the polarization electric fields due to ion-electron decoupling around strong
magnetic field gradients 62, 35, $B_{w}$ is the wave amplitude, $\phi$ is the
wave phase that determines the wave frequency $\omega=-\partial\phi/\partial
t$ and the local wave number $k(s)=\partial\phi/\partial s$. We use the
simplified cold plasma dispersion relation
$kc=\Omega_{pe}(s)\cdot\left(\Omega_{0}(s)/\omega-1\right)^{-1/2}$, see Ref.
70, where $\Omega_{pe}(s)$ is the plasma frequency.

The bow shock is described simply by a magnetic field ramp,
$\Omega_{0}(s)=\Omega_{\min}\cdot(1+3b(s))$, a compressional plasma density
increase
$\Omega_{pe}(s)/\Omega_{pe,\min}\propto\sqrt{\Omega_{0}(s)/\Omega_{\min}}$,
and an electrostatic potential $\Phi(s)=\Phi_{0}b(s)$ (see, e.g., Ref. 35).
Our choice of parameters are typical for the Earth’s bow shock:
$(\Omega_{pe}/\Omega_{0})_{\min}=100$, $\Phi_{0}\in[0,100]$ eV (see Refs. 37,
62). Function $b(s)=(1/2)\cdot\left(1+\tanh(s/L)\right)$ varies from 0 to 1
with a spatial scale $L\approx 1000$ km (the typical scale of magnetic field
variations at the bow shock, see Ref. 46). This spatial scale determines the
large system parameter $\eta=L/d_{e}\sim 10^{4}$, with
$d_{e}=c/\Omega_{pe,\min}$ being the electron inertial length, the wavelength
scale.

The foreshock magnetic field model corresponds to foreshock transients,
localized enhancements of the magnetic field that lead to electron magnetic
trapping (see Paper 1 and Refs. 66, 65). This magnetic field model has a
magnetic configuration akin to a magnetic field bottle,
$\Omega_{0}(s)=\Omega_{\min}\sqrt{1+(s/L)^{2}}$, a compressional plasma
density increase $\Omega_{pe}(s)=100\Omega_{0}(s)$, and a large system
parameter $\eta=L/d_{e}\sim 10^{4}$. We do not include any electrostatic
potential mostly because there is no observational study of such potential in
foreshock transients.

###  2.1 Hamiltonian of nonlinear pendulum

The large parameter $\eta$ and small wave amplitude
$B_{w}/B_{0}\in[10^{-4},10^{-2}]$ (see Refs. 42, 41, 65) make Equation (1) a
slow-fast (slow $s$, $p_{\parallel}$ variables and fast $\phi$, $\psi$ phases)
one with a small resonant perturbation $\sim U_{w}$. To demonstrate the main
properties of nonlinear resonant interactions in the Hamiltonian system (1),
we follow the procedure of typical slow-fast resonant system analysis (see
Refs. 53, 52, 14).

First, we introduce phase $\zeta=\phi+\psi$ as a new conjugate coordinate to a
new magnetic moment $\tilde{\mu}$. For this we use a generating function
$F(\phi,s;\tilde{\mu},p;t)=(\phi+\psi)\tilde{\mu}+ps$ with new coordinate-
momentum pairs $(\tilde{s},p)$ and $(\zeta,\tilde{\mu})$:

$\begin{split}\zeta=&\frac{\partial
F}{\partial\tilde{\mu}}=\phi+\psi,\;\;\mu=\frac{\partial
F}{\partial\psi}=\tilde{\mu},\\\ \tilde{s}=&\frac{\partial F}{\partial
p}=s,\;\;p_{\parallel}=\frac{\partial F}{\partial
s}=p+k\tilde{\mu}.\end{split}$ (2)

Because $\tilde{\mu}=\mu$ and $\tilde{s}=s$, the tilde sign will be omitted.
The new Hamiltonian $\mathcal{H}=H+\partial F/\partial t$ has the form:

$\mathcal{H}=\frac{(p+k\mu)^{2}}{2m_{e}}+\mu(\Omega_{0}-\omega)-e\Phi(s)+U_{w}(s,\mu)\cos\zeta$
(3)

This Hamiltonian describes a conservative system because
$\partial\mathcal{H}/\partial t=0$.

The resonance condition in new Hamiltonian variables is

$\dot{\zeta}=\frac{\partial\mathcal{H}}{\partial\mu}\approx
k\frac{(p+k\mu)}{m_{e}}+\Omega_{0}-\omega=0.$ (4)

Equation (4) determines the resonant momentum $\mu=\mu_{R}(s,p)$:

$\mu_{R}(s,p)=\frac{m_{e}}{k^{2}}\left(\omega-\Omega_{0}-k\frac{p}{m_{e}}\right).$
(5)

Expansion of the Hamiltonian near the resonance gives:

$\begin{split}\mathcal{H}&\approx\Lambda(s,p)+\frac{k^{2}}{2m_{e}}(\mu-\mu_{R})^{2}+U_{w}(s,\mu_{R})\cos\zeta,\\\
\Lambda&(s,p)=\frac{(p+k\mu_{R})^{2}}{2m_{e}}+\mu_{R}(\Omega_{0}-\omega)-e\Phi.\end{split}$
(6)

To introduce the canonical variable $P_{\zeta}=\mu-\mu_{R}$, we use the
generating function
$W(\zeta,s;P_{\zeta},\tilde{p})=(P_{\zeta}+\mu_{R})\zeta+\tilde{p}s$ with new
conjugate variables:

$\begin{split}\tilde{\zeta}=&\frac{\partial W}{\partial
P_{\zeta}}=\zeta,\;\;\mu=\frac{\partial
W}{\partial\zeta}=P_{\zeta}+\mu_{R},\\\ \tilde{s}=&\frac{\partial
W}{\partial\tilde{p}}=s+\frac{\partial\mu_{R}}{\partial\tilde{p}}\zeta,\;\;p=\frac{\partial
W}{\partial\tilde{s}}=\tilde{p}+\frac{\partial\mu_{R}}{\partial\tilde{s}}\zeta.\end{split}$
(7)

The first term $\Lambda(s,p)$ in the new Hamiltonian $\tilde{\mathcal{H}}$
will differ from the first term $\Lambda(s,p)$ from the Hamiltonian (6),
because of the difference between old variables $(s,p)$ and new variables
$(\tilde{s},\tilde{p})$. Because the second term in the expressions for the
old variables is small, it allows us to expand the $\Lambda$ term in the
Hamiltonian. Noticing that
$s=\tilde{s}+\\{\mu_{R},\tilde{s}\\}\zeta,\,p=\tilde{p}+\\{\mu_{R},\tilde{p}\\}\zeta$
where $\\{\cdot,\cdot\\}$ are the Poisson brackets, it can be shown that

$\Lambda(s,p)=\Lambda(\tilde{s},\tilde{p})+\\{\mu_{R},\Lambda\\}\zeta,$

and the new Hamiltonian can be written as:

$\tilde{H}\approx\Lambda(\tilde{s},\tilde{p})+\frac{k^{2}}{2m_{e}}P_{\zeta}^{2}+\\{\mu_{R},\Lambda\\}\zeta+U_{w}(s,\mu_{R})\cos\zeta.$
(8)

In this form, the first term $\Lambda(\tilde{s},\tilde{p})$ describes the slow
motion in the $(s,p)\approx(\tilde{s},\tilde{p})$ plane while the following
three are analogous to the nonlinear pendulum Hamiltonian:

$\mathcal{H}_{\zeta}=\frac{1}{{2M}}P_{\zeta}^{2}+{\rm A}\zeta+{\rm
B}\cos\zeta,$ (9)

which describes the fast motion near the resonance in the $(\zeta,P_{\zeta})$
plane. Coefficients $M$, ${\rm A}$, ${\rm B}$ depend on the coordinates in the
$(s,p)$ plane:

$M=\frac{m_{e}}{k^{2}},\;\;{\rm
B}=U_{w}(s,\mu_{R})=\sqrt{\frac{{2\mu_{R}\Omega_{0}}}{{m_{e}c^{2}}}}\frac{{eB_{w}}}{k},$

and

${\rm A}=-\frac{\partial\ln k}{\partial
s}\frac{p_{\parallel,R}^{2}}{km_{e}}-\frac{\partial\Omega_{0}}{\partial
s}\frac{p_{\parallel,R}-k\mu_{R}}{k^{2}}-\frac{e}{k}\frac{\partial\Phi}{\partial
s},$

where $p_{\parallel,R}=m_{e}\left(\omega-\Omega_{0}\right)/k$ is the solution
of equation $\dot{\phi}+\dot{\psi}=0$ for the Hamiltonian (1) and $\mu_{R}$ is
defined through the electron’s resonant energy and the coordinate $s$ of the
resonance.

The phase portrait of the Hamiltonian $\mathcal{H}_{\zeta}$ in
$(\zeta,P_{\zeta})$ is presented in Fig. 2(top panel). This portrait contains
two types of trajectories: transient trajectories cross the resonance
$P_{\zeta}=0$ once – electrons moving along such trajectories experience
scattering with small energy/pitch-angle change; and phase trapped
trajectories which are closed around the resonance $P_{\zeta}=0$ – electrons
moving along such trajectories stay around the resonance for a long time (see
details in Refs. 57, 68, 14). The trajectory demarcating the phase domains
with transient and phase-trapped trajectories is the separatrix. An important
system parameter is the area of the region enclosed by the separatrix:

$\begin{split}\mathcal{S}=2\sqrt{2M\rm
B}\int_{\zeta_{X}}^{\zeta_{max}}\sqrt{\cos\zeta_{X}-\cos\zeta+{\rm\frac{A}{B}}(\zeta_{X}-\zeta)}{\rm
d}\zeta,\end{split}$ (10)

where $\zeta_{X}$ is the solution of equations $P_{\zeta}=0$,
$\dot{P}_{\zeta}\rightarrow 0$. Area $\mathcal{S}$ depends on $(s,p)$
coordinates in the resonance, where $p=p_{\parallel,R}-k\mu_{R}$. The
combination of the conservation law $E-e\Phi-\omega\mu=h=const$ (see Eq. (3))
and the resonance condition makes $\mathcal{S}$ a function of energy $E$ only,
i.e., $\mathcal{S}=\mathcal{S}(E)$ (see Fig. 2(bottom) showing three examples
of $\mathcal{S}(E)$). This function characterizes the electron energy change
during resonant interactions, $\Delta E$, and the probability of electron
phase trapping, $\Pi$ (see Refs. 67, 69, 1). Such probability can be defined
as the ratio of resonant electrons that experience phase trapping for a single
resonant interaction to the total number of resonant electrons 51, 67.

The initial energy $E$ determines the phase portrait in $(\zeta,P_{\zeta})$
plane but cannot describe electron trajectories in the phase portrait, because
they also depend on the $\mathcal{H}_{\zeta}$ magnitude. The coefficients in
$\mathcal{H}_{\zeta}$ depend on $E$, whereas phase $\zeta$ is a fast
oscillating variable. Thus, we may introduce an normalized energy
$\mathcal{H}_{\zeta}$ near the resonance $P_{\zeta}=0$,
$2\pi\xi=\zeta_{R}+({\rm B}/{\rm A})\cos\zeta_{R}$, and treat $\xi$ as a
random variable. An important property of the Hamiltonian (9) is that the
$\xi$-distribution is uniform (see numerical tests in Refs. 43, 33).
Therefore, we can use $\mathcal{S}(E)$ and the random variable $\xi$ to
characterize the wave-particle resonant interactions 17, 16.

Figure 2: (a) A phase portrait of the Hamiltonian (9) with $|A/B|<1$. (b) The
separatrix area $\mathcal{S}$ as a function of the initial energy. We plot
$\mathcal{S}$ for bow shock and foreshock transient models. For the bow shock
model, we trace electrons that move from the upstream region, get reflected
from the shock, and then resonate with whistler-mode waves. Dashed lines show
the prohibited initial energy range: electrons with such initial energies
cannot move away from the shock, because these electrons are not reflected by
the shock and go downstream (for a given $h$ value, this corresponds to a
certain range of the initial pitch-angle). The maximal resonant energy
corresponds to a resonance location ($s$) at the minimum of the ambient
magnetic field.

##  3 Mapping technique for long packets

The $\xi$-averaged characteristics of nonlinear resonant interactions (energy
changes due to bunching $\langle\Delta E\rangle_{bunching}$ and trapping
$\langle\Delta E\rangle_{trapping}$, and the trapping probability $\Pi$) are
determined by the profile $\mathcal{S}(E)$ alone 17, 16:

$\displaystyle\left\langle{\Delta E}\right\rangle_{bunching}$ $\displaystyle=$
$\displaystyle\omega\left\langle{\Delta\mu}\right\rangle_{bunching}=-\frac{\omega}{{2\pi}}\mathcal{S}$
$\displaystyle\mathcal{S}\left({E+\left\langle{\Delta
E}\right\rangle_{trapping}}\right)$ $\displaystyle=$
$\displaystyle\mathcal{S}\left(E\right),\;\;\;\Pi=-\frac{\omega}{{2\pi}}\frac{{d\mathcal{S}}}{{dE}}$

with $E-e\Phi-\omega\mu=h=const$. Therefore, knowing $\mathcal{S}$ (see, e.g.,
Fig. 2(bottom)), we may construct a map for energy changes:

$E_{n+1}=E_{n}+\left\\{{\begin{array}[]{*{20}c}{\left\langle{\Delta
E}(E_{n})\right\rangle_{bunching},}&{\xi\in\Xi_{bunching}(E_{n})}\\\
{\left\langle{\Delta
E}(E_{n})\right\rangle_{trapping},}&{\xi\notin\Xi_{bunching}(E_{n})}\\\
\end{array}}\right.$

where $n$ is the number of resonant interactions (map iteration number),
$\Xi_{bunching}(E)$ determines the range of $\xi$ corresponding to bunched
particles, and $\xi$ is a random variable that is uniformly distributed over
$\xi\in[0,1]$ (see Refs. 43, 33). Although the function $\Xi_{bunching}(E)$
can be quite complicated (see details in, e.g., Ref. 2), it can be
approximated by a simple step-wise function16:

$E_{n+1}=E_{n}+\left\\{{\begin{array}[]{*{20}c}{\left\langle{\Delta
E}\right\rangle_{bunching},}&{\xi\in\left({\Pi(E_{n}),1}\right]}\\\
{\left\langle{\Delta
E}\right\rangle_{trapping},}&{\xi\in\left[{0,\Pi(E_{n})}\right]}\\\
\end{array}}\right.$ (11)

This map should be supplemented by the equation of pitch-angle changes
($\alpha_{0}$ is defined as the pitch angle at the minimum $\Omega_{0}$):
$\alpha_{0,n+1}=\alpha_{0,n}+\Delta\alpha$, with $\Delta\alpha$ determined
from
$E_{n+1}-e\Phi_{R}(E_{n+1},\alpha_{0,n+1})-\omega\mu(E_{n+1},\alpha_{0,n+1})=E_{n}-e\Phi_{R}(E_{n},\alpha_{0,n})-\omega\mu(E_{n},\alpha_{0,n})$;
here, $\Phi_{R}(E,\alpha_{0})=\Phi(s_{R}(E,\alpha_{0}))$ is the electrostatic
potential when electrons enter or escape from the resonance (which could be
treated as equal for bunching and for trapping and could be found from
$\mathcal{S}(E_{n})=\mathcal{S}(E_{n+1})$). Iteration number $n$ can be
substituted by time as $t_{n+1}=t_{n}+\tau(E_{n})$, with $\tau(E_{n})$ being
the time interval between two resonance (see Ref. 19).

Figure 3 depicts several examples of electron trajectories calculated by
direct integration of the Hamiltonian equation (1) and by evaluation of the
map (11). We use the magnetic field model of the foreshock transient to show
multiple resonant interactions, and we convert time to the number of resonant
interactions $n$. Although the trajectories of electrons with energy $E_{n}$
are not identical between the results from the test particle approach and the
mapping technique (the difference is due to the random $\xi$), these two
approaches show statistically similar results: electron energy often decreases
due to the phase bunching (small negative jumps) and more rarely increases due
to the phase trapping (large jumps).

Figure 3: Test particle (TP) trajectories obtained by numerical integration
approach for two wave amplitudes (panels a and b, respectively), in comparison
with trajectories obtained from the mapping technique (11). The electron
energy is plotted at the resonance as a function of the number of resonant
interactions, $n$.

##  4 Short wave packets

The majority of waves observed around the bow shock are propagating in short
wave-packets (or as a train of such short packets), which significantly
changes the properties of nonlinear resonant interactions, i.e., the
efficiency of phase trapping 9, 34, 5. In the first cyclotron resonant
interaction, electrons move in the opposite direction to the whistler-mode
waves, and thus the packet size controls the time-scale of the electron motion
in the trapping regime 71, 83, 49. However, the probability of trapping
depends on $d\mathcal{S}/dE\propto dB_{w}/ds$ gradient 51, 52, which is very
large around the wave-packet edges; this can significantly increase the number
of trapped particles 24, 11, 33. Moreover, for all resonance coordinates
(i.e., independent of the background magnetic field and plasma gradients),
electrons can interact with the wave-packet edge and thus be trapped. Although
all these effects seem to complicate the description of wave-particle
nonlinear interactions, they also randomize the interaction process and reduce
the importance of specific shapes of the $\mathcal{S}(E)$ profile. For
sufficiently short wave-packets, $\mathcal{S}(E)$ can be approximated by
$\mathcal{S}/2\pi=\omega^{-1}C\cdot\sqrt{\varepsilon}\cdot\left[1-(E/\delta
E)^{2}\right]^{5/4}$, where $\varepsilon=\max B_{w}/B_{0}$, $\delta E$ is the
resonance energy range for which $\mathcal{S}\neq 0$, and $C\sim 1$ is a
numerical factor of the order of one (see derivations of this approximation in
Ref. 15 and in Appendix of Ref. 49). Note that we have introduced the
normalization factor $\omega^{-1}$ for the $\mathcal{S}$ function, and thus
$C$ has a dimension of energy. Interactions with small wave-packets mean that
at each interaction we shall use
$\mathcal{S}_{n}/2\pi=\omega^{-1}C\cdot\sqrt{\varepsilon}\cdot\left[1-((E-E_{n}^{*})/\delta
E)^{2}\right]^{5/4}$, with $E_{n}^{*}$ determining the location of the wave-
packet relative to the electron in resonance. For fixed $h$, the resonant
energy $E$ determines the resonant coordinate $s$, and thus the relative
location of electrons and wave-packets can be modeled by parameter $E_{n}^{*}$
in the $\mathcal{S}_{n}$ equation. If $E_{n}\notin[E_{n}^{*}-\delta
E,E_{n}^{*}+\delta E]$, the electron in question will not meet the wave-packet
in resonance (see derivations of this approximation in Ref. 18).

To model electron resonant interactions with wave-packets, we numerically
integrate the Hamiltonian equations for system (1) with $B_{w}\to
B_{w}f(\phi)$, where $f(\phi)=\exp\left(-5\cos^{2}(\phi/2\pi\beta)\right)$ is
the function modulating the wave field and separating it into wave-packets
with the duration of $\sim\beta$ for each packet73. There are two scenarios of
electron resonant interactions with the wave-packet train (several wave-
packets). The first scenario assumes that the entire train has been generated
within the same wave source region that does not change significantly during
the wave generation, and thus the phase $\phi$ is coherent along the entire
train. In this case, electrons may be retrapped by the next wave-packet after
escaping from resonance with the previous wave-packet. This situation is
especially common for short wave-packets, i.e., with $\beta<10$, when the
distance between two consecutively moving wave-packets is small. Such multi-
trapping electron acceleration is quite effective 38, 31 and does not
principally differ from the electron trapping into an infinitely long wave-
packet (except that the trapping probability is higher due to strong wave
field gradients at the leading edges of wave-packets). Figure 4(a) shows the
distribution of the electron energy change, $\Delta E$, due to resonant
interactions with phase-coherent wave-packet trains. The second scenario
assumes that wave-modulation (separation of wave field into the wave-packets)
destroys the wave phase coherence, and leads to $\phi$ jumps between wave-
packets (see examples of in Refs. 60, 82, 54). Such jumps of $\phi$ prevent
the electrons from being trapped into multiple wave-packets 82. To model this
effect, we introduce a coherence length measured in wave-packet duration,
$N_{c}$: all electrons will escape from resonance after interaction with
$N_{c}$ packets (some electrons may escape sooner, but no electrons are
allowed to stay in resonance longer than this). Figure 4(b) shows the
distribution of the electron energy change, $\Delta E$, due to resonant
interactions with a wave-packet train having $N_{c}=2$. Comparison of Panels
(a) and (b) demonstrates the effect of wave phase jumps: even for $\beta=50$,
the range of $\Delta E>0$ due to trapping significantly shrinks for $N_{c}=2$,
in comparison with $N_{c}\to\infty$. Note that we use a long wave packet to
better illustrate the effect of a finite $N_{c}$.

We compare these numerically derived $\Delta E$-distributions with the results
obtained from electron trajectories integrated using the mapping technique for
different $\delta E$ parameters. Figure 4(c) shows that the mapping with
$\mathcal{S}_{n}/2\pi=\omega^{-1}C\cdot\sqrt{\varepsilon}\cdot\left[1-((E-E_{n}^{*})/\delta
E)^{2}\right]^{5/4}$ can reproduce the main properties of the numerically
obtained $\Delta E$-distributions: a significant population of electrons do
not meet the wave-packet at resonance (peak around $\Delta E\sim 0$). This
spreading of the distribution ($\Delta E>0$) is because of the randomization
of energy gain of the trapped electrons. Therefore, we propose utilizing a
combination of
$\mathcal{S}_{n}/2\pi=\omega^{-1}C\cdot\sqrt{\varepsilon}\cdot\left[1-((E-E_{n}^{*})/\delta
E)^{2}\right]^{5/4}$ functions in the mapping technique to accurately
replicate any particular $\Delta E$-distribution obtained from the numerical
integration of the electron resonant interactions with wave-packet trains.

Figure 4: $\Delta E$-distributions for a single resonant interaction with (a)
the coherent wave-packet train, (b) wave-packet train with $N_{c}=2$. Panel
(c) shows $\Delta E$-distribution obtained from the mapping technique.
Parameters of map
$\mathcal{S}/2\pi=\omega^{-1}C\cdot\sqrt{\varepsilon}\cdot\left[1-((E-E_{n}^{*})/\delta
E)^{2}\right]^{5/4}$ are given on the panel (c), top; see text for details.

##  5 Synthetic map

In this section, we further construct the mapping function as a sum of
$\mathcal{S}(E)$ functions. Such a synthetic map allows us to describe the
probability distribution ${\cal P}$ in $(\Delta E,E)$ space. We adopt the
distribution of wave-packet sizes, $\beta$, as derived from observations (see
Fig. 1(a)): $P_{\beta}=C_{0}\beta^{-1.67}$ for $\beta\in[2,100]$ and
$C_{0}=\left(\int P_{\beta}(\beta)d\beta\right)^{-1}$. Next, we numerically
integrate a large ensemble of electron trajectories, where each electron
undergoes resonance with wave-packets that have randomly chosen $\beta$
values.

To construct such a synthetic map (a sum of many maps with different
$\mathcal{S}(E)$), we first determine how $\mathcal{S}(E)$ parameters (i.e.,
$\delta E$, $C$, $E_{n}^{*}$ distributions) control the resulting $\Delta
E$-distribution. Figure 5(a) depicts a schematic of the role of $\delta E$ and
$C$ in determining the $\Delta E$-distribution characteristics. The magnitude
of $\mathcal{S}(E)$ controls the $\Delta E$ range of bunching with $\min\Delta
E=-\max\omega\mathcal{S}(E)/2\pi$, whereas $\delta E$ controls the $\Delta E$
range of trapping with $\max\Delta E=\delta E$. The maximum relative number of
trapped particles, $\Pi$, is determined by
$\max\Pi/\sqrt{\varepsilon}=(C/\delta E)\cdot(5/2^{1/2}3^{3/4})\approx
1.551\cdot(C/\delta E)$. Both bunching and trapping energy ranges $\Delta E$
depend on initial electron energy $E$ (see Fig. 4). As a result, we set
$C=C(E)$ and $\delta E=\delta E(E)$. If such dependencies are weak,
$|dC/dE|\ll C/\delta E$ and $d\delta E/dE\ll 1$, they will not significantly
change the value of $\Pi$.

The distribution of wave-packet positions, $E_{n}^{*}$, determines how often
electrons will resonate with waves. If $E_{n}^{*}$ is uniformly distributed
within $E\pm\delta E$, then each map iteration will have an energy change,
i.e., electrons will encounter wave-packets each time when crossing the
resonance. This describes a system with dense wave-packet trains, where one
packet moves right after another. If $E_{n}^{*}$ is uniformly distributed
within the entire resonant energy range $[E_{-},E_{+}]$ and $\delta
E<E_{+}-E_{-}$, then the probability of the particle energy change (the
probability to meet a wave-packet in resonance) is $\delta E/(E_{+}-E_{-})$.
For small $\delta E/(E_{+}-E_{-})$, the resonant interactions are rare, and
this describes a system with well-separated individual wave-packets. Figure
5(b) shows $\Delta E$-distributions for three different $E_{n}^{*}$
distributions: a smaller probability of resonant interactions means a higher
value of probability to have $\Delta E=0$ (see a peak of ${\cal P}$ at $\Delta
E=0$).

Figure 5: The basic principles to construct the synthetic map: (a) a schematic
view of $\delta E$ and $C$ roles in determining the $\Delta E$-distribution
characteristics (the resonant energy range is $E\in[E_{-},E_{+}]$ and $\delta
E\leq[E_{-},E_{+}]$); (b) three $\Delta E$-distributions for three uniform
$E_{n}^{*}$ distributions within $E\pm\delta E$ (black), within
$[E_{-},E_{+}]$ (red), and within $E\pm 2\delta E$ (blue); (c) cumulative
distribution functions for $\Delta E$-distributions from (b).

###  5.1 Single resonance system

Figure 6(a,b) shows the probability distribution functions ${\cal P}(\Delta
E,E)$ for a single resonant interaction with randomly chosen $\beta$,
$N_{c}=5$ and $N_{c}=10$. For $N_{c}=10$, the distribution ${\cal P}(\Delta
E,E)$ has a slightly higher probability for large $\Delta E>0$ at low energies
($E\leq 490$eV), but the difference between $N_{c}=5$ and $N_{c}=10$ is
significantly reduced due to a wide $\beta$ distribution dominated by small
$\beta$ (see Fig. 1). For $N_{c}=5$, the distribution ${\cal P}(\Delta E,E)$
is almost independent of $E$. In the first paper, we have examined the limit
of small $N_{c}$ when the probability distribution function ${\cal P}(\Delta
E,E)$ can be reduced to 1D ${\cal P}(\Delta E)$, without significant
dependence on $E$.

We fit two ${\cal P}(\Delta E,E)$ distributions from Fig. 6(a,b) by $G(\Delta
E,E)$ functions derived from the synthetic map consisting of a sum
($\sum_{k}\mathcal{S}_{k}P_{\delta E}(\delta E_{k})$) of

$\mathcal{S}_{k}/2\pi=\omega^{-1}C_{k}\cdot\sqrt{\varepsilon}\cdot\left(1-\left((E-E_{n}^{*})/\delta
E_{k}\right)^{2}\right)^{5/4}$

We use $P_{\delta E}\propto(\delta E)^{-1.67}$, which is based on the
$P_{\beta}$ distribution and the $\delta E\propto\beta$ relation (see
schematic in Fig. 5 and Ref. 18). Figure 6(c,d) shows two $G(\Delta E,E)$
distributions constructed with four $\mathcal{S}_{k}$ functions having
different $C_{k}$ and $E_{n}^{*}=E\pm\delta E_{k}$ (parameters are in the
figure caption). These synthetic $G(\Delta E,E)$ distributions reproduce the
main details of the numerically obtained ${\cal P}(\Delta E,E)$ distributions.
Importantly, the fitting procedure for $G(\Delta E,E)$ does not have a unique
solution, but should approach the distribution of energy changes $\Delta E$
that fully describes electron resonant dynamics. Thus, we will further test
whether the constructed $G(\Delta E,E)$ provides a correct long-term evolution
of the electron ensemble (see Sect. 6).

Figure 6: Panels (a,b) show two distributions ${\cal P}(\Delta E,E)$ for the
bow shock magnetic field model and $N_{c}=5$(a), $N_{c}=10$(b). For each
resonant interaction, we use a random value of $\beta$ determined by the
probability distribution function $P_{\beta}=C_{0}\cdot\beta^{-1.67}$ for
$\beta\in[2,100]$ and $C_{0}=\left(\int P(\beta)d\beta\right)^{-1}$ and the
same wave amplitude $B_{w}/\min B_{0}=10^{-2}$. Panels (c,d) show two
distributions $G(\Delta E,E)$ obtained from the mapping of a sum of
$\mathcal{S}_{k}$ with $P_{\delta E}(\delta E_{k})=a_{k}\cdot(\delta
E)^{-1.67}$. For panel (c) $\delta E_{k}=\\{1,9,15,25\\}$,
$\sqrt{\varepsilon}C_{k}=\\{9,14,18,22\\}$, $a_{k}=\\{6.3,17.7,5.7,9.4\\}$,
and for panel (d) $\delta E_{k}=\\{5,9,14,23\\}$,
$\sqrt{\varepsilon}C_{k}=\\{17,15,22,19\\}$, $a_{k}=\\{0.1,22.0,22.0,22.0\\}$.

###  5.2 Magnetically trapped electrons

Figure 7(a,c) repeats the results from Fig. 6(a,b) for a single resonant
interaction with wave-packets having $\beta$ distributions of
$P_{\beta}\propto\beta^{-1.67}$. However, in this case, the trajectories are
traced in the foreshock transient magnetic field model. For $N_{c}\to\infty$,
the shortness of wave-packets can be compensated by the effect of multiple
trapping, and the corresponding probability distribution function ${\cal
P}(\Delta E,E)$ contains a finite probability of large $\Delta E>0$ for small
$E$. Despite that the $\Delta E$-distribution depends on $E$, the synthetic
mapping technique reproduces the main features of the 2D distribution ${\cal
P}(\Delta E,E)$ (compare Fig. 7(a) and Fig. 7(b)).

For $N_{c}=3$, there is almost no effect of multiple trappings, and the
corresponding probability distribution function ${\cal P}(\Delta E,E)$ does
not contain large $\Delta E>0$ jumps. Moreover, due to the dominant role of
small $\beta$ in $P_{\beta}\propto\beta^{-1.67}$ distribution, the energy
change $\Delta E$ due to resonant interactions only weakly depends on the
initial energy $E$ (similar to the bow shock system; compare Fig. 7(c) and
Fig. 6(a)). Although the synthetic map can describe such a 1D distribution of
${\cal P}(\Delta E)$ (compare Fig. 7(c) and Fig. 7(d)), we note that a simpler
approach for such $E$-independent systems has been proposed in Paper 1.

Figure 7: Panels (a, b) show the probability distribution function ${\cal
P}(\Delta E,E)$ for $N_{c}\rightarrow\infty$ and the corresponding synthetic
map, starting from the initial distribution of $\sim\exp\left(-E/50\text{
eV}\right)$; panels (c, d) show the same but for $N_{c}=3$. Parameters for
(b): $\delta E_{k}=\\{67,47,30,26,18,15,14,5\\}$,
$\sqrt{\varepsilon}C_{k}=\\{10,16,20,16,14,8,11,12\\}$,
$a_{k}=\\{3.7,7.8,2.9,7.7,2.5,6.1,10.3,10.7\\}$; for (d): $\delta
E_{k}=\\{11,10,5,4\\}$, $\sqrt{\varepsilon}C_{k}=\\{21,17,14,12\\}$,
$a_{k}=\\{0.4,3.1,4.9,6.0\\}$. Panels (e, f) show the evolution of the energy
distribution with $N_{c}\rightarrow\infty$ and the corresponding evolution of
distribution with the use of the synthetic map from panel (b); panels (g, h)
show the same but for $N_{c}=3$ with the synthetic map from panel (d).

##  6 Verification of the mapping technique

We next verify the suggested approach for constructing synthetic maps in the
foreshock transient magnetic field model. This assumes multiple resonant
interactions for each electron, and thus includes the long-term dynamics of
the electron energy distribution. In the test particle simulation, we
numerically integrate $10^{4}$ electron trajectories with the same $h$ and
initial energy distributions with energy falloff $\sim\exp(-E/50{\rm eV})$.
Trajectories are trapped in the system with an ensemble of wave-packets
described by $P_{\beta}$, but for half of the bounce period each electron only
resonates with one wave-packet (new wave characteristics for the next half of
the bounce period are set when an electron crosses the $B_{0}$ minimum).
Electrons that undergo phase bunching tend to drift towards smaller energy and
pitch angles. This drift may cause them to enter the small pitch-angle range,
where the mapping technique should be modified due to the effects of anomalous
trapping 45. Moreover, such electrons should escape the system because their
mirror points are outside of the foreshock transient region (magnetic field
magnitude in the mirror points exceed $\sim 10\times$ of the minimum magnetic
field). Thus, we exclude electrons reaching the small pitch-angle boundary,
$\alpha_{0}\approx 20^{\circ}$. To keep the same number of particles during
the simulation, we substitute each escaping electron with a new one (e.g.,
trapped into the foreshock transient from the solar wind flow) of identical
initial energy as the lost electron. Modeling them this way is equivalent to
including a large trapping probability for small energy electrons, to reflect
them from the small-energy/small pitch-angle boundary (see Refs. 45, 3, 12 for
discussions of this effect for electron scattering by whistler-mode waves in
the radiation belts).

Figures 7(e,f) and (g,h) compare the results of test particle simulations and
the mapping technique using the synthetic maps shown in Fig. 7(b) and (d),
respectively. For a large $N_{c}$, we see a significant acceleration of
electrons, whereas, for small $N_{c}$, electrons mostly drift to small
energy/pitch-angle ranges and form a field-aligned population. These effects
are clearly evident in the results obtained from the test particle simulations
(panels (e, g)) as well as in the mapping technique results (panels (f, h)).
Thus, the presented comparison validates the proposed mapping technique and
synthetic map.

##  7 Discussion and Conclusions

This is the second paper of two accompanying studies on modeling the nonlinear
wave-particle resonant interactions around the Earth’s bow shock. Paper 1
described the probabilistic approach for resonant interactions dominated by
short wave-packets, while this (second) paper merges the probabilistic
approach and mapping technique to describe the rare but effective electron
acceleration by long wave-packets. This generalized approach is based on the
generation of a synthetic map, which reproduces the main statistical
properties of resonant wave-particle interactions, while keeping all important
relations analytical during the modeling of such interactions. Below is a
summary of our results::

  * •

We have demonstrated the technique of constructing a synthetic map that
superposes multiple analytical maps describing nonlinear resonant
interactions.

  * •

The main parameters of such a synthetic map can be determined by comparing
mapping results and the probability distribution function of electron energy
changes, derived, e.g., from test particle simulations.

  * •

The synthetic map approach allows the modeling of the long-term dynamics of
electron distributions with multiple nonlinear resonant interactions with
realistic wave-packet distributions in amplitude and size.

  * •

The proposed synthetic map has been verified with test particle simulations
for the magnetic field model in Earth’s foreshock transients.

This approach can be useful for modeling the electron resonant interaction
effects in large-scale simulations, where the long-term electron dynamics is
dominated by adiabatic heating and transport, whereas wave-particle resonant
effects should be responsible for scattering and relaxation of the electron
anisotropy (see discussions in Refs. 22, 61, 63, 81, 50, 36). Although we have
developed and verified this approach for electron resonant interactions with
whistler-mode waves, the same effects of nonlinear resonant phase trapping and
bunching can be found in systems of intense electrostatic waves 21 and
ion/electron holes 76. Moreover, this approach can be included in test
particle simulations of electron heating by low-frequency compressional waves
in the presence of whistler-mode wave scattering (i.e., the so-called pumping
effect, see Refs. 48, 23, 32). Systems with such a combination of adiabatic
heating and wave-particle scattering, which controls the long-term dynamics of
electrons, are not limited to just the solar wind or shocks: a similar
interplay of large-scale adiabatic processes and small-scale wave-particle
interactions are believed to control electron fluxes in planetary radiation
belts 30, 29 and reconnection outflow regions 74, 13. Therefore, the proposed
synthetic map can be further combined with test particle simulations in these
systems.

## Acknowledgments

X.S., A.V.A., X.-J.Z., and V.A. acknowledge THEMIS contract NAS5-02099, NASA
grants 80NSSC22K1634, 80NSSC21K0581 (spacecraft data analysis and numerical
simulations). A.V.A. and D.S.T. also acknowledge Russian Science Foundation
through grant No. 19-12-00313 (theoretical models).

## Data Availability

This is a theoretical study, and all figures are plotted using numerical
solutions of equations provided in the paper. The data used for figures and
findings in this study are available from the corresponding author upon a
reasonable request. Spacecraft data for the first figure are publicly
available in MMS data repository https://lasp.colorado.edu/mms/sdc/public.
Data access and processing were done using SPEDAS V4.1 10.

## References

  * Albert 1993 Albert, J.M., 1993.  Cyclotron resonance in an inhomogeneous magnetic field.  Physics of Fluids B 5, 2744–2750.  doi:10.1063/1.860715.
  * Albert et al. 2022 Albert, J.M., Artemyev, A., Li, W., Gan, L., Ma, Q., 2022.  Analytical results for phase bunching in the pendulum model of wave-particle interactions.  Frontiers in Astronomy and Space Sciences 9, 971358.  doi:10.3389/fspas.2022.971358.
  * Albert et al. 2021 Albert, J.M., Artemyev, A.V., Li, W., Gan, L., Ma, Q., 2021.  Models of resonant wave-particle interactions.  Journal of Geophysical Research: Space Physics 126, e2021JA029216.  doi:10.1029/2021JA029216.
  * Albert et al. 2013 Albert, J.M., Tao, X., Bortnik, J., 2013.  Aspects of Nonlinear Wave-Particle Interactions, in: Summers, D., Mann, I.U., Baker, D.N., Schulz, M. (Eds.), Dynamics of the Earth’s Radiation Belts and Inner Magnetosphere.  doi:10.1029/2012GM001324.
  * Allanson et al. 2021 Allanson, O., Watt, C.E.J., Allison, H.J., Ratcliffe, H., 2021\.  Electron Diffusion and Advection During Nonlinear Interactions With Whistler Mode Waves.  Journal of Geophysical Research (Space Physics) 126, e28793.  doi:10.1029/2020JA028793.
  * Allanson et al. 2020 Allanson, O., Watt, C.E.J., Ratcliffe, H., Allison, H.J., Meredith, N.P., Bentley, S.N., Ross, J.P.J., Glauert, S.A., 2020\.  Particle-in-Cell Experiments Examine Electron Diffusion by Whistler-Mode Waves: 2. Quasi-Linear and Nonlinear Dynamics.  Journal of Geophysical Research (Space Physics) 125, e27949.  doi:10.1029/2020JA027949.
  * Amano et al. 2020 Amano, T., Katou, T., Kitamura, N., Oka, M., Matsumoto, Y., Hoshino, M., Saito, Y., Yokota, S., Giles, B.L., Paterson, W.R., Russell, C.T., Le Contel, O., Ergun, R.E., Lindqvist, P.A., Turner, D.L., Fennell, J.F., Blake, J.B., 2020.  Observational Evidence for Stochastic Shock Drift Acceleration of Electrons at the Earth’s Bow Shock.  Phys. Rev. Lett. 124, 065101\.  doi:10.1103/PhysRevLett.124.065101, arXiv:2002.06787.
  * Amano et al. 2022 Amano, T., Matsumoto, Y., Bohdan, A., Kobzar, O., Matsukiyo, S., Oka, M., Niemiec, J., Pohl, M., Hoshino, M., 2022.  Nonthermal electron acceleration at collisionless quasi-perpendicular shocks.  Reviews of Modern Plasma Physics 6, 29.  doi:10.1007/s41614-022-00093-1, arXiv:2209.03521.
  * An et al. 2022 An, Z., Wu, Y., Tao, X., 2022.  Electron Dynamics in a Chorus Wave Field Generated From Particle-In-Cell Simulations.  Geophys. Res. Lett. 49, e97778.  doi:10.1029/2022GL097778.
  * Angelopoulos et al. 2019 Angelopoulos, V., Cruce, P., Drozdov, A., Grimes, E.W., Hatzigeorgiu, N., King, D.A., Larson, D., Lewis, J.W., McTiernan, J.M., Roberts, D.A., Russell, C.L., Hori, T., Kasahara, Y., Kumamoto, A., Matsuoka, A., Miyashita, Y., Miyoshi, Y., Shinohara, I., Teramoto, M., Faden, J.B., Halford, A.J., McCarthy, M., Millan, R.M., Sample, J.G., Smith, D.M., Woodger, L.A., Masson, A., Narock, A.A., Asamura, K., Chang, T.F., Chiang, C.Y., Kazama, Y., Keika, K., Matsuda, S., Segawa, T., Seki, K., Shoji, M., Tam, S.W.Y., Umemura, N., Wang, B.J., Wang, S.Y., Redmon, R., Rodriguez, J.V., Singer, H.J., Vandegriff, J., Abe, S., Nose, M., Shinbori, A., Tanaka, Y.M., UeNo, S., Andersson, L., Dunn, P., Fowler, C., Halekas, J.S., Hara, T., Harada, Y., Lee, C.O., Lillis, R., Mitchell, D.L., Argall, M.R., Bromund, K., Burch, J.L., Cohen, I.J., Galloy, M., Giles, B., Jaynes, A.N., Le Contel, O., Oka, M., Phan, T.D., Walsh, B.M., Westlake, J., Wilder, F.D., Bale, S.D., Livi, R., Pulupa, M., Whittlesey, P., DeWolfe, A., Harter, B., Lucas, E., Auster, U., Bonnell, J.W., Cully, C.M., Donovan, E., Ergun, R.E., Frey, H.U., Jackel, B., Keiling, A., Korth, H., McFadden, J.P., Nishimura, Y., Plaschke, F., Robert, P., Turner, D.L., Weygand, J.M., Candey, R.M., Johnson, R.C., Kovalick, T., Liu, M.H., McGuire, R.E., Breneman, A., Kersten, K., Schroeder, P., 2019\.  The Space Physics Environment Data Analysis System (SPEDAS).  Space Sci. Rev. 215, 9.  doi:10.1007/s11214-018-0576-4.
  * Artemyev et al. 2012 Artemyev, A.V., Krasnoselskikh, V., Agapitov, O., Mourenas, D., Rolland, G., 2012.  Non-diffusive resonant acceleration of electrons in the radiation belts.  Physics of Plasmas 19, 122901\.  doi:10.1063/1.4769726.
  * Artemyev et al. 2021a Artemyev, A.V., Neishtadt, A.I., Albert, J.M., Gan, L., Li, W., Ma, Q., 2021a.  Theoretical model of the nonlinear resonant interaction of whistler-mode waves and field-aligned electrons.  Physics of Plasmas 28, 052902\.  doi:10.1063/5.0046635.
  * Artemyev et al. 2022a Artemyev, A.V., Neishtadt, A.I., Angelopoulos, V., 2022a.  On the Role of Whistler-Mode Waves in Electron Interaction With Dipolarizing Flux Bundles.  Journal of Geophysical Research (Space Physics) 127, e30265.  doi:10.1029/2022JA030265.
  * Artemyev et al. 2018a Artemyev, A.V., Neishtadt, A.I., Vainchtein, D.L., Vasiliev, A.A., Vasko, I.Y., Zelenyi, L.M., 2018a.  Trapping (capture) into resonance and scattering on resonance: Summary of results for space plasma systems.  Communications in Nonlinear Science and Numerical Simulations 65, 111–160.  doi:10.1016/j.cnsns.2018.05.004.
  * Artemyev et al. 2019a Artemyev, A.V., Neishtadt, A.I., Vasiliev, A.A., 2019a.  Kinetic equation for nonlinear wave-particle interaction: Solution properties and asymptotic dynamics.  Physica D Nonlinear Phenomena 393, 1–8.  doi:10.1016/j.physd.2018.12.007, arXiv:1809.03743.
  * Artemyev et al. 2020 Artemyev, A.V., Neishtadt, A.I., Vasiliev, A.A., 2020.  Mapping for nonlinear electron interaction with whistler-mode waves.  Physics of Plasmas 27, 042902\.  doi:10.1063/1.5144477, arXiv:1911.11459.
  * Artemyev et al. 2018b Artemyev, A.V., Neishtadt, A.I., Vasiliev, A.A., Mourenas, D., 2018b.  Long-term evolution of electron distribution function due to nonlinear resonant interaction with whistler mode waves.  Journal of Plasma Physics 84, 905840206\.  doi:10.1017/S0022377818000260.
  * Artemyev et al. 2021b Artemyev, A.V., Neishtadt, A.I., Vasiliev, A.A., Mourenas, D., 2021b.  Transitional regime of electron resonant interaction with whistler-mode waves in inhomogeneous space plasma.  Phys. Rev. E 104, 055203\.  doi:10.1103/PhysRevE.104.055203, arXiv:2107.13511.
  * Artemyev et al. 2021c Artemyev, A.V., Neishtadt, A.I., Vasiliev, A.A., Zhang, X.J., Mourenas, D., Vainchtein, D., 2021c.  Long-term dynamics driven by resonant wave-particle interactions: from Hamiltonian resonance theory to phase space mapping.  Journal of Plasma Physics 87, 835870201\.  doi:10.1017/S0022377821000246, arXiv:2011.00208.
  * Artemyev et al. 2022b Artemyev, A.V., Shi, X., Liu, T.Z., Zhang, X.J., Vasko, I., Angelopoulos, V., 2022b.  Electron Resonant Interaction With Whistler Waves Around Foreshock Transients and the Bow Shock Behind the Terminator.  Journal of Geophysical Research (Space Physics) 127, e29820.  doi:10.1029/2021JA029820.
  * Artemyev et al. 2019b Artemyev, A.V., Vasiliev, A.A., Neishtadt, A.I., 2019b.  Charged particle nonlinear resonance with localized electrostatic wave-packets.  Communications in Nonlinear Science and Numerical Simulations 72, 392–406.  doi:10.1016/j.cnsns.2019.01.011.
  * Balikhin et al. 1993 Balikhin, M., Gedalin, M., Petrukovich, A., 1993.  New mechanism for electron heating in shocks.  Physical Review Letters 70, 1259–1262.  doi:10.1103/PhysRevLett.70.1259.
  * Borovsky et al. 2017 Borovsky, J.E., Horne, R.B., Meredith, N.P., 2017.  The Contribution of Compressional Magnetic Pumping to the Energization of the Earth’s Outer Electron Radiation Belt During High-Speed Stream-Driven Storms.  Journal of Geophysical Research (Space Physics) 122, 12,072–12,089.  doi:10.1002/2017JA024607.
  * Bortnik et al. 2008 Bortnik, J., Thorne, R.M., Inan, U.S., 2008.  Nonlinear interaction of energetic electrons with large amplitude chorus.  Geophys. Res. Lett. 35, 21102.  doi:10.1029/2008GL035500.
  * Cattell et al. 2021 Cattell, C., Breneman, A., Dombeck, J., Short, B., Wygant, J., Halekas, J., Case, T., Kasper, J.C., Larson, D., Stevens, M., Whittesley, P., Bale, S.D., Dudok de Wit, T., Goodrich, K., MacDowall, R., Moncuquet, M., Malaspina, D., Pulupa, M., 2021\.  Parker Solar Probe Evidence for Scattering of Electrons in the Young Solar Wind by Narrowband Whistler-mode Waves.  Astrophys. J. Lett. 911, L29.  doi:10.3847/2041-8213/abefdd, arXiv:2101.05098.
  * Cattell and Vo 2021 Cattell, C., Vo, T., 2021\.  Modeling Interactions of Narrowband Large Amplitude Whistler-mode Waves with Electrons in the Solar Wind inside 0.3 au and at 1 au Using a Particle Tracing Code.  Astrophys. J. Lett. 914, L33.  doi:10.3847/2041-8213/ac08a1, arXiv:2104.02824.
  * Davis et al. 2021 Davis, L.A., Cattell, C.A., Wilson, L. B., I., Cohen, Z.A., Breneman, A.W., Hanson, E.L.M., 2021\.  ARTEMIS Observations of Plasma Waves in Laminar and Perturbed Interplanetary Shocks.  Astrophys. J. 913, 144.  doi:10.3847/1538-4357/abf56a, arXiv:2006.01064.
  * Drummond and Pines 1962 Drummond, W.E., Pines, D., 1962\.  Nonlinear stability of plasma oscillations.  Nuclear Fusion Suppl. 3, 1049–1058.
  * Elkington et al. 2019 Elkington, S.R., Chan, A.A., Jaynes, A.N., Malaspina, D., Albert, J., 2019.  K2: Towards a Comprehensive Simulation Framework of the Van Allen Radiation Belts, in: AGU Fall Meeting Abstracts, pp. SM44B–01.
  * Elkington et al. 2018 Elkington, S.R., Chan, A.A., Li, Z., Hudson, M.K., Jaynes, A.N., Baker, D.N., 2018\.  Generalizing Global Simulations of the Radiation Belts: Addressing Advective and Diffusive Processes in a Common Simulation Framework, in: AGU Fall Meeting Abstracts, pp. SM11B–02.
  * Foster et al. 2021 Foster, J.C., Erickson, P.J., Omura, Y., 2021.  Subpacket structure in strong VLF chorus rising tones: characteristics and consequences for relativistic electron acceleration.  Earth, Planets and Space 73, 140\.  doi:10.1186/s40623-021-01467-4.
  * Fowler et al. 2020 Fowler, C.M., Agapitov, O.V., Xu, S., Mitchell, D.L., Andersson, L., Artemyev, A., Espley, J., Ergun, R.E., Mazelle, C., 2020.  Localized Heating of the Martian Topside Ionosphere Through the Combined Effects of Magnetic Pumping by Large-Scale Magnetosonic Waves and Pitch Angle Diffusion by Whistler Waves.  Geophys. Res. Lett. 47, e86408.  doi:10.1029/2019GL086408.
  * Frantsuzov et al. 2023 Frantsuzov, V.A., Artemyev, A.V., Zhang, X.J., Allanson, O., Shustov, P.I., Petrukovich, A.A., 2023\.  Diffusive scattering of energetic electrons by intense whistler-mode waves in an inhomogeneous plasma.  Journal of Plasma Physics 89, 905890101\.  doi:10.1017/S0022377822001271.
  * Gan et al. 2022 Gan, L., Li, W., Ma, Q., Artemyev, A.V., Albert, J.M., 2022\.  Dependence of Nonlinear Effects on Whistler-Mode Wave Bandwidth and Amplitude: A Perspective From Diffusion Coefficients.  Journal of Geophysical Research (Space Physics) 127, e30063.  doi:10.1029/2021JA030063.
  * Gedalin 1996 Gedalin, M., 1996.  Ion reflection at the shock front revisited.  J. Geophys. Res. 101, 4871–4878.  doi:10.1029/95JA03669.
  * Gedalin 2020 Gedalin, M., 2020.  Large-scale versus Small-scale Fields in the Shock Front: Effect on the Particle Motion.  Astrophys. J. 895, 59.  doi:10.3847/1538-4357/ab8af0.
  * Goodrich and Scudder 1984 Goodrich, C.C., Scudder, J.D., 1984\.  The adiabatic energy change of plasma electrons and the frame dependence of the cross-shock potential at collisionless magnetosonic shock waves.  J. Geophys. Res. 89, 6654–6662.  doi:10.1029/JA089iA08p06654.
  * Hiraga and Omura 2020 Hiraga, R., Omura, Y., 2020\.  Acceleration mechanism of radiation belt electrons through interaction with multi-subpacket chorus waves.  Earth, Planets, and Space 72, 21\.  doi:10.1186/s40623-020-1134-3.
  * Hsieh et al. 2020 Hsieh, Y.K., Kubota, Y., Omura, Y., 2020.  Nonlinear Evolution of Radiation Belt Electron Fluxes Interacting With Oblique Whistler Mode Chorus Emissions.  Journal of Geophysical Research (Space Physics) 125, e27465.  doi:10.1029/2019JA027465.
  * Hsieh and Omura 2017 Hsieh, Y.K., Omura, Y., 2017\.  Nonlinear dynamics of electrons interacting with oblique whistler mode chorus in the magnetosphere.  J. Geophys. Res. 122, 675–694.  doi:10.1002/2016JA023255.
  * Hull et al. 2020 Hull, A.J., Chaston, C.C., Bonnell, J.W., Damiano, P.A., Wygant, J.R., Reeves, G.D., 2020\.  Correlations Between Dispersive Alfvén Wave Activity, Electron Energization, and Ion Outflow in the Inner Magnetosphere.  Geophys. Res. Lett. 47, e88985.  doi:10.1029/2020GL088985.
  * Hull et al. 2012 Hull, A.J., Muschietti, L., Oka, M., Larson, D.E., Mozer, F.S., Chaston, C.C., Bonnell, J.W., Hospodarsky, G.B., 2012\.  Multiscale whistler waves within Earth’s perpendicular bow shock.  J. Geophys. Res. 117, 12104\.  doi:10.1029/2012JA017870.
  * Itin et al. 2000 Itin, A.P., Neishtadt, A.I., Vasiliev, A.A., 2000.  Captures into resonance and scattering on resonance in dynamics of a charged relativistic particle in magnetic field and electrostatic wave.  Physica D: Nonlinear Phenomena 141, 281–296.  doi:10.1016/S0167-2789(00)00039-7.
  * Karpman 1974 Karpman, V.I., 1974.  Nonlinear Effects in the ELF Waves Propagating along the Magnetic Field in the Magnetosphere.  Space Sci. Rev. 16, 361–388.  doi:10.1007/BF00171564.
  * Kitahara and Katoh 2019 Kitahara, M., Katoh, Y., 2019\.  Anomalous Trapping of Low Pitch Angle Electrons by Coherent Whistler Mode Waves.  J. Geophys. Res. 124, 5568–5583.  doi:10.1029/2019JA026493.
  * Krasnoselskikh et al. 2013 Krasnoselskikh, V., Balikhin, M., Walker, S.N., Schwartz, S., Sundkvist, D., Lobzin, V., Gedalin, M., Bale, S.D., Mozer, F., Soucek, J., Hobara, Y., Comisel, H., 2013\.  The Dynamic Quasiperpendicular Shock: Cluster Discoveries.  Space Sci. Rev. 178, 535–598.  doi:10.1007/s11214-013-9972-y, arXiv:1303.0190.
  * Kuramitsu and Krasnoselskikh 2005 Kuramitsu, Y., Krasnoselskikh, V., 2005\.  Gyroresonant Surfing Acceleration.  Physical Review Letters 94, 031102–+.  doi:10.1103/PhysRevLett.94.031102.
  * Lichko et al. 2017 Lichko, E., Egedal, J., Daughton, W., Kasper, J., 2017\.  Magnetic Pumping as a Source of Particle Heating and Power-law Distributions in the Solar Wind.  Astrophys. J. Lett. 850, L28.  doi:10.3847/2041-8213/aa9a33, arXiv:1710.02106.
  * Mourenas et al. 2018 Mourenas, D., Zhang, X.J., Artemyev, A.V., Angelopoulos, V., Thorne, R.M., Bortnik, J., Neishtadt, A.I., Vasiliev, A.A., 2018\.  Electron Nonlinear Resonant Interaction With Short and Intense Parallel Chorus Wave Packets.  J. Geophys. Res. 123, 4979–4999.  doi:10.1029/2018JA025417.
  * Mozer and Sundkvist 2013 Mozer, F.S., Sundkvist, D., 2013\.  Electron demagnetization and heating in quasi-perpendicular shocks.  Journal of Geophysical Research (Space Physics) 118, 5415–5420.  doi:10.1002/jgra.50534.
  * Neishtadt 1975 Neishtadt, A.I., 1975.  Passage through a separatrix in a resonance problem with a slowly-varying parameter.  Journal of Applied Mathematics and Mechanics 39, 594–605.  doi:10.1016/0021-8928(75)90060-X.
  * Neishtadt 2014 Neishtadt, A.I., 2014.  Averaging, passage through resonances, and capture into resonance in two-frequency systems.  Russian Mathematical Surveys 69, 771\.  URL: http://stacks.iop.org/0036-0279/69/i=5/a=771.
  * Neishtadt and Vasiliev 2006 Neishtadt, A.I., Vasiliev, A.A., 2006\.  Destruction of adiabatic invariance at resonances in slow fast Hamiltonian systems.  Nuclear Instruments and Methods in Physics Research A 561, 158–165.  doi:10.1016/j.nima.2006.01.008, arXiv:arXiv:nlin/0511050.
  * Nunn et al. 2021 Nunn, D., Zhang, X.J., Mourenas, D., Artemyev, A.V., 2021\.  Generation of Realistic Short Chorus Wave Packets.  Geophys. Res. Lett. 48, e92178.  doi:10.1029/2020GL092178.
  * Oka et al. 2019 Oka, M., Otsuka, F., Matsukiyo, S., Wilson, L. B., I., Argall, M.R., Amano, T., Phan, T.D., Hoshino, M., Le Contel, O., Gershman, D.J., Burch, J.L., Torbert, R.B., Dorelli, J.C., Giles, B.L., Ergun, R.E., Russell, C.T., Lindqvist, P.A., 2019.  Electron Scattering by Low-frequency Whistler Waves at Earth’s Bow Shock.  Astrophys. J. 886, 53.  doi:10.3847/1538-4357/ab4a81.
  * Oka et al. 2017 Oka, M., Wilson, III, L.B., Phan, T.D., Hull, A.J., Amano, T., Hoshino, M., Argall, M.R., Le Contel, O., Agapitov, O., Gershman, D.J., Khotyaintsev, Y.V., Burch, J.L., Torbert, R.B., Pollock, C., Dorelli, J.C., Giles, B.L., Moore, T.E., Saito, Y., Avanov, L.A., Paterson, W., Ergun, R.E., Strangeway, R.J., Russell, C.T., Lindqvist, P.A., 2017\.  Electron Scattering by High-frequency Whistler Waves at Earth’s Bow Shock.  Astrophys. J. Lett. 842, L11.  doi:10.3847/2041-8213/aa7759.
  * Omura et al. 1991 Omura, Y., Matsumoto, H., Nunn, D., Rycroft, M.J., 1991\.  A review of observational, theoretical and numerical studies of VLF triggered emissions.  Journal of Atmospheric and Terrestrial Physics 53, 351–368.
  * Omura et al. 2015 Omura, Y., Miyashita, Y., Yoshikawa, M., Summers, D., Hikishima, M., Ebihara, Y., Kubota, Y., 2015.  Formation process of relativistic electron flux through interaction with chorus emissions in the Earth’s inner magnetosphere.  J. Geophys. Res. 120, 9545–9562.  doi:10.1002/2015JA021563.
  * Page et al. 2021 Page, B., Vasko, I.Y., Artemyev, A.V., Bale, S.D., 2021\.  Generation of High-frequency Whistler Waves in the Earth’s Quasi-perpendicular Bow Shock.  Astrophys. J. Lett. 919, L17.  doi:10.3847/2041-8213/ac2748.
  * Santolík et al. 2014 Santolík, O., Kletzing, C.A., Kurth, W.S., Hospodarsky, G.B., Bounds, S.R., 2014.  Fine structure of large-amplitude chorus wave packets.  Geophys. Res. Lett. 41, 293–299.  doi:10.1002/2013GL058889.
  * Schwartz et al. 2011 Schwartz, S.J., Henley, E., Mitchell, J., Krasnoselskikh, V., 2011\.  Electron Temperature Gradient Scale at Collisionless Shocks.  Physical Review Letters 107, 215002\.  doi:10.1103/PhysRevLett.107.215002.
  * Scudder 1995 Scudder, J.D., 1995.  A review of the physics of electron heating at collisionless shocks.  Advances in Space Research 15, 181–223.  doi:10.1016/0273-1177(94)00101-6.
  * See et al. 2013 See, V., Cameron, R.F., Schwartz, S.J., 2013.  Non-adiabatic electron behaviour due to short-scale electric field structures at collisionless shock waves.  Annales Geophysicae 31, 639–646.  doi:10.5194/angeo-31-639-2013, arXiv:1304.4841.
  * Shapiro and Sagdeev 1997 Shapiro, V.D., Sagdeev, R.Z., 1997\.  Nonlinear wave-particle interaction and conditions for the applicability of quasilinear theory.  Physics Reports 283, 49–71.  doi:10.1016/S0370-1573(96)00053-1.
  * Shi et al. 2023 Shi, X., Liu, T., Artemyev, A., Angelopoulos, V., Zhang, X.J., Turner, D.L., 2023\.  Intense Whistler-mode Waves at Foreshock Transients: Characteristics and Regimes of Wave-Particle Resonant Interaction.  Astrophys. J. 944, 193.  doi:10.3847/1538-4357/acb543, arXiv:2211.05398.
  * Shi et al. 2020 Shi, X., Liu, T.Z., Angelopoulos, V., Zhang, X.J., 2020\.  Whistler Mode Waves in the Compressional Boundary of Foreshock Transients.  Journal of Geophysical Research (Space Physics) 125, e27758.  doi:10.1029/2019JA027758.
  * Shklyar 1981 Shklyar, D.R., 1981.  Stochastic motion of relativistic particles in the field of a monochromatic wave.  Sov. Phys. JETP 53, 1197–1192.
  * Shklyar and Matsumoto 2009 Shklyar, D.R., Matsumoto, H., 2009\.  Oblique Whistler-Mode Waves in the Inhomogeneous Magnetospheric Plasma: Resonant Interactions with Energetic Charged Particles.  Surveys in Geophysics 30, 55–104.  doi:10.1007/s10712-009-9061-7.
  * Solovev and Shkliar 1986 Solovev, V.V., Shkliar, D.R., 1986\.  Particle heating by a low-amplitude wave in an inhomogeneous magnetoplasma.  Sov. Phys. JETP 63, 272–277.
  * Stix 1962 Stix, T.H., 1962.  The Theory of Plasma Waves.
  * Tao et al. 2013 Tao, X., Bortnik, J., Albert, J.M., Thorne, R.M., Li, W., 2013.  The importance of amplitude modulation in nonlinear interactions between electrons and large amplitude whistler waves.  Journal of Atmospheric and Solar-Terrestrial Physics 99, 67–72.  doi:10.1016/j.jastp.2012.05.012.
  * Tong et al. 2019 Tong, Y., Vasko, I.Y., Artemyev, A.V., Bale, S.D., Mozer, F.S., 2019.  Statistical Study of Whistler Waves in the Solar Wind at 1 au.  Astrophys. J. 878, 41.  doi:10.3847/1538-4357/ab1f05, arXiv:1905.08958.
  * Tsai et al. 2022 Tsai, E., Artemyev, A., Zhang, X.J., Angelopoulos, V., 2022\.  Relativistic Electron Precipitation Driven by Nonlinear Resonance With Whistler-Mode Waves.  Journal of Geophysical Research (Space Physics) 127, e30338.  doi:10.1029/2022JA030338.
  * Ukhorskiy et al. 2022 Ukhorskiy, A.Y., Sorathia, K.A., Merkin, V.G., Crabtree, C., Fletcher, A.C., Malaspina, D.M., Schwartz, S.J., 2022.  Cross-scale energy cascade powered by magnetospheric convection.  Scientific Reports 12, 4446\.  doi:10.1038/s41598-022-08038-x.
  * Vainchtein et al. 2018 Vainchtein, D., Zhang, X.J., Artemyev, A.V., Mourenas, D., Angelopoulos, V., Thorne, R.M., 2018\.  Evolution of Electron Distribution Driven by Nonlinear Resonances With Intense Field-Aligned Chorus Waves.  Journal of Geophysical Research (Space Physics) 123, 8149–8169.  doi:10.1029/2018JA025654, arXiv:1806.00066.
  * Vasko et al. 2018 Vasko, I.Y., Mozer, F.S., Krasnoselskikh, V.V., Artemyev, A.V., Agapitov, O.V., Bale, S.D., Avanov, L., Ergun, R., Giles, B., Lindqvist, P.A., Russell, C.T., Strangeway, R., Torbert, R., 2018.  Solitary Waves Across Supercritical Quasi-Perpendicular Shocks.  Geophys. Res. Lett. 45, 5809–5817.  doi:10.1029/2018GL077835.
  * Vedenov et al. 1962 Vedenov, A.A., Velikhov, E., Sagdeev, R., 1962.  Quasilinear theory of plasma oscillations.  Nuclear Fusion Suppl. 2, 465–475.
  * Veltri and Zimbardo 1993 Veltri, P., Zimbardo, G., 1993\.  Electron-whistler interaction at the Earth’s bow shock: 2. Electron pitch angle diffusion.  J. Geophys. Res. 98, 13335–13346.  doi:10.1029/93JA01144.
  * Vo et al. 2022 Vo, T., Lysak, R., Cattell, C., 2022.  Stochastic diffusion of electrons interacting with whistler-mode waves in the solar wind.  Physics of Plasmas 29, 012904\.  doi:10.1063/5.0074474, arXiv:2111.06341.
  * Wilson et al. 2013 Wilson, L.B., Koval, A., Szabo, A., Breneman, A., Cattell, C.A., Goetz, K., Kellogg, P.J., Kersten, K., Kasper, J.C., Maruca, B.A., Pulupa, M., 2013.  Electromagnetic waves and electron anisotropies downstream of supercritical interplanetary shocks.  J. Geophys. Res. 118, 5–16.  doi:10.1029/2012JA018167, arXiv:1207.6429.
  * Wilson et al. 2014 Wilson, L.B., Sibeck, D.G., Breneman, A.W., Contel, O.L., Cully, C., Turner, D.L., Angelopoulos, V., Malaspina, D.M., 2014\.  Quantified energy dissipation rates in the terrestrial bow shock: 2. Waves and dissipation.  J. Geophys. Res. 119, 6475–6495.  doi:10.1002/2014JA019930.
  * Zhang et al. 2020 Zhang, X.J., Agapitov, O., Artemyev, A.V., Mourenas, D., Angelopoulos, V., Kurth, W.S., Bonnell, J.W., Hospodarsky, G.B., 2020\.  Phase Decoherence Within Intense Chorus Wave Packets Constrains the Efficiency of Nonlinear Resonant Electron Acceleration.  Geophys. Res. Lett. 47, e89807.  doi:10.1029/2020GL089807.
  * Zhang et al. 2018 Zhang, X.J., Thorne, R., Artemyev, A., Mourenas, D., Angelopoulos, V., Bortnik, J., Kletzing, C.A., Kurth, W.S., Hospodarsky, G.B., 2018.  Properties of Intense Field-Aligned Lower-Band Chorus Waves: Implications for Nonlinear Wave-Particle Interactions.  Journal of Geophysical Research (Space Physics) 123, 5379–5393.  doi:10.1029/2018JA025390.