# On a transitional regime of electron resonant interaction with whistler-mode
waves in inhomogeneous space plasma

Artemyev A.V.1,2<EMAIL_ADDRESS>Neishtadt A.I.2,3 Vasiliev A.A.2
Mourenas D.4 1 Institute of Geophysics and Planetary Physics, University of
California, Los Angeles, CA, USA; 2 Space Research Institute RAS, Moscow,
Russia; 3 Department of Mathematical Sciences, Loughborough University,
Loughborough LE11 3TU, UK;
4 LPC2E, CNRS, Orleans, France

###### Abstract

Resonances with electromagnetic whistler-mode waves are the primary driver for
the formation and dynamics of energetic electron fluxes in various space
plasma systems, including shock waves and planetary radiation belts. The basic
and most elaborated theoretical framework for the description of the integral
effect of multiple resonant interactions is the quasi-linear theory, that
operates through electron diffusion in velocity space. The quasi-linear
diffusion rate scales linearly with the wave intensity, $D_{QL}\sim
B_{w}^{2}$, which should be small enough to satisfy the applicability criteria
of this theory. Spacecraft measurements, however, often detect whistle-mode
waves sufficiently intense to resonate with electrons nonlinearly. Such
nonlinear resonant interactions imply effects of phase trapping and phase
bunching, which may quickly change the electron fluxes in a non-diffusive
manner. Both regimes of electron resonant interactions (diffusive and
nonlinear) are well studied, but there is no theory quantifying the transition
between these two regimes. In this paper we describe the integral effect of
nonlinear electron interactions with whistler-mode waves in terms of the time-
scale of electron distribution relaxation, $\sim 1/D_{NL}$. We determine the
scaling of $D_{NL}$ with wave intensity $B_{w}^{2}$ and other main wave
characteristics, such as wave-packet size. The comparison of $D_{QL}$ and
$D_{NL}$ provides the range of wave intensity and wave-packet sizes where the
electron distribution evolves at the same rates for the diffusive and
nonlinear resonant regimes. The obtained results are discussed in the context
of energetic electron dynamics in the Earth’s radiation belt.

###### pacs:

##  I Introduction

Wave-particle resonant interactions in space plasma play a crucial role in
momentum and energy exchange between different charged particle populations in
absence of particle collisions. Two main regimes of such interactions are
quasi-linear diffusion (Vedenov  _et al._ , 1962; Drummond and Pines, 1962)
and nonlinear resonant interactions (O’Neil, 1965; Mazitov, 1965). The most
advanced theoretical approaches for these two regimes (Kennel and Engelmann,
1966; Karpman, 1974) have been developed to model the dynamics of energetic
electron populations in the Earth magnetosphere, where electrons interact
resonantly with whistler-mode waves. Electron interactions with low-amplitude
broadband whistler-mode waves are well described by the diffusion model
proposed by (Kennel and Petschek, 1966), whereas the model of resonant
interactions with intense coherent waves describes such fast non-diffusive
effects as phase trapping Nunn (1971); Karpman  _et al._ (1974). Many aspects
of these two models have been confirmed and verified with spacecraft
observations of long-term (diffusive) electron flux evolution (e.g., Thorne
_et al._ , 2013; Li  _et al._ , 2014) or rapid (non-diffusive) electron
acceleration (e.g., Agapitov  _et al._ , 2015; Foster  _et al._ , 2014; Gan
_et al._ , 2020) and losses (e.g., Mourenas  _et al._ , 2016). The same two
model approaches (quasi-linear diffusion and nonlinear resonances) can
describe various wave-particle interactions in the solar wind (e.g.,
Voshchepynets  _et al._ , 2017; Kuzichev  _et al._ , 2019; Shaaban  _et al._ ,
2019), planetary and interplanetary shock waves (e.g., Veltri and Zimbardo,
1993; Bykov and Treumann, 2011), aurora acceleration region (e.g., Kletzing,
1994; Shen and Knudsen, 2020).

The basic equations for the quasi-linear diffusion model have been initially
derived for a homogeneous plasma (Andronov and Trakhtengerts, 1964; Kennel and
Engelmann, 1966), and then generalized for inhomogeneous magnetic traps, like
the Earth dipole field (Lyons and Williams, 1984; Trakhtengerts and Rycroft,
2008). An interesting and important aspect of such a generalization is the
significant relaxation of the broad wave spectrum requirement (see Karpman,
1974; Le Queau and Roux, 1987). Indeed, for coherent waves the resonance
width, a crucial parameter for the diffusive model, can be determined by the
magnetic field inhomogeneity (Karpman and Shkliar, 1977; Shklyar, 1981;
Albert, 1993) and/or the wave frequency drift Demekhov  _et al._ (2006, 2009);
Omura  _et al._ (2007). Electron resonant interactions with coherent whistler-
mode waves in the Earth’s magnetosphere can be described by a quasi-linear
diffusion model (Albert, 2010) if wave intensity is sufficiently small.
Therefore, both regimes of wave-particle interaction (quasi-linear diffusion
and nonlinear resonances) can operate for the same coherent monochromatic
wave, and only wave intensity determines the relevant regime. Thus, the
question arises of the transition between these two regimes (see, e.g.,
discussion in Tao  _et al._ , 2013; Allanson  _et al._ , 2020; Mourenas  _et
al._ , 2018).

The probabilistic distribution of whistler-mode wave intensities in the
Earth’s magnetosphere contains a large population of low-intensity waves
(presumably interacting with electrons diffusively), but this distribution is
not Gaussian and there is a significant population of whistler-mode waves
sufficiently intense to resonate with electrons nonlinearly (Cully  _et al._ ,
2008; Artemyev  _et al._ , 2016; Tyler  _et al._ , 2019). The quasi-linear
diffusion model relies on the average wave intensity presumably dominated by
low-intensity waves, whereas the nonlinear interaction model invokes the much
more rarely observed intense waves. How do such rare nonlinear resonances
alter the generally diffusive electron flux evolution? And can the
contribution of intense waves be described by a simple increase of diffusion
rates of the quasi-linear model? To address these questions, we should
describe the transition between these two regimes of wave-particle
interactions. This topic is the focus of our study. We start with basic
equations of wave-particle resonant interactions for the most widespread case
of field-aligned whistler-mode waves. Then we derive the main characteristics
of resonant electron dynamics for both regimes of resonant interactions. We
use the mapping technique for such interactions (Benkadda  _et al._ , 1996;
Artemyev  _et al._ , 2020a) to derive the main scaling laws of the transition
between quasi-linear and nonlinear interactions. Then we discuss the obtained
results and summarize them.

##  II Field-aligned whistler-mode waves

We start with the Hamiltonian of a relativistic electron (the charge is $-e$,
the rest mass is $m_{e}$, energy is $E=m_{e}c^{2}(\gamma-1)$ where $c$ is the
speed of light and $\gamma$ is the Lorentz factor) moving in the inhomogeneous
background magnetic field $B_{0}(s)$ (where $s$ is the field-aligned
coordinate) and whistler-mode wave field (Albert, 1993; Vainchtein  _et al._ ,
2018):

$\displaystyle H$ $\displaystyle=$ $\displaystyle
m_{e}c^{2}\gamma+\sqrt{\frac{{2I_{x}\Omega_{0}}}{{m_{e}c^{2}}}}\frac{{eB_{w}}}{{k\gamma}}\sin\left({\phi+\psi}\right)$
$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\sqrt{1+\left({\frac{{p_{\parallel}}}{{m_{e}c}}}\right)^{2}+\frac{{2I_{x}\Omega_{0}}}{{m_{e}c^{2}}}}$
(1)

where $p_{\parallel}$ is the field-aligned momentum (conjugate to $s$),
$\Omega_{0}=eB_{0}(s)/m_{e}c$ is the electron gyrofrequency, $I_{x}$ and
$\psi$ are magnetic moment (normalized in such a way that $2I_{x}\Omega_{0}$
is the perpendicular energy) and the conjugate gyrophase, $B_{w}(s)$ is the
wave magnetic field amplitude, $\phi$ is the wave phase, with
$\partial\phi/\partial s=k(s)$ the wave number and $-\partial\phi/\partial
t=\omega={\rm const}$ the wave frequency. We use the cold plasma dispersion
relation for $kc/\omega_{pe}=(\Omega_{0}(s)/\omega-1)^{-1/2}$ with a constant
plasma frequency $\omega_{pe}$. The spatial scale of the background magnetic
field inhomogeneity $R$ is much larger than the wavelength ($Rk\gg 1$),
whereas the wave amplitude is much smaller than the background magnetic field
($B_{w}/B_{0}\ll 1$).

Hamiltonian equations for (1) are

$\displaystyle\dot{p}_{\parallel}$ $\displaystyle=$
$\displaystyle-\frac{{I_{x}}}{\gamma}\frac{{\partial\Omega_{0}}}{{\partial
s}}-\sqrt{\frac{{2I_{x}\Omega_{0}}}{{m_{e}c^{2}}}}\frac{{eB_{w}}}{\gamma}\cos\left({\phi+\psi}\right),$
$\displaystyle\dot{s}$ $\displaystyle=$
$\displaystyle\frac{{p_{\parallel}}}{{m_{e}\gamma}},\quad\dot{\psi}=\frac{{\Omega_{0}}}{\gamma}+\sqrt{\frac{{2I_{x}\Omega_{0}}}{{m_{e}c^{2}}}}\frac{{eB_{w}}}{{2I_{x}k\gamma}}\sin\left({\phi+\psi}\right),$
$\displaystyle\dot{I}_{x}$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{{2I_{x}\Omega_{0}}}{{m_{e}c^{2}}}}\frac{{eB_{w}}}{{k\gamma}}\cos\left({\phi+\psi}\right)$
(2)

where the rates of variation of $I_{x}$ and $p_{\parallel}$ are much smaller
than the rate of variation of $\psi$, because $\Omega_{0}\gg c/R$ and
$B_{0}\gg B_{w}$. Therefore, $p_{\parallel}$, $s$, $I_{x}$ are slow variables,
and both the wave phase $\phi$ and gyrophase $\psi$ are fast variables.

In the absence of a wave ($B_{w}=0$) the electron moves with constant energy
$\gamma$ and magnetic moment $I_{x}$ (because the Hamiltonian does not depend
on $\psi$ and wave phase $\phi(s,t)$). Therefore, each trajectory can be
characterized by a pair of initial $(\gamma,I_{x})$. Instead of $I_{x}$, it is
often more convenient to use the electron pitch-angle $\alpha_{0}$ determined
at the $B_{0}$ minimum (at the equator, $s=0$):
$2I_{x}\Omega_{0}(0)/m_{e}c^{2}=(\gamma^{2}-1)\sin^{2}\alpha_{0}$. The
classical magnetic field configuration of the Earth magnetosphere includes
$B_{0}(s)$ growing away from the equator $s=0$ (e.g., the dipole field is
$B_{0}=B_{0}(0)\sqrt{1+3\sin^{2}\lambda}/\cos^{6}\lambda$ with magnetic
latitude $\lambda$ defined as $ds/R=\sqrt{1+3\sin^{2}\lambda}\cos\lambda
d\lambda$). In such a field, electrons are moving within a magnetic trap:
$p_{\parallel}=\pm\sqrt{1-\gamma^{2}-2I_{x}\Omega_{0}(s)/m_{e}c^{2}}$ is
oscillating between $\pm\sqrt{1-\gamma^{2}-2I_{x}\Omega_{0}(0)/m_{e}c^{2}}$
values and reaches zeros at the mirror coordinates $s_{\max,\min}$ defined as
$\Omega_{0}(s_{\max,\min})\sin^{2}\alpha_{0}/\Omega_{0}(0)=1$. The first-order
cyclotron resonance condition
$\dot{\phi}+\dot{\psi}=kp_{z}/m_{e}\gamma-\omega+\Omega_{0}/\gamma=0$
determines the position of resonance $s_{R}$ for given energy and pitch-angle
$\alpha_{0}$ with the resonant
$p_{z}=-\sqrt{1-\gamma^{2}-\Omega_{0}(s_{R})\sin^{2}\alpha_{0}/\Omega_{0}(0)}$.

The equation for $p_{\parallel}$ in system (2) contains two terms: the mirror
force and the wave force. For the diffusive regime of resonant interactions
the mirror force is much larger than the wave force, and thus wave effects can
be calculated using the integration of wave field over unperturbed electron
trajectory. For nonlinear resonant interactions, these two terms should be of
the same order, i.e. the wave force should be able to compete with the mirror
force and change the electron trajectory substantially. The ratio of the
magnitudes of these two forces is $(B_{w}/B_{0})R\Omega_{0}/c$. Thus, the
regime of nonlinear resonant interaction requires $B_{w}/B_{0}\sim
c/R\Omega_{0}$, whereas if $B_{w}/B_{0}\ll c/R\Omega_{0}$ the resonant
interaction should be diffusive. Figure 1 shows electron trajectories for
these two regimes: panel (a) shows the energy scattering with a mean zero for
$B_{w}/B_{0}\ll c/R\Omega_{0}$, whereas panels (b) shows a significant energy
change for $B_{w}/B_{0}\sim c/R\Omega$. The nonlinear regime is characterized
by two processes: a small number of particles experience phase trapping with a
large energy increase, whereas most of the particles experience nonlinear
scattering (phase bunching) with a small energy decrease. The long term
dynamics (multiple resonances) for a single electron would resemble the random
fluctuations of a diffusive regime (see panel (c)) with some sort of cycle for
the nonlinear regime (see panel (d)). This cycle consists in multiple drifts
to smaller energy due to the nonlinear scattering and rare trappings with
energy increase. We aim to describe the transition between these two regimes
for an ensemble of electrons. For this reason we shall start with the
quantification of the main characteristics of electron dynamics in both
regimes.

Figure 1:  Trajectories of electrons resonantly interacting with a parallel
whistler-mode wave (one resonant interaction is shown): wave amplitude is
sufficiently low and electrons are diffusively scattered (a), wave amplitude
is high and electrons experience phase bunching and phase trapping (b). Long-
term dynamics of electron trajectories ($\sim 100$ resonances) for low (c) and
high (d) wave amplitudes. All electrons have the same initial energy of $150$
keV and pitch-angle $45^{\circ}$. The background magnetic field is the Earth’s
dipole field with an equatorial magnitude corresponding to $6$ Earth radii
away from the planet (the outer radiation belt). Wave characteristics are
typical for parallel whistler-mode waves: $\omega/\Omega_{0}(0)=0.25$,
$B_{w}=20$ pT for (a,c) and $B_{w}=200$ pT for (b,d) (Li  _et al._ , 2011;
Zhang  _et al._ , 2018). The background plasma density is given by an
empirical model (Sheeley  _et al._ , 2001).

##  III Equations of particle motion around the resonance

The diffusive regime of resonant interaction can be described by the diffusion
coefficient $D_{QL}$, whereas the nonlinear regime is characterized by energy
changes due to the nonlinear scattering $\Delta\gamma_{scat}$, energy changes
due to trapping $\Delta\gamma_{trap}$, and the probability $\Pi$ of trapping
(Albert  _et al._ , 2013; Shklyar and Matsumoto, 2009; Artemyev  _et al._ ,
2018). In this section we demonstrate the relations between these different
characteristics. First, we should consider Hamiltonian (1) around the
resonance. We follow the procedure described in (Neishtadt, 1975; Itin  _et
al._ , 2000) and introduce the new phase $\zeta=\phi+\psi$ through the
generating function $W=(\phi+\psi)I+P_{\parallel}s$. In the new variables, the
Hamiltonian takes the form:

$\displaystyle H_{I}$ $\displaystyle=$ $\displaystyle m_{e}c^{2}\gamma-\omega
I+\sqrt{\frac{{2I\Omega_{0}}}{{m_{e}c^{2}}}}\frac{{eB_{w}}}{{k\gamma}}\sin\zeta$
$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\sqrt{1+\left({\frac{{P_{\parallel}+kI}}{{m_{e}c}}}\right)^{2}+\frac{{2I\Omega_{0}}}{{m_{e}c^{2}}}}$
(3)

where we keep the same notation for $s$ conjugate to
$P_{\parallel}=p_{\parallel}-kI$ and take into account that new Hamiltonian
$H_{I}=H+\partial W/\partial t$. New momentum $I$ conjugate to $\zeta$ equals
to $I_{x}$. Hamiltonian (3) does not depend on time, and thus $\gamma-\omega
I/m_{e}c^{2}=h$ is the integral of motion. This relation between energy and
$I$ for a given frequency $\omega$ makes the resonant interaction effectively
1D, i.e. any electron energy and pitch-angle changes obey
$\gamma-(\omega/2\Omega_{0}(0))(\gamma^{2}-1)\sin^{2}\alpha_{0}=h={\rm
const}$. Figure 2(a,b) shows such 1D motion in the $(\gamma,\alpha_{0})$ plane
for diffusive and nonlinear resonant interactions of electrons from Fig.
1(c,d).

Therefore, we can introduce the 1D electron distribution $f(\gamma)$ for
$h={\rm const}$ and follow the evolution of this distribution in these two
different regimes. Figure 2(c) shows that the initially localized $f(\gamma)$
is diffusively spread for low wave intensity. Phase trapping and nonlinear
scattering modify the initially localized $f(\gamma)$ distribution in a non-
diffusive manner: there is a drift of the entire distribution due to nonlinear
scattering and the appearance of a high-energy electron population due to
trappings (see Fig. 2(d)). An interesting and important property of the wave-
particle resonant interaction is that both regimes (diffusion and nonlinear
resonances) ultimately result in $f(\gamma)$ relaxation to a plateau with
$\partial f/\partial\gamma|_{h={\rm const}}\to 0$ (see Artemyev  _et al._ ,
2019). The energy ranges are different for the two regimes, and the diffusive
relaxation would work for a wider energy range because $B_{w}/B_{0}\sim
c/R\Omega_{0}$ can be satisfied only for a certain
$\gamma\in[\gamma_{\max},\gamma_{\min}]$, where $\gamma_{\max,\min}$ are
determined by $B_{0}(s)$ and $B_{w}(s)$ profiles. The time-scales of
$f(\gamma)$ relaxation are also different for the two regimes: the diffusive
relaxation requires more time and Fig. 2(d) shows how $f(\gamma)$ continues
spreading outside the energy range of nonlinear resonances where the plateau
forms first.

Figure 2:  Two trajectories from Fig. 1(c,d) are shown in (energy, pitch-
angle) plane for diffusive (a) and nonlinear (b) resonant interactions (color
shows time). Evolution of an initially localized peak of distribution
$f(\gamma)$ for the same
$h=\gamma-(\omega/2\Omega_{0}(0))(\gamma^{2}-1)\sin^{2}\alpha_{0}$ as from
panels (a,b): diffusive (c) and nonlinear (d) resonant interactions. Note that
time-scales are different for (c) and (d) panels.

The resonance condition for Hamiltonian (3) is given by equation
$\dot{\zeta}=0$ and this equation determines the resonant momentum $I_{R}$:

$\displaystyle\frac{{kI_{R}}}{{m_{e}c}}$ $\displaystyle=$
$\displaystyle-\frac{{P_{\parallel}}}{{m_{e}c}}-\frac{{\Omega_{0}}}{{kc}}+\left({\left({\frac{{kc}}{\omega}}\right)^{2}-1}\right)^{-1/2}$
(4) $\displaystyle\times$
$\displaystyle\left({1-\left({\frac{{\Omega_{0}}}{{kc}}}\right)^{2}-2\frac{{\Omega_{0}}}{{kc}}\frac{{P_{\parallel}}}{{m_{e}c}}}\right)^{1/2}$

We substitute $I_{R}$ into the $\gamma$ expression and obtain the resonant
energy

$\gamma_{R}=\frac{{kc/\omega}}{{\sqrt{\left({kc/\omega}\right)^{2}-1}}}\left({1-\left({\frac{{\Omega_{0}}}{{kc}}}\right)^{2}-2\frac{{\Omega_{0}}}{{kc}}\frac{{P_{\parallel}}}{{m_{e}c}}}\right)^{1/2}$
(5)

Substituting $P_{\parallel}$ from $h=\gamma_{R}-\omega I_{R}/m_{e}c^{2}$ into
Eq. (5) we can obtain $\gamma_{R}(s)$ that directly connects the energy of the
resonant electron and the resonant $s_{R}$, i.e. $h={\rm const}$ determines
$\gamma_{R}=\gamma_{R}(s_{R})$.

To determine the electron dynamics (energy change) around the resonance, we
expand Hamiltonian (3) as

$\displaystyle H_{I}$ $\displaystyle=$ $\displaystyle
m_{e}c^{2}\gamma_{R}-\omega I_{R}$ (6) $\displaystyle+$
$\displaystyle\frac{1}{2}m_{e}c^{2}g\left({I-I_{R}}\right)^{2}+\sqrt{\frac{{2I_{R}\Omega_{0}}}{{m_{e}c^{2}}}}\frac{{eB_{w}}}{{k\gamma_{R}}}\sin\zeta$
$\displaystyle g$ $\displaystyle=$
$\displaystyle\left.{\frac{{\partial^{2}\gamma}}{{\partial
I^{2}}}}\right|_{I=I_{R}}=\frac{{k^{2}}}{{m_{e}^{2}c^{2}}}$

Such expansion can be done for $2\Omega_{0}I/m_{e}c^{2}\gg B_{w}/B_{0}$,
whereas for smaller $I$ an alternative consideration would be required (see
Artemyev  _et al._ , 2021a; Albert  _et al._ , 2021).

We use the generating function $Q=(I-I_{R})\zeta+P_{\parallel}s$ to introduce
$P_{\zeta}=I-I_{R}$. The new Hamiltonian consists of two parts
$\Lambda=m_{e}c^{2}\gamma_{R}-\omega I_{R}$ and $H_{\zeta}$:

$\displaystyle H_{\zeta}$ $\displaystyle=$
$\displaystyle\frac{1}{2}m_{e}c^{2}gP_{\zeta}^{2}+\left\\{{\Lambda,I_{R}}\right\\}\zeta+\sqrt{\frac{{2I_{R}\Omega_{0}}}{{m_{e}c^{2}}}}\frac{{eB_{w}}}{{k\gamma_{R}}}\sin\zeta$
(7)

where $\Lambda$ depends on new variables $p=P_{\parallel}+\zeta\partial
I_{R}/\partial s$, $q=s-\zeta\partial I_{R}/\partial P_{\parallel}$, and we
expand $H_{I}$ over $\zeta\partial I_{R}/\partial s$, $-\zeta\partial
I_{R}/\partial P_{\parallel}$ to get the Poisson brackets $\\{\cdot,\cdot\\}$
(see, e.g., Artemyev  _et al._ , 2015a; Vainchtein  _et al._ , 2018).
Hamiltonian $H_{\zeta}$ describes the electron dynamics on the
$(\zeta,P_{\zeta})$ plane (around the resonance $P_{\zeta}=0$), and
coefficients of this Hamiltonian depend on slowly changing $(p,q)$. Note that
the dependence on $(p,q)$ can be rewritten as a dependence on $\gamma_{R}$,
because $(p,q)\approx(P_{\parallel},s)$ where $s=s_{R}$ and
$P_{\parallel}=P_{\parallel}(s_{R})$ due to $\gamma_{R}-\omega
I_{R}/m_{e}c^{2}=h$ conservation.

Hamiltonian (7) describes the classical system of a pendulum with a torque
(e.g., Arnold  _et al._ , 2006), and the phase portraits of this system with
different values of coefficients are shown in Fig. 3(a). To describe the main
properties of this Hamiltonian, let us rewrite it in a dimensionless form
$\mathcal{H}=H/m_{e}c^{2}$:

$\mathcal{H}=\frac{1}{2}wP^{2}+\varepsilon\left({r\zeta+u\sin\zeta}\right)$
(8)

where $P=P_{\zeta}/P_{0}$, $w=gP_{0}^{2}$ ($P_{0}$ is the $P_{\zeta}$
magnitude), $\max{B}_{w}/B_{0}(0)=\varepsilon$,
$u=\sqrt{2I_{R}\Omega_{0}/m_{e}c^{2}}\left(\Omega_{0}(0)/ck\gamma_{R}\right)(B_{w}/\max
B_{w})$, and $r=\\{\Lambda,I_{R}\\}/m_{e}c^{2}\varepsilon$, and time
renormalized to $1/P_{0}$ to keep Hamiltonian form of equations. Note
$\\{\Lambda,I_{R}\\}\propto c/R\Omega_{0}$ and we assume that
$c/R\Omega_{0}\leq B_{w}/B_{0}$.

For a typical wave field distribution along magnetic field lines ($B_{w}=0$ at
the equator $s=0$ and at high magnetic latitudes $s\to 1$, whereas there is a
$B_{w}$ maximum at the medium latitudes (see Agapitov  _et al._ , 2013,
2018)), there are three spatial regions with different phase portraits of
Hamiltonian (8). The wave is generated near the equator $s=0$ and moves to
higher $s$ (i.e., wave number $k>0$), whereas the resonant electron moves from
high $s$ to the equator because the resonant momentum
$p_{\parallel}=(\gamma\omega-\Omega_{0})/k$ is negative for not-too-high
$\gamma$. Thus, we consider the phase portrait evolution moving from large $s$
to the equator. At high $s$ wave amplitude is small and $|r|>u$. Thus, the
phase portrait totally consists of trajectories crossing the resonance $P=0$
once (open trajectories). As $s$ decreases, we obtain $u>|r|$. The
corresponding phase portrait contains closed trajectories and particles on
these trajectories are trapped around the resonance (i.e. cross $P=0$ many
times). The growth of the area $S$ filled by such trapped trajectories in
$(\zeta,P)$ plane implies that particles from the region with open
trajectories can be trapped into the resonance. The area $S$ reaches its
maximum at a value of $s$ where $|u/r|$ is maximal; as $s$ further decreases,
this area starts decreasing. Such area $S$ decrease corresponds to particles
detrapping (escape) from the resonance. Close to the equator $s=0$ all
particles will be detrapped and only open trajectories remain in the phase
portrait (see Fig. 3).

Figure 3:  The profile of $u/r$ ratio (a). Phase portraits of Hamiltonian (8)
for different $u/r$ ratios (b). Profiles of $\gamma_{R}$ and
$\bar{S}=S\cdot(R\Omega_{0}(0)/c)^{1/2}$ along $s$ coordinate (note
$R\Omega_{0}(0)/c$ is of the order of $1/\varepsilon$ for the system
parameters) (a). System parameters correspond to the trajectory shown in Fig.
1(b)

The evolution of the area $S$ with closed trajectories along $s$ is much
slower than trapped particles rotation around $P=0$, and thus this periodic
rotation has an adiabatic invariant $I_{\zeta}=(2\pi)^{-1}\oint{Pd\zeta}$. At
the moment of trapping $I_{\zeta}$ equals to $S/2\pi$ (and $S$ increases), and
due to $I_{\zeta}$ conservation the detrapping should appear when $S$
decreases and comes back to the value $2\pi I_{\zeta}$. However, during this
$S$ evolution the resonant energy would change (see the example of
$\gamma_{R}$ profile along $s$ in Fig. 3). Thus, between trapping and
detrapping electrons gain energy $\Delta\gamma_{trap}$, and this energy gain
is determined by $S(\gamma_{R})$ curve $S=\oint{P_{\zeta}d\zeta}$:

$S=2\sqrt{\frac{{2\varepsilon}}{w}}\int\limits_{\zeta_{-}}^{\zeta_{+}}{\sqrt{\left({r\zeta_{+}+u\sin\zeta_{+}}\right)-\left({r\zeta+u\sin\zeta}\right)}d\zeta}$
(9)

where $\zeta_{\pm}$ are shown in Fig. 3. The amount of particles that will be
trapped during a single resonant interaction (i.e., the probability of
trapping, $\Pi$) is determined by the gradient of $S$ (see Neishtadt, 1975;
Neishtadt and Vasiliev, 2005; Shklyar, 1981): $\Pi=(\omega/2\pi
m_{e}c^{2})\cdot(\partial S/\partial\gamma)$.

All particles that are not trapped cross the resonance $P=0$ once, moving
along open trajectories. These particles are scattered with the energy change
$\Delta\gamma=\omega\Delta I/m_{e}c^{2}$, which depends on the phase
$\zeta_{R}$ at the resonance:

$\displaystyle\Delta\gamma$ $\displaystyle=$
$\displaystyle-\int\limits_{-\infty}^{\zeta_{R}}{\frac{{2\sqrt{\frac{{2\omega^{2}}}{{m_{e}c^{2}g}}}\sqrt{\frac{{2I_{R}\Omega_{0}}}{{m_{e}c^{2}}}}\frac{{eB_{w}}}{{k\gamma_{R}m_{e}c^{2}}}\cos\zeta}}{{\sqrt{H_{\zeta}-\left\\{{\Lambda,I_{R}}\right\\}\zeta-\sqrt{\frac{{2I_{R}\Omega_{0}}}{{m_{e}c^{2}}}}\frac{{eB_{w}}}{{k\gamma_{R}}}\sin\zeta}}}}d\zeta$
(10) $\displaystyle=$
$\displaystyle-2\sqrt{\frac{{2\varepsilon\omega^{2}}}{{m_{e}^{2}c^{4}g}}}\int\limits_{-\infty}^{\zeta_{R}}{\frac{{u\cos\zeta}}{{\sqrt{2\pi\theta
r-r\zeta-u\sin\zeta}}}}d\zeta$

where $2\pi\theta=\zeta_{R}+(u/r)\sin\zeta_{R}\;({\rm mod}\;2\pi)$. The
function $\Delta\gamma(\theta)$ is periodic; its mean value over
$\theta\in[0,1]$ is equal to
$\Delta\gamma_{scat}=\langle\Delta\gamma(\theta)\rangle_{\theta}=-(\omega/2\pi
m_{e}c^{2})\cdot S$ (see Neishtadt, 1999; Artemyev  _et al._ , 2018). Thus,
for the phase portrait with $S\neq 0$ the energy change due to scattering is
also determined by the $S(\gamma_{R})$ profile. This relation has been
verified many times with numerical simulations for electron interactions with
whistler-mode waves (see Albert  _et al._ , 2013; Artemyev  _et al._ , 2015a;
Vainchtein  _et al._ , 2018). For the phase portrait with $S=0$ we get
$\langle\Delta\gamma(\theta)\rangle_{\theta}=0$ (i.e. no energy drift due to
the resonance), but
$\langle\left(\Delta\gamma(\theta)\right)^{2}\rangle_{\theta}\neq 0$ and there
is an energy diffusion. Equation (10) shows that for $|u/r|\ll 1$ we get

$\Delta\gamma\approx-2\varepsilon\sqrt{\frac{{2\omega^{2}}}{{m_{e}c^{2}g}}}\int\limits_{-\infty}^{\zeta_{R}}{\frac{{u\cos\zeta
d\zeta}}{{\sqrt{2\pi\theta-\left\\{{\Lambda,I_{R}}\right\\}\zeta/m_{e}c^{2}}}}}$
(11)

and
$D_{QL}=\langle\left(\Delta\gamma(\theta)\right)^{2}\rangle_{\theta}\sim\varepsilon^{2}$,
as it should be for quasi-linear diffusion coefficients proportional to the
wave intensity (see detailed comparison of $D_{QL}$ calculated with Eq. (11)
and quasi-linear diffusion rates in (Karpman, 1974; Albert, 2010)).

As electron energy changes only at resonances and remains constant between
resonances, its evolution for any trajectory can be described by the mapping
technique (e.g., Chirikov, 1978). For the nonlinear resonances such a map
takes the form (see Artemyev  _et al._ , 2020a):

$\gamma_{n+1}=\gamma_{n}+\left\\{{\begin{array}[]{*{20}c}{\Delta\gamma_{trap},}&{\theta\in\left[{0,\Pi}\right]}\\\
{\Delta\gamma_{scat},}&{\theta\in\left({\Pi,1}\right]}\\\ \end{array}}\right.$
(12)

where $n$ is the number of resonance crossings (the number of map iterations).
Equation (12) shows that new energy (after $n+1$ resonant interaction)
$\gamma_{n+1}$ is the sum of energy $\gamma_{n}$ after previous resonance and
energy change due to scattering or trapping. Variable $\theta$ is the rescaled
effective energy at the resonance (see Eq. (10)), and the variation of
$\theta$ between two resonance crossings is determined by the normalized phase
gain $\Delta\zeta\sim\varepsilon^{-1}\gg 1$ (see equations in Appendix of
Artemyev  _et al._ , 2020a). This energy $\theta$ can be treated as a random
variable with uniform distribution on $[0,1]$ (see discussion of possible
limitations of such a consideration in Artemyev  _et al._ , 2020b). Map (12)
shows that for fixed $h$ the entire dynamics of electron energy due to
nonlinear resonant interactions is described by a single curve $S(\gamma)$:
the energy change due to scattering is $\Delta\gamma_{scat}=-(\omega/2\pi
m_{e}c^{2})\cdot S$, the probability of trapping is $\Pi=(\omega/2\pi
m_{e}c^{2})\cdot(dS/d\gamma)$, and energy change due to trapping
$\Delta\gamma_{trap}$ is defined as the difference of $\gamma_{R}$ between two
$S(\gamma)$ values (see scheme in Fig. 4(b)).

To verify the mapping technique described by Eq. (12) we calculate $S(\gamma)$
for $h$ from Fig. 1(b) and iterate a large ensemble of trajectories. Figure
4(a) shows $S(\gamma)$, $\Pi(\gamma)$ profiles, several individual
trajectories, and dynamics of $f(\gamma)$ relaxation. Note that to transform
iterations $n$ into time, one should use $t_{n+1}=t_{n}+\tau_{b}(\gamma)$
where $\tau_{b}$ is the electron bounce period, i.e. the time interval between
two resonant interactions (we consider waves only in the $s>0$ hemisphere):
for the dipole field $\tau_{b}\approx
4(1-\gamma^{-2})^{-1/2}\left(1.3802-0.6397\sin^{3/4}\alpha_{0}\right)$ Orlova
and Shprits (2011). Comparison of Figs. 2 and 4 demonstrates the applicability
of the mapping technique for the description of electron resonant dynamics
(see more examples with this approach in Artemyev  _et al._ , 2020a, 2021b).

Figure 4:  Panel (a) shows profiles of $S(\gamma)$, $\Pi(\gamma)$ for the
trajectory from Fig. 1(b). Schematic view of electron motion along $\gamma$
for given $\bar{S}(\gamma)=S\cdot(R\Omega_{0}(0)/c)^{1/2}$ (b). (note
$R\Omega_{0}(0)/c$ is of the order of $1/\varepsilon$ for the system
parameters). An example of trajectory $\gamma(n)$ for $n=0..100$ resonant
iterations (c). The relaxation of the distribution function $f(\gamma)$ (d).
Plots in panels (c,d) are obtained with map (12).

To summarize, the nonlinear resonances provide trapping and nonlinear
scattering with energy changes described by $S(\gamma)$ curve. In this regime
of resonant interaction $S\sim\sqrt{\varepsilon}$ (see Eq. (9)). As $|u/r|$
decreases, the magnitude of $S$ tends to zero, and for sufficiently small
$|u/r|$ there is only diffusion of electrons with the diffusion rate
$D_{QL}=\langle(\Delta\gamma)^{2}\rangle\sim\varepsilon^{2}$ (see Eq. (11) and
(Albert, 2010)). We aim to investigate the transition between these two
regimes and to obtain estimates of the corresponding different time-scales of
$f(\gamma)$ relaxation.

##  IV Main system scalings

To consider the transition between nonlinear resonant interaction and quasi-
linear diffusion, we need to consider a $S\to 0$ limit for a fixed
$\varepsilon$. We start with the evaluation of Hamiltonian (8) with small $S$.
Next, we investigate time-scales of $f(\gamma)$ relaxation for $S\to 0$. Then
we explain the relation of these time-scales and the time-scale of electron
acceleration within the trapping. Finally, all obtained equations are combined
together to provide the relation between time-scales of $f(\gamma)$ relaxation
due to nonlinear resonances and quasi-linear diffusion.

###  IV.1 Threshold $S$ values

Let us consider Hamiltonian (8) for $|r/u|$ around one, i.e. when $|r|$
becomes smaller than $u$ (condition required for existence of $S\neq 0$) only
within a short $\gamma_{R}$ range. Then the $\zeta$ range of the area filled
with closed trajectories ($\zeta\in[\zeta_{-},\zeta_{+}]$ in Fig. 3(b)) is
small and Hamiltonian (8) can be expanded as:

$\displaystyle\mathcal{H}=\frac{1}{2}wP^{2}+\varepsilon
u\cdot\left({\frac{r}{u}\zeta+\sin\zeta}\right)$ (13)
$\displaystyle\approx\frac{1}{2}wP^{2}+\varepsilon
u\cdot\left({\left({\frac{r}{u}-1}\right)\zeta+\frac{1}{6}\zeta^{3}}\right)$

where $r/u$ slowly changes along the trajectory from $>1$ (no closed
trajectories in the phase portrait) to $\min r/u<1$ (maximum area filled by
closed trajectories), and then to $>1$ again. Figure 5(a) shows the phase
portrait of Hamiltonian (13) for such $r/u<1$. To model this evolution we can
write $r/u=1-\delta_{0}+(\varepsilon t)^{2}$ where the small parameter
$\delta_{0}>0$ determines how far $|r/u|$ is from $1$ and $(\varepsilon
t)^{2}$ models the slow evolution of $r/u$ along the trajectory (i.e. we
change the slow coordinate $s$ to a slow time here):

$\mathcal{H}\approx\frac{1}{2}wP^{2}-\varepsilon
u\cdot\left({\left({\delta_{0}-\left({\varepsilon
t}\right)^{2}}\right)\zeta-\frac{1}{6}\zeta^{3}}\right)$ (14)

and $w$, $u$ can be considered as constant along a short interval of
$\varepsilon t\in[-\sqrt{\delta_{0}},\sqrt{\delta_{0}}]$. The equation for $S$
for Hamiltonian (14) can be written as

$\displaystyle S$ $\displaystyle=$
$\displaystyle\sqrt{\frac{{8u\varepsilon}}{w}}\int\limits_{\zeta_{+}}^{\zeta-}{\left({\left({\delta_{0}-\left({\varepsilon
t}\right)^{2}}\right)\left({\zeta-\zeta_{+}}\right)-\frac{{\zeta^{3}-\zeta_{+}^{3}}}{6}}\right)^{1/2}d}\zeta$
(15) $\displaystyle=$
$\displaystyle\sqrt{\frac{{8u\varepsilon}}{w}}\delta_{0}^{5/4}\left({1-\frac{{t^{2}}}{{t_{0}^{2}}}}\right)^{5/4}$
$\displaystyle\times$
$\displaystyle\int\limits_{\bar{\zeta}_{+}}^{\bar{\zeta}-}{\left({\left({\bar{\zeta}-\bar{\zeta}_{+}}\right)-\frac{{\bar{\zeta}^{3}-\bar{\zeta}_{+}^{3}}}{6}}\right)^{1/2}d}\bar{\zeta}$
$\displaystyle=$
$\displaystyle\sqrt{\frac{{8u\varepsilon}}{w}}\delta_{0}^{5/4}\left({1-\left({t/t_{0}}\right)^{2}}\right)^{5/4}\cdot\frac{{12\cdot
2^{3/4}}}{5}$

where $\bar{\zeta}_{+}=-\sqrt{2}$, $\bar{\zeta}_{-}=\sqrt{8}$,
$t_{0}=\sqrt{\delta_{0}}/\varepsilon$ and $t/t_{0}$ is equivalent to
$\gamma_{R}$, the energy at resonance, because different slow time values
imply here different values of slow $s$, i.e. different values of resonant
$s_{R}$ related to $\gamma_{R}$ through Eq. (5) and $h={\rm const}$. Equation
(15) demonstrates that the $\delta_{0}$ parameter controls the magnitude of
$S$, and effects of nonlinear interactions should disappear as $\delta_{0}\to
0$. In the nonlinear regime, there are well separated populations of trapped
particles (a small number of particles gaining a large energy) and nonlinearly
scattered particles (a large number of particles losing energy). In contrast,
in the diffusive regime the numbers of particles gaining and losing energy are
(approximately) equal. Therefore, there is a threshold $\delta_{0}$ value (or
$S$ value) such that for $\delta_{0}$ below this threshold, we cannot separate
trapping and nonlinear scattering. Let us obtain this threshold $\delta_{0}$
(or $S$) value.

Hamiltonian equations for Hamiltonian (14) can be combined to get the second
order equation for $\zeta$:

$\frac{{d^{2}\zeta}}{{d\tilde{t}^{2}}}=\varepsilon\cdot\left({\delta_{0}-\left({\varepsilon\tilde{t}}\right)^{2}-\frac{1}{2}\zeta^{2}}\right)$
(16)

where $\tilde{t}=t\sqrt{uw}$ and $u$, $w$ are constants of the order of one
(not dependent on $\varepsilon$). Equation (16) can be rewritten in normalized
variables $\tau=\tilde{t}\varepsilon^{1/2}\delta_{0}^{1/4}$,
$\xi=\zeta/\sqrt{\delta_{0}}$

$\frac{{d^{2}\xi}}{{d\tau^{2}}}=\left({1-\frac{\varepsilon}{{\delta_{0}^{3/2}}}\tau^{2}}\right)-\frac{1}{2}\xi^{2}$
(17)

Equation (17) shows that for $\delta_{0}=\varepsilon^{2/3}$ there is no
separation of time-scales, i.e. the equation describing the electron motion
around the resonance does not contain a slow time. Thus, both trapped and
nonlinearly scattered particles should stay in the resonance approximately the
same time, and there is no separation between these two types of trajectories.
Figure 5(b) confirms this conclusion: we solve equation Eq. (17) with
$\delta_{0}=\varepsilon^{2/3}$ for set of trajectories and plot these
trajectories in the phase plane $(\xi,d\xi/d\tau)$. There are still some
trajectories similar to trapped trajectories of the original system, but
particles on these trajectories do not fulfil a single oscillation across the
resonance $d\xi/d\tau=0$, i.e. we cannot separate particles on trapped and
scattered trajectories.

Figure 5:  The phase portrait of Hamiltonian (13) for $r/u$ around $1$ (a).
Set of trajectories obtained by numerical integration of Eq. (17) with
$\delta_{0}=\varepsilon^{2/3}$. Color codes the time interval required to
cross the shown $d\xi/d\tau$ range.

###  IV.2 Time-scales of $f(\gamma)$ evolution

To investigate how the rate of $f(\gamma)$ evolution scales with $\delta_{0}$
we use the mapping technique. Equation (15) shows that the area $S$ can be set
as $\varepsilon^{1/2}\delta_{0}^{5/4}\cdot(1-y^{2})^{5/4}$ where $y$ is the
effective particle energy at the resonance (shifted relative to the energy
where $S$ reaches maximum), i.e. different values of slow time
$\sim\varepsilon t$ in Eq. (15) correspond to different $s_{R}$ values of the
original system and to different resonant energies. Therefore, we can rewrite
mapping (12) as

$y_{n+1}=y_{n}+\left\\{{\begin{array}[]{*{20}c}{-2y_{n},}&{\theta\in\left[{0,\Pi}\right]}\\\
{-\mathcal{S}\left({y_{n}}\right),}&{\theta\in\left({\Pi,1}\right]}\\\
\end{array}}\right.$ (18)

and

$\displaystyle\mathcal{S}$ $\displaystyle=$
$\displaystyle\varepsilon^{1/2+\kappa}\left({1-y^{2}}\right)^{5/4}$
$\displaystyle\Pi$ $\displaystyle=$
$\displaystyle\frac{{d\mathcal{S}}}{{dy}}=-\frac{5}{2}y\varepsilon^{1/2+\kappa}\left({1-y^{2}}\right)^{1/4}$

where we introduce $\kappa$ as $\delta_{0}=\varepsilon^{4\kappa/5}$, and take
into account $\mathcal{S}(-y)=\mathcal{S}(y)$ to calculate $y$ change due to
trapping. Figure 6(a) shows an example of $y_{n}$ trajectory described by map
(18). This trajectory repeats all features seen in the original system
trajectories (see Figs. 1(d), 4(c)). The relaxation of the initially localized
$f(y)$ distribution also repeats well the relaxation of the $f(\gamma)$
distribution: compare Fig. 6(b) and Figs. 2(d), 4(d).

To characterize the time-scale of $f(y)$ evolution as a function of $\kappa$,
we numerically integrate a large ensemble of trajectories described by map
(18) for various values of $\kappa$. The initial $y_{0,i}$ values for $i=0…N$
trajectories are uniformly distributed within $[-1,1]$ range, and we calculate
two characteristics:
$\mathcal{M}_{1}(n)=N^{-1}\sum_{i=0..N}\left(y_{n,i}-y_{0,i}\right)$ and
$\mathcal{M}_{2}(n)=N^{-1}\sum_{i=0..N}\left(y_{n,i}-y_{0,i}\right)^{2}-\mathcal{M}_{1}^{2}(n)$.
Figure 6(e) shows $\mathcal{M}_{1}(n)$, $\mathcal{M}_{2}(n)$ profiles for two
$\kappa$ values: $\kappa=0$ corresponds to the system with well distinguished
populations of trapped and scattered particles (i.e., the system is far from
the diffusive regime), whereas $\kappa=1/3$ corresponds to
$\delta_{0}=\varepsilon^{4\kappa/5}=\varepsilon^{4/15}$ and this is a case
with much closer time-scales of the dynamics of trapped and scattered particle
populations. For both values of $\kappa$ $\mathcal{M}_{2}$ grows with $n$ and
$\mathcal{M}_{1}$ oscillates around the zero.

To derive the time-scale of $f(y)$ relaxation we fit the growing fragment of
$\mathcal{M}_{2}(n)$ profile by the linear function $D_{NL}\cdot{n}$ with the
coefficient $D_{NL}$ playing the role of a diffusion coefficient in systems
with diffusive resonant interactions. Figure 6(f) shows the $D_{NL}$
dependence on $\kappa$ for three $\varepsilon$ values. There is a clear
scaling $D_{NL}\approx\varepsilon^{1/2+\kappa}$, i.e. $D_{NL}$ varies as
$\varepsilon^{1/2}$ for $\kappa=0$ and as $\varepsilon^{4/3}$ for
$\kappa=5/6$. This scaling may be explained by the dominating role of
trappings in the $\mathcal{M}_{2}$ increase. Most of nonlinearly scattered
particles change their energy only slightly, and the first order contribution
of this energy change goes into $\mathcal{M}_{1}$, whereas the contribution of
these particles to the growth of $\mathcal{M}_{2}$is about
$\sim\mathcal{S}^{2}\sim\varepsilon^{1+2\kappa}$. However, even a small
population of trapped particles would contribute to
$N^{-1}\sum_{i=0..N}\left(y_{n,i}-y_{0,i}\right)^{2}$ as
$\sim\Pi\sim\varepsilon^{1/2+\kappa}$ (the number of trapped particles is
about the probability of trapping, whereas these particles have
$y_{n,i}-y_{0,i}\sim O(1)$); on the other hand, their contribution to
$\mathcal{M}_{1}^{2}$ is much smaller. Thus, $\mathcal{M}_{2}\sim\Pi$ and this
confirms the scaling from Fig. 6(f).

Figure 6:  Examples of $y_{n}$ trajectories described by map (18) (a,b) and
evolution of $f(y)$ distribution (c,d) for two $\kappa$ values. Profiles of
$\mathcal{M}_{1}(n)$, $\mathcal{M}_{2}(n)$ for two $\kappa$ values (e).
Dependence of $D_{NL}$ on $\kappa$ for three $\varepsilon$ values (f).

###  IV.3 Time-scale of trapping acceleration

The scaling from Fig. 6(d) shows that $D_{NL}$ not only depends on
$\varepsilon$, but also scales with $\delta_{0}\sim\varepsilon^{4\kappa/5}$.
This parameter, $\delta_{0}$, controls the effective range of energy change
due to trapping, and should be related to the number of complete rotations
around the resonance of trapped particles, $N_{trap}$. Thus, we can derive the
$\delta_{0}(N_{trap})$ dependence and rewrite $D_{NL}$ as a function of
$\varepsilon$ and $N_{trap}$. We define $N_{trap}$ as the maximum number of
periods of a trapped particle’s rotation in the phase space (see phase
portrait in Fig. 3(b)). This parameter is quite universal: it can be
determined for any particular wave-field model (see, e.g., Zhang  _et al._
(2018) for a discussion of $N_{trap}$ values typical for whistler-mode waves
observed in the Earth’s radiation belts). The trapped period for Hamiltonian
(14) can be written as

$T_{trap}\left({\varepsilon
t}\right)=2\int\limits_{\zeta_{-}}^{\zeta_{{}_{+}}}{\frac{{d\zeta}}{{\dot{\zeta}}}}=\frac{C}{{\varepsilon^{1/2}\left({\delta_{0}-\varepsilon^{2}t^{2}}\right)^{1/4}}}$
(19)

where constant $C\sim O(1)$ is determined by the distance from the separatrix
in the phase portrait. The maximum time of trapped particle motion is of the
order of $\sim 2\delta_{0}^{1/2}/\varepsilon$. Therefore we can write for
$N_{trap}$

$N_{trap}=\frac{2}{\varepsilon}\int\limits_{0}^{\sqrt{\delta_{0}}/\varepsilon}{\frac{{d\varepsilon
t}}{{T_{trap}\left({\varepsilon
t}\right)}}}=\frac{{\delta_{0}^{3/4}}}{{\sqrt{\varepsilon}}}\tilde{C}\sim\varepsilon^{3\kappa/5-1/2}$
(20)

where constant $\tilde{C}\sim O(1)$. Equation (20) shows that
$N_{trap}\sim\varepsilon^{3\kappa/5-1/2}$. For the threshold value
$\kappa=5/6$, we get $N_{trap}\sim O(1)$, i.e. indeed for $\kappa=5/6$
($\delta_{0}\sim\varepsilon^{2/3}$) the number of trapping periods does not
depend on $\varepsilon$ and there is no separation between trapped and
scattered particles anymore.

Combining Eq. (20) and the $D_{NL}$ scaling, we obtain the final scaling of
the coefficient $D_{NL}$

$D_{NL}\sim\varepsilon^{1/2+\kappa}\sim\varepsilon^{4/3}N_{trap}^{5/3}\sim
D_{QL}\cdot\varepsilon^{-2/3}N_{trap}^{5/3}$ (21)

where we take into account that $D_{QL}\sim\varepsilon^{2}$. Therefore,
$D_{NL}/D_{QL}=\mathcal{Q}\varepsilon^{-2/3}N_{trap}^{5/3}$ where the factor
$\mathcal{Q}\sim O(1)$ is determined by the system parameters. According to
this scaling, the transition between the diffusive regime ($N_{trap}\to 0$)
and the nonlinear resonant regime ($N_{trap}\gg 1$) occurs when
$N_{trap}\sim\varepsilon^{2/5}\ll 1$, whereas for $\kappa=5/6$ ($N_{trap}\sim
1$) we get $D_{NL}/D_{QL}\sim\varepsilon^{-2/3}\gg 1$.

However, the scaling in Eq. (21) is obtained for systems where $\varepsilon$
and $N_{trap}$ (or more precisely $\kappa$, which determines $\delta_{0}$) are
assumed to be independent parameters that can be varied separately. But
$\varepsilon$ and $N_{trap}$ are not always independent. This is especially
the case in realistic systems, where whistler-mode waves often propagate in
the form of short wave packets (see examples of such wave packets in
observations Zhang  _et al._ (2018) and in numerical simulations Nunn  _et
al._ (2021); Zhang  _et al._ (2021)). We consider this realistic situation in
the next subsection.

###  IV.4 Systems with short wave packets: Shrinking of the resonant energy
range

In the Earth’s magnetosphere, intense whistler-mode waves contain a fine
structure of packets or subpackets, where packets/subpackets are formed by
strong wave amplitude modulations (by at least a factor of $\sim 2$)
accompanied by large and random wave phase jumps in-between successive
packets/subpackets, allowing to treat electron interactions with different
packets/subpackets independently (Tao  _et al._ , 2012, 2013; Mourenas  _et
al._ , 2018; Zhang  _et al._ , 2020a). In particular, most long wave
envelopes, characterized by an amplitude continuously remaining above $50$ pT,
actually consist of many shorter packets/subpackets when internal strong
amplitude modulations and wave frequency/phase jumps are taken into account
(Mourenas  _et al._ , 2018; Zhang  _et al._ , 2020b, a). The overwhelming
majority of such intense parallel propagating whistler-mode wave
packets/subpackets observed in the magnetosphere have a short length of
$\beta<10$ wave periods, with much more rare long packets/subpackets reaching
$\beta\sim 100-300$ periods (Zhang  _et al._ , 2018, 2019; Mourenas  _et al._
, 2018). Short packets of length $\beta<10$ are likely often produced by wave
superposition (Zhang  _et al._ , 2019, 2020b; Nunn  _et al._ , 2021) and their
length $\beta$ is mostly independent of their amplitude $\varepsilon$. Packets
of intermediate length $\beta\sim 10-30$ are probably partly formed by
trapping-induced amplitude modulation that depends on $\varepsilon^{-1/2}$,
but statistical observations show that all packets are distributed over a wide
range of $\beta$ values for any given $\varepsilon$ (Tao  _et al._ , 2013;
Zhang  _et al._ , 2020b; Nunn  _et al._ , 2021), allowing to treat $\beta$ and
$\varepsilon$ as independent parameters to leading order.

Taking into account that first-order cyclotron resonance between whistler-mode
waves and $<1$ MeV electrons occurs for waves and electrons moving in opposite
directions, the time interval of trapped particle motion is limited to $\sim
2\pi\beta/\Omega_{0}$, whereas one period of trapped electron oscillation is
$T_{trap}\sim 2\pi\varepsilon^{-1/2}/\Omega_{0}$ for typical resonant energies
Karpman (1974). Therefore, $N_{trap}$ can be expressed as a function of the
independent parameters $\beta$ and $\varepsilon$ of short wave packets, giving
$N_{trap}\sim\beta\varepsilon^{1/2}$ (see also Zhang  _et al._ , 2018).

In the case of short wave packets, although the magnitude of $S$ remains
$\sim\varepsilon^{1/2}$, the limited packet size can lead to important changes
as compared with the scaling obtained in Eq. (21) of the preceding subsection
in the ideal case of infinitely long wave packets or waves. Indeed, a given
short wave packet occupies only a limited latitudinal extent along a magnetic
field line. Eq. (5) and the conservation of $h$, which relate latitude of
resonance to electron energy, imply that this limited latitudinal extent
directly corresponds to a limited energy range for electron resonance with
this packet during one bounce period. This limited energy range of actually
resonant electrons represents a fraction $\ell<1$ of the total energy range of
particles potentially reaching resonance with a wave packet over the full
length of the magnetic field line. Electron trapping by a short packet leads
to a smaller energy change than trapping by an ideal infinitely long wave
packet, because the electron is released from trapping faster, corresponding
to a reduction of its energy change by a factor $\sim\ell$. Besides, short
wave packets may sometimes be rare and occur only once every bounce period,
corresponding to a reduction of their occurrence rate by a factor $\sim\ell<1$
as compared with both the case of a close succession of short packets and the
ideal case of infinitely long packets assumed in the preceding subsection.

Accordingly, let us consider a realistic situation where the magnitude of
$\mathcal{S}\sim\sqrt{\varepsilon}$ remains the same, but the range of
nonlinear resonant interactions (formally the $\varepsilon t$ range in Eq.
(14)) shrinks. There are two kinds of systems with an entire resonant range
$y\in[-1,1]$, but where $\mathcal{S}$ is not equal to zero only for
$y\in[-\ell,\ell]$, with $\ell=\sqrt{\delta_{0}}$. In systems of the first
kind, we center $\mathcal{S}$ around a $\bar{y}_{n}$ randomly generated at
each map iteration around $y_{n}$, i.e., for each iteration there is a finite
change of $y_{n}$. This type of mapping mimics electron resonant interactions
with a set of short wave packets that would fill the entire magnetic field
lines, but which cannot trap electrons for a long time because of the limited
packet duration. In the system of the second kind, we center $\mathcal{S}$
around a randomly generated $\bar{y}_{n}$, but we do not control the position
of $\bar{y}_{n}$ relative to $y_{n}$ and some map iterations can occur without
any change of $y_{n}$ because $y_{n}$ is outside the $\mathcal{S}\neq 0$
range. This type of mapping mimics electron resonant interaction with rare
short wave-packets propagating with some time separation. It corresponds to a
situation where, during each bounce period, only one wave packet is present
and many electrons reach the latitude of cyclotron resonance without
encountering this intense packet there. For both systems, the mapping (18) can
be rewritten as

$y_{n+1}=y_{n}+\left\\{{\begin{array}[]{*{20}c}{\bar{y}_{n}-2y_{n},}&{\theta\in\left[{0,\Pi}\right]}\\\
{-\mathcal{S}_{n}\left({y_{n}}\right),}&{\theta\in\left({\Pi,1}\right]}\\\
\end{array}}\right.$ (22)

and

$\displaystyle\mathcal{S}_{n}$ $\displaystyle=$
$\displaystyle\varepsilon^{1/2}\left({\ell^{2}-\left({y-\bar{y}_{n}}\right)^{2}}\right)^{5/4}$
$\displaystyle\Pi$ $\displaystyle=$
$\displaystyle\frac{{d\mathcal{S}_{n}}}{{dy}}=-\frac{5}{2}\left({y-\bar{y}_{n}}\right)\varepsilon^{1/2}\left({\ell^{2}-\left({y-\bar{y}_{n}}\right)^{2}}\right)^{1/4}$

where $\bar{y}_{n}=y_{n}+\ell\cdot R$ for the system of the first kind, and
$\bar{y}_{n}=-1+\ell+2(1-\ell)\cdot R$ for the system of the second kind ;
here $R$ is a random number with a uniform distribution within $[-1,1]$ (note
for $\bar{y}_{n}=y_{n}+\ell\cdot R$ we also control that $\mathcal{S}\neq 0$
range does not cross $y\pm 1$ boundaries). An important property of both
systems is that for small $\ell$ trappings are possible for any $y_{n}$
values, whereas for $\ell=1$ (the initial map given by Eq. (18)) trapping is
possible only for $y_{n}<0$. Figure 7(a,b) shows a set of sample trajectories
for each of the two kinds of systems. The rate of change of $y$ is going down
as $\ell$ decreases, and this effect is stronger for the system of the second
kind (Fig. 7(b)).

For these two kinds of systems we set the range of $\ell$ and for each $\ell$
value calculate $M_{2}(n)$. Figure 7(c,d) shows examples of $M_{2}$ profiles.
We fit the growing fragment of $\mathcal{M}_{2}(n)$ by $D_{NL}n$, and Fig.
7(e,f) shows the $D_{NL}$ dependence on $\ell$. For both systems, $D_{NL}$
scales with $\ell$ as $D_{NL}\sim\varepsilon^{1/2}\ell^{\eta}$. But
$\eta\approx 7/2$ for the system of the first kind where resonances occur at
each iteration, whereas $\eta\approx 9/2$ for the system of the second kind,
where resonance occurrence is decreased by a factor $\ell\ll 1$. Taking into
account that $\ell=\sqrt{\delta_{0}}$, we can rewrite Eq. (20) as
$N_{trap}\sim\ell^{3/2}/\sqrt{\varepsilon}$. Therefore, for $D_{NL}$ scaling
with $N_{trap}$ we have
$D_{NL}\sim\varepsilon^{1/2}\ell^{\eta}\sim\varepsilon^{1/2+\eta/3}N_{trap}^{2\eta/3}$
and

$D_{NL}\sim\left\\{{\begin{array}[]{*{20}c}{\varepsilon^{5/3}N_{trap}^{7/3},}&{\eta=7/2}\\\
{\varepsilon^{2}N_{trap}^{3},}&{\eta=9/2}\\\ \end{array}}\right.$ (23)

for the two kinds of systems.

A general scaling, including both the results in Eq. (23) and in Eq. (21), is

$D_{NL}\sim\varepsilon^{1/2+\eta/3}N_{trap}^{2\eta/3}\sim\varepsilon^{1/2+2\eta/3}\beta^{2\eta/3}.$
(24)

For $\eta=5/2$, Eq. (24) allows to recover the initial scaling from Eq. (21),
which corresponds to the ideal case of electron interactions with infinitely
long wave packets/waves present all the time along magnetic field lines. In
the more realistic situation of short wave packets, two different kinds of
systems are possible, corresponding to $\eta=7/2$ and $\eta=9/2$. For the
system of the first kind, we get
$D_{NL}/D_{QL}\sim\varepsilon^{-1/3}N_{trap}^{7/3}\sim\varepsilon^{5/6}\beta^{7/3}$.
In this case, $D_{NL}/D_{QL}\sim O(1)$ requires that
$N_{trap}\sim\varepsilon^{1/7}$ and $\beta\sim\varepsilon^{-5/14}$. For
realistic whistler-mode wave amplitudes $\varepsilon\in[3\times
10^{-4},3\times 10^{-3}]$ in the Earth’s radiation belts (Zhang  _et al._ ,
2020b), the transition between the regimes of quasi-linear diffusion and
nonlinear resonant interaction should therefore occur approximately for
$N_{trap}>0.35$ and $\beta>10$. In the system of the second kind, we have
$D_{NL}/D_{QL}\sim N_{trap}^{3}\sim\beta^{3}\varepsilon^{3/2}$. In this case,
$D_{NL}/D_{QL}\sim O(1)$ corresponds to $N_{trap}\sim 1$ and
$\beta\sim\varepsilon^{-1/2}$. Thus, for this system of very rare short
packets, the transition between the regimes of quasi-linear diffusion and
nonlinear resonant interaction should occur around $N_{trap}\sim 1$ and
approximately for $\beta>20$ – i.e. near the $\kappa=5/6$ threshold value when
$\delta_{0}\sim\varepsilon^{2/3}$ and $\ell\sim\varepsilon^{1/3}$.

Figure 7:  Three trajectories for different $\ell$ values (see color codes in
panels (c,d)) for systems of two different kinds (a,b). Examples of $M_{2}$
profiles for these two systems (c,d). Dependencies of $D_{NL}$ on $\ell$ for
the two systems and different $\varepsilon$ (d,f). Left column (a,c,e) shows
results for the system of the first kind: the entire resonant range
$y\in[-1,1]$, but $S$ is not equal to zero only for $[-\ell,\ell]$ appearing
always around the particle location, i.e. there is a resonant interaction at
each iteration. The right column (b,d,f) shows results for the system of the
second kind: the entire resonant range $y\in[-1,1]$, but $S$ is not equal to
zero only over $[-\ell,\ell]$ appearing at random locations, not necessarily
near the particle location, i.e., there are iterations without change of
particle $y$.

Note that the $D_{NL}$ scaling in Eq. (24) for the first kind of system with
short wave packets, corresponding to $\eta=7/2$, is valid only when cyclotron
resonance with wave packets is always available, i.e. when electrons with any
energy within the resonant range interact with the wave once (or more
frequently) per bounce period. In practice, however, intense wave packets are
not always present (Zhang  _et al._ , 2018; Mourenas  _et al._ , 2018).
Nevertheless, the overwhelming majority of wave packets interact with
electrons independently of the other packets (Zhang  _et al._ , 2020a), and
their very large population is mostly randomly distributed in time.
Consequently, over a sufficiently long time period, all latitudes of resonance
with all electron energies should be uniformly reached by packets in each
($\beta,\varepsilon$) range. In this limit, the time-averaged $\langle
D_{NL}\rangle$ can be simply obtained by a double integration over the full
measured distribution of $(\beta>0,\varepsilon>0)$ packets of $D_{NL}$,
weighted by the measured temporal occurrence rate (taking into account packet
duration $\beta$) of $(\beta,\varepsilon)$ packets (Zhang  _et al._ , 2018,
2019; Mourenas  _et al._ , 2018). Finally, $\langle D_{NL}\rangle/\langle
D_{NL}\rangle$ can be obtained by dividing $\langle D_{NL}\rangle$ by the
integral over $\varepsilon$ of $D_{QL}\sim\varepsilon^{2}$ weighted by the
measured temporal occurrence rate of $\varepsilon$ packets. In addition,
$D_{NL}/D_{QL}$ should be considered as $\neq 1$ in the above integration only
for $\varepsilon$ values above the threshold for nonlinear effects (Omura  _et
al._ , 2008; Zhang  _et al._ , 2019). For lower amplitudes, one can simply
assume $D_{NL}/D_{QL}\approx 1$.

##  V Discussion and Conclusion

In this study, we investigate the transition between two regimes of resonant
electron interactions with whistler-mode waves: quasi-linear diffusion and
nonlinear resonant interaction. The typical effects of these two regimes are
very different: diffusive scattering results in small energy changes with zero
mean value, whereas nonlinear resonant interactions result in energy drifts
(due to phase bunching) or large jumps (due to phase trapping). This
difference makes it impossible to directly compare energy changes for these
two regimes, i.e. energy drifts and jumps cannot be compared with the
diffusion rates. Therefore, a more general (nonlocal, i.e. independent on the
initial energy value) characteristic should be determined to compare these two
regimes. For such characteristic, we choose the typical time-scale of
relaxation of the energy distribution function $f(\gamma)$, which is
determined along a single resonant curve $h={\rm const}$. This characteristic
describes the entire energy, pitch-angle range for a given
$h=\gamma-\left(\omega/2\Omega(0)\right)(\gamma^{2}-1)\sin^{2}\alpha_{0}$ and
wave frequency, i.e. it is an integral characteristic that simplifies the
system description. The relaxation of the $f(\gamma)$ distribution depends on
the initial $f(\gamma)$ and contains many elements that are different for
these two regimes (e.g., formation of local maxima due to the phase trapping
acceleration). Therefore, to further simplify the description of this
evolution and reduce it to a single parameter, we restrict our consideration
to the dynamics of $f(\gamma)$ dispersion $\mathcal{M}_{2}$ that increases
with the number of resonance crossings and finally saturates. This parameter
$\mathcal{M}_{2}(n)$ is the unique characteristic for the diffusive resonant
regime, but for nonlinear resonant interactions there are also
$\mathcal{M}_{1}(n)$ and higher-order moments. Thus, some justification for
taking only $\mathcal{M}_{2}$ into account should be given. The first moment
$\mathcal{M}_{1}(n)=Vn$ scales with $\varepsilon$ as
$V\sim\mathcal{S}\sim\varepsilon^{1/2+\kappa}$, and the same scaling can be
obtained for $\mathcal{M}_{2}(n)=D_{NL}n$ with
$D_{NL}\sim\mathcal{S}\sim\varepsilon^{1/2+\kappa}$. The beginning of the
evolution (relaxation) of the distribution $f(\gamma)$ is dominated by
trapping effects contributing to $\mathcal{M}_{2}$, because
$\delta\gamma^{2}\sim D_{NL}n$ is larger than $\delta\gamma^{2}\sim
V^{2}n^{2}$ for small $n$. The threshold value of $n$ such that drifts become
more important than trappings is $n^{*}\sim
D_{NL}/V^{2}\sim\varepsilon^{-1/2-\kappa}$. During this number of resonant
interactions the first trapped particles would be drifted back to their
initial energies, because $\Delta\gamma_{trap}/\Delta\gamma_{scat}\sim
1/S\sim\varepsilon^{-1/2-\kappa}$ and
$n^{*}\cdot\left(\Delta\gamma_{trap}/\Delta\gamma_{scat}\right)\sim O(1)$. The
drift effects may control the relaxation of $f(\gamma)$ only after the main
relaxation stage is finished (when the initially trapped particles reach back
their initial energies, the first large cycle of trapping-scattering would be
finished; see, e.g., example of such cycles in Figs. 1(d)). Thus, we indeed
can use $\mathcal{M}_{2}(n)$ to characterize the main stage of $f(\gamma)$
relaxation. Note that for the regime of nonlinear resonant interaction
$\mathcal{M}_{2}(n)$ does not describe diffusion, but shows how quickly the
dispersion of $f(\gamma)$ grows due to trapping effects.

In the Earth’s magnetosphere, whistler-mode waves mainly propagate in the form
of wave packets, and most of the observed parallel propagating intense wave
packets are short, containing $\beta<10-30$ wave periods (Zhang  _et al._ ,
2018; Mourenas  _et al._ , 2018). The relations obtained here, connecting
$D_{NL}/D_{QL}$ with wave packet amplitude $\varepsilon$ and number of trapped
electron oscillations $N_{trap}$ or packet length $\beta$, can be considered
as inverse scaling laws for the time-scale of $f(\gamma)$ relaxation –
although such a consideration simplifies the details of $f(\gamma)$ evolution.
The main potential application of these scaling laws is to allow including the
leading order effect of nonlinear resonant interactions into global models of
radiation belt dynamics relying on the Fokker-Planck diffusion equation. This
equation describes the evolution of $f(\gamma)$ driven by diffusion on a
typical time-scale $\sim 1/D_{QL}$. This time-scale is close to the observed
time-scales of electron flux dynamics in the Earth’s radiation belts during
quiet time periods characterized by not-to-high wave intensity (e.g., Artemyev
_et al._ , 2013; Mourenas  _et al._ , 2017; Claudepierre  _et al._ , 2020).
However, during more disturbed periods with a high occurrence rate of intense
wave packets Thorne  _et al._ (2013), the observed electron flux evolution
could be faster than the evolution predicted by a diffusion model (see
discussions in, e.g., (Mourenas  _et al._ , 2018; Gan  _et al._ , 2020;
Allanson  _et al._ , 2021)). Our scaling laws of $D_{NL}/D_{QL}$ allow for a
simple renormalization of the quasi-linear diffusion rates based on the
observed occurrence rate of intense waves resonantly interacting with
electrons in the nonlinear regime. An important ingredient of such a
renormalization is to take into account not only the wave intensity
($\sim\varepsilon^{2}$), but also the actual fine structure of intense wave
packets (Tao  _et al._ , 2012, 2013; Zhang  _et al._ , 2020b, a). We can take
into account these fine structure effects via the parameter $N_{trap}$, the
number of periods of trapped particle motion. Although $N_{trap}$ is not a
directly measured wave characteristic, this parameter can include effects of
wave-packet shortness (Zhang  _et al._ , 2018, 2020a; Mourenas  _et al._ ,
2018; Hiraga and Omura, 2020) and effects of wave coherence destruction (Tao
_et al._ , 2013; Artemyev  _et al._ , 2015b; Zhang  _et al._ , 2020a).
Alternatively, we can take into account the fine structure of whistler-mode
waves via the parameter $\beta$, the wave packet length in number of wave
periods, which can be directly measured together with $\varepsilon$ (Zhang
_et al._ , 2018). For resonant interactions with $<1$ MeV electrons, we get
$D_{NL}/D_{QL}\sim\varepsilon^{-3/2+2\eta/3}\beta^{2\eta/3}$ with a value of
$\eta\in[5/2,9/2]$ that depends on assumptions concerning the wave packet
distribution in the system. For the rare very long wave packets with
$\beta\sim 10^{2}$ and high wave amplitude $\varepsilon\sim 10^{-3}$ we obtain
$D_{NL}/D_{QL}\sim 650$ for $\eta=5/2$ and $D_{NL}/D_{QL}\sim 150$ for
$\eta=7/2$, but the time-weighted occurrence rate of such long and intense
wave packets varies between $\sim 0.5\times 10^{-3}$ and $\sim 2\times
10^{-3}$ from mildly to strongly active geomagnetic conditions (Zhang  _et
al._ , 2018; Mourenas  _et al._ , 2018). In the most realistic case of a long
succession of short wave packets (corresponding to $\eta=7/2$) and for wave
amplitudes above an approximate threshold $\varepsilon\geq 10^{-3}$ for
nonlinear interaction (Omura  _et al._ , 2008; Zhang  _et al._ , 2019),
integrating over $\beta\sim 2-200$ the expression (24) of $D_{NL}/D_{QL}$,
weighted by the measured occurrence rate of such $\beta$ packets (Mourenas
_et al._ , 2018), gives $\langle D_{NL}/D_{QL}\rangle_{\beta}\sim 3$. For
smaller amplitudes $\varepsilon<10^{-3}$, representing $\approx 70$% of all
packets (Zhang  _et al._ , 2019), we should have $D_{NL}/D_{QL}\sim 1$, giving
a final approximate ratio $\langle D_{NL}\rangle_{\beta,\varepsilon}/\langle
D_{QL}\rangle_{\varepsilon,\beta}\sim 2$. These estimates suggest that over
relatively long time periods ($\sim$ days), nonlinear interactions should only
slightly speed up the evolution of electron fluxes in comparison with the
diffusive evolution for the same time-averaged wave intensity. However, a more
accurate parametrical study would be needed to determine for what energies,
pitch-angles, and geomagnetic conditions the contribution of nonlinear
interactions maximizes. It is also worth noting that the present comparison of
relatively long time-scales of electron distribution relaxation does not
contradict the fact that only the regime of nonlinear resonant interaction can
explain various short-lived events of rapid electron acceleration or loss,
which occur much faster than the diffusive evolution (e.g., Agapitov  _et al._
, 2015; Gan  _et al._ , 2020).

To conclude, we have investigated the transition between two regimes of
resonant interaction of electrons and whistler-mode waves in the Earth’s
radiation belts: quasi-linear diffusion and nonlinear resonant interaction
(including effects of phase trapping and phase bunching). To characterize both
regimes within the same framework, we have introduced the typical time-scale
$1/D$ of evolution of the electron energy distribution on the resonance curve.
This time-scale characterizes the increase of the distribution dispersion for
both diffusive ($1/D_{QL}$) and nonlinear ($1/D_{NL}$) relaxations of the
electron distribution. For the diffusive relaxation
$D_{QL}\sim\varepsilon^{2}$ where $\varepsilon=B_{w}/B_{0}$ is the normalized
wave amplitude. Thus, we investigated how $D_{NL}$ depends on $\varepsilon$
and on the characteristics of intense wave-packets, which are described by
$N_{trap}$, the number of periods in trapped motion, or $\beta$, the wave
packet size ($N_{trap}\sim\beta\varepsilon^{1/2}$ when $\beta<1/\varepsilon$)
and/or to the closeness of the wave packet amplitude to its threshold value
required for nonlinear resonances
($N_{trap}\sim\delta_{0}^{3/4}\varepsilon^{-1/2}$ with $\delta_{0}\to 1$ for
large amplitude packets). The obtained scaling is
$D_{NL}/D_{QL}\sim\varepsilon^{-3/2}\left(\varepsilon\beta\right)^{2\eta/3}$
with $\eta$ varying from $5/2$ (when we compare electron distribution
relaxation due to diffusion and nonlinear interactions over the same energy
range) to $9/2$ (when diffusion acts over a wide energy range, whereas
nonlinear interactions are very rare and occur in a narrower energy range).
These scalings can be used for generalizing the existing radiation belt models
and for an evaluation of the importance of nonlinear interactions for
realistic distributions of wave characteristics.

###### Acknowledgements.

The work of A.V.A., A.I.N., and A.A.V. was supported by the Russian Science
Foundation, Project No. 19-12-00313.

## References

  * Vedenov  _et al._ (1962) A. A. Vedenov, E. Velikhov, and R. Sagdeev, Nuclear Fusion Suppl. 2, 465 (1962).
  * Drummond and Pines (1962) W. E. Drummond and D. Pines, Nuclear Fusion Suppl. 3, 1049 (1962).
  * O’Neil (1965) T. O’Neil, Physics of Fluids 8, 2255 (1965).
  * Mazitov (1965) R. K. Mazitov, Journal of Applied Mechanics and Technical Physics 6, 22 (1965).
  * Kennel and Engelmann (1966) C. F. Kennel and F. Engelmann, Physics of Fluids 9, 2377 (1966).
  * Karpman (1974) V. I. Karpman,  Space Sci. Rev. 16, 361 (1974).
  * Kennel and Petschek (1966) C. F. Kennel and H. E. Petschek, J. Geophys. Res. 71, 1 (1966).
  * Nunn (1971) D. Nunn, Journal of Plasma Physics 6, 291 (1971).
  * Karpman  _et al._ (1974) V. I. Karpman, J. N. Istomin, and D. R. Shklyar, Plasma Physics 16, 685 (1974).
  * Thorne  _et al._ (2013) R. M. Thorne, W. Li, B. Ni, Q. Ma, J. Bortnik, L. Chen, D. N. Baker, H. E. Spence, G. D. Reeves, M. G. Henderson, C. A. Kletzing, W. S. Kurth, G. B. Hospodarsky, J. B. Blake, J. F. Fennell, S. G. Claudepierre, and S. G. Kanekal, Nature (London) 504, 411 (2013).
  * Li  _et al._ (2014) W. Li, R. M. Thorne, Q. Ma, B. Ni, J. Bortnik, D. N. Baker, H. E. Spence, G. D. Reeves, S. G. Kanekal, J. C. Green, C. A. Kletzing, W. S. Kurth, G. B. Hospodarsky, J. B. Blake, J. F. Fennell, and S. G. Claudepierre,  J. Geophys. Res. 119, 4681 (2014).
  * Agapitov  _et al._ (2015) O. V. Agapitov, A. V. Artemyev, D. Mourenas, F. S. Mozer, and V. Krasnoselskikh,  Geophys. Res. Lett. 42, 10 (2015).
  * Foster  _et al._ (2014) J. C. Foster, P. J. Erickson, D. N. Baker, S. G. Claudepierre, C. A. Kletzing, W. Kurth, G. D. Reeves, S. A. Thaller, H. E. Spence, Y. Y. Shprits, and J. R. Wygant,  Geophys. Res. Lett. 41, 20 (2014).
  * Gan  _et al._ (2020) L. Gan, W. Li, Q. Ma, A. V. Artemyev, and J. M. Albert,  Geophys. Res. Lett. 47, e90749 (2020).
  * Mourenas  _et al._ (2016) D. Mourenas, A. V. Artemyev, O. V. Agapitov, F. S. Mozer, and V. V. Krasnoselskikh,  J. Geophys. Res. 121, 4498 (2016).
  * Voshchepynets  _et al._ (2017) A. Voshchepynets, A. Volokitin, V. Krasnoselskikh, and C. Krafft, Journal of Geophysical Research (Space Physics) 122, 3915 (2017).
  * Kuzichev  _et al._ (2019) I. V. Kuzichev, I. Y. Vasko, A. Rualdo Soto-Chavez, Y. Tong, A. V. Artemyev, S. D. Bale, and A. Spitkovsky, Astrophys. J. 882, 81 (2019), arXiv:1907.04878 [physics.space-ph] .
  * Shaaban  _et al._ (2019) S. M. Shaaban, M. Lazar, P. H. Yoon, S. Poedts, and R. A. López,  Monthly Notices of the Royal Astronomical Society 486, 4498 (2019), arXiv:1903.08005 [astro-ph.SR] .
  * Veltri and Zimbardo (1993) P. Veltri and G. Zimbardo,  J. Geophys. Res. 98, 13335 (1993).
  * Bykov and Treumann (2011) A. M. Bykov and R. A. Treumann,  The Astronomy and Astrophysics Review 19, 42 (2011), arXiv:1105.3221 [astro-ph.HE] .
  * Kletzing (1994) C. A. Kletzing,  J. Geophys. Res. 99, 11095 (1994).
  * Shen and Knudsen (2020) Y. Shen and D. J. Knudsen, Journal of Geophysical Research (Space Physics) 125, e27291 (2020), arXiv:2007.00660 [physics.space-ph] .
  * Andronov and Trakhtengerts (1964) A. A. Andronov and V. Y. Trakhtengerts, Geomagnetism and Aeronomy 4, 233 (1964).
  * Lyons and Williams (1984) L. R. Lyons and D. J. Williams,  _Quantitative aspects of magnetospheric physics.. L. R. Lyons, D. J. Williams.D. Reidel Publishing Company, Dordrecht - Boston - Lancaster. 15+231 pp. (1984). ISBN 90-277-1663-3._ , edited by Lyons, L. R. & Williams, D. J. (1984).
  * Trakhtengerts and Rycroft (2008) V. Y. Trakhtengerts and M. J. Rycroft,  _Whistler and Alfvén Mode Cyclotron Masers in Space, by V. Y. Trakhtengerts and M. J. Rycroft. ISBN 978-0-521-87198-3. Published by Cambridge University Press, Cambridge, UK, 2008._ (Cambridge University Press, 2008).
  * Le Queau and Roux (1987) D. Le Queau and A. Roux,  Solar Physics 111, 59 (1987).
  * Karpman and Shkliar (1977) V. I. Karpman and D. R. Shkliar,  Plan. Sp. Sci. 25, 395 (1977).
  * Shklyar (1981) D. R. Shklyar, Sov. Phys. JETP 53, 1197 (1981).
  * Albert (1993) J. M. Albert, Physics of Fluids B 5, 2744 (1993).
  * Demekhov  _et al._ (2006) A. G. Demekhov, V. Y. Trakhtengerts, M. J. Rycroft, and D. Nunn, Geomagnetism and Aeronomy 46, 711 (2006).
  * Demekhov  _et al._ (2009) A. G. Demekhov, V. Y. Trakhtengerts, M. Rycroft, and D. Nunn, Geomagnetism and Aeronomy 49, 24 (2009).
  * Omura  _et al._ (2007) Y. Omura, N. Furuya, and D. Summers,  J. Geophys. Res. 112, A06236 (2007).
  * Albert (2010) J. M. Albert,  J. Geophys. Res. 115, A00F05 (2010).
  * Tao  _et al._ (2013) X. Tao, J. Bortnik, J. M. Albert, R. M. Thorne, and W. Li, Journal of Atmospheric and Solar-Terrestrial Physics 99, 67 (2013).
  * Allanson  _et al._ (2020) O. Allanson, C. E. J. Watt, H. Ratcliffe, H. J. Allison, N. P. Meredith, S. N. Bentley, J. P. J. Ross, and S. A. Glauert, Journal of Geophysical Research (Space Physics) 125, e27949 (2020).
  * Mourenas  _et al._ (2018) D. Mourenas, X.-J. Zhang, A. V. Artemyev, V. Angelopoulos, R. M. Thorne, J. Bortnik, A. I. Neishtadt, and A. A. Vasiliev,  J. Geophys. Res. 123, 4979 (2018).
  * Cully  _et al._ (2008) C. M. Cully, J. W. Bonnell, and R. E. Ergun,  Geophys. Res. Lett. 35, L17S16 (2008).
  * Artemyev  _et al._ (2016) A. V. Artemyev, O. Agapitov, D. Mourenas, V. Krasnoselskikh, V. Shastun, and F. Mozer,  Space Sci. Rev. 200, 261 (2016).
  * Tyler  _et al._ (2019) E. Tyler, A. Breneman, C. Cattell, J. Wygant, S. Thaller, and D. Malaspina,  Geophys. Res. Lett. 46, 2328 (2019).
  * Benkadda  _et al._ (1996) S. Benkadda, A. Sen, and D. R. Shklyar, Chaos 6, 451 (1996).
  * Artemyev  _et al._ (2020a) A. V. Artemyev, A. I. Neishtadt, and A. A. Vasiliev, Physics of Plasmas 27, 042902 (2020a), arXiv:1911.11459 [physics.plasm-ph] .
  * Vainchtein  _et al._ (2018) D. Vainchtein, X. J. Zhang, A. V. Artemyev, D. Mourenas, V. Angelopoulos, and R. M. Thorne, Journal of Geophysical Research (Space Physics) 123, 8149 (2018), arXiv:1806.00066 [physics.plasm-ph] .
  * Li  _et al._ (2011) W. Li, J. Bortnik, R. M. Thorne, and V. Angelopoulos,  J. Geophys. Res. 116, A12205 (2011).
  * Zhang  _et al._ (2018) X. J. Zhang, R. Thorne, A. Artemyev, D. Mourenas, V. Angelopoulos, J. Bortnik, C. A. Kletzing, W. S. Kurth, and G. B. Hospodarsky, Journal of Geophysical Research (Space Physics) 123, 5379 (2018).
  * Sheeley  _et al._ (2001) B. W. Sheeley, M. B. Moldwin, H. K. Rassoul, and R. R. Anderson,  J. Geophys. Res. 106, 25631 (2001).
  * Albert  _et al._ (2013) J. M. Albert, X. Tao, and J. Bortnik, in  _Dynamics of the Earth’s Radiation Belts and Inner Magnetosphere_, American Geophysical Union, edited by D. Summers, I. U. Mann, D. N. Baker, and M. Schulz (2013).
  * Shklyar and Matsumoto (2009) D. R. Shklyar and H. Matsumoto, Surveys in Geophysics 30, 55 (2009).
  * Artemyev  _et al._ (2018) A. V. Artemyev, A. I. Neishtadt, D. L. Vainchtein, A. A. Vasiliev, I. Y. Vasko, and L. M. Zelenyi, Communications in Nonlinear Science and Numerical Simulations 65, 111 (2018).
  * Neishtadt (1975) A. Neishtadt, Journal of Applied Mathematics and Mechanics 39, 594 (1975).
  * Itin  _et al._ (2000) A. P. Itin, A. I. Neishtadt, and A. A. Vasiliev, Physica D: Nonlinear Phenomena 141, 281 (2000).
  * Artemyev  _et al._ (2019) A. V. Artemyev, A. I. Neishtadt, and A. A. Vasiliev, Physica D Nonlinear Phenomena 393, 1 (2019), arXiv:1809.03743 [physics.plasm-ph] .
  * Artemyev  _et al._ (2021a) A. V. Artemyev, A. I. Neishtadt, J. M. Albert, L. Gan, W. Li, and Q. Ma, Physics of Plasmas 28, 052902 (2021a).
  * Albert  _et al._ (2021) J. M. Albert, A. V. Artemyev, W. Li, L. Gan, and Q. Ma, Journal of Geophysical Research: Space Physics 126, e2021JA029216 (2021).
  * Artemyev  _et al._ (2015a) A. V. Artemyev, A. A. Vasiliev, D. Mourenas, A. I. Neishtadt, O. V. Agapitov, and V. Krasnoselskikh, Physics of Plasmas 22, 112903 (2015a).
  * Arnold  _et al._ (2006) V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt,  _Mathematical Aspects of Classical and Celestial Mechanics_ , 3rd ed., Dynamical Systems III. Encyclopedia of Mathematical Sciences (Springer-Verlag, New York, 2006).
  * Agapitov  _et al._ (2013) O. V. Agapitov, A. Artemyev, V. Krasnoselskikh, Y. V. Khotyaintsev, D. Mourenas, H. Breuillard, M. Balikhin, and G. Rolland,  J. Geophys. Res. 118, 3407 (2013).
  * Agapitov  _et al._ (2018) O. V. Agapitov, D. Mourenas, A. V. Artemyev, F. S. Mozer, G. Hospodarsky, J. Bonnell, and V. Krasnoselskikh, Journal of Geophysical Research (Space Physics) 123, 297 (2018).
  * Neishtadt and Vasiliev (2005) A. Neishtadt and A. Vasiliev, Nonlinearity 18, 1393 (2005).
  * Neishtadt (1999) A. I. Neishtadt, in Hamiltonian Systems with Three or More Degrees of Freedom, ed. Simo C., NATO ASI Series C. Dordrecht: Kluwer Acad. Publ. 533, 193 (1999).
  * Chirikov (1978) B. V. Chirikov, Soviet Journal of Plasma Physics 4, 521 (1978).
  * Artemyev  _et al._ (2020b) A. V. Artemyev, A. I. Neishtadt, and A. A. Vasiliev, Regular and Chaotic Dynamics 25, 2 (2020b).
  * Orlova and Shprits (2011) K. G. Orlova and Y. Y. Shprits, Physics of Plasmas 18, 092904 (2011).
  * Artemyev  _et al._ (2021b) A. V. Artemyev, A. I. Neishtadt, A. A. Vasiliev, X.-J. Zhang, D. Mourenas, and D. Vainchtein, Journal of Plasma Physics 87, 835870201 (2021b), arXiv:2011.00208 [physics.space-ph] .
  * Nunn  _et al._ (2021) D. Nunn, X. J. Zhang, D. Mourenas, and A. V. Artemyev,  Geophys. Res. Lett. 48, e92178 (2021).
  * Zhang  _et al._ (2021) X. J. Zhang, A. G. Demekhov, Y. Katoh, D. Nunn, C. Tao, D. Mourenas, Y. Omura, A. V. Artemyev, and V. Angelopoulos, Journal of Geophysical Research (Space Physics) 126, submitted (2021).
  * Tao  _et al._ (2012) X. Tao, J. Bortnik, R. M. Thorne, J. M. Albert, and W. Li,  Geophys. Res. Lett. 39, L06102 (2012).
  * Zhang  _et al._ (2020a) X. J. Zhang, O. Agapitov, A. V. Artemyev, D. Mourenas, V. Angelopoulos, W. S. Kurth, J. W. Bonnell, and G. B. Hospodarsky,  Geophys. Res. Lett. 47, e89807 (2020a).
  * Zhang  _et al._ (2020b) X. J. Zhang, D. Mourenas, A. V. Artemyev, V. Angelopoulos, W. S. Kurth, C. A. Kletzing, and G. B. Hospodarsky,  Geophys. Res. Lett. 47, e88853 (2020b).
  * Zhang  _et al._ (2019) X. J. Zhang, D. Mourenas, A. V. Artemyev, V. Angelopoulos, J. Bortnik, R. M. Thorne, W. S. Kurth, C. A. Kletzing, and G. B. Hospodarsky,  Geophys. Res. Lett. 46, 7182 (2019).
  * Omura  _et al._ (2008) Y. Omura, Y. Katoh, and D. Summers,  J. Geophys. Res. 113, A04223 (2008).
  * Artemyev  _et al._ (2013) A. V. Artemyev, D. Mourenas, O. V. Agapitov, and V. V. Krasnoselskikh, Annales Geophysicae 31, 599 (2013).
  * Mourenas  _et al._ (2017) D. Mourenas, Q. Ma, A. V. Artemyev, and W. Li,  Geophys. Res. Lett. 44, 3009 (2017).
  * Claudepierre  _et al._ (2020) S. G. Claudepierre, Q. Ma, J. Bortnik, T. P. O’Brien, J. F. Fennell, and J. B. Blake,  Geophys. Res. Lett. 47, e86056 (2020).
  * Allanson  _et al._ (2021) O. Allanson, C. E. J. Watt, H. J. Allison, and H. Ratcliffe, Journal of Geophysical Research (Space Physics) 126, e28793 (2021).
  * Hiraga and Omura (2020) R. Hiraga and Y. Omura, Earth, Planets, and Space 72, 21 (2020).
  * Artemyev  _et al._ (2015b) A. V. Artemyev, D. Mourenas, O. V. Agapitov, D. L. Vainchtein, F. S. Mozer, and V. V. Krasnoselskikh, Physics of Plasmas 22, 082901 (2015b).