## 7 Incorporation of nonlinear resonant interactions into radiation belt models

The currently existing radiation belt models can be formally separated into two classes: test particle simulations in the global electromagnetic fields provided by MHD models, and solvers of the Fokker-Planck diffusion equations [see reviews in Shprits, Elkington, Meredith and Subbotin, 2008; Shprits, Subbotin, Meredith and Elkington, 2008]. Let us discuss the possible approaches for an incorporation of nonlinear resonant interactions into such existing models.

We start with test-particle simulation models, which build on advances in high-resolution global MHD simulations [Elkington et al., 2004; Hudson et al., 2012, 1999]. These models reproduce well many important details of energetic electron injections into the inner magnetosphere by fast plasma flows [Sorathia et al., 2020], losses across the realistic dynamical magnetopause [Hudson et al., 2014; Sorathia et al., 2017], and radial transport by ultra-low-frequency (MHD) waves [Hudson et al., 2015; Sorathia et al., 2018]. A quite obvious deficiency of such models is the absence of any wave-particle interaction effects corresponding to electron scattering by whistler-mode (and any other non-MHD mode) waves. This issue may be resolved if test electron trajectories can be evaluated within the SDE approach, i.e., wave-particle interactions can be included as perturbations of electron dynamics for prescribed quasi-linear diffusion coefficients [Chan et al., 2023]. Although this combined approach (test particle trajectories in MHD field and SDE for wave-particle interaction effects) is still under construction, we can suggest that it can be further generalized. The inclusion of nonlinear resonant effects can be performed by changing the probability distribution of electron energy/pitch-angle changes from the Gaussian one (describing diffusion) to a more complicated form including phase bunching and phase trapping effects. Section 6.3 describe such a generalization, which formally consists in replacing the SDE approach by the mapping technique [see also Artemyev, Neishtadt and Angelopoulos, 2022].

Finally, let us discuss the inclusion of nonlinear effects into existing radiation belt models based on Fokker-Planck diffusion equations. The numerical scheme of these models cannot be easily generalized for solving equations with integral operators, and it looks more reasonable to include nonlinear effects in the form of a renormalization of diffusion rates. Sections 6.1.1, 6.1.2 show that over the long term, electron interactions with intense waves indeed result in an effective non-linear diffusion, with a nonlocal diffusion rate, $\tilde{D}_{NL}$, which represents the rate of electron mixing in the energy, pitch-angle domain of $S \neq 0$. Such effective diffusion rates can be calculated for realistic wave-packet distributions in two regimes: for realistically high wave amplitudes, and for very small wave amplitudes, $B_{w,small}$. The second regime will mimic quasi-linear diffusion and provide an estimate of the corresponding nonlocal mixing rate, $\tilde{D}_{QL}$. This rate can be further renormalized to the actual wave amplitude, $\tilde{D}^*_{QL} = \tilde{D}_{QL} \cdot (B_w/B_{w,small})^2$. The ratio of these rates can be considered as a normalized factor $R = \tilde{D}_{NL}/\tilde{D}^*_{QL}$ indicating the relative contribution of nonlinear effects. Figure 34(a) shows examples of $R$ dependencies on wave amplitude $B_w$, in the case of short ($\ell = 10$) and long ($\ell = 300$) wave packets. For sufficiently small $B_w$, there is no contribution from nonlinear effects, and for both $\ell$ values we have $R \approx 1$. As the wave amplitude $B_w$ increases, the contribution of nonlinear effects increases, providing a larger $R$. This $R$ in-
<figure><img src="image_34.png" /><figcaption>**Fig. 34** (a) *R* as a function of *B*<sub>w</sub> in the case of short (ℓ = 10) and long (ℓ = 300) wave-packets. (b) 〈*R*〉 coefficient for two typical energies and two datasets of whistler-mode waves-packets (red and blue) [see details in Artemyev, Mourenas, Zhang and Vainchtein, 2022].</figcaption></figure>

crease is stronger for longer wave-packets, due to a more important contribution of phase trapping into $\tilde{D}_{NL}$.

The coefficients $\tilde{D}_{NL}$ and $\tilde{D}^*_{QL}$ can be derived for different wave amplitudes and wave-packet sizes, and then averaged over the $P(\mathcal{B}_w, \beta)$ distribution. Such averaging will provide the weighted factor $\langle R \rangle$ that should depend on the geomagnetic activity level, since $P$ varies with geomagnetic activity (see Fig. 30). This factor can be determined for some energy, pitch-angle domains (roughly corresponding to $S \neq 0$ domains for different constants $h$ given by Eq. (4)). Figure 34(b) shows examples of $\langle R \rangle$ evaluated for several electron energies such that $h$ remains constant. The factor $\langle R \rangle$ weakly increases with geomagnetic activity, but it varies significantly with energy. Depending on the specific procedure of $\tilde{D}_{NL}$ averaging over the distribution of wave-packet sizes and amplitudes, $\langle R \rangle$ can vary from 1.5 up to 6 for 100 keV, whereas for 1 MeV $\langle R \rangle$ can vary between 1.1 and 1.5 [see details in Artemyev, Mourenas, Zhang and Vainchtein, 2022]. By definition, $\langle R \rangle$ is determined from the statistics of $I_x$ variations. Therefore, this factor is equally applicable to pitch-angle and energy diffusion, i.e., the derived $\langle R \rangle$ will increase the rate of electron acceleration and precipitation.

Figure 34 shows $\langle R \rangle$ for intermediate equatorial pitch-angles at ~100 keV and for small pitch-angles at ~1 MeV. Simulations for a multiple pitch-angle, energy domains provide a set of $\langle R \rangle$ values that can be fitted by a function of energy and pitch-angle. The coefficients of such fitting depend on properties of the $P(\mathcal{B}_w, \beta)$ distribution. In particular, the relative contribution of long wave-packets in realistic situations is largely determined by the definition of the wave-packet edge, i.e., by the chosen method of determination of the wave-packet length in statistical observations [Artemyev, Mourenas, Zhang and Vainchtein, 2022; Zhang, Mourenas, Artemyev, Angelopoulos, Bortnik, Thorne, Kurth, Kletzing and Hospodarsky, 2019; Zhang, Mourenas, Artemyev, Angelopoulos, Kurth, Kletzing and Hospodarsky, 2020; Zhang, Thorne, Artemyev, Mourenas, Angelopoulos, Bortnik, Kletzing, Kurth and Hospodarsky, 2018].
For instance, if the wave-packet length is calculated between two consecutive minima of $B_w$ below ~50 pT but above 10 pT [Artemyev, Mourenas, Zhang and Vainchtein, 2022; Zhang, Mourenas, Artemyev, Angelopoulos, Kurth, Kletzing and Hospodarsky, 2020], $P(B_w, \beta)$ contains a significant fraction of long wave-packets, and nonlinear effects (phase trapping) significantly amplify electron mixing in the phase space. Then, we can use

$$ \langle R \rangle \approx 1 + A \cdot \frac{1 + \sin \alpha_{eq}}{1 + (E/E_0)} $$

where $E_0 \approx 50$ keV and $A \approx 7$, so that $\langle R \rangle \approx 3$ for $\alpha_{eq} = 0$ and $\langle R \rangle \approx 5.6$ for $\alpha_{eq} = \pi/2$ for 100 keV, whereas $\langle R \rangle \approx 1.3$ for $\alpha_{eq} = 0$ and $\langle R \rangle \approx 1.7$ for $\alpha_{eq} = \pi/2$ for 1 MeV.

If the fraction of long wave-packets is reduced in $P(B_w, \beta)$, nonlinear effects are also reduced. For example, if we now assume that when the wave amplitude reaches one half of the packet peak amplitude, the wave phase fluctuations [Hanzelka et al., 2020; Santolík et al., 2014] destroy nonlinear resonances, then the wave-packet size distribution will contain only a small population of long packets [Artemyev, Mourenas, Zhang and Vainchtein, 2022; Zhang, Mourenas, Artemyev, Angelopoulos, Bortnik, Thorne, Kurth, Kletzing and Hospodarsky, 2019]. Under such an assumption, we get $A \approx 3.5$, so that $\langle R \rangle \approx 2.2$ for $\alpha_{eq} = 0$ and $\langle R \rangle \approx 3.3$ for $\alpha_{eq} = \pi/2$ for 100 keV, whereas $\langle R \rangle \approx 1.17$ for $\alpha_{eq} = 0$ and $\langle R \rangle \approx 1.3$ for $\alpha_{eq} = \pi/2$ for 1 MeV.

To demonstrate the applicability of the proposed approach, which consists in the introduction of a renormalization factor $\langle R \rangle$, we consider a case study during the 31 August 2019 - 01 September 2019 storm [see details of this event in Hudson et al., 2021; Nasi et al., 2022]. Figure 35 shows examples of electron phase space density (PSD) derived from measurements of Van Allen Probe A MagEIS (for $\mu = 1000$ MeV/G) and REPT (for $\mu = 4000$ MeV/G) instruments [Baker et al., 2013; Blake et al., 2013; Claudepierre et al., 2021]. The fast, localized (in $L^*$ evaluated with the TS05 [Tsyganenko and Sitnov, 2005] magnetic field model) increase in phase space density indicates an effective, local acceleration via wave-particle resonant interactions [Chen et al., 2007; Thorne et al., 2013]. As an example, we analyze the electron acceleration at $L^* = 4.2$, where the phase space density increased ~20-fold from 6:35 UT to 11:10 UT on 01 September (Figure 35b).

During this event, the characteristics of quasi-parallel whistler-modes wave and cold plasma are available from measurements of Van Allen Probe A ($MLT \in [9, 15]$) and ERG ($MLT \in [3, 9]$) [see details of missions in Mauk et al., 2013; Miyoshi et al., 2017]. We use background magnetic field measurements from the Magnetic Field Experiment [Matsuoka et al., 2018] at 8-second resolution, and high resolution electric and magnetic field wave spectra from the Onboard Frequency Analyzer [see Matsuda et al., 2018] onboard on the Plasma Wave Experiment [Kasahara, Kasaba, Kojima, Yagitani, Ishisaka, Kumamoto, Tsuchiya, Ozaki, Matsuda, Imachi, Miyoshi, Hikishima, Katoh, Ota, Shoji, Matsuoka and Shinohara, 2018]. We also utilize measurements from the Electric and Magnetic Field Instrument Suite and Integrated Science Waves instrument [Kletzing et al., 2013] providing all components of the wave electric and magnetic fields. To exclude measurements within the plasmasphere, we use the electron plasma density inferred from the upper hybrid resonance frequency line in the 10-400 kHz range for Van Allen Probes [Kurth et al., 2015] and in the 10 kHz - 10 MHz range for
<figure><img src="image_35.png" /><figcaption>**Fig. 35** (a, b) Evolution of the electron phase space density for two magnetic moment values with a fixed second adiabatic invariant, as derived from Van Allen Probe A measurements on 01 September 2019, and (c, d) the corresponding electron energy as a function of $L^*$. Colors represent the times when Van Allen Probe A was at $L^* = 4.2$. The phase space density is derived with the TS05 [Tsyganenko and Sitnov, 2005] magnetic field model.</figcaption></figure>

ERG [Kumamoto et al., 2018]. To obtain the full MLT coverage of waves and total electron density, we use measured wave intensities to scale an empirical wave model [Meredith et al., 2012], whereas ambient plasma measurements are used to scale global empirical density models [Denton et al., 2006; Sheeley et al., 2001]. Measurements by Van Allen Probe A and ERG are combined at $L^* \sim 4.2$, binned and interpolated in time to an 1-h resolution, and then we calculate diffusion rates at each 1-h time step (see Fig. 36).

We use classical quasi-linear diffusion coefficients, without or with the nonlinear renormalization factor $\langle R \rangle > 1$, to solve the bounce-averaged Fokker-Planck equation [Glauert and Horne, 2005; Lyons et al., 1972] using University of California, Los Angeles (UCLA) full diffusion code [see Ma et al., 2018, 2015; Ni, Thorne, Meredith, Shprits and Horne, 2011; Ni et al., 2008]. The wave normal angle distributions and latitude ranges at different MLT sectors are from [Thorne et al., 2013]. After the diffusion coefficients are calculated, we perform 2D Fokker Planck simulation [Ma et al., 2012] to examine the electron phase space density evolution due to chorus waves. Figure 37 compares the observed electron flux dynamics (dashed) and the simulation results (solid). The first simulation, which includes only pure quasi-linear diffusion (with $\langle R \rangle = 1$), clearly underestimates the electron acceleration rate above 1 MeV, where the simulation fails to reproduce the observed rapid ~5-hour electron flux increase (Figure 37a). The second simulation, with renormalized diffusion rates (with $\langle R \rangle > 1$), leads to a much better agreement between the modelled increase of relativistic electron fluxes and actual spacecraft observations (Figure 37b). This benchmark comparison therefore confirms the validity of our approach for incorporating the effects of nonlinear resonant interactions into classical Fokker-Planck simulations of energetic electron dynamics. It also underlines the importance of such generalized radiation belt models for an accurate description of electron dynamics in the presence of intense wave-packets [see Kondrashov et al., 2024, for another example of the application of the same approach].
<figure><img src="image_36.png" /><figcaption>Fig. 36 Time, MLT and bounce-averaged pitch-angle (left) and momentum (right) diffusion rates for electron acceleration at $L^* = 4.2$ observed from 6:35 UT to 11:10 UT on 01 September 2019 [see equations of quasi-linear diffusion rates in Glauert and Horne, 2005]; (a, b) quasi-linear diffusion rates $D_{QL}$ (top), and (c, d) rescaled rates $\langle R \rangle D_{QL}$ (bottom) based on the multiplication factor $\langle R \rangle > 1$ obtained for the wave model in Fig. 34.</figcaption></figure>

# 8 Discussion and Conclusions

## 8.1 Effects beyond the classical resonance model

For simplicity, we may define wave-particle resonant interactions as processes satisfying the cyclotron resonance condition with $O(1)$ accuracy in the $B_w/B_0$ parameter:

$$ \frac{p_{\parallel}}{m_e} k_{\parallel} - \gamma\omega + \Omega_0 = 0 \qquad (36) $$

Then, we can discuss several effects going beyond this definition, calling them nonresonant, although some of them can still be called nonlinear resonant effects. Many of these effects have been investigated in details for electromagnetic ion cyclotron waves [Bortnik et al., 2022; Grach et al., 2022; Grach and Demekhov, 2018a, 2020] and for magnetosonic waves [Bortnik and Thorne, 2010; Bortnik et al., 2015], whereas the importance of these effects for whistler-mode waves is not always obvious. However, to offer a comprehensive picture of nonlinear interactions, we discuss below all these effects, as applied to electron resonant interaction with whistler-mode waves.
<figure><img src="image_37.png" /><figcaption>**Fig. 37** Evolution of pitch-angle distributions of electron PSD at different energies at $L^* = 4.2$: solid lines show simulation results after 265 min in the simulation, dotted lines show the PSD observed at 06:35 UT (used as the initial condition), and dashed lines show the PSD observed at 11:10 UT (used as the final state to verify the model results). (a) Simulation results using quasi-linear diffusion rates, and (b) simulation results obtained with the rescaled (using a renormalization factor $\langle R \rangle > 1$) diffusion rates (see Fig. 36).</figcaption></figure>

*Anomalous phase trapping* One example of such effects has already been considered in Section 3, where we discussed wave-particle interactions for small pitch-angle electrons, when the term $\propto (B_w/B)/\sqrt{I_x}$ should be included into the equation of phase variation

$$ \frac{p_{\parallel}}{m_e} k_{\parallel} - \gamma\omega + \Omega_0 = \sqrt{\frac{2\Omega_0}{Im_e c^2}} \frac{eB_w}{k} \cos\zeta \quad (37) $$

Such a generalization results in a new type of resonant dynamics, where electrons appear to be phase trapped by default; the so-called *autoresonance* effect [Fajans and Friedland, 2001; Friedland, 2009; Neishtadt, 1975; Neishtadt et al., 2013; Sinclair, 1972]. This effect is often called *anomalous trapping* [Gan et al., 2020; Kitahara and Katoh, 2019] and consists in nearly 100% of trapping probability for small pitch-angle electrons. So high trapping probability significantly exceeds estimates given in Section 3 for $\Pi \sim \sqrt{B_w/B_0}$. This discrepancy is due to the fact that the scaling $\Pi \sim \sqrt{B_w/B_0}$ has been derived for the classical resonance conditions given by Eq. (36) and the corresponding specific hierarchy of timescales of $(\zeta, I)$ and $(s, p)$ variations. However, an accurate description of a system with a significant wave amplitude impact on the variation of the phase $\zeta$ requires using different approaches and the evaluation of a new Hamiltonian system (see details
in Appendix B and [Albert et al., 2021; Artemyev, Neishtadt, Albert, Gan, Li and Ma, 2021]).

**Positive phase bunching** Another effect that cannot be explained within the classical framework of resonant interactions is the positive phase bunching, which corresponds to the situation when resonant electrons gain energy during phase bunching, instead of losing energy. To explain the positive bunching effect, we shall consider the dimensionality of initial particle coordinates in velocity space. With the introduction of the magnetic moment as an adiabatic invariant, the 3D electron velocity vector can be represented by energy, local pitch-angle, and gyrophase, whereas the magnetic moment conservation allows projecting the local pitch-angle to the equatorial one. Both energy and equatorial pitch-angle are integrals of motion and will change only at resonance with the waves. The electron gyrophase is the fast oscillating variable, and introducing the magnetic moment excludes (averages out) this variable from the equations [Cary and Brizard, 2009]. However, the energy and pitch-angle changes at the resonance depend on the gyrophase, and this dependence is usually treated as a dependence on some random variable. Therefore, for fixed initial electron energy and equatorial pitch-angle, the particle ensemble for a single resonant interaction will be characterized by some probability distribution of energy, pitch-angle changes (see examples in Section 4.4 and [Allanson et al., 2019; Bortnik et al., 2008; Frantsuzov et al., 2023; Tao, Bortnik, Thorne, Albert and Li, 2012]). The conventional approach is to determine the main characteristics of such distributions through phase averaging. However, such a phase averaging approach can result in a loss of information concerning the particular resonant particles that do not follow one of the two main trends, i.e., which do not drift with the main phase bunched population or do not experience energy, pitch-angle jumps with the phase trapped population. Test particle simulations show a population of electrons experiencing a positive phase bunching: such resonant electrons do not follow either of the two main trends, i.e., they do not lose energy (as phase bunched electrons do) nor gain a large amount of energy (as phase trapped electrons do) [Albert et al., 2021; Gan et al., 2022]. Theoretical interpretation of this positive bunching effect has been proposed in [Albert et al., 2022], where the distribution of energy change as a function of the phase in the resonance (or initial gyrophase) has been considered. This distribution indeed includes a small population of bunched particles with the energy gain. Although a contribution of this population into the net electron energy change is not large [Vargas et al., 2023], this effect may be important for short-scale precipitations (a couple of resonant interactions).

**Resonance broadening** The finite resonance width effect (also known as resonance broadening) is most important for large-amplitude waves, such that the particle momentum resonance range is proportional to the square root of the wave amplitude [Karney, 1978; Palmadesso, 1972]. This effect effectively extends the pitch-angle, energy range of electrons affected by resonant interactions with intense waves. An important consequence of this effect for whistler-mode waves is that it provides a potential solution (at least in the presence of very intense waves) to the so-called 90° problem: the problem of absence of resonant scattering for very high equatorial pitch-angle electrons [see discussion in Allanson et al., 2022; Cai et al., 2020]. This is a problem of quasi-linear theory excluding scattering of $\alpha_{eq} \approx 90^\circ$.
particles, and thus leaving in the numerical simulations a phase space density peak at large pitch-angles. In test particle simulations such problem disappears [Camporeale, 2015; Camporeale and Zimbardo, 2015], likely due to the resonance broadening effect. The basic theoretical description of the resonance broadening has been proposed in [Karimabadi et al., 1992; Karimabadi and Menyuk, 1991], and then adopted for description of diffusion coefficients for whistler-mode waves [Cai et al., 2020] and electromagnetic ion cyclotron waves [Tonoian et al., 2022].

*Scattering by short wave-packets* For the sake of simplicity, the resonance condition given by Eq. (36) is used in many studies for the main frequency characterizing the highest-amplitude plane wave component of the observed wave-packets. However, electron resonant interactions can also be affected by the other frequencies (and corresponding wave numbers) of the wave-packet envelop, even if these waves are usually much less intense than the main wave. This effect is stronger when the wave-packet is shorter, because the temporal/spatial scales of short wave-packets can become comparable to the main wave characteristics. Importantly, the wave-packet envelop may be formed by whistler-mode waves: either by the beating of two waves with close frequencies [see Nunn et al., 2021; Zhang, Mourenas, Artemyev, Angelopoulos, Kurth, Kletzing and Hospodarsky, 2020] or by oscillations of currents of electrons nonlinearly interacting with waves around the wave generation region [see Zhang et al., 2021, and references therein]. However, the envelope may also be formed by external effects, e.g. by a modulation of the whistler-mode wave source region by ultra-low-frequency waves [e.g., Xia et al., 2016, 2020], or by rapid wave damping or diffraction. Therefore, the contribution of the wave-packet envelop may not always be taken into account only via a simple broadening of the wave spectrum, especially in the case when the main wave interacts with electrons nonlinearly.

The short size of wave-packets can contribute to a widening of the energy range of electrons affected by the waves compared with the case of the main wave at peak power alone, an effect which was mostly investigated for electromagnetic ion cyclotron waves [An, Artemyev, Angelopoulos, Zhang, Mourenas and Bortnik, 2022; Grach and Demekhov, 2023; Hanzelka et al., 2023], where this effect can account for the observed sub-relativistic electron precipitation by such waves, well below the minimum resonant energy with the main wave [see discussions in An et al., 2024; Angelopoulos et al., 2023; Shi et al., 2024]. So far, there is no investigations of short wave-packet effect for electron scattering by whistler-mode waves [although numerical simulations may include such effect; see Camporeale, 2015; Camporeale and Zimbardo, 2015].

*Fractional resonances* The resonant condition given by Eq. (36) is written for unperturbed electron trajectory. his equation does not include the wave amplitude. Observed waves, however, can be sufficiently intense for modifying electron trajectories and change the resonance condition, but still not strong enough to result in nonlinear electron dynamics for these new conditions. A good example of this type of effects is the fractional resonance [Lewak and Chen, 1969; Smirnov and Frank-Kamenestkiï, 1968], recently reevaluated in details for electromagnetic ion cyclotron waves [Hanzelka et al., 2023]. The basic idea of fractional resonance for whistler-mode waves have been discussed in Hamiltonian formalism in [Fu
et al., 2015]. In the equation of phase (49) we should take into account a perturbation of electron gyrorotation by wave field: $\psi = \psi_0 + Q \sin(\psi_0 + \phi_\parallel)$ where $Q \propto B_w/B_0$ and $\dot{\psi}_0 = \Omega_0/\gamma$. Thus, after couple Jacobi-Anger expansions instead of sum $\sim \sum_{n_r} \sin(\phi_\parallel - n_r \psi)$ we obtain

$$ \sim \sum_{n_r, n_l} h_0^{(n_r, n_l)} \sin(\phi_\parallel n_l - (n_r - n_l) \psi_0) + \sum_{n_r, n_l} h_{+1}^{(n_r, n_l)} \sin(\phi_\parallel n_l - (n_r - n_l + 1) \psi_0) \\ + \sum_{n_r, n_l} h_{+2}^{(n_r, n_l)} \sin(\phi_\parallel n_l - (n_r - n_l + 2) \psi_0) $$

with a following resonant condition:

$$ \dot{\phi}_{\parallel} - n_q \dot{\psi}_0 = k_{\parallel} \frac{p_{\parallel}}{\gamma m_e} - \omega - n_q \frac{\Omega_0}{\gamma} = 0 $$

where $n_q = -1+(n_r+1)/n_l$, $-1+(n_r+2)/n_l$, $-1+n_r/n_l$, and $N_l \neq 0$. Therefore, besides the classical resonances with $n_q = n_r$ (for $n_l = 1$), we will have fractional resonances, e.g., $n_q = -1/2$ (for $n_l = 2$ and $n_r = -1$). Such fractional resonances can be considered within the approach proposed in [Hanzelka et al., 2023; Terasawa and Matsukiyo, 2012]. Note that a fractional resonance requires both a high wave amplitude and an oblique propagation of the wave (see in Eq. (49) term with the gyrophase has an amplitude proportional to $k_\perp$). The main importance of this effect is associated with the scattering of electrons at sub-cyclotron resonant energies, i.e., the fractional resonance can scatter electrons with energies insufficiently large for classical cyclotron resonant scattering. Such sub-cyclotron scattering may be important for electron interaction with electromagnetic ion cyclotron waves [because the minimum cyclotron resonant energy for these waves is quite high, see Summers and Thorne, 2003], but it seem to be less important for whistler-mode waves.

**Transient scattering** The last effect that we would like to discuss in this section is the transient scattering, when the resonance condition in Eq. (36) can be satisfied only within a limited spatial domain, smaller than the wavelength [Bortnik and Thorne, 2010; Bortnik et al., 2015]. The energy variation in the resonance is a periodic function of phase gain (see Section 3 and Fig. 9(a)), and in the non-resonant situation this integral is equal to zero. However, if the wave field varies within one period (one wavelength), this integral becomes finite even outside the resonance. This is some analog of the short wave-packet effect, but this transient scattering has been mainly considered for the extremely low frequency branch of whistler-mode waves, fast magnetosonic waves [Bortnik and Thorne, 2010], which are indeed strongly confined near the equator [Mourenas et al., 2013]. For chosen and hiss modes (propagating quasi field-aligned with a frequency exceeding the lower-hybrid frequency) such a spatial localization might occasionally be possible, but only in the case of wave scattering and trapping by small scale density fluctuations that would lead to a significant variation of wave characteristics within a typical scale of one wavelength [Hanzelka and Santolík, 2022; Streltsov and Bengtson, 2020; Streltsov et al., 2012].
## 8.2 Stability of nonlinear resonant interactions

In this review, we have discussed the importance of wave coherence for nonlinear wave-particle resonant interaction: for less coherent waves with a more modulated wave field (shorter wave-packets) and phase jumps at the wave-packet edges, nonlinear phase trapping becomes quite ineffective. The asymptotic limit of such very ineffective phase trapping is the nonlinear diffusion described in Section 5.2. However, even in the case of long wave-packets with potentially strong electron acceleration due to phase trapping, some additional low-coherency waves or non-resonant waves may interrupt this trapping and thereby strongly reduce the acceleration efficiency [Brinca, 1978, 1980]. The same effect of trapping destruction may be provided by resonant sideband waves [Nunn, 1986]. Let us consider the corresponding stability of nonlinear resonant interactions for two most extreme variants: (1) the destruction of phase trapping by the nonresonant noise-like waves, (2) the reduction of the efficiency of nonlinear resonant interactions in case of a broadband wave spectrum.

**Trapping destruction** The effect of trapping destruction by non-resonant waves involves the idea that the phase trapped electron motion can be affected by high-frequency fluctuations [Artemyev et al., 2011]. To describe this effect, let us introduce the distribution function of trapped particles $f_{tr}$. Trapping is characterized by a periodical motion in the $(\zeta, P_{\zeta})$ plane (see Fig. 7(b)), and the area surrounded by the trapped electron trajectory, $I_{\zeta} = (2\pi)^{-1} \oint P_{\zeta}d\zeta$ is conserved during the slow evolution of the coefficients of the Hamiltonian (6). These coefficients depend on the coordinate and momentum $(s, p)$ and vary with time along the trapped trajectory. Therefore, the trapped particles can be characterized by $I_{\zeta}$: when $I_{\zeta} = S/2\pi$, particles either get trapped (if $S$ grows) or escape from the trapping (if $S$ decreases). However, non-resonant particle scattering by magnetic field fluctuations with a time-scale smaller than the trapping period $\sim 1/\Omega_{tr}$ can interrupt the conservation of $I_{\zeta}$. This results in an escape of the particle from the resonance much earlier than in the case of a conserved $I_{\zeta}$. This process of destruction of the $I_{\zeta}$ conservation is described by the diffusion equation written for $f_{tr}(I_{\zeta})$ [Artemyev, Mourenas, Agapitov, Vainchtein, Mozer and Krasnoselskikh, 2015]:

$$ \frac{\partial f_{tr}}{\partial t} = \frac{\partial}{\partial I_{\zeta}} \left( Q(t) I_{\zeta} \frac{\partial f_{tr}}{\partial I_{\zeta}} \right) $$

where $t$ is the time along the trapped trajectory, and $Q$, which depends on $(s, p)$, can be rewritten as a function of time along the trapped trajectory. The diffusion coefficient $Q$ is proportional to the power density of nonresonant fluctuation, $Q \propto B_{w,nonres}^2 \nu_{nonres}$ where $\nu_{nonres}$ is the typical frequency of such fluctuations.

The above diffusion equation shows how $f_{tr}(I_{\zeta})$ spreads over the $I_{\zeta}$ space and reaches the boundary $I_{\zeta} = S/2\pi$, where particles escape from trapping and join transient particles. The slow diffusive character of the destruction of $I_{\zeta}$ conservation suggests that this effect should become important only for long wave-packets, because in such a case the particles are supposed to spend a sufficient time within each phase trapping. Therefore, the most important application of this trapping limitation mechanism is the destruction of the *turning acceleration*, which is due to a particularly prolonged electron acceleration through protracted phase trapping [Omura et al., 2007; Summers and Omura, 2007].
**Nonlinear resonant interaction in a broadband wave spectrum** There are two competitive processes for systems with a broadband wave ensemble. Firth, a broadband wave ensemble includes multiple resonant waves, and each of these waves may perturb electron trajectories and destroy/interrupt resonant trapping. Second, even during a quarter of the bounce period (the time scale of a single resonant interaction with a monochromatic wave), electrons can experience multiple interactions with waves constituting the broadband ensemble, and thus the overall electron energy, pitch-angle changes can be large despite trapping destruction. This situation is roughly similar to multiple trapping by a wave-packet train [Hiraga and Omura, 2020], except that trapping duration is very short. The effects of multiple resonant interactions with broadband intense waves has been numerically investigated in [Gan et al., 2022], where the electron pitch-angle, energy change due to multiple resonances has been called *successive resonant acceleration*. It can be shown that the maximum pitch-angle, energy change via this *successive resonant acceleration* is bounded by values of pitch-angle, energy change due to phase trapping by a monochromatic wave along a magnetic field line [Gan et al., 2022]. Overall, these changes result in an electron diffusion rate smaller than quasi-linear rates for fully destroyed nonlinear resonances. Therefore, similarly to the scaling shown in Fig. 26, numerical simulations demonstrate that electron scattering by intense broadband waves is less effective than the quasi-linear diffusion estimates for the same high wave intensity [see also, e.g., Allanson et al., 2021, 2020; Tao, Bortnik, Albert, Liu and Thorne, 2011; Tao, Bortnik, Albert and Thorne, 2012]. However, nonlinear drifts may then need to be taken into account, similarly to the case of short intense wave-packets.

## 8.3 Feedback to waves

This review focuses on dynamics of energetic electrons, which fluxes are usually low enough to neglect the electron feedback to waves. However, the energy range of electrons responsible for whistler-mode wave generation can cover 10–100keV particles [Fu et al., 2014], formally associated with energetic component of electron spectrum. A realistic feedback of such particles to waves scattering and accelerating them is quite important, because this feedback can limit the efficiency of acceleration and determine the wave characteristics. The most straightforward approach for accounting of electron feedback to waves is the self-consistent simulations [see Katoh and Omura, 2004, 2007; Katoh et al., 2008; Tao et al., 2014] where electrons are traced as individual particles [see alternatives in Demekhov, 2011; Demekhov and Trakhtengerts, 2008; Nunn et al., 2005, 2021; Tao et al., 2021]. This approach is mostly applicable to a small-scale and short simulation setups focused on the wave generation problem [e.g., Demekhov, Taubenschuss and Santolík, 2017; Katoh and Omura, 2013, 2016], whereas the simulation of long-term dynamics of energetic electron populations requires too large numerical resources within such approach.

The approach alternative to the full numerical simulations consists in theoretical determination of impact of energetic electrons on wave growth/damping processes controlling wave amplitude variation [Omura, 2021; Omura et al., 2008, 2013; Shklyar, 2011, 2017; Tao et al., 2020]. This approach reproduces many important characteristics of whistler-mode waves [see details in Cully et al., 2011;
Mourenas et al., 2015; Tao, Li, Bortnik, Thorne and Angelopoulos, 2012] and principally can provide self-consistent variation of wave amplitude [see examples in Luzhkovskiy and Shklyar, 2023; Shklyar and Luzhkovskiy, 2023; Shklyar, 2021]. The main advantage of such theoretical consideration is that it can be incorporated into evaluation of main characteristics of particle acceleration models based on mapping technique or the Green function approach. In perspective, this merging of particle feedback and acceleration models should provide more realistic results for wave-particle nonlinear interactions in the radiation belts and should naturally limit nonlinear effects.

## 9 Conclusions

This review aimed to provide a detailed overview of the effects of nonlinear resonant interactions of radiation belt electrons with intense whistler-mode waves. We focused on the idea that such interactions should change the timescales of electron acceleration and losses in comparison with quasi-linear diffusion, and then investigated different approaches for modeling this change. The most basic approach described here, the generalized Fokker-Planck equation, accounted for all effects of bunching and phase trapping for the evolution of the electron distribution, and, most importantly, allowed us to show that the asymptotic solutions for nonlinear interactions and for diffusion are identical. Although this is a simplified approach and many realistic generalizations would require significant modifications, the existence of this asymptotic solution remains a key for further constructions of more advanced techniques for the simulation of nonlinear wave-particle interactions. One of the most promising technique, including many realistic properties of whistler-mode waves, is the mapping technique, which allowed us to consider the effects of a wave ensemble as well as the effects of short wave-packets. This technique, carefully applied to observed wave and electron distributions, may model electron flux dynamics with a sufficient accuracy to be verified by comparisons with spacecraft observations (see examples in Appendix E). Two alternative techniques, the *Green function approach* and the *probabilistic approach*, may include many realistic wave details [see Hiraga and Omura, 2020; Hsieh and Omura, 2017a, 2023; Hsieh et al., 2022; Kubota and Omura, 2018; Omura et al., 2015], but they still require verification against spacecraft observations. Besides the mapping technique, we discussed a possible approach for the incorporation of the effects of nonlinear resonant interactions in the existing global numerical Fokker-Planck models of the radiation belts. Although this last approach takes wave-particle interactions into account only in a very simplified, somewhat integral way, it is particularly promising, because it directly brings nonlinear physics into already well developed state of the art simulations.

**Acknowledgements** This review incorporates several results obtained by Alexander Lukin, Victor Frantsuzov, and David Tonoian [Frantsuzov et al., 2023; Lukin et al., 2024; Tonoian et al., 2023]. We are deeply grateful to these researchers for providing datasets and for helpful discussion of their results.

The review summarizes Authors' current understanding of nonlinear resonant interactions and theoretical approaches for evaluation and description of these interactions. An invaluable contribution to this understanding has been provided by multiple fruitful discussions and joint research projects with leading specialists of radiation belt physics and nonlinear dynamical
systems, far from complete list of whom includes Adnane Osmane, Andrei Demekhov, David Shklyar, David Nunn, Li Li, Longzhi Gan, Lunjin Chen, Jay Albert, Miroslav Hanzelka, Oliver Allanson, Veronica Grach, Xin An, Xin Tao, Xuzhi Zhou, Yikai Hsieh, Yoshiharu Omura.

The work of O.V.A. was supported by NASA contracts 80NSSC20K0218, 80NSSC22K0433, 80NSSC22K0522, 80NSSC20K0697, and 80NSSC21K1770. Q.M. would like to acknowledge the NASA grants 80NSSC20K0196 and 80NSSC24K0572 and NSF grant AGS-2225445. J.B. gratefully acknowledges support from NASA 80NSSC22K1637, NASA/HTMS 80NSSC20K1270 subcontract through Boston University, NSF/GEM award 2225613 subcontract to UCLA through the University of Iowa, and NSF/GEM award AGS-2025706. The work of V.V.K. was supported by CNES Solar Orbiter and Parker Solar Probe projects, and by NASA contract 80NSSC21K1770. The work of A.V.A., X.-J.Z., and D.L.V. was supported by NASA contracts 80NSSC24K0138, 80NSSC23K0100, 80NSSC23K0108, 80NSSC22K0522, 80NSSC23K1038, 80NSSC24K0561, 80NSSC19K0844.

## Conflict of Interest

The authors have no relevant financial or non-financial interests to disclose.

## Appendix A

This Appendix is devoted to derivation of the main Hamiltonian equations for wave-particle resonant interaction in two systems (two types of resonances): cyclotron resonance with field-aligned whistler-mode waves and Landau resonance with highly-oblique (quasi-electrostatic) whistler-mode waves. The general approach for the derivation of these Hamiltonians repeats results presented in [Albert, 1993; Artemyev, Neishtadt, Vasiliev and Mourenas, 2018; Vainchtein et al., 2018]: we start with writing wave field through the vector and scalar potentials, then we introduce the magnetic moment, and an adiabatic invariant resulted from averaging of cyclotron rotations, and finally we consider a general Hamiltonian system around the resonance. We provide and expand results on the separation of Hamiltonians describing slow electron bounce motion and fast resonant phase oscillations. The latter Hamiltonian is the classical Hamiltonian of the pendulum with torque [Arnold et al., 2006]. We derive coefficients of this Hamiltonian for two resonances, and then describe a very important property of this Hamiltonian leading to the equation of particle energy change due to the nonlinear phase bunching [Neishtadt, 1975].

### 9.1 Wave equations

The cold plasma dispersion relation [Stix, 1962] provides the relationship between electric and magnetic field components of whistler mode waves. Let $B_y = B_{w0}e^{i\phi} = B_{w0} \cos \phi$ where $\phi$ is the wave phase, then [Tao and Bortnik, 2010]

$$ 
\begin{aligned}
\frac{B_x}{B_w} &= -\epsilon_2 \frac{N^2 \sin^2 \theta - \epsilon_3}{\epsilon_3 (\epsilon_1 - N^2)} \sin \phi, & \frac{B_z}{B_w} &= \tan \theta \epsilon_2 \frac{N^2 \sin^2 \theta - \epsilon_3}{\epsilon_3 (\epsilon_1 - N^2)} \sin \phi, \\
\frac{E_x}{B_w} &= -\frac{N^2 \sin^2 \theta - \epsilon_3}{\epsilon_3 N \cos \theta} \cos \phi, & \frac{E_y}{B_w} &= \frac{\epsilon_2}{\epsilon_1 - N^2} \frac{N^2 \sin^2 \theta - \epsilon_3}{\epsilon_3 N \cos \theta} \sin \phi \\
\frac{E_z}{B_w} &= -\frac{N \sin \theta}{\epsilon_3} \cos \phi
\end{aligned}
\qquad (38) $$
Functions $\varepsilon_1$, $\varepsilon_2$, and $\varepsilon_3$ are components of the cold plasma dielectric tensor:

$$ \varepsilon_1 = 1 - \frac{\Omega_{pe}^2}{\omega^2 - \Omega_0^2}, \quad \varepsilon_2 = \frac{\Omega_0}{\omega} \frac{\Omega_{pe}^2}{\omega^2 - \Omega_0^2}, \quad \varepsilon_3 = 1 - \frac{\Omega_{pe}^2}{\omega^2} \quad (39) $$

where $\omega$ is the wave frequency, $k_\perp$ and $k_\parallel$ are two components of the wave vector lying in the $(x, z)$ plane ($k_\perp = k \sin\theta$, $k_\parallel = k \cos\theta$), $\Omega_0 > 0$ is the electron gyrofrequency in the background magnetic field directed along the $z$ axis, $\Omega_{pe}$ the plasma frequency. Wave frequency and wave vector satisfy the wave dispersion relation $N = N(\omega, \theta)$ where $N = kc/\omega$ is the wave refractive index [Stix, 1962]. Equations (38) lead to the following expressions for the total wave magnetic and electric fields:

$$ \begin{aligned} \bar{B}_w^2 &= \frac{1}{2\pi} \int_0^{2\pi} (B_x^2 + B_y^2 + B_z^2) d\phi = \frac{1}{2} \bar{B}_w^2 (C_2 C_1 + 1) \\ \bar{\mathcal{E}}_w^2 &= \frac{1}{2\pi} \int_0^{2\pi} (E_x^2 + E_y^2 + E_z^2) d\phi = \frac{1}{2} \frac{\bar{B}_w^2}{N^2} \left( C_1 (1 + C_2) + (N^2 \varepsilon_3^{-1} \sin\theta)^2 \right) \\ C_1 &= \left( \frac{N^2 \varepsilon_3^{-1} \sin^2\theta - 1}{\cos\theta} \right)^2, \quad C_2 = \left( \frac{\varepsilon_2}{\varepsilon_1 - N^2} \right)^2 \end{aligned} \quad (40) $$

For parallel propagating waves ($\theta = 0$), we have $C_1 = 1$, $N^2 = \varepsilon_1 - \varepsilon_2$, $C_2 = 1$, and $\bar{B}_w = \bar{B}_w/k$, $\bar{\mathcal{E}}_w = \bar{B}_w/N$.

We use Coulomb gauge to write field components through components of vector potential $\mathbf{A}$ and scalar potential $\varphi$:

$$ \begin{aligned} A_y &= \frac{\mathcal{B}_w}{k} \sqrt{C_2 C_1} \cos \phi, \quad A_x = \frac{\mathcal{B}_w}{k} \cos \theta \sin \phi, \\ A_z &= -\frac{\mathcal{B}_w}{k} \sin \theta \sin \phi, \quad \varphi = \frac{\mathcal{B}_w}{k N} (N^2 \varepsilon_3^{-1} - 1) \tan \theta \sin \phi \end{aligned} \quad (41) $$

We introduce $A_w = \bar{B}_w/k$, $C = \sqrt{C_1 C_2}$, $C_3 = (N^2 \varepsilon_3^{-1} - 1)/N \cos\theta$ and write $A_x = A_w \cos\theta \sin\phi$, $A_y = A_w C \cos\phi$, $A_z = -A_w \sin\theta \sin\phi$, $\varphi = A_w C_3 \sin\theta \sin\phi$ where $\bar{B}_w = \bar{B}_w/\sqrt{2/(C^2 + 1)}$. Highly oblique waves, with $\theta$ close to the resonance cone angle $\theta_r$ [Stix, 1962], are characterised by a significant variation of vector and scalar potential amplitudes, whereas these amplitudes are almost constant for parallel propagating waves.

## 9.2 Electron dynamics

We start with the Hamiltonian of an electron (of charge $-e$ and mass $m_e$) moving in the background magnetic field (described by the $y$-component of the vector potential $A_0 = -xB_0(z)$ [Bell, 1984]) and the wave electromagnetic field (given by Eqs.(41)):

$$ H = \sqrt{m_e^2 c^4 + c^2 \left( \mathbf{p} + \frac{e}{c} \mathbf{A}(x, z, t) + \frac{e}{c} A_0(x, z) \mathbf{e}_y \right)^2} - e\varphi(x, z, t) \quad (42) $$
where **p** is the particle generalized momentum. Note that (x, y, z) are Cartesian coordinates, i.e., we do not take into account the magnetic field line curvature of the dipole field [see a more general consideration in Shklyar and Matsumoto, 2009]. Although we use a $B_0(z)$ function simulating the distribution of dipole field strength along a magnetic field line, the same consideration is valid for arbitrary ambient magnetic fields with a sufficiently large spatial scale of inhomogeneity.

The Hamiltonian (42) does not depend on y and the conjugated momentum is constant, $p_y = \text{const}$. We consider particles motion around field-line with $x = 0$, and thus $p_y$ can be taken equal to zero. There are two important assumptions here. First, we assume that the spatial scale of $B_0(z)$ variation is much larger than the wavelength $\sim (\partial\phi/\partial x)^{-1}$, $(\partial\phi/\partial z)^{-1}$. Therefore, the period of particle oscillations in the plane $(z, p_z)$ is much longer than the wave period $\phi$ (wave frequency $-\partial\phi/\partial t$ is about electron gyrofrequency $eB_0/m_ec$ and much larger than the inverse of the time-scale of $(z, p_z)$ oscillations). Second, we assume a small wave amplitude $eA_0/m_ec^2 \ll 1$ (for typical conditions in the Earth radiation belts, this limit is satisfied since it means that the wave magnetic field amplitude must be much smaller than 10 nT) and expand the Hamiltonian (42) as:

$$H = H_0 + \frac{ce}{H_0} \mathbf{A} \left( \mathbf{p} + \frac{e}{c} A_0 \mathbf{e}_y \right) - e\varphi \quad (43)$$

$$H_0 = \sqrt{m_e^2 c^4 + c^2 p_z^2 + c^2 p_x^2 + c^2 (m_e x \Omega_0(z))^2}$$

where $\Omega_0 = eB_0(z)/m_ec$. Considering the unperturbed system $H = H_0$, we introduce the adiabatic invariant (magnetic moment):

$$I_x = \frac{1}{2\pi} \oint p_x dx = \frac{H_0^2 - m_e^2 c^4 - c^2 p_z^2}{2m_e c^2 \Omega_0} \quad (44)$$

The corresponding coordinates $(x, p_x)$ can be rewritten as: $p_x = -\sqrt{2I_x\Omega_0 m_e} \sin\psi$ and $x = \sqrt{2I_x/\Omega_0 m_e} \cos\psi$ where $I_x, \psi$ are conjugated variables (i.e., the transformation $(x, p_x)$ to $(\psi, I_x)$ is canonical [Landau and Lifshitz, 1988]). To introduce these variables we use the generating function $W_0 = \int p_x dx + \tilde{p}_z z$ where

$$\tilde{p}_z = p_z - \frac{\partial W_0}{\partial z} = p_z - I_x \frac{1}{2} \frac{\partial \ln \Omega_0}{\partial z} \sin 2\psi \quad (45)$$

and $z$ does not change. Therefore, the Hamiltonian (43) can be rewritten in new variables:

$$H = m_e c^2 \sqrt{1 + \frac{1}{m_e^2 c^2} \left( \tilde{p}_z + I_x \frac{1}{2} \frac{\partial \ln \Omega_0}{\partial z} \sin 2\psi \right)^2 + \frac{\rho^2 \Omega_0^2}{c^2}} + H_1, \quad (46)$$

$$H_1 = \frac{1}{\gamma_0} \left( \frac{\rho \Omega_0}{c} (eA_y \cos \psi - eA_x \sin \psi) + \frac{eA_z}{m_e c} \left( \tilde{p}_z + I_x \frac{1}{2} \frac{\partial \ln \Omega_0}{\partial z} \sin 2\psi \right) \right) - e\varphi$$

where $\rho = \sqrt{2I_x/m_e\Omega_0}$. There is a fast oscillating term $\sim \sin 2\psi$. This term results in variation of adiabatic invariant $I_x$ ($\dot{I}_x = -\partial H/\partial\psi$). The amplitude of this term is $\sim \partial \ln \Omega_0/\partial z$, i.e. $I_x$ is conserved with the accuracy $\sim \partial \ln \Omega_0/\partial z$. In absence of wave perturbations (i.e., in absence of external forces depending on wave phase $\phi$) we can always introduce the improved adiabatic invariant $J_x$, which is conserved with the accuracy $\sim (\partial \ln \Omega_0/\partial z)^2$ [Arnold et al., 2006]. In the system with wave
depending on phase $\phi$, the variation of $I_x$ is determined by resonance of wave phase $n_r\psi + \phi$, $n_r = 0, \pm 1, \pm 2, \dots$. For resonant particles with $d(n_r\psi + \phi)/dt \approx 0$ the phase $2\psi$ is a fast oscillating nonresonant phase, and, after expansion of $\gamma_{new}$ over $\partial \ln \Omega_0/\partial z$, the corresponding term $\sim \sin 2\psi$ can be omitted as a fast oscillating term with mean zero value. Therefore, we can rewrite Hamiltonian (46) as

$$ 
\begin{align} 
H &= m_e c^2 \gamma + H_1, \quad \gamma = \sqrt{1 + \frac{p_z^2}{m_e^2 c^2} + \frac{\rho^2 \Omega_0^2}{c^2}} \tag{47} \\ 
H_1 &= \frac{1}{\gamma} \left( \frac{\rho \Omega_0}{c} (e A_y \cos \psi - e A_x \sin \psi) + \frac{e A_z}{m_e c} p_z \right) - e \varphi \nonumber 
\end{align} 
$$

were $\tilde{p}_z \approx p_z$.

Note that the typical spatial scale of electron dynamics in the strong magnetic field, of the order of the electron gyroradius $\rho_e$, is much smaller than the magnetic field curvature radius, $R_c = R_E L/3$ for a dipole field. This allows to omit effects of electron curvature drifts, $\sim \rho_e/R_c$, and consider electron motion along a magnetic field line as a straight trajectory, what makes $z$ coordinate and conjugated momentum $p_z$ equal to the field-aligned coordinate $s$ and the corresponding momentum $p_\parallel$. Therefore, we now can replace $(z, p_z)$ by $(s, p_\parallel)$, underlying that we deal with the field-aligned coordinate and momentum of electrons. Systems with a finite $\rho_e/R_c$ effect include the so-called curvature scattering [e.g., Artemyev, Orlova, Mourenas, Agapitov and Krasnoselskikh, 2013; Birmingham, 1984; Brizard, 2023; Büchner and Zelenyi, 1989] and for such systems the $(z, p_z) \rightarrow (s, p_\parallel)$ transition requires application of a quite sophisticated procedure, either based on non-canonical [e.g., Cary and Brizard, 2009, and references therein] or canonical [e.g., Neishtadt and Artemyev, 2020, and references therein] guiding center theory. Some examples of such transformations for systems with nonlinear wave-particle resonant interactions can be found in [Artemyev, Neishtadt and Angelopoulos, 2022; Neishtadt et al., 2011; Vainchtein et al., 2009], whereas the application of such transformations for charged particle diffusion by waves was described in details by [Brizard and Chan, 2022].

We then substitute Eqs. (41) into the $H_1$ expression to get:

$$ H_1 = \frac{eA_w}{\gamma} \frac{\rho\Omega_0}{c} \left( C \cos \phi \cos \psi - \cos \theta \sin \phi \sin \psi \right) - eA_w \left( \frac{p_\parallel}{\gamma m_e c} + C_3 \right) \sin \theta \sin \phi \quad (48) $$

where wave phase $\phi$ can be written as

$$ \phi = \phi_{\parallel} + k_{\perp} \rho \cos \psi, \quad \phi_{\parallel} = \int^{s} k_{\parallel}(\tilde{s}) d\tilde{s} - \omega t \quad (49) $$

Using the Jacobi-Anger expansion, we rewrite Eq. (48) as

$$ 
\begin{align} 
H_1 &= eA_w \sum_{n_r=-\infty}^{\infty} h^{(n_r)}(I_x, p_\parallel, s) \sin\left(\phi_\parallel + n_r \frac{\pi}{2} - n_r \psi\right) \tag{50} \\ 
h^{(n_r)} &= \frac{\rho\Omega_0}{2c\gamma} \left( (C - \cos\theta) J_{n_r-1}(k_\perp\rho) - (C + \cos\theta) J_{n_r+1}(k_\perp\rho) \right. \nonumber \\ 
&\qquad \left. - \left( \frac{p_\parallel}{\gamma m_e c} + C_3 \right) \sin\theta J_{n_r}(k_x\rho) \right) \nonumber 
\end{align} 
$$
where $J_{n_r}$ is the Bessel function. We consider waves with insufficiently strong amplitudes to allow an overlap of resonances, i.e., such that each term in the sum of resonances can be considered independently [i.e., the resonance width $\sim \sqrt{|eA_w h^{(n_r)}/m_e|}$ is much smaller than the smallest distance between two resonances $\sim \Omega_0/k \cos\theta$, see the corresponding discussion in Shklyar, 1981]. For one particular $n_r$, Eq. (47) takes the form

$$H^{(n_r)} = m_e c^2 \gamma + eA_w h^{(n_r)} \sin \left( \phi_{\parallel} + n_r \frac{\pi}{2} - n_r \psi \right)$$

and we rewrite this Hamiltonian as

$$H^{(n_r)} = m_e c^2 \gamma + eA_w h^{(n_r)} \cos \left( \phi_{\parallel} + (n_r + 1)\frac{\pi}{2} - n_r \psi \right) \quad (51)$$

Numerically solving the Hamiltonian equations for $(s, p_\parallel)$, $(\psi, I_x)$ variables for the Hamiltonian (51) allows to describe the resonant interaction of charged particles with a whistler mode wave. In the next subsection, we investigate the Hamiltonian (51) to determine the main characteristics of this resonant interaction.

## 9.3 Wave-particle resonance

First, we would like to consider a conservative system, with conserved energy, whereas the Hamiltonian (51) depends on time through phase $\phi_\parallel$. To exclude this temporal dependence, we introduce new conjugated variables $(\zeta_{n_r}, I)$ and $(s, p)$ with $\zeta_{n_r} = \phi_\parallel - (n_r + 1)\psi + n_r\pi/2$. This variable change excludes phase $\psi$ from the Hamiltonian and, thus, the new momentum $\tilde{I}_x$ is constant – i.e., instead of two pairs of conjugated variables $(s, p_\parallel)$, $(\psi, I_x)$ we introduce $(s, p)$, $(\zeta_{n_r}, I)$. To this aim, we use the generating function $W = (\phi_\parallel + n_r\psi + n_r\pi/2)I + sp + \psi \tilde{I}_x$ corresponding to:

$$\zeta_{n_r} = \phi_{\parallel} + n_r \frac{\pi}{2} - n_r \psi, \quad p_{\parallel} = k_{\parallel} I + p, \quad I_x = -n_r I + \tilde{I}_x \quad (52)$$

Therefore, the new Hamiltonian $\mathcal{H} = H^{(n_r)} + \partial W / \partial t$ takes the form

$$\begin{align}
\mathcal{H} &= -\omega I + m_e c^2 \gamma + eA_w h^{(n_r)} \cos \zeta_n \tag{53} \\
\gamma &= \sqrt{1 + \frac{(k_\parallel I + p)^2}{m_e^2 c^2} + \frac{2\Omega_0}{m_e c^2} (\tilde{I}_x - n_r I)} \nonumber
\end{align}$$

where $h^{(n_r)} = h^{(n_r)}(\tilde{I}_x - n_r I, k_\parallel I + p, s)$. Hamiltonian (53) does not depend on $\psi$, and thus $\tilde{I}_x$ is the constant. For $n_r = 0$ we have $I_x = \tilde{I}_x = \text{const}$, i.e. $\tilde{I}_x$ is the initial magnetic moment that is conserved for Landau resonance. For $n_r \neq 0$ we have $I_x = -n_r I + \tilde{I}_x$ and we can set $\tilde{I}_x = 0$ by choosing the initial $I$ value (far from the resonance). For example, for the first cyclotron resonance with $n_r = -1$ we can set initial $I$ equals to the initial magnetic moment $I_x$, and thus $\tilde{I}_x = 0$.

The resonance condition $\dot{\zeta}_{n_r} = 0$ can be written through Hamiltonian equations:

$$\dot{\zeta}_{n_r} = \frac{\partial \mathcal{H}}{\partial I} = -\omega + \frac{k_{\parallel} (k_{\parallel} I + p) - m_e \Omega_0 n_r}{m_e \gamma} = 0 \quad (54)$$
where we omit perturbations ~ $eA_w \ll m_e c^2$. The solution of Eq. (54) provides the resonant value $I_R$ of momentum $I$:

$$ \frac{k_{\parallel} I_R}{m_e c} = -\frac{p}{m_e c} + \frac{n_r \Omega_0}{\omega N_{\parallel}} + \frac{1}{\sqrt{N_{\parallel}^2 - 1}} \sqrt{1 + \beta_{\perp}^2 - \frac{n_r^2 \Omega_0^2}{\omega^2 N_{\parallel}^2} + 2 \frac{n_r \Omega_0}{\omega N_{\parallel}} \frac{p}{m_e c}} \quad (55) $$

where $N_{\parallel} = k_{\parallel}c/\omega$, and $\beta_{\perp} = \sqrt{2\tilde{I}_x\Omega_0/m_e c^2}$. Particles with $I = I_R$ are in resonance with the wave. Therefore, to consider such resonant particles we expand Hamiltonian (41) around $I_R$:

$$ \begin{aligned} \mathcal{H} &= \Lambda + \frac{1}{2M} (I - I_R)^2 + eA_w h_R^{(n)} \cos \zeta_n \\ \Lambda &= -\omega I_R + m_e c^2 \gamma_R = \frac{m_e c^2}{N_{\parallel}^2} \left( \frac{N_{\parallel} p}{m_e c} - \frac{n_r \Omega_0}{\omega} + (N_{\parallel}^2 - 1) \gamma_R \right) \\ \gamma_R &= \frac{N_{\parallel}}{\sqrt{N_{\parallel}^2 - 1}} \sqrt{1 + \beta_{\perp}^2 - \frac{n_r^2 \Omega_0^2}{\omega^2 N_{\parallel}^2} + 2 \frac{n_r \Omega_0}{\omega N_{\parallel}} \frac{p}{m_e c}} \\ \frac{1}{M} &= \left. \frac{\partial^2 \gamma}{\partial I^2} \right|_{I=I_R} = (N_{\parallel}^2 - 1) \frac{\omega^2}{m_e c^2 \gamma_R} \end{aligned} \quad (56) $$

where $h_R^{(n_r)} = h^{(n_r)}(nI_R + \tilde{I}_x, k_{\parallel}I_R + p, s)$. We introduce variables $P_{\zeta} = I - I_R$, $\tilde{p}$, $\tilde{s}$ through the generating function $W = (I - I_R)\zeta_{n_r} + p\tilde{s}$ (we keep the old notation for $\zeta_n$, because this transformation does not change the phase):

$$ P_{\zeta} = I - I_R, \quad \tilde{p} = p - \frac{\partial I_R}{\partial s} \zeta_{n_r}, \quad \tilde{s} = s + \frac{\partial I_R}{\partial p} \zeta_{n_r} \quad (57) $$

The new Hamiltonian $\mathcal{H} = \Lambda(\tilde{s}, \tilde{p}) + m_e c^2 P_{\zeta}^2 / 2M(\tilde{s}, \tilde{p}) + eA_0 h_R^{(n_r)}(\tilde{s}, \tilde{p}) \sin \zeta_{n_r}$ contains the $\Lambda$ function depending on $(\tilde{s}, \tilde{p})$. Terms ~ $\partial I_R / \partial s$, $\partial I_R / \partial p$ in Eq. (57) are much smaller than $(s, p)$ terms due to the condition $\partial / \partial s \ll k_{\parallel}$. Therefore, we can expand the function $\Lambda$:

$$ \begin{aligned} \Lambda(s, p) &= \Lambda \left( s - \frac{\partial I_R}{\partial \tilde{p}} \zeta_{n_r}, p + \frac{\partial I_R}{\partial \tilde{s}} \zeta_{n_r} \right) \\ &= \Lambda(s, p) + \left( \frac{\partial I_R}{\partial \tilde{p}} \frac{\partial \Lambda}{\partial \tilde{s}} - \frac{\partial I_R}{\partial \tilde{s}} \frac{\partial \Lambda}{\partial \tilde{p}} \right) \zeta_{n_r} = \Lambda - A \zeta_{n_r} \end{aligned} \quad (58) $$

where $A = \{\Lambda, I_R\}_{\tilde{s}, \tilde{p}} \approx \{\Lambda, I_R\}_{s, p}$. Therefore, the new Hamiltonian $\mathcal{H}$ can be split into two Hamiltonians $\mathcal{H} = \Lambda + \mathcal{H}_{\zeta}$, where $\Lambda = \Lambda(s, p) \sim m_e c^2$ describes the slow evolution of $(s, p)$ variables, while $\mathcal{H}_{\zeta} \sim eA_0 \ll m_e c^2$ depends on these variables as on parameters and describes the fast variations of $(\zeta_{n_r}, P_{\zeta})$:

$$ \begin{aligned} \mathcal{H}_{\zeta} &= \frac{1}{2M} P_{\zeta}^2 + A \zeta_n + B \cos \zeta_{n_r}, \quad B = eA_w h_R^{(n_r)} \\ A &= -\frac{m_e c^2 \mathcal{D} N_{\parallel}^2}{N_{\parallel}^2 - 1} \frac{1}{\gamma_R} \left( \frac{(\gamma_R + n_r \Omega_0 / \omega)^2}{N_{\parallel}^2} \frac{\partial \ln N_{\parallel}}{\partial \ln \Omega_0} - \frac{n_r \Omega_0}{\omega N_{\parallel}} \frac{p}{m_e c} - \frac{1}{2} \beta_{\perp}^2 \right) \end{aligned} \quad (59) $$

where $\mathcal{D} = c(\partial \ln \Omega_0 / \partial s) / N_{\parallel} \omega \ll 1$ is a dimensionless factor determining the scale ratio of the inhomogeneity scale ~ $(\partial / \partial s)^{-1}$ and wavelength scale ~ $1/k_{\parallel}$.
## 9.4 Parallel propagating waves: cyclotron resonance

Let us consider nonlinear resonant electron interaction with parallel propagating whistler-mode waves (i.e., with $k_⊥ = 0$ and $C_1 = C_2 = C = 1$ in Eq. (50)). Therefore, $n_r = -1$ and the Hamiltonian $H_1$ takes the form

$$H_1 = eA_w \frac{\rho\Omega_0}{c\gamma} \cos(\phi_\parallel + \psi) \quad (60)$$

and there are no other resonances (the corresponding Bessel functions in Eq. (50) are equal to zero). We consider a magnetic field model with $\Omega_0(s/R)$, where $R$ is the spatial scale of the field gradient. The coefficient $A$ from Eq. (59) takes the form:

$$ \begin{aligned} A &= -\frac{m_e c^2 D}{\gamma_R N_\parallel^2 - 1} \left( \left( \frac{p_{\parallel,R}}{m_e c} \right)^2 \frac{\partial \ln N_\parallel}{\partial \ln \Omega_0} + \frac{\Omega_0}{\omega N_\parallel} \frac{p}{m_e c} - \frac{1}{2} \beta_\perp^2 \right) \\ &= -\frac{m_e c^2 D}{\gamma_R N_\parallel^2 - 1} \left( \left( \frac{p_{\parallel,R}}{m_e c} \right)^2 \frac{\partial \ln N_\parallel}{\partial \ln \Omega_0} + \frac{\Omega_0}{\omega N_\parallel} \frac{p_{\parallel,R}}{m_e c} - \frac{\Omega_0 (I_R + \tilde{I}_x)}{m_e c^2} \right) \quad (61) \\ &= -\frac{m_e c^2 D}{\gamma_R N_\parallel^2 - 1} \left( \left( \frac{p_{\parallel,R}}{m_e c} \right)^2 \frac{\partial \ln N_\parallel}{\partial \ln \Omega_0} + \frac{\Omega_0}{\omega N_\parallel} \frac{p_{\parallel,R}}{m_e c} - \frac{\Omega_0 I_{x,R}}{m_e c^2} \right) \end{aligned} $$

where $p_{\parallel,R}/m_e c = (\gamma_R - n_r \Omega_0/\omega)/N_\parallel = (\gamma_R \omega - \Omega_0)/k_\parallel c$ is the normalized resonant momentum, and $I_{x,R}$ is the magnetic moment in the resonance. For $I_{x,R}$ we can use $I_R$ expression, because constant $\tilde{I}_x$ can be set to zero for $n_r \neq 0$. We also can use definition of $\gamma$ in the resonance, $\gamma_R = \sqrt{1 + (p_{\parallel,R}/m_e c)^2 + 2I_{x,R}\Omega_0/m_e c^2}$, and rewrite Eq. (61) as

$$A = -\frac{m_e c^2 D}{\gamma_R N_\parallel^2 - 1} \left( \left( \frac{p_{\parallel,R}}{m_e c} \right)^2 \left( \frac{\partial \ln N_\parallel}{\partial \ln \Omega_0} + \frac{1}{2} \right) + \frac{\Omega_0}{\omega N_\parallel} \frac{p_{\parallel,R}}{m_e c} + \frac{1}{2} \left( 1 - \gamma_R^2 \right) \right)$$

Using $\zeta = \zeta_{-1}$, we can rewrite general equation (59) for the first cyclotron resonance as:

$$\mathcal{H}_{\zeta} = \frac{1}{2M} P_{\zeta}^2 + A\zeta + B \cos \zeta \quad (62)$$

$$\frac{1}{M} = \frac{N_{\parallel}^2 - 1}{\gamma_R} \frac{\omega^2}{m_e^2 c^4}, \quad B = eA_0 \sqrt{\frac{2I_R \Omega_0}{m_e c^2}}$$

Note that for the simplified dispersion relation valid for parallel propagating waves [Stix, 1962], we can write:

$$N_{\parallel} = \frac{\Omega_{pe}}{\omega} \left( \frac{\Omega_0}{\omega} - 1 \right)^{-1/2}, \quad \frac{\partial \ln N_{\parallel}}{\partial \ln \Omega_0} + \frac{1}{2} = -\frac{1}{2} \frac{\Omega_0}{\Omega_0 - \omega} + \frac{1}{2} = -\frac{1}{2} \frac{\omega}{\Omega_0 - \omega} \quad (63)$$

where $\Omega_{pe} = \text{const}$ is the plasma frequency.
## 9.5 Strongly Oblique waves: Landau resonance

In this section, we consider electrons in Landau resonance $n_r = 0$ with strongly oblique whistler mode waves propagating with $\theta$ within one degree from the resonant cone angle $\theta_r = \arccos(\omega/\Omega_0)$ [Stix, 1962]. Therefore, the Hamiltonian $H_1$ from Eq. (50) takes the form

$$ 
\begin{aligned}
H_1 &= +eA_0h^{(0)} \cos \phi_{\parallel} \\
h^{(0)} &= -\frac{\rho\Omega_0}{c\gamma}CJ_1(k_{\perp}\rho) - \left(\frac{p_{\parallel}}{\gamma m_e c} + C_3\right) \sin\theta J_0(k_{\perp}\rho)
\end{aligned} 
\quad (64) $$

where $p_{\parallel} = k_{\parallel}I_R + p_{\parallel,R} = \gamma_R/N_{\parallel}$.

Note that using a simplified dispersion relation valid for obliquely propagating waves in a high-density plasma [Stix, 1962], we write:

$$ N_{\parallel} = \frac{\Omega_{pe} \cos \theta}{\sqrt{\Omega_0 \omega \cos \theta - \omega^2}}, \quad \frac{\partial \ln N_{\parallel}}{\partial \ln \Omega_0} = -\frac{1}{2} \frac{\Omega_0 \cos \theta}{\Omega_0 \cos \theta - \omega} \quad (65) $$

where $\Omega_{pe} = \text{const}$ is the plasma frequency.

## 9.6 Properties of pendulum equation

Hamiltonian (53) is conservative, i.e. $H = \text{const}$. Therefore, the energy change $m_e c^2 \Delta\gamma$ due to resonant wave-particle interaction is approximately equal to $\omega\Delta I$. For resonant scattering (crossing of the resonance), $\Delta I$ can be written as

$$ 
\begin{aligned}
\Delta I &= \int_{-\infty}^{+\infty} \dot{I} dt = \int_{-\infty}^{+\infty} \frac{\partial H}{\partial \zeta} dt = eA_w h^{(n_r)} \int_{-\infty}^{+\infty} \sin \zeta dt = 2eA_w h^{(n_r)} \int_{-\infty}^{\zeta_R} \frac{\sin \zeta d\zeta}{\zeta} \\
&= 2eA_w h^{(n_r)} M \int_{-\infty}^{\zeta_R} \frac{\sin \zeta d\zeta}{P_{\zeta}} = eA_w h^{(n_r)} \sqrt{\frac{2M}{A}} \int_{-\infty}^{\zeta_R} \frac{\sin \zeta d\zeta}{\sqrt{h_{\zeta} - \zeta - a \cos \zeta}}
\end{aligned} 
$$

where $a = |B/A|$, $h_{\zeta} = H_{\zeta}/A$, $\zeta_R$ is $\zeta$ value at the resonance. This equation gives $\Delta I$ as a function of the energy $h_{\zeta}$, but the precise $h_{\zeta}$ value depends on initial electron gyrophase and can be considered as a random number. Therefore, we should consider $\langle \Delta I \rangle_{h_{\zeta}}$. An important property of the $H_{\zeta}$ system is that the integral

$$ F(a, h_{\zeta}) = \int_{-\infty}^{\zeta_R} \frac{\sin \zeta d\zeta}{\sqrt{h_{\zeta} - \zeta - a \cos \zeta}} $$

averaged over $h_{\zeta}$ gives [Neishtadt, 1975]

$$ \langle F \rangle_{h_{\zeta}} = -\frac{1}{\pi} \int_{\zeta_{-}}^{\zeta_{+}} \sqrt{\zeta_{+} - \zeta + a(\cos \zeta_{+} - \cos \zeta)} d\zeta $$
Let us re-derive this equation. We introduce

$$ \tilde{h}_{\zeta} = \begin{cases} 0, & \zeta < \zeta_{-} \\ \zeta + a \cos \zeta, & \zeta > \zeta_{-} \end{cases} $$

and $h_{-} = \zeta_{-} + a \cos \zeta_{-}$, where $\zeta_{-}$ is the solution of $1 - a \sin \zeta = 0$. Then we rewrite the integral from

$$ \langle F \rangle = \frac{1}{2\pi} \int_{0}^{2\pi} F(a, h_{\zeta}) dh_{\zeta} = \frac{1}{2\pi} \int_{-\infty}^{\zeta_{R}} d\zeta \int_{0}^{2\pi} \frac{a \sin \zeta dh_{\zeta}}{\sqrt{h_{\zeta} - \zeta - a \cos \zeta}} $$

as a sum of two terms

$$ \int_{-\infty}^{\zeta_R} \int_{0}^{2\pi} \frac{a \sin \zeta dh_{\zeta} d\zeta}{\sqrt{h_{\zeta} - \zeta - a \cos \zeta}} = \int_{-\infty}^{\zeta_{-}+2\pi} \int_{\tilde{h}_{\zeta}+h_{-}}^{2\pi+h_{-}} \frac{a \sin \zeta dh_{\zeta} d\zeta}{\sqrt{h_{\zeta} - \zeta - a \cos \zeta}} - \int_{\zeta_{-}+h_{-}}^{\zeta_{+}} \int_{\tilde{h}_{\zeta}+h_{-}}^{h_{-}} \frac{a \sin \zeta dh_{\zeta} d\zeta}{\sqrt{h_{\zeta} - \zeta - a \cos \zeta}} $$

where $\zeta_{+}$ is the solution of $\zeta_{-} + a \cos \zeta_{0} - \zeta - a \cos \zeta = 0$. The first integral can be written as

$$ \begin{aligned} & \int_{-\infty}^{\zeta_{-}+2\pi} \int_{\tilde{h}_{\zeta}+h_{-}}^{2\pi+h_{-}} g^{-1}(h_{\zeta}) a \sin \zeta d\zeta dh_{\zeta} = 2 \int_{-\infty}^{\zeta_{-}+2\pi} a \sin \zeta (g(2\pi+h_{-}) - g(\tilde{h}_{\zeta}+h_{-})) d\zeta \\ &= 2 \int_{-\infty}^{\zeta_{-}} a \sin \zeta (g(2\pi+h_{-}) - g(h_{-})) d\zeta + 2 \int_{-\infty}^{\zeta_{-}+2\pi} a \sin \zeta g(2\pi+h_{-}) d\zeta \\ &= 2 \lim_{N \to \infty} \int_{\zeta_{-}-2\pi N}^{\zeta_{-}} a \sin \zeta (g(2\pi+h_{-}) - g(h_{-})) d\zeta + 2 \int_{-\infty}^{\zeta_{-}+2\pi} a \sin \zeta g(2\pi+h_{-}) d\zeta \\ &= -2 \int_{\zeta_{-}-2\pi}^{\zeta_{-}} a \sin \zeta g(2\pi+h_{-}) d\zeta + 2 \int_{-\infty}^{\zeta_{-}+2\pi} a \sin \zeta g(2\pi+h_{-}) d\zeta = 0 \end{aligned} $$

where $g(h_{\zeta}) = \sqrt{h_{\zeta} - \zeta - a \cos \zeta}$. The second integral has the form

$$ \begin{aligned} & \int_{\zeta_{-}+h_{-}}^{\zeta_{+}} \int_{\tilde{h}_{\zeta}+h_{-}}^{2\pi+h_{-}} g^{-1}(h_{\zeta}) a \sin \zeta d\zeta dh_{\zeta} = 2 \int_{\zeta_{-}}^{\zeta_{+}} g(h_{-}) a \sin \zeta d\zeta \\ &= -2 \int_{\zeta_{-}}^{\zeta_{+}} g(h_{-}) (1 - a \sin \zeta) d\zeta + 2 \int_{\zeta_{-}}^{\zeta_{+}} g(h_{-}) d\zeta = 2 \int_{\zeta_{-}}^{\zeta_{+}} g(h_{-}) d\zeta \end{aligned} $$

Therefore, for $\langle F \rangle$ we get

$$ \langle F \rangle_{h_{\zeta}} = -\frac{1}{\pi} \int_{\zeta_{-}}^{\zeta_{+}} \sqrt{h_{-} - \zeta - a \cos \zeta} d\zeta \quad (66) $$
## Appendix B

This Appendix is devoted to the problem of whistler-mode wave resonance with small pitch-angle electrons [Inan et al., 1978; Lundin and Shkliar, 1977]. Non-linear resonance in such a case can be very different from the classical version: the primary effect of small pitch-angles is that the probability of phase trapping increases significantly and can reach 100% [Kitahara and Katoh, 2019]. This is so-called auto-resonance [Fajans and Frièdland, 2001; Friedland, 2009; Neishtadt, 1975; Neishtadt et al., 2013; Sinclair, 1972], and we provide a basic theoretical description of this phenomenon for electrons resonating with field-aligned whistler-mode waves.

In Hamiltonian (3) the equation for phase $ζ$ can be written as

$$ \dot{\zeta} = \frac{\partial \mathcal{H}_I}{\partial I} = -\omega + \frac{k_{\parallel}(p + k_{\parallel}I)}{\gamma m_e} + \frac{\Omega_0}{\gamma} - \frac{\partial eU_w}{\partial I} \cos \zeta $$

where $I$ is equivalent to the magnetic moment $I_x$, and thus small pitch-angle electrons have small $I$. For field-aligned waves $U_w \propto \sqrt{I}$ and $\partial U_w / \partial I = U_w / 2I$. For sufficiently small $I$ the term

$$ \frac{eU_w}{2I} \cos \zeta \propto I^{-1/2} $$

will be larger than the main resonance term

$$ \frac{k_{\parallel}(p + k_{\parallel}I)}{m_e \gamma} - \omega + \frac{\Omega_0}{\gamma} $$

and this should change the applicability of expansion around resonance (small $I - I_R$) that we used to derive Hamiltonian (5). Indeed, numerical simulations show that resonant interactions in small-$I$ systems differs significantly from the theoretical predictions based on analysis of these Hamiltonians [Grach and Demekhov, 2018b, 2020; Kitahara and Katoh, 2019]. In this Appendix we describe an approach for analysis of small-$I$ systems (see details in [Albert et al., 2022, 2021; Artemyev, Neishtadt, Albert, Gan, Li and Ma, 2021]).

Let us start with Hamiltonian (3) for $n_r = -1$ (when constant $\tilde{I}_x = 0$ and $U_w$ is given by Eq. (60)). We expand this Hamiltonian around small $I$:

$$ \begin{aligned} \mathcal{H}_I &= m_e c^2 \gamma_0 + I \left( \frac{k_{\parallel} p}{m_e \gamma_0} + \frac{\Omega_0}{\gamma_0} - \omega \right) + \frac{1}{2M_0} I^2 - \sqrt{\frac{2I\Omega_0}{m_e c^2}} \frac{eA_w}{\gamma_0} \cos \zeta \\ &= \Lambda + \frac{1}{2M_0} K^2 (I - I_R)^2 - \sqrt{\frac{2I\Omega_0}{m_e c^2}} \frac{eA_w}{\gamma_0} \cos \zeta \end{aligned} \quad (67) $$

where

$$ \begin{aligned} \frac{1}{M_0} &= \frac{k_{\parallel}^2}{m_e \gamma_0^3} \left( 1 - 2 \frac{p}{m_e c^2} \frac{\Omega_0}{k_{\parallel}} - \left( \frac{\Omega_0}{k_{\parallel} c^2} \right)^2 \right), \quad \gamma_0 = \sqrt{1 + \left( \frac{p}{m_e c} \right)^2} \\ \Lambda &= m_e c^2 \gamma_0 - \frac{1}{2M_0} I_R^2, \quad I_R = M_0 \left( \omega - \frac{\Omega_0}{\gamma_0} - \frac{k_{\parallel} p}{m_e \gamma_0} \right) \end{aligned} $$
Note that $I_R$ is the resonant momentum, because $\partial\mathcal{H}_I/\partial I \propto I - I_R$ and the resonant condition is $\partial\mathcal{H}_I/\partial I = 0$.

Hamiltonian equations for $I$ and $\zeta$ can be written as

$$ \dot{I} = -\frac{\partial \mathcal{H}_I}{\partial \zeta} = -\sqrt{\frac{2I\Omega_0}{m_e c^2}} \frac{eA_w}{\gamma_0} \sin \zeta \quad (68) $$

$$ \dot{\zeta} = \frac{\partial \mathcal{H}_I}{\partial I} = \frac{1}{2M_0}(I - I_R) - \frac{1}{2I}\sqrt{\frac{2I\Omega_0}{m_e c^2}} \frac{eA_w}{\gamma_0} \cos \zeta $$

and $\zeta$ changes much faster than $(s, p, I)$. Thus, for frozen $(s, p)$ we can rewrite Eqs. (68) as

$$ \frac{\partial Y}{\partial \tau} = -\sqrt{2Y} \sin \zeta, \quad \frac{\partial \zeta}{\partial \tau} = Y - Y_R - \frac{u}{\sqrt{2Y}} \cos \zeta \quad (69) $$

where

$$ \tau = \Omega_0 t \cdot \left( \frac{eA_w}{m_e c^2} \right)^{2/3} \left( \frac{Kc}{\Omega_0} \right)^2, \quad Y = \frac{\Omega_0 I}{m_e c^2} \left( \frac{eA_w}{m_e c^2} \right)^{2/3}, \quad u = \frac{1}{\gamma_0} \left( \frac{\Omega_0}{Kc} \right)^2 $$

and we introduce $K = \sqrt{m_e/M_0}$, which has the dimensionality of a wave number.

Equations (68) does not contain a small parameter $\propto eA_w/m_e c^2$, i.e. if $\Omega_0 I/m_e c^2$ is of the order of $(eA_w/m_e c^2)$, time-scales of $\zeta$ and $I$ variations become comparable. This makes this system different from the one described by Hamiltonian (6), where $(\zeta, P_\zeta)$ are fast variables, but $(s, p, I)$ are slow variables.

Let us write the Hamiltonian for $(\zeta, Y)$, i.e., such that Eqs. (69) will be Hamilton's equations:

$$ H_Y = \frac{1}{2} (Y - Y_R)^2 - \sqrt{2Y} u \cos \zeta \quad (70) $$

To describe dynamics for Hamiltonian (71) we will introduce new variables $(q, P)$, such that

$$ q = -\sqrt{2Y} \cos \zeta, \quad P = \sqrt{2Y} \sin \zeta, \quad \frac{\partial q}{\partial \zeta} \frac{\partial P}{\partial Y} - \frac{\partial q}{\partial Y} \frac{\partial P}{\partial \zeta} = 1 $$

The latter equation shows that $(q, P)$ are new canonical variables, whereas a new Hamiltonian is

$$ \mathcal{L} = \frac{1}{2} \left( \frac{P^2 + q^2}{2} - Y_R \right)^2 + uq \quad (71) $$

Hamiltonian (71) describes second type resonant systems and has been investigated in [Henrard and Lemaitre, 1983; Neishtadt, 1975; Sinclair, 1972].

Let us consider a profile of Hamiltonian (71) on the axis $P = 0$: $U = \mathcal{L}_{P=0} = (1/2)(q^2/2 - Y_R)^2 + uq$. The equation determining extrema of function $U(q)$ is $dU/dq = (1/2)q^3 - Y_Rq + u = 0$, which can be rewritten as $(1/2)\tilde{q}^3 - (3/2)\tilde{q}(Y_R/Y_R^*) + 1 = 0$ with $\tilde{q} = q/u^{1/3}$ and $Y_R^* = (3/2)u^{2/3}$. Figure 38(a) shows that for $Y_R < Y_R^*$ there is only one extremum and for $Y_R > Y_R^*$ there are three extrema of $U(q)$. Therefore, the phase portraits of Hamiltonian (71) have two types shown in Fig. 38(b): for $Y_R < Y_R^*$ there is only one O-point in the phase plane and phase trajectories rotate around this point, whereas for $Y_R > Y_R^*$ there are two O-points and
<figure><img src="image_38.png" /><figcaption>**Fig. 38** (a) Profiles of $dU/dq = \tilde{q}^3/2 - (3/2)\tilde{q}(Y_R/Y_R^*) + 1$ for different $Y_R$; $Y_R^* = (3/2)u^{2/3}$ and $\tilde{q} = q/u^{1/3}$. (b) Phase portraits of Hamiltonian (71) with $u = 2$ for $Y_R > Y_R^*$. Bold red curve shows the separatrices $\ell_{1,2}$.</figcaption></figure>

one X-point (saddle point), and two separatrices $\ell_{1,2}$ demarcate the phase portrait onto three domains. Ratio $Y_R/Y_R^*$ can be written as

$$ \frac{Y_R}{Y_R^*} = \frac{2 \kappa^{2/3} k^2 c^2}{3 \gamma_0^{4/3} \Omega_0^2} \frac{I_R \Omega_0}{m_e c^2 (B_w / B)^{2/3}} $$

where

$$ \kappa = 1 - \frac{\Omega_0^2}{k^2 c^2} - 2 \frac{\Omega_0}{k c} \frac{p}{m_e c} $$

and at the resonance $I_R = I_x$, $\kappa = \gamma_0^2 (1 - (\omega/kc)^2)$.

For constant $u$, the $Y_R$ system with Hamiltonian (71) is integrable, whereas for slowly changing $u$, $Y_R$ (slowly changing $(s, p)$) we can introduce an adiabatic invariant $I_P = (2\pi)^{-1} \oint Pdq$ (because all phase trajectories in the portrait shown in Fig. 38(b) are closed). In absence of separatrix (for $Y_R < Y_R^*$), $I_P$ would be conserved with the exponential accuracy $\sim \exp(-(B_w/B_0)^{-1/3})$ where $(B_w/B_0)^{1/3} \ll 1$ separates time-scales of $u$, $Y_R$ change $(s, p)$ change) and $P, q$ change. For conserved $I_P$ the system becomes integrable and $Y$ well before the resonance is equal to $Y$ well after the resonance. Note that $I_P = (2\pi)^{-1} \oint Pdq = 2Y(2\pi)^{-1} \oint \cos^2 \zeta d\zeta = Y$ far from $\ell_{1,2}$ or in absence of $\ell_{1,2}$.

Figure 12 shows the relation between phase portraits in Fig. 38(b) and Fig. 7(b). Hamiltonian $H_\zeta$ from Eq. (6) describes the charged particle motion from the domain $G_1$ to the domain $G_2$, and this motion may include a temporary particle trapping into the domain $G_0$. The trapping probability is typically quite small and most of particles will move from $G_1$ to $G_2$ via a single resonance $\zeta \propto I - I_R = 0$ crossing. However, if initial particle $I_x(I)$ is small, we should consider Hamiltonian $L$ to describe particle motion. Small $I$ means small $Y$, and thus small area $I_P$ in the phase portrait of Hamiltonian $L$. Thus, particles will small $I$ (small $I_P$) appear within the $G_0$ domain in the phase portrait, whereas particles with large $I_P$ will
move along the orbits outside the $ℓ_1$ separatrix (see Fig. 38(b)), within the $G_1$ domain. This predicts that most of particles with small $I$ experience the phase trapping.

Particles having $I$ sufficiently large to start motion in $G_1$ domain, but sufficiently small to be described by $\mathcal{L}$ Hamiltonian, can experience the phase trapping (transition from $G_1$ to $G_0$) or phase bunching (scattering; transition from $G_1$ to $G_2$). The latter transition happens when crossing the separatrix $ℓ_1$, which will result in $I_P$ jump, $\Delta I_P = \Delta S/2\pi$, where $\Delta S$ is the difference of areas of $G_1$ and $G_2$ domains. This $I_P$ jump is directly related to the jump of $I$ (and $I_x$)

$$ \Delta I_x = \Delta I = \frac{mc^2 \Delta S}{2\pi \Omega_0} \left( \frac{eA_w}{m_e c^2} \right)^{2/3} \qquad (72) $$

This scaling $\Delta I_x \propto A_w^{2/3} \propto B_w^{2/3}$ differs from the phase bunching scaling $\Delta I_x \propto B_w^{1/2}$ derived in Section 3. Therefore, two main features of small $I$ (small pitch-angle) systems are: (a) a large probability of trapping, when electrons are already within the trapping region when the separatrix appears, (b) a different scaling of momentum (energy) jump due to phase bunching.

## Appendix C

In this Appendix we describe the limit of a small area $S$ system, i.e., such systems where parameter $a = |B/A|$ from Hamiltonian (6) is about one. To make nonlinear resonant interaction possible, we assume that $|A|$ becomes smaller than $|B|$, but to keep $S$ small the inequality $a > 1$ holds only within a short range of magnetic latitudes, i.e., over a short range of resonant energies, $\gamma_R$. Then, the $\zeta$ range of the area filled with closed trajectories (where $\zeta \in [\zeta_-, \zeta_+]$ in Fig. 7(b), $\zeta_-$ is the saddle point, and $\zeta_+$ is the coordinate of separatrix crossing $P_\zeta = 0$, right from the $\zeta_-$) is small and Hamiltonian (6) can be expanded as:

$$ \mathcal{H}_{\zeta} = \frac{1}{2M} P_{\zeta}^2 + B \cdot \left( \frac{1}{a}\zeta + \cos\zeta \right) \approx \frac{1}{2M} P_{\zeta}^2 + B \cdot \left( \left(\frac{1}{a} - 1\right)\zeta + \frac{1}{6}\zeta^3 \right) \quad (73) $$

where $1/a$ slowly changes along the trajectory from $> 1$ (no closed trajectories in the phase portrait) to $\min 1/a < 1$ (maximum area filled by closed trajectories), and then to $> 1$ again. This expansion is around $\zeta = \pi/2$, because $\zeta_\pm \to \pi/2$ with $a \to 1$ accordingly to equation $\sin\zeta = 1/a$. Figure 39(a) shows the phase portrait of Hamiltonian (73) for such $1/a < 1$. There are two small system parameters: $B \propto D \propto k^{-1}(\partial \ln \Omega_0 / \partial t) \ll 1$ and $A \propto B_w / B_0 \ll 1$. When these two parameters are of the same order of magnitude, $a = |B/A| \sim 1$. In this Appendix we introduce $\epsilon \ll 1$ such that $B \propto \epsilon$ and $A \propto \epsilon$, so that $\tilde{A} = A/\epsilon$ is of the order of one. To model the system evolution we can write $1/a = 1 - \delta_0 + (\epsilon t)^2$ where the small parameter $\delta_0 > 0$ determines how far $1/a$ is from 1 and $(\epsilon t)^2$ models the slow evolution of $1/a$ along the trajectory (i.e., we change here the slow coordinate $s$ to a slow time):

$$ \mathcal{H}_{\zeta} \approx \frac{1}{2M} P_{\zeta}^2 - \epsilon \tilde{A} \cdot \left( \left( \delta_0 - (\epsilon t)^2 \right) \zeta - \frac{1}{6} \zeta^3 \right) \qquad (74) $$
and M, ˜A can be considered as constant along a short interval of $\epsilon t \in [-\sqrt{\delta_0}, \sqrt{\delta_0}]$. The equation for S for Hamiltonian (6) can be written as

$$ 
\begin{aligned}
S &= \sqrt{8\epsilon\tilde{A}M} \int_{\zeta_+}^{\zeta_-} \left( (\delta_0 - (\epsilon t)^2)(\zeta - \zeta_+) - \frac{\zeta^3 - \zeta_+^3}{6} \right)^{1/2} d\zeta \\
&= \sqrt{8\epsilon\tilde{A}M} \delta_0^{5/4} \left(1 - \frac{t^2}{t_0^2}\right)^{5/4} \int_{\zeta_+}^{\bar{\zeta}_-} \left( (\bar{\zeta} - \bar{\zeta}_+) - \frac{\bar{\zeta}^3 - \bar{\zeta}_+^3}{6} \right)^{1/2} d\bar{\zeta} \\
&= \sqrt{8\epsilon\tilde{A}M} \delta_0^{5/4} \left(1 - (t/t_0)^2\right)^{5/4} \cdot \frac{12 \cdot 2^{3/4}}{5} \quad (75)
\end{aligned}
$$

where $\bar{\zeta}_+ = -\sqrt{2}$, $\bar{\zeta}_- = \sqrt{8}$, $t_0 = \sqrt{\delta_0}/\epsilon$ 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 the resonant $s_R$ related to $\gamma_R$ through Eq. (4). Equation (75) demonstrates that the $\delta_0$ parameter controls the magnitude of S, and effects of nonlinear interactions should disappear when $\delta_0 \to 0$. In the nonlinear regime, there are well separated populations of trapped particles (a small number of particles gaining a large energy) and phase bunched 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 derive an expression for this threshold $\delta_0$ (or S) value.

Hamiltonian equations for Hamiltonian (74) can be combined to get the second order equation for $\zeta$:

$$ \frac{d^2\zeta}{dt^2} = \epsilon \cdot \left( \delta_0 - (\epsilon t)^2 - \frac{1}{2}\zeta^2 \right) \quad (76) $$

where $\tilde{t} = t\sqrt{\tilde{A}M}$ and ˜A, M are constants of the order of one (not dependent on $\epsilon$). Equation (76) can be rewritten in normalized variables $\tau = \tilde{t}\epsilon^{1/2}\delta_0^{1/4}$, $\xi = \zeta/\sqrt{\delta_0}$

$$ \frac{d^2\xi}{d\tau^2} = \left(1 - \frac{\epsilon}{\delta_0^{3/2}}\tau^2\right) - \frac{1}{2}\xi^2 \quad (77) $$

Equation (77) shows that for $\delta_0 = \epsilon^{2/3}$ there is no separation of time-scales, i.e. the equation describing 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 39(b) confirms this conclusion: we solve equation Eq. (77) with $\delta_0 = \epsilon^{2/3}$ 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.

The 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. We define
<figure><img src="image_39.png" /><figcaption>Fig. 39 The phase portrait of Hamiltonian (73) for r/u around 1 (a). Set of trajectories obtained by numerical integration of Eq. (77) with $\delta_0 = \epsilon^{2/3}$.</figcaption></figure>

$N_{trap}$ as the maximum number of periods of a trapped particle’s rotation in the phase space (see phase portrait in Fig. 39(b)). This parameter is quite universal: it can be determined for any particular wave-field model (see, e.g., [Zhang, Thorne, Artemyev, Mourenas, Angelopoulos, Bortnik, Kletzing, Kurth and Hospodarsky, 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 (74) can be written as

$$T_{trap}(\epsilon t) = 2 \int_{\zeta_{-}}^{\zeta_{+}} \frac{d\zeta}{\dot{\zeta}} = \frac{C}{\epsilon^{1/2} (\delta_0 - \epsilon^2 t^2)^{1/4}} \quad (78)$$

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 $\sim 2\delta_0^{1/2}/\epsilon$. Therefore we can write for $N_{trap}$

$$N_{trap} = \frac{2}{\epsilon} \int_{0}^{\sqrt{\delta_{0}}} \frac{d\epsilon t}{T_{trap}(\epsilon t)} = \frac{\delta_{0}^{3/4}}{\sqrt{\epsilon}} \tilde{C} \sim \epsilon^{3\kappa/5-1/2} \quad (79)$$

where constant $\tilde{C} \sim O(1)$. Equation (79) shows that $N_{trap} \sim \epsilon^{3\kappa/5-1/2}$. For the threshold value $\kappa = 5/6$, we get $N_{trap} \sim O(1)$, i.e., for $\kappa = 5/6$ ($\delta_0 \sim \epsilon^{2/3}$) the number of trapping periods does not depend on $\epsilon$ and there is no separation between trapped and scattered particles anymore.

## Appendix D

In this Appendix we consider an important property of resonant systems described by the Hamiltonian $H_I = H_0(I, s, p) + \epsilon H_1(I, \zeta, s, p)$ with $\epsilon = \text{const} \ll 1$: the gain of phase $\zeta$ between two successive resonances is a random value. The rigorous
proof of this property can be found in [Gao et al., 2023], whereas a simplified version of this proof is provided below [this derivation is somewhat similar to the derivation provided in Neishtadt and Vasiliev, 2005]. Examples with $\mathcal{H}_0 = -\omega I + \gamma(s, p, I)$, $\varepsilon\mathcal{H}_1 = -eU_w(s, I)\cos\zeta$, and $\varepsilon \sim B_w/B_0$ are provided by Eq. (3), but within this Appendix we do not use explicit forms for $H_0, H_1$. In this Hamiltonian $(s, \varepsilon^{-1}p), (\zeta, I)$ are pairs of conjugate variables (hence $(\zeta, I)$ are fast variables, $(q, p)$ are slow variables), and $\mathcal{H}_1$ is periodic in $\zeta$. Momentum $I$ is an adiabatic invariant: $\dot{I} = -\varepsilon\partial\mathcal{H}_1/\partial\zeta$, and $I$ is constant in the averaged over $\zeta$ system. There is no explicit dependence on time, and thus $\mathcal{H}_I = h = \text{const}$. The resonance condition is determined by the equation $\partial\mathcal{H}_0/\partial I = 0$. Solving this equation for $I$ gives the equation $I = I_R(s, p)$ of the surface of resonance. Denote $\Lambda(s, p) = \mathcal{H}_0(I_R(s, p), s, p)$. The Hamiltonian can be expanded around the resonance surface similarly to Eq. (6). For variables $\zeta$ and $P_\zeta = I - I_R$ we get Hamiltonian

$$ \mathcal{H}_{\zeta} = \frac{1}{2M} P_{\zeta}^2 + A\zeta + \mathcal{H}_1 $$

where

$$ M^{-1} = \left. \frac{\partial^2 \mathcal{H}_0}{\partial I^2} \right|_{I=I_R} \approx \text{const, } \quad A = \{\Lambda, I_R\} \approx \text{const.} $$

We assume that the phase portrait of the Hamiltonian $\mathcal{H}_\zeta$ looks like one shown in Fig. 40 (we put $I$ instead of $P_\zeta$ onto the vertical axis there).

We introduce the improved adiabatic invariant $J$ with the variable transformation $(I, \zeta, p, s) \mapsto (J, \theta, P, Q)$ such that the new Hamiltonian is $\mathcal{H} = \mathcal{H}_0(J, P, Q) + \varepsilon\bar{\mathcal{H}}_1(J, P, Q)$, where $\bar{\mathcal{H}}_1$ is the average of $\mathcal{H}_1$ over $\zeta$ (in the leading approximation). Far from the resonance $\theta$ changes with the frequency

$$ \dot{\theta} = \frac{\partial \mathcal{H}_0}{\partial J} + \varepsilon \frac{\partial \bar{\mathcal{H}}_1}{\partial J} \qquad (80) $$

with $J = \text{const}$, and

$$ \dot{Q} = \varepsilon \frac{\partial \mathcal{H}_0}{\partial P} + \varepsilon^2 \frac{\partial \bar{\mathcal{H}}_1}{\partial P}, \quad \dot{P} = -\varepsilon \frac{\partial \mathcal{H}_0}{\partial Q} - \varepsilon^2 \frac{\partial \bar{\mathcal{H}}_1}{\partial Q} \qquad (81) $$

We introduce $\omega_0(J, P, Q) = \partial\mathcal{H}_0/\partial J$, $\omega_1(J, P, Q) = \partial\bar{\mathcal{H}}_1/\partial J$, and consider a large number $N \gg 1$ of rounds of $\zeta$ from $t = t_0$ (when phase point is far from the resonance and moves towards the resonance) to $t = t_N$; the last round is sufficiently far from the resonance and $\theta \approx \zeta$ in the leading approximation. The last round ends at $\zeta = \zeta_{cN} = \zeta_c \mod 2\pi$; (see Fig. 40 for the definition of $\zeta_c$). Then

$$ \zeta_c = \zeta_0 + \int_{t_0}^{t_N} (\omega_0 + \varepsilon\omega_1) dt \quad \mod 2\pi. \qquad (82) $$

We introduce $t_*$ as the time of crossing the resonance, i.e. $\omega_0(J, P, Q)|_{t=t_*} = 0$, and rewrite Eq. (82):

$$ \zeta_c = \zeta_0 + \int_{t_0}^{t_*} (\omega_0 + \varepsilon\omega_1) dt - \int_{t_N}^{t_*} (\omega_0 + \varepsilon\omega_1) dt \quad \mod 2\pi \qquad (83) $$
<figure><img src="image_40.png" /><figcaption>Fig. 40 Schematic of phase portrait.</figcaption></figure>

Because $t_* - t_N \ll 1/\epsilon$ we can use $\dot{Q} = \epsilon \partial \mathcal{H}_0 / \partial P$, $\dot{P} = -\epsilon \partial \mathcal{H}_0 / \partial P$ and $Q \approx q$, $P \approx p$ in the last integral in Eq. (83). We also assume that $\zeta_c \approx \zeta_{c*} = \zeta_c|_{t=t_*}$. Thus, we can replace $\zeta_c$ with $\zeta_{c*}$ in (83).

To describe system dynamics for $t \in [t_N, t_*]$ we use the expansion of the Hamiltonian around the resonance: $H = \Lambda + \Upsilon$ and

$$ \Upsilon = \frac{1}{2M} (I - I_R)^2 + \epsilon \mathcal{H}_1. \qquad (84) $$

The Hamiltonian in new variables $(J, \theta, P, Q)$ can be expanded as

$$ \mathcal{H} = \Lambda + \frac{1}{2M} (J - I_R)^2 + \epsilon \bar{H}_1 \qquad (85) $$

We introduce $\epsilon = (J - I_R)^2 / 2M$ and write

$$ \begin{aligned} \dot{\epsilon} &= -\epsilon M^{-1} (J - I_R) A, \quad A = \{\Lambda, I_R\} \approx \text{const} \\ \omega_0 &= \frac{\partial \mathcal{H}}{\partial J} = M^{-1} (J - I_R) \end{aligned} \qquad (86) $$

Using $dt = d\epsilon / (d\epsilon / dt)$ we rewrite integral

$$ \int_{t_N}^{t_*} (\omega_0 + \epsilon\omega_1) dt \approx -\frac{1}{\epsilon} \int_{\epsilon_N}^{0} \frac{\omega_0 d\epsilon}{g(J - I_R)A} = -\frac{1}{\epsilon} \int_{\epsilon_N}^{0} \frac{d\epsilon}{A} = \frac{\epsilon_N}{\epsilon A} \qquad (87) $$

where $\epsilon_N$ is the value $\epsilon$ along the trajectory at $t = t_N$, and we omit $\epsilon\omega_1$ because $t_* - t_N \ll 1/\epsilon$. Using $\epsilon + \epsilon\bar{H}_1 = \Upsilon$, we write

$$ \epsilon_N = \Upsilon_N - \epsilon \bar{H}_1 \approx \Upsilon_N - \epsilon \bar{H}_{1*} \qquad (88) $$

where $\Upsilon_N$ is the value $\Upsilon$ along the trajectory at $t = t_N$, and $\bar{H}_{1*}$ is the resonant value of $\bar{H}_1$. Substituting Eqs. (87, 88) to Eq. (83), we obtain

$$ \zeta_{c*} = \zeta_0 + \int_{t_0}^{t_*} (\omega_0 + \epsilon\omega_1) dt - \frac{\Upsilon_N - \epsilon\bar{H}_{1*}}{\epsilon A} \quad \text{mod } 2\pi \qquad (89) $$
or

$$ \frac{\Upsilon_N}{2\pi\varepsilon A} = \frac{\bar{\mathcal{H}}_{1*}}{2\pi A} + \frac{1}{2\pi} \left( \zeta_0 - \zeta_{c*} + \int_{t_0}^{t_*} (\omega_0 + \varepsilon\omega_1) dt \right) \mod 1 \quad (90) $$

We define $\Upsilon_{last}$ as the value of $\Upsilon$ along the trajectory at the last crossing of the line $\zeta = \zeta_c$ before crossing the resonance. Thus, $\Upsilon_{last} = \Upsilon_N \mod 2\pi\varepsilon A$, because the change of $\Upsilon$ for one round of $\zeta$ equals $2\pi\varepsilon A$. We introduce $\xi = (\Upsilon_{last} - \varepsilon\mathcal{H}_{1c*})/(2\pi\varepsilon A)$, where $\mathcal{H}_{1c*}$ is value of $\mathcal{H}_1$ at $\zeta = \zeta_c$, $t = t_*$ and write

$$ \xi = \text{Frac} \left( \frac{\bar{\mathcal{H}}_{1*} - \mathcal{H}_{1c*}}{2\pi A} + \frac{\zeta_0 - \zeta_{c*}}{2\pi} + \frac{1}{2\pi\varepsilon} \int_{\tau_0}^{\tau_*} (\omega_0 + \varepsilon\omega_1) d\tau \right) \quad (91) $$

Here $\tau = \varepsilon t$, $\tau_* = \varepsilon t_*$. Note that $\xi$ can be written as

$$ \xi = \frac{\Upsilon_{last} - \varepsilon H_{1c*}}{2\pi\varepsilon r} = \frac{\Upsilon_{last} + \varepsilon A\zeta_{c*} - (\varepsilon H_{1c*} + \varepsilon A\zeta_{c*})}{2\pi\varepsilon A} = \frac{\mathcal{E}_{last} - \mathcal{E}_{c*}}{2\pi\varepsilon A} \quad (92) $$

where

$$ \mathcal{E} = \frac{1}{2M} (I - I_R)^2 + \varepsilon A \zeta + \varepsilon \mathcal{H}_1, \quad (93) $$

and $\mathcal{E}_{c*}$ is the value of $\mathcal{E}$ at $\zeta = \zeta_c$, $t = t_*$, $\mathcal{E}_{last}$ is the value of $\mathcal{E}$ at the last crossing of the line $\zeta = \zeta_c$ before crossing the resonance.

Let us use Eq. (91) to consider two successive crossings of the same resonance in the same direction. During the period of a slow motion a particle crosses the same resonance twice in opposite directions. We consider crossing the resonance in direction from positive to negative value of frequency, as in Fig.40, assuming that for crossing in the opposite direction we have $H_1 \equiv 0$. Then there is only one resonant interaction for one period of slow motion, i.e. two successive resonance crossings are separated by one slow period. Far from the resonance the improved adiabatic invariant $J$ can be considered as a constant. Denote $\tau_-$ and $\tau_+$ slow time moments of the resonance crossings ($\tau = \varepsilon t$). Let $\xi_\pm$ be values of the variable $\xi$ corresponding to these two crossings. We are looking for a relation between $\xi_+$ and $\xi_-$. Due to periodicity of the slow motion, values of $\zeta_{c*}$, $\mathcal{H}_{1c*}$, $\bar{\mathcal{H}}_{1*}$, $A$ are the same at $\tau = \tau_+$ and $\tau = \tau_-$.

We consider value $\xi_-$ for the first of the resonance crossings, and the corresponding value $\mathcal{E} = \mathcal{E}_{last-}$. At $\tau = \tau_-$ the phase point is on the line $\zeta = \zeta_{c*}$ with $I > I_R$ at the position indicated by the symbol $I_-$ in Fig. 40. We assume that this phase point crosses the resonance without trapping. Thus, at some $\tau = \tau'$ it arrives again to the line $\zeta = \zeta_c = \zeta_{c*}'$ with the value $\mathcal{E} = \mathcal{E}'$ and $I < I_R$. The phase point position is indicated by the symbol $I'$ in Fig. 40. We denote $\xi' = (\mathcal{E}' - \mathcal{E}_{c*})/(2\pi\varepsilon A)$.

At some moment of the slow time $\tau_0 \in (\tau_-, \tau_+)$ the phase point is far from the resonance and has $\zeta = \zeta_0$. Then Eq. (91) gives

$$ \xi_+ = \text{Frac} \left( \frac{\bar{\mathcal{H}}_{1*} - \mathcal{H}_{1c*}}{2\pi A} + \frac{\zeta_0 - \zeta_{c*}}{2\pi} + \frac{1}{2\pi\varepsilon} \int_{\tau_0}^{\tau_+} (\omega_0 + \varepsilon\omega_1) d\tau \right) \quad (94) $$
Similarly, considering the backward motion on the time interval from $\tau_0$ to $\tau'$ we get

$$ \xi' = \text{Frac} \left( \frac{\bar{H}_{1*} - \bar{H}_{1c*}}{2\pi A} + \frac{\zeta_0 - \zeta_{c*}}{2\pi} - \frac{1}{2\pi\epsilon} \int_{\tau'}^{\tau_0} (\omega_0 + \epsilon\omega_1) d\tau \right) \quad (95) $$

In this expression we can replace in the leading approximation $\xi'$ with $\xi_-$ and $\tau'$ with $\tau_-$ (note that

$$ \frac{1}{2\pi\epsilon} \int_{\tau_-}^{\tau'} (\omega_0 + \epsilon\omega_1) d\tau $$

is small, because $\omega_0$ vanishes on the resonant surface). Thus for the value $\Delta\xi = \xi_+ - \xi_-$ we get

$$ \Delta\xi = \xi_+ - \xi_- = \frac{1}{2\pi\epsilon} \int_{\tau_-}^{\tau_+} (\omega_0 + \epsilon\omega_1) d\tau \mod 1 \quad (96) $$

and this describes the phase $\zeta$ gain between two resonance crossings (between moments $\tau_-$ and $\tau_+$) normalized on $2\pi$. In the main text $\bar{H}_1 = 0$ and thus $\omega_1 = \partial\bar{H}_1/\partial J = 0$.

## Appendix E

In this Appendix, we demonstrate the applicability of the mapping technique for two types of events observed in the outer radiation belts. Both events are associated with energetic electron precipitations via scattering by whistler-mode waves. Such scattering corresponds to a decrease of the electron equatorial pitch-angle $\alpha_{eq}$ and a change of electron energy. Field-aligned chorus waves interact with electrons through the first cyclotron resonance. The nonlinear regime of this resonant interaction includes phase trapping (with $\alpha_{eq}$ and energy increase, see, e.g., [Omura et al., 2015]) and phase bunching (with $\alpha_{eq}$ and energy decrease, see, e.g., [Albert, 2000; Vainchtein et al., 2018]). The competition between phase trapping and phase bunching [Artemyev, Neishtadt, Vasiliev and Mourenas, 2016; Istomin et al., 1973; Shklyar, 2011; Solovev and Shkliar, 1986] controls the net electron transport into the loss-cone, but all precipitating electrons lose energy, because precipitation is provided only by phase bunching. Thus, both linear and non-linear electron interaction with chorus via the cyclotron resonance is associated with electron deceleration. Such effective scattering and deceleration of electrons by field-aligned waves is modeled with the mapping technique in Subsection Appendix E.2.

Although field-aligned chorus waves are the most intense whistler-mode wave population in the radiation belt [Agapitov et al., 2013; Li, Bortnik, Thorne and Angelopoulos, 2011], there is also a significant population of very oblique whistler-mode waves (almost electrostatic mode, see [Artemyev, Agapitov, Mourenas, Krasnoselskikh, Shastun and Mozer, 2016]) propagating near the resonance cone angle [Agapitov et al., 2013, 2018; Li, Santolik, Bortnik, Thorne, Kletzing, Kurth and Hospodarsky, 2016]. These oblique waves can interact resonantly with electrons through Landau resonance [e.g., Nunn and Omura, 2015; Shklyar and Matsumoto,
2009] and can accelerate electrons efficiently via phase trapping [Agapitov et al., 2014; Artemyev et al., 2012; Hsieh et al., 2020; Hsieh and Omura, 2017a]. The corresponding Landau resonant energies are much lower than cyclotron resonant energies with quasi-parallel waves, allowing very oblique waves to resonate with 100 eV - 10 keV electrons [Artemyev, Agapitov, Mourenas, Krasnoselskikh, Shastun and Mozer, 2016; Artemyev, Agapitov, Mourenas, Krasnoselskikh and Mozer, 2015; Mourenas, Artemyev, Agapitov and Krasnoselskikh, 2014] which can almost never be scattered by field-aligned lower-band chorus waves [Li et al., 2010]. Moreover, in contrast to the cyclotron resonance associated with an energy decrease of precipitating electrons, the Landau resonant trapping is associated with energy increase (see Figs. 6 and 20). Thus, precipitating electrons (those with decreasing $\alpha_{eq}$) are simultaneously accelerated, corresponding to an increase of low-$\alpha_{eq}$ energetic electron flux [Agapitov et al., 2015b]. Since very oblique whistler-mode waves can be quite intense [Agapitov et al., 2014; Cattell et al., 2008; Cully, Bonnell and Ergun, 2008; Wilson et al., 2011], these waves may in principle interact nonlinearly with electrons and efficiently trap them through Landau resonance [Agapitov et al., 2015b; Mourenas et al., 2016]. Such trapping results in intense, short-lived precipitation of $\lesssim 200$ keV electrons [Artemyev, Zhang, Zou, Mourenas, Angelopoulos, Vainchtein, Tsai and Wilkins, 2022; Zhang, Artemyev, Angelopoulos, Tsai, Wilkins, Kasahara, Mourenas, Yokota, Keika, Hori, Miyoshi, Shinohara and Matsuoka, 2022]. An important feature of this precipitation mechanism is the *loss-cone overfilling*, when precipitating fluxes at low altitudes appear to be larger than quasi-trapped fluxes just outside of the loss-cone [Zhang, Artemyev, Angelopoulos, Tsai, Wilkins, Kasahara, Mourenas, Yokota, Keika, Hori, Miyoshi, Shinohara and Matsuoka, 2022]. In Subsection Appendix E.1 we provide a simulation of this effect using the mapping technique.

The loss-cone size $\alpha_{LC}$ (the maximum pitch-angle range of precipitating electrons) in the outer radiation belt around the magnetic equator is only a few degrees, which makes it challenging for near-equatorial spacecraft to directly measure such electron losses [Kasahara, Miyoshi, Yokota, Kasahara, Matsuda, Kumamoto, Matsuoka, Kazama, Frey, Angelopoulos, Kurita, Keika, Seki and Shinohara, 2018]. Spacecraft at low altitudes (where magnetic field is large and $\alpha_{LC}$ reaches 60–70°), however, can easily measure electron fluxes within the loss cone. Conjugate spacecraft measurements of near-equatorial waves responsible for electron scattering into the loss cone and low-altitude electron fluxes within it, are optimal for testing theoretical models of electron scattering and precipitation by waves. [Li, Ni, Thorne, Bortnik, Green, Kletzing, Kurth and Hospodarsky, 2013]. Thus, we use such a combination of observations from the equatorial THEMIS-E spacecraft [Angelopoulos, 2008] and the low-altitude ($\sim 400–450$ km) ELFIN spacecraft [Angelopoulos et al., 2020].

The two identical CubeSats of the Electron Losses and Fields Investigation (ELFIN) mission were launched on September 15th, 2018, with onboard energetic particle detectors of ions (EPD-I) and electrons (EPD-E) [Angelopoulos et al., 2020]. Electrons are measured in an energy range from 50 keV to $\sim 6$ MeV with an energy resolution ($\Delta E/E$) less than 40 % and a time resolution ($T_{spin}$) of $\sim 2.85$ s [Angelopoulos et al., 2023; Tsai and et al., 2024]. The high resolution of the pitch-angle of the EPD-E ($\Delta\alpha \simeq 22.5°$) enables it to distinguish trapped electrons (outside the local bounce loss cone) from precipitation electrons (within the local bounce loss cone) [Angelopoulos et al., 2023; Zhang, Angelopoulos, Moure-
nas, Artemyev, Tsai and Wilkins, 2022, see details]. The ratio of the precipitating electron flux and the trapped electron flux, $j_{prec}/j_{trap}$, indicates the efficiency of equatorial electron scattering by waves [Capannolo et al., 2023; Mourenas et al., 2021; Shen et al., 2022; Zhang, Artemyev, Angelopoulos, Tsai, Wilkins, Kasahara, Mourenas, Yokota, Keika, Hori, Miyoshi, Shinohara and Matsuoka, 2022] or of the magnetic field line curvature process [Wilkins et al., 2023; Zou et al., 2024]. In this Appendix, we consider two ELFIN events with electron scattering by whistler-mode waves [additional examples are provided in Chen et al., 2022; Gan et al., 2023; Kang et al., 2024; Tsai et al., 2022; Zhang, Angelopoulos, Artemyev, Mourenas, Agapitov, Tsai and Wilkins, 2023].

## Appendix E.1: Nonlinear Landau resonance: loss-cone overfilling effect

Figure 41(a) is an overview of THEMIS E measurements on 2021-01-01 from 22:20 to 23:30 UT in the dayside inner magnetosphere, when the spacecraft moved outward from $L \sim 7$ to $L \sim 9$ ($L$ is evaluated with the [Tsyganenko, 1989] magnetic field model). There is strong whistler-mode wave activity within the frequency range $f/f_{ce} \in [0.2, 0.4]$, where the electron cyclotron frequency $f_{ce}$ is calculated from in-situ measurements from the fluxgate magnetometer [Auster et al., 2008] and the wave spectrum is obtained from the search coil [Cully, Ergun, Stevens, Nammari and Westfall, 2008; Le Contel et al., 2008]. Using plasma density estimates from the spacecraft potential [Nishimura et al., 2013], we can estimate the plasma frequency to electron gyrofrequency ratio, $f_{pe}/f_{ce} \sim 3$. Figure Figure 41(b) shows several whistler-mode wave packets with a large parallel electric field component, suggesting the presence of very oblique whistler-mode waves [Artemyev, Agapitov, Mourenas, Krasnoselskikh, Shastun and Mozer, 2016]. Using $f_{pe}/f_{ce}$ and 3D electric fields, we estimate the wave normal angle $\theta$ for these wave-packets: $\theta \in [60, 70]$ (see details of $\theta$ estimation method in [Agapitov et al., 2014; Ni, Thorne, Meredith, Shprits and Horne, 2011]), with a Gendrin angle $\theta_g = \arccos(2f/f_{ce}) \approx 45^{\circ}$ and a resonance cone angle $\theta_r = \arccos(f/f_{ce}) \approx 69^{\circ}$. Thus, the observed waves indeed propagate obliquely with $\theta \in [\theta_g, \theta_r]$, i.e., in the quasi-electrostatic mode [Agapitov et al., 2013; Li, Santolik, Bortnik, Thorne, Kletzing, Kurth and Hospodarsky, 2016], and can interact with electrons through the Landau resonance [Artemyev, Agapitov, Mourenas, Krasnoselskikh, Shastun and Mozer, 2016; Shklyar and Matsumoto, 2009]. An interesting and important feature of the observed whistler-mode wave activity is the transient nature of wave bursts, which may be due to quasi-periodic wave generation modulated by compressional ultra-low-frequency waves [Li, Bortnik, Thorne, Nishimura, Angelopoulos and Chen, 2011; Li, Thorne, Bortnik, Nishimura and Angelopoulos, 2011; Xia et al., 2016; Zhang, Chen, Artemyev, Angelopoulos and Liu, 2019]. Indeed, measurements of precipitating electrons by the Precipitating Electron and Ion Spectrometer onboard two DMSP satellites [Hardy et al., 1984; Rich et al., 1985] show clear spatially/temporally localized enhancements of $\sim 5$ keV electron precipitation, consistent with the presence of a transient (quasi-periodic) equatorial generation of very oblique whistler-mode waves [see details in Artemyev, Zhang, Zou, Mourenas, Angelopoulos, Vainchtein, Tsai and Wilkins, 2022].

There are a couple of orbits of ELFIN A and B within the same time interval and L-shell, MLT range [see Artemyev, Zhang, Zou, Mourenas, Angelopoulos,
Vainchtein, Tsai and Wilkins, 2022, for THEMIS, DMSP, and ELFIN orbits as a function of MLT and L-shell]. Figure 41(c, d) shows that within the L-shell range corresponding to the outer radiation belt, ELFIN observed transient, strong precipitation of < 200 keV electrons. The timescale of these precipitation bursts is about (or even less than) ELFIN's half-spin-period, 1.5s. Therefore, we infer that these precipitation events are probably driven by electron scattering by transient whistler-mode waves which, due to their amplitude modulation by ULF waves, do not maintain the same spatial distribution at the equator over time intervals longer than ~1-2 minutes. Most of the precipitation bursts reach a ratio ~1 (and even exceed it) at energies < 200 keV, i.e., they represent very strong precipitation with an entirely filled loss-cone, roughly corresponding to the so-called strong diffusion limit [Kennel, 1969] (or the loss-cone overfilling [Zhang, Artemyev, Angelopoulos, Tsai, Wilkins, Kasahara, Mourenas, Yokota, Keika, Hori, Miyoshi, Shinohara and Matsuoka, 2022]).

To model electron precipitation by very oblique whistler-mode waves, we use the mapping technique as described by Eqs. (25). To evaluate the trapping probability, $\Pi$, and energy/pitch-angle changes due to trapping and bunching, we specify several properties of the wave model. We use a simplified model that fits the observed $B_w(\lambda)$ distribution obtained from satellite statistics by [Agapitov et al., 2018]:

$$B_w = B_{w,eq} \cdot \frac{1}{2} (1 + \tanh(\lambda/\delta\lambda_1)) \cdot \exp(-\lambda^2/\delta\lambda_2^2) \quad (97)$$

where $\delta\lambda_1$ determines the spatial extent of the wave generation region, and $\delta\lambda_2$ determines the latitude range of wave propagation before damping at middle latitudes. This model describes waves in the $\lambda > 0$ hemisphere, and we assume a symmetric wave field distribution relative to the equator, i.e., $B_w(\lambda) = B_w(-\lambda)$. Wave frequency can be assumed to be fixed along the wave propagation (or slowly varying in time, if we deal with chorus waves), whereas the wave number profile $k(\lambda)$ can be obtained from the cold plasma dispersion [Stix, 1962] for a given wave normal angle distribution $\theta(\lambda)$ at this particular frequency. For very oblique whistler-mode waves, we use a $\theta = \theta_r(\lambda) - \Delta\theta$ model, with $\theta_r = \arccos(\omega/\Omega_{ce})$ being the resonant cone angle and $\Delta\theta = \text{const}$ the model parameter specifying the wave normal angle deviation from $\theta_r$. This wave normal angle model is based on previous observations of very oblique whistler-mode wave propagation around the resonance cone angle [Agapitov et al., 2013; Li, Santolik, Bortnik, Thorne, Kletzing, Kurth and Hospodarsky, 2016]. The cold plasma dispersion relation does not work for $\Delta\theta \to 0$, where the thermal electron contribution to the wave dispersion properties becomes crucial [Sazhin and Horne, 1990]. Theoretical estimates [Artemyev, Agapitov, Mourenas, Krasnoselskikh, Shastun and Mozer, 2016; Mourenas, Artemyev, Agapitov and Krasnoselskikh, 2014] and spacecraft observations [Ma et al., 2017] suggest that the minimum $\Delta\theta$ can be determined from the limitation of the wave refractive index $N = kc/\omega$ to values $N < N_{hot} \approx 100-300$ for typical thermal electron energies in the inner magnetosphere.

During Landau resonance ($\omega\gamma - k_{\parallel}c\sqrt{\gamma^2 - 1} \cos\alpha = 0$), the electron propagates in the same direction as the wave, in contrast to cyclotron resonance characterized by opposite propagations of wave and electron. For a fixed wave frequency at Landau resonance, the resonance curve ($\omega/\Omega_{ce,eq})(\gamma^2 - 1) \sin^2\alpha_{eq}/2 = \text{const}$ (see Fig. 6) is given by the equation of the constant magnetic moment (we use the normalized moment $I_x = (\omega/\Omega_{ce,eq})(\gamma^2 - 1) \sin^2\alpha_{eq}/2$). Combining this equation
<figure><img src="image_41.png" /><figcaption>Fig. 41 Overview of the event with the effect of nonlinear Landau resonance: near-equatorial THEMIS observations of whistler-mode electric field (a, b), low altitude ELFIN observations of trapped fluxes (c) and precipitating-to-trapped flux ratio (d).</figcaption></figure>

with the resonance condition, we obtain the equation for dependence of the Landau resonant energy on the magnetic latitude:

$$ \gamma_R = \frac{1}{\sqrt{1 - (\omega/ck_{\parallel})^2}} \sqrt{1 + 2I_x \Omega_{ce}} \quad (98) $$

The wave intensity increases with latitudes $dB_w/d\lambda > 0$ around the equator [Agapitov et al., 2018; Omura et al., 2008]. This condition suggests that electrons may be trapped by waves. Being trapped at small resonant latitudes, electrons are transported by the wave to higher latitudes (with a decrease of their pitch-angle) and escape from the resonance at a latitude where $dB_0/d\lambda$ becomes sufficiently strong (for a precise description of the trapping and de-trapping conditions,
see, Section 2.1 and e.g., [Artemyev et al., 2012; Artemyev, Vasiliev, Mourenas, Agapitov and Krasnoselskikh, 2013]). Such an electron transport is associated with an energy increase, e.g., a 5 keV equatorial electron can gain 100 keV before reaching resonant latitudes ~ 30° and finishing within the loss-cone [Artemyev, Zhang, Zou, Mourenas, Angelopoulos, Vainchtein, Tsai and Wilkins, 2022]. Therefore, the trapping moves lower energy/larger pitch-angle electrons toward the larger energy/lower pitch-angle region in phase space (see schematic in Fig. 6). The Landau resonance scattering moves electrons along the same resonance curve $I_x = const$, but in the direction opposite to the trapping motion. Therefore, Landau resonance results in electron drifts toward smaller energy/larger pitch-angle (the phase bunching effect) and more rare jumps to larger energy/smaller pitch-angle (the phase trapping effect). Figure 42(a) shows three examples of such electron dynamics evaluated with the mapping technique (25) for typical wave characteristics (see also Fig. 20).

Near-equatorial spacecraft measurements of whistler-mode waves provide distributions of wave characteristics, e.g., the average wave intensity for various frequency ranges, L-shells, MLT, and $\lambda$ [Agapitov et al., 2013, 2015a; Meredith et al., 2001, 2012]. Such time-averaged intensities can be directly applied to the evaluation of quasi-linear diffusion rates [e.g., Agapitov et al., 2018; Horne, Kersten, Glauert, Meredith, Boscher, Sicard-Piet, Thorne and Li, 2013; Ma et al., 2018], but they are not suitable for the evaluation of nonlinear wave-particle interactions. Instead of time-averaged intensities, the models of nonlinear resonant wave-particle interactions require a knowledge of the distributions (occurrence rates) of intense waves that can interact with electrons nonlinearly [Zhang, Mourenas, Artemyev, Angelopoulos, Bortnik, Thorne, Kurth, Kletzing and Hospodarsky, 2019; Zhang, Thorne, Artemyev, Mourenas, Angelopoulos, Bortnik, Kletzing, Kurth and Hospodarsky, 2018]. Therefore, we use THEMIS E measurements during the event in Fig. 41 to obtain the distribution of wave amplitudes $\mathcal{E}_w$, as inferred from the wave frequency spectrum $\hat{E}_w^2(f): \mathcal{E}_w^2 = \int_{f_{ce}/20}^{f_{ce}/2} \hat{E}_w^2(f) df$. As shown in Figure 42(b), most waves have amplitudes < 3 mV/m. However, $E_w$ here is inferred from the 1s-averaged spectrum [Cully, Ergun, Stevens, Namari and Westfall, 2008] and the actual magnitude of individual wave-packets is often significantly larger (see Fig. 41). To account for such an underestimation of $E_w$ due to time averaging, we introduce a multiplicative factor $K$, i.e., we assume that electrons interact with waves of amplitudes $K \times E_w$. To keep the same average intensity as without the $K$ factor, we assume that electrons resonate with waves only a fraction $(1/K^2)$ of the simulation time, whereas during the remaining fraction $(1-1/K^2)$ of the time there is no resonant interaction.

The wave-particle resonant interaction is not determined solely by $E_w$, but also by the wave frequency, wave normal angle, and the $\mathcal{E}_w(\lambda)$ profile along the magnetic field line. The wave frequency can be determined from the wave frequency spectrum: $\langle f \rangle = \int_{f_{ce}/20}^{f_{ce}/2} f \cdot \hat{E}_w^2(f) df / \mathcal{E}_w^2$ (with $f_{ce} = \Omega_0/2\pi$ and $f = \omega/2\pi$). Figure 42(c) shows the $P(B_w, \langle f \rangle)$ distribution for the event in Fig. 41, with $B_w$ recalculated from $E_w$ using the cold plasma dispersion [see examples in, e.g., Agapitov et al., 2014; Ni, Thorne, Meredith, Horne and Shprits, 2011]. Using this $P(B_w, \langle f \rangle)$ distribution, we can rewrite the mapping in Eqs. (25) as a two-step equation. During each half of the bounce period, we generate a random number to select a $(B_w, \langle f \rangle)$ bin from the $P$ distribution. According to this number, we choose the wave am-
<figure><img src="image_42.png" /><figcaption>**Fig. 42** Panels (a) shows energy and equatorial pitch-angle for three trajectories evaluated with the mapping technique (26). Panel (b) shows the distribution of electric field amplitudes derived from the wave frequency spectra (fff data) shown in Fig. 41. Panel (c) shows the distribution of wave amplitudes and frequencies for the event in Fig. 41. Wave magnetic field amplitudes are converted from the wave electric field amplitudes using cold plasma dispersion relation [Tao and Bortnik, 2010]. Panel (d) shows pitch-angle distributions for different energies after ~10R/c ~2s of interactions (sufficient to make at least one bounce period for all considered particles). The initial (equatorial) electron phase space density is obtained by fitting THEMIS E observations. We use a K = 3 factor for this simulation, with the corresponding probability of wave-particle interactions 1/K².</figcaption></figure>

plitude $B_w$ and wave frequency $\langle f \rangle/f_{ce}$ to calculate three main characteristics of wave-particle interactions: $\Delta\gamma_{trap}$, $\Delta\gamma_{scat}$, and $\Pi$. Then, we make one iteration for energy and pitch-angle using Eq. (28). Thus, the observed distribution of wave characteristics, $P(B_w, \langle f \rangle)$, determines $\Delta\gamma_{trap}$, $\Delta\gamma_{scat}$, and $\Pi$ for each resonant interaction. Figure 42(a) shows several examples of electron trajectories calculated with the observed $P$ distribution.

Using the mapping technique with the observed distribution $P(B_w, \langle f \rangle)$, we integrate a large ensemble of electron trajectories having an initial (energy, pitch-angle) distribution measured by THEMIS E around equator. As we are interested
in precipitation patterns, we restrict the integration time to ~1 bounce period of 100 keV electrons. All electrons with $\alpha_{eq} < 2^\circ$ are considered to be precipitating (as appropriate at $L < 8$). Figure 42(d) shows several pitch-angle distributions for different energies after wave-particle resonant interactions. There is a clear peak of small $\alpha_{eq}$ electron fluxes, and this peak is most pronounced for energies higher than ~30 keV. The formation of such a field-aligned (and partially precipitating) electron population is produced by the electron trapping acceleration in the Landau resonance. Despite the small probability of electron Landau trapping II [see, e.g., Artemyev et al., 2012; Artemyev, Vasiliev, Mourenas, Agapitov and Krasnoselskikh, 2014], this acceleration mechanism is quite effective, because waves trap particles with initially smaller energies (~ a few keV) and accelerate them up to ~30–100 keV. Therefore, the large difference of initial phase space densities in the energy of trapping and release compensates the small probability of such trapping (II) and allows for the formation of a very substantial field-aligned electron population at ~30–100 keV.

The time-scale of such electron acceleration and precipitation is about a bounce period, that is, the time-scale of electron transport in the Landau trapping. This represents an insufficiently long time for nonlinear wave-particle interactions to establish a fine balance between trapping and phase bunching (nonlinear scattering) and, thus, electron transport toward the loss-cone (due to trapping) may not be fully compensated by electron transport to higher pitch-angles (due to bunching). After a sufficiently long wave-particle interaction, such a fine balance will establish a new distribution function without strong gradients along the resonance curve $I_x = \text{const}$ (see Section 4.2 and [Artemyev, Neishtadt and Vasiliev, 2019]). Thus, electron precipitation associated with the Landau trapping acceleration should be provided by bursts of very oblique whistler-mode waves, and they can share properties of microbursts. Therefore, intense (microburst) precipitation events can indicate an effective electron acceleration and an increase of ~50–200 keV electron fluxes at small pitch-angles, rather than being a signal of electron flux depletion. The accelerated electrons ending up outside the loss-cone, however, have already small pitch-angles, and can be later scattered into the loss-cone by much weaker field-aligned whistler-mode waves [see discussions of such double-resonance mechanism of electron losses due to Landau trapping and cyclotron scattering in Hsieh et al., 2022; Ma et al., 2016; Mourenas et al., 2016].

## Appendix E.2: Nonlinear cyclotron resonance: electron precipitation from the injection region

In this Subsection we consider the mapping modeling of electron nonlinear resonant interaction with field-aligned whistler-mode waves. The Subsection structure repeats Appendix E.1: we combine the mapping technique from Section 4.4, near-equatorial THEMIS E measurements, and low-altitude ELFIN measurements. We examine the event with plasma sheet injection observed by THEMIS E [see details in Artemyev, Neishtadt and Angelopoulos, 2022]. We use magnetic field measurements by the fluxgate magnetometer [Auster et al., 2008] with 1/3 s resolution, and magnetic field fluctuations in the 10–4000 Hz frequency range measured by the search-coil [Le Contel et al., 2008]. We use both waveforms measured in
the wave burst mode and wave spectra evaluated on board at 1 s time resolution [Cully, Ergun, Stevens, Nammari and Westfall, 2008].

Figure 43 shows observations of the plasma injection, associated with sporadic and transient bursts of whistler-mode waves (see panel (a)). The wave spectrum $B_w^2(f)$ with $f = \omega/2\pi$ shows intense bursts around $f \in [0.1, 0.4]f_{ce}$ (with $f_{ce} = \Omega_0/2\pi$ and $f = \omega/2\pi$), in the typical frequency range of whistlers captured around dipolarizing flux bundles [Breuillard et al., 2016; Grigorenko et al., 2020; Le Contel et al., 2009; Zhang, Angelopoulos, Artemyev and Liu, 2018]. For each 1 s spectrum we calculate the root-mean-square time-averaged wave amplitude $B_w = \left( \int_{f_{ce}/10}^{f_{ce}/2} \hat{B}_w^2 df \right)^{1/2}$ and mean wave frequency $f_m = \int_{f_{ce}/10}^{f_{ce}/2} \hat{B}_w^2 df / B_w^2$. These wave characteristics are input parameters for the mapping technique. $B_w^2$ is the time-averaged wave intensity calculated from wave spectra. The amplitude of individual wave-packets can be significantly larger than $B_w$. This difference is not important for the quasi-linear diffusion describing wave-particle resonant interaction by diffusion rates that depend only on the time-averaged $B_w$ [Lyons and Williams, 1984; Schulz and Lanzerotti, 1974]. For nonlinear resonant interactions, however, the electron dynamics are determined by peak wave amplitudes and the occurrence rate of intense waves. Therefore, we need to recalculate the $B_w$ distribution derived from the wave spectra integration to estimate the actual occurrence rate of intense waves. This is done by using simultaneous measurements of whistler wave packets (see Figure 43(b)). Typical wave packet amplitudes during this event are ~100 pT, i.e., a factor $K \approx 20$ larger than the average $B_w \sim 5$ pT and a factor $K \approx 5$ larger than the level $B_w \sim 20$ pT of the most intense wave population (see the $P(B_w, f/f_{ce})$ distribution in Fig. 44(a)). Accordingly, we multiply the time-averaged $B_w$ by $K = 5$. To keep the same time-averaged wave intensity $B_w^2$ we assume that such intense wave packets are observed only a fraction $1/K^2$ of the time, whereas during the rest $1 - 1/K^2$ of the time there are no whistlers. This may be a simplistic approach, but nonetheless it captures the essential features of the wave packet for the purpose of nonlinear interactions. Thus, during each half of the bounce period, there is a probability $1/K^2$ that an electron resonates with one of the waves from the $P$ distribution, and a probability $1 - 1/K^2$ that there is no resonant interaction and thus electron characteristics remain unchanged. To set the initial electron distribution function we use THEMIS E electron measurements [Angelopoulos et al., 2008; McFadden et al., 2008].

Electron scattering by whistlers may significantly decrease electron pitch-angles and move particles to the loss-cone. An accurate estimate of the loss-cone size for the near-Earth magnetotail (injection region) is almost impossible, because empirical magnetic field models [Andreeva and Tsyganenko, 2019; Tsyganenko, 1995; Tsyganenko and Sitnov, 2007], with only few exceptions [Sitnov et al., 2021; Stephens et al., 2019; Tsyganenko et al., 2021], do not include the effects of dipolarization associated with plasma injection and strong variations of the equatorial magnetic field (such variations may significantly change the loss-cone size and precipitating electron fluxes, see, e.g. [Eshetu et al., 2018]). Typical loss-cone angle $\alpha_{LC}$ estimates give $\le 2^\circ$ in the magnetotail (see Fig. 3(d) in [Zhang et al., 2015]), and this value can be larger closer to the Earth. For a simulation of precipitating electron fluxes, we use $\alpha_{LC} = 3^\circ$. This value could overestimate the precipitating flux magnitude, but there is almost no energy change due to the wave-particle resonant interaction at pitch-angles $< 3$ degrees. Therefore, the energy spectrum of
<figure><img src="image_43.png" /><figcaption>**Fig. 43** Overview of the event with the effect of nonlinear cyclotron resonance: near-equatorial THEMIS observations of whistler-mode magnetic field (a, b), low altitude ELFIN observations of trapped fluxes (c) and precipitating-to-trapped flux ratio (d). The precipitation burst due to electron scattering by whistler-mode waves is embedded into the dispersive precipitation structure (higher energies of precipitating electrons closer to the Earth) due to electron scattering by the magnetotail current sheet (CS) [see details in Artemyev et al., 2023; Wilkins et al., 2023].</figcaption></figure>

precipitating electrons derived from our simulation should reproduce the expected spectrum of electrons precipitating from the dipolarizing flux bundle region. To check this, we compare model results with observations from the low-altitude, polar-orbiting CubeSats ELFIN [Angelopoulos et al., 2020] that provide conjugate measurements for the event in Fig. 43. Figure 43 shows an overview of ELFIN A trapped and precipitating fluxes: pattern of strong $j_{trap}$ increase at the inner edge of the plasma sheet (05:39:00-05:39:45 for ELFIN A) is also associated with a high ratio $j_{loss}/j_{trap} \sim 1$ showing strong energy dispersion: larger energies with
<figure><img src="image_44.png" /><figcaption>**Fig. 44** Panels (a) shows the distribution of wave amplitudes and frequencies for the event in Fig. 43. Panel (b) shows energy spectra from ELFIN A for the time interval associated with enhanced electron precipitations. The red curve shows the model result.</figcaption></figure>

$j_{loss}/j_{trap} \sim 1$ are closer to the Earth). This pattern most likely corresponds to the electron isotropic boundary [Imhof et al., 1979; Sergeev et al., 2012; Wilkins et al., 2023; Yahnin et al., 1997], i.e., energetic electron scattering in the current sheet due to the magnetic field line curvature [Birmingham, 1984; Büchner and Zelenyi, 1989; Lukin et al., 2021; Young et al., 2002]. The curvature scattering efficiency depends on energy and on the equatorial magnetic field curvature radius and intensity. These dependencies are responsible for the observed energy dispersion [Dubyagin et al., 2021; Sergeev et al., 2018]. Closer to the Earth, ELFIN shows both a secondary burst of $j_{loss}/j_{trap} \sim 1$ and an increase of $j_{trap}$ (around 05:39:55), and this second precipitation pattern is not due to electron curvature scattering, because the energies of precipitating electrons are lower than energies of the pattern associated with the isotropic boundary observed farther away from the Earth. For ELFIN A this second precipitation pattern is observed around its conjunction to the equatorial injection region observed by THEMIS. We thus attribute this pattern to electron scattering from the injection region. Figure 44(b) shows spectra of precipitating electrons. Their average spectral shape is similar to the precipitating electron fluxes derived from the simulation driven by THEMIS equatorial measurements of electron fluxes and whistlers (compare red and grey curves). Therefore, our simulation confirms that the mapping technique can reproduce electron resonant interactions with field-aligned whistler-mode waves.

References

Åsnes, A., Friedel, R. W., Stadsnes, J., Thomsen, M., Østgaard, N. and Cayton, T. [2005], ‘Statistical pitch angle properties of substorm-injected electron clouds and their relation to dawnside energetic electron precipitation’, *J. Geophys. Res.* **110**, 5207.
Agapitov, O. V., Artemyev, A., Krasnoselskikh, V., Khotyaintsev, Y. V., Mourenas, D., Breuillard, H., Balikhin, M. and Rolland, G. [2013], ‘Statistics of whistler mode waves in the outer radiation belt: Cluster STAFF-SA measurements’, *J. Geophys. Res.* **118**, 3407–3420.

Agapitov, O. V., Artemyev, A., Mourenas, D., Krasnoselskikh, V., Bonnell, J., Le Contel, O., Cully, C. M. and Angelopoulos, V. [2014], ‘The quasi-electrostatic mode of chorus waves and electron nonlinear acceleration’, *J. Geophys. Res.* **119**, 1606–1626.

Agapitov, O. V., Artemyev, A. V., Mourenas, D., Mozer, F. S. and Krasnoselskikh, V. [2015a], ‘Empirical model of lower band chorus wave distribution in the outer radiation belt’, *J. Geophys. Res.* **120**, 10.

Agapitov, O. V., Artemyev, A. V., Mourenas, D., Mozer, F. S. and Krasnoselskikh, V. [2015b], ‘Nonlinear local parallel acceleration of electrons through Landau trapping by oblique whistler mode waves in the outer radiation belt’, *Geophys. Res. Lett.* **42**, 10.

Agapitov, O. V., Mourenas, D., Artemyev, A. V. and Mozer, F. S. [2016], ‘Exclusion principle for very oblique and parallel lower band chorus waves’, *Geophys. Res. Lett.* **43**(21), 11,112–11,120.
URL: http://dx.doi.org/10.1002/2016GL071250

Agapitov, O. V., Mourenas, D., Artemyev, A. V., Mozer, F. S., Hospodarsky, G., Bonnell, J. and Krasnoselskikh, V. [2018], ‘Synthetic Empirical Chorus Wave Model From Combined Van Allen Probes and Cluster Statistics’, *Journal of Geophysical Research (Space Physics)* **123**(1), 297–314.

Albert, J. M. [1993], ‘Cyclotron resonance in an inhomogeneous magnetic field’, *Physics of Fluids B* **5**, 2744–2750.

Albert, J. M. [2000], ‘Gyroresonant interactions of radiation belt particles with a monochromatic electromagnetic wave’, *J. Geophys. Res.* **105**, 21191.

Albert, J. M. [2001], ‘Comparison of pitch angle diffusion by turbulent and monochromatic whistler waves’, *J. Geophys. Res.* **106**, 8477–8482.

Albert, J. M. [2010], ‘Diffusion by one wave and by many waves’, *J. Geophys. Res.* **115**, 0.

Albert, J. M. [2017], ‘Quasi-linear diffusion coefficients for highly oblique whistler mode waves’, *J. Geophys. Res.* **122**, 5339–5354.

Albert, J. M. [2018], ‘Diagonalization of diffusion equations in two and three dimensions’, *Journal of Atmospheric and Solar-Terrestrial Physics* **177**, 202–207.

Albert, J. M., Artemyev, A., Li, W., Gan, L. and Ma, Q. [2022], ‘Analytical results for phase bunching in the pendulum model of wave-particle interactions’, *Frontiers in Astronomy and Space Sciences* **9**, 971–975.

Albert, J. M., Artemyev, A. V., Li, W., Gan, L. and Ma, Q. [2021], ‘Models of resonant wave-particle interactions’, *Journal of Geophysical Research: Space Physics* **126**(6), e2021JA029216.

Albert, J. M. and Bortnik, J. [2009], ‘Nonlinear interaction of radiation belt electrons with electromagnetic ion cyclotron waves’, *Geophys. Res. Lett.* **36**, 12110.

Albert, J. M., Tao, X. and Bortnik, J. [2013], Aspects of Nonlinear Wave-Particle Interactions, in D. Summers, I. U. Mann, D. N. Baker and M. Schulz, eds, ‘Dynamics of the Earth’s Radiation Belts and Inner Magnetosphere’, American Geophysical Union.

Alekhin, J. K. and Shklyar, D. R. [1980], ‘Some questions of VLF wave propagation in the magnetosphere’, *Geomagnetism and Aeronomy/Geomagnetism*
*Aeronomiia* **20**, 501–507.

Allanson, O., Ma, D., Osmane, A., Albert, J. M., Bortnik, J., Watt, C. E. J., Chapman, S. C., Spencer, J., Ratliff, D. J., Meredith, N. P., Elsden, T., Neukirch, T., Hartley, D. P., Black, R., Watkins, N. W. and Elvidge, S. [2024], ‘The challenge to understand the zoo of particle transport regimes during resonant wave-particle interactions for given survey-mode wave spectra’, *Frontiers in Astronomy and Space Sciences* **11**, 1332931.

Allanson, O., Thomas, E., Watt, C. E. J. and Neukirch, T. [2022], ‘Weak Turbulence and Quasilinear Diffusion for Relativistic Wave-Particle Interactions Via a Markov Approach’, *Frontiers in Physics* **8**:805699.

Allanson, O., Watt, C. E. J., Allison, H. J. and Ratcliffe, H. [2021], ‘Electron Diffusion and Advection During Nonlinear Interactions With Whistler Mode Waves’, *Journal of Geophysical Research (Space Physics)* **126**(5), e28793.

Allanson, O., Watt, C. E. J., Ratcliffe, H., Allison, H. J., Meredith, N. P., Bentley, S. N., Ross, J. P. J. and 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**(7), e27949.

Allanson, O., Watt, C. E. J., Ratcliffe, H., Meredith, N. P., Allison, H. J., Bentley, S. N., Bloch, T. and Glauert, S. A. [2019], ‘Particle-in-cell Experiments Examine Electron Diffusion by Whistler-mode Waves: 1. Benchmarking With a Cold Plasma’, *Journal of Geophysical Research (Space Physics)* **124**(11), 8893–8912.

Allison, H. J. and Shprits, Y. Y. [2020], ‘Local heating of radiation belt electrons to ultra-relativistic energies’, *Nature Communications* **11**, 4533.

An, X., Artemyev, A., Angelopoulos, V., Zhang, X.-J., Mourenas, D., Bortnik, J. and Shi, X. [2024], ‘Nonresonant scattering of energetic electrons by electromagnetic ion cyclotron waves: Spacecraft observations and theoretical framework’, *Journal of Geophysical Research: Space Physics* **129**(3), e2023JA031863.

An, X., Artemyev, A., Angelopoulos, V., Zhang, X., Mourenas, D. and Bortnik, J. [2022], ‘Nonresonant Scattering of Relativistic Electrons by Electromagnetic Ion Cyclotron Waves in Earth’s Radiation Belts’, *Phys. Rev. Lett* **129**(13), 135101.

An, Z., Wu, Y. and Tao, X. [2022], ‘Electron Dynamics in a Chorus Wave Field Generated From Particle-In-Cell Simulations’, *Geophys. Res. Lett.* **49**(3), e97778.

Andreeva, V. A. and Tsyganenko, N. A. [2019], ‘Empirical Modeling of the Geomagnetosphere for SIR and CME-Driven Magnetic Storms’, *Journal of Geophysical Research (Space Physics)* **124**(7), 5641–5662.

Angelopoulos, V. [2008], ‘The THEMIS Mission’, *Space Sci. Rev.* **141**, 5–34.

Angelopoulos, V., Sibeck, D., Carlson, C. W., McFadden, J. P., Larson, D., Lin, R. P., Bonnell, J. W., Mozer, F. S., Ergun, R., Cully, C., Glassmeier, K. H., Auster, U., Roux, A., Lecontel, O., Frey, S., Phan, T., Mende, S., Frey, H., Donovan, E., Russell, C. T., Strangeway, R., Liu, J., Mann, I., Rae, J., Raeder, J., Li, X., Liu, W., Singer, H. J., Sergeev, V. A., Apatenkov, S., Parks, G., Fillingim, M. and Sigwarth, J. [2008], ‘First Results from the THEMIS Mission’, *Space Sci. Rev.* **141**, 453–476.

Angelopoulos, V., Tsai, E., Bingley, L., Shaffer, C., Turner, D. L., Runov, A., Li, W., Liu, J., Artemyev, A. V., Zhang, X. J., Strangeway, R. J., Wirz, R. E., Shprits, Y. Y., Sergeev, V. A., Caron, R. P., Chung, M., Cruce, P., Greer, W., Grimes, E., Hector, K., Lawson, M. J., Leneman, D., Masongsong, E. V., Russell, C. L., Wilkins, C., Hinkley, D., Blake, J. B., Adair, N., Allen, M., Anderson, M.,
Arreola-Zamora, M., Artinger, J., Asher, J., Branchevsky, D., Capitelli, M. R., Castro, R., Chao, G., Chung, N., Cliffe, M., Colton, K., Costello, C., Depe, D., Domae, B. W., Eldin, S., Fitzgibbon, L., Flemming, A., Fox, I., Frederick, D. M., Gilbert, A., Gildemeister, A., Gonzalez, A., Hesford, B., Jha, S., Kang, N., King, J., Krieger, R., Lian, K., Mao, J., McKinney, E., Miller, J. P., Norris, A., Nuesca, M., Palla, A., Park, E. S. Y., Pedersen, C. E., Qu, Z., Rozario, R., Rye, E., Seaton, R., Subramanian, A., Sundin, S. R., Tan, A., Turner, W., Villegas, A. J., Wasden, M., Wing, G., Wong, C., Xie, E., Yamamoto, S., Yap, R., Zarifian, A. and Zhang, G. Y. [2020], ‘The ELFIN Mission’, *Space Sci. Rev.* **216**(5), 103.

Angelopoulos, V., Zhang, X. J., Artemyev, A. V., Mourenas, D., Tsai, E., Wilkins, C., Runov, A., Liu, J., Turner, D. L., Li, W., Khurana, K., Wirz, R. E., Sergeev, V. A., Meng, X., Wu, J., Hartinger, M. D., Raita, T., Shen, Y., An, X., Shi, X., Bashir, M. F., Shen, X., Gan, L., Qin, M., Capannolo, L., Ma, Q., Russell, C. L., Masongsong, E. V., Caron, R., He, I., Iglesias, L., Jha, S., King, J., Kumar, S., Le, K., Mao, J., McDermott, A., Nguyen, K., Norris, A., Palla, A., Roosnovo, A., Tam, J., Xie, E., Yap, R. C., Ye, S., Young, C., Adair, L. A., Shaffer, C., Chung, M., Cruce, P., Lawson, M., Leneman, D., Allen, M., Anderson, M., Arreola-Zamora, M., Artinger, J., Asher, J., Branchevsky, D., Cliffe, M., Colton, K., Costello, C., Depe, D., Domae, B. W., Eldin, S., Fitzgibbon, L., Flemming, A., Frederick, D. M., Gilbert, A., Hesford, B., Krieger, R., Lian, K., McKinney, E., Miller, J. P., Pedersen, C., Qu, Z., Rozario, R., Rubly, M., Seaton, R., Subramanian, A., Sundin, S. R., Tan, A., Thomlinson, D., Turner, W., Wing, G., Wong, C. and Zarifian, A. [2023], ‘Energetic Electron Precipitation Driven by Electromagnetic Ion Cyclotron Waves from ELFIN’s Low Altitude Perspective’, *Space Sci. Rev.* **219**(5), 37.

Arnold, V. I., Kozlov, V. V. and Neishtadt, A. I. [2006], *Mathematical Aspects of Classical and Celestial Mechanics*, Dynamical Systems III. Encyclopedia of Mathematical Sciences, 3rd edn, Springer-Verlag, New York.

Artemyev, A. V., Agapitov, O., Mourenas, D., Krasnoselskikh, V., Shastun, V. and Mozer, F. [2016], ‘Oblique Whistler-Mode Waves in the Earth’s Inner Magnetosphere: Energy Distribution, Origins, and Role in Radiation Belt Dynamics’, *Space Sci. Rev.* **200**(1-4), 261–355.

Artemyev, A. V., Agapitov, O. V., Mourenas, D., Krasnoselskikh, V. V. and Mozer, F. S. [2015], ‘Wave energy budget analysis in the Earth’s radiation belts uncovers a missing energy’, *Nature Communications* **6**, 8143.

Artemyev, A. V., Agapitov, O. V., Mourenas, D., Krasnoselskikh, V. and Zelenyi, L. M. [2013], ‘Storm-induced energization of radiation belt electrons: Effect of wave obliquity’, *Geophys. Res. Lett.* **40**, 4138–4143.

Artemyev, A. V., Angelopoulos, V., Zhang, X. J., Chen, L. and Runov, A. [2023], ‘Dispersed Relativistic Electron Precipitation Patterns Between the Ion and Electron Isotropy Boundaries’, *Journal of Geophysical Research (Space Physics)*** **128**(12), e2023JA032200.

Artemyev, A. V., Demekhov, A. G., Zhang, X. J., Angelopoulos, V., Mourenas, D., Fedorenko, Y. V., Maninnen, J., Tsai, E., Wilkins, C., Kasahara, S., Miyoshi, Y., Matsuoka, A., Kasahara, Y., Mitani, T., Yokota, S., Keika, K., Hori, T., Matsuda, S., Nakamura, S., Kitahara, M., Takashima, T. and Shinohara, I. [2021], ‘Role of Ducting in Relativistic Electron Loss by Whistler-Mode Wave Scattering’, *Journal of Geophysical Research (Space Physics)*** **126**(11), e29851.
Artemyev, A. V., Krasnoselskikh, V., Agapitov, O., Mourenas, D. and Rolland, G. [2012], ‘Non-diffusive resonant acceleration of electrons in the radiation belts.’, *Physics of Plasmas* **19**, 122901.

Artemyev, A. V. and Mourenas, D. [2020], ‘On Whistler Mode Wave Relation to Electron Field-Aligned Plateau Populations’, *Journal of Geophysical Research (Space Physics)* **125**(3), e27735.

Artemyev, A. V., Mourenas, D., Agapitov, O. V., Vainchtein, D. L., Mozer, F. S. and Krasnoselskikh, V. V. [2015], ‘Stability of relativistic electron trapping by strong whistler or electromagnetic ion cyclotron waves’, *Physics of Plasmas* **22**, 082901.

Artemyev, A. V., Mourenas, D., Zhang, X. J. and Vainchtein, D. [2022], ‘On the Incorporation of Nonlinear Resonant Wave-Particle Interactions Into Radiation Belt Models’, *Journal of Geophysical Research (Space Physics)* **127**(9), e30853.

Artemyev, A. V., Neishtadt, A. I., Albert, J. M., Gan, L., Li, W. and Ma, Q. [2021], ‘Theoretical model of the nonlinear resonant interaction of whistler-mode waves and field-aligned electrons’, *Physics of Plasmas* **28**(5), 052902.

Artemyev, A. V., Neishtadt, A. I. and Angelopoulos, V. [2022], ‘On the Role of Whistler-Mode Waves in Electron Interaction With Dipolarizing Flux Bundles’, *Journal of Geophysical Research (Space Physics)* **127**(4), e30265.

Artemyev, A. V., Neishtadt, A. I., Vainchtein, D. L., Vasiliev, A. A., Vasko, I. Y. and Zelenyi, L. M. [2018], ‘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.

Artemyev, A. V., Neishtadt, A. I. and Vasiliev, A. A. [2019], ‘Kinetic equation for nonlinear wave-particle interaction: Solution properties and asymptotic dynamics’, *Physica D Nonlinear Phenomena* **393**, 1–8.

Artemyev, A. V., Neishtadt, A. I. and Vasiliev, A. A. [2020a], ‘A Map for Systems with Resonant Trappings and Scatterings’, *Regular and Chaotic Dynamics* **25**(1), 2–10.

Artemyev, A. V., Neishtadt, A. I. and Vasiliev, A. A. [2020b], ‘Mapping for nonlinear electron interaction with whistler-mode waves’, *Physics of Plasmas* **27**(4), 042902.

Artemyev, A. V., Neishtadt, A. I. and Vasiliev, A. A. [2021], *Kinetic Equation for Systems with Resonant Captures and Scatterings*, Springer Singapore, Singapore, pp. 171–180.

Artemyev, A. V., Neishtadt, A. I., Vasiliev, A. A. and Mourenas, D. [2016], ‘Kinetic equation for nonlinear resonant wave-particle interaction’, *Physics of Plasmas* **23**(9), 090701.

Artemyev, A. V., Neishtadt, A. I., Vasiliev, A. A. and Mourenas, D. [2017a], ‘Kinetic equation for systems with resonant captures and scatterings’, *ArXiv e-prints*.

Artemyev, A. V., Neishtadt, A. I., Vasiliev, A. A. and Mourenas, D. [2017b], ‘Probabilistic approach to nonlinear wave-particle resonant interaction’, *Phys. Rev. E* **95**(2), 023204.

Artemyev, A. V., Neishtadt, A. I., Vasiliev, A. A. and Mourenas, D. [2018], ‘Long-term evolution of electron distribution function due to nonlinear resonant interaction with whistler mode waves’, *Journal of Plasma Physics* **84**, 905840206.

Artemyev, A. V., Neishtadt, A. I., Vasiliev, A. A. and Mourenas, D. [2021], ‘Transitional regime of electron resonant interaction with whistler-mode waves in
inhomogeneous space plasma', *Phys. Rev. E* **104**(5), 055203.

Artemyev, A. V., Neishtadt, A. I., Vasiliev, A. A., Zhang, X.-J., Mourenas, D. and Vainchtein, D. [2021], 'Long-term dynamics driven by resonant wave-particle interactions: from Hamiltonian resonance theory to phase space mapping', *Journal of Plasma Physics* **87**(2), 835870201.

Artemyev, A. V., Orlova, K. G., Mourenas, D., Agapitov, O. V. and Krasnoselskikh, V. V. [2013], 'Electron pitch-angle diffusion: resonant scattering by waves vs. nonadiabatic effects', *Annales Geophysicae* **31**, 1485-1490.

Artemyev, A. V., Vainchtein, D., Neishtadt, A. and Zelenyi, L. [2011], 'Resonant acceleration of charged particles in the presence of random fluctuations', *Phys. Rev. E* **84**(4), 046213.

Artemyev, A. V., Vasiliev, A. A., Mourenas, D., Agapitov, O. and Krasnoselskikh, V. [2013], 'Nonlinear electron acceleration by oblique whistler waves: Landau resonance vs. cyclotron resonance.', *Physics of Plasmas* **20**, 122901.

Artemyev, A. V., Vasiliev, A. A., Mourenas, D., Agapitov, O., Krasnoselskikh, V., Boscher, D. and Rolland, G. [2014], 'Fast transport of resonant electrons in phase space due to nonlinear trapping by whistler waves', *Geophys. Res. Lett.* **41**, 5727-5733.

Artemyev, A. V., Vasiliev, A. A., Mourenas, D., Agapitov, O. V. and Krasnoselskikh, V. V. [2014], 'Electron scattering and nonlinear trapping by oblique whistler waves: The critical wave intensity for nonlinear effects', *Physics of Plasmas* **21**(10), 102903.

Artemyev, A. V., Vasiliev, A. A., Mourenas, D., Neishtadt, A. I., Agapitov, O. V. and Krasnoselskikh, V. [2015], 'Probability of relativistic electron trapping by parallel and oblique whistler-mode waves in Earth's radiation belts', *Physics of Plasmas* **22**(11), 112903.

Artemyev, A. V., Vasiliev, A. A. and Neishtadt, A. I. [2019], 'Charged particle nonlinear resonance with localized electrostatic wave-packets', *Communications in Nonlinear Science and Numerical Simulations* **72**, 392-406.

Artemyev, A. V., Zhang, X. J., Angelopoulos, V., Mourenas, D., Vainchtein, D., Shen, Y., Vasko, I. and Runov, A. [2020], 'Ionosphere Feedback to Electron Scattering by Equatorial Whistler Mode Waves', *Journal of Geophysical Research (Space Physics)* **125**(9), e28373.

Artemyev, A. V., Zhang, X. J., Demekhov, A. G., Meng, X., Angelopoulos, V. and Fedorenko, Y. V. [2024], 'Relativistic Electron Precipitation Driven by Mesoscale Transients, Inferred From Ground and Multi-Spacecraft Platforms', *Journal of Geophysical Research (Space Physics)* **129**(2), e2023JA032287.

Artemyev, A. V., Zhang, X. J., Zou, Y., Mourenas, D., Angelopoulos, V., Vainchtein, D., Tsai, E. and Wilkins, C. [2022], 'On the Nature of Intense Sub-Relativistic Electron Precipitation', *Journal of Geophysical Research (Space Physics)* **127**(6), e30571.

Auster, H. U., Glassmeier, K. H., Magnes, W., Aydogar, O., Baumjohann, W., Constantinescu, D., Fischer, D., Fornacon, K. H., Georgescu, E., Harvey, P., Hillenmaier, O., Kroth, R., Ludlam, M., Narita, Y., Nakamura, R., Okafka, K., Plaschke, F., Richter, I., Schwarzl, H., Stoll, B., Valavanoglou, A. and Wiedemann, M. [2008], 'The THEMIS Fluxgate Magnetometer', *Space Sci. Rev.* **141**, 235-264.

Baker, D. N., Erickson, P. J., Fennell, J. F., Foster, J. C., Jaynes, A. N. and Verronen, P. T. [2018], 'Space Weather Effects in the Earth's Radiation Belts',
*Space Sci. Rev.* **214**(1), 17.

Baker, D. N., Jaynes, A. N., Hoxie, V. C., Thorne, R. M., Foster, J. C., Li, X., Fennell, J. F., Wygant, J. R., Kanekal, S. G., Erickson, P. J., Kurth, W., Li, W., Ma, Q., Schiller, Q., Blum, L., Malaspina, D. M., Gerrard, A. and Lanzerotti, L. J. [2014], ‘An impenetrable barrier to ultrarelativistic electrons in the Van Allen radiation belts’, *Nature* **515**(7528), 531–534.

Baker, D. N., Jaynes, A. N., Kanekal, S. G., Foster, J. C., Erickson, P. J., Fennell, J. F., Blake, J. B., Zhao, H., Li, X., Elkington, S. R., Henderson, M. G., Reeves, G. D., Spence, H. E., Kletzing, C. A. and Wygant, J. R. [2016], ‘Highly relativistic radiation belt electron acceleration, transport, and loss: Large solar storm events of March and June 2015’, *Journal of Geophysical Research (Space Physics)* **121**(7), 6647–6660.

Baker, D. N., Kanekal, S. G., Hoxie, V. C., Batiste, S., Bolton, M., Li, X., Elkington, S. R., Monk, S., Reukauf, R., Steg, S., Westfall, J., Belting, C., Bolton, B., Braun, D., Cervelli, B., Hubbell, K., Kien, M., Knappmiller, S., Wade, S., Lamprecht, B., Stevens, K., Wallace, J., Yehle, A., Spence, H. E. and Friedel, R. [2013], ‘The Relativistic Electron-Proton Telescope (REPT) Instrument on Board the Radiation Belt Storm Probes (RBSP) Spacecraft: Characterization of Earth’s Radiation Belt High-Energy Particle Populations’, *Space Sci. Rev.* **179**, 337–381.

Bell, T. F. [1984], ‘The nonlinear gyroresonance interaction between energetic electrons and coherent VLF waves propagating at an arbitrary angle with respect to the earth’s magnetic field’, *J. Geophys. Res.* **89**, 905–918.

Bell, T. F. [1986], ‘The wave magnetic field amplitude threshold for nonlinear trapping of energetic gyroresonant and Landau resonant electrons by nonducted VLF waves in the magnetosphere’, *J. Geophys. Res.* **91**, 4365–4379.

Bell, T. F., Inan, U. S., Bortnik, J. and Scudder, J. D. [2002], ‘The Landau damping of magnetospherically reflected whistlers within the plasmasphere’, *Geophys. Res. Lett.* **29**, 1733.

Benkadda, S., Sen, A. and Shklyar, D. R. [1996], ‘Chaotic dynamics of charged particles in the field of two monochromatic waves in a magnetized plasma’, *Chaos* **6**(3), 451–460.

Birmingham, T. J. [1984], ‘Pitch angle diffusion in the Jovian magnetodisc’, *J. Geophys. Res.* **89**, 2699–2707.

Blake, J. B., Carranza, P. A., Claudepierre, S. G., Clemmons, J. H., Crain, W. R., Dotan, Y., Fennell, J. F., Fuentes, F. H., Galvan, R. M., George, J. S., Henderson, M. G., Lalic, M., Lin, A. Y., Looper, M. D., Mabry, D. J., Mazur, J. E., McCarthy, B., Nguyen, C. Q., O’Brien, T. P., Perez, M. A., Redding, M. T., Roeder, J. L., Salvaggio, D. J., Sorensen, G. A., Spence, H. E., Yi, S. and Zakrzewski, M. P. [2013], ‘The Magnetic Electron Ion Spectrometer (MagEIS) Instruments Aboard the Radiation Belt Storm Probes (RBSP) Spacecraft’, *Space Sci. Rev.* **179**, 383–421.

Bonnell, J. W., Mozer, F. S., Delory, G. T., Hull, A. J., Ergun, R. E., Cully, C. M., Angelopoulos, V. and Harvey, P. R. [2008], ‘The Electric Field Instrument (EFI) for THEMIS’, *Space Sci. Rev.* **141**, 303–341.

Bortnik, J., Albert, J. M., Artemyev, A., Li, W., Jun, C.-W., Grach, V. S. and Demekhov, A. G. [2022], ‘Amplitude Dependence of Nonlinear Precipitation Blocking of Relativistic Electrons by Large Amplitude EMIC Waves’, *Geophys. Res. Lett.* **49**(12), e98365.
Bortnik, J., Inan, U. S. and Bell, T. F. [2006], ‘Landau damping and resultant unidirectional propagation of chorus waves’, *Geophys. Res. Lett.* **33**, 3102.

Bortnik, J. and Thorne, R. M. [2010], ‘Transit time scattering of energetic electrons due to equatorially confined magnetosonic waves’, *J. Geophys. Res.* **115**, 7213.

Bortnik, J., Thorne, R. M. and Inan, U. S. [2008], ‘Nonlinear interaction of energetic electrons with large amplitude chorus’, *Geophys. Res. Lett.* **35**, 21102.

Bortnik, J., Thorne, R. M., Meredith, N. P. and Santolik, O. [2007], ‘Ray tracing of penetrating chorus and its implications for the radiation belts’, *Geophys. Res. Lett.* **34**, L15109.

Bortnik, J., Thorne, R. M., Ni, B. and Li, J. [2015], ‘Analytical approximation of transit time scattering due to magnetosonic waves’, *Geophys. Res. Lett.* **42**, 1318–1325.

Breuillard, H., Agapitov, O., Artemyev, A., Krasnoselskikh, V., Le Contel, O., Cully, C. M., Angelopoulos, V., Zaliznyak, Y. and Rolland, G. [2014], ‘On the origin of falling-tone chorus elements in Earth’s inner magnetosphere’, *Annales Geophysicae* **32**, 1477–1485.

Breuillard, H., Le Contel, O., Retino, A., Chasapis, A., Chust, T., Mirioni, L., Graham, D. B., Wilder, F. D., Cohen, I., Vaivads, A., Khotyaintsev, Y. V., Lindqvist, P.-A., Marklund, G. T., Burch, J. L., Torbert, R. B., Ergun, R. E., Goodrich, K. A., Macri, J., Needell, J., Chutter, M., Rau, D., Dors, I., Russell, C. T., Magnes, W., Strangeway, R. J., Bromund, K. R., Plaschke, F., Fischer, D., Leinweber, H. K., Anderson, B. J., Le, G., Slavin, J. A., Kepko, E. L., Baumjohann, W., Mauk, B., Fuselier, S. A. and Nakamura, R. [2016], ‘Multispacecraft analysis of dipolarization fronts and associated whistler wave emissions using MMS data’, *Geophys. Res. Lett.* **43**, 7279–7286.

Breuillard, H., Zaliznyak, Y., Krasnoselskikh, V., Agapitov, O., Artemyev, A. and Rolland, G. [2012], ‘Chorus wave-normal statistics in the Earth’s radiation belts from ray tracing technique’, *Ann. Geophys.* **30**, 1223–1233.

Brinca, A. L. [1978], ‘Turbulence effects in the cyclotron resonance of monochromatic whistlers’, *Geophys. Res. Lett.* **5**, 839–842.

Brinca, A. L. [1980], ‘On the evolution of the geomagnetospheric coherent cyclotron resonance in the midst of noise’, *J. Geophys. Res.* **85**, 4711–4714.

Brizard, A. J. [2023], ‘Variational formulation of higher-order guiding-center Vlasov-Maxwell theory’, *Physics of Plasmas* **30**(10), 102106.

Brizard, A. J. and Chan, A. A. [2022], ‘Hamiltonian formulations of quasilinear theory for magnetized plasmas’, *Frontiers in Astronomy and Space Sciences* **9**, 1010133.

Büchner, J. and Zelenyi, L. M. [1989], ‘Regular and chaotic charged particle motion in magnetotaillike field reversals. I - Basic theory of trapped motion’, *J. Geophys. Res.* **94**, 11821–11842.

Burton, R. K. and Holzer, R. E. [1974], ‘The Origin and Propagation of Chorus in the Outer Magnetosphere’, *J. Geophys. Res.* **79**, 1014–1023.

Cai, B., Wu, Y. and Tao, X. [2020], ‘Effects of Nonlinear Resonance Broadening on Interactions Between Electrons and Whistler Mode Waves’, *Geophys. Res. Lett.* **47**(11), e87991.

Camporeale, E. [2015], ‘Resonant and nonresonant whistlers-particle interaction in the radiation belts’, *Geophys. Res. Lett.* **42**(9), 3114–3121.

Camporeale, E. and Zimbardo, G. [2015], ‘Wave-particle interactions with parallel whistler waves: Nonlinear and time-dependent effects revealed by particle-in-cell
simulations', *Physics of Plasmas* **22**(9), 092104.

Capannolo, L., Li, W., Ma, Q., Qin, M., Shen, X. C., Angelopoulos, V., Artemyev, A., Zhang, X. J. and Hanzelka, M. [2023], 'Electron Precipitation Observed by ELFIN Using Proton Precipitation as a Proxy for Electromagnetic Ion Cyclotron (EMIC) Waves', *Geophys. Res. Lett.* **50**(21), e2023GL103519.

Cary, J. R. and Brizard, A. J. [2009], ‘Hamiltonian theory of guiding-center motion’, *Reviews of Modern Physics* **81**, 693–738.

Cattell, C., Wygant, J. R., Goetz, K., Kersten, K., Kellogg, P. J., von Rosenvinge, T., Bale, S. D., Roth, I., Temerin, M., Hudson, M. K., Mewaldt, R. A., Wiedenbeck, M., Maksimovic, M., Ergun, R., Acuna, M. and Russell, C. T. [2008], ‘Discovery of very large amplitude whistler-mode waves in Earth’s radiation belts’, *Geophys. Res. Lett.* **35**, 1105.

Chan, A. A., Elkington, S. R., Longley, W. J., Aldhurais, S. A., Alam, S. S., Albert, J. M., Jaynes, A. N., Malaspina, D. M., Ma, Q. and Li, W. [2023], ‘Simulation of radiation belt wave-particle interactions in an mhd-particle framework’, *Frontiers in Astronomy and Space Sciences* **10**.

Chen, H., Wang, X., Chen, L., Zhang, X.-J., Omura, Y., Chen, R., Hsieh, Y.-K., Lin, Y. and Xia, Z. [2024], ‘Nonlinear Electron Trapping Through Cyclotron Resonance in the Formation of Chorus Subpackets’, *Geophys. Res. Lett.* **51**(11), e2024GL109481.

Chen, L., Breneman, A. W., Xia, Z. and Zhang, X.-j. [2020], ‘Modeling of Bouncing Electron Microbursts Induced by Ducted Chorus Waves’, *Geophys. Res. Lett.* **47**(17), e89400.

Chen, L., Thorne, R. M., Li, W. and Bortnik, J. [2013], ‘Modeling the wave normal distribution of chorus waves’, *J. Geophys. Res.* **118**, 1074–1088.

Chen, L., Zhang, X.-J., Artemyev, A., Angelopoulos, V., Tsai, E., Wilkins, C. and Horne, R. B. [2022], ‘Ducted Chorus Waves Cause Sub-Relativistic and Relativistic Electron Microbursts’, *Geophys. Res. Lett.* **49**(5), e97559.

Chen, R., Gao, X., Lu, Q., Chen, L., Tsurutani, B. T., Li, W., Ni, B. and Wang, S. [2021], ‘In Situ Observations of Whistler Mode Chorus Waves Guided by Density Ducts’, *Journal of Geophysical Research (Space Physics)* **126**(4), e28814.

Chen, R., Gao, X., Lu, Q., Tsurutani, B. T. and Wang, S. [2021], ‘Observational Evidence for Whistler Mode Waves Guided/Ducted by the Inner and Outer Edges of the Plasmapause’, *Geophys. Res. Lett.* **48**(6), e92652.

Chen, R., Gao, X., Lu, Q. and Wang, S. [2019], ‘Unraveling the Correlation Between Chorus Wave and Electron Beam-Like Distribution in the Earth’s Magnetosphere’, *Geophys. Res. Lett.* **46**(21), 11,671–11,678.

Chen, Y., Reeves, G. D. and Friedel, R. H. W. [2007], ‘The energization of relativistic electrons in the outer Van Allen radiation belt’, *Nature Physics* **3**, 614–617.

Chirikov, B. V. [1987], *Particle dynamics in magnetic traps*, Vol. 13, 1st edn, Consultants Bureau, New York.

Claudepierre, S. G., Blake, J. B., Boyd, A. J., Clemmons, J. H., Fennell, J. F., Gabrielse, C., Looper, M. D., Mazur, J. E., O’Brien, T. P., Reeves, G. D., Roeder, J. L., Spence, H. E. and Turner, D. L. [2021], ‘The Magnetic Electron Ion Spectrometer: A Review of On-Orbit Sensor Performance, Data, Operations, and Science’, *Space Science Reviews* **217**, 80.

Cohen, R. H., Rowlands, G. and Foote, J. H. [1978], ‘Nonadiabaticity in mirror machines’, *Physics of Fluids* **21**, 627–644.
Collier, A. B., Lichtenberger, J., Clilverd, M. A., Rodger, C. J. and Steinbach, P. [2011], 'Source region for whistlers detected at Rothera, Antarctica', *Journal of Geophysical Research (Space Physics)* **116**(A3), A03219.

Cully, C. M., Angelopoulos, V., Auster, U., Bonnell, J. and Le Contel, O. [2011], 'Observational evidence of the generation mechanism for rising-tone chorus', *Geophys. Res. Lett.* **38**, 1106.

Cully, C. M., Bonnell, J. W. and Ergun, R. E. [2008], 'THEMIS observations of long-lived regions of large-amplitude whistler waves in the inner magnetosphere', *Geophys. Res. Lett.* **35**, 17.

Cully, C. M., Ergun, R. E., Stevens, K., Nammari, A. and Westfall, J. [2008], 'The THEMIS Digital Fields Board', *Space Sci. Rev.* **141**, 343-355.

Demekhov, A. G. [2011], 'Generation of VLF emissions with the increasing and decreasing frequency in the magnetospheric cyclotron maser in the backward wave oscillator regime', *Radiophysics and Quantum Electronics* **53**, 609-622.

Demekhov, A. G., Manninen, J., Santolík, O. and Titova, E. E. [2017], 'Conjugate Ground-Spacecraft Observations of VLF Chorus Elements', *Geophys. Res. Lett.* **44**(23), 11,735–11,744.

Demekhov, A. G., Taubenschuss, U. and Santolík, O. [2017], 'Simulation of VLF chorus emissions in the magnetosphere and comparison with THEMIS spacecraft data', *J. Geophys. Res.* **122**, 166–184.

Demekhov, A. G. and Trakhtengerts, V. Y. [2008], 'Dynamics of the magnetospheric cyclotron ELF/VLF maser in the backward-wave-oscillator regime. II. The influence of the magnetic-field inhomogeneity', *Radiophysics and Quantum Electronics* **51**, 880–889.

Demekhov, A. G., Trakhtengerts, V. Y., Rycroft, M. J. and Nunn, D. [2006], 'Electron acceleration in the magnetosphere by whistler-mode waves of varying frequency', *Geomagnetism and Aeronomy* **46**, 711–716.

Denton, R. E., Takahashi, K., Galkin, I. A., Nsumei, P. A., Huang, X., Reinisch, B. W., Anderson, R. R., Sleeper, M. K. and Hughes, W. J. [2006], 'Distribution of density along magnetospheric field lines', *J. Geophys. Res.* **111**, 4213.

Drummond, W. E. and Pines, D. [1962], 'Nonlinear stability of plasma oscillations', *Nuclear Fusion Suppl.* **3**, 1049-1058.

Dubyagin, S., Apatenkov, S., Gordeev, E., Ganushkina, N. and Zheng, Y. [2021], 'Conditions of Loss Cone Filling by Scattering on the Curved Field Lines for 30 keV Protons During Geomagnetic Storm as Inferred From Numerical Trajectory Tracing', *Journal of Geophysical Research (Space Physics)* **126**(1), e28490.

Dudley, R. M. [2014], *Uniform Central Limit Theorems*, Cambridge Studies in Advanced Mathematics, 2 edn, Cambridge University Press.

Dykhne, A. M. [1960], 'Quantum transitions in the adiabatic approximation', *Soviet JETP* **11**, 411-415.

Elkington, S. R., Wiltberger, M., Chan, A. A. and Baker, D. N. [2004], 'Physical models of the geospace radiation environment', *Journal of Atmospheric and Solar-Terrestrial Physics* **66**(15-16), 1371-1387.

Elliott, P. D. T. A. [1980], *Probabilistic Number Theory II: Central Limit Theorems*, Springer New York, NY.

Eshetu, W. W., Lyon, J. G., Hudson, M. K. and Wiltberger, M. J. [2018], 'Pitch Angle Scattering of Energetic Electrons by BBFs', *Journal of Geophysical Research (Space Physics)* **123**(11), 9265-9274.
Fajans, J. and Frièdland, L. [2001], ‘Autoresonant (nonstationary) excitation of pendulums, Plutinos, plasmas, and other nonlinear oscillators’, *American Journal of Physics* **69**(10), 1096–1102.

Foster, J. C., Erickson, P. J., Omura, Y., Baker, D. N., Kletzing, C. A. and Claude-pierre, S. G. [2017], ‘Van allen probes observations of prompt mev radiation belt electron acceleration in nonlinear interactions with vlf chorus’, *Journal of Geophysical Research: Space Physics* **122**(1), 324–339.
URL: http://dx.doi.org/10.1002/2016JA023429

Frantsuzov, V. A., Artemyev, A. V., Zhang, X.-J., Allanson, O., Shustov, P. I. and Petrukovich, A. A. [2023], ‘Diffusive scattering of energetic electrons by intense whistler-mode waves in an inhomogeneous plasma’, *Journal of Plasma Physics* **89**(1), 905890101.

Friedland, L. [2009], ‘Autoresonance in nonlinear systems’, *Scholarpedia* **4**(1), 5473.

Fu, X., Cowee, M. M., Friedel, R. H., Funsten, H. O., Gary, S. P., Hospodarsky, G. B., Kletzing, C., Kurth, W., Larsen, B. A., Liu, K., MacDonald, E. A., Min, K., Reeves, G. D., Skoug, R. M. and Winske, D. [2014], ‘Whistler anisotropy instabilities as the source of banded chorus: Van Allen Probes observations and particle-in-cell simulations’, *Journal of Geophysical Research (Space Physics)* **119**, 8288–8298.

Fu, X., Guo, Z., Dong, C. and Gary, S. P. [2015], ‘Nonlinear subcyclotron resonance as a formationmechanism for gaps in banded chorus’, *Geophys. Res. Lett.* **42**, 3150–3159.

Furuya, N., Omura, Y. and Summers, D. [2008], ‘Relativistic turning acceleration of radiation belt electrons by whistler mode chorus’, *J. Geophys. Res.* **113**, 4224.

Galeev, A. A. and Sagdeev, R. Z. [1979], *Nonlinear Plasma Theory*, in A. M. A. Leontovich, ed., ‘Reviews of Plasma Physics, Volume 7’, Vol. 7 of *Reviews of Plasma Physics*, p. 1.

Gan, L., Artemyev, A., Li, W., Zhang, X.-J., Ma, Q., Mourenas, D., Angelopoulos, V., Tsai, E. and Wilkins, C. [2023], ‘Bursty Energetic Electron Precipitation by High-Order Resonance With Very-Oblique Whistler-Mode Waves’, *Geophys. Res. Lett.* **50**(8), e2022GL101920.

Gan, L., Li, W., Albert, J. M., Hanzelka, M., Ma, Q. and Artemyev, A. V. [2024], ‘nhomogeneity Ratio for Nearly Field-Aligned Electrons Interacting with Whistler-Mode Waves’, *Geophys. Res. Lett.* .

Gan, L., Li, W., Ma, Q., Albert, J. M., Artemyev, A. V. and Bortnik, J. [2020], ‘Nonlinear Interactions Between Radiation Belt Electrons and Chorus Waves: Dependence on Wave Amplitude Modulation’, *Geophys. Res. Lett.* **47**(4), e85987.

Gan, L., Li, W., Ma, Q., Artemyev, A. V. and 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**(5), e30063.

Gao, L., Vainchtein, D., Artemyev, A. V. and Zhang, X. J. [2022], ‘Statistics of Whistler-Mode Waves in the Near-Earth Plasma Sheet’, *Journal of Geophysical Research (Space Physics)* **127**(8), e30603.

Gao, X., Mourenas, D., Li, W., Artemyev, A. V., Lu, Q., Tao, X. and Wang, S. [2016], ‘Observational evidence of generation mechanisms for very oblique lower band chorus using THEMIS waveform data’, *Journal of Geophysical Research (Space Physics)* **121**(7), 6732–6748.
Gao, Y., Neishtadt, A. and Okunev, A. [2023], 'On phase at a resonance in slow-fast Hamiltonian systems', *Reguar and Chaotic Dynamics* **28**(4-5), 585-612.

Glauert, S. A. and Horne, R. B. [2005], 'Calculation of pitch angle and energy diffusion coefficients with the PADIE code', *J. Geophys. Res.* **110**, 4206.

Glauert, S. A., Horne, R. B. and Meredith, N. P. [2014], 'Three-dimensional electron radiation belt simulations using the BAS Radiation Belt Model with new diffusion models for chorus, plasmaspheric hiss, and lightning-generated whistlers', *J. Geophys. Res.* **119**, 268-289.

Glauert, S. A., Horne, R. B. and Meredith, N. P. [2018], 'A 30-Year Simulation of the Outer Electron Radiation Belt', *Space Weather* **16**, 1498-1522.

Grach, V. S., Artemyev, A. V., Demekhov, A. G., Zhang, X.-J., Bortnik, J., Angelopoulos, V., Nakamura, R., Tsai, E., Wilkins, C. and Roberts, O. W. [2022], 'Relativistic Electron Precipitation by EMIC Waves: Importance of Nonlinear Resonant Effects', *Geophys. Res. Lett.* **49**(17), e99994.

Grach, V. S. and Demekhov, A. G. [2018a], 'Resonance Interaction of Relativistic Electrons with Ion-Cyclotron Waves. I. Specific Features of the Nonlinear Interaction Regimes', *Radiophysics and Quantum Electronics* **60**(12), 942-959.

Grach, V. S. and Demekhov, A. G. [2018b], 'Resonant Interaction of Relativistic Electrons with Electromagnetic Ion-Cyclotron Waves. II. Integral Parameters of Interaction Regimes', *Radiophysics and Quantum Electronics* **61**(6), 389-401.

Grach, V. S. and Demekhov, A. G. [2020], 'Precipitation of Relativistic Electrons Under Resonant Interaction With Electromagnetic Ion Cyclotron Wave Packets', *Journal of Geophysical Research (Space Physics)* **125**(2), e27358.

Grach, V. S. and Demekhov, A. G. [2023], 'Interaction of Relativistic Electrons with Packets of the Electromagnetic Ion Cyclotron Waves of Finite Length and Low Amplitude', *Plasma Physics Reports* **49**(7), 901-911.

Grigorenko, E. E., Malykhin, A. Y., Shklyar, D. R., Fadanelli, S., Lavraud, B., Panov, E. V., Avanov, L., Giles, B. and Le Contel, O. [2020], 'Investigation of Electron Distribution Functions Associated With Whistler Waves at Dipo-larization Fronts in the Earth's Magnetotail: MMS Observations', *Journal of Geophysical Research (Space Physics)* **125**(9), e28268.

Hanzelka, M., Li, W. and Ma, Q. [2023], 'Parametric analysis of pitch angle scattering and losses of relativistic electrons by oblique EMIC waves', *Frontiers in Astronomy and Space Sciences* **10**, 1163515.

Hanzelka, M. and Santolík, O. [2019], 'Effects of Ducting on Whistler Mode Chorus or Exohiss in the Outer Radiation Belt', *Geophys. Res. Lett.* **46**(11), 5735-5745.

Hanzelka, M. and Santolík, O. [2022], 'Effects of Field-Aligned Cold Plasma Density Filaments on the Fine Structure of Chorus', *Geophys. Res. Lett.* **49**(24), e2022GL101654.

Hanzelka, M., Santolík, O., Omura, Y., Kolmašová, I. and Kletzing, C. A. [2020], 'A Model of the Subpacket Structure of Rising Tone Chorus Emissions', *Journal of Geophysical Research (Space Physics)* **125**(8), e28094.

Hardy, D. A., Schmitt, L. K., Gussenhoven, M. S., Marshall, F. J. and Yeh, H. C. [1984], 'Precipitating electron and ion detectors (SSJ/4) for the block 5D/Flights 6-10 DMSP (Defense Meteorological Satellite Program) satellites: Calibration and data presentation', Unknown.

Helliwell, R. A. [1965], *Whistlers and Related Ionospheric Phenomena*, Stanford University Press, USA.
Helliwell, R. A. [1967], ‘A theory of discrete VLF emissions from the magnetosphere’, *J. Geophys. Res.* **72**, 4773–4790.

Helliwell, R. A. and Crystal, T. L. [1973], ‘A feedback model of cyclotron interaction between whistler-mode waves and energetic electrons in the magnetosphere’, *J. Geophys. Res.* **78**(31), 7357–7371.

Henrard, J. and Lemaitre, A. [1983], ‘A Second Fundamental Model for Resonance’, *Celestial Mechanics* **30**(2), 197–218.

Hiraga, R. and Omura, Y. [2020], ‘Acceleration mechanism of radiation belt electrons through interaction with multi-subpacket chorus waves’, *Earth, Planets, and Space* **72**(1), 21.

Horne, R. B. [2007], ‘Plasma astrophysics: Acceleration of killer electrons’, *Nature Physics* **3**, 590–591.

Horne, R. B., Glauert, S. A., Meredith, N. P., Boscher, D., Maget, V., Heynderickx, D. and Pitchford, D. [2013], ‘Space weather impacts on satellites and forecasting the Earth’s electron radiation belts with SPACECAST’, *Space Weather* **11**, 169–186.

Horne, R. B., Kersten, T., Glauert, S. A., Meredith, N. P., Boscher, D., Sicardi-Piet, A., Thorne, R. M. and Li, W. [2013], ‘A new diffusion matrix for whistler mode chorus waves’, *J. Geophys. Res.* **118**, 6302–6318.

Horne, R. B., Thorne, R. M., Shprits, Y. Y., Meredith, N. P., Glauert, S. A., Smith, A. J., Kanekal, S. G., Baker, D. N., Engebretson, M. J., Posch, J. L., Spasojevic, M., Inan, U. S., Pickett, J. S. and Decreau, P. M. E. [2005], ‘Wave acceleration of electrons in the Van Allen radiation belts’, *Nature* **437**, 227–230.

Hsieh, Y.-K., Kubota, Y. and Omura, Y. [2020], ‘Nonlinear Evolution of Radiation Belt Electron Fluxes Interacting With Oblique Whistler Mode Chorus Emissions’, *Journal of Geophysical Research (Space Physics)* **125**(2), e27465.

Hsieh, Y.-K. and Omura, Y. [2017a], ‘Nonlinear dynamics of electrons interacting with oblique whistler mode chorus in the magnetosphere’, *J. Geophys. Res.* **122**, 675–694.

Hsieh, Y.-K. and Omura, Y. [2017b], ‘Study of wave-particle interactions for whistler mode waves at oblique angles by utilizing the gyroaveraging method’, *Radio Science* **52**(10), 1268–1281. 2017RS006245.
URL: http://dx.doi.org/10.1002/2017RS006245

Hsieh, Y.-K. and Omura, Y. [2023], ‘Precipitation Rates of Electrons Interacting With Lower-Band Chorus Emissions in the Inner Magnetosphere’, *Journal of Geophysical Research (Space Physics)* **128**(6), e2023JA031307.

Hsieh, Y.-K., Omura, Y. and Kubota, Y. [2022], ‘Energetic Electron Precipitation Induced by Oblique Whistler Mode Chorus Emissions’, *Journal of Geophysical Research (Space Physics)* **127**(1), e29583.

Hudson, M., Brito, T., Elkington, S., Kress, B., Li, Z. and Wiltberger, M. [2012], ‘Radiation belt 2D and 3D simulations for CIR-driven storms during Carrington Rotation 2068’, *Journal of Atmospheric and Solar-Terrestrial Physics* **83**, 51–62.

Hudson, M. K., Baker, D. N., Goldstein, J., Kress, B. T., Paral. J., Toffoletto, F. R. and Wiltberger, M. [2014], ‘Simulated magnetopause losses and Van Allen Probe flux dropouts’, *Geophys. Res. Lett.* **41**, 1113–1118.

Hudson, M. K., Elkington, S. R., Li, Z., Patel, M., Pham, K., Sorathia, K., Boyd, A., Jaynes, A. and Leali, A. [2021], ‘MHD-Test Particles Simulations of Moderate CME and CIR-Driven Geomagnetic Storms at Solar Minimum’, *Space Weather* **19**(12), e02882.
Hudson, M. K., Elkington, S. R., Lyon, J. G., Goodrich, C. C. and Rosenberg, T. J. [1999], ‘Simulation of radiation belt dynamics driven by solar wind variations’, *Washington DC American Geophysical Union Geophysical Monograph Series* **109**, 171–182.

Hudson, M. K., Paral, J., Kress, B. T., Wiltberger, M., Baker, D. N., Foster, J. C., Turner, D. L. and Wygant, J. R. [2015], ‘Modeling CME-shock-driven storms in 2012-2013: MHD test particle simulations’, *J. Geophys. Res.* **120**, 1168–1181.

Imhof, W. L., Reagan, J. B. and Gaines, E. E. [1979], ‘Studies of the sharply defined L dependent energy threshold for isotropy at the midnight trapping boundary’, *J. Geophys. Res.* **84**, 6371–6384.

Inan, U. S. [1987], ‘Gyroresonant pitch angle scattering by coherent and incoherent whistler mode waves in the magnetosphere’, *J. Geophys. Res.* **92**(A1), 127–142.

Inan, U. S., Bell, T. F. and Helliwell, R. A. [1978], ‘Nonlinear pitch angle scattering of energetic electrons by coherent VLF waves in the magnetosphere’, *J. Geophys. Res.* **83**(A7), 3235–3254.

Istomin, Y. N., Karpman, V. I. and Shklyar, D. R. [1973], ‘Nonlinear theory of a quasimonochromatic packet of electrostatic waves in an inhomogeneous plasma’, *Soviet Journal of Experimental and Theoretical Physics* **37**, 1045.

Itin, A. P., Neishtadt, A. I. and 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.

Kang, N., Artemyev, A. V., Bortnik, J., Zhang, X.-J. and Angelopoulos, V. [2024], ‘The principal role of chorus ducting for night-side relativistic electron precipitation’, *ESS Open Archive*.

Kang, N., Bortnik, J., Zhang, X., Claudepierre, S. and Shi, X. [2022], ‘Relativistic Microburst Scale Induced by a Single Point-Source Chorus Element’, *Geophys. Res. Lett.* **49**(23), e2022GL100841.

Karimabadi, H., Akimoto, K., Omidi, N. and Menyuk, C. R. [1990], ‘Particle acceleration by a wave in a strong magnetic field - Regular and stochastic motion’, *Physics of Fluids B* **2**, 606–628.

Karimabadi, H., Krauss-Varban, D. and Terasawa, T. [1992], ‘Physics of pitch angle scattering and velocity diffusion 1. Theory’, *J. Geophys. Res.* **97**(A9), 13853–13864.

Karimabadi, H. and Menyuk, C. R. [1991], ‘A fast and accurate method of calculating particle diffusion: Application to the ionosphere’, *J. Geophys. Res.* **96**(A6), 9669–9687.

Karney, C. F. F. [1978], ‘Stochastic ion heating by a lower hybrid wave’, *Physics of Fluids* **21**, 1584–1599.

Karpman, V. I. [1974], ‘Nonlinear Effects in the ELF Waves Propagating along the Magnetic Field in the Magnetosphere’, *Space Sci. Rev.* **16**, 361–388.

Karpman, V. I., Istomin, I. N. and Shkliar, D. R. [1975], ‘Effects of nonlinear interaction of monochromatic waves with resonant particles in the inhomogeneous plasma’, *Physica Scripta* **11**, 278–284.

Karpman, V. I., Istomin, J. N. and Shklyar, D. R. [1974], ‘Nonlinear theory of a quasi-monochromatic whistler mode packet in inhomogeneous plasma’, *Plasma Physics* **16**, 685–703.

Karpman, V. I. and Kaufman, R. N. [1982], ‘Whistler wave propagation in density ducts’, *Journal of Plasma Physics* **27**(2), 225–238.
Karpman, V. I. and Shklyar, D. R. [1977], 'Particle precipitation caused by a single whistler-mode wave injected into the magnetosphere', *Plan. Sp. Sci.* **25**, 395-403.

Kasahara, S., Miyoshi, Y., Yokota, S., Kasahara, Y., Matsuda, S., Kumamoto, A., Matsuoka, A., Kazama, Y., Frey, H. U., Angelopoulos, V., Kurita, S., Keika, K., Seki, K. and Shinohara, I. [2018], 'Pulsating aurora from electron scattering by chorus waves', *Nature* **554**, 337-340.

Kasahara, Y., Kasaba, Y., Kojima, H., Yagitani, S., Ishisaka, K., Kumamoto, A., Tsuchiya, F., Ozaki, M., Matsuda, S., Imachi, T., Miyoshi, Y., Hikishima, M., Katoh, Y., Ota, M., Shoji, M., Matsuoka, A. and Shinohara, I. [2018], 'The Plasma Wave Experiment (PWE) on board the Arase (ERG) satellite', *Earth, Planets, and Space* **70**(1), 86.

Katoh, Y. and Omura, Y. [2004], 'Acceleration of relativistic electrons due to resonant scattering by whistler mode waves generated by temperature anisotropy in the inner magnetosphere', *J. Geophys. Res.* **109**, 12214.

Katoh, Y. and Omura, Y. [2007], 'Relativistic particle acceleration in the process of whistler-mode chorus wave generation', *Geophys. Res. Lett.* **34**, L13102.

Katoh, Y. and Omura, Y. [2013], 'Effect of the background magnetic field inhomogeneity on generation processes of whistler-mode chorus and broadband hiss-like emissions', *J. Geophys. Res.* **118**, 4189-4198.

Katoh, Y. and Omura, Y. [2016], 'Electron hybrid code simulation of whistler-mode chorus generation with real parameters in the Earth's inner magnetosphere', *Earth, Planets, and Space* **68**(1), 192.

Katoh, Y., Omura, Y. and Summers, D. [2008], 'Rapid energization of radiation belt electrons by nonlinear wave trapping', *Annales Geophysicae* **26**, 3451-3456.

Ke, Y., Chen, L., Gao, X., Lu, Q., Wang, X., Chen, R., Chen, H. and Wang, S. [2021], 'Whistler Mode Waves Trapped by Density Irregularities in the Earth's Magnetosphere', *Geophys. Res. Lett.* **48**(7), e92305.

Ke, Y., Gao, X., Lu, Q., Wang, X., Chen, R., Chen, H. and Wang, S. [2022], 'Defor- mation of Electron Distributions Due to Landau Trapping by the Whistler-Mode Wave', *Geophys. Res. Lett.* **49**(3), e96428.

Kennel, C. F. [1966], 'Low-Frequency Whistler Mode', *Physics of Fluids* **9**, 2190-2202.

Kennel, C. F. [1969], 'Consequences of a magnetospheric plasma.', *Reviews of Geophysics and Space Physics* **7**, 379-419.

Kennel, C. F. and Engelmann, F. [1966], 'Velocity Space Diffusion from Weak Plasma Turbulence in a Magnetic Field', *Physics of Fluids* **9**, 2377-2388.

Kennel, C. F. and Petschek, H. E. [1966], 'Limit on Stably Trapped Particle Fluxes', *J. Geophys. Res.* **71**, 1-28.

Khazanov, G. V., Tel'nikhin, A. A. and Kronberg, T. K. [2013], 'Radiation belt electron dynamics driven by large-amplitude whistlers', *Journal of Geophysical Research (Space Physics)* **118**(10), 6397-6404.

Khazanov, G. V., Tel'nikhin, A. A. and Kronberg, T. K. [2014], 'Stochastic electron motion driven by space plasma waves', *Nonlinear Processes in Geophysics* **21**(1), 61-85.

Kitahara, M. and Katoh, Y. [2019], 'Anomalous Trapping of Low Pitch Angle Electrons by Coherent Whistler Mode Waves', *J. Geophys. Res.* **124**(7), 5568-5583.
Kletzing, C. A., Kurth, W. S., Acuna, M., MacDowall, R. J., Torbert, R. B., Averkamp, T., Bodet, D., Bounds, S. R., Chutter, M., Connerney, J., Crawford, D., Dolan, J. S., Dvorsky, R., Hospodarsky, G. B., Howard, J., Jordanova, V., Johnson, R. A., Kirchner, D. L., Mokrzycki, B., Needell, G., Odom, J., Mark, D., Pfaff, R., Phillips, J. R., Piker, C. W., Remington, S. L., Rowland, D., Santolik, O., Schnurr, R., Sheppard, D., Smith, C. W., Thorne, R. M. and Tyler, J. [2013], ‘The Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS) on RBSP’, *Space Sci. Rev.* **179**, 127–181.

Kondrashov, D., Drozdov, A. Y. and Shprits, Y. [2024], ‘Nonlinear wave-particle interaction effects on radiation belt electron dynamics in 9 october 2012 storm’, *Journal of Geophysical Research: Space Physics* **129**(9), e2024JA032898. e2024JA032898.

Kong, Z., Gao, X., Chen, H., Lu, Q., Chen, R., Ke, Y. and Wang, S. [2021], ‘The Correlation Between Whistler Mode Waves and Electron Beam-Like Distribution: Test Particle Simulations and THEMIS Observations’, *Journal of Geophysical Research (Space Physics)* **126**(11), e29834.

Kubota, Y. and Omura, Y. [2018], ‘Nonlinear Dynamics of Radiation Belt Electrons Interacting With Chorus Emissions Localized in Longitude’, *Journal of Geophysical Research (Space Physics)* **123**, 4835–4857.

Kulsrud, R. M. [1957], ‘Adiabatic Invariant of the Harmonic Oscillator’, *Physical Review* **106**, 205–207.

Kumamoto, A., Tsuchiya, F., Kasahara, Y., Kasaba, Y., Kojima, H., Yagitani, S., Ishisaka, K., Imachi, T., Ozaki, M., Matsuda, S., Shoji, M., Matsuoka, A., Katoh, Y., Miyoshi, Y. and Obara, T. [2018], ‘High Frequency Analyzer (HFA) of Plasma Wave Experiment (PWE) onboard the Arise spacecraft’, *Earth, Planets and Space* **70**(1), 82.

Kurth, W. S., De Pascuale, S., Faden, J. B., Kletzing, C. A., Hospodarsky, G. B., Thaller, S. and Wygant, J. R. [2015], ‘Electron densities inferred from plasma wave spectra obtained by the Waves instrument on Van Allen Probes’, *J. Geophys. Res.* **120**, 904–914.

Laird, M. J. and Nunn, D. [1975], ‘Full-wave VLF modes in a cylindrically symmetric enhancement of plasma density’, *Plan. Sp. Sci.* **23**(12), 1649–1657.

Landau, L. D. and Lifshitz, E. M. [1960], *Vol. 1: Mechanics*, Course of Theoretical Physics, 1st edn, Oxford: Pergamon Press.

Landau, L. D. and Lifshitz, E. M. [1988], *Vol. 1: Mechanics*, Course of Theoretical Physics, Oxford: Pergamon.

Le Contel, O., Roux, A., Jacquey, C., Robert, P., Berthomier, M., Chust, T., Grison, B., Angelopoulos, V., Sibeck, D., Chaston, C. C., Cully, C. M., Ergun, B., Glassmeier, K.-H., Auster, U., McFadden, J., Carlson, C., Larson, D., Bonnell, J. W., Mende, S., Russell, C. T., Donovan, E., Mann, I. and Singer, H. [2009], ‘Quasi-parallel whistler mode waves observed by THEMIS during near-earth dipolarizations’, *Annales Geophysicae* **27**, 2259–2275.

Le Contel, O., Roux, A., Robert, P., Coillot, C., Bouabdellah, A., de La Porte, B., Alison, D., Ruocco, S., Angelopoulos, V., Bromund, K., Chaston, C. C., Cully, C., Auster, H. U., Glassmeier, K. H., Baumjohann, W., Carlson, C. W., McFadden, J. P. and Larson, D. [2008], ‘First Results of the THEMIS Search Coil Magnetometers’, *Space Sci. Rev.* **141**, 509–534.

Lenard, A. [1959], ‘Adiabatic invariance to all orders’, *Annals of Physics* **6**, 261–276.
Leoncini, X., Vasiliev, A. and Artemyev, A. [2018], ‘Resonance controlled transport in phase space’, *Physica D Nonlinear Phenomena* **364**, 22–26.

Lewak, G. J. and Chen, C. S. [1969], ‘Higher order resonances in a plasma’, *Journal of Plasma Physics* **3**(3), 481–497.

Li, J.-H., Liu, Z.-Y., Zhou, X.-Z., Li, L., Omura, Y., Yue, C., Zong, Q.-G., Pu, Z.-Y., Fu, S.-Y., Xie, L., Russell, C. T., Pollock, C. J., Le, G. and Burch, J. L. [2022], ‘Anomalous resonance between low-energy particles and electromagnetic plasma waves’, *Communications Physics* **5**(1), 300.

Li, W., Bortnik, J., Thorne, R. M. and Angelopoulos, V. [2011], ‘Global distribution of wave amplitudes and wave normal angles of chorus waves using THEMIS wave observations’, *J. Geophys. Res.* **116**, 12205.

Li, W., Bortnik, J., Thorne, R. M., Cully, C. M., Chen, L., Angelopoulos, V., Nishimura, Y., Tao, J. B., Bonnell, J. W. and Lecontel, O. [2013], ‘Characteristics of the Poynting flux and wave normal vectors of whistler-mode waves observed on THEMIS’, *J. Geophys. Res.* **118**, 1461–1471.

Li, W., Bortnik, J., Thorne, R. M., Nishimura, Y., Angelopoulos, V. and Chen, L. [2011], ‘Modulation of whistler mode chorus waves: 2. Role of density variations’, *J. Geophys. Res.* **116**, A06206.

Li, W., Mourenas, D., Artemyev, A. V., Bortnik, J., Thorne, R. M., Kletzing, C. A., Kurth, W. S., Hospodarsky, G. B., Reeves, G. D., Funsten, H. O. and Spence, H. E. [2016], ‘Unraveling the excitation mechanisms of highly oblique lower band chorus waves’, *Geophys. Res. Lett.* **43**, 8867–8875.

Li, W., Ni, B., Thorne, R. M., Bortnik, J., Green, J. C., Kletzing, C. A., Kurth, W. S. and Hospodarsky, G. B. [2013], ‘Constructing the global distribution of chorus wave intensity using measurements of electrons by the POES satellites and waves by the Van Allen Probes’, *Geophys. Res. Lett.* **40**, 4526–4532.

Li, W., Santolik, O., Bortnik, J., Thorne, R. M., Kletzing, C. A., Kurth, W. S. and Hospodarsky, G. B. [2016], ‘New chorus wave properties near the equator from Van Allen Probes wave observations’, *Geophys. Res. Lett.* **43**, 4725–4735.

Li, W., Thorne, R. M., Bortnik, J., Baker, D. N., Reeves, G. D., Kanekal, S. G., Spence, H. E. and Green, J. C. [2015], ‘Solar wind conditions leading to efficient radiation belt electron acceleration: A superposed epoch analysis’, *Geophys. Res. Lett.* **42**, 6906–6915.

Li, W., Thorne, R. M., Bortnik, J., Nishimura, Y. and Angelopoulos, V. [2011], ‘Modulation of whistler mode chorus waves: 1. Role of compressional Pc4-5 pulsations’, *J. Geophys. Res.* **116**, A06205.

Li, W., Thorne, R. M., Ma, Q., Ni, B., Bortnik, J., Baker, D. N., Spence, H. E., Reeves, G. D., Kanekal, S. G., Green, J. C., Kletzing, C. A., Kurth, W. S., Hospodarsky, G. B., Blake, J. B., Fennell, J. F. and Claudepierre, S. G. [2014], ‘Radiation belt electron acceleration by chorus waves during the 17 March 2013 storm’, *J. Geophys. Res.* **119**, 4681–4693.

Li, W., Thorne, R. M., Nishimura, Y., Bortnik, J., Angelopoulos, V., McFadden, J. P., Larson, D. E., Bonnell, J. W., Le Contel, O., Roux, A. and Auster, U. [2010], ‘THEMIS analysis of observed equatorial electron distributions responsible for the chorus excitation’, *J. Geophys. Res.* **115**.

Lichtenberg, A. J. and Lieberman, M. A. [1983], *Regular and stochastic motion*.

Lieberman, M. A. and Lichtenberg, A. J. [1973], ‘Theory of electron cyclotron resonance heating. II. Long time and stochastic effects’, *Plasma Physics* **15**(2), 125–150.
Lukin, A. S., Artemyev, A. V., Petrukovich, A. A. and Zhang, X. J. [2021], 'Charged particle scattering in dipolarized magnetotail', *Physics of Plasmas* **28**(10), 102901.

Lukin, A. S., Artemyev, A. V., Zhang, X. J., Allanson, O. and Tao, X. [2024], 'On the Two Approaches to Incorporate Wave-Particle Resonant Effects Into Global Test Particle Simulations', *Journal of Geophysical Research (Space Physics)* **129**(2), e2023JA032163.

Lundin, B. V. and Shkliar, D. R. [1977], 'Interaction of electrons with low transverse velocities with VLF waves in an inhomogeneous plasma', *Geomagnetism and Aeronomy* **17**, 246-251.

Luzhkovskiy, A. A. and Shklyar, D. R. [2023], 'Energy Transfer Between Various Electron Populations Via Resonant Interaction With Whistler Mode Wave', *Journal of Geophysical Research (Space Physics)* **128**(12), e2023JA031962.

Lyons, L. R., Thorne, R. M. and Kennel, C. F. [1972], 'Pitch-angle diffusion of radiation belt electrons within the plasmasphere.', *J. Geophys. Res.* **77**, 3455-3474.

Lyons, L. R. and Williams, D. J. [1984], *Quantitative aspects of magnetospheric physics*.

Ma, Q., Artemyev, A. V., Mourenas, D., Li, W., Thorne, R. M., Kletzing, C. A., Kurth, W. S., Hospodarsky, G. B., Reeves, G. D., Spence, H. E. and Wygant, J. [2017], 'Very Oblique Whistler Mode Propagation in the Radiation Belts: Effects of Hot Plasma and Landau Damping', *Geophys. Res. Lett.* **44**(24), 12,057-12,066.

Ma, Q., Li, W., Bortnik, J., Thorne, R. M., Chu, X., Ozeke, L. G., Reeves, G. D., Kletzing, C. A., Kurth, W. S., Hospodarsky, G. B., Engebretson, M. J., Spence, H. E., Baker, D. N., Blake, J. B., Fennell, J. F. and Claudepierre, S. G. [2018], 'Quantitative Evaluation of Radial Diffusion and Local Acceleration Processes During GEM Challenge Events', *Journal of Geophysical Research (Space Physics)* **123**(3), 1938-1952.

Ma, Q., Li, W., Thorne, R. M., Ni, B., Kletzing, C. A., Kurth, W. S., Hospodarsky, G. B., Reeves, G. D., Henderson, M. G., Spence, H. E., Baker, D. N., Blake, J. B., Fennell, J. F., Claudepierre, S. G. and Angelopoulos, V. [2015], 'Modeling inward diffusion and slow decay of energetic electrons in the Earth's outer radiation belt', *Geophys. Res. Lett.* **42**, 987-995.

Ma, Q., Mourenas, D., Artemyev, A., Li, W., Thorne, R. M. and Bortnik, J. [2016], 'Strong enhancement of 10-100 keV electron fluxes by combined effects of chorus waves and time domain structures', *Geophys. Res. Lett.* **43**(10), 4683-4690.

Ma, Q., Ni, B., Tao, X. and Thorne, R. M. [2012], 'Evolution of the plasma sheet electron pitch angle distribution by whistler-mode chorus waves in non-dipole magnetic fields', *Annales Geophysicae* **30**, 751-760.

MacDonald, W. M. and Walt, M. [1961], 'Distribution function of magnetically confined electrons in a scattering atmosphere', *Annals of Physics* **15**(1), 44-62.

Martinez-Calderon, C., Katoh, Y., Manninen, J., Kasahara, Y., Matsuda, S., Kumamoto, A., Tsuchiya, F., Matsuoka, A., Shoji, M., Teramoto, M., Shinohara, I., Shiokawa, K. and Miyoshi, Y. [2020], 'Conjugate Observations of Dayside and Nightside VLF Chorus and QP Emissions Between Arase (ERG) and Kannuslehto, Finland', *Journal of Geophysical Research (Space Physics)* **125**(1), e26663.

Martinez-Calderon, C., Shiokawa, K., Miyoshi, Y., Ozaki, M., Schofield, I. and Connors, M. [2015], 'Statistical study of ELF/VLF emissions at subauroral lat-
itudes in Athabasca, Canada', *Journal of Geophysical Research (Space Physics) 120*(10), 8455-8469.

Matsoukis, S. K., Chapman, S. and Rowlands, G. [2000], 'Does stochasticity due to whistler mode wave coupling persist in self-consistent systems?', *Physica D Nonlinear Phenomena* **138**, 251-266.

Matsuda, S., Kasahara, Y., Kojima, H., Kasaba, Y., Yagitani, S., Ozaki, M., Imachi, T., Ishisaka, K., Kumamoto, A., Tsuchiya, F., Ota, M., Kurita, S., Miyoshi, Y., Hikishima, M., Matsuoka, A. and Shinohara, I. [2018], 'Onboard software of Plasma Wave Experiment aboard Arase: instrument management and signal processing of Waveform Capture/Onboard Frequency Analyzer', *Earth, Planets, and Space* **70**(1), 75.

Matsumoto, H. and Omura, Y. [1981], 'Cluster and channel effect phase bunchings by whistler waves in the nonuniform geomagnetic field', *J. Geophys. Res. 86*(A2), 779-791.

Matsuoka, A., Teramoto, M., Nomura, R., Nosé, M., Fujimoto, A., Tanaka, Y., Shinohara, M., Nagatsuma, T., Shiokawa, K., Obana, Y., Miyoshi, Y., Mita, M., Takashima, T. and Shinohara, I. [2018], 'The ARASE (ERG) magnetic field investigation', *Earth, Planets, and Space* **70**(1), 43.

Mauk, B. H., Fox, N. J., Kanekal, S. G., Kessel, R. L., Sibeck, D. G. and Ukhorskiy, A. [2013], 'Science Objectives and Rationale for the Radiation Belt Storm Probes Mission', *Space Sci. Rev.* **179**, 3-27.

McFadden, J. P., Carlson, C. W., Larson, D., Ludlam, M., Abiad, R., Elliott, B., Turin, P., Marckwordt, M. and Angelopoulos, V. [2008], 'The THEMIS ESA Plasma Instrument and In-flight Calibration', *Space Sci. Rev.* **141**, 277-302.

Meredith, N. P., Horne, R. B. and Anderson, R. R. [2001], 'Substorm dependence of chorus amplitudes: Implications for the acceleration of electrons to relativistic energies', *J. Geophys. Res.* **106**, 13165-13178.

Meredith, N. P., Horne, R. B., Sicard-Piet, A., Boscher, D., Yearby, K. H., Li, W. and Thorne, R. M. [2012], 'Global model of lower band and upper band chorus from multiple satellite observations', *J. Geophys. Res.* **117**, 10225.

Miyoshi, Y., Kasaba, Y., Shinohara, I., Takashima, T., Asamura, K., Matsumoto, H., Higashio, N., Mitani, T., Kasahara, S., Yokota, S., Wang, S., Kazama, Y., Kasahara, Y., Yagitani, S., Matsuoka, A., Kojima, H., Katoh, Y., Shiokawa, K., Seki, K., Fujimoto, M., Ono, T. and ERG project Group [2017], Geospace exploration project: Arase (ERG), in 'Journal of Physics Conference Series', Vol. 869 of *Journal of Physics Conference Series*, p. 012095.

Mourenas, D., Artemyev, A. V., Agapitov, O. V. and Krasnoselskikh, V. [2013], 'Analytical estimates of electron quasi-linear diffusion by fast magnetosonic waves', *J. Geophys. Res.* **118**, 3096-3112.

Mourenas, D., Artemyev, A. V., Agapitov, O. V. and Krasnoselskikh, V. [2014], 'Consequences of geomagnetic activity on energization and loss of radiation belt electrons by oblique chorus waves', *J. Geophys. Res.* **119**, 2775-2796.

Mourenas, D., Artemyev, A. V., Agapitov, O. V., Krasnoselskikh, V. and Li, W. [2014], 'Approximate analytical solutions for the trapped electron distribution due to quasi-linear diffusion by whistler mode waves', *J. Geophys. Res.* **119**, 9962-9977.

Mourenas, D., Artemyev, A. V., Agapitov, O. V., Krasnoselskikh, V. and Mozer, F. S. [2015], 'Very oblique whistler generation by low-energy electron streams', *J. Geophys. Res.* **120**, 3665-3683.
Mourenas, D., Artemyev, A. V., Agapitov, O. V., Mozer, F. S. and Krasnoselskikh, V. V. [2016], ‘Equatorial electron loss by double resonance with oblique and parallel intense chorus waves’, *J. Geophys. Res.* **121**, 4498–4517.

Mourenas, D., Artemyev, A. V., Zhang, X. J., Angelopoulos, V., Tsai, E. and Wilkins, C. [2021], ‘Electron Lifetimes and Diffusion Rates Inferred From ELFIN Measurements at Low Altitude: First Results’, *Journal of Geophysical Research (Space Physics)* **126**(11), e29757.

Mourenas, D., Zhang, X.-J., Artemyev, A. V., Angelopoulos, V., Thorne, R. M., Bortnik, J., Neishtadt, A. I. and Vasiliev, A. A. [2018], ‘Electron Nonlinear Resonant Interaction With Short and Intense Parallel Chorus Wave Packets’, *J. Geophys. Res.* **123**, 4979–4999.

Mourenas, D., Zhang, X. J., Nunn, D., Artemyev, A. V., Angelopoulos, V., Tsai, E. and Wilkins, C. [2022], ‘Short Chorus Wave Packets: Generation Within Chorus Elements, Statistics, and Consequences on Energetic Electron Precipitation’, *Journal of Geophysical Research (Space Physics)* **127**(5), e30310.

Mozer, F. S., Artemyev, A., Agapitov, O. V., Mourenas, D. and Vasko, I. [2016], ‘Near-relativistic electron acceleration by Landau trapping in time domain structures’, *Geophys. Res. Lett.* **43**, 508–514.

Nasi, A., Katsavrias, C., Daglis, I. A., Sandberg, I., Aminalragia-Giamini, S., Li, W., Miyoshi, Y., Evans, H., Mitani, T., Matsuoka, A., Shinohara, I., Takashima, T., Hori, T. and Balasis, G. [2022], ‘An event of extreme relativistic and ultra-relativistic electron enhancements following the arrival of consecutive corotating interaction regions: Coordinated observations by Van Allen Probes, Arase, THEMIS and Galileo satellites’, *Frontiers in Astronomy and Space Sciences* **9**, 949788.

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.

Neishtadt, A. I. [2000], ‘On the accuracy of persistence of adiabatic invariant in single-frequency system’, *Regular and chaotic dynamics* **5**, 213–218.

Neishtadt, A. I. [2014], ‘Averaging, passage through resonances, and capture into resonance in two-frequency systems’, *Russian Mathematical Surveys* **69**(5), 771. URL: http://stacks.iop.org/0036-0279/69/i=5/a=771

Neishtadt, A. I. and Artemyev, A. V. [2020], ‘Hamiltonian in Guiding Center Theory: A Symplectic Structure Approach’, *Proc. Steklov Inst. Math.* **310**, 214–219.

Neishtadt, A. I., Vainchtein, D. and Vasiliev, A. [2011], ‘Dynamics of electrons in a parabolic magnetic field perturbed by an electromagnetic wave’, *Plasma Physics and Controlled Fusion* **53**(8), 085014.

Neishtadt, A. I. and Vasiliev, A. A. [2005], ‘Phase change between separatrix crossings in slow fast Hamiltonian systems’, *Nonlinearity* **18**, 1393–1406.

Neishtadt, A. I., Vasiliev, A. A. and Artemyev, A. V. [2013], ‘Capture into resonance and escape from it in a forced nonlinear pendulum’, *Regular and Chaotic Dynamics* **18**(6), 686–696.

Ni, B., Thorne, R. M., Meredith, N. P., Horne, R. B. and Shprits, Y. Y. [2011], ‘Resonant scattering of plasma sheet electrons leading to diffuse auroral precipitation: 2. Evaluation for whistler mode chorus waves’, *J. Geophys. Res.* **116**, 4219.

Ni, B., Thorne, R. M., Meredith, N. P., Shprits, Y. Y. and Horne, R. B. [2011], ‘Diffuse auroral scattering by whistler mode chorus waves: Dependence on wave
normal angle distribution', *J. Geophys. Res.* **116**, 10207.

Ni, B., Thorne, R. M., Shprits, Y. Y. and Bortnik, J. [2008], 'Resonant scattering of plasma sheet electrons by whistler-mode chorus: Contribution to diffuse auroral precipitation', *Geophys. Res. Lett.* **35**, 11106.

Nishimura, Y., Bortnik, J., Li, W., Thorne, R. M., Ni, B., Lyons, L. R., Angelopoulos, V., Ebihara, Y., Bonnell, J. W., Le Contel, O. and Auster, U. [2013], 'Structures of dayside whistler-mode waves deduced from conjugate diffuse aurora', *Journal of Geophysical Research (Space Physics)* **118**(2), 664–673.

Nunn, D. [1974], 'A self-consistent theory of triggered VLF emissions', *Plan. Sp. Sci.* **22**, 349–378.

Nunn, D. [1986], 'A nonlinear theory of sideband stability in ducted whistler mode waves', *Plan. Sp. Sci.* **34**, 429–451.

Nunn, D. and Omura, Y. [2015], 'A computational and theoretical investigation of nonlinear wave-particle interactions in oblique whistlers', *J. Geophys. Res.* **120**, 2890–2911.

Nunn, D., Rycroft, M. and Trakhtengerts, V. [2005], 'A parametric study of the numerical simulations of triggered VLF emissions', *Annales Geophysicae* **23**(12), 3655–3666.

Nunn, D., Zhang, X. J., Mourenas, D. and Artemyev, A. V. [2021], 'Generation of Realistic Short Chorus Wave Packets', *Geophys. Res. Lett.* **48**(7), e92178.

Omura, Y. [2021], 'Nonlinear wave growth theory of whistler-mode chorus and hiss emissions in the magnetosphere', *Earth, Planets and Space* **73**(1), 95.

Omura, Y., Furuya, N. and Summers, D. [2007], 'Relativistic turning acceleration of resonant electrons by coherent whistler mode waves in a dipole magnetic field', *J. Geophys. Res.* **112**, 6236.

Omura, Y., Hikishima, M., Katoh, Y., Summers, D. and Yagitani, S. [2009], 'Non-linear mechanisms of lower-band and upper-band VLF chorus emissions in the magnetosphere', *J. Geophys. Res.* **114**, 7217.

Omura, Y., Katoh, Y. and Summers, D. [2008], 'Theory and simulation of the generation of whistler-mode chorus', *J. Geophys. Res.* **113**, 4223.

Omura, Y. and Matsumoto, H. [1982], 'Computer simulations of basic processes of coherent whistler wave-particle interactions in the magnetosphere', *J. Geophys. Res.* **87**(A6), 4435–4444.

Omura, Y., Miyashita, Y., Yoshikawa, M., Summers, D., Hikishima, M., Ebihara, Y. and 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.

Omura, Y., Nunn, D. and Summers, D. [2013], Generation Processes of Whistler Mode Chorus Emissions: Current Status of Nonlinear Wave Growth Theory, in D. Summers, I. U. Mann, D. N. Baker and M. Schulz, eds, 'Dynamics of the Earth's Radiation Belts and Inner Magnetosphere', American Geophysical Union, pp. 243–254.

Omura, Y. and Zhao, Q. [2012], 'Nonlinear pitch angle scattering of relativistic electrons by EMIC waves in the inner magnetosphere', *J. Geophys. Res.* **117**, 8227.

Palmadesso, P. J. [1972], 'Resonance, Particle Trapping, and Landau Damping in Finite Amplitude Obliquely Propagating Waves', *Physics of Fluids* **15**, 2006–2013.
Rich, F. J., Hardy, D. A. and Gussenhoven, M. S. [1985], ‘Enhanced ionosphere-magnetosphere data from the DMSP satellites’, *EOS Transactions* **66**(26), 513–514.

Roberts, C. S. [1969], ‘Pitch-angle diffusion of electrons in the magnetosphere.’, *Reviews of Geophysics and Space Physics* **7**, 305–337.

Sagdeev, R. Z. and Shafranov, V. D. [1961], ‘On the Instability of a Plasma with an Anisotropic Distribution of Velocities in a Magnetic Field’, *Soviet Phys. JETP* **12**(1), 130–132.

Sagdeev, R. Z., Usikov, D. A. and Zaslavsky, G. M. [1988], *Nonlinear Physics. From the Pendulum to Turbulence and Chaos*.

Santolík, O., Kletzing, C. A., Kurth, W. S., Hospodarsky, G. B. and Bounds, S. R. [2014], ‘Fine structure of large-amplitude chorus wave packets’, *Geophys. Res. Lett.* **41**, 293–299.

Sazhin, S. [1993], *Whistler-mode Waves in a Hot Plasma*, Cambridge University Press.

Sazhin, S. S. and Horne, R. B. [1990], ‘Quasilongitudinal approximation for whistler-mode waves in the magnetospheric plasma’, *Plan. Sp. Sci.* **38**, 1551–1553.

Schulz, M. and Lanzerotti, L. J. [1974], *Particle diffusion in the radiation belts*, Springer, New York.

Sergeev, V. A., Gordeev, E. I., Merkin, V. G. and Sitnov, M. I. [2018], ‘Does a Local B-Minimum Appear in the Tail Current Sheet During a Substorm Growth Phase?’, *Geophys. Res. Lett.* **45**, 2566–2573.

Sergeev, V. A., Nishimura, Y., Kubyshkina, M., Angelopoulos, V., Nakamura, R. and Singer, H. [2012], ‘Magnetospheric location of the equatorward prebreakup arc’, *Journal of Geophysical Research (Space Physics)* **117**(A1), A01212.

Shapiro, V. D. and Sagdeev, R. Z. [1997], ‘Nonlinear wave-particle interaction and conditions for the applicability of quasilinear theory’, *Physics Reports* **283**, 49–71.

Sheeley, B. W., Moldwin, M. B., Rassoul, H. K. and Anderson, R. R. [2001], ‘An empirical plasmasphere and trough density model: CRRES observations’, *J. Geophys. Res.* **106**, 25631–25642.

Shen, Y., Artemyev, A. V., Ma, Q., Zhang, X.-J., Mourenas, D., Tsai, E., Wilkins, C., Wu, J. and Angelopoulos, V. [2022], ‘Inner Belt Wisp Precipitation Measured by ELFIN: Regimes of Energetic Electron Scattering by VLF Transmitter Waves’, *Journal of Geophysical Research (Space Physics)* **127**(11), e2022JA030968.

Shi, X., Artemyev, A., Zhang, X.-J., Mourenas, D., An, X. and Angelopoulos, V. [2024], ‘Properties of Intense H-Band Electromagnetic Ion Cyclotron Waves: Implications for Quasi-Linear, Nonlinear, and Nonresonant Wave-Particle Interactions’, *Journal of Geophysical Research (Space Physics)* **129**(1), e2023JA032179.

Shi, X., Tonoian, D. S., Artemyev, A. V., Zhang, X.-J. and Angelopoulos, V. [2023], ‘Electron resonant interaction with whistler-mode waves around the Earth’s bow shock I: The probabilistic approach’, *Physics of Plasmas* **30**(12), 122902.

Shklyar, D. and Luzhkovskiy, A. [2023], ‘Self-consistent amplitude profile of ducted VLF transmitter signal due to resonant interaction with energetic electrons in the magnetosphere’, *Advances in Space Research* **71**(1), 228–243.

Shklyar, D. R. [1981], ‘Stochastic motion of relativistic particles in the field of a monochromatic wave’, *Sov. Phys. JETP* **53**, 1197–1192.