Resonant interaction of relativistic electrons with realistic electromagnetic ion–cyclotron wave packets

Abstract

We study the influence of real structure of electromagnetic ion-cyclotron wave packets in the Earth’s radiation belts on precipitation of relativistic electrons. Automatic algorithm is used to distinguish isolated elements (wave packets) and obtain their amplitude and frequency profiles from satellite observations by Van Allen Probe B. We focus on rising-tone EMIC wave packets in the proton band, with a maximum amplitude of 1.2–1.6 nT. The resonant interaction of the considered wave packets with relativistic electrons 1.5–9 MeV is studied by numerical simulations. The precipitating fluxes are formed as a result of both linear and nonlinear interaction; for energies 2–5 MeV precipitating fluxes are close to the strong diffusion limit. The evolution of precipitating fluxes is influenced by generation of higher-frequency waves at the packet trailing edge near the equator and dissipation of lower-frequency waves in the $\text {He}^+$ cyclotron resonance region at the leading edge. The wave packet amplitude modulation leads to a significant change of precipitated particles energy spectrum during short intervals of less than 1 minute. For short time intervals about 10–15 s, the approximation of each local amplitude maximum of the wave packet by a Gaussian amplitude profile and a linear frequency drift gives a satisfactory description of the resonant interaction.

Introduction

The dynamics of Earth radiation belts has been studied experimentally and theoretically for many years (Kennel and Petschek 1966; Tverskoy 1969; Lyons and Thorne 1973; Bespalov and Trakhtengerts 1986; Trakhtengerts and Rycroft 2000; Millan and Thorne 2007; Morley et al. 2010; Li and Hudson 2019). Among various phenomena of radiation belt physics, an important place is occupied by precipitation of relativistic electrons. The precipitating fluxes of relativistic electrons play an important role in depletion of radiation belts that pose a danger for the operation of geosynchronous satellites; precipitating relativistic electrons can also significantly affect atmospheric chemistry (Krivolutsky and Repnev 2012; Mironova et al. 2015).

One of the possible mechanisms of relativistic electron precipitation is their resonant interaction with electromagnetic ion-cyclotron (EMIC) waves (Thorne and Kennel 1971). First studies of resonant interaction of relativistic electrons with EMIC waves have been within the framework of the quasi-linear theory (Summers and Thorne 2003; Jordanova et al. 2008; Shprits et al. 2009). In this approach, the waves are assumed to be noise-like broadband emissions, i.e., have incoherent phases, which allows one to describe the wave–particle interaction in terms of velocity-space diffusion. However, observation of quasi-monochromatic wave packets with short durations and large amplitudes from 1 to 14 nT (Kangas et al. 1998; Demekhov 2007; Engebretson et al. 2007, 2008; Pickett et al. 2010; Nakamura et al. 2019) inspired analysis of possible nonlinear resonant interaction and its influence on precipitation (Albert and Bortnik 2009; Artemyev et al. 2015; Omura and Zhao 2012, 2013; Kubota and Omura 2017; Grach and Demekhov 2018a, b, 2020a). Also, there are observations of rapid loss of the outer radiation belt (Morley et al. 2010; Nakamura et al. 2019), which is too fast to be explained by quasi-linear diffusion rates, and thus requires nonlinear analysis.

Nonlinear theory of wave–particle interaction has been studied extensively for various wave modes (Karpman et al. 1974; Albert 1993, 2000; Shklyar and Matsumoto 2009; Albert and Bortnik 2009; Albert et al. 2012; Artemyev et al. 2015, 2017). The features of various interaction regimes were described analytically, including trapping by the wave field (Karpman et al. 1974; Albert 1993; Demekhov et al. 2006, 2009; Artemyev et al. 2015), phase bunching or nonlinear scattering (Albert 1993, 2000; Artemyev et al. 2017) and force bunching (Lundin and Shkliar 1977; Suvorov and Tokman 1988).

Wave–particle interaction with finite wave packets with various amplitude and frequency profiles has been studied mostly by test particle simulations (Tao et al. 2012; Omura and Zhao 2012, 2013; Chen et al. 2016; Kubota and Omura 2017; Zhang et al. 2018; Grach and Demekhov 2018a, b, 2020a; Zhang et al. 2020; Hiraga and Omura 2020). Regimes like directed scattering (Kubota and Omura 2017; Grach and Demekhov 2020a, b) or nonlinear shift of the resonance point (Grach and Demekhov 2020a) were revealed and analyzed. For whistler mode waves, it was shown that for realistic wave packets, nonlinear effects are much weaker than for single-frequency waves with a constant wave amplitude, because of the effects of amplitude modulation and short packet length (Tao et al. 2012; Zhang et al. 2018, 2020). On the other hand, the interaction with a chorus element, consisting of multiple subpackets, can still provide an effective acceleration (Hiraga and Omura 2020).

For EMIC waves, test particle simulations showed that resonant interaction of relativistic electrons with rising-tone EMIC wave packets can be very effective (Omura and Zhao 2012, 2013; Kubota and Omura 2017; Grach and Demekhov 2018b, 2020a). Precipitation flux in this case is heavily influenced and increased by nonlinear effects (Grach and Demekhov 2020a). (Chen et al. 2016) showed that sharp edges of the EMIC wave packets can lead to an effective nonresonant scattering of electrons with low energies.

Test particle simulations for resonant interaction with EMIC waves use model wave packets with main parameters based on observations. In particular, our previous paper (Grach and Demekhov 2020a) used model wave packets with linear frequency profile and two amplitude profiles, flat and Gaussian shaped. (Kubota and Omura 2017) used a more complicated structure with several subpackets but without direct correspondence to observational data.

Fine structure of a wave packet can play important role in the resonant interaction, both in linear and nonlinear regimes. In this paper, we study wave packets directly corresponding to Van Allen Probes (Mauk et al. 2013) observations. Specifically, we use the data for an event of 14 September 2017 observed by Van Allen Probe B.

Wave packet modeling

Data processing

Fig. 1

a Dynamic spectrum from 11:50 to 13:50 UT 2017-09-14; b obtained wave amplitude profile of discrete elements and c obtained wave frequency profile of discrete elements. The elements are numbered in chronological order, corresponding to the first data point

We use high-resolution magnetic data of the Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS) (Kletzing et al. 2013). The EMIC dynamic spectrum for Van Allen Probe B observations on 14 September 2017 is shown in Fig. 1a. We focus on rising-tone EMIC wave packets in proton band. Wave normal angles (not shown) are low, which is typical for EMIC waves (Loto’Aniu et al. 2005; Engebretson et al. 2007).

Discrete elements were identified using an algorithm developed by (Larchenko et al. 2019). The parameters of the algorithm were chosen so as to ensure 90 % of the wave packet energy $\int |B_{\text {w}f}|^2 \text {d} f/(8 \pi )$ to be confined in the detected elements (here $|B_{\text {w}f}|^2$ is the power spectral density (PSD) of the wave magnetic field and f is the wave frequency). The integration that is effectively replaced by summation over FFT frequencies is performed over the identified element range. Using such integration, we obtain instant amplitude $B_\text {w}$ and frequency f of the wave packet: $B_\text {w} = \sqrt{\int |B_{\text {w}f}|^2\text {d}f}$, $f = \int |B_{\text {w}f}|^2f\text {d}f / B_\text {w}^2$.

We distinguished 31 isolated element (wave packet) and obtained their amplitude and frequency profiles (see Fig. 1b, c). The maximum wave amplitude is about 1.2–1.6 nT. Some of these elements can be considered isolated, some are overlapping in time and space. We divide 31 element into 12 groups, containing from 1 to 4 elements each and with duration from 40 to 250 s. We assume that each group corresponds to a fixed satellite location, and the plasma parameters stay constant during the generation, propagation and dissipation of elements in the group.

We assume that EMIC wave packets are generated in a small region near the equator, propagate along the geomagnetic field line and then dissipate in the $\text {He}^+$ cyclotron resonance region located father from the equator than the spacecraft. We use the dipole geomagnetic field model and obtain McIlwain parameter L using the satellite geomagnetic latitude $\lambda _\text {S}$ and the measured geomagnetic field $B_0$. The gyrotropic model of the field-aligned profile of plasma density is used ($N_\text {e} \propto B_0$), and the measured local density is averaged over the group duration.

We study the wave–particle interaction with a single packet. For this study, we choose elements 14 and 22, which amplitudes are high enough for nonlinear interaction. The element 14 is overlapped with elements 15 and 16, but we neglect element 15 because of its small amplitude (the ratio of amplitudes $B_{\text {w}14}/B_{\text {w}15}=10\div 100$) and we will focus our study of element 14 on times before element 16 is generated.

The values of plasma parameters $L,\lambda _\text {S}, N_\text {e}$, corresponding to elements 14 and 22, are shown in Table 1. Hereafter, the subscripts L and S denote the values at the equator and at the spacecraft location, respectively. The other parameters, shown in Table 1, are discussed below.

Table 1 Parameters for wave packet modeling

Modeling of wave packet propagation

Propagation properties

We model the wave packet propagation in the geometrical-optics approximation, i.e., assume that each point propagates with a group velocity $V_{\text {gr}}$ corresponding to the local dispersion relation and has its own frequency f that does not vary during the propagation: $\partial f/\partial t + V_\text {gr}(f) \partial f/\partial z = 0$ and amplitude $B_{\text {w}}$, which is discussed below.

We use the following additional simplifying assumptions for the wave generation and dissipation. (1) Each point of the packet starts at a single point $z_{\text {start}}<0$ near the equator and propagates with increasing coordinate z along the geomagnetic field line ($z=0$ corresponds to the equator). (2) The wave packet dissipates when it approaches the $\text {He}^+$ cutoff $z = Z_X(f)$ as detailed below. (3) We restrict the simulation parameters in such a way that the wave packet is not broken, i.e., the trajectories of its points do not intersect.

To obtain the wave amplitude and frequency at an arbitrary point within the wave packet, we trace discrete points $j=1\ldots M$ (where $j=1, M$ correspond to the leading and trailing edges of the packet, respectively) and interpolate between these points. Thus, we have the frequency profile $f_j(T_j)$ at the spacecraft location (see Figs. 2e and 3e). For each data point, we calculate the propagation “backwards” to $z_{\text {start}}$ and thus obtain start times $t_j(f_j)$ for each traced point. Then, for each point, we calculate the propagation “forward” until the packet dissipates near cutoff location $Z_{X}(f_{\,j}) \equiv Z_{X\,j}$.

For this approach to work, we have to place some restrictions on plasma composition, i.e., the $\text {He}^+$ density, to ensure the absence of wave packet breaking during the propagation. The dependence of group velocity $V_{\text {gr}}$ on $f/f_{\text {He}^+}$ ($f_{\text {He}^+}$ is the local helium gyrofrequency) is non monotonic: $V_{\text {gr}}$ decreases with $f/f_{\text {He}^+}$ far from cutoff frequency and increases with $f/f_{\text {He}^+}$ close to it. For a wave packet with strong enough negative frequency gradient (rising tone), the intersection of the trajectories (i.e, the amplitude profile breaking) is possible for the points with higher frequencies (trailing edge of the wave packet) when we calculate the propagation “backward” from the satellite to the equator. This happens if cutoff locations are too far away from the satellite location ($N_{\text {He}^+}$ too low), thus decreasing of $V_{\text {gr}}$ with $f/f_{\text {He}^+}$ is too abrupt. On the other hand, when a wave packet with rising tone propagates “forward” away from the equator and nears the cutoff locations, the trajectories of the points with lower frequencies (leading edge of the wave packet) intersect, which leads to the wave packet distortion. This distortion cannot be correctly described by geometrical optics, so we need to gradually dissipate the wave packet, starting from the leading edge, before this distortion happens. For the model wave packet to correspond to satellite data, this dissipation should begin after the satellite location, which means that cutoff locations should not be too close to the satellite location ($N_{\text {He}^{+}}$ cannot be too high). Thus, the value of $N_{\text {He}^{+}}$ should be high enough that “backward” propagation from the satellite to the equator is possible without distortion and at the same time $N_{\text {He}^+}$ should be low enough for the correct “forward” propagation from the equator to the satellite. The values for two considered wave packets are shown in Table 1. We also assume $N_{\text {O}^+}=N_{\text {He}^+}$.

Due to the properties of wave packet propagation, discussed above, “forward” propagation to the equator (from $z_{\text {start}}$ to $z=0$) without distortion is possible only in a small region. The values of $z_{\text {start}}$ (chosen as the maximum possible $|z_{\text {start}}|$) are also shown in Table 1 (hereafter, $R_\text {E}$ denotes the Earth radius).

Amplitude and phase of the wave packet

We assume that there is a small region $\Delta z_{\text {gen}}$ where the wave packet is generated (its energy increases), a small region $\Delta z_{\text {damp}}$ where wave packet dissipates (its energy decreases, this region is located farther from the equator than the satellite) and in between for each point of the packet its energy flux ${\mathcal {E}}\propto B_\text {w}^2 V_{\text {gr}}/(8\pi )$ remains constant and corresponds to the satellite data. We choose the following model:

$$\begin{aligned} {\mathcal {E}}(f_{\,j})=\left\{ \begin{matrix} {\mathcal {E}}_{\text {S}\,j}\delta _{\text {gen}} \exp [{\eta _{\text {gen}j}}(z-z_{\text {start}})], ~~~~z_\text {start}\le z\le z_{\text {c}\,j}; \\ {\mathcal {E}}_{\text {S}\,j},~~~~z_{\text {c}\,j}<z \le z_{\text {damp}j};\\ {\mathcal {E}}_{\text {S}\,j}\exp [-{\eta ^2_{\text {damp}\,j}} (z-z_{\text {damp}\,j})^2], ~~~~z_{\text {damp}\,j}<z< \min \{0.99Z_{X\,j}, z_{\text {cr}\,j}\}. \end{matrix} \right. \end{aligned}$$

(1)

Here, $j=1...M$, $z_{\text {cr}j}$ is the location of possible trajectory intersection, $\delta _{\text {gen}}=1/50$, ${\eta _{\text {gen}j}}=-\ln {\delta _{\text {gen}}}/(z_{\text {c}j} -z_{\text {start}})$, $z_{\text {c}j}=z_{\text {start}}+\Delta z_{\text {gen}j}$, ${\eta _{\text {damp}j}^2}=5 \ln 10/(Z_{Xj}-z_{\text {damp}j})^2$.

We choose $\Delta z_{\text {gen}j}=0.1Z_{Xj}$, thus the generation region takes up to 10 % of the area of wave packet existence. The relation between $z_{\text {damp}}$ and $Z_X$ is chosen empirically for each wave packet, to ensure that wave amplitude is small enough at the dissipation point. The values of $z_{\text {damp}}$ are shown in Table 1.

We also calculate the change in the wave phase for the leading and trailing edges of the packet:

$$\begin{aligned} \vartheta _j=\int k_j\text {d}z-\int \omega _j \text {d}t=\frac{\omega _j}{c}\int n_j\text {d}z - \omega _j t. \end{aligned}$$

(2)

Here, $\omega _j=2\pi f_j$, $k_j$ and $n_j$ are wave number and refractive index for the frequency $f_j$, respectively. We set $\vartheta _j=0$ at $z=z_{\text {start}}$ for any j.

While the packet is generated, the trailing edge of the packet is located at $z_{\text {start}}$ and so the phase of the trailing edge $\vartheta _{\text {te}}=0$. Once the generation is finished, $\vartheta _{\text {te}}=\vartheta _M$ and changes smoothly. On the contrary, the phase of the leading edge changes smoothly at first, while $\vartheta _{\text {le}}=\vartheta _1$, then, once the packet starts to dissipate, $\vartheta _{\text {le}}=\vartheta _j$, where j increases (the leading edge frequency increases until the packet dissipates completely).

The results of wave packets modeling are shown in Figs. 2 and 3. Hereafter, time t starts at the generation moment of the leading edge of the packet.

As one can see from Figs. 2,3, all the frequencies which can be seen in satellite data do not exist at the same time. By the time the wave with the highest frequency in the packet (trailing edge) is generated, the waves with lowest frequencies (leading edge) have been already dissipated at the $\text {He}^+$ resonance. This means that any spatial distribution will contain only part of the frequency spectrum.

Fig. 2

Wave packet propagation for element 14: amplitude (a–c) and frequency (d–f) profiles. Panels b and e show spatial profiles at $t=10$ s (dashed red line), 25 s (solid red line), 45 s (dashed green line) and 60 s (solid green line). Panels c and f show temporal profiles at the satellite location $z=z_\text {S}$ (black solid line), black markers represent the data from Fig. 1. Times 10; 25; 45 and 60 s are shown in panels a, d by horizontal lines and in panels c, f by vertical lines. Location $z=z_\text {S}$ is shown in panels a, d, b, e by vertical lines

Fig. 3

Wave packet propagation for element 22: amplitude (a–c) and frequency (d–f) profiles. Panels b and e show spatial profiles at $t=25$ s (dashed red line) and 40 s (solid red line). Panels c and f show temporal profiles at the satellite location $z=z_\text {S}$ (black solid line), black markers represent the data from Fig. 1. Times 25 and 40 s are shown in panels a, d by horizontal lines and in c, f by vertical lines. Location $z=z_\text {S}$ is shown in panels a, d, b, e by vertical lines

Wave–particle interaction: theory

Basic equations

The resonant interaction with parallel-propagating EMIC waves is possible only for relativistic electrons and at the anomalous cyclotron resonance. The resonance condition is written as follows:

$$\begin{aligned} \Delta =\omega -kv_{||}+\Omega _\text {c}/\gamma =0, \end{aligned}$$

(3)

where $\omega$ and k are wave frequency and number, respectively, $v_{||}$ is field-aligned velocity, $\Omega _\text {c}=eB_{0}/mc$, $B_0$ is geomagnetic field, $e>0$ is elementary charge, $\gamma =\sqrt{1+\left[ {p}/({mc})\right] ^2}$, m and p are the electron rest mass and momentum, respectively.

We use the same equations of motion for test electrons interacting with EMIC waves as (Grach and Demekhov 2020a):

$$\begin{aligned}\frac{\text {d} W}{\text {d}t}= - e v_\bot |E_\text {w}| \sin \Psi ; \end{aligned}$$

(4)

$$\begin{aligned}\frac{\text {d}{I_\bot }}{\text {d}t}= -\frac{2e}{mB_0}p_\bot (1-n_{||}\beta _{||})|E_\text {w}|\sin \Psi ; \end{aligned}$$

(5)

$$\begin{aligned}\frac{\text {d} \Psi }{\text {d}t} =-\Delta -\frac{e}{p_{\bot }} (1-n_{||}\beta _{||})|E_\text {w}|\cos \Psi ; \end{aligned}$$

(6)

$$\begin{aligned}\frac{\text {d}z}{\text {d}t}=\frac{p_{||}}{m\gamma }. \end{aligned}$$

(7)

Here, the subscripts || and $\bot$ denote projections to the parallel and transverse directions with respect to ${\mathbf {B}}_0$, respectively, $E_\text {w}$ is slowly changing wave electric field amplitude, $n_{||}=kc/\omega$, $\Psi =\vartheta -\varphi$, $\varphi$ is the gyrophase in the geomagnetic field ${\mathbf {B}}_0$, $\vartheta$ is the wave phase, $\beta _{||}=v_{||}/c$, $W=(\gamma -1)mc^2$ and $I_\bot = {p_\bot ^2}/({mB_0})$ are the electron kinetic energy and the first adiabatic invariant respectively, and z is coordinate along the geomagnetic field with $z=0$ corresponding to the equator. In the right-hand side of equation (5), the first term represents inertial, or kinematic bunching, while the second one represents the direct influence of Lorentz force on the particle phase (force bunching).

In (4)–(7) it is assumed that the external field inhomogeneity is smooth, the wave magnetic field amplitude is small ($B_\text {w} \ll B_0$) and wave characteristics vary slowly in time and space on the scales of $2 \pi /\Omega _\text {c}$ and $2\pi /k$, respectively.

For EMIC waves $\omega \ll \Omega _\text {c}$, and thus the resonant interaction is possible only for $k_\Vert v_\Vert > 0$ and the change in electron energy W will be insignificant: $\gamma \approx \text {const}$ (Bespalov and Trakhtengerts 1986; Albert and Bortnik 2009). The interaction result is described by the change in the adiabatic invariant $I_\bot$ or equatorial pitch angle $\Theta _L$, $\mu =\sin ^2{\Theta _L}=(p_\bot ^2/p^2)(B_L/B_0)$.

Summary of earlier analytical results

For the reader’s convenience, we briefly summarize earlier results of various authors (Karpman et al. 1974; Albert 1993, 2000; Albert and Bortnik 2009; Kubota and Omura 2017; Grach and Demekhov 2018a, 2020a).

Particle behavior during the interaction (interaction regime) is determined by the inhomogeneity parameter ${\mathcal {R}}=\sigma _R R$ (Karpman et al. 1974; Albert 1993, 2000; Albert and Bortnik 2009; Kubota and Omura 2017; Grach and Demekhov 2018a), where $\sigma _R=\pm 1$ determines the effective inhomogeneity sign, and

$$\begin{aligned} R=\frac{|\text {d}\Delta /\text {d}t|}{\Omega _{\text {tr}}^2}. \end{aligned}$$

(8)

Here, the time derivative $\text {d}/\text {d}t=\partial /\partial t + V_{||} \partial /\partial z$ represents the full change along the electron orbit with differentiating only on functions z and t, excluding W and $I_\bot$; $\Omega _{\text {tr}}^2=(1-n_{||}^{-2}){e\omega n_{||}^{2} v_\bot |E_\text {w}|}/({mc^2\gamma })$ is the frequency of electron oscillations in the wave field near the effective potential minimum (Grach and Demekhov 2018a; Demekhov et al. 2006). Under real conditions, the parameter R changes both in time and in space. These changes are associated both with medium inhomogeneity (including changes in the wave packet frequency and amplitude) and nonlinear changes in the particle parameters during the interaction. However, the main features of the particle motion can be categorized based on the R values calculated at the resonance point in the linear approximation. For $R>1$, the trajectories of all particles on the phase plane are open (all particles are untrapped), and for $R<1$, there is a minimum of the wave effective potential, i.e., particle trapping by the wave field is possible. The phase trajectories of the trapped particles are closed. For resonant interaction of electrons with EMIC wave packet, which is generated near the equator and propagates away from it, the effective inhomogeneity is negative. Hereafter, we assume $\sigma _R=-1$.

The case of $R \gg 1$ corresponds to the linear regime. In this case, the change in particle equatorial pitch angle depends on initial phase. For an ensemble of particles with the same initial equatorial pitch angle and initial phases uniformly distributed in $[0, 2\pi )$ pitch angle diffusion takes place (Albert 2000; Albert and Bortnik 2009; Grach and Demekhov 2018a, 2020a):

$$\begin{aligned} \langle \Delta \mu \rangle ^{\text {lin}}=0; \end{aligned}$$

(9)

$$\sqrt{\langle (\Delta \mu ^{\text {lin}}-\langle \Delta \mu \rangle ^{\text {lin}})^2 \rangle }=\sigma _{\mu }^{\text {lin}}>0.$$

(10)

Hereafter, angle brackets denote averaging over initial phases of the considered particle ensemble. The root mean square deviation $\sigma _\mu$ determines the diffusion coefficients in the linear regime.

For $R \le 1$, the resonant interaction is nonlinear, which leads to drift in pitch angles for both trapped and untrapped particles. For relativistic electrons interacting with EMIC waves, propagating away from the equator, trapping can lead to a significant decrease in pitch angle (Albert and Bortnik 2009; Kubota and Omura 2017; Grach and Demekhov 2018a, 2020a). For most of the untrapped particles, phase bunching without trapping or nonlinear scattering takes place (Albert 1993, 2000; Artemyev et al. 2017), which leads to pitch angle increase (Albert and Bortnik 2009; Grach and Demekhov 2018b, 2020a). When $R<1$ is not too small, directed scattering, which leads to a significant pitch angle decrease, is possible for a small group of untrapped particles (Kubota and Omura 2017; Grach and Demekhov 2018a, 2020a).

Nonlinear effects can also take place for not too high $R \ge 1$. If the wave amplitude is high enough, then the region of resonant interaction is large and the exact resonance point is shifted during the interaction. It leads to particles with the same initial pitch angle but different initial phases having different resonance points, and, consequently, different values of R. For an individual particle, $|\Delta \mu |$ in the linear regime depends on both the R value and the initial phase, while the sign of $\Delta \mu$ is determined only by phase. Thus, for example, if R decreases with $\mu$, then for particles with initially increasing $\mu$ the change $|\Delta \mu |$ will be bigger than for particles with initially decreasing $\mu$. So, when the dependence $R(\mu )$ is significant, the nonlinear shift of the resonance point causes drift in $\mu$ ($\langle \Delta \mu \rangle \ne 0$, see (Grach and Demekhov 2020a) for more details).

Force bunching (the Lorentz force term in Eq. (5) for the particle phase, which is neglected in linear approximation) becomes significant for particles with low $\Theta _L$. It was shown analytically by (Lundin and Shkliar 1977) that force bunching leads to systematic increase in electron pitch angle for resonance electrons with low transverse velocities in the field of a whistler mode parallel propagating wave. Lately, numerical simulations showed similar results for electrons interacting with EMIC waves (Grach and Demekhov 2020a), right-hand extraordinary mode (Grach and Demekhov 2020b), and whistler waves (Kitahara and Katoh 2019; Gan et al. 2020). The significance of force bunching term has also been proven for electron dynamics upon cyclotron resonance breakdown and heating in laboratory mirror traps by high-power microwaves (Suvorov and Tokman 1988). For EMIC waves, it was showed in (Grach and Demekhov 2020a) that force bunching can completely block precipitation from low pitch angles close to the loss cone.

Wave–particle interaction: numerical calculations

The system (4)–(7) was solved numerically by Bogacki–Shampine variant of the Runge–Kutta method. The wave field $E_\text {w}(z,t)=B_\text {w}(z,t)/n_{||}(z,t)$ and frequency $\omega (z,t)=2\pi f(z,t)$ are calculated beforehand according to the wave packet model described above (see Figs. 2, 3). When the particle is outside the wave packet, its energy W and adiabatic invariant $I_\bot$ remain constant and only gyrophase $\varphi$ and particle location z change. The phase $\Psi$ on the particle next entrance in the packet is calculated taking into account the increments both in $\varphi$ and in the wave phase $\vartheta$ for the packet edges ($\vartheta _{\text {le}}$ and $\vartheta _{\text {te}}$, see (2)). The wave packet at any given time is short enough, so that its evolution (generation, propagation and dissipation) during the single pass of the particle through it is negligible, but it becomes significant over several bounce oscillations.

For every energy, calculations were done for 82 values of equatorial pitch angle (range $4^\circ$–$85^\circ$, step of 1 degree) and 360 values of the initial phase (uniformly in $[0,2\pi )$). Thus, for every energy, the trajectories of 29520 particles were calculated. We assume particle energies $1~\text {MeV} \le W_0 \le 10~\text {MeV}$. At the beginning of simulation $t=t_{\text {begin}}$ all particles are placed at the point $z=-0.1R_\text {E}$ with positive longitudinal velocities. As is shown below, this is insignificant for the results, since the particles are spread over the field line in 4–8 bounces. If a particle is in the loss cone after leaving the packet ($\Theta _L< \Theta _{L\text {c}}$), then the simulation for this particle is stopped. For both considered sets of plasma parameters, the equatorial pitch angle, corresponding to the loss cone, $\Theta _{L\text {c}} \approx 3.4^\circ$.

Preliminary analysis of resonance conditions

Fig. 4

Resonance conditions. Distribution of resonance points with $R<10$ over time (a) and dependencies of $R_{\text {min}} \le 5$ on time (b, c) for the whole packet. Dependencies $R_{\text {min}}(\Theta _{L0},W_0)$ for the chosen time intervals: element 14, 10–25 s (case I, d), element 14, 45–60 s (case II, e), element 22, 25–40 s (case III, f). On the bottom panels the $R_{\text {min}}$ scale is limited by $R=10$

Our simulation model, based on Eqs. (4)–(7) and independently calculated wave packets, shown in Figs. 2 and 3, does not take into account the effect of magnetic drift on the electron distribution function in a given flux tube. To justify this approximation, we must limit our simulation to times which do not exceed drift times across the wave packet. These times are significantly shorter than 100–120 s of wave packets existence (specific values are discussed below). To determine simulation time and range of test particle energies, we first analyze resonant conditions for the considered wave packets and the behavior of inhomogeneity parameter R unperturbed values.

If the frequency range of the wave packet remains constant during the simulation, we can analyze resonant conditions for the single pass through the wave packet near the equator and then predict their behavior during wave packet propagation (Grach and Demekhov 2018a, 2020a). With lower-frequency waves dissipating and higher-frequency waves generating during possible simulation time (see bottom panels of Figs. 2 and 3 ), we have to analyze resonance points and R values at these points for the whole time when the wave packet exists.

In Fig. 4a, the distribution of resonance points with $R<10$ over time is shown for both wave packets. In Fig. 4b and c, we show time dependencies of minimum R values over the trajectory (for $R_{\text {min}}\le 5$). Based on these calculations and several test simulations we choose three cases for our further simulations: (I) element 14, time interval 10–25 s; (II) element 14, time interval 45–60 s; (III) element 22, time interval 25–40 s. Dependencies of $R_\text {min}(\Theta _{L0},W_0)$ for these three cases are shown at bottom panels of Fig. 4. These intervals correspond to the most effective wave–particle interaction and represent all the possible interaction regimes. Note that taking into account wave packet dissipation in the $\text {He}^+$ cyclotron resonance region allowed us to use longer simulation time than in (Grach and Demekhov 2020a) (15 s instead of 6.5 s).

These time intervals are shown in Figs. 2 and 3, as well as spatial profiles of wave amplitude and frequency at the start and end of the intervals (hereafter, $t_\text {begin}$ and $t_\text {end}$, respectively). As one can see from the Figures, Case I (element 14, time interval 10–25 s) corresponds to the biggest changes in the packet length during the interaction interval. We can also see that in Case I there is an active generation of higher-frequency waves at the trailing edge during the simulation time but dissipation of lower-frequency waves at the leading edge does not take place. Case II (element 14, time interval 45–60 s) corresponds to the widest frequency range (1.4–2.35 Hz) and to the longest wave packet spatial structure (close to $2R_\text {E}$). Generation of the packet is mostly finished by the beginning of Case II, but dissipation of lower-frequency waves has started. Case III (element 22, time interval 25–40 s) corresponds to the strongest increase in frequency; both generation and dissipation take place during the simulation time.

The highest frequencies in Case II result in the lowest resonant energies (see Fig. 4e). We also can see that Case II has the largest range of resonant pitch angles and the widest area of possible nonlinear interaction (corresponding to lower R values). It is noteworthy that the zone of resonant interaction is determined by the effective wave packet length for the considered packets, since the zone determined by the phase mismatch is wider.

The calculations were done in the following energy ranges: $3.5\div 8.0$ MeV for Case I, $1.5\div 5.5$ MeV for Case II and $3.5 \div 9.0$ MeV for Case III, with steps of 0.5 MeV. Here, the lower limit corresponds to the lowest energy for which the range of resonant pitch angles exceeds $20^\circ$, the upper limit is chosen based on test estimates on precipitation flux (see below).

For particles with maximum considered energy 9 MeV the chosen simulation duration of 15 s corresponds to the drift time across an arc of $5^\circ$, which seems reasonable transverse size for an EMIC wave packet. For longer time intervals, drift effects have to be taken into account. The chosen time interval 15 s corresponds to 25–50 bounce periods.

Interaction regimes

First, we study which interaction regimes are possible for the considered wave packets and particle ensemble, and how their features depend on particle parameters and wave packet evolution.

In this section, we analyze the interaction regimes based on phase averaged change in $\mu =\sin ^2\Theta _L$ after a single pass through wave packet for given $W_0$, $\Theta _L$ and time. Significant $\langle \Delta \mu \rangle \ne 0$ will allow us to determine a predominant nonlinear regime: $\langle \Delta \mu \rangle > 0$ corresponds to either force bunching, nonlinear shift of the resonance point (lower $\Theta _L$) or phase bunching, $\langle \Delta \mu \rangle <0$ corresponds to either nonlinear shift of the resonance point (higher $\Theta _L$) or trapping (Grach and Demekhov 2020a).

To calculate $\langle \Delta \mu \rangle$ at a time t for every $\Theta _L=4^\circ , 5^\circ ,..., 85^\circ$ we find all test particles outside the wave packet with equatorial pitch angle $\Theta _L^* \in [\Theta _L-0.5^\circ ,\Theta _L+0.5^\circ ]$. Then, change in $\mu$ for these particle after their next pass through the wave packet is averaged over initial phases, and thus we obtain $\langle \Delta \mu \rangle$ and rms $\sigma _\mu$ as a function of $\Theta _L$.

In Fig. 5, we plot $\langle \Delta \mu \rangle (\Theta _L)$ and $\sigma _\mu (\Theta _L)$ for two moments during the simulation and three values of electron energy $W_0$ for all three cases (here $\Theta _L$ is the equatorial pitch angle before the pass through the packet).

The dependencies $\langle \Delta \mu \rangle (\Theta _L,W_0)$, $\sigma _\mu (\Theta _L,W_0)$ are similar for all three cases and are also in agreement with previous results for model wave packets with Gaussian amplitude profile (Grach and Demekhov 2020a).

For lower energies (close to the lower limit of the resonant range, top row in Fig. 5) small positive maximum of $\langle \Delta \mu \rangle$ at smaller $\Theta _L$ is related to force bunching, with small influence of nonlinear shift of the resonance point. The negative minimum of $\langle \Delta \mu \rangle$ at higher $\Theta _L$ is also caused by nonlinear shift of the resonance point (trapping by the wave field is not possible at these energies, see Fig. 4). For higher energies (close to the upper limit of the resonant range, bottom row in Fig. 5), positive maximum of $\langle \Delta \mu \rangle$ is located at intermediate $\Theta _L$ and caused by a nonlinear shift of the resonance point with a small influence of phase bunching. A strong negative minimum near the upper limit of resonant range of $\Theta _L$ is related to the particle trapping by the wave field. The nonlinear effects are the strongest ($|\langle \Delta \mu \rangle |\ge \sigma _\mu$) for lower energies and low pitch angles close to the loss cone and for trapping (Case II, higher energies). For higher energies and relatively wide range of $\Theta _L<25^\circ$–$40^\circ$, the interaction is linear ($\langle \Delta \mu \rangle (\Theta _L) \approx 0$) with small rms deviation $\sigma _\mu$ (and, consequently, small diffusion coefficients).

With increasing energies (from top row to bottom in Fig. 5), the extrema of $\langle \Delta \mu \rangle (\Theta _L)$ are shifted to the higher values of $\Theta _L$.

The influence of complicated packet structure (in comparison with simpler models (Grach and Demekhov 2018b, 2020a)) can be seen in the temporal dynamics of the dependencies $\langle \Delta \mu \rangle (\Theta _L), \sigma _\mu (\Theta _L)$, because this dynamics is mostly determined by the wave packet evolution. For dependencies $\langle \Delta \mu \rangle (\Theta _L)$, only Case II shows roughly the same temporal dynamics as the model wave packets, considered in (Grach and Demekhov 2018b, 2020a): $|\langle \Delta \mu \rangle (\Theta _L)|$ decreases with time, and the extrema of $\langle \Delta \mu \rangle (\Theta _L)$ are shifted to the lower values of $\Theta _L$. For Cases I and III, $|\langle \Delta \mu \rangle (\Theta _L)|$ can either increase or decrease and the extrema of $\langle \Delta \mu \rangle (\Theta _L)$ are shifted to the higher values of $\Theta _L$.

This dynamics is explained as follows. For electrons with lower energies and/or higher pitch angles (corresponding to exrema in $\langle \Delta \mu \rangle (\Theta _L)$), the resonance points are located closer to the trailing edge. Therefore, the resonant interaction with these particles will be strongly influenced by generation of higher-frequency waves at the packet trailing edge near the equator. For Case II, the generation of higher-frequency waves during the simulation time is not significant (see Fig. 2): generation stops in the middle of the simulation, the increase of trailing edge frequency is small and the amplitudes of the generated waves are low. Thus, evolution of $\langle \Delta \mu \rangle (\Theta _L)$ in Case II is caused mostly by wave packet propagation, like in (Grach and Demekhov 2020a). On the contrary, for Cases I and III, generation of higher-frequency waves at the equator during the simulation time plays an important role in the packet evolution and temporal dynamics of wave–particle interaction (see Figs. 2,3).

Temporal dynamics of $\sigma _\mu (\Theta _L)$ in the linear regime at low $\Theta _L$ is also different for different cases (see bottom row of Fig. 5): $\sigma _\mu$ increases with time in Case I, decreases only slightly in Case II and remains constant in Case III. The results of (Grach and Demekhov 2020a), on the contrary, show only an increase in $\sigma _\mu$ with time for low pitch angles and higher energies (see Appendix there). The reason for such a difference stems from the fact that the resonance points of such electrons are located near the leading edge of the packet (Grach and Demekhov 2018b, 2020a). So, the resonant interaction with these particles will be influenced by the dissipation of lower-frequency waves once the packet nears $\text {He}^+$ resonance. In Case I, there is no dissipation during the simulation time; so, $\sigma _\mu$ increasing is due to wave packet propagation, like in (Grach and Demekhov 2020a). In Cases II and III, $\sigma _\mu (\Theta _L)$ temporal dynamics is determined by the wave packet dissipation, though amplitude modulation also can have a quantitative effect.

We should also note that particle trapping by the wave field leads to a significant change in $\Theta _L$ (up to $40^\circ$) only in Case II (the case with minimum values of inhomogeneity parameter R and the longest packet). In Cases I and III, pitch angle change as a result of trapping does not exceed $20^\circ$, mostly because effective packet lengths are shorter in these cases. The similar results of particles detrapping due to short packet effective lengths were observed in test particle simulations for whistler mode packets (Tao et al. 2012).

Fig. 5

Phase averaged change in $\mu$ (solid lines) and root mean square deviation $\sigma _\mu$ (dashed lines) for different energies for Case I (a), II (b) and III (c) for $t-t_\text {begin}=4$ s and $t-t_\text {begin}=12$ s (blue and red lines, respectively)

Precipitation mechanisms

Let us analyze particle precipitation mechanisms during the entire simulation time. Following (Grach and Demekhov 2020a), we plot the scattering pitch angle $\Theta _{L\text {sc}}$ (equatorial pitch angles of precipitating electrons before the last interaction with the wave packet) as a function of time. Figure 6 shows the results for Case II, and the results for the other two cases are qualitatively similar.

As one can see from Fig. 6, the maximum scattering pitch angle $\Theta _{L\text {sc}}$ is about $30^\circ$. According to previous analysis (see Figs. 4, 5 and relevant discussion), for these pitch angles $R>1$. Thus, of nonlinear regimes only force bunching and/or nonlinear shift of the resonance point are possible.

Force bunching blocks precipitation from low pitch angles for lower energies (about one half of the considered energy range). At the same time, when precipitation from low $\Theta _L$ is blocked by force bunching, it is possible from higher $\Theta _L$. Specifically, at $W_0=2$ MeV (Fig. 6b), precipitation is blocked from $\Theta _{L\text {sc}}\le 20^\circ$ (at early times) but is possible from $\Theta _{L\text {sc}}$ up to $30^\circ$. This precipitation occurs in the regime which is close to linear (with small influence of nonlinear shift of the resonance point). For higher energies, when the effect of force bunching is absent (Fig. 6e and f), $\Theta _{L\text {sc}}\le 10^\circ$. The range of blocked pitch angles and maximum $\Theta _{L\text {sc}}$ have a maximum over energy; for entirely linear precipitation the range of $\Theta _{L\text {sc}}$ decreases with energy.

Trapping by the wave field cannot directly cause precipitation, because it occurs at high pitch angles and $|\Delta \Theta _L| < \Theta _{L0}-\Theta _{L\text {c}}$. To analyze indirect influence of trapping, we plot the maximum change (decrease) in $\Theta _L$ of precipitating particles over their trajectories $|\Delta \Theta _L|_\text {max}^\text {lost}$. The results for Case II are shown in Fig. 7. For lower energies (Fig. 7a and b) the trapping is impossible (see Fig. 4e), and $|\Delta \Theta _L|_\text {max}^\text {lost}$ has roughly the same value as the change which leads directly to precipitation. For intermediate energies (Fig. 7c and d) particles that have been trapped along their trajectory can make up a significant part of precipitating particles. For higher energies (Fig. 7f), there are no trapped particles in the loss cone, despite the fact that for these energies trapping is the most effective. It happens because trapping takes place for $\Theta _L\ge 60^\circ$, while in rather wide range $\Theta _L\le 40^\circ$ the wave–particle interaction is linear with small $\sigma _\mu$ (see bottom row of Fig. 5). Thus, trapped particles do not have enough time to reach the loss cone due to diffusion after they leave the trapping region.

Figures 6 and 7 also demonstrate the influence of the initial particle distribution in space at the beginning of the simulation. As one can see, this influence disappears after $4-8$ bounce oscillations (approximately $2-5$ s).

Temporal dynamics of $\Theta _{L\text {sc}}$ is complicated, different for different energies even within one case and more diverse than for a model packets considered in (Grach and Demekhov 2020a). When nonlinear effects are strong ($W_0<4$ MeV, Fig. 6a–d), the difference from the earlier results is mostly due to longer simulation time. When precipitation is linear or almost linear ($W_0\ge 4$ MeV, Fig. 6e and f), the range of precipitation pitch angles has a slight minimum or decreases with time; for a model wave packet with Gaussian amplitude profile, precipitation pitch angle range increases with time for linear precipitation (Grach and Demekhov 2020a).

This difference is explained by the fact that for particles with higher energies and low pitch angles resonant interaction is influenced by dissipation of the lower-frequency waves once the packet nears the $\text {He}^+$ resonance. Also, spatial amplitude profiles for Case II are different from Gaussian one (see Fig. 2b); this amplitude modulation is not strong enough to cause qualitative difference for interaction regimes at any given time, but can have quantitative effect.

For Cases I and III, generation of higher-frequency waves might influence $\Theta _{L\text {sc}}$ temporal dynamics. These dynamics is roughly the same as temporal dynamics of precipitating flux, that is discussed below.

Fig. 6

Equatorial pitch angles $\Theta _{L\text {sc}}$ of precipitating electrons before the last interaction with the wave packet as a function of time for Case II. Time is counted from the beginning of simulation and corresponds to particle exiting the wave packet

Fig. 7

The maximum change in equatorial pitch angle over the trajectory for precipitating electrons. Case II, time is counted from the beginning of simulation and corresponds to particle exiting the wave packet after the maximum change has occurred. Horizontal line shows the maximum change for the precipitation itself

Evolution of the distribution function

To analyze the simulation results in terms of particle distribution function and for correct analysis of precipitating fluxes, we have to establish the connection between the distribution function $\Phi _{\Theta _L}(\Theta _L)$ and the distribution of the test particles in the phase space. As in (Grach and Demekhov 2020a) we use the following normalization (Bespalov and Trakhtengerts 1986; Trakhtengerts and Rycroft 2008):

$$\begin{aligned} N=\int n(z) \frac{B_\text {0m}}{B_0(z)}\text {d}z. \end{aligned}$$

(11)

Here, N is the number of particles in a geomagnetic flux tube with a unit cross section at the ionosphere, $n(z)=\int f d^3{\mathbf {p}}=\int f \sin {\Theta } \text {d}\Theta \, p^2\text {d}p\,\text {d}\Psi$ is the local number density, f is the local particle distribution function averaged over gyrophases, $\Theta$ is the local pitch angle, p is particle momentum and $B_{0\text {m}}$ is the maximum field for the given geomagnetic field line. The distribution function f can be averaged over bounce oscillations and integrated over particle momentum (taking into account that the electron energy is conserved with a high accuracy) and over phases and thus, we obtain particles distribution in equatorial pitch angles $\Phi _{\Theta _L}(\Theta _L)$.

We can also express N via the number of test particles in the simulation, $N_\text {p}$:

$$\begin{aligned} N=\frac{v_0 {\overline{T}}_B}{2 \mu _{\text {c}}} \beta _\text {V} N_\text {p}, \end{aligned}$$

(12)

where $v_0$ is particle velocity (which stays constant during the interaction), $\mu _\text {c}=\sin ^2\Theta _{L\text {c}}$ corresponds to the loss cone, ${\overline{T}}_B=\int T_\text {B}(\mu ) \text {d}\mu$, $\beta _\text {V}$ is the normalization constant, and $N_\text {p}$ is the number of test particles in the simulation.

Using (11) and (12), we can write the connection between distribution function ${\Phi }_{\Theta _L}$ and distribution of the test particles in the phase space as follows:

$$\begin{aligned} {\Phi }_{\Theta _L}=\frac{\Delta N_\text {p}}{\Delta \Theta _L}\frac{ {\overline{T}}_B}{T_\text {B}}\frac{\beta _\text {V}}{\sin {(2\Theta _L)}}. \end{aligned}$$

(13)

Here $\Delta N_\text {p}$ is the number of test particles having the pitch angle $\Theta _L$ within the range $\Delta \Theta _L$.

We also assign a specific weight to each test particle to ensure that the initial particle distribution $\Phi _{\Theta _L}|_{t=t_{\text {begin}}}$ is constant. More details can be found in the Appendix of (Grach and Demekhov 2020a).

We divide the total simulation time 15 s in 26 intervals $\{\Delta t_{i}\} = t_{i+1}-t_i$, $i=0,1,...,26$, $t_0 = t_{\text {begin}}$, $t_{26} = t_{\text {end}}$, where $\Delta t_0 = \Delta t_{25}= 0.3$ s and the other intervals $\Delta t_{0<i<25}=0.6$ s. The value 0.6 s corresponds to the bounce period of particles close to the loss cone: $T_\text {B}(\Theta _L=\Theta _{Lc}) \approx 0.6$–0.62 s. The first interval is chosen shorter, because at $t_0 = t_{\text {begin}}$ the particle ensemble is located near the equator; the time $\Delta t_0 = 0.3$ s is long enough that all particles precipitated after the first pass through the packet will reach the ionosphere and short enough that all other particles during $\Delta t_0$ will pass the wave packet in the resonant direction only once. In the further analysis, both $\Delta t_0$ and $\Delta t_{25}$ will be ignored.

We average the particle distribution function over the intervals $\Delta t_ {i}$ and attribute the obtained result to the time $\tau _i=(t_ {i + 1} + t_i)/2$ (the middle of the interval $\Delta t_ {i}$). We use the grid in $\Theta _L$ which is not fine enough to ensure distribution function resolution within the loss cone, so in the loss cone the distribution function has one value, $\Phi _{\Theta _L}^{\text {c}}$.

Fig. 8

Evolution of distribution function. Case II, time is counted from the beginning of simulation. Initial distribution is shown in black, the magenta line corresponds to the loss cone

Figure  8 shows the evolution of distribution function for 3 values of energy in Case II. These energies represent three typical patterns of wave–particle interaction in Case II; Cases I and III also demonstrate the similar patterns.

For $W_0=2$ MeV (lower energies), the distribution function is close to isotropic for $\Theta _L\le 60^\circ$ (there is no resonant interaction for higher $\Theta _L$, so the distribution function remains undisturbed). The value of $\Phi _{\Theta _L}$ slowly decreases with time from the initial value $\Phi _{\Theta _L}^0$ to approximately $0.8\Phi _{\Theta _L}^0$.

For $W_0=3$ MeV (intermediate energies), the distribution function is close to isotropic up to $\Theta _L\approx 20^\circ$–$30^\circ$ and then there are noticeable variations with a maximum in the vicinity of $40^\circ$ and a minimum in the vicinity of $50^\circ$. These variations are caused by phase bunching and trapping.

For $W_0=5$ MeV (higher energies), the distribution function increases from $\Phi _{\Theta _L}^{\text {c}}\approx (0$–$0.5)\Phi _{\Theta _L}^0$ in the loss cone to the initial value $\Phi _{\Theta _L}^0$ at $\Theta _L \approx 20^\circ$. Noticeable variations, associated with phase bunching and trapping by the wave field, take place in the area $\Theta _L \approx 40^\circ$–$70^\circ$ and thus do not influence the precipitation.

The behavior of the distribution function in the vicinity of the loss cone is roughly the same as for the model Gaussian packet in (Grach and Demekhov 2020a). Trapping by the wave field is not effective for Gaussian packet and particle ensemble considered in (Grach and Demekhov 2020a), so noticeable variations, associated with trapping, were present only for a model packet with flat amplitude profile.

Precipitating flux

For the further analysis, we normalize the precipitating fluxes $S_\text {pr}^\text {num}$, directly corresponding to the numerical simulation results, to the flux $S_\text {pr}^\text {SD}$ in the limiting case of strong diffusion. In this case the loss cone is filled continuously and distribution function is isotropic; the precipitating flux takes the limiting value equal to the trapped flux (Kennel and Petschek 1966; Bespalov and Trakhtengerts 1986; Trakhtengerts and Rycroft 2008):

$$\begin{aligned} S_\text {pr}^\text {SD}=\frac{N\mu _\text {c}}{{\overline{T}}_B}. \end{aligned}$$

(14)

Here N is the number of particles in geomagnetic field tube with a unit cross section at the ionosphere (11), (12). The simulated precipitating flux is evaluated as:

$$\begin{aligned} S_\text {pr}^\text {num}(\tau _i)=\frac{N\delta N_\text {p}}{\Delta t_i}. \end{aligned}$$

(15)

Here, $\delta N_\text {p}=N_{\text {p}\,\text {lost}}/N_\text {p}$ is the relative number of test particles, precipitated during time interval $\Delta t_i$, $\tau _i$ and $\Delta t_i$ are described above.

Dependencies of the normalized precipitating fluxes ${\tilde{S}}=S_\text {pr}^\text {num}/S_\text {pr}^\text {SD}$ on the time and energy are shown in Fig. 9. Precipitating fluxes ${\tilde{S}}_{\text {av}}$, averaged over the whole simulation time, as well as maximum and minimum values, are shown in Fig. 10.

Fig. 9

Temporal dynamics of normalized precipitating fluxes for Case I (a), II (b) and III (c)

Fig. 10

Normalized precipitating fluxes, averaged (solid lines), maximized (dotted lines) and minimized (dotted lines) over the simulation time, for Case I (a), II (b) and III (c)

The energy dependence of time-averaged fluxes is similar for all three cases. At the lowest energy, ${\tilde{S}}_{\text {av}} \approx 0.4$–0.6, then it increases to ${\tilde{S}}_{\text {av}} \approx 1$ and is nearly constant for an interval about 1 MeV ($W_0=4$–5 MeV for Case I; $W_0=2$–3 MeV for Case II; $W_0=4.5$–5.5 MeV for Case III) and then decreases, to values ${\tilde{S}}_{\text {av}} \le 0.25$ at the right boundary of the energy range.

The maximum values of precipitating fluxes correspond to the case of strong diffusion, i.e., to an almost isotropic distribution function in the vicinity of the loss cone (see Fig. 8). These cases correspond to the strongest interaction at $\Theta _L< 40^\circ$, i.e., the most effective force bunching, the highest values of precipitating pitch angles $\Theta _{L\text {sc}}$ and the widest range of $\Theta _{L\text {sc}}$ (see Fig. 6b–d). In Case II, there is an energy range (2.5 MeV$\le W_0 \le 3.0$ MeV), for which $S_\text {pr}^\text {num}/S_\text {pr}^\text {SD} \sim 1$ and trapping by the wave influences the precipitation (see Fig. 7c and 7d). In Cases I and III for energies corresponding to maximum fluxes trapping is not possible (see Fig. 4e and  4f).

The time dependence for maximum precipitating fluxes is not significant: fluxes oscillate near the average value. For lower and higher energies, when $S_\text {pr}^\text {num}/S_\text {pr}^\text {SD} < 1$, temporal dynamics of the fluxes is determined by generation of higher-frequency waves at the trailing edge, dissipation of the lower-frequency waves at the leading edge and propagation effects. For particles with lower energies, resonance points are located closer to the trailing edge (for rising tone packets, see (Grach and Demekhov 2018a, 2020a)), thus the resonant interaction is influenced by the generation of higher-frequency waves. When the generation takes place during the simulation, precipitating fluxes for lower energies increase with time (Cases I and III); when the generation is finished and wave packet propagates away from the equator, precipitating fluxes for lower energies decrease with time (Case II and model Gaussian packet in (Grach and Demekhov 2020a)). For particles with higher energies and low pitch angles (which determine the precipitation), resonant interaction, on the contrary, is influenced by the dissipation of lower-frequency waves at the leading edge. Thus, before the dissipation starts, precipitating fluxes for higher energies increase with time (Case I and model Gaussian packet in (Grach and Demekhov 2020a)) due to propagation effects; when the dissipation takes place precipitating fluxes for higher energies either fluctuate near an average value (Case III) or decrease with time (Case II).

It is also important to note that the temporal dynamics of the precipitating fluxes in the linear regime (higher energies) agrees with temporal dynamics of $\sigma _\mu$ (see Fig. 5 and relevant discussion) and thus with temporal dynamics of diffusion coefficients. The decrease of ${\tilde{S}}_{\text {av}}$ with energy once the precipitation becomes linear is slowest for the case with lowest frequencies at the leading edge (Case III). Under similar wave amplitudes lower frequencies at the leading edge lead to smaller values of R for low pitch angles and higher energies, which in turn leads to larger $\sigma _\mu$.

Note that Cases I and II actually belong to the same wave packet (element 14) with strong amplitude modulation and simulation for Case II starts 20 s after simulation for Case I ends (see Fig. 2). Thus, for element 14, we can assume a significant change in energy spectrum of precipitated particles on a time scale about 30 s. Within one simulation (15 s), there also can be a slight change of precipitated particles energy spectrum, caused by the wave packet evolution: the flux maximum on energy can become smoother (Case I), more pronounced (Case II) or shift to lower energies (Case III).

The temporal dynamics of precipitated fluxes and the energy spectrum of precipitating particles is generally in qualitative agreement with the results of (Kubota and Omura 2017), corresponding to the case when trapping does not cause direct precipitation (the case with low cold plasma density).

Summary

We have studied the resonant interaction of relativistic electrons with EMIC wave packets within one event (11:50–13:50 UT, 14 September 2017, Van Allen Probe B). The considered wave packets have rising tone within proton band and amplitudes up to 1.2 nT.

As a result of interaction with the wave packets under consideration, electrons with energies of 1.5–9 MeV can effectively precipitate into the loss cone. For particles with energies 2–5 MeV (depending on the wave packet and time interval), the precipitating flux is close to the limiting value corresponding to the strong diffusion regime.

The influence of a realistic wave packet structure brings the following specific features in the interaction, compared with idealized cases considered earlier (Grach and Demekhov 2020a).

For the considered short time intervals, the approximation of each local amplitude maximum of the wave packet by a Gaussian amplitude profile and a linear frequency drift gives a satisfactory description of the resonant interaction dynamics. At the same time, generation of higher-frequency waves at the packet trailing edge near the equator and dissipation of lower-frequency waves in the $\text {He}^+$ gyroresonance region at the leading edge can play an important role.

Generation of higher-frequency waves mostly influences interaction for electrons with lower energies and/or higher equatorial pitch angles, i.e. the particles, for which nonlinear interaction takes place. As long as the higher-frequency parts of the wave packet are generated near the equator with high enough amplitudes, the precipitating flux at lower energies increases. Once the generation stops and the wave propagates away from the equator, the corresponding precipitating flux decreases.

Dissipation of lower-frequency waves mostly affects interaction for particles with higher energies and low equatorial pitch angles, i.e., linear interaction. Once the dissipation starts, precipitating flux for particles with higher energies decreases with time. If there is no dissipation in the considered time interval, precipitating flux for higher energy particles (linear precipitation) will increase with time.

The amplitude modulation of the wave packet leads to a significant change of energy spectrum of precipitated particles during short time. Specifically, for element 14 of the considered event (three local amplitude maxima), the energy with maximum precipitating flux decreases from $W_0\approx 4.5$ MeV to $W_0\approx 2.5$ MeV during 30 s, while the element itself exists for about 120 s.

The main nonlinear effects, which affect the precipitation, are the force bunching and nonlinear shift of the resonance point. Force bunching blocks precipitation for particles with low pitch angles, up to $\Theta _L\approx 20^\circ$. At the same time, the precipitation can exist from a noticeable range of higher $\Theta _L\approx 10^\circ$–$30^\circ$. This situation corresponds to maximum precipitating fluxes.

Particle trapping by the wave field can indirectly influence precipitation for some energies, but this influence is not crucial for the considered parameters. Effective precipitation due to trapping and directed scattering is possible for longer and higher-amplitude wave packets (amplitudes up to 14 nT are observed (Nakamura et al. 2019)) and also for higher cold plasma density (Kubota and Omura 2017; Grach and Demekhov 2018a, b).

In summary, for EMIC wave packets with amplitudes around 1 nT nonlinear effects play an important role in the formation of precipitating fluxes, even in the cases when wave packets are short and trapping by the wave field is not effective. Model wave packets with main parameters based on observations give a satisfactory description of precipitation dynamics, but real fine structure of a wave packet influences actual values of precipitating fluxes very significantly. It may be of interest to study other specific cases corresponding to real observations, and this will be a subject of future work.