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

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).

### 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.

### 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, \(\Phi _{\Theta _L}^{\text {c}}\).

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.

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).