Transitional regime of electron resonant interaction with whistler-mode waves in inhomogeneous space plasma

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

Figure 1

Trajectories of electrons resonantly interacting with a parallel whistler-mode wave (one resonant interaction is shown): Wave amplitude is sufficiently low and electrons are diffusively scattered (a) and wave amplitude is high and electrons experience phase bunching and phase trapping (b). Long-term dynamics of electron trajectories ($\sim 100$ resonances) for low (c) and high (d) wave amplitudes. All electrons have the same initial energy of 150 keV and pitch angle ${45}^{\circ }$. The background magnetic field is the Earth's dipole field with an equatorial magnitude corresponding to 6 Earth radii away from the planet (the outer radiation belt). Wave characteristics are typical for parallel whistler-mode waves: $\omega /{\mathrm{\Omega }}_0\left(0\right)=0.25, B_w=20$ pT for (a) and (c) and $B_w=200$ pT for (b) and (d) [43, 44]. The background plasma density is given by an empirical model [45].

Figure 2

Two trajectories from Figs. 1 and 1 are shown in the (energy, pitch angle) plane for diffusive (a) and nonlinear (b) resonant interactions (color shows time). Evolution of an initially localized peak of distribution $f\left(\gamma \right)$ for the same $h=\gamma -[\omega /2{\mathrm{\Omega }}_0\left(0\right)]({\gamma }^2-1){sin}^2{\alpha }_0$ as from panels (a) and (b): diffusive (c) and nonlinear (d) resonant interactions. Note that timescales are different for (c) and (d) panels.

Figure 3

The profile of $u/r$ ratio (a). Phase portraits of Hamiltonian (8) for different $u/r$ ratios (b). Profiles of ${\gamma }_R$ and $\overline{S}=S{[R{\mathrm{\Omega }}_0\left(0\right)/c]}^{1/2}$ along $s$ coordinate [note $R{\mathrm{\Omega }}_0\left(0\right)/c$ is of the order of $1/\varepsilon$ for the system parameters] (a). System parameters correspond to the trajectory shown in Fig. 1.

Figure 4

Panel (a) shows profiles of $S\left(\gamma \right), \mathrm{\Pi }\left(\gamma \right)$ for the trajectory from Fig. 1. Schematic view of electron motion along $\gamma$ for given $\overline{S}\left(\gamma \right)=S{(R{\mathrm{\Omega }}_0\left(0\right)/c)}^{1/2}$ (b) [note $R{\mathrm{\Omega }}_0\left(0\right)/c$ is of the order of $1/\varepsilon$ for the system parameters]. An example of trajectory $\gamma \left(n\right)$ for $n=0..100$ resonant iterations (c). The relaxation of the distribution function $f\left(\gamma \right)$ (d). Plots in panels (c) and (d) are obtained with map (12).

Figure 5

The phase portrait of Hamiltonian (13) for $r/u$ around 1 (a). Set of trajectories obtained by numerical integration of Eq. (17) with ${\delta }_0={\varepsilon }^{2/3}$. Color codes the time interval required to cross the shown $d\xi /d\tau$ range.

Figure 6

Examples of $y_n$ trajectories described by map (18) [(a) and (b)] and evolution of $f\left(y\right)$ distribution [(c) and (d)] for two $\kappa$ values. Profiles of ${\mathcal{M}}_1\left(n\right)$ and ${\mathcal{M}}_2\left(n\right)$ for two $\kappa$ values (e). Dependence of $D_{\mathrm{NL}}$ on $\kappa$ for three $\varepsilon$ values (f).

Figure 7

Three trajectories for different $\ell$ values [see color codes in panels (c) and (d)] for systems of two different kinds [(a) and (b)]. Examples of $M_2$ profiles for these two systems [(c) and (d)]. Dependencies of $D_{\mathrm{NL}}$ on $\ell$ for the two systems and different $\varepsilon$ [(d) and (f)]. Left column [(a), (c), and (e)] shows results for the system of the first kind: the entire resonant range $y\in [-1,1]$, but $S$ is not equal to zero only for $[-\ell ,\ell ]$ appearing always around the particle location, i.e., there is a resonant interaction at each iteration. The right column [(b), (d), and (f)] shows results for the system of the second kind: the entire resonant range $y\in [-1,1]$, but $S$ is not equal to zero only over $[-\ell ,\ell ]$ appearing at random locations, not necessarily near the particle location, i.e., there are iterations without change of particle $y$.