On a 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, ${D}_{\mathrm{QL}}\ensuremath{\sim}{B}_{w}^{2}$, 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, $\ensuremath{\sim}1/{D}_{\mathrm{NL}}$. 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.