Simulation and Quasi‐linear Theory of Magnetospheric Bernstein Mode Instability

Abstract

Multiple‐harmonic electron cyclotron emissions, often known in the literature as the (n + 1/2)fce emissions, are a common occurrence in the magnetosphere. These emissions are often interpreted in terms of the Bernstein mode instability driven by the electron loss cone velocity distribution function. Alternatively, they can be interpreted as quasi‐thermal emission of electrostatic fluctuations in magnetized plasmas. The present paper carries out a one‐dimensional relativistic electromagnetic particle‐in‐cell simulation and also employs a reduced quasi‐linear kinetic theoretical analysis in order to compare against the simulation. It is found that the Bernstein mode instability is indeed excited by the loss cone distribution of electrons, but the saturation level of the electrostatic mode is quite low, and that the effects of instability on the electrons is rather minimal. This supports the interpretation of multiple‐harmonic emission in the context of the spontaneous emission and reabsorption in quasi‐thermal magnetized plasma in the magnetosphere.

Whistler Instability Driven by Electron Thermal Ring Distribution With Magnetospheric Application

“…In our previous paper (Lee et al, ) we modeled the loss cone‐like energetic electron distribution by a DGH model (Dory et al, ), but for the present purpose, we simply adopt a thermal ring velocity distribution function for the hot electrons, especially in the PIC simulation, to be described later, where v ⊥ and v ‖ represent velocity component perpendicular and parallel to the ambient magnetic field vector, α represents the velocity spread, v 0 denotes the degree of perpendicular velocity population inversion, and is the error function. The DGH model and the thermal ring model share many properties, so we expect similar results from either distribution.…”

“…In our previous paper (Lee et al, 2018a) we modeled the loss cone-like energetic electron distribution by a DGH model (Dory et al, 1965), but for the present purpose, we simply adopt a thermal ring velocity distribution function for the hot electrons, especially in the PIC simulation, to be described later,…”

Quasilinear Model of Jovian Whistler Mode Emission

“…Note that perpendicular and parallel wave numbers, defined with respect to the direction of the ambient magnetic field, are given by k ⊥ = k sin θ and k ∥ = k cos θ , respectively. The (quasi) linear growth rate for the whistler mode waves is given for, a general hot electron distribution function f , as (Lee, Yoon, et al., 2018; Melrose, 1986) $$$\begin{array}{rll}\hfill \gamma & =\hfill & \frac{\pi }{2}\frac{{({{\Omega}}_{e}\mathrm{cos}\theta \hspace*{.5em}-\hspace*{.5em}\omega )}^{2}}{{{\Omega}}_{e}\mathrm{cos}\theta }\int d\mathbf{v}\,{v}_{\perp }^{2}\sum\limits _{n=-\infty }^{\infty }{[(\frac{{\omega }^{2}{\mathrm{sin}}^{2}\theta \hspace*{.5em}+\hspace*{.5em}n{{\Omega}}_{e}({{\Omega}}_{e}\mathrm{cos}\theta \hspace*{.5em}-\hspace*{.5em}\omega )}{({{\Omega}}_{e}\mathrm{cos}\theta -\omega ){{\Omega}}_{e}\mathrm{cos}\theta })\frac{{J}_{n}(b)}{b}+{J}_{n}^{\prime }(b)]}^{2}\hfill \\ \hfill & \hfill & \times \hspace*{.5em}\delta (\omega -n{{\Omega}}_{e}-k{v}_{{\Vert} }\mathrm{cos}\theta )(\frac{n{{\Omega}}_{e}}{{v}_{\perp }}\frac{\partial }{\partial {v}_{\perp }}+k\hspace*{.5em}\mathrm{cos}\theta \...$$…”

“…In Equation 8T and K are the coefficients that define the unit electric field vector, or equivalently, the polarization vector (Melrose, 1986). For the whistler mode these are given by (Lee, Yoon, et al, 2018)…”

“…In Equation T and K are the coefficients that define the unit electric field vector, or equivalently, the polarization vector (Melrose, 1986). For the whistler mode these are given by (Lee, Yoon, et al., 2018) $$$T=-1,\qquad K=-\frac{(1+{q}^{2}){q}_{\perp }}{{q}_{{\Vert} }},$$$ so that we may write Equation , in appropriate dimensionless quantities and variables, as …”