Stochastic Electron Acceleration by the Whistler Instability in a Growing Magnetic Field

We use 2D particle-in-cell (PIC) simulations to study the effect of the saturated whistler instability on the viscous heating and nonthermal acceleration of electrons in a shearing, collisionless plasma with a growing magnetic field, $\mathbf{\text{B}}$. In this setup, an electron pressure anisotropy with $p_{\perp ,e}>p_{\mathrm{\mid }\mathrm{\mid },e}$ naturally arises due to the adiabatic invariance of the electron magnetic moment ($p_{\mathrm{\mid }\mathrm{\mid },e}$ and $p_{\perp ,e}$ are the pressures parallel and perpendicular to $\mathbf{\text{B}}$). If the anisotropy is large enough, the whistler instability arises, efficiently scattering the electrons and limiting $\mathrm{\Delta }{p}_e$ ($\equiv p_{\perp ,e}-p_{\mathrm{\mid }\mathrm{\mid },e}$). In this context, $\Delta p_e$ taps into the plasma velocity shear, producing electron heating by the so called anisotropic viscosity. In our simulations, we permanently drive the growth of $\mathrm{\mid }\mathbf{\text{B}}\mathrm{\mid }$ by externally imposing a plasma shear, allowing us to self-consistently capture the long-term, saturated whistler instability evolution. We find that besides the viscous heating, the scattering by whistler modes can stochastically accelerate electrons to nonthermal energies. This acceleration is most prominent when initially ${\beta }_e\sim 1$, gradually decreasing its efficiency for larger values of ${\beta }_e$ ($\equiv 8\pi p_e/|$\mathbf{\text{B}}$|^2$)\mathrm{.}Ifinitially$\beta_e \sim 1$, the final electron energy distribution can be approximately described by a thermal component, plus a power-law tail with spectral index $\sim 3.7$. In these cases, the nonthermal tail accounts for $\sim 5\mathrm{\%}$ of the electrons, and for $\sim 15\mathrm{\%}$ of their kinetic energy. We discuss the implications of our results for electron heating and acceleration in low-collisionality astrophysical environments, such as low-luminosity accretion flows.