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, \textbf{B}. In this setup, an electron pressure anisotropy with $p_{\perp,e} > p_{||,e}$ naturally arises due to the adiabatic invariance of the electron magnetic moment ($p_{||,e}$ and $p_{\perp,e}$ are the pressures parallel and perpendicular to \textbf{B}). If the anisotropy is large enough, the whistler instability arises, efficiently scattering the electrons and limiting $\Delta p_e$ ($\equiv p_{\perp,e}-p_{||,e}$). 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 $|\textbf{B}|$ 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 $\beta_e$ ($\equiv 8\pi p_e/|\textbf{B}|^2$). If initially $\beta_e \sim 1$, 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 $\sim 5\%$ of the electrons, and for $\sim 15\%$ 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.