Numerical Simulation of the Harris Instability in Two Dimensions

The electrostatic Harris instability arising from a bi‐Maxwellian electron velocity distribution in the presence of a uniform magnetic field is studied linearly by numerically solving the electrostatic dispersion relation and nonlinearly by two dimensional electrostatic numerical simulation. Contour plots of growth rate in $k_{\perp }{\mathrm{\hspace{0.17em}-\hspace{0.17em}k}}_{\Vert }$ space for a range of ${\omega }_p{\mathrm{/\Omega }}_c$ and $T_{\perp }{\mathrm{/T}}_{\Vert }$ are used to summarize the linear theory. With the initial conditions $T_{\perp }\text{\hspace{0.17em}≠\hspace{0.17em}0}$ and $T_{\Vert }\text{\hspace{0.17em}=\hspace{0.17em}0}$ the numerical simulations show the nonlinear development with $T_{\Vert }$ increasing the $T_{\perp }$ decreasing until $T_{\perp }{\mathrm{/T}}_{\Vert }\text{∼3}$ which is the practical stability boundary from linear theory. The saturation level of the electrostatic field energy is very small being 0.06% of the total kinetic energy consistent with the predictions of a simple nonlinear theory.