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⊥ − k‖ space for a range of ωp/Ωc and T⊥/T‖ are used to summarize the linear theory. With the initial conditions T⊥ ≠ 0 and T‖ = 0 the numerical simulations show the nonlinear development with T‖ increasing the T⊥ decreasing until T⊥/T‖∼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.