Weibel-like instability in magnetohydrodynamics

Abstract

In magnetohydrodynamics (MHD), a density perturbation perpendicular to an electron temperature gradient generates a magnetic field around itself that acts to increase the perturbation, which can lead to instability. An MHD dispersion relation is obtained for perturbations perpendicular to a fixed electron temperature gradient with an initial in-plane magnetic field, including resistivity, viscosity, and the electrothermal coefficient. Instability occurs for sufficiently small electron temperature-gradient scale lengths determined by the ion collisionless skin depth. Both viscosity and resistivity are required to prevent growth at arbitrarily small spatial scales and to give a physical result for the fastest growing mode. The perpendicular electrothermal coefficient is only significant for a narrow range of low electron Hall parameters, causing a modest reduction in magnetic field growth and modifying the criteria for instability in the presence of viscosity. If the definition of the Weibel instability [E. S. Weibel, Phys. Rev. Lett. 2, 83 (1959)] is extended to include all instabilities due to anisotropy in the electron velocity distribution, then this is a Weibel-like instability because an electron temperature gradient implies an anisotropic electron velocity distribution. The implications for the formation of filaments in laser-produced plasmas and for the verification of MHD codes are considered.