Hartmann Flow with Braginsky Viscosity: A Test Problem for Plasma in the Intracluster Medium

Abstract

We consider a Hartmann layer, stationary flow of a viscose and resistive fluid between two plates with superimposed transverse magnetic field, in the limit of gyrotropic plasma, when viscosity across the field is strongly suppressed. For zero cross-field viscosity, the problem is not well posed, since viscosity then vanishes on the boundaries and in the middle of the layer, where there is no longitudinal field. An additional arbitrarily small isotropic viscosity allows one to find magnetic field and velocity profiles that are independent of this viscosity floor and different from flows with isotropic viscosity. Velocity sharply rises in a thin boundary layer, and the thinness of this boundary layer depends both on the Hartmann number and on the Lundquist number of the flow. The implication of the work is that, in simulating ICM dynamics, it is imperative to use numerical schemes that take into account anisotropic viscosity. Although magnetic fields are dynamically subdominant in the ICM, they do determine its dissipative properties, the stability of embedded structures, and the transition to turbulence.