Magnetohydrodynamic stability thresholds as a function of Hartmann number and pinch ratio

The linear instabilities of an incompressible, uniform-density magnetofluid are considered in the periodic straight cylinder approximation. Spatially-dependent resistivity and viscosity profiles are regarded as fixed, but their magnitudes and the magnitudes of the currents driven in the plasma are not. It is shown that the stability boundary is determined by only two dimensionless numbers: the Hartmann number and the axisymmetric, zero-flow pinch ratio, which is inversely proportional to the safety factor at the wall. The result is analogous to the well known result from the theory of hydrodynamic shear flows that the Reynolds number governs the onset of hydrodynamic activity. (Author)