Magneto-rotational instability in differentially rotating liquid metals

We study the stability of Couette flow between two cylinders in the presence of axial magnetic field in local WKB approximation. We find the analytical expression of the critical angular velocity minimized over the wave number and the imposed magnetic field as a function of the measure of deviation of the rotation law from the Rayleigh line. The result found is in a good agreement with the previously known numerical results based on the global analysis. We perform a minimization of the critical Reynolds number over the wave number at fixed magnetic field both analytically and numerically. We show that a compromise between resistive suppression of magneto-rotational instability at weak magnetic field and the increase of the critical Reynolds number with the increase of magnetic field is possible. It takes place at moderate values of magnetic field of order 3 × 102 gauss giving the critical Reynolds number of order 4×104.