Universal stability of hydromagnetic flows

Abstract

The non-linear stability of hydromagnetic flows is investigated by applying energy methods. A universal stability estimate, namely a stability limit for motions subject to arbitrary non-linear disturbances, is obtained for bounded or periodic domains. Our analysis is restricted to fluids possessing constant density and electrical conductivity and we do not take into account temperature or Hall effects. This result establishes the existence of an open region of certain stability near the origin of the $(\tilde{R}_e, \tilde{R}_m)$ Cartesian plane for every fixed Pm (where $(\tilde{R}_e, \tilde{R}_m)$ and Pm are the Reynolds number, magnetic Reynolds number and magnetic Prandtl number, respectively). The universal stability limit can then be improved by suitably defining a maximum problem using variational techniques, and obtaining the relevant Euler–Lagrange equations. The tentative solution to this problem gives a stability limit which enlarges the universal stability region. Our results are then compared with linear and experimental ones, with special emphasis given to the role played by the magnetic field.