The instability of the states of rest of a viscous ideally conducting compressible medium containing a magnetic field

The problem of the linear stability of the states of rest of a viscous compressible medium with infinite conductivity in a magnetic field is investigated. It is shown, using Lyapunov's direct method, that these states of rest are unstable to small spatial perturbations, which reduce the effective potential energy, that is the sum of the internal energy of the medium and the magnetic-field energy. A priori bilateral exponential estimates of the increase in the perturbations are obtained, where the exponents in these estimates are calculated from the parameters of the state of rest and the initial data for the perturbations. A class of the most rapidly growing perturbations is obtained and an exact formula for determining their growth rate is derived. An example of the states of rest and of the initial perturbations which evolve in accordance with the estimates obtained so long as the linear approximation holds is constructed. (C) 2000 Elsevier Science Ltd. All rights reserved.