THE CONDITIONS FOR THE NONLINEAR STABILITY OF PLANE AND HELICAL MHD FLOWS

The stability of the steady flows of an ideal incompressible fluid of uniform density in a magnetic field is investigated. Only those MHD fl ows are considered which possess one of the types of symmetry (transla tional, axial, rotational or helical). The sufficient conditions for n on-linear stability of the flows in question with respect to perturbat ions of this symmetry are obtained. These conditions are proved by the method of coupling the integrals of motion [1, 2] in the form [3-8], based on constructing functionals having absolute minima on specified steady solutions. Each of the functionals constructed is the sum of th e kinetic energy, the integral of an arbitrary function of the Lagrang ian coordinate and another integral, specific for the hows being inves tigated. The use of Lagrangian coordinate fields leads to a whole fami ly of new definitions of stability. According to these definitions, de viations of the perturbed flows from the unperturbed ones are measured by the integrals of the squares of the velocity-field and Lagrangian- coordinate perturbations. The stability conditions obtained are extend ed to existing results [5-7, 9] on new types of flows. These condition s are of an a priori nature since the corresponding theorems of existe nce of the solutions are not proved.