The conditions for the non-linear 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 flows are considered which possess one of the types of symmetry (translational, axial, rotational or helical). The sufficient conditions for non-linear stability of the flows in question with respect to perturbations of this symmetry are obtained. These conditions are proved by the method of coupling the integrals of motion [1, 21 in the form [3-81, based on constructing functionals having absolute minima on specified steady solutions. Each of the functionals constructed is the sum of the kinetic energy, the integral of an arbitrary function of the Lagrangian coordinate and another integral, specific for the flows being investigated. The use of Lagrangian ooordinate fields leads to a whole family of new definitions of stability. According to these definitions, deviations of the perturbed flows from the unperturbed ones are measured by the integrals of the squares of the velocity-field and Lagrangiancoordinate perturbations. The stability conditions obtained are extended to existing results [5-7, 9] on new types of flows. These conditions are of an a priori nature since the corresponding theorems of existence of the solutions are not proved.