Abstract
The equilibrium and Lyapunov stability properties for two-dimensional ideal magnetohydrodynamic (MHD) plasmas with incompressible and homogeneous (i.e., constant density) flows are investigated. In the unperturbed steady state, both the velocity and magnetic field are nonzero and have three components in a Cartesian coordinate system with translational symmetry (i.e., one ignorable spatial coordinate). It is proved that (a) the solutions of the ideal MHD steady state equations with incompressible and homogeneous flows in the plane are also valid for equilibria with the axial velocity component being a free flux function and the axial magnetic field component being a constant, (b) the conditions of linearized Lyapunov stability for these MHD flows in the planar case (in which the fields have only two components) are also valid for symmetric equilibria that have a nonplanar velocity field component as well as a nonplanar magnetic field component. On using the method of convexity estimates, nonlinear stability conditions are established.
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