Steady axisymmetric toroidal equilibrium in ideal magnethohydrodynamics

A self-consistent theory of steady axisymmetric toroidal equilibrium in ideal magnetohydrodynamics with poloidal rotation about the magnetic axis and toroidal flow around the torus is described. An application of the characteristic theory of hyperbolic partial differential equations to the steady hydromagnetic equations yields critical rotational velocities (poloidal) for the formation of standing hydromagnetic waves, which depend on the direction of the wave normal. For low β and large aspect ratio, the critical velocities for the slow (magnetosonic) mode are, for any direction, nearly equal to a velocity found by Stringer for the onset of anomalous diffusion. Flows with poloidal motion about the critical slow velocities are investigated by analogy with nozzle flow in conventional gasdynamics, the distance along the axis of nozzle corresponding to the poloidal angle. The periodicity of entropy with respect to the poloidal angle excludes shock transitions and, as a result, trans-slow-magnetosonic flows. A class of continuous, self-consistent solutions in the standard tokamak ordering is obtained in terms of the expansion of the square root of the inverse aspect ratio, in which the poloidal flow approaches, at a point corresponding to the throat of nozzle, the critical slow velocity in the poloidal direction; the solutions are found to be approximated by continuous solutions in the usual low-β approximation which does not make use of Ampere's law.