Axisymmetric ideal magnetohydrodynamic equilibria with incompressible flows

Abstract

It is shown that the ideal magnetohydrodynamic (MHD) equilibrium states of an axisymmetric plasma with incompressible flows are governed by an elliptic partial differential equation for the poloidal magnetic flux function ψ containing five surface quantities along with a relation for the pressure. Exact equilibria are constructed including those with nonvanishing poloidal and toroidal flows and differentially varying radial electric fields. Unlike the case in cylindrical incompressible equilibria with isothermal magnetic surfaces which should have necessarily circular cross sections [G. N. Throumoulopoulos and H. Tasso, Phys. Plasmas 4, 1492 (1997)], no restriction appears on the shapes of the magnetic surfaces in the corresponding axisymmetric equilibria. The latter equilibria satisfy a set of six ordinary differential equations which for flows parallel to the magnetic field B can be solved semianalytically. In addition, it is proved the nonexistence of incompressible axisymmetric equilibria with (a) purely poloidal flows and (b) nonparallel flows with isothermal magnetic surfaces and |B|=|B|(ψ) (omnigenous equilibria).