Magnetohydrodynamic equilibrium. I - Exact solutions of the equations

The steady equations of magnetohydrodynamics appropriate for an inviscid fluid of high electrical conductivity are considered for the case that the physical quantities are independent of one Cartesian coordinate or the angle φ in cylindrical and spherical coordinates. By using the magnetic vector potential as a variable, the vector equations are integrated, and the system is reduced to a single, scalar, elliptic, partial differential equation, containing arbitrary functions of the magnetic potential. The result includes, as special cases, Ferraro's law of isorotation, Chandrasekhar's equivalent differential equations for axisymmetric fields and toroidal fluid motions, the theorems of Chandrasekhar and Prendergast for the equilibrium of axisymmetric magnetic fields, the known theorems on force-free fields, and other conditions for magnetostatic and hydrodynamic equilibrium. Some simple solutions of the differential equation are presented to illustrate the applications.