Equilibrium properties and exact solutions for two-dimensional nonlinear force-free magnetic fields with mass flow

The steady state equilibrium properties of force-free magnetic fields (FFMFs) with mass flow in Cartesian geometry and invariant in a given direction are investigated. For constant Mach number (defined in Sec. II) flows, several classes of exact analytic solutions associated with solitonlike, antisolitonlike, kinklike, and antikinklike configurations are obtained. For nonconstant Mach number flows, it is shown that the equilibrium flow is impossible in some cases of FFMFs, and exact equilibria can be obtained in the other cases. The magnetic surfaces have a circular cross section for two-dimensional flows