General formulation and exact solution for two-dimensional field-aligned magnetohydrodynamic equilibrium flows

A general formulation is presented for steady field-aligned magnetohydrodynamic (MHD) equilibrium flows with isotropic or gyrotropic pressures. Closure to the anisotropic MHD model is provided by a pair of double-polytropic energy equations, for which double-adiabatic and double-isothermal conditions are special limits of the model. For the latter case, a MHD counterpart of Bernoulli’s equation is derived. The study is then focused on the two-dimensional (∂/∂y=0 but By≠0) problems, for which a generalized Grad–Shafranov equation is developed for field-aligned MHD flow equilibria with isotropic or gyrotropic pressures. The formulation is put in a form that allows self-consistent solutions to be constructed numerically in a way similar to the static case; examples of such MHD equilibria are shown. An asymptotic formulation is also developed for stretched gyrotropic plasma configurations, which, however, is not applicable to two-dimensional planar configurations with regions of weak magnetic field strength, such as the geomagnetic tail.