Transonic magnetohydrodynamic flows

Abstract

Stationary flows of an ideal plasma with translational symmetry along the (vertical) z axis are considered, and it is demonstrated how they can be described in the intrinsic (natural) coordinates (ξ, η, &), where ξ is a label of flux and stream surfaces, η is the total pressure and ϑ is the angle between the horizontal magnetic (and velocity) field and the x axis. Three scalar nonlinear equilibrium equations of mixed elliptic–hyperbolic type for ϑ(ξ, η), ξ(η, ϑ) and η(ϑ, ξ) respectively are derived. The equilibrium equation for ϑ(ξ, η) is especially useful, and has considerable advantages compared with the coupled system of algebraic–differential equations that are conventionally used for studying plasma flows. In particular, for this equation the location of the regions of ellipticity and hyperbolicity can be determined a priori. Relations between the equilibrium equation for ϑ(ξ, η) and the nonlinear hodograph equation for ξ(η, ϑ) are elucidated. Symmetry properties of the intrinsic equilibrium equations are discussed in detail and their self-similar solutions are described. In particular, magnetohydrodynamic counterparts of several classical flows of an ideal fluid (the Prandtl–Meyer flows around a corner, the spiral flows and the Ringleb flows around a plate, etc.) are found. Stationary flows described in this paper can be used for studying both astrophysical and thermonuclear plasmas.