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, ϑ), 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.

Stationary Two-Dimensional Magnetohydrodynamic Flows with Shocks: Characteristic Analysis and Grid Convergence Study

“…Hence the class of stationary field-aligned flow is an important class of MHD flows. One could even argue that 2D stationary flow problems in a finite domain with the magnetic field not aligned to the plasma flow are rare [19,23,30]. It is hard to define the boundary conditions consistently in that case.…”

Trigonometric and hyperbolic functions method for constructing analytic solutions to nonlinear plane magnetohydrodynamics equilibrium equations

“…MHD equilibria of ideal plasmas with incompressible flows and translational as well as axial symmetry were investigated by many authors. [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] The fundamental concept behind MHD equilibria with flow is that magnetic fields can induce currents in a moving conductive fluid, which in turn creates forces on the fluid and also changes the magnetic field itself. The set of equations which describe the MHD equilibria with flow are a combination of the Navier-Stokes equations of fluid dynamics and Maxwell's equations of electromagnetism.…”

Time reversal duality of magnetohydrodynamic shocks

“…At the end of this section, the jump conditions for ideal MHD discontinuities and shocks are presented in a particular form, that has been exploited by Goedbloed and Lifshitz 17,18 for transonic flows in nontrivial geometries. The ensuing different types of discontinuities are reviewed in Sec.…”