STEADY MHD FLOWS WITH AN IGNORABLE COORDINATE AND THE POTENTIAL TRANSONIC FLOW EQUATION

The paper explores the interrelationship between the generalized Grad- Shafranov equation, or trans-field force balance equation, for steady MI-ID flows with an ignorable co-ordinate, and work by Imai on field-a ligned MHD hows. The development of Imai, assumes at the outset that t he fluid velocity V is parallel to the magnetic field B, and exploits an analogy with steady compressible irrotational flow in ordinary flui d dynamics. In Imai's analysis the magnetic induction B is written in the form B = sigma b, where sigma proportional to 1/(M(A)(2)-1), and M (A) is the appropriate Alfven Mach number. Gauss' law del.B = del.(sig ma b) = 0 then plays a role analogous to the mass continuity equation in ordinary fluid dynamics, where sigma corresponds to the density of the pseudo-fluid. Imai's analysis leads to a transonic equation for th e field potential phi defined by b = del phi. For a restricted class o f flows the trans-field force balance equation formulation also leads to the transonic potential flow equation, but the assumption of an ign orable co-ordinate allows for the possibility of non-field-aligned flo ws with non-zero electric field potential Phi(E). The characteristics of the generalized Grad-Shafranov equation are related to the Mach con e and the group velocity surface for linear magnetosonic waves. The co rresponding forms of the characteristics for the potential transonic f low equation in the (x, y) plane and in the (b(x), b(y)) hodograph pla ne are discussed. Sample solutions of the potential transonic flow equ ation for radial, helical and spiral flows are obtained by means of th e hodograph transformation, and are used to illustrate the differences between hyperbolic and elliptic flows. The potential transonic flow e quation is obtained for the case of an ignorable co-ordinate x of a re ctangular Cartesian co-ordinate system (x, y, z), and also for the cas e of flows with an ignorable co-ordinate phi of a spherical polar co-o rdinate system (r,theta,phi). Astrophysical applications are briefly d iscussed.