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-aligned MHD hows. The development of Imai, assumes at the outset that the fluid velocity V is parallel to the magnetic field B, and exploits an analogy with steady compressible irrotational flow in ordinary fluid 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.(sigma 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 the field potential phi defined by b = del phi. For a restricted class of flows the trans-field force balance equation formulation also leads to the transonic potential flow equation, but the assumption of an ignorable co-ordinate allows for the possibility of non-field-aligned flows with non-zero electric field potential Phi(E). The characteristics of the generalized Grad-Shafranov equation are related to the Mach cone and the group velocity surface for linear magnetosonic waves. The corresponding forms of the characteristics for the potential transonic flow equation in the (x, y) plane and in the (b(x), b(y)) hodograph plane are discussed. Sample solutions of the potential transonic flow equation for radial, helical and spiral flows are obtained by means of the hodograph transformation, and are used to illustrate the differences between hyperbolic and elliptic flows. The potential transonic flow equation is obtained for the case of an ignorable co-ordinate x of a rectangular Cartesian co-ordinate system (x, y, z), and also for the case of flows with an ignorable co-ordinate phi of a spherical polar co-ordinate system (r,theta,phi). Astrophysical applications are briefly discussed.