Solutions to the flow equilibrium problem in elliptic regions

Abstract

The existence of poloidal flow transforms the elliptic Grad4hafranovSchliitr (GSS) equation into an EGSS system (Extented GSS) of partial differential equation and an algebraic Bernoulli's equation. The EGSS System becomes alternatively elliptic and hyperbolic as the Mach numkr of the poloidal flow increases with.respect to the Alfven Epeed of the paloidal magnetic field. A computer p r o p m for solving EGSS equations in elliptic res;ons ushg Ihe iovene method and Fourier decomposition'has been prepared. The solutions in the first and second elliptic regions have been found for different plasma cross-sections, not necessarily u p down-symmiric. The solulions in different elliptic regions exibit the significart differences in the poloidal magnetic held configuration and the shifts between magnetic axis and density axis differ in sign and magnitude,

On resistive magnetohydrodynamic equilibria of an axisymmetric toroidal plasma with flow

“…It is pointed out that, unlike to the usual procedure followed in equilibrium studies with flow [10,11,12,13,14,15] in the present work an equation of state is not included in the above set of equations from the outset and therefore the equation of state independent Eqs. (15) and (16) below are first derived.…”

“…(21) and (22), which are coupled through the density ρ, should be solved numerically under appropriate boundary conditions. This can be accomplished by the existing ideal MHD equilibrium codes [13,14,15]. The problem of compressible flows with isothermal magnetic surfaces [Eqs.…”

On axisymmetric double adiabatic MHD equilibria with plasma flow

“…The problem of axisymmetric magnetohydrodynamic (MHD) equilibria with isotropic plasma pressure and general flow has been studied by several authors (Zehrfeld and Green 1970, Morozov and Solov'ev 1980, Hameiri 1983, Maschke and Perrin 1984, Kerner and Tokuda 1987, Zelazny et al 1993, Tasso and Throumoulopoulos 1998. In general, the problem is reduced to two coupled equations, one nonlinear partial differential equation for the poloidal magnetic flux function, containing hypotheses on five surface quantities and a nonlinear algebraic Bernoulli equation defining plasma density.…”

Detailed derivation of axisymmetric double adiabatic MHD equilibria with general plasma flow

“…Later, Maschke and Perrin 16 h a ve solved such equilbrium equations in the limit of small ratio of poloidal to toroidal magnetic eld and small beta. Further developments were due to Kerner and Tokuda 17 , Zelazny et al 19 , and Tasso and Thromoulopoulos 18 . A common feature of all these models is the need of supplementing the partial di erential equation for the magnetic ux with a Bernoulli-like algebraic equation for the density which contains several other surface quantities.…”