A class of resistive axisymmetric magnetohydrodynamic equilibria in a periodic cylinder

Abstract

We consider a model of viscoresistive incompressible magnetohydrodynamics in a periodic cylinder, with boundary conditions meant to idealize in a tractable way those of a laboratory plasma. The resistivity is described by a tensor presenting a field-dependent anisotropic part suggested by kinetic theory, controlled by a certain anisotropy parameter. An explicit analytical description of the corresponding axisymmetric zero-flow equilibria is given, and it is shown how tokamak-like or paramagnetic-pinch-like field profiles are obtained a's the anisotropy parameter is changed. The study of the stability properties of such equilibria is deferred to a later paper

Toroidal steady states in visco-resistive magnetohydrodynamics

“…The idea that finite transport coefficients (resistivity in particular) must affect confined MHD steady states is an old one, and goes back at least to the unpublished but influential manuscript of Pfirsch and Schlüter (1962). A variety of subsequent papers have appeared over the years which have addressed the matter (see for example Grad 1967;Grad and Hogan 1970;Grad et al 1977;Rosen and Greene 1977;Ponno et al 2002;Throumoulopoulos and Tasso 2003). One important feature is that a resistive steady state can only be maintained against Ohmic decay by the presence of some externally-applied driving mechanism that achieves energy balance and does work on the plasma.…”

“…Second, Ohm's law (irrelevant for an ideal MHD steady state of the Grad-Shafranov variety) needed to be satisfied by whatever current density and electric field were present, with some basis for the choice of profile for the transport coefficients beyond simple algebraic convenience. (We cannot defend energetically the practice of choosing a resistivity profile spatial dependence, for example, solely to make the arithmetic 'come out right' (see for example Ponno et al 2002;Throumoulopoulos and Tasso 2003).) Third, Faraday's law, which demands curl-free electric fields in the steady state, needed to be taken seriously.…”