# On resistive magnetohydrodynamic equilibria of an axisymmetric toroidal plasma with flow

## Abstract

It is shown that the magnetohydrodynamic (MHD) equilibrium states of an axisymmetric toroidal plasma with finite resistivity and flows parallel to the magnetic field, are governed by a second-order partial differential equation for the poloidal magnetic flux function 1/F coupled with a Bernoulli-type equation for the plasma density (which are identical in form to the corresponding ideal MHD equilibrium equations) along with the relation delta/star epsilon=Vc sigma (here delta/star is the Grad-Schltiter-Shafranov operator, sigma is the conductivity and 1fc is the constant toroidal-loop voltage divided by 2 pi). In particular, for incompressible flows, the above-mentioned partial differential equation becomes elliptic and decouples from the Bernoulli equation [H. Tasso and G. N. Throumoulopoulos, Phys. Plasma 5,2378 (1998)].

For a conductivity of the form sigma=sigma (R, epsilon) (where R is the distance from the axis of symmetry), several classes of analytic equilibria with incompressible flows can be constructed having qualitatively plausible s profiles, i.e. profiles with s taking a maximum value close to the magnetic axis and a minimum value on the plasma surface. For sigma=sigma (epsilon), consideration of the relation delta/star epsilon=Vc sigma (epsilon) in the vicinity of the magnetic axis leads then to a proof of the non-existence of either compressible or incompressible equilibria. This result can be extended to the more general case of non-parallel flows lying within the magnetic surfaces. .

For a conductivity of the form sigma=sigma (R, epsilon) (where R is the distance from the axis of symmetry), several classes of analytic equilibria with incompressible flows can be constructed having qualitatively plausible s profiles, i.e. profiles with s taking a maximum value close to the magnetic axis and a minimum value on the plasma surface. For sigma=sigma (epsilon), consideration of the relation delta/star epsilon=Vc sigma (epsilon) in the vicinity of the magnetic axis leads then to a proof of the non-existence of either compressible or incompressible equilibria. This result can be extended to the more general case of non-parallel flows lying within the magnetic surfaces. .

### Additional information

Bibliographic Reference: An article published in: J. Plasma Physics (2000), vol.64, part 5, pp.601-612