Adiabatic Plasma Rotations in Orthogonal Coordinate Systems

Azimuthal rotation in Tokamaks and other fusion machines is observed, for example, when the con ned plasma is subjected to neutral beam heating. The impacts of the beam particles with plasma electrons and ions amounts to a net momentum transfer with causes rotation in the toroidal direction [1, 2]. Plasma rotation with high Mach numbers have been observed in almost all operating regimes of Tokamaks [3, 4], as well as in reversedeld pinches [5]. A key problem in the theoretical study of azimuthal rotation is whether such a plasma ow could coexist with a state of MHD (stationary) equilibrium. The answer turns to be positive provided some requirements are ful lled by the system. Without resistivity, Alfv en's theorem says that magnetic eld lines rotate rigidly with the plasma. If axissymetry exists, eld lines lie on magnetic ux surfaces with topology of tori and characterized by surface quantities, like the transversal magnetic ux. The set of ideal MHD equations allows us to derive a partial di erential equation for it [6, 7]. Maschke and Perrin [8, 9] obtained a MHD equilibrium equation for azimuthal plasma ows supposing that either the temperature or the entropy were surface quantities. They have considered only cylindrical coordinates, having obtained exact analytical solutions for the transversal magnetic ux. There are a few other solved cases in cylindrical [10] and spherical [11] geometries, but considering the temperature as a surface quantity. The case where the plasma ow is adiabatic, however, demands the use of the entropy as a surface quantity. This is particularly important in the case of anisotropic plasmas, where a double-adiabatic theory is necessary to describe the situation [12, 13]. To apply the MHD equilibrium theory for realistic magnetic con nement schemes, one would need an equilibrium equation in a general curvilinear coordinate system. In a previous paper [14] an equilibrium equation for plasmas with azimuthal rotation was derived, assuming that the temperature was a surface quantity, in an orthogonal curvilinear coordinate system. In this paper we will derive a similar equation, but with the plasma entropy as a surface quantity, according the methodology introduced by Maschke and Perrin [8]. This paper is organized as follows: in the second section we outline the basic equations and thermodynamical relations to be used, the magnetic eld and velocity representations. In section III we use these equations to obtain a pressure equilibrium equation, which is supplemented by a Bernoulli-like algebraic equation. Section IV presents a particular form of this equation, obtained by a special choice of some surface quantities. Section V discusses how the, general equations look like in some coordinate systems, like cylindrical, spherical and prolate spheroidal ones.