Axisymmetric toroidal equilibrium with flow and anisotropic pressure

Abstract

Axisymmetric toroidal plasma equilibria with mass flows and anisotropic pressure are investigated. The equilibrium system is derived for a general functional form of the pressures, which includes both fluid models, such as the magnetohydrodynamic (MHD) and the double-adiabatic models, and Grad’s guiding center model [Proceedings of the Symposium on Electromagnetics and Fluid Dynamics of Gaseous Plasmas, edited by J. Fox (Polytechnic Inst. of Brooklyn, New York, 1961), p. 37]. This allows for detailed comparisons between the models and clarifies how the ‘‘first hyperbolic region,’’ occurring in fluid theory when the poloidal flow is of the order of the poloidal sound speed, can be eliminated in guiding center theory. In the case of a pure toroidal rotation, macroscopic equations of state are derived from the guiding center model, characterized by a parallel temperature that is constant on each magnetic surface and a perpendicular temperature that varies with the magnetic field. The outward centrifugal shifts of the magnetic axis and of the mass density profile, caused by toroidal rotation, are increased by anisotropy if p∥<p⊥ or decreased (and can even be inverted) if p∥>p⊥. In the guiding center model poloidal flow produces an inward shift of the density profile, in contrast with the MHD result.

Sheared flow effects on ballooning instabilities in three-dimensional equilibria

“…Previous authors have noted the issue of modeling flow effects with ideal MHD since the effects of Landau damping on the sound wave are not properly accounted for. 21,22 While this effect is not addressed here, a correct treatment of this effect would presumably enhance stabilization. Finally, we note that a similar averaging process may also apply to the question of drift wave stability in threedimensions.…”

Equilibrium beta limits in a dipole configuration

“…[4]. The details of the formulation are described in references [4] and [5]. In the present section we will only state the results needed for the discussion in the rest of this work.…”

“…In the analysis, the effects of toroidal flow and anisotropy are also included, leading to a modified GS equation, given by Ref. [4]. The details of the formulation are described in references [4] and [5].…”

Plasma Internal Energy for Toroidal Elliptic Plasmas with Triangularity

“…The analysis of plasma equilibrium performed by several authors [3][4][5][6][7][8][9][10][11] is much more complicated than those of plasma confinement with no rotation. The Grad-Shafranov equation has to be analyzed coupled with a Bernoulli-type equation and furthermore there are regions where that equation is of hyperbolic type instead of elliptic [5]. As it is well known, if nonlinear convective terms are included in the momentum equation, pressure is no longer a constant on the magnetic surfaces, but those terms cannot be neglected when significant poloidal or toroidal flows occur in tokamaks.…”

“…The time independent MHD momentum equation including viscosity and non-linear convective terms is [5] …”