Equilibria of toroidal plasmas with toroidal and poloidal flow in high-beta reduced magnetohydrodynamic models

Abstract

A reduced set of magnetohydrodynamic equilibrium equations for high-beta tokamaks is derived from the fluid moment equations for collisionless, magnetized plasmas. Effects of toroidal and poloidal flow comparable to the poloidal-sound velocity, two-fluid, ion finite Larmor radius (FLR), pressure anisotropy and parallel heat fluxes are incorporated into the Grad–Shafranov equation by means of asymptotic expansions in terms of the inverse aspect ratio of a torus. The two-fluid effects induce the diamagnetic flows, which result in asymmetry of the equilibria with respect to the sign of the E × B flow. The gyroviscosity and other FLR effects cause the so-called gyroviscous cancellation of the convection due to the ion diamagnetic flow. The qualitative difference between the equilibria with and without the parallel heat fluxes is shown to stem from characteristics of the sound waves. Higher order terms of quantities like the pressures and the stream functions show the shift of their isosurfaces from the magnetic surfaces due to effects of flow, two-fluid and pressure anisotropy. The reduced form of the diamagnetic current associated with pressure anisotropy is also obtained.

Grad–Shafranov reconstruction of magnetohydrostatic equilibria with nonisotropic plasma pressure: the theory

“…There are other general formulations of the GS equation with pressure anisotropy, for example, for ideal magnetohydrodynamic flows with the double adiabatic relations (e.g., Beskin and Kuznetsova 2000) and for high-beta tokamaks (e.g., Ito and Nakajima 2011). From the reconstruction point of view, such GS formulations remain challenged for reconstruction.…”

Magnetic flux coordinates for analytic high-beta tokamak equilibria with flow

“…where U 2 * is a free function of ψ 1 . The derivation of the reduced equilibrium equations by using Ψ and Φ, including the effects of two-fluid, ion finite Larmor radius and pressure anisotropy, is shown in [19]. The first order density n 1 is obtained from the expansion of ( 2) with (20) as…”

“…By these orderings, the fast magnetosonic wave is excluded while the slow magnetosonic wave characterized by the poloidal sound velocity is retained. The derivation of equlibrium equations was extended to the two-fluid MHD with finite Larmor radius effects [19,17,18]. An analytic solution for the reduced MHD equilibrium was found [20,21].…”

Plasma Physics Researches in Asia Pacific Region

“…Analytic solutions of the Grad-Shafranov equation in the high β limit and numerical solutions of a generalized model that included two-fluid effects, flows and pressure anisotropy were compared in [4]. These modifications were seen to cause higher order terms of quantities like the pressures and the stream functions to display shifts of their isosurfaces from the magnetic surfaces.…”