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.
Energetically resolved multiple-fluid equilibria of tokamak plasmas