Three-scale expansion of the solution of the magnetohydrodynamic equations and the Reynolds equation for a tokamak

Abstract

An asymptotic solution of the equations of magnetohydrodynamics(MHD) at large Reynolds numbers is constructed. The threescale asymptotic solution describes the evolution of small rapidly varying perturbations of the equilibrium state with allowance for nonlinear interaction. It is shown that in the linearly unstable situation there arises an anJsotropic coherent structure whose evolution leads to energy exchange between the high- and low-frequency waves. A closed system of MHD Reynolds equations for the anisotropic structure that permits calculation of the Reynolds stresses is derived. In this paper we construct an asymptotic solution of the equations of magnetohydrodynamics (MHD) at large Reynolds numbers Rem-1/2-Re-l/2=e ,~ 1. It is assumed that the volume ~ occupied by the plasma is a deformed toms, whose major radius R is much less than the minor radius ao, 2~rR/ao=l/6>> I with ~>>e. The three-scale (with respect to the parameters and e) asymptotic solution describes the evolution of small rapidly varying perturbations of the equilibrium state with allowance for the nonlinear interaction. It is shown that in the presence of resonant surfaces in fI an anisotropic coherent structure arises in the linearly unstable situation, and the evolution of this structure leads to energy exchange between the high- and lowfrequency waves. We derive a closed system that is the MHD analog of the Reynolds equations for the anisotropic structure, and this enables us to calculate the Reynolds stresses. We show that in the given problem the Reynolds equations are a generalization of the Kadomtsev--Pogutse equations that takes into account the toroidal effects and the influence of the highfrequency components of the solution in dissipative layers. We consider the MHD equations