Geometric properties of plasma equilibrium near a given magnetic surface
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ETIC CONFINEMENT SYSTEMS
Geometric Properties of Plasma Equilibrium near a Given Magnetic Surface A. A. Skovoroda Nuclear Fusion Institute, Russian Research Centre Kurchatov Institute, pl. Kurchatova 1, Moscow, 123182 Russia Received December 23, 2005; in final form, March 21, 2006
Abstract—In order to describe plasma equilibrium near a given magnetic surface, it is sufficient to specify the shape of the surface, the distribution of the magnetic field strength on it, and two profile coefficients (the derivatives of the plasma pressure and current). Geometrically, this means that all the basis vectors of the flux coordinate system should be determined on the magnetic surface. Expressions for these vectors in an invariant basis are obtained. The maximum possible value of the pressure profile coefficient consistent with equilibrium is described by a universal geometric relationship that expresses the limiting value of the torsion of the magnetic field line on the magnetic surface as a function of the curvature of the surface. The relationships obtained are used to show that the stability of a system with closed magnetic field lines is governed by perturbations of the anti-Mercier type. PACS numbers: 52.55.Dy DOI: 10.1134/S1063780X06120014
1. INTRODUCTION Good confinement of a hot plasma generally implies that there is a finite three-dimensional region of “good” (i.e., nonintersecting) equilibrium toroidal magnetic surfaces. Knowledge of the equilibrium magnetic field B makes it possible to unambiguously answer the question about the existence of such a regular family of magnetic surfaces. The solution to the equation B ⋅ —Ψ = 0
(1)
(where Ψ is the poloidal magnetic flux) determines the family of magnetic surfaces Ψ = const. In the practice of magnetic confinement of a hot plasma, however, the equilibrium magnetic field is usually not known a priori and the existence of equilibrium magnetic surfaces is postulated. This is why an inverse problem of determining the equilibrium field B from given magnetic surfaces arises. It is known that the equilibrium magnetic field B in the entire confinement region [1] is unambiguously determined by specifying the shape of the outermost magnetic surface r = r(θ, ζ) (where θ and ζ are arbitrary angular coordinates on the surface) and by defining the plasma pressure p and the total poloidal current F on the outside of the surface as functions of the toroidal magnetic flux Φ. The objective of the present paper is to study the properties of plasma equilibrium in the vicinity of a given (reference) magnetic surface. Knowing these properties, one can solve problems concerning the plasma stability and also the maximum possible values of the profile coefficients consistent with equilibrium (see below).
In order to unambiguously determine plasma equilibrium near a given magnetic surface, it is insufficient to specify only the shape of this surface, r = rr(θ, ζ), and the profile coefficients—namely, the derivatives of the plasma pressure, p', and total poloidal (or toroidal) current, F ' (
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