Choice of coordinates on a toroidal magnetic surface
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ETIC CONFINEMENT SYSTEMS
Choice of Coordinates on a Toroidal Magnetic Surface A. A. Skovoroda Nuclear Fusion Institute, Russian Research Centre Kurchatov Institute, pl. Kurchatova 1, Moscow, 123182 Russia Received June 21, 2007; in final form, March 17, 2008
Abstract—Construction of global angular coordinates on an arbitrarily shaped toroidal surface is considered. It is shown that global orthogonal, isothermal, and semigeodesic geometric coordinates can always be introduced on a toroidal surface. Such coordinates can be rather efficient in solving problems of plasma equilibrium and stability in a magnetic field. At the same time, it is impossible to introduce global geodesic coordinates and coordinates based on curvature lines. It is proposed to use a magnetic analogy to search for transformations of global angular geometric coordinates that simplify the expression for the length element on an arbitrary toroidal surface. An algorithm for the computation of such coordinates is offered. With this approach, a “virtual” magnetic field such that its force lines, as well as the lines orthogonal to them, are closed is searched for on the toroidal surface. These lines comprise a geometric coordinate grid on an actual magnetic surface formed by the actual magnetic field. PACS numbers: 52.55.Dy DOI: 10.1134/S1063780X08110019
1. INTRODUCTION Long-term plasma confinement can be provided by a system of nested toroidal magnetic surfaces. The shape of toroidal surfaces is described by three periodical radius vector functions r = r(θ, ζ) of angular (geometric) coordinates x2 = θ and x3 = ζ. The squared length element dl 2 on the surface (the first quadratic form) is expressed as dl 2 = g22dθ2 + 2g23dθdζ + g33dζ2, where gij = (∂r/∂x i )(∂r/∂x j ) are the metric components. In practice, quasi-cylindrical geometric coordinates θ and ζ in which all three components gij are nonzero are widely used. The so-called flux angular coordinates on a toroidal magnetic surface, θF and ζF, have been successfully used for more than 50 years. In these coordinates, the vector of the magnetic field B forming nested magnetic surfaces is expressed functionally in the most natural way [1]. Of the two conditions that can be specified for the two unknown functions θF(r) and ζF(r), the first is usually reduced to the requirement that the field lines be “straightened,” whereas the second is used to choose the form of the Jacobian. These functions are described by second-order equations in partial derivatives with respect to the geometric angular coordinates θ and ζ on the chosen magnetic surface [1]. The solution to these equations is a special coordinate transformation θn = θn(θ, ζ) and ζn = ζn(θ, ζ) after which functions on the surface remain periodic (periodic transformation). The main goal of this work is to simplify the form of the length element. Local simplifications of the first quadratic form on a fragment of an arbitrary surface are well known [2]. These are the so-called orthogonal, isothermal, semigeodesic, and geodesic coordinates, as
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