Relationship between the shape of equilibrium magnetic surfaces and the magnetic field strength
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ETIC CONFINEMENT SYSTEMS
Relationship between the Shape of Equilibrium Magnetic Surfaces and the Magnetic Field Strength A. A. Skovoroda Nuclear Fusion Institute, Russian Research Centre Kurchatov Institute, pl. Kurchatova 1, Moscow, 123182 Russia Received October 18, 2006
Abstract—A local analysis of the magnetic field near an equilibrium magnetic surface shows that there is generally no relationship between the magnetic field strength and the shape of the surface. However, the relationship exists under additional requirements such as the absence of the toroidal current, symmetry conservation, and the conservation of the magnetic field strength distribution on the nearest surface. An equilibrium magnetic surface can be calculated by specifying three functions of two angular variables—the magnetic field strength, the periodic component of the magnetic potential, and the mean curvature of the surface. PACS numbers: 52.55.Dy DOI: 10.1134/S1063780X0707001X
1. INTRODUCTION A high-temperature plasma is usually confined in a magnetic field B that forms nested toroidal magnetic surfaces B · ∇ψ = –µB · ∇Φ = 0, where ψ is the poloidal magnetic flux, Φ is the toroidal magnetic flux, and µ is the rotational transform. The particle confinement quality is largely governed by the topography of isomagnetic lines |B | = B = const on magnetic surfaces. One of the necessary geometric conditions for optimum plasma confinement is the pseudosymmetry of a magnetic configuration, when all the isomagnetic lines on equilibrium magnetic surfaces are closed either around the magnetic axis (poloidal pseudosymmetry) or around the major axis of the torus (toroidal pseudosymmetry) [1, 2]. In this context, the problem arises of finding pseudosymmetric confinement configurations among all possible systems, their further optimization, and their practical implementation by using currentcarrying windings. The goal of the present paper is to consider the general question of the relationship between the shape of a magnetic surface and the distribution of the magnetic field strength on it. We will use the flux and current representations of the magnetic field that generates a family of nested magnetic surfaces [3]: 2πB = ∇Φ × ∇ ( θ + η ) – µ∇Φ × ∇ζ ∂η ⎛ 1 + ∂η ------⎞ e 3 + ⎛ µ – ------⎞ e 2 ⎝ ⎝ ∂ζ ⎠ ∂θ ⎠ = ------------------------------------------------------------ , g 2πB = F∇ζ + J∇θ – ν∇Φ + ∇ϕ.
(1)
Here, θ and ζ are periodic angular coordinates1 on the magnetic surface r = r(Φ0, θ, ζ); e2 = ∂r/∂θ and e3 = ∂r/∂ζ are the basic vectors; the functions η, ϕ, and ν are periodic in the angular coordinates and have a zero mean value; F and J are the poloidal (external) and toroidal currents; and g is the Jacobian. The magnetic surfaces are labeled by the toroidal magnetic flux Φ. The angular coordinates on the surface, θ and ζ, can be chosen arbitrarily, e.g., so as to simplify either the metric coefficients or the periodic functions η, ϕ, and ν. The first possibility is provided by orthogonal coordinates2 with g23 = 0 and canonical coordinates with g22
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