Eigenvalues associated with relaxed states of toroidal plasmas

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Abstract
In the theory of relaxed states of toroidal plasmas certain eigenvalues of the equation ∇×B=μB play a crucial role. These eigenvalues are associated with vanishing toroidal flux and determine the onset of current limitation in a toroidal discharge. In axisymmetric systems there are both periodic and axisymmetric eigenfunctions and it is important to know whether the eigenmode associated with the lowest eigenvalue is periodic or axisymmetric. This depends on the shape of the poloidal cross section and determines the nature of the current‐limited discharge. The eigenvalues of periodic and axisymmetric modes have been computed in rectangular and elliptical cross sections and in reentrant Multipinch‐like cross sections. The reentrant case required new numerical techniques, which are described. It is found that in rectangular and elliptic cross sections the lowest mode is always periodic. However, in the Multipinch a transition occurs in which the lowest eigenmode changes from periodic to axisymmetric as the ‘‘waist’’ in the cross section is made narrower. The critical width is determined. These calculations suggest that in the GA Multipinch experiment [Nucl. Fusion 2 6, 255 (1986)] the current saturated discharge should be axisymmetric—unlike all other existing pinch experiments where it is periodic.