Hydromagnetic equilibria and their proper coordinates

Abstract

Proper coordinate systems are constructed in hydromagnetic equilibria and their properties are studied. First, the contra-gradient components of magnetic field and of current density are surface quantities. Second, the equi-pressure surfaces which have no singularity within a finite volume must be topologically torus-shaped. Third, a general condition of no charge separation is deduced as follows: There must be a simple closed loop on every equi-pressure surface having the property that the integral is constant for the variable A. Here, the integral is carried out along a magnetic line from a point A on the loop to the returning point B on the same loop, and dl is the magnetic line element. An expression of the shape of current lines is obtained from the condition in the cases of unclosed field and of twisted field. A method of determining the magnetic surfaces which coincide with equi-pressure surfaces is obtained in the case of closed field. We examine the successive approximation method developed by M. Kruskal et al. with the help of these methods. It fails in the first approximation in almost all cases of twisted fields. It can be used in the case of unclosed field if the rotational transform ratio is one of the continued fractions constructed in this paper. It can be used in the case of the closed field with mirror symmetry when the plasma pressure gradient is not too steep. An effect of the closed magnetic lines in a twisted field is considered. The diffusing velocity of plasma is infinite in the neighbourhood of magnetic surfaces which are made of closed lines not satisfying the condition of no charge separation. The ratio of the measure of the highly diffusing region to the measure of the whole system is estimated in an easy case. The result suggests that the confinement time of plasma may be considerably shorter than that of plasma in the field compatible with the condition of no charge separation.