Nonaxisymmetric equilibria in tokamaks and Ohmically heated stellarators

An asymptotic expansion scheme is set up, which is appropriate to deal with the deviations from axisymmetry occurring in tokamaks and Ohmically heated stellarators. Solutions of the magnetohydrostatic equations are sought to all orders in the expansion parameter, which is related both to the magnitude of the deviations from axisymmetry and to the ratio of poloidal vs toroidal components of current and magnetic field. Arguments are presented for the existence of the solution to any finite order, the pressure and current profiles across the plasma column, in general, being determined by the nonaxisymmetric part of the external magnetic field. If the latter, however, has certain symmetries, then a degeneracy occurs, and the pressure and current profiles can again be given with some degree of arbitrariness. It is shown that for tokamaks and stellarators of standard design, viz., including the perturbations associated with the ’’ripples’’ in the toroidal field, the ’’gaps’’ in the copper shell, and the helical windings, the degree of arbitrariness in the pressure and current profiles is the same as in the axisymmetric theory. Thus, in such machines the above profiles are expected to be governed entirely by transport effects, a conclustion which is borne out by the experiments.