On a representation of toroidal surfaces. Applications to magnetohydrodynamic equilibria

By a study of the representation of a class of toroidal surfaces, the following problem can be solved for plasmas of small diameter: Given a toroidal surface, what are the magnetohydrodynamic (MHD) equilibria which have this surface as a magnetic surface (or constant pressure surface)? The solution of this problem gives a physical interpretation of the quantities involved in the local stability criterion developed in the neighbourhood of the magnetic axis and yields a qualitative understanding of what happens when a toroidal MHD system is such that its angle of rotational transformation on the magnetic axis ι cO is in the neighbourhood of 2kπ where k is a whole number. A study of the MHD equations in the neighbourhood of a magnetic axis shows that, in general, equilibrium is not possible for ι cO /2π = k. If ι cO /2π approaches k, it is shown that the magnetic axis moves toward the external surface and tends towards a helix form turning around the central axis of the configuration with a pitch equal to L/k . It could be postulated as a condition for the existence of such states that the magnetic axis does not move too far from the centre of the configuration. This requirement limits the stability zones which a study of the local stability criterion produces for values of ι cO /2π ≥ k. Finally, a fairly strict limitation of the quantity β = 2p/B 2 is obtained at least in cases where k ≠ 0. The calculations are developed for the case of a plane magnetic axis (l/T(s) = 0) defined by l/R(s) = a 0 + 2a k cos 2kπs/L. This intrinsic equation represents a closed curve of length L = 2 π/a 0 irrespective of what a k may be if k is even.