Nonlinear stability of doublets against axisymmetric resistive MHD modes

Abstract

Linear stability of doublets against axisymmetric modes in the approximation of resistive MHD was studied earlier; two modes were identified as those most likely to be unstable, namely the ‘up-down’ mode, which tends to move the plasma up or down and the ‘droplet-ellipse’ mode which tends to deform the doublet into either a droplet or an ellipse. In the treatment described in this paper, quadratic terms in the perturbation are retained. For an equilibrium which is symmetric about the midplane it is easy to see that the quadratic terms vanish for the ‘up-down’ mode. For the ‘droplet-ellipse’ mode this is not the case. This mode allows two different deformations according to linear theory; the effect of the quadratic terms is found always to be stabilizing for one of these deformations, and destabilizing for the other. Thus, owing to the effects of such nonlinear terms, one may expect in experiments to observe unstable developments in a preferred direction, namely that of destabilization, and one expects to observe a faster deterioration of an equilibrium than that predicted from linear theory. These tendencies have been observed in doublet experiments. In this paper the formalism for including quadratic terms in the perturbation is discussed and results from numerical examples are given.