Perturbation analysis of a simple model of magnetic island structures

Singular perturbation theory is used to construct a class of stationary solutions to the resistive plasma equilibrium equations in a plane slab geometry. These are nonlinearly saturated magnetic island structures which bifurcate from the basic Ohmic state when Δ′(k) is positive and small in some sense. The relationship between the nondimensional wavenumber k and the size of the islands is obtained by a self-consistent, matched asymptotic expansion. The assumed resisitivity profile is found to determine the relationship uniquely. It is found that either two bifurcating branches exist or none at all depending upon the profile chosen. The structure of the plasma current due to the islands is evaluated in terms of two functions which are independent of the resistivity profile. It is found that for sufficiently large Δ′(k) and suitable resistivity, a self-consistent, steady magnetic island state cannot be found. A possible connection between this phenomenon and disruptive instabilities observed in experimental plasmas is discussed very briefly. The proposed model complements the work done on magnetic islands by various groups using powerful numerical codes. In contrast to such codes, the current distributions due to a wide class of resistivity profiles have been calculated using the model. The solutions constructed shed some light on how nonlinear saturation in the island states can occur.