A new approach to the matching problem in MHD stability analysis

Full text: The linear ideal MHD stability of a tokamak plasma can be modelled by d/dx [(λ +f (x)) dy/dx]+V(x; β) y(x)=0, x element of (0,1). Here y(0) = y(1) = 0, and it is assumed that f(x) is non-negative but there exists the point x0 in (0,1) such that f(x0) = 0 (the rational surface) and f(x)=f0(x-x0)2 + ..., f0>0 The parameter β in the function V describes the performance of the plasma confinement such as the normalized pressure. If Eq.(1) has an eigenvalue λ > 0, then the plasma is unstable against ideal MHD perturbations. Equation (1) with λ = 0 is called the Newcomb equation and x0 is the regular singular point for it. The stability analysis can be reduced to the following problems: (1) to find the critical value βcr such that an eigenvalue λ > 0 emerges for β > βcr, (2) to find the β dependence of λ when β is close to βcr. The problem (1) has been solved by making an eigenvalue problem with the Strum- Liouville property from the Newcomb equation. For the problem (2) we have proposed a new matching method. It introduces a thin layer (inner layer) with finite width and solves Eq.(1) there, while it solves the Newcomb equation in the regions outside the inner layer. The eigenvalue is determined from the condition that the both solutions should be matched smoothly. The present matching method gives us a new approach to solving analytically Eq.(1) by using the property Eq.(2) in the inner layer. For solving the equation in the layer, we propose the Hamilton-Lie perturbation method together with the conjugate variable method, which has recently been formulated. (author)