By use of computer math programs, the MHD stability properties of fusion confinement plasmas can now be determined analytically, without further approximations, to high orders in time and space coordinates. We here present results from expanding the linearized ideal and resistive MHD eigenvalue equations for cylinder geometry in radius r. The effect of sheared fluid flow on stability is also included. Not only analytical growth rates, but also analytical forms of the eigenfunctions are obtained. As expected, we find that moderately large expansions provide high accuracy for mildly localized eigenfunctions.
The method, for linearized eigenvalue problems, is based on expansions of the dependent perturbed variables in powers of the spatial independent variables; e.g. ur, uq, uz, br, bq, bz as functions of rn, with n ≤ nmax, the maximum order of the expansion. The basic, scalar MHD equations can thus be retained without further elimination of variables. The equilibrium state is also expanded in r, retaining desired control coefficients.
By successively solving the MHD equations at higher orders of r, solutions being exact up to this order are obtained. The indicial equation (at r = 0) is automatically solved in this method. By applying boundary conditions at r = rwall, an implicit equation for the eigenvalue w as function of the mode numbers (m,k) and the equilibrium parameters is obtained. This is similar to the often used, however numerical, shooting method. The number of solutions wi are, as implied by the oscillation theorem, dependent on nmax. The roots are obtained by implicit root solvers or graphically. Compact analytical forms of eigenvalues and eigenfunctions can also be obtained by retaining dominating terms in specified parameter limits.
Our method can be used for any geometry, and also for nonlinear systems of equations. It thus has a very large potential for efficient and exact analytical investigations of (one- or many-fluid) MHD equations, also including additional physical effects. The calculations require efficient math program routines and high speed and memory computer platforms, wherefore they were difficult to perform earlier.
28th EPS Conference on Controlled Fusion and Plasma Physics, Madeira, 18-22 June 2001, P4.106, p. 516