Efficient numerical method for analyzing the spectrum of toroidal dissipative plasma

Abstract

For a plasma with high but finite conductivity, the parameter v is of the order of 10-6-10 -9. This imposes essential restrictions on the time step r and is an obstacle to numerical integration of the time problem. The difficulty is the exact description of the narrow dissipative layers surrounding the resonance surfaces in plasma. We have to allow for finite conductivity only in these interior layers, while outside these layers the plasma is essentially ideally conducting. The width of these interior boundary layers is proportional to v 1/3 and requires a substantially higher density of the space grid. A method was proposed in [1] for solving the MHD problem in plane and cylindrical geometry. This method efficiently resolves the difficulty with higher resolution in narrow boundary layers. The method constructs directly for a chain of onedimensional partial differential equations a difference scheme that converges uniformly in the parameter v. The uniform convergence is accomplished by allowing for the exact asymptotic behavior inside the layer. In this paper, we extend the method to the problem of the spectrum of MHD-instabilities in a toroidal plasma with pressure. Section 2 presents the system of equations of MHD-instability. Sections 3 and 4 examine the construction of asymptotic expansions in the neighborhood of the singular point and formulate a purely implicit difference scheme. Section 5 discusses the method of spectrum determination. Section 6 presents the numerical results.