Linear and nonlinear stability in resistive magnetohydrodynamics

A sufficient stability condition with respect to purely growing modes is derived for resistive magnetohydrodynamics. Its 'nearness' to necessity is analysed. It is found that for physically reasonable approximations the condition is in some sense necessary and sufficient for stability and against all modes. This together with hermiticity makes its analytical numerical evaluation worthwhile for the optimization of magnetic configurations. Physically motivated test functions are introduced. This leads to simplified versions of the stability functional, which makes its evaluation and minimization more tractable. In the case of special force-free fields the simplified functional reduces to a good approximation of the exact stability functional derived by other means. It turns out that in this case the condition is also sufficient for nonlinear stability. Nonlinear stability in hydrodynamics and magnetohydrodynamics is discussed especially in connection with 'unconditional' stability and with severe limitations of the Reynolds number. Two examples in magnetohydrodynamics show that the limitations on the Reynolds numbers can be removed but unconditional stability is preserved. Practical stability needs to be treated for limited levels of perturbations or for conditional stability. This implies some knowledge of the basin of attraction of the unperturbed solution, which is a very difficult problem. Finally, a special inertia-caused Hopf bifurcation is identified and the nature of the resulting attractors is discussed. The purpose of this lecture is to report recent progress on resistive or dissipative magnetohydrodynamics (MHD) in general geometry. Though the problem of linear stability has not been completely settled analytically, it has reached the mature level of a general quadratic and Hermitean form ready to be minimized numerically in a way similar to that of the ideal energy principle. In contrast, nonlinear stability analysis still falls far short of the stage of maturity attained by the linear theory. It was recently possible, however, to prove general statements concerning unconditional stability of three-dimensional force-free fields and unconditional stability with respect to two-dimensional perturbations. The paper is organized as follows: Sections 1-3 are devoted to the linear stability condition. The topics are sufficiency and necessity of the condition and simplified versions of it. Nonlinear stability is analyzed in section 4. The topics are force-free fields, two-dimensional perturbations and a special Hopf bifurcationAvailable from TIB Hannover: RA 71(6/314) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman