New approach to magnetohydrodynamics spectral theory of stationary plasma flows

While the basic equations of MHD spectral theory date back to 1958 for static plasmas (Bernstein et al 1958 Proc. R. Soc. A 244 17) and to 1960 for stationary plasma flows (Frieman and Rotenberg 1960 Rev. Mod. Phys. 32 898), progress on the latter subject has been slow since it suffers from lack of analytical insight concerning the structure of the spectrum. One of the reasons is the usual misnomer of 'non-self adjointness' of the stationary flow problem. Actually, self-adjointness of the occurring operators, namely the generalized force operator and the Doppler-Coriolis gradient operator-i rho v.del, was proved right away by Frieman and Rotenberg. Based on the reality of the two quadratic forms corresponding to these operators, we here construct (a) an effective method to compute the solution paths in the complex omega plane on which the eigenvalues are situated, (b) the counterpart of the oscillation theorem for eigenvalues of static equilibria (Goedbloed and Sakanaka 1974 Phys. Fluids 17 908) for the eigenvalues of stationary flows, based on the monotonicity of the alternating ratio, or alternator, of the boundary values of the displacement. and the total pressure perturbation Pi. This enables one to map out the complete spectrum of eigenvalues in the complex.-plane. The intricate topology of the solution paths is discussed for the fundamental examples of Rayleigh-Taylor, Kelvin-Helmholtz and combined instabilities.