New construction of the magnetohydrodynamic spectrum of stationary plasma flows. I. Solution path and alternator

Abstract

A new method of systematically constructing the full structure of the complex magnetohydrodynamic spectra of stationary flows is presented. It is based on the self-adjointness of the generalized force operator G and the Doppler–Coriolis shift operator U, and the associated quadratic forms for the normalized energy W¯ and the normalized Doppler–Coriolis shift V¯, which may be constructed for all complex values of ω if the original eigenvalue problem is converted into a one-sided boundary value problem. This turns W¯ into a complex expression, while V¯ remains real. Whereas the solution path Ps of stable modes is just the real axis, the solution path Pu of unstable modes in the complex ω plane is found by requiring that the solution-averaged Doppler–Coriolis shifted real part of the frequency vanishes, σ−V¯[ξ(ω)]=0, or that the energy is real, Im W¯[ξ(ω)]=0. The location of the eigenvalues on these solution paths is determined by two quadratic forms, which may straightforwardly be evaluated in any of the finite element spectral codes in existence. A new oscillation theorem is proved about the monotonicity of complex eigenvalues for one-dimensional systems. Instead of counting internal nodes of the real displacement vector ξ (as in static plasmas), it is based on counting the zeros of the alternating ratio, or alternator, R≡ξ/Π of the boundary values of the complex functions ξ and the total pressure perturbation Π, which is real on the solution path. This finally provides the generalization of the basic structural properties of the magnetohydrodynamic spectrum of static plasmas, which has been known for a long time, to stationary plasmas.