The Spectral Web of stationary plasma equilibria. I. General theory

A new approach to computing the complex spectrum of magnetohydrodynamic waves and instabilities of moving plasmas is presented. It is based on the concept of the Spectral Web, exploiting the self-adjointness of the generalized Frieman–Rotenberg force operator, G, and the Doppler–Coriolis gradient operator parallel to the velocity, U. The problem is solved with an open boundary, where the complementary energy Wcom represents the amount of energy to be delivered to or extracted from the system to maintain a harmonic time-dependence. The eigenvalues are connected by a system of curves in the complex ω-plane, the solution path and the conjugate path (where Wcom is real or imaginary) which together constitute the Spectral Web, having a characteristic geometry that has to be clarified yet, but that has a deep physical significance. It is obtained by straightforward contour plotting of the two paths. The complex eigenvalues, within a specified rectangle of the complex ω-plane, are found by fast, reliable, and accurate iterations. Real and complex oscillation theorems, replacing the familiar tool of counting nodes of eigenfunctions, provide an associated mechanism of mode tracking along the two paths. The Spectral Web method is generalized to toroidal systems and extended to include a resistive wall by accounting for the dissipation in such a wall. It is applied in an accompanying Paper II [J. P. Goedbloed, Phys. Plasmas 25, 032110 (2018).] to a multitude of the basic fundamental instabilities operating in cylindrical plasmas.