Bifurcation of the resistive Alfvén wave spectrum

Abstract

A phase integral (WKBJ) theory of damped modes in a resistive cylindrical plasma column is developed. The theory predicts several phenomena observed previously in numerical studies: (i) the complex frequency ω has a point spectrum lying on a locus which is approximately independent of the resistivity, and which intersects the ideal continuum only at its end points (at angles of 45° and 30°); (ii) the qualitative features are independent of current profile and m–number, provided |k.B| is monotonic and strictly positive; and (iii) the two branches of the spectral locus joining the two end points of the ideal continuum correspond to precisely defined ‘internal’ and ‘wall’ modes, and bifurcate from a ‘global’ mode at a certain complex frequency which can be predicted from the topological features of the Stokes diagram in the complex radial position plane. In order to derive consistent connexion formulae for reflexions from the wall, the magnetic axis, and from turning points, we reduce the resistive hydromagnetic equations to a scalar wave equation, uniformly valid over the entire plasma to two orders in an expansion in the square root of the resistivity. Good numerical agreement is found between eigenvalues from the full equations, the reduced equation, and from the WKBJ method.