Once more: The continuous spectrum of ideal magnetohydrodynamics

Abstract

A controversy on the existence of continuous spectra of ideal magnetohydrodynamics in addition to the well-known Alfvén and slow continua, dating back to a conjecture by Grad [Proc. Natl. Acad. Sci. USA 70, 3377 (1973)] and revived by Lashmore-Davies, Thyagaraja, and Cairns [Phys. Plasmas 4, 3243 (1997)], is once more resolved by demonstrating that the resolvent operator is bounded in the relevant domain: There are no additional continua. In addition, the solution of the initial value problem is constructed in terms of the three-dimensional Green's dyadic, which is free of apparent singularities and clearly exhibits the classical continua as δ functions on the diagonal. This construction provides the connection with the proper and improper normal modes and shows that the local dynamics on the magnetic surfaces is described by the classical continua.