The critical layers and other singular regions in ideal hydrodynamics and magnetohydrodynamics

Abstract

The governing singular differential equations are stated for certain systems in ideal hydrodynamics and magnetohydrodynamics. The solutions of these equations are studied in the neighborhood of the singular regions, and can be conveniently characterised by the roots of the corresponding indicial equation. The nature of the solution determines the nature of the dispersion relation (relating frequency ω and wavenumberk) and of the Green's function for the problem. The differences between the various classes of problems are discussed, and exemplified by considering the initial value problem for three cases: (a) unstratified shear flow, (b) stratified shear flow, and (c) a static magnetofluid. The latter case is typical of a number of problems of astrophysical interest, and possesses a rich mathematical and physical structure.