Mathematics
Scientific paper
Jan 1995
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1995nyasa.773...80b&link_type=abstract
Waves in Astrophysics, vol. Volume 773, p. 80-94
Mathematics
4
Continuous Spectra, Plasma Equilibrium, Eigenvalues, Magnetohydrodynamic Stability, Space Plasmas, Plasma Oscillations, Incompressible Flow, Differential Equations, Singularity (Mathematics), Shear Flow, Vortices, Shallow Water, Rotating Fluids
Scientific paper
In theory of fluids, plasmas, and stellar systems, we frequently encounter the question of the stability of equilibria. The answer is provided in part on determining the evolution of an infinitesimal disturbance away from equilibrium, an approach that usually goes by way of a normal mode expansion. This approach can at times be very powerful, and amounts to solving an eigenvalue problem. It can, however, run into difficulty in circumstances for which that eigenvalue problem is, in some sense, irregular. What we might call regular eigenvalue problems involve the solution of a set of ordinary differential equations with regular coefficients on a domain of finite size. Here we are concerned with situations for which the eigenvalue problem is irregular and the resulting spectrum is at least partly continuous. This kind of a spectrum can arise as a result of solving the problem on an infinite domain, in which case there is simply no quantization condition. Of more interest are problems in which the set of ordinary differential equations is not autonomous and contains coefficients that become singular at points within the domain. In physical situations, singularities in the equations governing the evolution of an infinitesimal disturbance can result from a variety of effects, and they do not always affect the form of the eigenspectrum. An important class of problems for which the singularity has direct repercussions on the eigenspectrum occurs in fluids, plasmas, and stellar systems. These are ideal problems in which there are wave-mean flow or wave-particle resonances that result in the creation of a continuous eigenvalue spectrum. In these circumstances, coefficients in the differential problem are formally singular at the point at which resonance occurs. Moreover, that point is determined by the speed of a wavelike perturbation or, equivalently, the eigenvalue. In this paper we follow the directions indicated by Van Kampen for more general problems than the relatively simple plasma and fluid equilibria considered by Van Kampen and Case. We first describe the general method. Then, in the general context, the problem of plasma oscillations is reviewed. The remaining sections on parallel shear flow, shear flow in shallow water theory, incompressible circular vortices, and differentially rotating disks, are the bulk of the paper. We conclude with a discussion of the uses of singular eigenfunctions.
BALMFORTH Neil J.
Morrison Philip J.
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