On the generalized eigenvalue problem for the Rossby wave vertical velocity in the presence of mean flow and topography

摘要

In a series of papers, Killworth and Blundell have proposed to study the effects of a background mean flowudand topography on Rossby wave propagation by means of a generalized eigenvalue problem formulated inudterms of the vertical velocity, obtained from a linearization of the primitive equations of motion. However, itudhas been known for a number of years that this eigenvalue problem contains an error, which Killworth wasudprevented from correcting himself by his unfortunate passing and whose correction is therefore taken up inudthis note. Here, the author shows in the context of quasigeostrophic (QG) theory that the error can ulti-udmately be traced to the fact that the eigenvalue problem for the vertical velocity is fundamentally a non-udlinear one (the eigenvalue appears both in the numerator and denominator), unlike that for the pressure.udThe reason that this nonlinear term is lacking in the Killworth and Blundell theory comes from neglectingudthe depth dependence of a depth-dependent term. This nonlinear term is shown on idealized examples toudalter significantly the Rossby wave dispersion relation in the high-wavenumber regime but is otherwiseudirrelevant in the long-wave limit, in which case the eigenvalue problems for the vertical velocity and pressureudare both linear. In the general dispersive case, however, one should first solve the generalized eigenvalueudproblem for the pressure vertical structure and, if needed, diagnose the vertical velocity vertical structureudfrom the latter.