On the generalized eigenvalue problem for the Rossby wave vertical velocity in the presence of mean flow and topography
Tailleux, R. (2012) On the generalized eigenvalue problem for the Rossby wave vertical velocity in the presence of mean flow and topography. Journal of Physical Oceanography, 42 (6). pp. 1045-1050. ISSN 0022-3670
To link to this article DOI: 10.1175/JPO-D-12-010.1
In a series of papers, Killworth and Blundell have proposed to study the effects of a background mean ﬂow and topography on Rossby wave propagation by means of a generalized eigenvalue problem formulated in terms of the vertical velocity, obtained from a linearization of the primitive equations of motion. However, it has been known for a number of years that this eigenvalue problem contains an error, which Killworth was prevented from correcting himself by his unfortunate passing and whose correction is therefore taken up in this note. Here, the author shows in the context of quasigeostrophic (QG) theory that the error can ulti- mately be traced to the fact that the eigenvalue problem for the vertical velocity is fundamentally a non- linear one (the eigenvalue appears both in the numerator and denominator), unlike that for the pressure. The reason that this nonlinear term is lacking in the Killworth and Blundell theory comes from neglecting the depth dependence of a depth-dependent term. This nonlinear term is shown on idealized examples to alter signiﬁcantly the Rossby wave dispersion relation in the high-wavenumber regime but is otherwise irrelevant in the long-wave limit, in which case the eigenvalue problems for the vertical velocity and pressure are both linear. In the general dispersive case, however, one should ﬁrst solve the generalized eigenvalue problem for the pressure vertical structure and, if needed, diagnose the vertical velocity vertical structure from the latter.