On the Generalized Eigenvalue Problem for the Rossby Wave Vertical Velocity in the Presence of Mean Flow and Topography

Abstract

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 ultimately be traced to the fact that the eigenvalue problem for the vertical velocity is fundamentally a nonlinear 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.

Advection of baroclinic eddies by depth mean flow

“…Perhaps the most significant limitation is the neglect of any topographic variations. Nevertheless, aside from specific local topographic impacts [e.g., Hallberg, 1997;Killworth and Blundell, 1999;Tailleux and McWilliams, 2002;Marshall, 2011;Tailleux, 2012] and potential systematic biases through bottom pressure decoupling [Tailleux and McWilliams, 2001], our observational analysis suggests that the present arguments remain relevant even in the presence of variable bottom topography. This result is also consistent with the planetary geostrophic Rossby wave equation derived by Salmon [1994] under an assumption of uniform potential vorticity on isopycnals.…”

Storm tracks in the Southern Hemisphere subtropical oceans

“…Further, Maharaj et al . [] showed that there is no first baroclinic mode present at the Chile Rise but that Rossby wave speed estimates in such locations can only be roughly approximated by the extended linear theory baroclinic modes 2 and 3 of Killworth and Blundell [] (see also Tailleux [] for a correction to the theory of Killworth and Blundell). In our Figure d (Peak 3), we show no significant interactions at the Chile Rise (1800 km) for disturbances traveling at planetary wave speeds but clear resonant interactions, in particular at month 500, for the slower baroclinically unstable waves (Figure c; Peak 2) in this region.…”

Diagnosis of Large‐Scale, Low‐Frequency Sea Level Variability in the Northeast Pacific Ocean

“…Extended theories that consider the influence of mean flow and topography on the propagation speed change the standard dispersion relation and can account for the faster observed speed than that predicted by the standard linear theory (e.g., Aoki et al., 2009; Dewar, 1998; Killworth et al., 1997; Tailleux, 2012; Tailleux & McWilliams, 2001). The faster phase speed or more agreement between spectral peaks and the dispersion relation can be achieved by adjusting the Rossby radius of deformation in the standard dispersion relation (Osychny & Cornillon, 2004).…”