## 1. Introduction

A central question in the theory of ocean variability is how the barotropic and baroclinic normal modes of the standard linear theory (SLT) (e.g., Gill 1982) are modified by the presence of a background mean flow and variable topography. To address this issue, progress over the past decades has principally come from investigating the nature of the solutions of the equations of motion linearized around a background mean flow, most often in the context of quasigeostrophic (QG) theory. Under the Wentzel–Kramers–Brillouin approximation, which assumes that the scales over which the background mean flow and topography vary are large compared to that of the waves considered, approximately separable wave solutions still exist, whose vertical structure can be obtained as the eigenmodes of a nonself-adjoint eigenvalue problem. When the problem is formulated by linearizing the quasigeostrophic potential vorticity (PV) evolution equation around a background zonal mean flow, for instance, the generalized eigenvalue problem thus obtained is generally naturally formulated in terms of the vertical structure for the pressure [e.g., see Fu and Chelton (2001) and Aoki et al. (2009) for recent examples].

*M*function, which once inserted into the continuity equation led to the following expression for the vertical derivative of

*w*:

*ω*;

**k**

_{dim}= (

*k*,

_{x}*k*) is the dimensional wave vector; (

_{y}*k*,

*l*) are the angular zonal and meridional wavenumbers, which are related to the dimensional wavenumbers (

*k*,

_{x}*k*) by

_{y}*k*=

_{x}*k*/(

*a*cos

*θ*) and

*l*=

*k*/

_{y}*a*;

*a*is the earth’s radius;

*θ*is the latitude; Ω is the earth’s rotation rate; and

*f*= 2Ω sin

*θ*is the local Coriolis parameter. At this point, KB04 sought to derive an expression for

*w*by vertically integrating Eq. (1), which they took to be given by

*R*can be assumed to be independent of

*z*but, as noted earlier,

*R*is given by

*z*because the background mean flow depends on

*z*.

The above error was first identified by R. Samelson (2007, personal communication), who had pointed it out to Peter Killworth at the time. However, despite his best efforts, Peter Killworth passed away before he could find a cure to the problem. The main objective of this paper is twofold: 1) to clarify the nature of the error and show how to redress it and 2) to understand how the error affects some of KB04’s conclusions regarding the nature of the dispersion relation of Rossby waves in the presence of a background mean flow and topography. The issue is important to clarify because it also affects KB05’s results and KB07’s discussion of forced modes and baroclinic instability and is expected to alter some of the conclusions of the comparison of the KB04 theoretical dispersion relations against observations recently carried out by Maharaj et al. (2007, 2009).

In this paper, we note that the KB04 theory appears to rely on the same scaling arguments as those underlying the construction of QG theory and, hence, seek to derive a generalized eigenvalue problem for the vertical velocity directly from QG theory, which is done in section 2. The main result is that the eigenproblem thus obtained only differ from that of KB04 by a term that makes the QG eigenvalue problem for the vertical velocity nonlinear and hence intractable. Section 3 illustrates on some idealized examples that such a term affects the KB04 dispersion relations mostly in the high wavenumbers regime. Section 4 summarizes the results and discusses some of its consequences.

## 2. Generalized eigenvalue problem for *w* in QG theory

*D*/

_{g}*Dt*= ∂

_{t}+

*J*(Ψ, ·) is advection by the geostrophic velocity (

*u*,

_{g}*υ*) = −Ψ

_{g}_{y}, Ψ

_{x}, with Ψ being the geostrophic streamfunction, and

*q*is the potential vorticity,

*b*, pressure

*p*, and vertical velocity

*w*through the following relations:

*β*) admits separable wavelike solutions

*F*can be regarded as the eigenmodes of the eigenvalue problem:

*c*=

*ω*/

*k*is the zonal phase speed.

_{x}*w*. To that end, linearizing Eq. (7) yields the following linearized expression for

*w*:

*F*through

*W*, we successively differentiate the latter expression with respect to

*z*by making use of the links between

*F*and

*W*to successively eliminate terms involving

*F*and its derivatives. Thus, differentiating the latter equation a first time yields

*F*satisfies the eigenvalue problem (8), it is possible to remove the second-order term in

*F*to simplify this expression as follows:

*F*have been eliminated. Taking the vertical derivative a second time yields, this time,

*dF*/

*dz*in terms of

*F*and

*W*as

*F*in terms of

*dW*/

*dz*. As a result, the following eigenproblem is obtained:

*c*both in the numerator and denominator. In contrast, the eigenvalue problem derived by KB04 (in absence of meridional mean flow) is given by

*f*-plane approximation, it is in terms of the vertical velocity that the problem is most conveniently formulated as it is the problem in terms of pressure that becomes nonlinear [e.g., Gill 1982, Eq. (8.4.10)].

## 3. Particular example of the differences

*K*→ 0 so that we expect it to affect the eigensolutions only at large wavenumbers. That this is indeed the case is illustrated here in the particular case of the following idealized mean flow and stratification:

In the absence of a background mean flow, both QG and KB theories should be strictly equivalent. This is found to be the case, as illustrated in the left panels of Figs. 2 and 3, corresponding respectively to use of the standard flat-bottom boundary condition and that of bottom-pressure compensation theory of Tailleux and McWilliams (2001), which can be regarded as a limiting case of the effect of bottom topography in the infinitely steep slope limit (e.g., Tailleux 2003). As discussed above, we expect the two theories to yield dissimilar results in the presence of mean flow primarily at large wavenumbers. This is illustrated in the right panels of Figs. 2 and 3, which show that, for wavenumbers larger than the Rossby radius of deformation, the two theories may start to differ dramatically, demonstrating the importance of the corrective term overlooked by KB04 in such a region of the wavenumber space, both for a flat bottom and the bottom-pressure compensation boundary conditions. Note that, in the asymptotic limit *k _{x}* → −∞, the dispersion relationship becomes quasi nondispersive and given by

*ω*= −

*u*

_{min}

*k*, where

_{x}*u*

_{min}is the absolute minimum value of

*k*→ −∞.

_{x}Comparison of QG (solid line) and the KB (crosses) flat-bottom theories. (left) Dispersion relation in absence of mean flow as predicted by the classical QG theory (solid line) and the KB theory (crosses). (right) Dispersion relation for Rossby waves affected by the idealized Gaussian mean flow illustrated in Fig. 1, as predicted by the classical QG theory (continuous line) and KB theory (crosses). The dashed–dotted line represents the nondispersive dispersion relationship *ω* = *u*_{min}*k _{x}*; the dashed line represents the nondispersive relationship tangent at

*k*= 0: that is,

_{x}*ω*=

*c*(

*k*= 0)

_{x}*k*, where

*u*

_{min}is the absolute minimum of the horizontal zonal velocity along the vertical (which is strictly negative and located at

*z*= −500 m, according to Fig. 1). Frequency is normalized by the maximum frequency of the flat-bottom, no-mean-flow standard linear theory; the zonal wavenumber is normalized by the inverse of the Rossby radius of deformation.

Citation: Journal of Physical Oceanography 42, 6; 10.1175/JPO-D-12-010.1

Comparison of QG (solid line) and the KB (crosses) flat-bottom theories. (left) Dispersion relation in absence of mean flow as predicted by the classical QG theory (solid line) and the KB theory (crosses). (right) Dispersion relation for Rossby waves affected by the idealized Gaussian mean flow illustrated in Fig. 1, as predicted by the classical QG theory (continuous line) and KB theory (crosses). The dashed–dotted line represents the nondispersive dispersion relationship *ω* = *u*_{min}*k _{x}*; the dashed line represents the nondispersive relationship tangent at

*k*= 0: that is,

_{x}*ω*=

*c*(

*k*= 0)

_{x}*k*, where

*u*

_{min}is the absolute minimum of the horizontal zonal velocity along the vertical (which is strictly negative and located at

*z*= −500 m, according to Fig. 1). Frequency is normalized by the maximum frequency of the flat-bottom, no-mean-flow standard linear theory; the zonal wavenumber is normalized by the inverse of the Rossby radius of deformation.

Citation: Journal of Physical Oceanography 42, 6; 10.1175/JPO-D-12-010.1

Comparison of QG (solid line) and the KB (crosses) flat-bottom theories. (left) Dispersion relation in absence of mean flow as predicted by the classical QG theory (solid line) and the KB theory (crosses). (right) Dispersion relation for Rossby waves affected by the idealized Gaussian mean flow illustrated in Fig. 1, as predicted by the classical QG theory (continuous line) and KB theory (crosses). The dashed–dotted line represents the nondispersive dispersion relationship *ω* = *u*_{min}*k _{x}*; the dashed line represents the nondispersive relationship tangent at

*k*= 0: that is,

_{x}*ω*=

*c*(

*k*= 0)

_{x}*k*, where

*u*

_{min}is the absolute minimum of the horizontal zonal velocity along the vertical (which is strictly negative and located at

*z*= −500 m, according to Fig. 1). Frequency is normalized by the maximum frequency of the flat-bottom, no-mean-flow standard linear theory; the zonal wavenumber is normalized by the inverse of the Rossby radius of deformation.

Citation: Journal of Physical Oceanography 42, 6; 10.1175/JPO-D-12-010.1

Comparison of the QG (solid line) and KB (crosses) theories using the bottom boundary condition of the bottom-pressure compensation theory of Tailleux and McWilliams (2001). (left) Dispersion relation in absence of mean flow as predicted by the classical QG theory (solid line) and KB theory (crosses). (right) Dispersion relation for Rossby waves affected by the idealized Gaussian mean flow illustrated in Fig. 1 as predicted by the classical QG theory (continuous line) and KB theory (crosses). The dashed–dotted line represents the nondispersive dispersion relationship *ω* = *u*_{min}*k _{x}*, as in Fig. 2; the dashed line represents the nondispersive relationship tangent at

*k*= 0: that is,

_{x}*ω*=

*c*(

*k*= 0)

_{x}*k*. Frequency is normalized by the maximum frequency of the flat-bottom, no-mean-flow theory; the zonal wavenumber is normalized by the inverse of the Rossby radius of deformation.

_{x}Citation: Journal of Physical Oceanography 42, 6; 10.1175/JPO-D-12-010.1

Comparison of the QG (solid line) and KB (crosses) theories using the bottom boundary condition of the bottom-pressure compensation theory of Tailleux and McWilliams (2001). (left) Dispersion relation in absence of mean flow as predicted by the classical QG theory (solid line) and KB theory (crosses). (right) Dispersion relation for Rossby waves affected by the idealized Gaussian mean flow illustrated in Fig. 1 as predicted by the classical QG theory (continuous line) and KB theory (crosses). The dashed–dotted line represents the nondispersive dispersion relationship *ω* = *u*_{min}*k _{x}*, as in Fig. 2; the dashed line represents the nondispersive relationship tangent at

*k*= 0: that is,

_{x}*ω*=

*c*(

*k*= 0)

_{x}*k*. Frequency is normalized by the maximum frequency of the flat-bottom, no-mean-flow theory; the zonal wavenumber is normalized by the inverse of the Rossby radius of deformation.

_{x}Citation: Journal of Physical Oceanography 42, 6; 10.1175/JPO-D-12-010.1

Comparison of the QG (solid line) and KB (crosses) theories using the bottom boundary condition of the bottom-pressure compensation theory of Tailleux and McWilliams (2001). (left) Dispersion relation in absence of mean flow as predicted by the classical QG theory (solid line) and KB theory (crosses). (right) Dispersion relation for Rossby waves affected by the idealized Gaussian mean flow illustrated in Fig. 1 as predicted by the classical QG theory (continuous line) and KB theory (crosses). The dashed–dotted line represents the nondispersive dispersion relationship *ω* = *u*_{min}*k _{x}*, as in Fig. 2; the dashed line represents the nondispersive relationship tangent at

*k*= 0: that is,

_{x}*ω*=

*c*(

*k*= 0)

_{x}*k*. Frequency is normalized by the maximum frequency of the flat-bottom, no-mean-flow theory; the zonal wavenumber is normalized by the inverse of the Rossby radius of deformation.

_{x}Citation: Journal of Physical Oceanography 42, 6; 10.1175/JPO-D-12-010.1

## 4. Summary and conclusions

In this paper, we derived the generalized QG eigenvalue problem for the vertical velocity normal mode structure of Rossby waves in presence of a background mean flow and topography. Previously, such an eigenproblem had been formulated only for the pressure. Surprisingly, the vertical velocity eigenproblem appears to be nonlinear (the eigenvalue appears both in the numerator and denominator), which is very uncommon and not easily anticipated given that the eigenproblem for the pressure is linear. Such a result shows how the KB04 derivation might be corrected; at the same time, it also shows that the actual eigenproblem for the vertical velocity is not easily tractable and that the investigation of the propagation properties of Rossby waves in the presence of mean flow and topography is more easily addressed by solving the pressure eigenvalue problem. Whenever the vertical velocity structure *W* is needed, it is most conveniently diagnosed a posteriori from the knowledge of the pressure vertical structure *F* by using (the discretized version of) Eq. (10).

The results also show that the error made by KB04 is, in fact, equivalent to neglecting the term making the QG eigenproblem for the vertical velocity nonlinear. Such a term is small in the long-wave limit *K* → 0 so that the error is mostly of consequence for understanding the behavior of the Rossby wave dispersion relation at high wavenumbers. The study of the latter regime is an old problem, which was investigated in significant detail by Gnevyshev and Shrira (1989) and is therefore not discussed in more detail here.

## Acknowledgments

The author gratefully acknowledges financial support from CNES and Dudley Chelton and Roger Samelson for encouragements in pursuing this study. This paper is dedicated to the memory of Peter Killworth, who was a constant source of inspiration in the study of oceanic Rossby waves. We hope that he would have agreed on the solution proposed.

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