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  • View in gallery

    Schematic illustrating the role of boundary waves in the adjustment of the upper limb of the MOC to localized forcing through changes in deep water formation at high latitudes (downward arrow), or through westward-propagating Rossby waves and eddies impinging on the western boundary (rings and solid arrow). The boundary waves propagate cyclonically around each hemispheric basin (dashed arrows).

  • View in gallery

    Schematic showing the domain considered for the western and eastern boundary wave solutions.

  • View in gallery

    Schematic showing the energy fluxes involved when westward-propagating long Rossby waves excite short Rossby waves at a western boundary. The short wave is greatly reduced in amplitude equatorward of the incident long wave. The mathematical variables are defined in the text.

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    Schematic showing the solution for the Fofonoff gyre with uniform potential vorticity, about which linearized wave solutions are obtained in section 7. Sketched are the layer thickness contours, which serve as approximate streamlines for the flow. The layer thickness varies slightly along the boundary because the no-normal flow boundary condition applies to the total, rather than geostrophic, velocity.

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Propagation of Meridional Circulation Anomalies along Western and Eastern Boundaries

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  • 1 Department of Physics, University of Oxford, Oxford, United Kingdom
  • | 2 Department of Earth Sciences, University of Oxford, Oxford, United Kingdom
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Abstract

Motivated by the adjustment of the meridional overturning circulation to localized forcing, solutions are presented from a reduced-gravity model for the propagation of waves along western and eastern boundaries. For wave periods exceeding a few months, Kelvin waves play no role. Instead, propagation occurs through short and long Rossby waves at the western and eastern boundaries, respectively: these Rossby waves propagate zonally, as predicted by classical theory, and cyclonically along the basin boundaries to satisfy the no-normal flow boundary condition. The along-boundary propagation speed is cLd/δ, where c is the internal gravity/Kelvin wave speed, Ld is the Rossby deformation radius, and δ is the appropriate frictional boundary layer width. This result holds across a wide range of parameter regimes, with either linear friction or lateral viscosity and a no-slip boundary condition. For parameters typical of contemporary ocean climate models, the propagation speed is coincidentally close to the Kelvin wave speed. In the limit of weak dissipation, the western boundary wave dissipates virtually all of its energy as it propagates toward the equator, independent of the dissipation coefficient. In contrast, virtually no energy is dissipated in the eastern boundary wave. The importance of background mean flows is also discussed.

Corresponding author address: Dr. David P. Marshall, Clarendon Laboratory, Department of Physics, Parks Road, University of Oxford, Oxford OX1 3PU, United Kingdom. E-mail: marshall@atm.ox.ac.uk

Abstract

Motivated by the adjustment of the meridional overturning circulation to localized forcing, solutions are presented from a reduced-gravity model for the propagation of waves along western and eastern boundaries. For wave periods exceeding a few months, Kelvin waves play no role. Instead, propagation occurs through short and long Rossby waves at the western and eastern boundaries, respectively: these Rossby waves propagate zonally, as predicted by classical theory, and cyclonically along the basin boundaries to satisfy the no-normal flow boundary condition. The along-boundary propagation speed is cLd/δ, where c is the internal gravity/Kelvin wave speed, Ld is the Rossby deformation radius, and δ is the appropriate frictional boundary layer width. This result holds across a wide range of parameter regimes, with either linear friction or lateral viscosity and a no-slip boundary condition. For parameters typical of contemporary ocean climate models, the propagation speed is coincidentally close to the Kelvin wave speed. In the limit of weak dissipation, the western boundary wave dissipates virtually all of its energy as it propagates toward the equator, independent of the dissipation coefficient. In contrast, virtually no energy is dissipated in the eastern boundary wave. The importance of background mean flows is also discussed.

Corresponding author address: Dr. David P. Marshall, Clarendon Laboratory, Department of Physics, Parks Road, University of Oxford, Oxford OX1 3PU, United Kingdom. E-mail: marshall@atm.ox.ac.uk

1. Introduction

Wave propagation along western and eastern boundaries is a fundamental process through which the ocean adjusts to changes in its boundary conditions (e.g., Wajsowicz 1986; Kawase 1987; Fig. 1). High-frequency variability can propagate large distances along meridional boundaries through coastal Kelvin waves, topographic waves, or some combination thereof (Huthnance 1978). However, variability in the meridional overturning circulation (MOC), heat content, and regional sea level change also occurs on seasonal-to-decadal time scales, on which other forms of boundary wave may dominate the adjustment process.

Fig. 1.
Fig. 1.

Schematic illustrating the role of boundary waves in the adjustment of the upper limb of the MOC to localized forcing through changes in deep water formation at high latitudes (downward arrow), or through westward-propagating Rossby waves and eddies impinging on the western boundary (rings and solid arrow). The boundary waves propagate cyclonically around each hemispheric basin (dashed arrows).

Citation: Journal of Physical Oceanography 43, 12; 10.1175/JPO-D-13-0134.1

The adjustment of the ocean to a change in high-latitude buoyancy forcing varies widely across ocean general circulation models (OGCMs) and coupled climate models. The time scale on which anomalies propagate from high to low latitudes, the signal amplitude, and the inferred mechanisms all differ significantly. In the subpolar gyre north of about 40°N, several studies have suggested that advection—rather than boundary wave propagation—dominates the response of the MOC (e.g., Gerdes and Köberle 1995; Eden and Greatbatch 2003; Getzlaff et al. 2005; Zhang et al. 2011). This is perhaps not surprising given the strong mean boundary currents in the surface ocean and at depth, which are both directed equatorward here, and the recently reported interior pathways for North Atlantic Deep Water (Bower et al. 2009). Farther south, however, many studies see equatorward propagation of boundary anomalies at speeds significantly faster than the advective velocities associated with the deep western boundary current [e.g., Döscher et al. 1994; Capatondi 2000; Eden and Willebrand 2001; Goodman 2001; Dong and Sutton 2002; Getzlaff et al. 2005; Roussenov et al. 2008; Chiang et al. 2008; Hawkins and Sutton 2008; Köhl and Stammer 2008; Zhang 2010; Zhang et al. 2011; Hodson and Sutton 2012; for exceptions, see Marotzke and Klinger (2000) and Buckley et al. (2012)] and infer that these are Kelvin wave signals, traveling slower than the first baroclinic gravity wave speed (which is on the order of 1 m s−1) owing to numerical effects. Anomalies in these models take between a month (e.g., Dong and Sutton 2002) and a decade (e.g., Capatondi 2000) to reach the tropics.

The results of Hsieh et al. (1983) are often invoked to explain these boundary wave propagation differences in terms of the effect of model grid and resolution. Other authors cite the different forcing mechanisms (e.g., Marotzke and Klinger 2000; Johnson and Marshall 2002a) as explanation. In this paper we analyze the propagation characteristics of waves along meridional boundaries in a reduced-gravity model, at both high and low frequency and in the presence of both linear and lateral friction. Motivated by the disparate results of OGCMs in response to variable buoyancy forcing at high latitudes, and building on the work previously applied to coastal and tropical applications (e.g., Clarke 1983; Grimshaw and Allen 1988; Clarke and Shi 1991), we aim to present a unified view of first-mode baroclinic wave propagation relevant to large-scale ocean adjustment on time scales longer than a few months. We show that the propagation speed is not the Kelvin wave speed, but in fact varies inversely with the pertinent viscous boundary layer width; because the latter differs among OGCMs, this may offer a partial explanation for the spread in the propagation speed of Atlantic MOC anomalies observed in the studies above and others like them.

There are four outstanding questions concerning wave propagation along meridional boundaries that we plan to address in this paper (Fig. 1):

  • What is the relevant property that propagates along the boundary?
  • At what speed does it propagate?
  • How much energy is dissipated?
  • What is the amplitude of western boundary waves generated by incident long Rossby waves from the ocean interior, and how much of the incident wave energy is dissipated?

In the remainder of this section, we take each of these questions in turn and provide some background context and motivation. We also discuss the implications and relevance for the detection, monitoring and prediction of MOC change.

a. What propagates?

First-mode baroclinic waves propagate an anomaly in pycnocline depth along the boundary, with an associated signal in pressure, density, velocity, and often temperature. However, the amplitude of the anomaly need not be constant along the propagation path. Wajsowicz and Gill (1986) show that Kelvin waves on an f plane are attenuated by friction, and that their amplitude decays as a result. Johnson and Marshall (2002a) illustrate in a reduced-gravity model that, even in the absence of friction, when f varies with latitude the amplitude of a western boundary wave is reduced during equatorward propagation. On the eastern boundary, they see no corresponding increase in wave amplitude with latitude; instead, long Rossby waves are radiated into the ocean interior. This asymmetry between eastern and western boundaries is a key element of their “equatorial buffer” mechanism.

There is a substantial body of literature on the nature of eastern boundary wave propagation and the radiation of long Rossby waves. This has originated largely from the coastal oceanography community, motivated by the analysis of coastal sea level observations, and the propagation of ENSO-related signals out of equatorial regions. Clarke (1983) showed that motion at the eastern boundary of an ocean basin can be described in terms of either westward-propagating long Rossby waves, or coastal Kelvin waves, with a critical latitude-dependent frequency below which energy is radiated into the interior, rather than coastally trapped. He confirmed the result of Moore (1968) that the amplitude of a Kelvin wave propagating on a β plane is proportional to at high frequencies, where y is the meridional distance north of the equator.

Wajsowicz (1986), Grimshaw and Allen (1988), and Clarke and Shi (1991) built on these results. Taking into account the coastline orientation (Schopf et al. 1981), Clarke and Shi (1991) calculate critical frequencies for the World Ocean’s boundaries, which range from a month or two in the tropics to a year at midlatitudes. Grimshaw and Allen (1988) note that a consequence of these two regimes is that, while at high frequencies the amplitude of coastal disturbances varies as on both eastern and western boundaries, at lower frequencies this dependence changes. McCalpin (1995) finds that on eastern boundaries at low frequency the amplitude of the wave disturbance is in fact constant along the eastern boundary, consistent with the earlier findings of Clarke (1983) and Grimshaw and Allen (1988).

A change in the amplitude of wave signals as they propagate along meridional boundaries has important consequences for the diagnosis of waves in ocean models. Because of the coarse spatial and temporal resolution of typical OGCM output, together with the presence of a background mean flow and mesoscale eddy field, the propagation of anomalies along the boundary can be hard to detect. Many OGCM studies calculate propagation speed based on the arrival time of a given amplitude of anomaly at a particular latitude (e.g., Marotzke and Klinger 2000). This may well be a dangerous strategy because one cannot infer wave speed through tracking a contour of density if the amplitude is changing as the wave propagates. An understanding of how we expect wave amplitudes to change along eastern and western boundaries, as a function of frequency, may help us to more clearly identify these signals in models. It will also have implications for the detection and monitoring of change in the ocean.

Note that basin-mode theory (e.g., Liu et al. 1999; Cessi and Louazel 2001; Cessi and Paparella 2001; Cessi and Primeau 2001; Primeau 2002) assumes the amplitude of anomalies to be uniform along the eastern boundary at all times. This will be discussed in more detail in the following section.

b. At what speed?

Baroclinic coastal Kelvin waves propagate at the internal gravity wave speed , where gr is the reduced gravity, and h is the equivalent depth of the baroclinic mode. Any signal propagation occurring on the western boundary of ocean models on time scales from months to a decade is generally assumed to be the result of Kelvin waves, slowed by low stratification (Greatbatch and Peterson 1996) or numerical effects, especially on a B grid (Hsieh et al. 1983). At high frequencies this may well be the correct interpretation, but the physical argument outlined in the previous section suggests that at frequencies significantly lower than ω = βLd = βc/f (i.e., on time scales longer than about 2 months) coastally trapped Kelvin wave propagation is no longer expected on either eastern or western boundaries. This does not mean that propagation along the boundary is prohibited; simply that it need not occur at the gravity wave speed and need not be coastally trapped.

Primeau (2002) suggested that Kelvin waves play no role in the adjustment of the ocean to a change in forcing on all but the shortest of time scales. He concluded that, instead, internal gravity waves act to distribute pressure anomalies instantaneously along basin boundaries. This is consistent with the results of Clarke (1983) and Grimshaw and Allen (1988), both of whom point out that, in the absence of friction, there is no phase propagation along eastern boundaries, with all points on the boundary responding to an incident disturbance simultaneously. Cessi and Otheguy (2003) come to similar conclusions; in fact, this instantaneous redistribution of anomalies uniformly along the eastern boundary is central to the theory of basin modes. Clarke (1983) also considered the inclusion of linear damping in each of the momentum and continuity equations, obtaining a finite propagation speed proportional to the damping coefficient with linear friction in the continuity equation, but still synchronous variation along the boundary with linear friction in the momentum equation.

Note that, in closed-basin quasigeostrophic (QG) models, a “consistency condition” (McWilliams 1977) is used to ensure global mass conservation, determining the boundary value of the streamfunction, which is constant along the boundary but varies in time. This instantaneous adjustment is the QG model equivalent of the coastal Kelvin wave and allows for the radiation of long Rossby waves into the interior (see, e.g., Milliff and McWilliams 1994; McCalpin 1995).

Other boundary propagation mechanisms besides numerical Kelvin waves and advection have of course been proposed. Killworth (1985) and Winton (1996) both discuss propagation by viscous boundary waves, associated with the effect of friction at the sidewall. In Killworth (1985) the divergence associated with the wave at the coast occurs over one grid box due to the no-normal flow condition, and the wave propagates at a speed c2/fΔx, where Δx is the grid resolution in the zonal direction. In the presence of a sloping boundary and/or a continental shelf, topographically trapped wave modes as well as hybrid waves become possible (Huthnance 1978; Elipot et al. 2013) and can travel significantly faster than first-mode baroclinic Kelvin waves (O’Rourke 2009). We restrict our attention to the vertical sidewall case in this paper, because even this exhibits rich behavior.

We will show analytically in section 2, using the same equation set that results in pure Kelvin wave propagation at high frequencies, that at low frequencies the along-boundary propagation speed depends strongly on the frictional parameters of the model, and hence varies among OGCMs. These low-frequency waves may in some models travel along the boundary at approximately Kelvin wave speeds (e.g., Yang 1999; Johnson and Marshall 2002a) but this is simply fortuitous. This model dependence is a worrying state of affairs because we rely on these OGCMs, coupled to atmospheric circulation models, to predict the rate at which the ocean will respond to future forcing changes associated with increased greenhouse gas concentrations. Correctly representing the ocean adjustment mechanisms to anticipated changes at high latitudes is essential if we are to make accurate projections of global and particularly regional climate.

A proper representation of boundary wave propagation speeds is also important for our understanding of internal climate variability. Several recent studies suggest that the ocean exhibits damped, decadal oscillations that are stochastically excited by atmospheric forcing (e.g., Greatbatch and Peterson 1996; Cessi and Louazel 2001; Cessi and Paparella 2001; Eden and Greatbatch 2003; Dong and Sutton 2005; Danabasoglu 2008; Frankcombe et al. 2009; Czeschel et al. 2010; Danabasoglu et al. 2012; Kwon and Frankignoul 2012; Delworth and Zeng 2013). These modes of variability can have significant climate fingerprints (e.g., Knight et al. 2005; Hurrell et al. 2006; Zhang and Delworth 2006; Mahajan et al. 2011). The mechanisms and time scales underlying the oscillations differ from one model to another, with boundary adjustment processes often implicated (e.g., Greatbatch and Peterson 1996; Hawkins and Sutton 2007). As a result of the diversity of model behavior, there is little consensus on the causes of decadal-to-multidecadal variability in the real climate system. Understanding the mechanisms and time scales involved would facilitate attempts to develop the capacity for decadal climate prediction in the Atlantic sector (e.g., Latif et al. 2006; Msadek et al. 2010; Robson et al. 2012).

Note that the speed of boundary wave adjustment has implications beyond the ocean basin in which a change in forcing occurs. In response to high-latitude forcing in the North Atlantic, for example, fast boundary waves, together with equatorial Kelvin waves, carry the signal of a thermocline displacement to all the major ocean basins on relatively short time scales (e.g., Huang et al. 2000; Goodman 2001; Cessi and Otheguy 2003; Johnson and Marshall 2004). While the amplitude of variability on anything shorter than multidecadal time scales may be small outside the hemispheric basin of forcing (Johnson and Marshall 2002b), the implications for sea level are significant (Hsieh and Bryan 1996; Landerer et al. 2007). Fast atmospheric teleconnections via tropical air–sea interactions may also result in a global response (e.g., Dong and Sutton 2002).

c. Energetics and western boundary interactions

The mechanical energy budget of the global ocean circulation has received a lot of attention in recent years (e.g., Munk and Wunsch 1998; Wunsch and Ferrari 2004; Tailleux 2009; Hughes et al. 2009), with mechanisms of energy loss from the geostrophic modes including bottom drag (Sen et al. 2008), loss of balance (Molemaker et al. 2005), exchange of energy with pre-existing internal waves (Polzin 2008), and generation of internal waves by geostrophic flow over rough topography (Marshall and Naveira Garabato 2008; Nikurashin and Ferrari 2010; Nikashurin et al. 1998). Recently, Zhai et al. (2010) demonstrated in idealized model calculations and altimetric data that there is a significant sink of geostrophic eddy energy when westward-propagating eddies (e.g., Chelton et al. 2007) reach the western boundary—the so-called Rossby graveyard. In the model calculations, the amount of energy dissipation appears to be independent of model resolution and of the magnitude and nature of momentum dissipation. In the ocean, it is unclear how much of the energy is merely backscattered to the mean flow (cf. Starr 1968) or dissipated through vertically breaking boundary waves (Dewar and Hogg 2010).

Irrespective of how energy is dissipated in the ocean, in coarse-resolution models, it is likely that a significant amount of energy is dissipated in western boundary waves. In section 4, we show that this rate of energy dissipation is independent of the magnitude of the local friction and, in section 5, we show that the vast majority of the energy incident on the western boundary in the form of long Rossby waves is dissipated locally on reaching the western boundary.

The energy dissipation implicit in long waves reaching a western boundary is intimately related to the strong reduction in dynamic height variability at the western margin of an ocean basin. This reduction is crucial in preventing MOC variations inferred from boundary hydrographic and other data (Cunningham et al. 2007; Kanzow et al. 2007) from being swamped by eddy variability (Wunsch 2008). This issue has been previously discussed using a stripped-down version of the linear wave solutions presented in this manuscript (Kanzow et al. 2009) but nevertheless serves as an important motivation for the present study.

d. Structure of the paper

The paper is structured as follows. In section 2, we solve for the linear boundary waves in the presence of a meridional boundary and linear friction. In section 3, we discuss the behavior of meridional transport anomalies along western and eastern boundaries. In section 4, we analyze the energetics of western and eastern boundary waves. In section 5, we study the western boundary waves generated through incident long waves and the location and magnitude of energy dissipation associated with these interactions. In section 6, we solve for linear boundary waves in the presence of lateral friction and a no-slip boundary condition. In section 7, we investigate the impact of background mean flow on the boundary wave solutions. Finally, in section 8 we present a brief concluding discussion.

2. Wave solutions with linear friction

We consider solutions to the reduced-gravity equations, linearized about a state of rest. We adopt a semigeostrophic approximation (Hoskins 1975) in which we assume that geostrophic balance holds in the zonal momentum equation, but not in the meridional momentum equation where we know that the Coriolis acceleration vanishes at the boundary.

Thus,
e2.1
e2.2
e2.3
The velocities in the zonal x and meridional y directions are u and υ, respectively, h is the layer thickness, and r is the coefficient of linear friction. We adopt a β-plane approximation such that the Coriolis parameter f = βy. Primes indicate linear perturbations and zero subscripts the background mean state.
For simplicity, we restrict our attention to north–south coastlines located at x = 0 (Fig. 2), along which a no-normal flow boundary condition is applied
e2.4
Thus, western boundary solutions correspond to the half-plane x ≥ 0 and eastern boundary solutions correspond to the half-plane x ≤ 0 (Fig. 2). The orientation of the boundaries has some influence on the solution (e.g., Grimshaw and Allen 1988).
Fig. 2.
Fig. 2.

Schematic showing the domain considered for the western and eastern boundary wave solutions.

Citation: Journal of Physical Oceanography 43, 12; 10.1175/JPO-D-13-0134.1

From (2.1) to (2.3), we can derive a vorticity equation:
e2.5
where
e2.6
The no-normal flow boundary condition (2.4), when substituted in (2.2) and using (2.1), takes the form:
e2.7
Following Clarke and Shi (1991), we now seek solutions of the form
e2.8
where real parts is understood, and we anticipate that the zonal wavenumber k(y) varies with latitude because of the variation of the Coriolis parameter and deformation radius with latitude; for example, to allow the zonal decay scale of coastal Kelvin waves to vary with latitude.
Substituting this trial solution into (2.5) gives
e2.9
to which the general solution is
e2.10
where
e2.11
The convention we adopt in this and subsequent equations is that the first root corresponds to the western boundary solution and the second root to the eastern boundary solution.
Substituting the trial solution into the boundary condition (2.7) gives an equation for the amplitude variation along the boundary:
e2.12

a. Kelvin wave limit (λ ≫ 1, r ≪ ω)

First we consider the weakly damped Kelvin wave limit (λ ≫ 1, rω) equivalent to ωβLd/2. Typical values of β around 2 × 10−11 m−1 s−1 and Ld around 5 × 104 m gives ω ≪ 5 × 10−7 s or periods much shorter than 3–4 months (on which time-scale damping can safely be assumed to be weak).

1) Zonal structure

The leading-order solution for the zonal wavenumber is
e2.13
where we have neglected O(λ−1) and O(r2/ω2) terms in the expansion.
Thus, to leading order
e2.14
the Kelvin wave decays away from the boundary over Ld as expected, but note that the deformation radius decreases with increasing latitude.

2) Meridional structure

The real component of k is of little consequence for the zonal structure of the wave, but has an important impact on its meridional structure and propagation. Substituting (2.13) into the boundary condition (2.12) gives
eq1
The leading-order solution for A is
e2.15
In contrast to the f-plane limit, the wave amplitude (in h) varies as the square root of latitude y; this result was first derived by Moore (1968). The wave propagates cyclonically around the coastline at c and decays because of friction over a length scale 2c/r.

b. Rossby wave limit (λ ≪ 1, r ≪ ω)

Now consider the low-frequency limit λ ≪ 1. The analysis presented below applies irrespective of the relative magnitudes of ω and r. We first discuss the Rossby wave limit ωr and defer discussion of the Stommel boundary layer limit ωr until section 2c.

1) Zonal structure

The leading-order expression for the zonal wavenumber (2.10) is now
e2.16
where (2.11) has been used to substitute for λ in the final expression. The first, western boundary root corresponds to a short Rossby wave solution, damped over a zonal scale ω2/. The second, eastern boundary root corresponds to a long Rossby wave solution, independent of the size of r at leading order. Physically, as the angular frequency is reduced, short Rossby waves become shorter and eventually find the frictional boundary layer scale r/β (Stommel 1948), whereas long Rossby waves become longer and friction remains unimportant.

2) Meridional structure

The meridional structure of the waves is determined by substituting (2.16) into the boundary condition (2.12).

First for the western boundary short-wave solution, we find
eq2
The solution is
e2.17
Here we have set because ω2y2/c2 ≪ 1, and Taylor expanded in y about a mean latitude y0 absorbing the leading term in the Taylor expansion into the complex phase A0.
Thus the amplitude of the western boundary short-wave solution decays linearly with decreasing latitude and propagates equatorward along the boundary at a speed
e2.18
where δS = r/β is the Stommel frictional boundary layer width (Stommel 1948). Thus, the boundary wave propagates at the classical Kelvin wave speed, scaled by the parameter Ld/δS. We write (2.18) in this form because, as we shall see in due course, the same result emerges across many regimes, including with lateral friction and a no-slip boundary condition, provided that δS is replaced by the pertinent frictional boundary layer width.

For a typical ocean climate model the frictional boundary layer width is chosen to be comparable to the grid spacing, which in turn is coincidentally close to the Rossby deformation radius. Thus the boundary anomalies can be expected to propagate at roughly the classical Kelvin wave speed, but the boundary propagation speed is likely to vary between different models and with latitude.

For the eastern boundary wave, we instead have
eq3
to which the leading-order solution is
e2.19
e2.20
We have again assumed ω2y2/c2 ≪ 1 and Taylor expanded the solution about a reference latitude. In contrast to the western boundary solution, the amplitude A does not vary, at leading order, along the boundary. The wave propagates poleward at the same speed as the western boundary solution,
e2.21

c. Stommel boundary layer limit (λ ≪ 1, r ≫ ω)

The preceding analysis in section 2b also applies in the limit ωr, which we briefly comment on here. As the frequency decreases, short Rossby waves become increasingly shorter, such that friction becomes increasingly important, whereas long Rossby waves become increasingly longer and friction remains unimportant. This is clear when one considers the solution for the western boundary wave in section 2b. If rω, then
e2.22
and
e2.23
The solution is a frictional western boundary layer of width δS (Stommel 1948), with a flow direction that oscillates in time [also see Pedlosky (1965)]. As in section 2b, the solution propagates along the boundary at the speed given in (2.18).

3. Meridional transport anomalies

Meridional volume transport anomalies are related to the layer thicknesses anomalies on the western and eastern boundaries by integrating the geostrophic velocity anomaly multiplied by the mean layer thickness across the basin:
e3.1
These can be interpreted as meridional overturning transport anomalies associated with each baroclinic mode in a basin of uniform depth and stratification.
Western boundary wave solutions all decay rapidly in the zonal direction. Hence, meridional transport anomalies associated with these western boundary waves
e3.2
are confined to the western boundary region.
Conversely, eastern boundary waves do not decay (to leading order) as they cross the basin. Nevertheless, it makes sense mathematically to define the meridional transport anomalies associated with the eastern boundary waves as
e3.3
this decomposition also makes physical sense if we anticipate a result from section 5 that the layer thickness (or pressure) anomalies associated with long waves are vastly reduced in the vicinity of the western boundary [also see Kanzow et al. (2009)].

a. Kelvin wave limit

In the Kelvin wave limit, layer thickness anomalies on both western and eastern boundaries are proportional to . Thus the meridional transport anomalies vary as
e3.4
that is, the meridional transport anomalies are largest at low latitudes.

b. Rossby wave and Stommel boundary layer limits

In both the Rossby wave and Stommel boundary layer limits, there is an asymmetry in the variation of the layer thickness anomalies with latitude.

On the western boundary, the layer thickness anomalies are proportional to y, and hence
e3.5
In contrast, on the eastern boundary, the layer thickness anomalies are independent of y, and hence
e3.6
that is, they decay with increasing latitude as the wave propagates poleward.

This asymmetry is consistent with the theoretical model of Johnson and Marshall (2002a) for the adjustment of the MOC to forcing anomalies and, in particular, to the “equatorial buffer” mechanism introduced to explain the asymmetry between the response of the MOC to high-latitude localized forcing on each side of the equator, and between the western boundary layers and basin interior.

4. Energetics

We now turn to the energy budget of the wave solutions. The purpose of this short section is to demonstrate a fundamental asymmetry between the western and eastern boundary waves, with the majority of the energy being dissipated in the former but conserved in the latter. Subsequently in section 5, we consider the generation of short waves at the western boundary by long waves incident from the ocean interior.

The energy equation is derived from the momentum Eqs. (2.1)(2.3):
e4.1
where the overbar represents a time average over a wave period. Thus, the wave energy flux in (4.1) is . This energy flux is degenerate in the sense that any rotational gauge can be added to the flux without changing the energy equation (e.g., Longuet-Higgins 1964; Pedlosky 1987; Orlanski and Sheldon 1993; Chang and Orlanski 1994). For statistically steady wave solutions, the time derivative vanishes and the divergence of the energy flux is balanced by frictional energy dissipation.
We now consider the dominant terms in this energy balance for each of the wave solutions derived in section 2. In the derivation of these wave energy fluxes, recall that real parts of h′, u′, and υ′ are understood before taking quadratic products. From the momentum Eqs. (2.1) and (2.2), the form of the trial solution (2.8), and the boundary condition (2.12), it follows that
e4.2
and
e4.3
In the case of the meridional energy flux, integrating h′ × (2.1) further gives:
e4.4
where the integral (with square brackets indicating the limits of integration) is across the relevant western or eastern boundary wave. These relations are valuable in identifying the components of u′ and υ′ that are in phase (or antiphase) with h′ and hence contribute to the energy flux.

a. Kelvin wave

For Kelvin waves in the limit of vanishing friction (r = 0), the net meridional energy flux in (4.1) is
e4.5
independent of latitude. Hence, in the Kelvin wave solution, the variation of the wave amplitude on the boundary as the square root of latitude can be interpreted as being precisely what is needed to conserve wave energy (Moore 1968). Finite friction merely modifies this flux to account for the energy dissipated.
The zonal energy flux, obtained by substituting (2.13) in (4.3), is
e4.6
Note that this vanishes both on the boundary at x = 0 and as x → ±∞. Thus, the zonal energy flux is precisely that required to broaden/narrow the Kelvin wave as it propagates equator-/poleward but vanishes in the zonally integrated energy budget.

b. Short Rossby wave/Stommel boundary layer

The zonal energy flux, obtained by substituting the western boundary solution in (2.16) into (4.3), is
e4.7
because h′ and u′ are out of phase. In contrast to the Kelvin wave, the wavelength of the short Rossby wave/width of the Stommel boundary layer is independent of latitude and hence there is no need for a zonal energy flux.
The net meridional energy flux, using (4.4) and noting h′ → 0 as x → ∞, is
e4.8
The meridional energy flux is equatorward and decreases linearly with decreasing latitude. This energy is lost to dissipation at a rate
e4.9
independent of latitude. Note that the dissipation is also independent of the drag coefficient r. Physically, as r is reduced, the pointwise energy dissipation reduces, but occurs over a proportionately larger area as the short Rossby waves propagate further away from the boundary; these effects compensate at leading order.

In reality, this expression must break down close to the equator, when y approaches the equatorial deformation radius. Thus, a small amount of energy can be expected to leak into an equatorial Kelvin wave, but this is a small residual of the meridional energy flux within the short Rossby wave at higher latitudes.

c. Long Rossby wave

In the long Rossby wave solution, both u′ and υ′ are out of phase with h′ and hence both the zonal and meridional energy fluxes vanish, as does the dissipation at leading order. As mentioned above, the energy flux is degenerate to an arbitrary rotational flux; thus, the result that the energy fluxes vanish in the long-wave solution should be interpreted as indicating only that there is no divergence of the energy flux.

5. Long waves incident on a western boundary

a. Statement of problem

We now consider a problem of particular relevance to the ocean energy budget and to monitoring the meridional overturning circulation: the fate of long Rossby waves incident on a western boundary and the amplitude of the boundary wave anomaly excited on the western boundary. This problem is relevant to the ocean energy budget because analysis of satellite altimeter data and numerical model calculations suggests that the western boundary current acts an eddy graveyard and eddy energy sink (Zhai et al. 2010). Moreover, associated with this eddy energy sink is a reduction in the amplitude of dynamic height variability at the western boundary, also consistent with altimetric and hydrographic observations. The latter turns out to be crucial for end-point monitoring of the MOC (Cunningham et al. 2007), as discussed by [Kanzow et al. 2009; also see Wunsch (2008)].

We consider a long Rossby wave incident on a western boundary over a confined latitude band ysyyn. Because the incident wave is long relative to the short wave it will excite, it suffices to model the incident wave through a prescribed (complex) wave amplitude that varies with latitude and time
eq4
where vanishes for yys and yyn by assumption.
On reaching the western boundary, a short-wave or frictional boundary layer solution
eq5
is excited to satisfy the no-normal flow western boundary condition. Note that while vanishes for yyn (because the short-wave meridional energy flux is equatorward), need not vanish for yys.

A preliminary version of this analysis, valid in the limit of vanishing friction, was presented in Kanzow et al. (2009).

b. Boundary amplitude condition

The boundary condition (2.7) now applies to the sum of the long- and short-wave solutions and can be written
e5.1
Note that the term involving Al on the right-hand side of (5.1) is neglected because we have assumed that the wavenumber of the incident long wave is vanishingly small, |kl| ≪ |ks| (and we anticipate that As and Al are of similar magnitude).
Substituting for ks gives
e5.2
As in Kanzow et al. (2009), we can use the identity
eq6
to rewrite this result as
e5.3
Finally, integrating (5.3) over the latitude of the incident long wave gives
e5.4
where Δy = ynys. Thus, the net wave amplitude on the boundary As + Al is a factor Δy/y = βΔy/f smaller than the amplitude of the incident long wave Al.

Note that equatorward of the incident long wave, (5.3) reduces to the result that As/y is constant as found for a solitary short wave in (2.17).

c. Energetics

The foregoing analysis raises a number of questions about the energetics of the interaction between the incident long wave and the short wave generated at the western boundary:

  • What fraction of the incident energy reflects zonally as a short wave?
  • What fraction of the energy is fluxed equatorward along the boundary?
  • What fraction of the energy is dissipated by friction?

The natural way to pose the problem is to evaluate the zonal energy flux of the incident long wave grh0, the (zonal and meridional) energy flux of the reflected short wave grh0, and the energy dissipation in the short wave . However, for two waves of the same frequency, as considered here, both the wave energy and energy flux are modified by the cross terms representing the interference of the two waves.1 Thus, even the concept of short- and long-wave energies, and short- and long-wave fluxes, is problematic. However, these difficulties can be avoided if one integrates over a much larger area than the zonal decay scale of the short waves, as sketched in Fig. 3, such that the cross-interaction energy fluxes can be neglected.

Fig. 3.
Fig. 3.

Schematic showing the energy fluxes involved when westward-propagating long Rossby waves excite short Rossby waves at a western boundary. The short wave is greatly reduced in amplitude equatorward of the incident long wave. The mathematical variables are defined in the text.

Citation: Journal of Physical Oceanography 43, 12; 10.1175/JPO-D-13-0134.1

Firstly, the incident long-wave energy flux is
e5.5
Here, we have used the fact that (2.2) reduces to geostrophy in the limit → 0 (equivalently kl → 0) and integrated the resultant expression by parts with Al = 0 at the limits of integration.
As in section 4b, the reflected short-wave zonal energy flux vanishes equatorward of the incident long wave:
e5.6
Over the latitude band of the incident long wave, the short-wave energy flux is complicated by the modified boundary condition (5.3) and by the need to account for the interference between the short and long waves. Nevertheless, it is clear that these terms decay as one moves away from the boundary and hence do not affect the integral energy budget.

So where does the incident long-wave energy go?

The first option is that the energy is fluxed equatorward by the short waves:
e5.7
as in (4.8).
The second option is that the energy is dissipated. Because the meridional velocity associated with the long waves is vanishingly small by assumption, the net energy dissipation integrated across the boundary layer is simply:
e5.8
Note, again, that the energy dissipation is independent of the linear drag coefficient.
Now suppose that the incident long-wave energy flux is mostly confined to a narrow latitude band, as suggested by satellite altimeter measurements of sea surface height variance (e.g., Zhai et al. 2010). Then (5.4) implies that, to leading order,
eq7
over the latitude band of the incident long waves. In contrast, equatorward of the incident long waves, As is greatly reduced following (5.4). Thus, it follows that most of the dissipation occurs within the latitude band of the incident long waves. Only a small fraction of the incident energy is fluxed equatorward, and most of that is dissipated within the western boundary layer, at a rate that is independent of latitude. Only a very small amount of energy ultimately escapes into the equatorial waveguide where the preceding analysis breaks down.

These results are consistent with the Rossby graveyard mechanism proposed and diagnosed by Zhai et al. (2010) using altimetric data and numerical calculations. Because the wave energy flux in (4.1) is proportional to the bolus transport, an interesting corollary is that there is a convergence of eddy bolus transport at the western boundary, requiring offshore Eulerian-mean currents, analogous to the rip currents generated at a beach (see Marshall et al. 2013).

6. Wave solutions with lateral friction

In section 2, we derived the properties of Kelvin and boundary Rossby waves in the presence of linear friction, the main advantage being that the mathematics is relatively straightforward and solutions can be categorized in terms of a single nondimensional parameter. In this section, we summarize the results of incorporating lateral friction and a no-slip boundary condition. After a brief statement of the mathematical problem and derivation of the governing equations, we state the results for the boundary propagation speed and variation of wave amplitude along the western and eastern boundaries in three limiting cases corresponding to those studied in section 2. Details of the derivations, which involve the application of singular perturbation theory, are sketched in an appendix.

a. Governing equations

The meridional momentum equation is now
e6.1
and the associated vorticity equation is
e6.2
where v is the viscosity.
For boundary conditions, we impose no slip that, exploiting geostrophy for the meridional flow, can be written
e6.3
and the no-normal flow boundary condition, setting u = 0 in (6.1), takes the form
e6.4
We now seek trial solutions of the form
eq8
The form of this solution, with the sum of two exponentials in x, is required to satisfy the two boundary conditions and may be anticipated from classical boundary current theory (e.g., Munk 1950). Division by the coefficient (1 + γ) is to ensure that A(y) is equal to the wave amplitude on the boundary (at x = 0).
Substituting this trial solution into the vorticity Eq. (6.2) gives the depressed quartic equation:
e6.5
There are four roots, two each for the western and eastern boundary solutions. This can be solved analytically following the method described by Cardano (1968). However, the full analytical solution is complicated and provides little physical intuition. Instead we resort to approximate solutions obtained using singular perturbation theory (see the appendix).
Once we have the solutions for ka and kb, substituting the trial solution into the boundary conditions (6.3) and (6.4) yields
eq9
and
e6.6
The latter condition allows us to determine both the propagation and amplitude variations along the boundary.

b. Summary of solutions

In the appendix, we derive the solutions in three limiting cases:

  1. a Kelvin wave limit in which propagation is cyclonic around each hemispheric ocean basin and at the classical Kelvin wave speed c;
  2. a Rossby wave limit in which propagation is cyclonic and at the speed
    eq10
    where is the width of an oscillatory Stokes viscous boundary layer; and
  3. a Munk boundary layer limit in which propagation is cyclonic and at the speed
    eq11
    where δM = (ν/β)1/3 is the width of the Munk boundary layer (Munk 1950).

In all other respects, the solutions follow those obtained with linear friction in section 2, with the same variations of the wave amplitude along the western and eastern boundaries.

Thus, in the low-frequency limit, the waves are slowed down from the Kelvin wave speed by the ratio of the deformation radius and the pertinent frictional boundary layer width.

c. Energetics and western boundary interactions

We do not present any details here, but note that all of the key results from sections 4 and 5 for the energy fluxes, energy dissipation, and interactions between long and short waves carry over to the case with lateral friction and no-slip boundary conditions. This is not surprising because the amplitudes and the phases of the waves, and hence the energy fluxes, are the same outside the viscous boundary layers.

7. Effect of time-mean flow

To this point, we have analyzed wave solutions to the equations linearized about a state of rest. While it is tempting to suppose that mean flows simply Doppler shift the solutions described above, this need not be the case. For example, the non-Doppler shifting of long Rossby waves by background mean flows in the reduced-gravity model is well established (Liu 1999): physically, a mean flow both Doppler shifts the waves, but also modifies the mean potential vorticity gradient, thereby modifying the intrinsic Rossby wave speed; in the case of long Rossby waves in the reduced-gravity model, these two effects are equal and opposite.

A detailed treatment of mean flows lies beyond the scope of the present manuscript, both because it would require a large number of additional calculations, and equally because the technical challenge of including general mean flows is nontrivial. Nevertheless, here we include one simple, if somewhat extreme, scenario, mainly to illustrate the potential importance of mean flows.

Specifically, we consider the case of a mean, inertial western boundary current within which the potential vorticity is uniform. For this purpose, we employ a geostrophic vorticity model (Schär and Davies 1988; Tansley and Marshall 2000). The equations of motion, including nonlinear accelerations but neglecting dissipation, are as follows:
e7.1
e7.2
e7.3
These can be combined to obtain a potential vorticity conservation equation
e7.4
where
eq12
is the potential vorticity.
Setting the potential vorticity to a uniform value Q0 gives an elliptic equation for the time-mean layer thickness
e7.5
the solution to which is
e7.6
where
e7.7
is closely related to the conventional Rossby deformation radius. The time-mean layer thickness on the boundary is a weak function of latitude because the no-normal flow boundary condition applies to the total, rather than geostrophic, velocity. The solution, sketched in Fig. 4, is the classical Fofonoff gyre (Fofonoff 1954), slightly modified by the nonuniform value of on the boundary.
Fig. 4.
Fig. 4.

Schematic showing the solution for the Fofonoff gyre with uniform potential vorticity, about which linearized wave solutions are obtained in section 7. Sketched are the layer thickness contours, which serve as approximate streamlines for the flow. The layer thickness varies slightly along the boundary because the no-normal flow boundary condition applies to the total, rather than geostrophic, velocity.

Citation: Journal of Physical Oceanography 43, 12; 10.1175/JPO-D-13-0134.1

Because the potential vorticity is uniform, layer thickness anomalies satisfy
e7.8
to which the solution is
e7.9
Note that we have lost the zonal propagation of the short Rossby wave solution and instead have a solution that decays over a scale that is closely related to the classical deformation radius, as in the classical Kelvin wave. This is not surprising for two reasons: (i) Rossby waves rely on the presence of a background potential vorticity gradient that is absent here; (ii) both the present solution and the classical Kelvin wave are associated with vanishing potential vorticity anomalies. Time dependence is left arbitrary in this solution.
To solve for the meridional propagation of the wave, we need to apply the no-normal flow boundary condition linearized about the mean state:
e7.10
where is the mean boundary velocity and
eq13
is the perturbation geostrophic velocity.
Substituting for h′ gives:
e7.11
The general solution is nontrivial. Nevertheless, (7.11) is in flux form where we can identify
e7.12
as the relevant, nondispersive western boundary wave speed. To leading order, this is the classical Kelvin wave speed, Doppler shifted by the mean velocity on the boundary.

While the results of this section apply in just one extreme limit in which the potential vorticity is completely homogenized, the fact that the Kelvin wave solution is recovered should provide a further cautionary note about the propagation speeds of meridional circulation anomalies in OGCMs. In particular, models that fail to resolve eddies and hence the Rossby deformation radius, will also fail to resolve the inertial boundary layers and the mean flows described in this section. It remains to be determined whether this solution is of any relevance to more realistic cases in which there are finite potential vorticity gradients and Rossby waves and also when a no-slip boundary condition is incorporated.

8. Concluding remarks

Boundary waves play a fundamental role in the adjustment of the MOC to changes in surface wind and buoyancy forcing. Yet, this adjustment varies widely across different OGCMs, as reviewed in section 1. In this manuscript, we have analyzed boundary wave solutions along western and eastern boundaries in a reduced-gravity model. Contrary to what is often assumed in studies of MOC adjustment, but consistent with results from the coastal oceanography literature (e.g., Clarke and Shi 1991), boundary propagation occurs not through Kelvin waves, but through short and long Rossby waves at the western and eastern boundaries respectively: these Rossby waves propagate zonally, as predicted by classical theory, and cyclonically along the basin boundaries to satisfy the no-normal flow boundary condition. The along-boundary propagation speed is cLd/δ, where c is the internal gravity/Kelvin wave speed, Ld is the Rossby deformation radius, and δ is the appropriate frictional boundary layer width. This result holds across a wide range of parameter regimes, with either linear friction or lateral viscosity and a no-slip boundary condition. However, we note that this result may not carry over to numerical models in which the lateral resolution is insufficient to resolve the boundary layers; for example, Hsieh et al. (1983) has found sensitivity of the along-boundary propagation speed to the choice of lateral boundary condition in coarsely resolved models.

An important corollary is that the boundary propagation speed in OGCMs is likely to be sensitive to model parameters, offering a plausible explanation for the wide range of model behavior, as well as some of the discrepancies between models run at eddy-permitting and coarser resolution. These results are likely to be modified by the inclusion of realistic bottom topography, higher baroclinic modes, background mean flows (as shown in section 7), and will also differ for the adjustment of the deep limb of the MOC (e.g., Elipot et al. 2013). Nevertheless, we suggest that a cautious approach is apposite when discussing the boundary propagation of MOC anomalies in any one particular model. A proper representation and dynamical understanding of boundary wave propagation speeds is important for understanding internal climate variability, as reviewed in section 1, and may facilitate attempts to develop the capacity for decadal climate prediction in the Atlantic sector (e.g., Latif et al. 2006; Msadek et al. 2010; Robson et al. 2012).

One obvious extension of this work is to explain the lack of coherence of meridional circulation anomalies observed in models across the Gulf Stream over the past several decades (e.g., Bingham et al. 2007; Biastoch et al. 2008; Lozier et al. 2010). While this lack of coherence may be a consequence of the spatial pattern of wind forcing anomalies (e.g., Zhai et al. 2013, manuscript submitted to J. Climate), rather than boundary waves per se, a plausible explanation is that the change in potential vorticity encountered by a western boundary wave as it propagates across the Gulf Stream excites an eastward-propagating frontal wave (e.g., Cushman-Roisin et al. 1993; Cushman-Roisin 1993), to which it loses energy. Results from such a study will be reported in a future manuscript.

Acknowledgments

We are grateful to Sergey Danilov and two anonymous reviewers for helpful comments on a preliminary draft. Financial supported was provided by the U.K. Natural Environment Research Council and a Royal Society University Research Fellowship (HLJ).

APPENDIX

Derivation of Solutions with Lateral Friction and No-Slip Boundary Condition

a. Kelvin wave limit

Nondimensionalize the wavenumber by the deformation radius:
ea1
where henceforth tildes indicate nondimensional variables. The depressed quartic (6.5) becomes
eq14
where
eq15
is the nondimensional Munk viscous boundary layer width with δM = (ν/β)1/3. The nondimensional parameter
eq16
is analogous to λ1/2 in section 2. For Kelvin waves, we require ϵ ≫ 1 analogous to λ ≫ 1 in section 2a.
To ensure that the Kelvin waves are not swamped by lateral friction, we also require . The appropriate choice turns out to be , giving
ea2
First, to obtain the Kelvin wave roots, we expand (A2) in powers of ϵ−1,
eq17
giving at the leading two orders:
eq18
The solutions are
eq19
or, in dimensional form,
ea3
As in section 1, the first root corresponds to the western boundary and the second root to the eastern boundary.
Second, to obtain the viscous boundary layer roots, we expand (A2) in the distinguished limit,
eq20
giving
eq21
The solutions are
eq22
or, in dimensional form,
ea4
This is the classical solution for a viscous Stokes boundary layer in an oscillatory flow.
Finally, we determine the variation of the wave amplitude along the western and eastern boundaries by substituting for ka and kb in (6.6):
eq23
The solution, retaining just the leading terms for the boundary wave propagation and amplitude variation, is
ea5
that is, the wave propagates cyclonically along the boundaries at the classical Kelvin wave speed, c, and the wave amplitude varies as . The latter term represents the frictional decay of the Kelvin wave and is latitude dependent because the importance of friction depends on the relative magnitude of the deformation radius and the viscous boundary layer width, with the deformation radius being larger at low latitudes and thus frictional damping being less important.

b. Rossby wave limit

An alternative expansion that admits a short Rossby wave solution is obtained by nondimensionalizing the wavenumber by the short Rossby wavenumber:
ea6
Then
ea7
where now ϵ ≪ 1,
eq24
and
eq25
is the width of an oscillatory Stokes viscous boundary layer. We also set N = O(ϵ2); clearly the results are only valid when δωδM.
First, to obtain the Rossby wave roots, we expand in ϵ2,
eq26
giving
eq27
The leading terms in the solutions for the western and eastern boundaries are
ea8
that is, the short and long Rossby wave solutions, respectively.
To obtain the remaining roots, we expand in the distinguished limit:
eq28
giving
eq29
The leading terms in the solutions for the western and eastern boundaries are
ea9
corresponding to an oscillatory viscous boundary of width δω. The leading-order real and imaginary terms in the no-normal flow boundary condition for the western solution are
eq30
The solution is
ea10
that is, the propagation speed along the boundary is
ea11
At the eastern boundary, we find
eq31
The solution, exploiting ϵ2 ≪ 1, is
ea12
that is, the propagation speed along the boundary is again
ea13

c. Munk boundary layer limit

Finally, we consider a regime in which a viscous boundary layer solution is obtained at the western boundary. Here, we nondimensionalize the wavenumber by the deformation radius:
ea14
The depressed quartic (6.5) becomes
ea15
where
eq32
is the nondimensional Munk viscous boundary layer width with δM = (ν/β)1/3. We set δ = ~1, which ensures that viscosity has a leading-order impact at the deformation scale.
We now expand in the small parameter ϵ:
eq33
giving
eq34
The solutions for the western boundary are
ea16
ea17
and for the eastern boundary,
ea18
ea19
The western boundary solutions represent the classical viscous boundary layer solution of Munk (1950), extended to include the leading-order correction due to the oscillatory time dependence. The eastern boundary solutions represent the long Rossby wave and no-slip viscous boundary layer solutions, respectively.
The boundary condition gives for the western boundary:
eq35
which takes the same form as the equivalent equation with linear friction. The solution, expanded about a reference latitude, is again
ea20
that is, the propagation speed along the boundary is again
ea21
The boundary condition gives for the eastern boundary:
eq36
to which the solution is
ea22
that is, poleward propagation along the boundary, without any amplitude change, at a speed
ea23

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1

The cross-interaction terms in the wave energy budget only vanish when the wave frequencies are different. Consider, for example, two equal and opposite waves that destructively interfere. Each wave has finite energy, yet the sum of the two waves has zero energy. This paradox is resolved when one realizes that the cross-interaction terms provide a negative energy that exactly cancels the energies of each of the constituent waves.

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