propagation of meridional circulation anomalies along

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Propagation of meridional circulation anomalies along western and eastern boundaries

David P. Marshall ∗

Department of Physics, University of Oxford, United Kingdom

Helen L. JohnsonDepartment of Earth Sciences, University of Oxford, United Kingdom

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 westernand 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 easternboundaries respectively: these Rossby waves propagate zonally, as predicted by classical theory, andcyclonically along the basin boundaries in order 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 lateralviscosity and a no-slip boundary condition. For parameters typical of contemporary ocean climatemodels, the propagation speed is coincidentally close to the Kelvin wave speed. In the limit ofweak dissipation, the western boundary wave dissipates virtually all of its energy as it propagatestowards the equator, independent of dissipation coefficient. In contrast, virtually no energy isdissipated in the eastern boundary wave. The importance of background mean flows is also discussed.

1. Introduction

Wave propagation along western and eastern bound-aries is a fundamental process through which the ocean ad-justs to changes in its boundary conditions (e.g., Wajsow-icz 1986; Kawase 1987, Fig. 1). High frequency variabilitycan propagate large distances along meridional boundariesthrough coastal Kelvin waves, topographic waves or somecombination thereof (Huthnance 1978). However, vari-ability in the meridional overturning circulation (MOC),heat content and regional sea-level change also occurs onseasonal to decadal time scales, on which other forms ofboundary wave may dominate the adjustment process.

The adjustment of the ocean to a change in high lat-itude buoyancy forcing varies widely across ocean generalcirculation models (OGCMs) and coupled climate models.The timescale on which anomalies propagate from high tolow latitudes, the signal amplitude and the inferred mech-anisms all differ significantly. In the subpolar gyre north ofabout 40oN several studies have suggested that advection,rather than boundary wave propagation, dominates the re-sponse of the MOC (e.g., Gerdes and Koberle 1995; Edenand Greatbatch 2003; Getzlaff et al. 2005; Zhang et al.2011). This is perhaps not surprising given the strongmean 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). Further south, however, many stud-ies see equatorward propagation of boundary anomaliesat speeds significantly faster than the advective velocitiesassociated with the deep western boundary current (e.g.,Doscher et al. 1994; Capatondi 2000; Eden and Willebrand2001; Goodman 2001; Dong and Sutton 2002; Getzlaff et al.2005; Roussenov et al. 2008; Chiang et al. 2008; Hawkinsand Sutton 2008; Kohl and Stammer 2008; Zhang 2010;Zhang et al. 2011; Hodson and Sutton 2012, for exceptionssee Marotzke and Klinger (2000) and Buckley et al. (2012))and infer that these are Kelvin wave signals, travellingslower than the first baroclinic gravity wave speed (whichis of order 1 ms−1) due to numerical effects. Anomalies inthese models take between a month (e.g., Dong and Sut-ton 2002) and a decade (e.g., Capatondi 2000) to reach theTropics.

The results of Hsieh et al. (1983) are often invokedto explain these boundary wave propagation differences interms of the effect of model grid and resolution. Other au-thors cite the different forcing mechanisms (e.g., Marotzkeand Klinger 2000; Johnson and Marshall 2002a) as expla-nation. In this paper we analyse the propagation charac-teristics of waves along meridional boundaries in a reduced-gravity model, at both high and low frequency and in thepresence of both linear and lateral friction. Motivated by

1

the disparate results of OGCMs in response to variablebuoyancy forcing at high latitudes, and building on thework previously applied to coastal and tropical applications(e.g., Clarke 1983; Grimshaw and Allen 1988; Clarke andShi 1991), we aim to present a unified view of first baro-clinic mode wave propagation relevant to large-scale oceanadjustment on time scales longer than a few months. Weshow that the propagation speed is not the Kelvin wavespeed, but in fact varies inversely with the pertinent vis-cous boundary layer width; since the latter differs betweenOGCMs, this may offer a partial explanation for the spreadin the propagation speed of Atlantic MOC anomalies ob-served in the studies above and others like them.

Monday, 25 February 13

Fig. 1. Schematic diagram illustrating the role of bound-ary waves in the adjustment of the upper limb of the MOCto localized forcing through changes in deep water forma-tion at high latitudes, or through westward-propagatingRossby waves and eddies impinging on the western bound-ary. The boundary waves propagate cyclonically aroundeach hemispheric basin.

There are four outstanding questions concerning wavepropagation along meridional boundaries that we plan toaddress in this paper (Fig. 1):

• What is the relevant property that propagates alongthe boundary?

• At what speed does it propagate?

• How much energy is dissipated?

• What is the amplitude of western boundary wavesgenerated by incident long Rossby waves from theocean interior, and how much of the incident waveenergy is dissipated?

In the remainder of this section we take each of thesequestions in turn and provide some background contextand motivation. We also discuss the implications and rele-vance for the detection, monitoring and prediction of MOCchange.

a. What propagates?

First mode baroclinic waves propagate an anomaly inpycnocline depth along the boundary, with an associatedsignal in pressure, density, velocity and often temperature.However, the amplitude of the anomaly need not be con-stant along the propagation path. Wajsowicz and Gill(1986) show that Kelvin waves on an f plane are atten-uated by friction, and that their amplitude decays as a re-sult. Johnson and Marshall (2002a) illustrate in a reduced-gravity model that, even in the absence of friction, whenf varies with latitude the amplitude of a western bound-ary wave is reduced during equatorward propagation. Onthe eastern boundary they see no corresponding increase inwave amplitude with latitude; instead long Rossby wavesare radiated into the ocean interior. This asymmetry be-tween eastern and western boundaries is a key element oftheir ”equatorial buffer” mechanism.

There is a substantial body of literature on the natureof eastern boundary wave propagation, and the radiationof long Rossby waves. This has originated largely from thecoastal oceanography community, motivated by the anal-ysis of coastal sea level observations, and the propagationof ENSO-related signals out of equatorial regions. Clarke(1983) showed that motion at the eastern boundary of anocean basin can be described in terms of either westwardpropagating long Rossby waves, or coastal Kelvin waves,with a critical latitude-dependent frequency below whichenergy is radiated into the interior, rather than coastally-trapped. He confirmed Moore’s (1968) result that the am-plitude of a Kelvin wave propagating on a beta plane isproportional to

√y 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 intoaccount the coastline orientation (Schopf et al. 1981), Clarkeand Shi (1991) calculate critical frequencies for the world’socean boundaries which range from a month or two in thetropics to a year at mid-latitudes. Grimshaw and Allen(1988) note that a consequence of these two regimes isthat, while at high frequencies the amplitude of coastaldisturbances varies as

√y on both eastern and western

boundaries, at lower frequencies this dependence changes.McCalpin (1995) finds that on eastern boundaries at lowfrequency the amplitude of the wave disturbance is in factconstant along the eastern boundary, consistent with theearlier findings of Clarke (1983) and Grimshaw and Allen(1988).

A change in amplitude of wave signals as they propagate

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along meridional boundaries has important consequencesfor the diagnosis of waves in ocean models. Due to thecoarse spatial and temporal resolution of typical OGCMoutput, together with the presence of a background meanflow and mesoscale eddy field, the propagation of anomaliesalong the boundary can be hard to detect. Many OGCMstudies calculate propagation speed based on the arrivaltime of a given amplitude of anomaly at a particular lat-itude (e.g., Marotzke and Klinger 2000). This may wellbe a dangerous strategy since one cannot infer wave speedthrough tracking a contour of density if the amplitude ischanging as the wave propagates. An understanding ofhow we expect wave amplitudes to change along easternand western boundaries, as a function of frequency, mayhelp us to more clearly identify these signals in models. Itwill also have implications for the detection and monitoringof change in the ocean.

Note that basin mode theory (e.g., Liu et al. 1999; Cessiand Louazel 2001; Cessi and Paparella 2001; Cessi andPrimeau 2001; Primeau 2002) assumes the amplitude ofanomalies to be uniform along the eastern boundary at alltimes. This will be discussed in more detail in the followingsection.

b. At what speed?

Baroclinic coastal Kelvin waves propagate at the inter-nal gravity wave speed c =

√grh, where gr is the reduced

gravity and h the equivalent depth of the baroclinic mode.Any signal propagation occurring on the western boundaryof ocean models on timescales of months to a decade is gen-erally assumed to be the result of Kelvin waves, slowed bylow stratification (Greatbatch and Peterson 1996) or nu-merical effects, especially on a B grid (Hsieh et al. 1983).At high frequencies this may well be the correct interpre-tation, but the physical argument outlined in the previoussection suggests that at frequencies significantly lower thanω = βLd = βc/f (i.e. on timescales longer than about 2months) coastally trapped Kelvin wave propagation is nolonger expected on either eastern or western boundaries.This does not mean that propagation along the boundaryis prohibited; simply that it need not occur at the gravitywave speed, and need not be coastally trapped.

Primeau (2002) suggested that Kelvin waves play norole in the adjustment of the ocean to a change in forcingat all but the shortest of timescales. He concluded that,instead, internal gravity waves act to distribute pressureanomalies instantaneously along basin boundaries. This isconsistent with the results of Clarke (1983) and Grimshawand Allen (1988), both of whom point out that, in theabsence of friction, there is no phase propagation alongeastern boundaries, with all points on the boundary re-sponding to an incoming disturbance simultaneously. Cessiand Otheguy (2003) come to similar conclusions; in fact,this instantaneous redistribution of anomalies uniformly

along the eastern boundary is central to the theory of basinmodes. Clarke (1983) also considered the inclusion of lineardamping in each of the momentum and continuity equa-tions, obtaining a finite propagation speed proportional tothe damping coefficient with linear friction in the conti-nuity equation, but still synchronous variation along theboundary with linear friction in the momentum equation.

Note that in closed basin quasi-geostrophic (QG) mod-els, a ”consistency condition” (McWilliams 1977) is used toensure global mass conservation, determining the bound-ary value of the streamfunction which is constant alongthe boundary but varies in time. This instantaneous ad-justment is the QG model equivalent of the coastal Kelvinwave and allows for the radiation of long Rossby wavesinto the interior (see, e.g., Milliff and McWilliams 1994;McCalpin 1995).

Other boundary propagation mechanisms besides nu-merical Kelvin waves and advection have of course beenproposed. Killworth (1985) and Winton (1996) both dis-cuss propagation by viscous boundary waves, associatedwith the effect of friction at the side wall. In Killworth(1985) the divergence associated with the wave at the coastoccurs over one grid box due to the no normal flow condi-tion, and the wave propagates at a speed c2/f∆x, where∆x is the grid resolution in the zonal direction. In the pres-ence of a sloping boundary and/or a continental shelf, to-pographically trapped wave modes as well as hybrid wavesbecome possible (Huthnance 1978; Elipot et al. 2013) andcan travel significantly faster than first baroclinic modeKelvin waves (O’Rourke 2009). We restrict our attentionto the vertical sidewall case in this paper, since even thisexhibits rich behaviour.

We will show analytically in Section 2, using the sameequation set that results in pure Kelvin wave propaga-tion at high frequencies, that at low frequencies the along-boundary propagation speed depends strongly on the fric-tional parameters of the model, and hence varies betweenOGCMs. These low frequency waves may in some modelstravel along the boundary at approximately Kelvin wavespeeds (e.g., Yang 1999; Johnson and Marshall 2002a) butthis is simply fortuitous. This model dependence is a wor-rying state of affairs since we rely on these OGCMs, cou-pled to atmospheric circulation models, to predict the rateat which the ocean will respond to future forcing changesassociated with increased greenhouse gas concentrations.Correctly representing the ocean adjustment mechanismsto anticipated changes at high latitudes is essential if weare to make accurate projections of global and particularlyregional climate.

A proper representation of boundary wave propagationspeeds is also important for our understanding of inter-nal climate variability. Several recent studies suggest thatthe ocean exhibits damped, decadal oscillations that arestochastically excited by atmospheric forcing (e.g., Great-

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batch and Peterson 1996; Cessi and Louazel 2001; Cessiand Paparella 2001; Eden and Greatbatch 2003; Dong andSutton 2005; Danabasoglu 2008; Frankcombe et al. 2009;Czeschel et al. 2010; Danabasoglu et al. 2012; Kwon andFrankignoul 2012; Delworth and Zeng 2013)). These modesof variability can have significant climate fingerprints (e.g.,Knight et al. 2005; Hurrell et al. 2006; Zhang and Del-worth 2006; Mahajan et al. 2011). The mechanisms andtimescales underlying the oscillations differ from one modelto another, with boundary adjustment processes often im-plicated (e.g., Greatbatch and Peterson 1996; Hawkins andSutton 2007). As a result of the diversity of model be-haviour there is little consensus on the causes of decadal tomulti-decadal variability in the real climate system. Un-derstanding the mechanisms and timescales involved wouldfacilitate attempts to develop the capacity for decadal cli-mate 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 hasimplications beyond the ocean basin in which a change inforcing occurs. In response to high latitude forcing in theNorth Atlantic, for example, fast boundary waves, togetherwith equatorial Kelvin waves, carry the signal of a thermo-cline displacement to all the major ocean basins on rel-atively short timescales (e.g., Huang et al. 2000; Good-man 2001; Cessi and Otheguy 2003; Johnson and Mar-shall 2004). While the amplitude of variability on anythingshorter than multi-decadal timescales may be small outsidethe initial basin (Johnson and Marshall 2002b), the impli-cations for, e.g., sea level (Hsieh and Bryan 1996; Landereret al. 2007) are significant. Fast atmospheric teleconnec-tions via tropical air-sea interactions may also result in aglobal response (e.g., Dong and Sutton 2002).

c. Energetics and western boundary interactions

The mechanical energy budget of the global ocean cir-culation 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 ofenergy loss from the geostrophic modes including bottomdrag (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 geostrophicflow over rough topography (Marshall and Naveira Gara-bato 2013; Nikurashin and Ferrari 2010; Nikashurin et al.1998). Recently Zhai et al. (2010) have demonstrated inidealized model calculations and altimetric data that thereis a significant sink of geostrophic eddy energy when west-ward propagating eddies (e.g., Chelton et al. 2007) reachthe western boundary — the so-called “Rossby graveyard”.In the model calculations, the amount of energy dissipa-tion appears to be independent of model resolution and ofthe magnitude and nature of momentum dissipation. Inthe ocean, it is unclear how much of the energy is merely

backscattered to the mean flow (cf. Starr 1968) or whetherthe energy is dissipated, e.g., through vertically-breakingboundary waves (Dewar and Hogg 2010).

Irrespective of how energy is dissipated in the ocean,in coarse-resolution models, it is likely that a significantamount of energy is dissipated in western boundary waves.In section 4, we show that this rate of energy dissipationis independent of the magnitude of the local friction and,in section 5, we show that the vast majority of the en-ergy incident on the western boundary in the form of longRossby waves is dissipated locally on reaching the westernboundary.

The energy dissipation implicit in long waves reachinga western boundary is intimately related to the strong re-duction in dynamic height variability at the western marginof an ocean basin. This reduction is crucial in preventingMOC variations inferred from boundary hydrographic andother data (Cunningham and Coauthors 2007; Kanzow andCoauthors 2007) from being swamped by eddy variability(Wunsch 2008). This issue has been previously discussedusing a stripped-down version of the linear wave solutionspresented in this manuscript (Kanzow et al. 2009) but nev-ertheless serves as an important motivation for the presentstudy.

d. Structure of the paper

The paper is structured as follows. In section 2 we solvefor the linear boundary waves in the presence of a merid-ional boundary and linear friction. In section 3 we discussthe behavior of meridional transport anomalies along west-ern and eastern boundaries. In section 4 we analyse the en-ergetics of western and eastern boundary waves. In section5 we study the western boundary waves generated throughincident long waves and the location and magnitude of en-ergy dissipation associated with these interactions. In sec-tion 6 we solve for linear boundary waves in the presenceof lateral friction and a no-slip boundary condition. In sec-tion 7 we investigate the impact of background mean flowon the boundary wave solutions. Finally, in section 8 wepresent a brief concluding discussion.

2. Wave solutions with linear friction

We consider solutions to the reduced-gravity equations,linearised about a state of rest. We adopt a semigeostrophicapproximation (Hoskins 1975) in which we assume geostrophicbalance holds in the zonal momentum equation, but not inthe meridional momentum equation where we know thatthe Coriolis acceleration vanishes at the boundary.

4

Thus:

−βyv′ + gr∂h′

∂x= 0, (2.1)

∂v′

∂t+ βyu′ + gr

∂h′

∂y= −rv′, (2.2)

∂h′

∂t+ h0

(∂u′

∂x+∂v′

∂y

)= 0. (2.3)

Here u and v are the velocities in the zonal (x) and merid-ional (y) directions, h is the layer thickness, gr is the re-duced gravity and r is the coefficient of linear friction. Weadopt a β-plane approximation such that the Coriolis pa-rameter is f = βy. Primes indicate linear perturbationsand zero subscripts the background mean state.

For simplicity, we restrict our attention to north-southcoastlines located at x = 0 (Fig. 2), along which a no-normal flow boundary condition is applied:

u′ = 0 (x = 0). (2.4)

Thus western boundary solutions correspond to the half-plane x ≥ 0 and eastern boundary solutions to the half-plane x ≤ 0 (Fig. 2). The orientation of the boundarieshas some influence on the solution (e.g., Grimshaw andAllen 1988).


We consider solutions to the reduced-gravity equations, linearised about a state of rest.We adopt a semigeostrophic approximation (Hoskins and Bretherton, 1972) in whichwe assume geostrophic balance holds in the zonal momentum equation, but not in themeridional momentum equation where we know that the Coriolis acceleration vanishesat the boundary. Thus:

−βyv + g′∂h

∂x= 0, (1)

∂v

∂t+ βyu + g′∂h

∂y= −rv, (2)

∂h

∂t+ h0

(∂u

∂x+

∂v

∂y

)= 0. (3)

Here u and v are the velocities in the zonal (x) and meridional (y) directions, h is thelayer thickness, g′ is the reduced gravity and r is the coefficient of linear friction. Weadopt a β-plane approximation such that the Coriolis parameter is f = βy.

For simplicity, we will restrict out attention a north-south coastlines which we place atx = 0. We apply a no-normal flow boundary condition:

u = 0 (x = 0). (4)

Western boundary solutions correspond to the half-plane x ≥ 0 and eastern boundarysolutions to the half-plane x ≤ 0.

From these equations, we can derive the vorticity equation:

∂

∂t

(∂2h

∂x2− h

L2D

)+ β

∂h

∂x+ r

∂2h

∂x2= 0, (5)

where

LD(y) =

√g′H

βy

is the Rossby deformation radius.

The no-normal flow boundary condition, substituted in (2), takes the form:

∂2h

∂t∂x+ βy

∂h

∂y+ r

∂h

∂x= 0 (x = 0). (6)

Following Clarke and Shi (1987), we now seek solutions of the form

h = A(y) ek(y)xe−iωt, (7)

1

Generalised semi-geostrophic model (e.g., Clarke 1983, Clarke and Shi 1991)

- include linear friction: analytically simple and sets boundary propagation speed

- have also derived singular perturbation solutions with lateral friction and no-slip

Boundary condition:



−βyv + g′∂h

∂x= 0, (1)

∂v


∂y= −rv, (2)

∂h

∂t+ h0

(∂u

∂x+

∂v

∂y

)= 0. (3)



u = 0 (x = 0). (4)



∂

∂t

(∂2h

∂x2− h

L2D

)+ β

∂h

∂x+ r

∂2h

∂x2= 0, (5)

where

LD(y) =

√g′H

βy



∂2h

∂t∂x+ βy

∂h

∂y+ r

∂h

∂x= 0 (x = 0). (6)



1

3.4 Kelvin waves

In a Rossby wave, the flow is in geostrophic balance to first approximation.

However, there are two circumstances under which a pressure gradient

cannot be balanced by a Coriolis force:

• along a coastline

• along the equator

!hu =0

!

hf=0

Instead, such pressure gradients result in Kelvin waves which propagate

along coastlines and the equator.

a. Coastal Kelvin waves

Close to a meridional coastline, we can anticipate that u ≈ 0, giving:

−f0v + g′∂h

∂x= 0, (3.10)

∂v

∂t+ g′∂h

∂y= 0, (3.11)

∂h

∂t+ h0

∂v

∂y= 0. (3.12)

Note that we have assumed f = f0 is constant.

Firstly taking∂

∂t(3.12) − h0

∂

∂y(3.11)

FGDF 3-8

3.4Kelvinwaves

InaRossbywave,theflowisingeostrophicbalancetofirstapproximation.

However,therearetwocircumstancesunderwhichapressuregradient

cannotbebalancedbyaCoriolisforce:

•alongacoastline

•alongtheequator

!h u=0

!h

f=0

Instead,suchpressuregradientsresultinKelvinwaveswhichpropagate

alongcoastlinesandtheequator.

a.CoastalKelvinwaves

Closetoameridionalcoastline,wecananticipatethatu≈0,giving:

−f0v+g′∂h

∂x=0,(3.10)

∂v

∂t+g′∂h

∂y=0,(3.11)

∂h

∂t+h0

∂v

∂y=0.(3.12)

Notethatwehaveassumedf=f0isconstant.

Firstlytaking∂

∂t(3.12)−h0

∂

∂y(3.11)

FGDF3-8



−βyv + g′∂h

∂x= 0, (1)

∂v


∂y= −rv, (2)

∂h

∂t+ h0

(∂u

∂x+

∂v

∂y

)= 0. (3)



u = 0 (x = 0). (4)



∂

∂t

(∂2h

∂x2− h

L2D

)+ β

∂h

∂x+ r

∂2h

∂x2= 0, (5)

where

LD(y) =

√g′H

βy



∂2h

∂t∂x+ βy

∂h

∂y+ r

∂h

∂x= 0 (x = 0). (6)



1



−βyv + g′∂h

∂x= 0, (1)

∂v


∂y= −rv, (2)

∂h

∂t+ h0

(∂u

∂x+

∂v

∂y

)= 0. (3)



u = 0 (x = 0). (4)



∂

∂t

(∂2h

∂x2− h

L2D

)+ β

∂h

∂x+ r

∂2h

∂x2= 0, (5)

where

LD(y) =

√g′H

βy



∂2h

∂t∂x+ βy

∂h

∂y+ r

∂h

∂x= 0 (x = 0). (6)



1

westernboundary

easternboundary


Fig. 2. Schematic diagram showing the domain consideredfor the western and eastern boundary wave solutions.

From (2.1-2.3) we can derive a vorticity equation:

∂

∂t

(∂2h′

∂x2− h′

L2d

)+ β

∂h′

∂x+ r

∂2h′

∂x2= 0, (2.5)

where

Ld(y) =

√grh0βy

=c

βy(2.6)

is the Rossby deformation radius and c is the gravity wavespeed.

The no-normal flow boundary condition (2.4), whensubstituted in (2.2) and using (2.1), takes the form:

∂2h′

∂t∂x+ βy

∂h′

∂y+ r

∂h′

∂x= 0 (x = 0). (2.7)

Following Clarke and Shi (1991), we now seek solutionsof the form

h′ = A(y) eik(y)xe−iωt, (2.8)

where real parts is understood, and we anticipate that thezonal wave number, k(y), varies with latitude due to thevariation of the Coriolis parameter and deformation radiuswith latitude; for example, to allow the zonal decay scaleof coastal Kelvin waves to vary with latitude.

Substituting this trial solution into (2.5) gives

(ω + ir)k2 + βk +ω

L2d

= 0, (2.9)

to which the general solution is:

k =β

2(ω + ir)

(−1∓

√1− λ

)(2.10)

where

λ =4ω(ω + ir)

β2L2d

=4ω(ω + ir)

c2y2. (2.11)

The convention we adopt in this and subsequent equationsis that the first root corresponds to the western bound-ary solution and the second root to the eastern boundarysolution.

Substituting the trial solution into the boundary con-dition (2.7) gives an equation for the amplitude variationalong the boundary:

dA

dy= − (ω + ir)

βyk A. (2.12)

a. Kelvin wave limit (λ� 1, r � ω)

First we consider the weakly-damped Kelvin wave limit,λ � 1, r � ω, equivalent to ω � βLd/2. Typical valuesof β ∼ 2 × 10−11 m−1 s−1 and Ld ∼ 5 × 104 m gives ω �5 × 10−7 s or periods much shorter than 3-4 months (onwhich time-scales damping can safely be assumed weak).

1) Zonal structure

The leading order solution for the zonal wavenumber is:

k =iβ

2(ω + ir)λ1/2

{±√

1− λ−1 + iλ−1/2}

=iβ

2(ω + ir)λ1/2

{±1 + iλ−1/2 + · · ·

}

=i

Ld

(1 +

ir

ω

)−1/2 {±1 + iλ−1/2 + · · ·

}

= ± i

Ld± r

2ωLd− β

2ω+ · · · , (2.13)

where we have neglected O(λ−1) and O(r2/ω2) terms inthe expansion.

Thus, to leading order:

h′ ≈ A(y)e∓x/Lde−iωt; (2.14)

the Kelvin wave decays away from the boundary over thedeformation radius, Ld, as expected, but note that the de-formation radius decreases with increasing latitude.

5

2) Meridional structure

The real component of k is of little consequence for thezonal structure of the wave, but has an important impacton its meridional structure and propagation. Substituting(2.13) into the boundary condition (2.12) gives:

dA

dy=

{∓ iωc± r

2c+

1

2y+ · · ·

}A.

The leading-order solution for A is:

A ≈ A0

√y

y0e∓iωy/ce±ry/2c. (2.15)

In contrast to the f -plane limit, the wave amplitude (inh) varies as the square root of latitude, y; this result wasfirst derived by Moore (1968). The wave propagates cy-clonically around the coastline at the internal gravity wavespeed, c, and decays due to friction over a lengthscale 2c/r.

b. Rossby wave limit (λ� 1, r � ω)

Now consider the low frequency limit, λ � 1. Theanalysis presented below applies irrespective of the rela-tive magnitudes of ω and r. We first discuss the Rossbywave limit, ω � r, and defer discussion of the Stommelboundary layer limit, ω � r, until section 2c.

1) Zonal structure

The leading-order expression for the zonal wavenumber,(2.10), is now:

k =β

2(ω + ir)

{−1∓

(1− λ

2+ · · ·

)}

≈ − β

(ω + ir)

(1− λ

4

), − ω

βL2d

, (2.16)

where (2.11) has been used to substitute for λ in the finalexpression. The first, western boundary root correspondsto a short Rossby wave solution, damped over a zonal scaleω2/rβ. The second, eastern boundary root corresponds toa long Rossby wave solution, independent of the size of rat leading order. Physically, as the angular frequency isreduced, short Rossby waves become shorter and eventu-ally find the frictional boundary layer scale, r/β (Stommel1948), whereas long Rossby waves become longer and fric-tion remains unimportant.

2) Meridional structure

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

First for the western boundary short-wave solution, wefind:

dA

dy=

{1

y− ω(ω + ir)y

c2+ · · ·

}A.

The solution is

A = A0y

y0e−ω

2y2/2c2e−iωry2/2c2 + · · ·

≈ A0y

y0e−iωry0(y−y0)/c

2

. (2.17)

Here we have set e−ω2y2/2c2 ≈ 1 since ω2y2/c2 � 1, and

Taylor expanded in y about a mean latitude, y0, absorbingthe leading term in the Taylor expansion into the complexphase, A0.

Thus the amplitude of the western boundary short wavesolution decays linearly with decreasing latitude and prop-agates equatorward along the boundary at a speed

cw =c2

βy0δS= c

LdδS

(2.18)

where δS = r/β is the Stommel frictional boundary layerwidth (Stommel 1948). Thus, the boundary wave propa-gates at the i classical Kelvin wave speed, scaled by param-eter the parameter Ld/δS . We write (2.18) in this form be-cause, as we shall see in due course, the same result emergesacross many regimes, including with lateral friction and ano-slip boundary condition, provided that δS is replacedby the pertinent frictional boundary layer width.

For a typical ocean climate model the frictional bound-ary layer width is chosen to be comparable to the gridspacing, which in turn is coincidentally close to the Rossbydeformation radius. Thus the boundary anomalies can beexpected to propagate at roughly the classical Kelvin wavespeed, but the boundary propagation speed is likely to varybetween different models and with latitude.

For the eastern boundary wave, we instead have

dA

dy=ω(ω + ir)y

c2A+ · · · ,

to which the leading-order solution is

A = A0eiωry2/2c2 + · · · (2.19)

≈ A0eiωry0(y−y0)/c2 . (2.20)

We have again assumed ω2y2/c2 � 1 and Taylor expandedthe solution about a reference latitude. In contrast to thewestern boundary solution, the amplitude A does not vary,at leading order, along the boundary. The wave propagatespolewards at the same speed as the western boundary so-lution,

ce = cLdδS. (2.21)

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

The preceding analysis in section b also applies in thelimit ω � r, which we briefly comment on here. As the fre-quency decreases, short Rossby waves become increasingly

6

shorter, such that friction becomes increasingly important,whereas long Rossby waves become increasingly longer andfriction remains unimportant. This is clear when one con-siders the solution for the western boundary wave in section2b. If r � ω, then

k ≈ iβ

r − iω

{1− λ

4+ · · ·

}≈ iβ

r, (2.22)

andh′ ≈ A(y)e−x/δSe−iωt. (2.23)

The solution is a frictional western boundary layer of widthδS (Stommel 1948), with a flow direction that oscillatesin time (also see Pedlosky 1965). As in section 2b, thesolution propagates along the boundary at the speed givenin (2.18).

3. Meridional transport anomalies

Meridional volume transport anomalies are related tothe layer thicknesses on the western, hw, and eastern, he,boundaries by

T ′ =

∫h′v′ dx =

grh0βy

(h′e − h′w).

These can be interpreted as meridional overturning trans-port anomalies associated with each baroclinic mode in abasin of uniform depth and stratification.

Western boundary wave solutions all decay rapidly inthe zonal direction. Hence meridional transport anomaliesassociated with these western boundary waves,

T ′w = − cβ

h′wy, (3.1)

are confined to the western boundary region.Conversely, eastern boundary waves do not decay (to

leading order) as they cross the basin. Nevertheless, itmakes sense mathematically to define the meridional trans-port anomalies associated with the eastern boundary wavesas

T ′e =c

β

h′ey

; (3.2)

this decomposition also makes physical sense if we antic-ipate a result from section 5 that the layer thickness (orpressure) anomalies associated with long waves are vastlyreduced in the vicinity of the western boundary (also seeKanzow et al. 2009).

a. Kelvin wave limit

In the Kelvin wave limit, layer thickness anomalies onboth western and eastern boundaries are proportional to√y. Thus the meridional transport anomalies vary as

|T ′w|, |T ′e| ∝1√y, (3.3)

i.e., the meridional transport anomalies are largest at lowlatitudes.

b. Rossby wave and Stommel boundary layer limits

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

On the western boundary, the layer thickness anomaliesare proportional to y, and hence

|T ′w| ≈ constant. (3.4)

In contrast, on the eastern boundary, the layer thicknessanomalies are independent of y, and hence

|T ′e| ∝1

y, (3.5)

i.e., they decay with increasing latitude as the wave prop-agates poleward.

This asymmetry is consistent with the theoretical modelof Johnson and Marshall (2002) for the adjustment of theMOC to forcing anomalies and, in particular, to the “equa-torial buffer’ mechanism introduced to explain the asym-metry between the response of the MOC to high latitudelocalized forcing on each side of the equator, and betweenthe western boundary layers and basin interior.

4. Energetics

We now turn to the energy budget of the wave solu-tions. The purpose of this short section is to demonstratea fundamental asymmetry between the western and east-ern boundary waves, with the majority of the energy beingdissipated in the former but conserved in the latter. Sub-sequently in section 5, we consider the generation of shortwaves at the western boundary by long waves incident fromthe ocean interior.

The energy equation is derived from the equations ofmotion (2.1-2.3):

∂

∂t

(h0v′2

2+grh′2

2

)+∇ · (c2 h′u′) = −h0rv′2. (4.1)

Thus wave energy flux in (4.1) is c2h′u′. This energy fluxis degenerate in the sense that any rotational gauge canbe added to the flux without changing the energy equation(e.g., Longuet-Higgins 1964; Pedlosky 1987; Orlanski andSheldon 1993; Chang and Orlanski 1994). For statistically-steady wave solutions, the time derivative vanishes and thedivergence of the energy flux is balanced by frictional en-ergy dissipation.

We now consider the dominant terms in this energybalance for each of the wave solutions derived in section2. In the derivation of these wave energy fluxes, recall

7

that real parts of h′, u′ and v′ are understood before tak-ing quadratic products. From the momentum equations(2.1) and (2.2), the form of the trial solution (2.8), and theboundary condition (2.12), it follows that:

v′ =ikgrβy

h′, (4.2)

and

u′ = − grβy

(1

A

dA

dy+ ix

dk

dy+

(ω + ir)

βyk

)h

= − igrxβy

dk

dyh′. (4.3)

In the case of the meridional energy flux, integrating h′×(2.1) further gives:

∫h′v′ dx =

gr2βy

[h′2], (4.4)

where the integral is across the relevant western or easternboundary wave. These relations are valuable in identifyingthe components of u′ and v′ that are in phase (or anti-phase) 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:

∫c2 h′v′ dx = ∓c

2gr|A0|24βy0

, (4.5)

independent of latitude. Hence, in the Kelvin wave solu-tion, the variation of the wave amplitude on the boundaryas the square root of latitude can be interpreted as beingprecisely what is needed to conserve wave energy (Moore1968). Finite friction merely modifies this flux to accountfor the energy dissipated.

The zonal energy flux, obtained by substituting (2.13)in (4.3), is:

c2h′u′ = ±cgr|A0|22y

xe∓2x/Ld . (4.6)

Note that this vanishes both on the boundary at x = 0 andas x → ±∞. Thus the zonal energy flux is precisely thatrequired to broaden/narrow the Kelvin wave as it propa-gates equatorward/poleward but vanishes in the zonally-integrated energy budget.

b. Short Rossby wave/Stommel boundary layer

The zonal energy flux, obtained by substituting thewestern boundary solution in (2.16) into (4.3), is:

c2h′u′ = 0, (4.7)

since h′ and u′ are out of phase. In contrast to the Kelvinwave, the wavelength of the short Rossby wave/width ofthe Stommel boundary layer is independent of latitude andhence there is no need for a zonal energy flux.

The net meridional energy flux, using (4.4) and notingh′ → 0 as x→∞, is:

∫c2 h′v′ dx = −c

2gr|As|24βy

= −c2gr|A0|24βy20

y. (4.8)

The meridional energy flux is equatorward and decreaseslinearly with decreasing latitude. This energy is lost todissipation at a rate

∫h0rv′2 dx =

c2gr|A0|24βy20

, (4.9)

independent of latitude. Note that the dissipation is alsoindependent of the drag coefficient, r. Physically, as r isreduced, the pointwise energy dissipation reduces, but oc-curs over a proportionately larger area as the short Rossbywaves propagate further away from the boundary; theseeffects compensate at leading order.

In reality, this expression must break down close tothe equator, when y approaches the equatorial deforma-tion radius. Thus, a small amount of energy can be ex-pected to leak into an equatorial Kelvin wave, but this isa small residual of the meridional energy flux within theshort Rossby wave at higher latitudes.

c. Long Rossby wave

In the long Rossby wave solution, both u′ and v′ are outof phase with h′ and hence both the zonal and meridionalenergy fluxes vanish, as does the dissipation at leading or-der. As mentioned above, the energy flux is degenerateto an arbitrary rotational flux; thus the result that theenergy fluxes vanish in the long wave solution should beinterpreted an indicating only that there is no divergenceof the energy flux.

5. Long waves incident on a western boundary

a. Statement of problem

We now consider a problem of particular relevance tothe ocean energy budget and to monitoring the meridionaloverturning circulation: the fate of long Rossby waves in-cident on a western boundary and the amplitude of theboundary wave anomaly excited on the western boundary.This problem is relevant to the ocean energy budget sinceanalysis of satellite altimeter data and numerical model cal-culations suggests that the western boundary current actsan “eddy graveyard” and eddy energy sink (Zhai et al.2010). Moreover, associated with this eddy energy sink isa reduction in the amplitude of dynamic height variabilityat the western boundary, also consistent with altimetric

8

and hydrographic observations. The latter turns out to becrucial for end-point monitoring of the MOC (Cunninghamand Coauthors 2007), as discussed by (Kanzow et al. 2009,also see Wunsch 2008).

We consider a long Rossby wave incident on a westernboundary over a confined latitude band, ∆y. Since thewave is long relative to the short wave it will excite, itsuffices to model the incoming wave through a prescribed(complex) wave amplitude that varies with latitude andtime,

h′l = Al(y)e−iωt.

On reaching the western boundary, a short wave or fric-tional boundary layer solution,

h′s = As(y)e(iksx−ωt).

is excited in order to satisfy the no-normal flow westernboundary condition.

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

b. Boundary amplitude condition

The boundary condition (2.7) now applies to the sumof the long and short wave solutions and can be written:

d

dy(As +Al) = − (ω + ir)

βyksAs. (5.10)

Note that the term involving Al on the right hand sideof (5.10) is neglected because we have assumed that thewavenumber of the incoming long wave is vanishingly small,|kl| � |ks| (and we anticipate As and Al are of similarmagnitude).

Substituting for ks gives:

d

dy(As +Al) =

Asy

+ · · · . (5.11)

As in Kanzow et al. (2009) we can use the identity

d

dy

(As +Al

y

)=

1

y

d

dy(As +Al)−

(As +Al)

y2

to rewrite this result as

d

dy

(As +Al

y

)= −Al

y2+ · · · . (5.12)

Finally integrating (5.12) over the latitude of the incominglong wave and integrating the result by parts, we obtain

As +Al = y

∫Aly2dy + · · ·

= y

∫dAly

+ · · ·

∼ ∆y

yAl. (5.13)

Thus the wave amplitude on the boundary, As + Al, is afactor ∆y/y = β∆y/f smaller than the amplitude of theincoming long wave, Al.

Note that equatorward of the incoming long wave, (5.12)reduces to the result that As/y is constant as found for asolitary short wave.

c. Energetics

The foregoing analysis raises a number of questionsabout the energetics of the interaction between the incom-ing long wave and the short wave generated at the westernboundary:

• What fraction of the incoming energy reflects zonallyas a short wave?

• What fraction of the energy is fluxed equatorwardalong the boundary?

• What fraction of the energy is dissipated by friction?

The natural way to pose the problem is to evaluate theenergy flux of the incoming long wave, grh0hlul, the (zonaland meridional) energy flux of the reflected short wave,grh0hsus, and the energy dissipation in the short wave,−h0rv2s . However, for two waves of the same frequency,as considered here, both the wave energy and energy fluxare modified by the cross terms representing the interfer-ence of the two waves.1 Thus, even the concept of shortand long wave energies, and short and long wave fluxes, isproblematic. However, these difficulties can be avoided ifone integrates over suitable areas such that the problematicterms vanish; this is the approach we shall follow here.

Firstly, the incident long wave energy flux is

F(x)l =

∫c2gr4βy

d

dy|Al|2 dy

=c2gr4β

∫ |Al|2y2

dy. (5.14)

Here we have used the fact that (2.2) reduces to geostrophyin the limit vl → 0 (equivalently kl → 0) and integrated theresultant expression by parts with Al = 0 at the limits ofintegration. We also note that the final expression is equalto the integral of the long wave energy multipled by thelong Rossby group velocity: the latter does not generallyhold for this form of the energy flux, but falls out here asa consequence of Al vanishing at the limits of integration.Note that the first line of this expression is also consistent

1The cross-interaction terms in the wave energy budget only van-ish when the wave frequencies are different. Consider, for example,two equal and opposite waves that destructively interfere. Each wavehas finite energy, yet the sum of the two waves has zero energy. Thisparadox is resolved when one realises that the cross interaction termsprovide a negative energy that exactly cancels the energies of each ofthe constituent waves.

9

reflected short wave

yΔ(x) F(x)

F(x)

s

s

l

Fs(y)

F

Firstly, the incoming long wave energy flux is

F(x)l = −

∫g′h0hlul dy =

∫ c2g′

4βy

d

dy|Al|2 dy =

c2g′

4β

∫ |Al|2y2

dy.

Here we have used the fact that Al = 0 at the limits of the integration. We also notethat the final expression is equal to the integral of the long wave energy multipled by thelong Rossby group velocity (the latter does not generally hold for this form of the energyflux, but falls out here as a consequence of Al vanishing at the limits of integration).

The reflected short-wave energy flux can be shown to be

F(x)s =

∫g′h0hsus dy = −

∫c2g′y

4β

d

dy

(|As|2y2

)e−2βrx/(ω2+r2) dy

=c2g′

4βe−2βrx/(ω2+r2)

∫ |As|2y2

dy −[c2g′|As|2

4βye−2βrx/(ω2+r2)

].

Again, we caution that one must also calculate the energy flux associated with the crossterms between the short and long waves. Nevertheless it is clear that as x increases,

F(x)s → 0.

(The cross interaction terms also tend to zero.) Physically, this is simply stating thatit the short waves are confined to a small region close to the western boundary, hereof order (ω2 + r2)/βr, then outside of this region the short waves, and the short waveenergy flux, becomes vanishingly small.

However, we you calculate the energy fluxes, they are all zero to leading order, exceptfor the southward flux in the short wave solution. This appears to be a well-documented

3

incident long wave

F(x)s

F(x)l

F(y)s

�y


Fig. 3. Schematic diagram showing the energy fluxes in-volved when westward-propagating long Rossby waves ex-cite short Rossby waves at a western boundary. The shortwave is greatly reduced in amplitude equatorward of theincident long wave. The mathematical symbols are definedin the text.

with the result from section 4c that the energy flux vanisheswhen |Al| is constant.

As in section 4b, the reflected short-wave zonal energyflux vanishes equatorward of the incident long wave.

F(x)s = 0. (5.15)

Over the latitude band of the incident long wave, the shortwave energy flux is complicated by the modified boundarycondition (5.12) and by the need to account for the inter-ference between the short and long waves. Nevertheless,it is clear that these terms decay as one moves away fromthe boundary and hence do not affect the integral energybudget.

So where does the incident long wave energy go?The first option is that it is fluxed equatorward by the

short waves and their interference with the long waves.Here the cross interaction terms are easily taken into ac-count using (4.4), giving:

F(y)s+l =

c2gr4βy

(|Al|2 − |As +Al|2

)

= −c2gr

4βy

(|As|2 +AsA

∗l +A∗sAl

), (5.16)

where ∗ indicates a complex conjugate. Note that (5.16)reduces to (4.8) equatorward of the incident long wave.

The second option is that the energy is dissipated. Sincethe meridional velocity associated with the long waves isvanishingly small by assumption, the net energy dissipationintegrated across the boundary layer is simply:

D = −∫h0rv′

2s dx

=c2gr4β

|As|2y2

. (5.17)

Note, again, that the energy dissipation is independent ofthe linear drag coefficient.

Now suppose that the incident long wave energy fluxis mostly confined to a narrow latitude band, as suggestedby satellite altimeter measurements of sea surface heightvariance (e.g., Zhai et al. 2010). Then (5.13) implies that,to leading order,

As ≈ −Alover the latitude band of the incident long waves. In con-trast, equatorward of the incident long waves, As is greatlyreduced following (5.13). Thus it follows that most of thedissipation occurs within the latitude band of the incidentlong waves. Only a small fraction of the incident energyis fluxed equatorward, and most of that is itself dissipatedwithin the western boundary layer, at a rate that is inde-pendent of latitude. Only a very small amount of energyultimately escapes into the equatorial wave guide where thepreceding analysis breaks down. These results are fullyconsistent with the “Rossby graveyard” mechanism pro-posed and diagnosed by Zhai et al. (2010) using altimetricdata and numerical calculations.

6. Wave solutions with lateral friction

In section 2, we derived the properties of Kelvin andboundary Rossby waves in the presence of linear friction,the main advantage being that the mathematics is rela-tively straightforward and solutions can be categorized interms of a single nondimensional parameter. In this sec-tion, we summarise the results of incorporating lateral fric-tion and a no-slip boundary condition. After a brief state-ment of the mathematical problem and derivation of thegoverning equations, we state the results for the boundarypropagation speed and variation of wave amplitude alongthe western and eastern boundaries in three limiting casescorresponding to those studied in section 2. Details of thederivations, which involve the application of singular per-turbation theory, are sketched in an appendix.

a. Governing equations

The meridional momentum equation is now

∂v′

∂t+ βyu′ + gr

∂h′

∂y= ν

∂2v′

∂x2, (6.1)

and the associated vorticity equation is

∂

∂t

(∂2h′

∂x2− h′

L2d

)+ β

∂h′

∂x− ν ∂

4h′

∂x4= 0. (6.2)

For boundary conditions, we impose no-slip which, ex-ploiting geostrophy for the meridional flow, can be written

∂h′

∂x= 0 (x = 0), (6.3)

and the no-normal flow boundary condition, setting u = 0in (6.1), takes the form

βy∂h′

∂y− ν ∂

3h′

∂x3= 0 (x = 0). (6.4)

10

We now seek trial solutions of the form:

h′ = A(y)

(eika(y)x + γeikb(y)x

1 + γ

)e−iωt.

The form of this solution, with the sum of two exponentialsin x, is required in order to satisfy the two boundary con-ditions, and may be anticipated from classical boundarycurrent theory (e.g., Munk 1950). Division by the coeffi-cient (1 + γ) is to ensure that A(y) is equal to the waveamplitude on the boundary (at x = 0).

Substituting this trial solution into the vorticity equa-tion (6.2) gives the depressed quartic equation:

−νk4c + iωk2c + iβkc +iω

L2d

= 0 (c = a, b). (6.5)

There are four roots, two each for the western and easternboundary solutions. This can be solved analytically follow-ing the method described by Cardano (1545). However,the full analytical solution is complicated and provides lit-tle physical intuition. Instead we resort to approximatesolutions obtained using singular perturbation theory (seeAppendix).

Once we have the solutions for ka and kb, substitutingthe trial solution into the boundary conditions (6.3) and(6.4) yields

γ = −kakb,

anddA

dy=iν

βykakb(ka + kb)A. (6.6)

The latter condition allows us to determine both the prop-agation and amplitude variations along the boundary.

b. Summary of solutions

In the appendix, we derive the solutions in three limit-ing cases:

(i) a Kelvin wave limit in which propagation is cyclonicaround each hemispheric ocean basin and at the classicalKelvin wave speed, c;

(ii) a Rossby wave limit in which propagation is cyclonicand at the speed

cLdδω,

where δω =√ν/2ω is the width of an oscillatory Stokes

viscous boundary layer;(iii) a Munk boundary layer limit in which propagation

is cyclonic and at the speed

cLdδM

,

where δM = (ν/β)1/3 is the width of the Munk boundarylayer Munk (1950).

In all other respects, the solutions follow those obtainedwith linear friction in section 2, with the same variations ofthe wave amplitude along the western and eastern bound-aries.

Thus, in the low frequency limit, the waves are sloweddown from the Kelvin wave speed by the ratio of the defor-mation radius and the pertinent frictional boundary layerwidth.

c. Energetics and western boundary interactions

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

7. Effect of time-mean flow

To this point, we have analysed wave solutions to theequations linearized about a state of rest. While it istempting to suppose that mean flows simply Doppler shiftthe solutions described above, this need not be the case.For example, the non-Doppler shifting of long Rossby wavesby background mean flows in the reduced-gravity model iswell established (Liu 1999): physically, a mean flow bothDoppler shifts the waves, but also modifies the mean po-tential vorticity gradient, thereby modifying the intrinsicRossby wave speed; in the case of long Rossby waves inthe reduced gravity model, these two effects are equal andopposite.

A detailed treatment of mean flows lies beyond thescope of the present manuscript, both because it would re-quire a large number of additional calculations, and equallybecause the technical challenge of including general meanflows is non-trivial. Nevertheless, here we include one sim-ple, if somewhat extreme, scenario, mainly to illustrate thepotential importance of mean flows.

Specifically, we consider the case of a mean, inertialwestern boundary current within which the potential vor-ticity is uniform. For this purpose, we employ a geostrophicvorticity model (Schar and Davies 1988; Tansley and Mar-shall 2000). The equations of motion, including nonlinearaccelerations but neglecting dissipation, are:

−βyv + gr∂h

∂x= 0, (7.1)

∂v

∂t+

(βy +

∂v

∂x

)u+

∂

∂y

(grh+

v2

2

)= 0, (7.2)

∂h

∂t+

∂

∂x(hu) +

∂

∂y(hv) = 0. (7.3)

These can be combined to obtain a potential vorticity con-

11

servation equation,

(∂

∂t+ u

∂

∂x+ v

∂

∂y

)Q = 0 (7.4)

where

Q =1

h

(βy +

∂v

∂x

)

is the potential vorticity.Setting the potential vorticity to a uniform value Q0

gives an elliptic equation for the time-mean layer thickness,

βy +grβy

∂2h

∂x2= Q0h, (7.5)

the solution to which is

h =βy

Q0

(1− e−x/Ld

)+ hw(y)e−x/Ld (7.6)

where

Ld =

(gr

βyQ0

)1/2

(7.7)

is closely related to the conventional Rossby deformationradius. The time-mean layer thickness on the boundary,hw, is a weak function of latitude because the no-normalflow boundary condition applies to the total, rather thangeostrophic, velocity. The solution, sketched in Fig. 4, isthe classical Fofonoff gyre (Fofonoff 1954), slightly modifiedby the non-uniform value of hw on the boundary.

Since the potential vorticity is uniform, layer thicknessanomalies satisfy

grβy

∂2h′

∂x2= Q0h′, (7.8)

to which the solution is

h′ = A(y, t)e−x/Ld . (7.9)

Note that we have lost the zonal propagation of the shortRossby wave solution and instead have a solution that de-cays over a scale that is closely related to the classical de-formation radius, as in the classical Kelvin wave. This isnot surprising for two reasons: (i) Rossby waves rely onthe presence of a background potential vorticity gradientthat is absent here; (ii) both the present solution and theclassical Kelvin wave are associated with vanishing poten-tial vorticity anomalies. Time-dependence is left arbitraryin this solution.

To solve for the meridional propagation of the wave,we need to apply the no-normal flow boundary conditionlinearized about the mean state:

∂v′

∂t+

∂

∂y(vwv

′ + grh′) = 0 (x = 0) (7.10)

yΔ(x) F(x)

F(x)

s

s

l

Fs(y)

F

Firstly, the incoming long wave energy flux is

F(x)l = −

∫g′h0hlul dy =

∫ c2g′

4βy

d

dy|Al|2 dy =

c2g′

4β

∫ |Al|2y2

dy.

Here we have used the fact that Al = 0 at the limits of the integration. We also notethat the final expression is equal to the integral of the long wave energy multipled by thelong Rossby group velocity (the latter does not generally hold for this form of the energyflux, but falls out here as a consequence of Al vanishing at the limits of integration).

The reflected short-wave energy flux can be shown to be

F(x)s =

∫g′h0hsus dy = −

∫c2g′y

4β

d

dy

(|As|2y2

)e−2βrx/(ω2+r2) dy

=c2g′

4βe−2βrx/(ω2+r2)

∫ |As|2y2

dy −[c2g′|As|2

4βye−2βrx/(ω2+r2)

].

Again, we caution that one must also calculate the energy flux associated with the crossterms between the short and long waves. Nevertheless it is clear that as x increases,

F(x)s → 0.

(The cross interaction terms also tend to zero.) Physically, this is simply stating thatit the short waves are confined to a small region close to the western boundary, hereof order (ω2 + r2)/βr, then outside of this region the short waves, and the short waveenergy flux, becomes vanishingly small.

However, we you calculate the energy fluxes, they are all zero to leading order, exceptfor the southward flux in the short wave solution. This appears to be a well-documented

3


Fig. 4. Schematic showing the solution for the Fofonoffgyre with uniform potential vorticity, about which lin-earized wave solutions are obtained in section 7. Sketchedare the layer thickness contours, which serve as approx-imate streamlines for the flow. The layer thicknessvaries slightly along the boundary since the no-normalflow boundary condition applies to the total, rather thangeostrophic, velocity.

where vw(y) is the mean boundary velocity and

v′ =g

βy

∂h′

∂x

is the perturbation geostrophic velocity.Substituting for h′ gives:

∂

∂t

(A(y, t)√

y

)+

∂

∂y

({vw(y)− βyLd(y)

} A(y, t)√y

)= 0.

(7.11)The general solution is non-trivial. Nevertheless, (7.11) isin flux form where we can identify

cw(y) = vw − βyLd ≈ vw − c (7.12)

as the relevant, non-dispersive western boundary wave speed.To leading order, this is the classical Kelvin wave speed,Doppler-shifted by the mean velocity on the boundary.

Whilst the results of this section apply in just one ex-treme limit in which the potential vorticity is completelyhomogenized, the fact that the Kelvin wave solution is re-covered should provide a further cautionary note aboutthe propagation speeds of meridional circulation anoma-lies in OGCMs. In particular, models that fail to resolveeddies and hence the Rossby deformation radius, will alsofail to resolve the inertial boundary layers and the meanflows described in this section. It remains to be determinedwhether this solution is of any relevance to more realisticcases in which there are finite potential vorticity gradients

12

and Rossby waves, and also when a no-slip boundary con-dition is incorporated.

8. Concluding remarks

Boundary waves play a fundamental role in the adjust-ment of the MOC to changes in surface wind and buoyancyforcing. Yet, this adjustment varies widely across differentOGCMs, as reviewed in section 1. In this manuscript, wehave analysed boundary wave solutions along western andeastern boundaries in a reduced-gravity model. Contraryto what is often assumed in studies of MOC adjustment,but consistent with results from the coastal oceanographyliterature (e.g., Clarke and Shi 1991), boundary propaga-tion occurs not through Kelvin waves, but through shortand long Rossby waves at the western and eastern bound-aries respectively: these Rossby waves propagate zonally,as predicted by classical theory, and cyclonically along thebasin boundaries in order to satisfy the no-normal flowboundary condition. The along-boundary propagation speedis cLd/δ where c is the internal gravity/Kelvin wave speed,Ld is the Rossby deformation radius and δ is the appro-priate frictional boundary layer width. This result holdsacross a wide range of parameter regimes, with either lin-ear friction or lateral viscosity and a no-slip boundary con-dition.

An important corollary is that the boundary propaga-tion speed in OGCMs is likely to be sensitive to modelparameters, offering a plausible explanation for the widerange of model behavior, as well as some of the discrep-ancies between models run at eddy-permitting and coarserresolution. While these results are likely to be modified bythe inclusion of realistic bottom topography, higher baro-clinic modes and background mean flows (as shown in sec-tion 7), and will also differ for the adjustment of the deeplimb of the MOC (e.g., Elipot et al. 2013), we suggest thata cautious approach is apposite when discussing the bound-ary propagation of MOC anomalies in any one particularmodel. Nevertheless, a proper representation and dynam-ical understanding of boundary wave propagation speedsis important for understanding internal climate variability,as reviewed in section 1, and may facilitate attempts todevelop the capacity for decadal climate prediction in theAtlantic 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 lackof coherence of meridional circulation anomalies observedin models across the Gulf Stream over the past severaldecades (e.g., Bingham et al. 2007; Biastoch et al. 2008;Lozier et al. 2010). While this lack of coherence may be aconsequence of the spatial pattern of wind forcing anoma-lies (e.g., Zhai et al. 2013), rather than boundary wavesper se, a plausible explanation is that the change in poten-tial vorticity encountered by a western boundary wave as

it propagates across the Gulf Stream excites an eastwardpropagating frontal wave (e.g., Cushman-Roisin et al. 1993;Cushman-Roisin 1993), to which it loses energy. Resultsfrom such a study will be reported in a future manuscript.

Acknowledgments.

We are grateful to Sergey Danilov for helpful commentson a preliminary draft. Financial supported was providedby the U.K. Natural Environment Research Council. HLJis also grateful to the Royal Society for providing a Uni-versity Research Fellowship.

APPENDIX

Derivation of solutions with lateral friction andno-slip boundary condition

a. Kelvin wave limit

Nondimensionalize the wavenumber by the deformationradius:

kc = Ldkc, (A1)

where henceforth tildes indicate nondimensional variables.The depressed quartic (6.5) becomes:

iε−1δ3k4c + (k2c + 1) + ε−1kc = 0,

where

δ =δMLd

is the nondimensional Munk viscous boundary layer widthwith δM = (ν/β)1/3. The nondimensional parameter

ε =ω

βLd

is analogous to λ1/2 in section 2. For Kelvin waves werequire ε� 1 analogous to λ� 1 in section 2a.

To ensure that the Kelvin waves are not swamped bylateral friction, we also require δ � 1. The appropriatechoice turns out to be δ = O(ε−1/3), giving:

iε−2(δ3ε)k4c + (k2c + 1) + ε−1kc = 0. (A2)

Firstly, to obtain the Kelvin wave roots, we expand(A2) in powers of ε−1,

kc = k0 + ε−1k1 + · · · ,

giving at the leading two orders:

k20 + 1 = 0,

2k0k1 + k0 = 0.

The solutions are

k0 = ±i, k1 = − 1

2ε,

13

or, in dimensional form,

ka = ± i

Ld− β

2ω+ · · · . (A3)

As in section 1, the first root corresponds to the westernboundary and the second root to the eastern boundary.

Secondly, to obtain the viscous boundary layer roots,we expand (A2) in the distinguished limit,

kc = εk0 + k1 + · · · ,

giving:

i(δ3ε)k40 + k20 = 0,

4i(δ3ε)k30 k1 + 2k0k1 = 0.

The solutions are

k0 = ± (1 + i)√2

δ−3/2ε−1/2, k1 = 0,

or, in dimensional form,

kb = ± (1 + i)√2

√ω

ν+ · · · . (A4)

This is the classical solution for a viscous Stokes boundarylayer in an oscillatory flow.

Finally we determine the variation of the wave ampli-tude along the western and eastern boundaries by substi-tuting for ka and kb in (6.6):

dA

dy=

(∓ iωc

+1

2y± (1− i)√

2

ν1/2ω1/2βy

c2+ · · ·

)A.

The solution, retaining just the leading terms for the bound-ary wave propagation and amplitude variation, is:

A√y≈ A0√

y0e∓iωy/ce±ωδωβy

2/4c2 , (A5)

i.e., the wave propagates cyclonically along the boundariesat the classical Kelvin wave speed, c, and the wave am-plitude varies as

√y. The latter term represents the fric-

tional decay of the Kelvin wave, and is latitude dependentbecause the importance of friction depends on the rela-tive magnitude of the deformation radius and the viscousboundary layer width, with the deformation radius beinglarger at low latitudes and thus frictional damping beingless important.

b. Rossby wave limit

An alternative expansion that admits a short Rossbywave solution is obtained by nondimensionalizing the wavenum-ber by the short Rossby wavenumber:

kc =ω

βk. (A6)

Then:

ε2(N

ε2

)k4c − ik2c − ikc − iε2 = 0, (A7)

where now ε� 1,

N =β2ν

ω3=

(√2δωδM

)6

,

and

δω =

√ν

2ω

is the width of an oscillatory Stokes viscous boundary layer.We also set N = O(ε2); clearly the results are only validwhen δω � δM .

First, to obtain the Rossby wave roots, we expand inε2,

kc = k0 + ε2k1 + · · · ,giving:

k20 + k0 = 0,(N

ε2

)k40 − 2ik0k1 − ik1 − i = 0.

The leading terms in the solutions for the western and east-ern boundaries are:

ka = −1 + ε2 + iN + · · · , −ε2 + · · · , (A8)

i.e., the short and long Rossby wave solutions respectively.To obtain the remaining roots, we expand in the dis-

tinguished limit:

kc = ε−1k0 + k1 + · · · ,

giving(N

ε2

)k40 − ik20 = 0,

4

(N

ε2

)k20 k1 − 2ik1 − i = 0.

The leading terms in the solutions for the western and east-ern boundaries are:

kb = ± (1 + i)√2N

+1

2+ · · · , (A9)

corresponding to an oscillatory viscous boundary of widthδω. The leading-order real and imaginary terms in the nonormal flow boundary condition for the western solutionare:

dA

dy=

(1

y− iωβδωy

c2+ · · ·

)A,

The solution is:

A

y≈ A0

y0e−i(βδω/f0)(ω/βL

2d)y

′, (A10)

14

i.e., the propagation speed along the boundary is

cw = cLdδω. (A11)

At the eastern boundary, we find:

dA

dy=

(ω2y

c2+ i

ωβδωy

c2+ · · ·

)A.

The solution, exploiting ε2 � 1, is:

A ≈ A0e−i(βδω/f0)(ω/βL2

d)y′, (A12)

i.e., the propagation speed along the boundary is again

ce = cLdδω. (A13)

c. Munk boundary layer limit

Finally we consider a regime in which a viscous bound-ary layer solution is obtained at the western boundary.Here we nondimensionalize the wavenumber by the defor-mation radius:

kc = Ldkc. (A14)

The depressed quartic (6.5) becomes:

δ3k4c − iε(k2c − 1)− ikc = 0, (A15)

where

δ =δMLd

is the nondimensional Munk viscous boundary layer widthwith δM = (ν/β)1/3. We set δ ∼ 1, which ensures thatviscosity has a leading order impact at the deformationscale.

We now expand in the small parameter ε:

kc = k0 + εk1 + · · · ,

giving:

δ3k40 − ik0 = 0,

4δ3k30 k1 − ik1 − i(k20 + 1) = 0.

The solutions for the western boundary are

ka = − ie2πi/3

δ+ε

3

(1− e−2πi/3

δ2

)+ · · · , (A16)

kb = − ie−2πi/3

δ+ε

3

(1− e2πi/3

δ2

)+ · · · , (A17)

and for the eastern boundary,

ka = −ε+ · · · , (A18)

kb = − iδ

+ε

3

(1− 1

δ

)+ · · · . (A19)

The western boundary solutions represent the classical vis-cous boundary layer solution of Munk (1950), extended toinclude the leading order correction due to the oscillatorytime-dependence. The eastern boundary solutions repre-sent the long Rossby wave and no-slip viscous boundarylayer solutions respectively.

The boundary condition gives for the western bound-ary:

dA

dy=

(1

y− δMωβy

c2i+ · · ·

)A,

which takes the same form as the the equivalent equationwith linear friction. The solution, expanded about a refer-ence latitude, is again:

A

y=A0

y0e−i(βδM/f0)(ω/βL

2d)y

′, (A20)

i.e., the propagation speed along the boundary is again

cw = cLdδM

. (A21)

The boundary condition gives for the eastern boundary:

dA

dy=iδMωβy

c2A+ · · · ,

to which the solution is

A = A0e−i(βδM/f0)(ω/βL

2d)y

′, (A22)

i.e., poleward propagation along the boundary, without anyamplitude change, at a speed

ce = cLdδM

. (A23)

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propagation of meridional circulation anomalies along

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