1604 journal of physical oceanography volumepordlabs.ucsd.edu/pcessi/pantxi.pdf · 1604 journal of...

1604 VOLUME 33J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y

q 2003 American Meteorological Society

Oceanic Teleconnections: Remote Response to Decadal Wind Forcing

PAOLA CESSI

Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California

PANTXIKA OTHEGUY

E.N.S.T.A., Paris, France

(Manuscript received 15 July 2002, in final form 4 February 2003)

ABSTRACT

The transhemispheric and interbasin response to time-dependent wind forcing localized in the Northern Hemi-sphere of a single basin is examined using the reduced-gravity shallow-water equations in domains of simplegeometry. On decadal timescales, the pressure on the eastern boundary fluctuates synchronously in both hemi-spheres and thus communicates a signal to latitudes distant from the forcing. The signal then penetrates intothe interior through westward radiation of Rossby waves. Associated with the eastern boundary pressure fluc-tuation is a time-dependent mass flux across the equator that, in a single basin, is balanced by a storage of massin the unforced hemisphere. Two oceanic basins connected by a reentrant channel at the high-latitude edge ofthe Southern Hemisphere are then considered. Again the forcing is confined to the Northern Hemisphere of onebasin only. In this geometry the time-dependent mass flux across the equator of the forced basin is not entirelybalanced within the same basin, but induces a mass flux into the unforced basin, while the mass heaving withinthe periodic channel is negligible. This process is illustrated by considering winds oscillating at a period on thesame order as the Rossby wave transit time in high latitudes. The interhemispheric and interbasin teleconnectionis achieved by a combination of long Rossby waves and large-scale, low-frequency gravity waves forced by theRossby signal. These disturbances share no characteristics of Kelvin waves; that is, they are not boundarytrapped.

1. Introduction

This work concerns the decadal and longer responseof the oceanic circulation to changes in the wind forcing.Decadal timescales have a special significance in ocean-ography, since they coincide with the time for baroclinicadjustment of geostrophic flow, mediated by oceanicRossby waves.

For example, a surprisingly large fraction of the sealevel signal at decadal frequencies can be explained bythe Rossby wave response of a single baroclinic verticalmode to wind-curl forcing [see Sturges and Hong (2000)for a review of decadal variability of sea level]. Thedecadal sea level signal is of order a few centimeters,of similar magnitude to the global trend of 2 cm (10yr)21.

The notion that the upper-ocean signal at a givenlocation might be the result of forcing to the east andsubsequent westward propagation as Rossby waves, andnot just due to local forcing, is a familiar one. It is lessintuitive that in the low-frequency band, where Rossby

Corresponding author address: Paola Cessi, Scripps Institution ofOceanography, UCSD-0213, La Jolla, CA 92093-0213.E-mail: [email protected]

waves control the dynamics, a remote response is elic-ited not just from forcing at different longitudes, butalso from forcing at different latitudes.

Some of the previous work on latitudinally distantresponse has focused on the role of baroclinic Kelvinwaves, which operate on timescales of months. For ex-ample, Kawase (1987) and Huang et al. (2000) studiedthe adjustment of the thermocline depth to the switch-on of a localized source of dense water using a reduced-gravity, shallow-water model. In these initial valueproblems, the steady state is reached through rapid prop-agation of Kelvin waves followed by a slower adjust-ment due to Rossby waves. In both Kawase (1987) andHuang et al. (2000) the sink of thermocline water as-sociated with the high-latitude forcing is balanced by adistributed upward flux of mass provided by parame-terized vertical mixing between the thermocline and thequiescent abyss. Although the amplitude of the ther-mocline depth anomaly is inversely proportional to thevertical mixing parameter, the pattern of circulation be-comes independent of the mixing as long as the mixingtimescale is longer than the Rossby wave transit time.This is the realistic limit. In this weakly diffusive re-gime, a steady response is obtained far away from themass source: in the single basin calculation of Kawase

AUGUST 2003 1605C E S S I A N D O T H E G U Y

(1987) there is a steady circulation in the southern, un-forced, hemisphere as well as in the northern one; inthe global geometry considered by Huang et al. (2000)there is flow in every basin, although the source is con-fined to the North Atlantic.

While the steady state is well understood, at least inthe context of a simple baroclinic model, the time-de-pendent global response of the upper ocean at low fre-quencies has received less attention, and this is the focusof the present work.

A notable exception is the work of Hsieh and Bryan(1996), who examined the response of the global upperocean to constant heating localized in the high latitudes(either in the North Atlantic or in the Southern Ocean).In this case there is no mixing with the abyssal oceanthat compensates the heating source so that the both thethickness of the thermocline and the global sea levelincrease in time. The sea level rise is not spatially uni-form, and the instantaneous pattern obtained after 70 yris qualitatively similar to the steady solution of Huanget al. (2000), especially insofar as the remote responseis concerned. Both Hsieh and Bryan (1996) and Huanget al. (2000) find that the interior remote response isslow: because of the global scale of the adjustment eventhe directly forced subbasin responds on a multidecadaltimescale; that is, adjustment of the directly forced re-gion requires much longer than the local transit time ofbaroclinic Rossby waves.

The conclusion of Hsieh and Bryan (1996) and Huanget al. (2000) that there is a substantial remote responseis at odds with the recent result of Johnson and Marshall(2002) that only a small fraction of the signal reachesbeyond the forced hemisphere: our interpretation of thiscontradiction is that Johnson and Marshall (2002) arein a regime where the vertical mixing timescale is fasterthan the transit time of baroclinic Rossby waves.

In the present study, we examine the response to pe-riodic forcing and inquire whether there is a preferredpattern and frequency that is especially effective at ex-citing a remote signal. This quest is motivated by recentresults (Cessi and Primeau 2001; Cessi and Louazel2001; Primeau 2002) showing that at decadal frequen-cies there are basin modes with characteristic frequen-cies: the timescale defined by the lowest eigenfrequencyis an instrinsic property of the ocean. The dynamics ofthese large-scale basin modes involves only mass con-servation and geostrophy. Because of their large scale,most of the modal energy is in the form of availablepotential energy so that the modal damping timescaleis decadal even though the friction in the momentumequations corresponds to a few days. The hallmark ofthese basin modes is a synchronous oscillation of theeastern boundary pressure, which at low frequencies isuniform in space. The time-dependent eastern boundarypressure radiates a field of long Rossby waves that prop-agate westward. When the long Rossby waves reach thewestern wall, some of the energy is reflected into shorteastward Rossby waves that are rapidly dissipated, but

some energy excites a pressure signal on the boundary.At low frequencies, this boundary pressure is coherentall along the basin’s perimeter, and thus transmits energyback to the eastern boundary, initiating more Rossbywaves. As detailed in Primeau (2002), this low-fre-quency boundary pressure signal is not a free Kelvinwave since the latter is characterized by a boundarypressure that varies along the boundary, is associatedwith cyclonic propagation, and has a characteristicboundary trapping that scales as the deformation radius.None of these three signatures are present in low-fre-quency basin modes because at a decadal period thewavelength of a free baroclinic Kelvin wave would beof the order of 106 km, much larger than the circum-ference of the earth. Instead Primeau (2002) proves thatthe boundary signal is associated with low-frequencygravity waves forced by the interior Rossby waves.

We show in this paper that the boundary pressurefeedback on the interior flow is the vehicle throughwhich the interior circulation at one latitude inducesflow in regions as remote as the opposite hemisphere,or even another basin connected through a circumpolarchannel.

2. Model formulation

Our point of departure is the reduced-gravity shallow-water equations with friction and wind forcing. This isa minimal model for baroclinic flow in planetary basinssince it also represents the evolution of the first baro-clinic mode. The model excludes processes such as non-linear coupling of different vertical modes. The stron-gest coupling affecting the first vertical mode is due tothe wind-driven steady component of the second bar-oclinic mode (Dewar 1998; Liu 1999). However, thisnonlinearity is weak so that properties of the first bar-oclinic mode are largely unchanged, and the reduced-gravity one layer is a surprisingly accurate approxi-mation of the first baroclinic mode. It is also the casethat most of the SSH perturbations, h, can be inferredfrom the thermocline displacement, h of the first bar-oclinic mode, through h 5 2(g9/g)h. Thus, remarkably,despite the exclusion of important processes such asproduction of baroclinic eddies, this simple formulationreproduces the low-frequency wind-driven pressurefluctuations observed in long time series (Sturges andHong 2000). The resilience of baroclinic Rossby wavesis confirmed by satellite observations of the interannualsea surface height (SSH) signal, which reveals a bas-inwide coherent pattern of Rossby waves in all oceans.The latter, although most prominent on the eastern sideof the basins, survives westward all the way to theboundary current extension regions (Fu and Chelton2001).

We consider a rectangular basin with 2Ly , y , Ly

as the latitudinal coordinate and 0 , x , Lx as thecartesian longitudinal one, as shown in Fig. 1. In section


FIG. 1. A schematic figure of the basin geometry. The lower layeris at rest.

4, two basins connected at high latitudes by a periodicchannel will be considered (cf. Fig. 8).

We parameterize the eddy fluxes through linear Ray-leigh friction so that the oceanic flow is governed bythe momentum equations:

tu 1 uu 1 yu 2 f y 5 2g9h 2 ru 1 , (1)t x y x r(H 1 h)

y 1 uy 1 yy 1 fu 5 2g9h 2 ry , (2)t x y y

and the mass conservation equation:

h 1 [(H 1 h)u] 1 [(H 1 h)y] 5 0.t x y (3)

In (1)–(3)

f 5 by is the Coriolis parameter,u and y are the zonal and meridional velocities,r is the coefficient of linear drag,g9 is the reduced gravity,r is the mean density,h is the interface displacement around the mean con-

stant value H, andt is the zonal component of the wind stress.

Boundary conditions of no-normal flow are imposed onthe solid walls, that is, u · n 5 0, so that total mass isconserved

L Lx y

dx h dy 5 0. (4)E E0 2Ly

The integral constraint in (4) is obtained by integratingthe mass conservation equation (3) over the domain anddefining the mean layer thickness H as the initial averagedepth of the layer.

The simple geometries considered here omit the sphe-ricity of the earth and the detailed crenellations of thecoastline. As detailed in Cessi and Louazel (2001), thedominant dynamics at low frequency is that of long

Rossby waves and the essential timescale is their transittime. On the sphere b } cosf, where f is the latitude,but cosf also appears as the metric term that convertslongitude l to distance x: Dx 5 Dla cosf, where a isthe spherical radius. Consequently there is a remarkablecancellation of the metric term when considering Dxdivided by b. Thus the long Rossby wave transit timeis identical in Cartesian and spherical geometry. As forthe details of the coastline, Primeau (2002) has shownthat they are irrelevant for the large-scale, low-frequen-cy dynamics of baroclinic Rossby waves in a basin.

The low-frequency, weakly dissipated limit

Because we are concerned with forcing at periodsdecadal and longer, the equations can be simplified byneglecting the inertial terms. Moreover, for small-am-plitude flows, the nonlinear terms can also be neglected,and the flow is in geostrophic balance except in thinfrictional boundary layers. As detailed in Cessi and Lou-azel (2001), this implies that the interior dynamics isgoverned by combining the planetary geostrophic limitof (1) and (2) with the linearized mass conservation (3).Thus, for small friction, the interior solution satisfies

th 2 ch 5 , (5)t x 1 2f r

y

2c(y) [ bg9H/ f , (6)

where we have indicated with c the speed of long Rossbywaves. The velocities are related to h by the diagnosticequations

t2 f y 5 2g9h 1 ; fu 5 2g9h . (7)x yrH

The single boundary condition admitted by (5), u 5 0at x 5 Lx, is satisfied by the condition on h

h 5 h (t) at x 5 L .0 x (8)

Here h0(t) is an arbitrary function of time determinedby the supplemental constraint of total mass conser-vation (4). The condition (4) can be applied to the in-terior solution in (5) because the frictional correctionsmodify h only in thin boundary layers, whose contri-bution to the mass integral is negligible.

An analysis of the equatorial boundary layer (omittedhere) reveals that the condition (8) approximately ap-plies even across the equator so that on decadal time-scales the pressure is approximately independent of po-sition all along the eastern boundary of the basin. Thusthe interior dynamics that in Cessi and Louazel (2001)was confined to the extra-equatorial region extendsacross the equator into the other hemisphere. It is pre-cisely the time-varying pressure on the eastern boundarythat acts as a source of motion in latitudes that are notdirectly forced by the wind stress. In particular, if onlythe Northern Hemisphere is forced, then this boundary


signal freely crosses the equator and excites motion inthe Southern Hemisphere.

3. Periodic forcing in a closed basin

In this section we specialize to forcing that is periodicin time and confined to a single hemisphere. The goalis to examine the response in the hemisphere that is notdirectly forced. Because the response to low-frequency,large-scale wind-driven forcing is well described by thelinearized vorticity equation (5), the consideration ofmore complicated time dependence (i.e., stochastic pro-cess) can be achieved by linear superposition of thesolutions at each frequency. Thus we consider a windstress curl of the form

t Re[G(x, y) exp(ivt)] if y . 0,5 (9)1 2 5r f 0 if y # 0,

y

so that G is the Fourier transform of the wind stresscurl. Neglecting an initial transient, the response willalso be of the form

h(x, y, t) 5 Re[h(x, y) exp(ivt)]. (10)

Substituting (10) into (5), we find that h is given by

iv(x 2 L )xh 5 h exp0 [ ]c

x iv(x 2 x)212 c G(x, y) exp dx. (11)E [ ]cLz

The solution (11) satisfies the eastern boundary con-dition (8) and in the latitudinal bands where the forcingvanishes h(x, y) is entirely determined by the time-de-pendent variation on the eastern boundary, proportionalto h0. The latter constant is determined by enforcingmass conservation (4), which results in

1 1221 2iv xyh 5 D dx G(x, y )(e 2 1) dy, (12)0 E E

0 21

1g9H 222 2iv yD [ y (1 2 e ) dy. (13)E2bL Lx y 21

We have defined the nondimensional frequency and spa-tial coordinates

2bL L x yx yv [ v , (x, y ) 5 , , (14)1 2g9H L Lx y

in order to highlight the dependence of h0 on the non-dimensional frequency, . Notice that the relevant time-vscale in the problem is the transit time of long Rossbywaves at the high-latitude boundary:

2T [ L /c(L ) 5 bL L /(g9H).RW x y x y (15)

The expression for the eastern boundary displacement(12) demonstrates that all latitudes are coupled in con-trast with the usual interpretation that, in the long wave

approximation, the dynamics of Rossby waves at eachlatitude are independent (e.g., Frankignoul et al. 1997).

The denominator, D in (13), vanishes only for com-vplex. The zeroes of D approximate the discrete low-frequency eigenvalues of the linear homogeneous equa-tion associated with (1)–(3) in the limit of small friction.In particular, the least damped basin mode straddles bothhemispheres. Its interior spatial structure is

h 5 h exp[iv(x 2 L )/c],0 x (16)

with v given by the smallest zero of D (i.e., the zeroclosest to the origin in the complex v plane). Becausec becomes infinite at the equator, the pressure along theequator equals that on the eastern boundary, and thedepth anomaly is symmetric about the equator. As anexample, the least damped eigenmode obtained by find-ing numerically the zeroes of D in (13) for the parametervalues in (25) is

v 5 7.54 1 1.70i,eig (17)

corresponding to a period of 16.4 yr and an e-foldingdecay time of 11.6 yr. The essential point is that ad-justment to steady state is achieved in a time that mustexceed the longest eigenperiod.

The large-scale approximation (16) compares well tothe least-damped eigenmode obtained solving the un-forced, linearized shallow-water equation associatedwith the system (1) through (3), assuming a time de-pendence of the form (10), as illustrated in Fig. 2. Noticethat in the shallow water system the depth is indeedsymmetric around the equator, it is approximately con-stant on the eastern boundary and equal to the value atthe equator, as predicted by the interior approximation(16).

For v real, the denominator D does not vanish andit is a measure of the mass flux across the equator be-cause the storage rate in the Southern Hemisphere isgiven by

L 0x

dx h dy 5 Re[h D exp(ivt)]L L /2. (18)E E t 0 x y

0 2Ly

At least for the simple geometry considered here, (12)shows that the mass flux across the equator depends onthe frequency and shape of the wind stress curl over thewhole hemisphere.

a. Optimal forcing: An upper bound on the remoteresponse

Given that the least damped basin mode is interhem-ispherical, it is expected that forcing localized in onehemisphere can excite a distant response. However, be-cause the damping of the basin modes is not weak, itis not clear what forcing frequency would optimallyexcite a remote response. In this section we address thisquestion and determine both the spatial pattern and thefrequency of the forcing that give rise to the largest


FIG. 2. The (left) real and (right) imaginary part of the height field, h(x, y), for the least dampedeigenmode of the linearized, unforced system associated with (1)–(3). All the parameters are asin (25), and the friction is r 5 5.7 3 1027 s21. The eigenvalue is v 5 (1.15 1 i0.78) 3 1028

s21, corresponding to a period of 17.3 yr and an e-folding decay time of 4 yr. The amplitude ofthe eigenmode is arbitrary and negative values are shaded.

response in the hemisphere that is not directly forced,subject to a constraint on its norm.

From (11), the response in the unforced region isgiven by

h 5 Re{h exp[iv(x 2 L )/c 1 ivt]}.remote 0 x (19)

This is a long Rossby wave emanating synchronouslyfrom the eastern boundary and with crests and troughsarriving at the western wall at times that increase withthe square of the latitude.

As a measure of the response in the unforced hemi-sphere we use the average of over one period of2hremote

the oscillation, which coincides with | h0 | 2/2, and isindependent of x and y. This measure is proportional tothe potential energy in the unforced hemisphere, whichis the dominant form of energy in the long-scale limit.The maximization of the energy is subject to a constrainton the norm of the forcing, namely

1 1

dx GG* dy 5 const. (20)E E0 21

The optimization is obtained by maximizing the func-tional F (G, G*), defined as

1 1

F [ h h* 1 l dx GG* dy 2 const , (21)0 0 E E1 20 21

where l is a Lagrange multiplier. Given the expression(12) for h0 as a function of G, we find that the variationof F with respect to G*, for v fixed, is

1 1221 iv xydF | 5 h D dx (e 2 1)dG* dydG* 0 E E

0 21

1 1

1 l dx GdG* dy. (22)E E0 21

For a maximum, dF (and its complex conjugate) mustvanish for an arbitrary variation of dG* (and dG), andso

221 ivxylG 5 h D (1 2 e ).0 (23)

The optimal wind stress curl has a component that isconstant in space and a component that is a pattern ofwaves propagating at the local Rossby wave speed, sim-ilar to the eigenmodes of the systems, (16). While thefree basin modes are constant at the eastern boundary,there the optimal wind stress curl varies the most whilevanishing at the western boundary. Because the freemodes are Rossby waves generated at the eastern wallby the boundary pressure, the most efficient way toexcite them is to maximize the Ekman pumping at theeastern boundary and then force each free wave coher-ently by propagating the wind stress curl at the same


FIG. 3. The eastern boundary depth | h0 | predicted by the interiorapproximation (12) (smooth solid curve) as a function of the fre-quency v of the optimal forcing (24). The x marks show the time-and space-averaged in the unforced hemisphere obtained by2Ï2^h&numerically integrating the nonlinear shallow-water equations withfriction, r 5 8 3 1027 s21, for the same forcing.

speed. The pattern of optimal forcing is not realistic,but it provides an upper bound on the amplitude of anyocean remote response to atmospheric forcing subjectto the normalization (20).

Comparing (18) with (12) shows that a wind stresscurl of the shape (23) also maximizes the flux of massacross the equator.

The Lagrange multiplier l is determined by the con-dition (20). Because the spatial structure of the optimalforcing depends on v, the normalization condition re-quires the amplitude of the forcing to depend on thefrequency. The final expression for the optimal forcingas a function of the spatial coordinates and of the fre-quency is thus

21/21 2t sin(vs )0G(x, y ) 5 1 2 dsE 22 [ ]vsrbL Ï2 0y

2iv xy3 (1 2 e ). (24)

In the following we choose t0 5 0.1 N m22. For thisand subsequent calculations we use the following val-ues:

211 21 21b 5 2.3 3 10 m s , H 5 500 m,21 6g9 5 0.02 m s L 5 7.5 3 10 m,x

6L 5 6 3 10 m. (25)y

With this choice, the area of the basin is 2LyLx 5 9 31013 m2, which is 60% of the surface area of the PacificOcean. The b term is evaluated at the equator, and themean depth and stratification are chosen to be typicalof the subtropical thermocline in the Pacific.

Figure 3 shows the eastern boundary depth, | h0 | , asa function of frequency for the optimal forcing (24).

The offset between the curves in Fig. 3 is because thetheoretical result (12) is based only on interior dynamicsand does not account for the frictional boundary layers.Specifically, the northern boundary layer contains a sub-stantial amount of mass.

The forcing frequency is a parameter in the optimalvforcing (24). Thus we can now maximize the optimalresponse by varying . We find that the largest responsevis at 5 4.9. For the values quoted in (25) this cor-vresponds to a period of 26 yr. Thus the optimal forcingis peaked at a period that is quite different from that ofthe least damped eigenmode: this is typical of non-nor-mal systems such as (1) through (3) (Farrell and Ioannou1996).

Figure 4 (left panel) shows a snapshot of the solution(11) for the optimal wind forcing (24). Also shown(right panel) is h obtained by numerically integratingthe reduced-gravity shallow-water equations (1)–(3) in-cluding friction and nonlinearity. Because the optimalforcing (24) has its maximum along the northern bound-ary, where the analytic approximation must be correctedin a frictional boundary layer, the response of the fullshallow-water system is smaller than that predicted bythe inviscid approximation. In all the numerical calcu-lations that we use the following value for the frictionalparameter, which guarantees an adequate resolution ofthe western boundary layer:

27 21r 5 8 3 10 s , (26)

corresponding to a decay time of 14.4 days. An im-portant point is that, even though the Ekman spindowntimescale is about 14 days, the basin-scale oscillationsdamp only on several decades. Physically this is becauseonly a fraction (R/L)2 K 1 of the oscillation energy iskinetic energy (R is the baroclinic deformation radius,

/ f , and L the basin length scale). The largest dis-Ïg9Hcrepancies between the large-scale interior approxima-tion and the shallow-water solutions are due to boundarylayer corrections along the northern and southern walls,whose widths are of order O( ) ø 440 km. ThisÏrL /bx

is illustrated in Fig. 5 where a snapshot of the zonallyaveraged interface displacement predicted by (11)(dashed line) is compared to that obtained from the nu-merical solution (solid line). The overestimate by theanalytic approximation to h in the northern and southernboundary layers is compensated in the interior by ananomaly of the opposite sign (recall that the area undereach curve must integrate to zero due to mass conser-vation).

The analytic approximations (12) and (24) also pre-dict the dependence of the response on the optimal forc-ing’s frequency. In Fig. 3, we show (x marks),2Ï2^h &that is, the average over a period and over the unforcedhemisphere of the rms interface displacement obtainedfrom the numerical solution of (1)–(3). The wind stressis the optimal forcing (23) for different frequencies.Despite the disagreement in the frictional boundary lay-ers the optimal response is obtained at approximately


FIG. 4. (left) A snapshot of the analytic approximation to h for the optimal forcing (24) oscillating witha period of 26 yr. The maximum value of the wind is 0.085 N m22. The range in the Northern Hemisphereis from 80 to 267 m, while in the Southern Hemisphere it is 210 to 10 m. (right) The solution of h obtainedsolving (1)–(3) numerically at the same time. The range in the Northern Hemisphere is from 46 to 243 m,while in the Southern Hemisphere it is 6.0 to 26.7 m. The contour level is 3 m and negative contours areshaded.

FIG. 5. The zonally averaged interface displacement shown in theright panel of Fig. 4 (solid line) is compared with the zonal averageof h from the left panel of Fig. 4 (dashed line).

the same period (between 20 and 26 yr) as the analyticalprediction.

Figure 6 shows four snapshots of h in the unforcedhemisphere during half a period to illustrate the south-westward propagation of the remote response. Noticethat the maximum displacement migrates from the equa-tor and the eastern boundary toward the pole and thewestern wall during one period, so all of the SouthernHemisphere experiences a remote response. In both theanalytic approximation and the shallow-water solutionwith small dissipation the remote thermocline displace-ment is approximately 15% of the locally forced depthvariation. Notice that, although the optimal forcing ex-cites a response in the shape of the least damped ei-genmode, its pattern is nothing like it.

b. Suboptimal forcing

Because of the peculiar propagating structure of theoptimal forcing, we enquire whether a substantial re-mote response is also elicited with a more generic shapeof the wind stress. We thus explore the following familyof wind stress patterns:


FIG. 6. Four snapshots of the analytic approximation to h in the Southern Hemisphere during 0.5 period for theoptimal forcing (24), oscillating with a period of 26 yr. Notice that the maximum and minimum propagate polewardand westward. The contour level is 1 m and negative contours are shaded.

4 4t 5 t exp[2(y 2 y ) /l ] cos(vt),0 0 (27)

where both the central latitude y0 and the scale l can bevaried. This pattern of wind stress is standing, whilethat considered in the previous section is propagating.The hyper–Gaussian profile in (27) ensures that the forc-ing is localized in a strip y0 2 l , y , y0 1 l. Thisstrip might, for instance, correspond to a storm track.At any rate, the localized standing waves pattern (27)is very different from the broad traveling pattern in (24),thus providing an assessment of the robustness of con-clusions based on the optimal shape (24).

For the range of y0 and l such that the hyper–Gaussianforcing is localized in the Northern Hemisphere, themaximum response in the Southern Hemisphere, subjectto the constraint (20), is obtained when the central lat-itude of the forcing is y0 5 Ly/2, and its width is l 50.3Ly. In order for (27) to have the same norm as (24)the amplitude of the wind stress in (27) must be t0 50.015 N m22, which is much smaller than the wind stressamplitude of the optimal forcing. An analysis of theremote response, | h0 | 2, as a function of frequency re-veals that the largest response is obtained at a periodof 22 yr, rather close to the period of the optimal forcing(26 yr). However, the amplitude is smaller by about a

factor of 3 than that produced by the optimal forcing(24), indicating that the efficiency of the optimizationprocedure considered in the previous section is highwhen the norm of the wind stress curl is kept fixed asin (20).

Figure 7 (left panel) shows a snapshot of the solution(11) for the wind stress (27), with v 5 2p/22 yr. Alsoshown (right panel) is h obtained by numerically inte-grating the reduced-gravity shallow-water equations.The agreement between the theoretical prediction basedon interior dynamics and the numerical solution withsmall friction is much better than that obtained with theoptimal forcing because the response to a hyper-Gauss-ian forcing does not entail a northern boundary layer.Consequently, there is no ‘‘missing-mass’’ error. Theagreement between the analytic inviscid approximationand the numerical solution with friction is good, not justas far as the amplitude of the response, but also as faras the optimal frequency is concerned. A series of nu-merical calculation, not shown here, varying the fre-quency of the wind stress, confirms that the largest re-sponse is indeed obtained when the period is 22 yr.Although the maximum remote response obtained withthe family (27) is smaller than with the optimal forcing,


FIG. 7. (left) A snapshot of the analytic approximation to h for the hyper-Gaussian forcing (27), oscillatingwith a period of 22 yr. The range in the Northern Hemisphere is from 18 to 211 m, while in the SouthernHemisphere it is 3.6 to 23.5 m. (right) The interface displacement, h, obtained solving (1)–(3) numerically.The range in the Northern Hemisphere is from 15 to 27 m, while in the Southern Hemisphere it is 2.7 to23.0 m. The contour level is 1 m and negative contours are shaded. For both the large-scale approximation(left) and the full shallow-water solution (right) the ratio of the remote to the locally forced response isabout 25%.

FIG. 8. The geometry of two basins with a circumpolar connectionat the southern edge.

the ratio of the displacement amplitude in the unforcedto that in the forced region is larger with the suboptimalforcing in (27).

For both wind stress patterns considered in this sec-tion, there is an appreciable response in the unforcedSouthern Hemisphere. The remote response is maximum

when the period is about 26 yr. In the following weassess the robustness of the solution to the geometry ofthe basin.

4. Two basins with a circumpolar connection

In this section we examine the coupled problem oftwo basins connected by a periodic channel at the south-ernmost edge. With reference to Fig. 8, the region 2Ly

2 l , y , 2Ly is occupied by a reentrant channel thatconnects two basins of equal latitudinal extent, one ofzonal width Lx and the other of zonal width LA. Thetwo basins are separated by a landmass of width Ll andby an infinitesimally thin solid boundary. In the follow-ing we will focus on a configuration with

6 6L 5 10.56 3 10 m, L 5 6.6 3 10 mx y

L 5 L /2, L 5 L /4, l 5 0.16L (28)A x l x y

so that the area of the basin P, of the basin A, and ofthe channel approximate those of the Pacific, of theAtlantic, and of the Southern Ocean, respectively.

The interior dynamics (5) still holds away from the


boundaries and in each enclosed basin the pressure onthe eastern boundary is a function of time only. In thechannel the periodicity condition h(x, y, t) 5 h(x 1 Lx

1 Ll 1 LA, y, t) applies.In the following we will consider forcing that is pe-

riodic in time and confined to the Northern Hemisphereof basin P so that the Southern Hemisphere of P, thechannel, and basin A are all unforced. Using the notationof section 3, the solution in basin P is given by

h 5 Re[h(x, y) exp(ivt)],P

iv(x 2 L )xh(x, y) 5 h exp0 [ ]c

x iv(x 2 x)212 c G(x, y) exp dx, (29)E [ ]cLx

where h0 is a constant, characterizing the amplitude ofthe oscillating depth at x 5 Lx.

Since basin A is unforced, the solution there is giv-en by

h 5 Re{h exp[iv(x 2 L 2 L 2 L 1 ct)/c},A 1 x l A (30)

where h1 is the amplitude of the oscillating depth at theeastern boundary of basin A.

The constants h0 and h1 are determined by enforcingmass conservation in each basin, or alternatively massconservation in one basin and globally, as detailed short-ly. Because the constraint of global mass conservationleads to consideration of the total mass in the periodicchannel, we only need to examine the area average ofh there.

Neglecting friction, the evolution equation for thezonally averaged depth in the channel ish

ivh 5 0, (31)

so, in the absence of local forcing, the circumpolar re-gion does not contribute to the mass displacement, andonly the two basins need to be considered. The ap-proximate result (31) does not imply that the channelis at rest: free Rossby waves propagate through the chan-nel, but they have zero zonal average and do not con-tribute to mass storage.

The mass budget in basin A isL 1L 1Lx l A

h dx dy 5 H y | dx, (32)E t E y52Ly

Area(A) L 1Lx l

stating that the total storage rate is balanced by the massflux at the boundary between the basin and the channel.

The meridional velocity is in geostrophic balanceeven in the western boundary current so that

L 1L 1Lx l A

H y | dxE y52Ly

L 1Lx l

g9H5 [h(L 1 L 1 L , 2L , t)x l A yf |y52Ly

2 h(L 1 L , 2L , t)]. (33)x l y

The depth on the eastern boundary of basin A is in-dependent of y and is obtained from (30). The depth onthe southwest corner of basin A is within a westernboundary layer and cannot be obtained from the interiorapproximation (30). However, the requirement that thegeostrophic velocity normal to the southern edge of thelandmass vanishes1 leads to constant pressure on thatboundary. Thus, the depth at the southwest corner ofbasin A equals h at the southeast corner of basin P, whichin turn is the depth all along the eastern boundary x 5Lx. We thus have

h(L 1 L , 2 L , t) 5 h(L , t)x l y x (34)

so that the leading-order depth is constant all along thewestern and southern boundary of the landmass.

Substituting (29), (30), and (34) into (32) we find that

h 5 h S(v), (35)0 1

1 L /LA x LA 2S(v) [ 1 1 iv dy exp iv x 2 y dx,E E 1 2[ ]Lx21 0

(36)

where the nondimensional frequency and coordinatesdefined in (14) have been used. To summarize, the massbudget in basin A, together with geostrophy and thecondition of no flow through the southern boundary ofthe landmass, provides the connection (35) between theeastern boundary pressures of the two basins. Remark-ably, the width of the landmass does not matter as longas there is no forcing at its southern tip.

The requirement that the area integrals of (29) and(30) add up to zero, so as to conserve total mass, leadsto the following condition on h0, analogous to (12):

1 1221 2iv xyh 5 D9 dx G(x, y )(e 2 1) dy, (37)0 E E

0 21

1g9H 221 22 2iv yD9 [ 1 2 S(v) 1 y (1 2 e ) dy .E2 [ ]bL Lx y 21

(38)

Notice that neither the width of the continent nor thesize of the channel enter the expression for h0 becausethese subdomains do not contribute to the global massbalance. The eastern boundary pressure here differsfrom that in the closed basin of section 3 only throughdenominators D and D9 [cf. (13) and (38)]. The twodenominators become identical when the width of basinA vanishes, and so the results of section 3 apply alsoto a semienclosed basin whose southern solid boundaryis replaced by a reentrant channel.

Analogous to (18), the denominator D9 is proportional

1 As discussed in CL01, the thickness of the frictional boundarylayer on a southern boundary is O( ), leading to negligibleÏrL /bx

correction on the geostrophically balanced depth.


FIG. 9. The eastern boundary depth in basin P, | h0 | , and the easternboundary depth in basin A, | h1 | , (both in meters) predicted by theinterior approximations (29) and (30) as a function of the dimensionalfrequency, v, of the optimal forcing (24).

to the total mass flux across the equator for the twobasins combined. A simple calculation shows that thestorage rate of the combined Southern Hemispheres isgiven by

L 1L 1L 0x l A

dx h dyE E t

0 2Ly

5 Re[h D9 exp(ivt)]L L /2. (39)0 x y

Because the expressions for h0 in (12) and (37) onlydiffer through the denominators D and D9, the spatialstructure of the forcing (24) optimally excites the re-sponse also when considering several interconnected ba-sins.

Figure 9 shows the amplitudes of the two easternboundary pressures for basin P (solid line) and basin A(dashed line), as a function of the forcing frequency.For the value of parameters in (28), the Southern Hemi-sphere of basin P is maximally excited at a period of45 yr. This period is longer that the optimal value foundin section 3 because Lx and Ly are larger in the presentexample [cf. (28) with (25)]. In fact, the optimal non-dimensional frequency for | h0 | is 5 5.2, which isvslightly higher than that obtained with a single basin insection 3 ( 5 4.9). The essential point is that the op-vtimal forcing period is comparable to the transit time ofbaroclinic Rossby waves in the subpolar region of theforced basin, both in a single basin and in two basins(and presumably also three basins) connected by a pe-riodic high-latitude channel.

The response in basin A increases with increasingperiod and becomes equal to that of basin P in steadystate when the pressure difference between the easternand western boundary vanishes, and so no net flux ofmass enters or leaves each basin. Of course, when the

frequency is very close to zero, the approximation (29)is no longer valid because the neglected frictional effectsbecome essential.

As an example, we show in Fig. 10 a snapshot of hforced by a wind stress curl given by (24) oscillatingat a period of 40 yr for the geometry of Fig. 8. Thesolution is obtained by solving the shallow water system(1)–(3) with the parameters in (28) and r 5 8 3 1027.A comparison with the solution obtained in the closedbasin, whose forcing oscillates at 26 yr (cf. Figs. 4 and10), shows that the spatial pattern of the response in theSouthern Hemisphere of the forced basin is very similarin the two cases, confirming the interpretation that aproper mode of the system is excited. The response inbasin A is 15% of the response in the Southern Hemi-sphere of basin P and it is symmetric around the equator,as predicted by the interior approximation.

To quantify the role of each subasin in the globalmass balance, we plot in Fig. 11 the volume anomalystored in the various regions of the domain as a functionof time. The storage rate is proportional to the flux ofmass across the open portion of each subregion and canbe approximately obtained by multiplying the storageby the frequency v (and shifting the time series by onequarter period). The largest storages are in the Northernand Southern Hemispheres of the forced basin, P. Forthe values used here they imply a flux of mass acrossthe equator which peaks at 1.7 Sv (Sv [ 106 m3 s21).Because the storages in the two hemispheres of basinP do not cancel, there is a flux of mass at the boundarybetween basin P and the channel that peaks at 0.4 Svin the example of Fig. 11. The storage in the periodicchannel is negligible, and so all the mass leaving basinP is fluxed into basin A, whose storage is equally par-titioned between the two hemispheres. In basin A, themass flux across the equator peaks at 0.2 Sv, about 12%of the corresponding maximum flux in basin P.

5. Summary and discussion

We have examined the remote response to time-de-pendent wind forcing confined to a single hemispherein domains of simplified geometry. A time-dependentflow is obtained in unforced regions through the Rossbywaves that radiate from the eastern boundary. The east-ern boundary pressure fluctuates in time so as to guar-antee that the mass of the active layer is conserved.Combining this with simple dynamics leads to analyticresults that enable us to survey the parameter space andto identify the controlling nondimensional factors.

In the absence of alongshore winds, the fluctuationsare synchronous all along the eastern coasts on the de-cadal timescales considered here. Unlike Kelvin waves,these disturbances are robust to coastline irregularities(Primeau 2002).

We find that for a closed basin the remote responsein the unforced hemisphere is maximally excited bywinds oscillating at a period longer than, but of the same


FIG. 10. A snapshot of the interface displacement, h, obtained solving (1)–(3) numerically in the geometryshown in Fig. 8. The wind stress is obtained from the optimal forcing (24), oscillating with a period of 40yr. The maximum wind-stress is 0.053 N m22. The range in basin P is from 241 to 39 m in the NorthernHemisphere and from 25 to 6 m in the Southern Hemisphere. In basin A the range is 21 to 4 m. Thecontour level is 2 m and negative contours are shaded.

order as, the period of the least damped eigenmode ofthe system. This eigenmode is characterized by depthanomalies that are maximum in the poleward-west cor-ner of the basin and minimum on the eastern boundaryand the equator and have a period roughly equal to TRW

in (15). The pressure on the equator for these basinmodes equals that along the eastern boundary, sug-gesting that the low-frequency oceanic variability gen-erated in high latitude can influence tropical dynamicswithout any atmospheric mediation.

For basins that are symmetric around the equator thelow-frequency basin modes have no net mass flux acrossthe equator. However, when periodic wind forcing isapplied to a single hemisphere, the storage rate in thathemisphere does not vanish, and there is a time-depen-dent mass flux passing across the hemispheres, even ifthe local forcing vanishes there. Replacing the southernboundary with a reentrant channel connecting the high-latitude edge of the unforced hemisphere leaves the so-lution unchanged.

When two basins connected at the southern boundaryby a periodic channel are considered, forcing in theNorthern Hemisphere of one basin induces a time-de-pendent response in the other basin. This oceanic te-leconnection is characterized by a mass flux across theboundary between the forced basin and the channel. The

channel stores a negligible amount of mass, instead act-ing as a conduit between the two basins whose storagesare out of phase. The essential point is that, on theselong timescales, mass balance is not achieved withinone basin, but globally and with negligible contributionfrom the reentrant portion.

The lack of signal in the channel when the forcing isin the Northern Hemisphere is in agreement with thefinding by Hsieh and Bryan (1996) of a weak SouthernOcean sea level rise in response to heating in the Arctic.In contrast, Huang et al. (2000) find a sizeable signalin the Antarctic Circumpolar Current region sustainedby a balance between lateral eddy exchanges and ver-tical upwelling due to mixing (both parameterized). Thelatter process is absent in both Hsieh and Bryan (1996)and in our model, and it obviously has a qualitativerepercussion in reentrant geometries.

We have considered wind stresses whose amplituderanges from 0.01 to 0.05 N m22 oscillating over severaldecades: the upper range of values is larger by about afactor of 2 from those reported by Roemmich et al.(2001) on interannual scales. With such wind stresses,the remote response of the interface displacement is ofthe order of several meters. For the choice of parameters(25), a thermocline depression h of 10 m is associatedwith a sea level change, h 5 2(g9/g)h, of 2 cm, which


FIG. 11. The mass storage in each subregion is plotted as a functionof time for the calculation shown in Fig. 10. Notice that the storageis equipartitioned between the two hemispheres in basin A, and sothe two solid lines almost coincide. The storage rate in meters cubedper second can be obtained by multiplying the storages by v 5 5 31029 s21.

is of the same order as the decadal trend currently ob-served (Douglas 2000), indicating that sea level changesobserved at one location might be generated by verydistant forcing.

Within the confines of a highly idealized geometry,we have performed a calculation with two basins havingan area comparable to that of the surface Pacific andAtlantic, respectively. We obtain a mass flux on mul-tidecadal timescales across the equator of the modelPacific that is of magnitude comparable to that measuredby Roemmich et al. (2001) at 248N using geostrophicflow above 800 m. The observed geostrophic flow inthe thermocline is somewhat compensated by Ekmantransport, but presumably the compensation decreaseson longer timescales.

Besides the hemisphere-integrated transport fluxes,our calculation shows that the low-frequency globalthermocline displacement obtained from localized forc-ing has a nontrivial large-scale pattern that reflects thedynamics of baroclinic Rossby waves coupled throughthe pressure along the solid boundaries. Regardless ofthe shape of the forcing and the basin geometry, theinterface displacements within the semienclosed basinsall share a minimum in response in the eastern tropicalregion and a gyrelike pattern in both hemispheres atmidlatitudes (cf. Figs. 4, 7, and 10). These patterns arevery similar to those found in the steady-state calcu-lations of Huang et al. (2000) and in the initial valueproblem of Hsieh and Bryan (1996). A similar patternin transient sea level rise is found by Bryan (1996) ina calculation using the Geophysical Fluid DynamicsLaboratory coupled model with enhanced greenhousewarming, about 45 years after the warming starts.

Our analytic approximation clarifies that the pressure

on the boundaries is the vehicle for communication be-tween basins and that Rossby waves transmit the in-formation to the interior. We also hope to dispell thecommonly received opinion that free Kelvin waves arean essential element in the low-frequency oceanic fluc-tuations: there are no free Kelvin waves at decadal pe-riods that can fit in any terrestrial basin. Instead the fieldof Rossby waves forces low-frequency, large-scale grav-ity waves that redistribute the mass coherently withinthe global ocean. Because of the large-scale nature ofthe motions involved, the details of the coastline areirrelevant.

Here, we have limited our study to a single verticalbaroclinic mode, or equivalently to one reduced-gravitylayer. The inclusion of several vertical modes (or layers)would allow for more complex processes, for example,the interaction with time-mean currents and the spon-taneous generation of time-dependent flows. Althoughadvection by mean currents is bound to be important,we are very encouraged by some recent results by D.Stammer et al. (2003, unpublished manuscript: concernsthe response of the global ocean to radiative forcinganomalies over Asia) showing very similar propagatingpatterns of remote SSH anomalies in the Atlantic andPacific in response to anomalies of heat and wind stressforcing confined to the Indian Ocean in a multiyearcalculation using the Massachusetts Institute of Tech-nology global model (at 18 horizontal resolution, with23 vertical levels).

At the moment, we are not in a position to determinewhether the explicit presence of mesoscale-eddy vari-ability alters qualitatively the picture presented here.Our analysis of optimal forcing reveals that decadal re-mote response is most sensitive to forcing near the east-ern boundaries in high latitudes, while mesoscale-eddyenergy is largest near the western boundaries in mid-latitudes. Furthermore, our calculations show a lack ofsensitivity to the actual value of dissipation, also mostimportant in the western boundary current, and to thedynamics in the model Antarctic circumpolar region,another eddy-rich region. We thus anticipate our resultsmight be robust to the inclusion of eddy processes.

Acknowledgments. Funding for this research is pro-vided by the U.S. Department of Energy. We benefitedfrom discussions with Bill Young, and we thank Fran-cesco Paparella for developing the numerical shallowwater code employed here. The comments of the anon-ymous reviewers are gratefully acknowledged.

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