VOLUME 32 APRIL 2002J 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 2002 American Meteorological Society 1121

A Theory for the Surface Atlantic Response to Thermohaline Variability

HELEN L. JOHNSON AND DAVID P. MARSHALL

Department of Meteorology, University of Reading, Reading, United Kingdom

(Manuscript received 2 January 2001, in final form 17 July 2001)

ABSTRACT

The response of the upper, warm limb of the thermohaline circulation in the North Atlantic to a rapid changein deep-water formation at high latitudes is investigated using a reduced-gravity ocean model. Changes in deep-water formation rate initiate Kelvin waves that propagate along the western boundary to the equator on a timescaleof months. The response in the North Atlantic is therefore rapid. The Southern Hemisphere response is muchslower, limited by a mechanism here termed the ‘‘equatorial buffer.’’ Since to leading order the flow is ingeostrophic balance, the pressure anomaly decreases in magnitude as the Kelvin wave moves equatorward, wherethe Coriolis parameter is lower. Together with the lack of sustained pressure gradients along the eastern boundary,this limits the size of the pressure field response in the Southern Hemisphere. Interior adjustment is by thewestward propagation of Rossby waves, but only a small fraction of the change in thermohaline circulationstrength is communicated across the equator to the South Atlantic at any one time, introducing a much longertimescale into the system.

A new quantitative theory is developed to explain this long-timescale adjustment. The theory relates thewestward propagation of thermocline depth anomalies to the net meridional transport and leads to a ‘‘delayequation’’ in a single parameter—the thermocline depth on the eastern boundary—from which the time-varyingcirculation in the entire basin can be calculated. The theory agrees favorably with the numerical results. Im-plications for predictability, abrupt climate change, and the monitoring of thermohaline variability are discussed.

1. Introduction

The thermohaline overturning circulation transportsapproximately 1 PW of heat northward in the Atlanticbasin (Macdonald and Wunsch 1996). When released tothe atmosphere in the region of the North Atlantic jetstream, this heat intensifies the storm track and is re-sponsible for warming western Europe by as much as58–108C (Rahmstorf and Ganopolski 1999). The ther-mohaline circulation is global in scale, transporting ap-proximately 20 Sv (Sv [ 106 m3 s21) of warm waternorthward across the equator. It may therefore affecttropical sea surface temperatures to which the entireglobal atmosphere is particularly sensitive. Changes inthermohaline circulation strength could therefore havea significant impact on climate (Vellinga and Wood2002, hereafter VW).

In principle we should be able to observe the strengthof the meridional overturning circulation at high lati-tudes. For example, Curry et al. (1998) monitor Lab-rador Sea Water thickness as a proxy for deep-waterformation rate, and Marotzke et al. (1999) propose usingthe basinwide zonal pressure gradient. A key challenge,

Corresponding author address: Helen L. Johnson, Department ofMeteorology, University of Reading, P.O. Box 243, Reading RG6 6BB,United Kingdom.E-mail: h.l.johnson@reading.ac.uk

given these observations, is to determine the responseof the ocean at other latitudes, as well as the responseof the atmosphere.

Previous work has largely focused on the steady-statethermohaline problem, concerning either the structureof the mean circulation [beginning with Stommel andArons (1960)] or its multiple steady states (Stommel1961). Kawase (1987) and Cane (1989) looked at thespinup problem and showed that the abyssal ocean re-sponds to changes in the deep-water formation rate viathe propagation of Kelvin waves around the North At-lantic, on timescales of less than a year. More recentwork by Yang (1999) demonstrates that the surfaceocean responds in a similar way. Marotzke and Klinger(2000) suggest, however, that the ocean responds muchmore slowly, with a timescale set by advection in thedeep western boundary current. General circulationmodel (GCM) studies (e.g., McDermott 1996; Sugi-nohara and Fukasawa 1988) identify Kelvin waves dur-ing the spinup of the thermohaline circulation but findthat adjustment takes place over decades to centuries.The resolution of this issue is key. If we want to un-derstand the impact on climate of a change in ther-mohaline circulation strength, the first step is to under-stand how the surface ocean itself responds.

The thermohaline circulation can influence the at-mosphere only by rearrangement of the sea surface tem-

1122 VOLUME 32J 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

perature (SST) field (or the sea ice distribution). Yang(1999) shows a correlation between the deep-water for-mation rate at high latitudes and tropical patterns of SST.Because of the deep convective boundary layer in theTropics, the atmosphere here responds strongly to evensmall SST anomalies, so the potential for feedback onthe atmosphere is large. Yang (1999) demonstrates therelevance of oceanic teleconnections to climate. [Theimportance of atmospheric teleconnections is well es-tablished: see, e.g., the review by Trenberth et al.(1998)]. We follow Yang’s lead, looking at the telecon-nections associated with the thermohaline circulationusing a dynamical approach rather than the more con-ventional statistical one.

During the course of preparing this paper we havebecome aware of similar work carried out by Huang etal. (2000). They study the global adjustment of the ther-mocline in response to a change in the deep-water for-mation rate. Their emphasis, like that of the thermo-haline literature in general, is very much on the millenialtimescale solution to the problem. While they commenton the shorter term, transient response, they present noquantitative analysis. It is this decadal and shorter time-scale response, however, that is crucial for the rapidclimate change problem and is the focus of this study.

In this paper our aims are to

• describe the dynamical response of the upper, warmlimb of the thermohaline circulation, on decadal andshorter timescales, to a change in deep-water forma-tion rate at high latitudes;

• develop and test a quantitative theory of this response;and

• establish the importance of the equator in restrictingabrupt changes in thermohaline overturning to theNorth Atlantic, through a mechanism that we term the‘‘equatorial buffer.’’

To achieve these aims we use a highly idealized reduced-gravity model, similar to that of Huang et al. (2000). Itrepresents only the upper, warm branch of the ther-mohaline circulation, which is likely to be most im-portant for the atmospheric response on timescalesshorter than a few years. The model’s simple geometryallows us to obtain a detailed mechanistic understand-ing, which we expect to carry over in essence (if notin detail) to a continuously stratified ocean. Note thatwe do not attempt to model the deep-water formationprocess itself. We simply prescribe this at the northern-most edge of the domain and aim to establish the re-sponse at lower latitudes.

The paper will be structured as follows. The keyconcepts of the equatorial buffer are laid out briefly insection 2. Section 3 illustrates the dynamical responseusing a shallow-water model. In section 4 the theoryfor the longer timescale Rossby wave adjustment is de-veloped. Results and their implications are discussed insection 5.

2. The equatorial buffer

The equator plays two important roles in the adjust-ment of the surface ocean to a change in deep-waterformation. First, as noted by Kawase (1987), the equatoracts as a barrier to the propagation of Kelvin waves onthe western boundary. Here, however, we wish to in-troduce a second important mechanism, the equatorialbuffer, which limits the magnitude of the response inthe South Atlantic on short timescales.

Suppose the rate of deep-water formation in the high-latitude North Atlantic is suddenly perturbed:

• Mass continuity leads to a meridional velocity in thesurface layers of the ocean on the western boundary,in geostrophic balance with an anomaly in pressure,which then propagates southward along the westernboundary as a Kelvin wave at a speed cg of order 1m s21. This response is essentially the surface equiv-alent of that demonstrated by Kawase (1987) for thedeep ocean.

• To leading order the anomaly is in geostrophic bal-ance:

1u 5 k 3 =p, (1)

r f

where u is the velocity, r is the density, f 5 2V sinfis the Coriolis parameter, p is the pressure, and k isa unit vertical vector. Thus a smaller pressure gradientis required to support a given transport anomaly aslatitude decreases. The pressure anomaly on the west-ern boundary therefore reduces in amplitude as thesignal moves equatorward and is small at the equator.Equatorial Kelvin waves transmit the resulting smallthermocline displacement across the equator, and itthen propagates both northward and southward alongthe eastern boundary as a coastal Kelvin wave. Theamplitude of this signal is tiny compared to that onthe western boundary because it has passed throughthe equatorial region.

• Along the eastern boundary a meridional pressure gra-dient cannot be sustained, since to balance it a Coriolisforce would require a velocity across the coastline,and all other terms in the momentum equation aresmall. Pressure along the eastern boundary is thereforeuniform on timescales longer than the transit time ofa Kelvin wave. On the western boundary, where timedependence, inertia, and friction all become signifi-cant, a finite pressure gradient can be maintained.

• The thermocline displacement is communicated intothe interior by the radiation of Rossby waves from theeastern boundary. Propagation speed is a strong func-tion of latitude, so the thermocline displacement isspread over a large area, keeping its amplitude small.It is therefore the radiation of Rossby waves that pre-vents the reamplification of the pressure signal as theeast coast Kelvin waves move poleward. Meridionaltransport south of the equator responds only as a resultof adjustment by these Rossby waves.

APRIL 2002 1123J O H N S O N A N D M A R S H A L L

FIG. 1. Shallow-water model with moving surface layer and infi-nitely deep, motionless, lower layer. Thermohaline overturning isrepresented by a prescribed outflow from the surface layer on thenorthern boundary. Model domain is a sector ocean 508 wide, ex-tending from 458S to 658N.

Because of the restriction on zonal pressure gradientsimposed by the equatorial region, the forcing commu-nicated to the ocean interior via Rossby waves is con-stantly changing: we might think of the Tropics as abuffer zone that limits the size of the pressure fieldresponse (and hence the response in meridional trans-port) in the South Atlantic. This is analogous to a chem-ical buffer that prevents large changes in the concen-tration of a species. Only a small portion of the changein thermohaline circulation strength is communicated tothe South Atlantic at any one time. The long adjustmenttimescale in the system may therefore be considerablylonger than the Rossby wave transit time. A similarsituation arises in delay-oscillator models of the El Ninosystem, where the period of the oscillation is longerthan the propagation time of the individual equatorialwaves (e.g., Suarez and Schopf 1988).

The implication, then, is that the North Atlantic re-sponds very rapidly to changes in deep-water formationrate, but that the rest of the global ocean adjusts onmuch longer timescales. If deep-water formation wereto shut down completely, the northward heat transportassociated with the meridional overturning circulationwould cease in a few months in the North Atlantic butwould continue for several years south of the equator.This implies a convergence of heat in the Tropics, wherethe atmosphere is particularly sensitive to sea surfacetemperature anomalies.

3. Shallow-water model

As a test bed for our ideas we use a simple reduced-gravity shallow-water model. Our aim is to elucidatethe mechanisms of the ocean’s response. The model istherefore chosen to be as simple as possible, while stillrepresenting at a reasonable resolution the dynamics webelieve to be important.

a. Model formulation

The warm limb of the thermohaline circulation is rep-resented by a shallow-water model with a moving sur-face layer (initial depth 500 m) and an infinitely deep,motionless lower layer (Fig. 1). Deep-water formationis represented by a prescribed outflow from the surfacelayer on the northern boundary (the sinking is assumedto take place outside the model domain). Experiments(not shown) reveal that our results are unaffected by thepresence of mean wind-driven gyres, so in the spirit ofkeeping the model as simple as is reasonable we neglectwind-driven dynamics.

There has been much debate in the literature on thelocation of the upwelling required to balance the deep-water formation at high latitudes. Opinion is divided onwhether it is distributed evenly throughout the globalocean [as assumed for convenience in the original Stom-mel and Arons (1960) model], is associated with en-hanced mixing along ocean boundaries and rough to-

pography (Polzin et al. 1997), or occurs mainly in theSouthern Ocean (Doos and Webb 1994; Gnanadesikan1999). Here we sidestep this issue and set the diapycnalupwelling to zero. We feel that this is reasonable, giventhat the Atlantic represents only a small fraction of theglobal ocean area, so different upwelling prescriptionswill have a relatively small effect on the net meridionaltransport in the South Atlantic. In our model the up-welling required to balance the thermohaline outflow onthe northern boundary is therefore represented using a‘‘sponge’’ region to the south of the domain in whichsurface layer thickness is relaxed to a uniform value.The results are not sensitive to the details of this sponge,except in that the thickness value we relax to sets thesurface layer thickness in the southwest corner of thedomain.

Our restriction to a single basin is justified by thebuffering effect of the Tropics. The response south ofthe equator is small, and we expect the response in otherbasins to be even smaller. The model domain is thereforean idealized sector ocean 508 wide and stretching from458S to 658N. (Similar experiments have been conductedin an Atlantic domain with realistic coastlines and showessentially the same results. They will not be discussedhere.)

The model equations are the standard nonlinear shal-low-water equations:

]u21 ( f 1 j)k 3 u 1 =B 5 A¹ u (2)

]t

]h1 = · (hu) 5 0, (3)

]t

where j 5 ]y/]x 2 ]u/]y, B 5 g9h 1 (u2 1 y 2)/2, Ais the lateral friction coefficient, f 5 2V sinf, V is the

1124 VOLUME 32J 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

rotation rate of the earth, f is the latitude, u is thevelocity, h is the surface layer thickness, r0 is the av-erage density, x is the horizontal position vector, k is aunit vertical vector, and t is time. The initial layer thick-ness h0 is 500 m and the reduced gravity g9 is 0.02 ms22. No normal flow and no slip boundary conditionsare applied.

The equations are discretized on a C grid, with aresolution of 0.258. On a C grid, the Kelvin wave speedis unaffected by the spatial resolution (Hsieh et al.1983), although wave structure and damping are altered.Previous studies of the transient response to a changein thermohaline forcing have been run at far too coarsea resolution to resolve Kelvin waves. For example, Mar-otzke and Klinger (2000) conduct their experiments ona B grid at 3.758 by 48 resolution, and gravity wavespeed is slowed considerably as a result. Here the mid-latitude deformation radius / f ; 50 km is resolvedÏg9hby approximately two grid points.

b. Response

Our interest lies in the rapid climate change problem:what would be the response on timescales of months todecades if the meridional overturning circulation wereto suddenly weaken or shut down? In our model thedynamical response is well described by linear theory,so we consider here the cleaner problem of a suddenswitch on of 10 Sv deep-water formation in an oceaninitially at rest. The evolution of the system is followedafter an instantaneous change from 0 to 10 Sv in theprescribed thermohaline outflow from the surface layeron the northern boundary.

Figure 2 shows a series of snapshots of the thicknessfield after a thermohaline outflow of 10 Sv is switchedon at time t 5 0 in the northwest corner of the domain.Similar to Kawase (1987), after the initial Kelvin wavesignal on the western boundary reaches the Tropics (;1month), further Kelvin waves can be clearly seen prop-agating along the equator and both northward and south-ward along the eastern boundary. The thickness signalis then carried westward into the basin interior by Ross-by waves radiated from the eastern boundary. Equatorialwaves are apparent after 3 and 4 months. Note that nosignal continues southward into the Southern Hemi-sphere on the western boundary and that only a smallfraction of the thickness signal at high latitudes reachesthe eastern side of the basin (due to the buffering effectof the equator). Huang et al. (2000) and Yang (1999)find similar coastal and Rossby wave propagation time-scales in their models. Marotzke and Klinger (2000)report a transient signal that follows the same pathwaysbut takes much longer (45 years) to reach the equator.

More careful study of the model thickness field showsthat, because information is transmitted so quickly alongthe equator and eastern boundary, surface layer thick-ness is approximately uniform along these Kelvin wavepaths and shoals slowly as the interior adjusts (Fig. 3).

This is in contrast to the western boundary where astrong meridional gradient persists throughout. Physi-cally, the surface layer is shoaling because more wateris being ‘‘sucked’’ into the western boundary currentthan is being returned from the interior, due to the delayinvolved in the propagation of information across thebasin. Figure 4 shows a time series of the meridionaltransport as a function of latitude. Again the initialKelvin wave can be seen and takes about one month toreach the equator. The equator’s role as a buffer is thenclear, with adjustment to equilibrium taking place overa period of about 50 years.

These results suggest a timescale separation approach,splitting the response into a very fast Kelvin wave re-sponse, which acts to make the pressure uniform alongthe equator and eastern boundary, and a much slowerRossby wave response, buffered at the equator, whichpropagates the eastern boundary pressure anomalies intothe basin interior.

4. Theory for longer timescale adjustment

We now develop a theory for the longer timescaleadjustment through the propagation of Rossby waves.As noted above, there is a large separation between thetimescale of the initial Kelvin wave response (months)and the annual–decadal timescale of interest here. If weassume that the Kelvin waves are, in fact, infinitely fastso that the thickness of the surface layer along the equa-tor and eastern boundary adjusts instantly to the signalarriving at the western side of the equator, then we cansimplify the problem considerably.

a. Dynamical assumptions

We begin by listing the underlying assumptions. Inthe interior we assume geostrophic balance,

g9u 5 k 3 =h, (4)

f

and a linearized continuity equation:

]h1 H= · u 5 0, (5)

]t

where H is the initial surface layer thickness. Otherprocesses, such as nonlinearity and dissipation, are as-sumed to be important only in the western boundarycurrent region where they are required in order to closethe circulation. Dissipation is only of note to the theoryin that it allows a meridional gradient in surface layerthickness to be maintained on the western boundary; asin Marotzke (1997), on the eastern boundary where he

is uniform (see Fig. 3). The inclusion of wind forcingwould lead to extra (Ekman) terms in these equationsbut would not alter the essential dynamics of the re-sponse.

The volume of the western boundary current region

APRIL 2002 1125J O H N S O N A N D M A R S H A L L

FIG. 2. Surface layer thickness after a thermohaline overturning of 10 Sv is switched on at time t 5 0 in the northwestcorner of an ocean initially at rest. There is no wind forcing, and the surface layer is initially 500 m deep. The contour intervalis 2 m, and thicknesses less than 499 m are shaded. Note that the thickness anomaly on the western boundary is much greaterthan that in the interior (the layer thickness is approximately 350 m in the northwest corner of the domain; see Fig. 3 fordetails), but extra contours are not plotted here. The southernmost 108 of the domain comprises the sponge region, where wemight expect the dynamics to be somewhat unrealistic.

1126 VOLUME 32J 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

FIG. 3. Evolution of surface layer thickness on (a) eastern and (b)western boundaries. In (a) the lines show layer thickness at intervalsof 3 months. Note that eastern boundary thickness is approximatelyuniform in y, and shallows with time, while there is a meridionalgradient in western boundary layer thickness throughout. The south-ernmost 108 of the domain comprises the sponge region, where wemight expect the dynamics to be somewhat unrealistic.

is considered to be negligible compared with the rest ofthe ocean, that is,

b e

h dx K h dx, (6)E Ew w

where the integral limits w, b, and e represent the west-ern boundary, the outside of the western boundary cur-rent, and the eastern boundary, respectively. For aboundary current of width 50–100 km in a basin 5000km wide this introduces an error of less than 5%.

b. Meridional mass budget

The net northward volume transport T at any givenlatitude is defined as

e g92 2T(y) 5 hy dx 5 (h 2 h (y) ) (7)E e w2 fw

where he and hw are the surface layer thickness on the

eastern and western boundaries. We aim to predict thistransport as a function of latitude and time.

Zonally integrating the continuity equation we findthat

e] ]h dx 5 2 T(y). (8)E]t ]yw

Physically, the divergence of transport depends uponthe rate at which the zonally integrated layer thicknessis shallowing in any given latitude band.

c. Rossby wave mass budget

Substituting (4) into (5) gives

]h ]h2 c 5 0, (9)

]t ]x

where c is the long baroclinic Rossby wave speed:

bg9Hc 5 , (10)

2f

and b 5 (2V/R) cosf is the gradient in the Coriolisparameter. The interior adjustment process is governed,then, by the propagation of long nondispersive baro-clinic Rossby waves. Equation (9) only holds within thebasin interior. In the western boundary current the im-portance of inertia means it is no longer valid. Zonallyintegrating (9) from just outside the western boundarycurrent gives

e]h dx 5 c(h (t) 2 h (y, t)), (11)E e b]t b

where hb is the thickness of the surface layer immedi-ately outside the western boundary current. The twoterms on the right-hand side can be interpreted as vol-ume fluxes associated with the westward propagatingRossby waves (the zonal component of the mass budget;see Fig. 5). The change in zonally integrated thicknessat any given latitude is determined by the differencebetween the thickness anomaly communicated by Ross-by waves into the western boundary current (hb) andthat propagated into the interior on the eastern side ofthe basin (he).

Because hb is set by Rossby wave propagation fromthe eastern boundary, it is given by the history of he:

Lh 5 h t 2 , (12)b e1 2c

where L is the width of the domain, and L/c is the timetaken for a Rossby wave to cross the basin.

d. Equation for he

Since the volume of the boundary current is negli-gible, the right-hand sides of (8) and (11) can be equat-ed. Substituting for hb then gives

APRIL 2002 1127J O H N S O N A N D M A R S H A L L

FIG. 4. Net northward volume transport (Sv) as a function of latitude and time after a ther-mohaline overturning of 10 Sv is switched on at time t 5 0 in the northwest corner of an oceaninitially at rest: (a) the initial Kelvin wave response and (b) the longer timescale adjustment. Notethat the southernmost 108 of the domain comprises the sponge region, where we might expectthe dynamics to be somewhat unrealistic.

1128 VOLUME 32J 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

FIG. 5. Rossby wave mass budget. Westward propagating Rossbywaves communicate a thickness anomaly of he into the interior fromthe eastern boundary and of hb into the western boundary. The surfacelayer shoals with time as a result.

FIG. 6. Surface layer thickness on the eastern boundary. The solidline represents the theory, and the dotted line shows average easternboundary layer thickness from the shallow-water model.

]T(y, t) 5 c[h (t 2 L /c) 2 h (t)]. (13)e e]y

The y dependence on the right-hand side enters throughthe lag since both the speed of long baroclinic Rossbywaves and the width of the basin vary with latitude.

This gives us a delay equation from which, given thethickness of the layer on the eastern boundary, we cancalculate the net transport across any latitude circle. Allthat remains is to establish how he varies as a functionof time.

When (13) is integrated over the latitudinal extent ofthe domain we obtain

1 Lh (t) 5 h t 2 c dy 2 T 1 T . (14)e E e N S1 2[ ]c

c dyEHere TS is given by the zonally integrated momentumbalance at the southern boundary:

g92 2T 5 (h 2 h ), (15)S e 02 fS

where h0 is the prescribed layer thickness on the westernboundary at the south of the domain (one might imaginethat this is set by Kelvin waves propagating around Cape

Horn from the Pacific); TN is given by the imposednorthern boundary condition. Hence only TN, h0, andthe previous time history of he must be specified in orderto obtain he(t) and T(y, t).

e. Comparison of model and theory

The delay equation is solved numerically to obtain atime series of he. To prevent the wave speed c frombecoming infinite at the equator and to reflect the im-portance of equatorial dynamics in the real system, c iscapped at the speed of the fastest equatorial Rossbywaves. All westward propagating signals within a fewdegrees of the equator travel at one-third of the gravitywave speed, that is, at a speed /3. The easternÏg9hboundary thicknesses from the theoretical model and thefull shallow-water model are shown in Fig. 6. The agree-ment between the two is remarkably good. Over the first50 years the surface layer shoals by about 90 m.

If the southern boundary condition is linearized, then(14) has an exact analytical solution of the form

2t/th (t) 5 Dhe 1 h ,e eqbm (16)

where Dh is the total change in eastern boundary thick-ness during the adjustment to equilibrium, and the ad-justment timescale t is given by

f ASt 5 2 , (17)g9H

where A is the area of the domain, f S is the value ofthe Coriolis parameter at the southern boundary, and His the average layer thickness. For the parameters usedhere we find t 5 16 yr. Note that the adjustment timedepends upon the latitude of the southern boundary ofthe domain. Shoaling of the surface layer continues untilthe zonal gradient in thickness at the southern boundarycan support a meridional transport of 10 Sv. The fartherthis boundary is from the equator, the greater the changein he that is required to obtain a 10 Sv transport, in-

APRIL 2002 1129J O H N S O N A N D M A R S H A L L

FIG. 7. Net northward volume transport (Sv) as a function of lat-itude and time from (a) the shallow-water model and (b) theory. Thesponge region to the south of the shallow water domain is not shown.

creasing the adjustment timescale. However, if we setTS equal to zero at time t 5 0, then the rate at whichthe layer initially shallows is given by

dh Te N5 2 (18))dt At50

and is independent of the southern boundary condition.Details of the analytical solution are given in the ap-pendix.

The structure of the circulation throughout the entirebasin (both interior geostrophic circulation and bound-ary current strength) can be obtained relatively simplyfrom the time series of he shown in Fig. 6. As an il-lustration, Fig. 7 shows the net meridional transport asa function of latitude and time. Figure 7a shows theresults from the shallow-water model, and Fig. 7b showsthe transport calculated using Eq. (13). The two are inbroad agreement. This suggests that the dynamical ad-justment of the surface ocean can be explained to a largedegree by the simple theory developed in section 4.

5. Discussion

As in Kawase (1987), we have demonstrated that theocean responds to a change in thermohaline forcing viathe propagation of Kelvin waves along the western

boundary to the equator. Upon reaching the equator thesignal is deflected across the basin as an equatorialKelvin wave and then travels in both directions alongthe eastern boundary, radiating Rossby waves that prop-agate westward into the interior. We have then continuedwhere Kawase left off to demonstrate that, through theequatorial buffer mechanism, the equator acts to limitthe size of the response in the South Atlantic, preventhigh-frequency variability in thermohaline forcing frombeing transmitted to the Southern Hemisphere and in-troduce long adjustment timescales into the system. Wehave gone on to develop a new quantitative theory,based on a single variable—the thermocline thicknesson the eastern boundary—to describe this surface oceanresponse.

Our simple, mechanistic, shallow water approach hasmany limitations. The thermocline is only very crudelyrepresented by one active layer of uniform initial depth.The change in stratification with latitude in the realNorth Atlantic could substantially reduce the speed ofcoastal Kelvin waves at high latitudes (Greatbatch andPeterson 1996) and consequently lengthen the time-scales involved. Our model also assumes vertical side-walls and neglects all topographic influences on the sur-face dynamics. In reality, the signal is likely to be carriedby some kind of hybrid coastally trapped wave propa-gating along the topographic shelf (Huthnance 1978)and interacting with overflows and other topographicfeatures. We also neglect the effects of wind forcing andassume no background flow.

While the above limitations will undoubtedly modifythe details of the response in a more realistic system,the theory relies on just a few fundamental ingredientsthat hold in general and not just in the simple modelused here. As a consequence of geostrophy: (i) largepressure gradients are not possible in the Tropics, and(ii) pressure gradients cannot be maintained along east-ern boundaries (where the existence of a Coriolis forceto balance the pressure gradient force would require aflow through the boundary). These are the two key fea-tures of the equatorial buffer. The only other essentialelements of the adjustment process are the existence ofa rapid boundary wave that acts to communicate trans-port anomalies in the western boundary current fromhigh to low latitudes, and a westward propagating Ross-by wave mechanism in the interior. We therefore believethe essence of the ideas presented here to be robust.

The implications of our study are broad, but can beloosely grouped under three headings.

a. Heat transport

If the thermohaline circulation were to suddenlyweaken significantly or shut down, then less heat wouldbe transported northward by the surface ocean, and wemight expect North Atlantic SSTs to cool. Meridionalheat transport in the North Atlantic would cease veryquickly, with heat transport south of the equator affected

1130 VOLUME 32J 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

much more slowly. In our model, the ocean would adjustto this heat convergence by a deepening of the ther-mocline and an accumulation of heat in the ocean in-terior. Huang et al. (2000) suggest that this thermoclinedeepening may compensate for or even offset the neg-ative SST anomalies associated with a reduced north-ward heat advection. For a 10 Sv change in thermohalineoverturning, they find that the thermocline depth altersby between 50 and 100 m, with the largest changesoccurring in the Tropics and along the eastern boundaryof each basin. Surface layer thickness in our modelshoals by 80–90 m during the 70 years over whichadjustment takes place. To establish whether the dom-inant effect is likely to be a net warming of the upperocean, or a large change in the amount of heat ex-changed with the atmosphere, a detailed representationof the mixed layer and air–sea fluxes would be required.GCM experiments show a decrease in SST over mostof the North Atlantic, but often show a much smallerwarming over large parts of the Southern Hemisphere(e.g., VW), which could perhaps be explained by ther-mocline deepening.

b. Predictability and monitoring

For a known, single, instantaneous change in ther-mohaline forcing at high latitudes we have shown that,given a history of thermocline thickness on the easternboundary, the surface ocean adjustment at every latitudecan be predicted. If the thermohaline circulation wereto suddenly shut down completely, the theory outlinedin section 4 would therefore give a first-order estimateof the response over the next few decades. In reality,however, thermohaline forcing is unlikely to undergo astep change on less than decadal timescales. With aforcing that varies in time (TN ± const), the rapid re-sponse in the North Atlantic means that the lead timefor predictability is limited to the time it takes a Kelvinwave to propagate from the forcing region, along thewestern boundary, and then along the equator.

While this need to know the recent thermohaline forc-ing at high latitudes results in little real predictive power,our theory has much potential for thermohaline moni-toring and nowcasting. Given a time series of surfacelayer thickness at any point on the eastern boundary,together with a measure of thermohaline circulationstrength at high latitudes, the structure of the entire bas-inwide circulation can be recovered. From measure-ments made at just two points [e.g., dynamic height oneastern and western boundaries at a single latitude, assuggested by Marotzke et al. (1999)] monitoring of theoverturning circulation strength throughout the entirebasin could be achieved. Since we can compute the netvolume flux across each latitude circle in the surfaceocean (with an implicit return flow at depth), the theorysuggests that measurements of hydrographic propertiesalong the boundaries should provide strong constraintsfor inverse calculations.

c. Modeling abrupt climate change

The shallow-water model experiments conducted hereare at a resolution of 0.258, just enough to resolve Kelvinwaves, which decay exponentially away from theboundary over an internal deformation radius. LikeHsieh et al. (1983) though, we find that the Kelvin wavespeed is unaffected by changes in resolution on our Cgrid; for example, at as coarse a resolution as 48, whilethe structure of the waves is somewhat unrealistic, thespeed at which the pressure anomalies propagate is un-affected (not shown). This implies that we may not needto rely on very high resolution models for future studiesof the abrupt climate change problem. The work ofHsieh et al. (1983) does suggest, however, that weshould be careful in our interpretation of results fromB-grid models, where resolution does become a factorin wave speed.

While the shallow-water model used here is fully non-linear, its response to thermohaline variability is welldescribed by our linear theory. This is a remarkableresult and suggests that progress can be made on therapid climate change issue using, for example, a plan-etary geostrophic model.

The recent work of Marotzke and Klinger (2000) onthermohaline adjustment has been something of a puz-zle. In their idealized GCM experiments, adjustmenttakes place via a propagating signal that travels alongthe same Kelvin wave pathways but takes 45 years toreach the equator. The signal shows up clearly in thevertically integrated vertical velocity field. The equiv-alent diagnostic in our model is the Lagrangian rate ofchange of surface layer thickness (Dh/Dt), in which wesee a rapid Kelvin wave signal reaching the equator after1 month (not shown). The reason for the discrepancybetween the two timescales is not clear. The Marotzkeand Klinger (2000) model is run at coarse resolution ona B grid, with the Kelvin wave speed slowed consid-erably as a result, but not by enough to reconcile thedifferences. It is forced by a strong relaxation to a pre-scribed surface density field, unlike the ‘‘mass extrac-tion’’ style of forcing used here, which could be a factor.Further work is needed to resolve this issue.

6. Concluding remarks

In order to take these ideas further, there is a needfor simple process-based studies using GCMs to see ifthe theory carries over to a more complex stratification,geometry, and forcing structure. While several experi-ments have been conducted in which a salinity/fresh-water anomaly has been imposed in the North Atlantic(e.g., VW), more analysis of the initial stages of theseruns is needed in order to establish the importance ofcoastally trapped waves and the equatorial buffer. Interms of observations, the next logical step forward isto establish monitoring stations at the eastern and west-

APRIL 2002 1131J O H N S O N A N D M A R S H A L L

ern boundaries of the Atlantic basin, as suggested byMarotzke et al. (1999).

This study is motivated in part by the potential forfeedback on the atmosphere. Given some knowledge ofchanges in thermohaline overturning circulationstrength, can we determine the response of the atmo-sphere? It would be interesting to pursue these ideas bycoupling the kind of shallow-water model used here toa mixed layer model and a simple atmosphere.

Acknowledgments. We are grateful to Ted Johnson forsuggesting the analytical solution outlined in the ap-pendix, and to Joel Hirschi, Chris Hughes, Jochem Mar-otzke, Michael Vellinga, Carl Wunsch, and two anon-ymous reviewers for comments on a preliminary draft.This work was funded by the U.K. Natural EnvironmentResearch Council.

APPENDIX

Analytical Solution

If we assume an exponential solution of the form,2t/th (t) 5 Dhe 1 h ,e eqbm (A1)

and a linearized southern boundary condition,

g9HT 5 (h 2 h ), (A2)S e 0fS

where Dh is the total change in eastern boundary thick-ness during the adjustment to equilibrium, H is the av-erage layer thickness, f S is the value of the Coriolisparameter at the south of the domain, and h0 is the initiallayer thickness, then Eq. (14) becomes

g9HL /ctA (1 2 e )c dy 2E5 6fS

g9Ht /t5 (h 2 h ) 2 T e , (A3)eqbm 0 N5 6fS

where the integral is calculated over the full latitudinalextent of the domain.

Equating terms that are exponential in time gives

T fN Sh 5 1 h (A4)eqbm 0g9H

and sets the new equilibrium value of he.Balancing steady terms gives

g9HL /ct(1 2 e )c dy 5 . (A5)E fS

The exponent L/ct is the ratio of the time taken for aRossby wave to cross the basin (at a given latitude) tothe adjustment timescale. Since propagation time rangesfrom months to years, while adjustment to equilibriumoccurs over decades, it can be assumed to be small, andthe exponential can be approximated by

LL /cte ø 1 1 . (A6)

ct

This holds only marginally at the northernmost andsouthernmost extents of our domain, but works well inpractice for the parameters used here. Substituting (A6)into (A5) leads to

f ASt 5 2 , (A7)g9H

where A 5 # L dy is simply the area of the domain.Note the sensitivity of the timescale to the southernboundary position. For the shallow-water model dis-cussed in section 2 the parameters are g9 5 0.02 m s21,H 5 500 m, and f S 5 21024 s21, and the domain areais 5 3 1013 m2, which gives an adjustment timescale ofabout 16 years.

If we set TS 5 0 initially, then the initial rate ofadjustment is given by

dh Dhe 5 2 . (A8))dt tt50

Since the initial value of he is h0 and in the light of(A4),

T fN SDh 5 2 . (A9)g9H

So,

dh Te N5 2 . (A10))dt At50

For TN 5 10 Sv this is an initial shoaling of 6 m yr21.It should be noted that this analytical solution is some-

what contrived in that it assumes an exponential decayin eastern boundary thickness prior to t 5 0.

REFERENCES

Cane, M. A., 1989: A mathematical note on Kawase’s study of thedeep-ocean circulation. J. Phys. Oceanogr., 19, 548–550.

Curry, R. G., M. S. McCartney, and T. M. Joyce, 1998: Oceanictransport of subpolar climate signals to mid-depth subtropicalwaters. Nature, 391, 575–577.

Doos, K., and D. J. Webb, 1994: The Deacon cell and other meridionalcells of the southern ocean. J. Phys. Oceanogr., 24, 429–442.

Gnanadesikan, A., 1999: A simple predictive model for the structureof the oceanic pycnocline. Science, 283, 2077–2079.

Greatbatch, R. J., and K. A. Peterson, 1996: Interdecadal variabilityand oceanic thermohaline adjustment. J. Geophys. Res., 101(C9), 20 467–20 482.

Hsieh, W. W., M. K. Davey, and R. C. Wajsowicz, 1983: The freeKelvin wave in finite-difference numerical models. J. Phys.Oceanogr., 13, 1383–1397.

Huang, R. X., M. A. Cane, N. Naik, and P. Goodman, 2000: Globaladjustment of the thermocline in response to deepwater for-mation. Geophys. Res. Lett., 27, 759–762.

Huthnance, J. M., 1978: On coastal trapped waves: Analysis andnumerical calculation by inverse iteration. J. Phys. Oceanogr.,8, 74–92.

Kawase, M., 1987: Establishment of deep ocean circulation drivenby deep-water production. J. Phys. Oceanogr., 17, 2294–2317.

1132 VOLUME 32J 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

Macdonald, A. M., and C., Wunsch, 1996: An estimate of globalocean circulation and heat fluxes. Nature, 382, 436–439.

Marotzke, J., 1997: Boundary mixing and the dynamics of three-dimensional thermohaline circulations. J. Phys. Oceanogr., 27,1713–1728.

——, and B. A. Klinger, 2000: The dynamics of equatorially asym-metric thermohaline circulations. J. Phys. Oceanogr., 30, 955–970.

——, R. Giering, K. Q. Zhang, D. Stammer, C. Hill, and T. Lee,1999: Construction of the MIT ocean general circulation modeland application to Atlantic heat transport sensitivity. J. Geophys.Res., 104 (C12), 29 529–29 547.

McDermott, D. A., 1996: The regulation of northern overturning bySouthern Hemisphere winds. J. Phys. Oceanogr., 26, 1234–1255.

Polzin, K. L., J. M. Toole, G. R. Ledwell, and R. W. Schmitt, 1997:Spatial variability of turbulent mixing in the abyssal ocean. Sci-ence, 276, 93–96.

Rahmstorf, S., and A. Ganopolski, 1999: Long-term global warmingscenarios computed with an efficient coupled climate model.Climatic Change, 43, 353–367.

Stommel, H. M., 1961: Thermohaline convection with two stableregimes of flow. Tellus, 13, 224–230.

——, and A. B. Arons, 1960: On the abyssal circulation of the worldocean II. Idealized model of the circulation pattern and amplitudein oceanic basins. Deep-Sea Res., 6, 217–233.

Suarez, M. J., and P. S. Schopf, 1988: A delayed action oscillator forENSO. J. Atmos. Sci., 45, 3283–3287.

Suginohara, N., and M. Fukasawa, 1988: Set-up of deep circulationin multi-level numerical models. J. Oceanogr. Soc. Japan, 44,315–336.

Trenberth, K. E., G. W. Branstator, D. Karoly, A. Kumar, N. C. Lau,and C. Ropelewski, 1998: Progress during TOGA in understandingand modeling global tele-connections associated with tropical seasurface temperatures. J. Geophys. Res., 103 (C7), 14 291–14 324.

Vellinga, M., and R. A. Wood, 2002: Global climatic impacts of acollapse of the Atlantic thermohaline circulation. ClimaticChange, in press.

Yang, J., 1999: A linkage between decadal climate variations in theLabrador Sea and the tropical Atlantic Ocean. Geophys. Res.Lett., 26, 1023–1026.