2428 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME › bibliography › related_files › cec0401.pdf · 2428 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34 q 2004 American Meteorological Society

2428 VOLUME 34J 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 2004 American Meteorological Society

The Effects of Mesoscale Eddies on the Main Subtropical Thermocline

CARA C. HENNING AND GEOFFREY K. VALLIS

Geophysical Fluid Dynamics Laboratory, Princeton University, Princeton, New Jersey

(Manuscript received 8 July 2003, in final form 6 May 2004)

ABSTRACT

The effects of mesoscale eddies on the main subtropical thermocline are explored using a simply configuredwind- and buoyancy-driven primitive equation numerical model in conjunction with transformed Eulerian meandiagnostics and simple scaling ideas and closure schemes. If eddies are suppressed by a modest but nonnegligiblehorizontal diffusion and vertical diffusion is kept realistically small, the model thermocline exhibits a familiartwo-regime structure with an upper, advectively dominated ventilated thermocline and a lower, advective–diffusive internal thermocline, and together these compose the main thermocline. If the horizontal resolution issufficiently high and the horizontal diffusivity is sufficiently low, then a vigorous mesoscale eddy field emerges.In the mixed layer and upper-mode-water regions, the divergent eddy fluxes are manifestly across isopycnalsand so have a diabatic effect. Beneath the mixed layer, the mean structure of the upper (i.e., ventilated) thermoclineis still found to be dominated by mean advective terms, except in the ‘‘mode water’’ region and close to thewestern boundary current. The eddies are particularly strong in the mode-water region, and the low-potential-vorticity pool of the noneddying case is partially eroded away as the eddies try to flatten the isopycnals andreduce available potential energy. The intensity of the eddies decays with depth more slowly than does the meanflow, leading to a three-way balance among eddy flux convergence, mean flow advection, and diffusion in theinternal thermocline. Eddies subduct water along isopycnals from the surface into the internal thermocline,replenishing its water masses and maintaining its thickness. Just as in the noneddying case, the dynamics of theinternal thermocline can be usefully expressed as an advective–diffusive balance, but where advection is nowby the residual (eddy-induced plus Eulerian mean) circulation. The eddy-induced advection partially balancesthe mean upwelling through the base of the thermocline, and this leads to a slightly thicker thermocline thanin the noneddying case. The results suggest that as the diffusivity goes to zero, the residual circulation will goto zero but the thickness of the internal thermocline may remain finite, provided eddy activity persists.

1. Introduction

Understanding the structure of the ocean stratificationis a fundamental problem in ocean circulation dynamics,and the main thermocline is the single most obviousfeature of this stratification. Yet even the simpler prob-lem of understanding the structure of the thermoclineof an idealized ocean consisting of a single, isolatedbasin with a single subtropical and subpolar gyre is notfully understood. The particular geography of the oceanbasins and the circulation between them will surely playan important role in modifying the structure revealedby such a simple model. However, the dynamics of thethermocline of an idealized basin will likely transcendsuch geographical modifications and be relevant to thereal ocean, especially for the main Northern Hemispherebasins where the influence of the Southern Ocean onthe main thermocline is limited. Certainly, it is unlikelythat the dynamics of the thermocline in the real ocean

Corresponding author address: Geoffrey K. Vallis, Princeton Uni-versity, AOS Program, Sayre Hall, Princeton, NJ 08544.E-mail: [email protected]

can be understood without understanding the dynamicsof such models.

Consider first an idealized extraequatorial ocean, withwind forcing such as to produce both a subtropical andsubpolar gyre, and buoyancy forcing such that the sub-tropical gyre is warmed and the subpolar gyre cooled,effectively prescribing the surface temperature exceptin regions of strong western boundary currents. Supposefurther that the diapycnal thermal diffusivity is (real-istically) small, and that baroclinic instability is (lessrealistically) suppressed via the use of the planetarygeostrophic equations and a small but nonnegligible hor-izontal diffusivity. In these circumstances Samelson andVallis (1997, henceforth SV) showed that the main sub-tropical thermocline has two main dynamical regimes.The upper regime, comprising isopycnals that outcropin the region of Ekman downwelling (the subtropicalgyre), is advectively dominated, forming a ventilatedthermocline similar to that appearing in the calculationof Luyten et al. (1983)—a regime that was partly an-ticipated by Welander (1959), and analyzed further byHuang (1988). The lower thermocline, comprising is-opycnals that outcrop in the region of Ekman suction(the subpolar gyre), is characterized by an advective–

NOVEMBER 2004 2429H E N N I N G A N D V A L L I S

diffusive balance, forming an internal thermocline sim-ilar to those suggested by Robinson and Stommel(1959), Stommel and Webster (1963), and Salmon(1990) and analyzed further by Samelson (1999). Themain thermocline consists of the ventilated and internalthermoclines. In each case, the dominant balance in thethermodynamic equation along with the assumptions ofthermal wind and continuity give scaling predictions forhow the average depth of the ventilated thermocline andthickness of the internal thermocline will vary withmodel parameters such as the wind stress and diffusivity,and SV verified these predictions in their model.

The primary shortcoming of the SV model is its lackof mesoscale eddies—in fact, such eddies were pur-posely omitted from that model. However, the eddy ki-netic energy of the ocean is at least an order of mag-nitude larger than the mean kinetic energy, and it haslong been recognized that eddy effects may be importantin the oceanic general circulation. Rhines and Young(1982) suggested that eddies would be an effective wayof producing subsurface motion and of homogenizingpotential vorticity in closed gyres, and Cox (1985), us-ing a primitive equation numerical model, investigatedthe effect of eddies on the ventilated thermocline (al-though his integration length was too short to draw de-finitive conclusions). Vallis (2000) and, more explicitly,Marshall et al. (2002) wondered whether mesoscale ed-dies might indeed be an important factor in determiningthe stratification itself, rather than being a perturbationabout it. Since then Karsten et al. (2002) and Radkoand Marshall (2003) have found eddies to be importantin very idealized simulations.

Our goal in this paper is to explore the effects ofmesoscale eddies on the ocean stratification using aprimitive equation numerical model, configured in anidealized but oceanographically relevant domain. Ourfocus is on the main subtropical thermocline (in a sub-sequent paper we will report on the effects of a simplecircumpolar channel), and we will run at both noned-dying and eddy-permitting resolutions. In particular, wewish to answer the following questions:

1) How does the mean stratification depth and thicknessdiffer in the two cases, and can this result be capturedby simple scaling theories?

2) How do the details of the stratification in the eddy-permitting case differ from those in the case withouteddies?

Answering these involves both a qualitative look at thedifferences between individual isopycnals and a moredetailed evaluation of the overall balance of terms inthe thermodynamic equation. We perform a number ofexperiments in which we change model parameters (e.g.,wind strength, diffusivity) and investigate if and howthe overall structure of the thermocline changes as aconsequence. The numerical experiments are performedin a relatively small domain (somewhat smaller than theNorth Atlantic basin), but are all integrated for a long

period (.100 yr at eddying resolution, after a lower-resolution spinup of thousands of years) in order to en-sure that their thermodynamic structure properly equil-ibrates. Section 2 discusses the experimental setup, andthis is followed by a rather crude scaling analysis (sec-tion 3) in order to grasp the orders of magnitude in-volved in the eddy effects. Section 4 describes thechanges in the details and overall character of the strat-ification, and section 5 describes the effects of changingmodel parameters. This is followed by a general sum-mary and discussion.

2. Numerical model and experimental design

Equations of motion

We use a primitive equation ocean model, the Geo-physical Fluid Dynamics Laboratory (GFDL) ModularOcean Model (MOM)-3 (Pacanowski and Griffies 1999)in an enclosed, flat-bottomed, rectangular basin 238 lon-gitude in width and stretching from 128 to 488S.1 Theequations of motion solved by the model are

Du 1 ]p2 f y 5 2 , (1a)

Dt r ]xo

Dy 1 ]p1 fu 5 2 , (1b)

Dt r ]yo

]p5 2rg, (1c)

]z

Du5 k u , and (1d)y zzDt

= · v 5 0, (1e)

as well as a linear equation of state connecting densitywith potential temperature. (The viscous and forcingterms in the momentum equations and the experiment-dependent subgrid parameterizations are omitted.)These are the usual hydrostatic, Boussinesq equationswith no salinity. Thus, p is the pressure; u is potentialtemperature; r 5 r0 1 r9 is the density; u, y, and w arethe usual horizontal and vertical velocity components;and ky is the vertical diffusivity. If we write p 5 p0(z)1 r0f, where p0 is in hydrostatic balance with r0, thenthe hydrostatic relation becomes

]f5 b, (2)

]z

where b 5 2r9g/r0 is the buoyancy. For later theoreticaland scaling work we note that the thermodynamic equa-tion may be written in terms of buoyancy as

1 The experiments are carried out in a nominal Southern Hemi-spheric domain to provide comparison with later work in which weadd a circumpolar channel near the southern boundary.


FIG. 1. (top) The standard wind stress (N m22) and (bottom) thecurl of the wind stress (N m23) used in the base case. The wind stresscurl has a maximum near 238S.

TABLE 1. A summary of the experiments run with varying diffu-sivity and wind. ‘‘Standard’’ refers to the wind profile given inFig. 1.

ky (m2 s21) Wind stressLength of eddysimulation (yr)

Base caseWind 3 0.75Wind 3 1.25Wind 3 1.5Diff 3 0.5Diff 3 1.5Diff 3 2

2 3 1025

2 3 1025

2 3 1025

2 3 1025

1 3 1025

3 3 1025

4 3 1025

StandardStandard 3 0.75Standard 3 1.25Standard 3 1.5StandardStandardStandard

150150105140150100150

Db5 k b . (3)y zzDt

The model is mechanically forced using an idealized‘‘two gyre’’ zonal wind stress, which is a function oflatitude only. The wind stress curl, shown in Fig. 1, hasa maximum near 248S and it changes sign at 378S, givinga rather larger subtropical gyre than subpolar gyre. Theonly active tracer in the model is temperature and themodel is thermodynamically forced by relaxing the up-permost grid value of temperature to a prescribed valuethat falls linearly with latitude. Specifically, we use arestoring coefficient of 50 W m22 (8C)21, giving a re-storing time scale of 10 days with an the upper-layerthickness of 10 m. A simple convective adjustmentscheme and a bottom drag with a constant, dimension-less drag coefficient of cd 5 5 3 1023 are also employed,where the bottom stress is given by r0cd | u | u, where uis the horizontal velocity and | u | is its magnitude.

The simulations were carried out in several stages.First, integrations were completed at relatively coarseresolution, with a grid size of approximately 28. Thisscale is too coarse to resolve the radius of deformation,and these runs represent the noneddying cases. A uni-form Laplacian horizontal diffusion of potential tem-perature [kh(uxx 1 uyy)] was used, with a diffusivity of50 m2 s21, which is the smallest value that maintained

numerical stability. This diffusivity is two orders ofmagnitude smaller than those typically used to captureeddy mixing in low-resolution models, implying the cas-es are as close in interpretation to ‘‘noneddying’’ aspossible. Also, away from the western boundary current,the horizontal diffusion resulting from this small dif-fusivity is negligible in comparison with the verticaldiffusion, implying the Veronis effect (Veronis 1975) issmall and the cross-isopycnal mixing induced by thesubgrid-scale parameterizations is substantially capturedby the vertical diffusion. The horizontal viscosity is alsoLaplacian and is adjusted so that the Munk westernboundary layer is marginally resolved (5 3 104 m2 s21).

The vertical resolution is kept as high as computa-tional resources allow, with the same vertical resolutionused in all experiments. The resolution is 10 m at thesurface, stretched to 30 m at 500-m depth, and increasedto 890 m at 4000-m depth (the domain bottom), with atotal of 50 vertical layers. The model is first run as abase case with the given winds and with a uniformLaplacian vertical diffusivity of ky 5 2 3 1025 m2 s21

and Laplacian viscosity of 1 3 1024 m2 s21. In addition,four other cases are run with varying winds and verticaldiffusion parameters, as summarized in Table 1. Theselow-resolution cases are run at little expense; therefore,we started them from a uniform potential temperatureand integrated them for 20 000–30 000 yr to reach afully equilibrated state, and the thermocline depth andthickness are found to follow the SV scalings (notshown).

The end states of these integrations then become theinitial condition for the integrations with much higherhorizontal resolution, which develop vigorous meso-scale eddies. Thus, the final state at a 28 resolution wasinterpolated onto a 1/68 grid and the integration contin-ued.2 The vertical grid, diffusivity, and viscosity wereleft unchanged. However, the horizontal viscosity pa-rameterization was changed to a biharmonic schemeand, following a suggestion in Griffies et al. (2000), theviscosity was set at an intermediate value of 1 3 1011

m4 s21. This value is large enough to minimize indirect

2 The resolution refers to the longitudinal resolution. The latitudinalresolution varies with latitude to keep the grid nearly square and isalways finer than this resolution in terms of degrees.


dispersion errors in the tracer field due to the advectionscheme, but not so large as to overly damp the eddykinetic energy. Last, the horizontal diffusion scheme inthe thermodynamic equation was eliminated in favor ofthe Gent–McWilliams (GM) scheme (Gent et al. 1995)to eliminate tracer variance at the grid scale. Usuallythis scheme is used to mix thickness along isopycnalsand thus to mimic eddy effects in noneddy resolvingmodels; however, we use it merely as a device to adi-abatically eliminate small-scale noise. Thus, the GMcoefficient is kGM 5 5 3 1021 m2 s21, 3–4 orders ofmagnitude smaller than values typically chosen to sim-ulate eddy effects (see, e.g., Nakamura and Chao 2000;Kamenovich et al. 2000). Again, the resulting magni-tude is small compared to the vertical diffusion. Becausethe vertical diffusivity is kept constant between the low-resolution and eddy-permitting runs, then for identicalstratification profiles, both cases will have substantiallythe same cross-isopycnal subgrid-scale diffusion (al-though the eddy case may have an additional eddy-induced cross-isopycnal mixing).

The high-resolution experiments were typically runfor about 120–150 yr, which was found sufficient toequilibrate the thermocline. Much of our analysis fo-cuses on the subtropical gyre just east of the westernboundary current, and our primary ‘‘diagnostics region’’extends from about 68 to 88 east of the western boundaryand from 258 to 178S. This is an eddy-rich region wherethe source of available potential energy from the Ekmanpumping is large, but it is not within the western bound-ary layer.

3. Scaling analysis

Equations of motion

Dividing the buoyancy equation into time mean andeddying component in the usual way gives the time-averaged thermodynamic equation,

ub 1 y b 1 wbx y z

5 2(u9b9) 2 (y9b9) 2 (w9b9) 1 k b , (4)x y z y zz

where ( ) indicates a quantity averaged in time and( )9 indicates the deviation of that quantity from thetime average. The importance of mesoscale eddies tothis balance may be roughly estimated by comparingthe sizes of the two terms

= · u9b9 and = · (ub). (5)

If we take b9 ; dx · = ; LeDb/L, where Le is an eddybscale and Db and L refer to the scales of mean quantities,then the ratio of the magnitude of the eddy terms to themean advection is

U Le em ; , (6)VT UL

which is the inverse of an eddy Peclet number Pe 5

UL/UeLe. If mVT is O(1), then the eddies will participatein the buoyancy equation balance.

To understand how the parameter m might vary indifferent oceanic regimes, we can try to estimate itsmagnitude a priori. Suppose that the eddy scale is pro-portional to the deformation radius, Ldef ; ND/ f, whereN ; (Db/D)1/2 is the buoyancy frequency and D is adepth scale. If we take D to be the depth of the ther-mocline given by the classical ventilated thermoclinescaling,

1/22fL1/2D ; W , (7)a E 1 2Db

we obtain an a priori estimate of the deformation scale,

1/2 1/42DDb W DbLEL ; ø . (8)d 2 31 2 1 2f f

Using Db ; gaDT and g ø 10 m2/s, a ø 1024 K21,L ø 106 m, DT ø 20 K, and D ø 103 m, then Ld ø50 km and Ld/L ø 0.05. Observations suggest a similarvalue for the value of the deformation radius in themidlatitude ocean. Chelton et al. (1998), for example,find that the first deformation radius varies betweenabout 20 and 100 km in midlatitudes, with values ofabout 50 km being typical in the subtropical gyres. Ed-dies themselves tend to be rather larger than this, prob-ably reflecting both a weak inverse cascade and also thetendency of the length scale of maximum instability tobe larger than the deformation radius. Given this, wehave

U L Ue d em 5 ; 0.05 . (9)UL U

Observations (and our simulations, described in latersections) indicate that the eddy kinetic energy is muchlarger than the mean kinetic energy; in the main gyre,the eddy energy can be 20 times the mean energy (Wyrt-ki et al. 1976; Stammer 1997), suggesting that the eddyvelocity is 4 or 5 times the mean velocity in these cases.Supposing Ue/U to be O(1)–O(10) then m varies be-tween 0.05 and 0.5, indicating that although eddies willcertainly have some effect on the upper thermocline,they need not completely overwhelm the structure thatwould be apparent in their absence.

Other scalings are possible. If we choose the eddylength scale to be the size of the baroclinic zone, thenm 5 O(1) and eddy effects are obviously important(assuming Ue $ U), but this likely overestimates theirimportance. Geostrophic turbulence theory suggests thateddies might, by way of nonlinear effects, grow untiltheir size is limited by Rossby wave radiation (Rhines1975; Vallis and Maltrud 1993). In that case

U DbDaeL ; ; , (10)e 2! !b f L

where we have used b ; f /a where a is the radius of


FIG. 2. Relevant length scales for the thermocline problem, whereD is the depth of the thermocline, L is the length of the domain, dis the thickness of the internal thermocline, and Li is the internalhorizontal length scale in the internal thermocline.

FIG. 3. The time-averaged surface height for the (top) low-reso-lution and (bottom) eddy-permitting cases. The solid lines show pos-itive heights, and the dashed lines show negative heights. The contourinterval in both panels is 10 cm.

the earth and Ue ; U, where we have assumed U isgiven by the thermal wind relation. Using (7) for D,then

1/4 1/22W DbL aEL ; . (11)e 31 2 1 2f L

In comparison with (8), using the Rhines scale adds afactor of (a/L)1/2 to the prediction for m. This ratio isalso O(1) or somewhat larger, so that m may approachunity.

Notwithstanding the uncertainty of the scaling theory,we might expect eddy effects to be more important inthe lower thermocline than in the ventilated thermocline,at least away from the surface. This may seem coun-terintuitive since eddies tend to be strongest near thesurface; however, over their life cycle, eddies will tendto cascade energy to graver vertical scales (Smith andVallis 2002), distributing some eddy energy over theentire depth of the thermocline. The intensity of themean flow, however, falls off fairly rapidly with depthso that the ratio Ue/U may actually increase with depth.In the classical (noneddying) model the balance in thelower subtropical thermocline is between mean flow anddiffusion; the presence of mesoscale eddies suggests thepossibility of a subsurface balance involving eddy,mean, and diffusive terms. Below the mixed layer, anyeddy effects are likely to be predominantly adiabatic—that is, even though they may certainly affect the ther-mocline structure, they do so without directly changingthe census of water masses. This is not the case at thesurface, where we expect (and as we shall see) eddiescan have an explicitly diabatic effect, because of thedirect contact with the overlying atmosphere. This isbecause in the ocean interior the explicitly diabatic term,(kbzz), is small and effective only on a time scale muchlonger than an eddy turnover time, whereas at the sur-

face the exchange of sensible heat with the atmosphereoccurs on time scales comparable to an eddy time, andwater mass properties are not preserved on an advectivetime scale.

Although such general properties as the above can beinferred by analytic and intuitive reasoning, to gain morequantitative results we turn to numerical simulation.

4. Eddying and noneddying simulations

a. Qualitative overview

Figure 2 illustrates the relevant scales of motion, andFig. 3 shows the mean surface height for the low-res-olution (upper) and eddy-permitting (lower) cases in thecontrol case (see Table 1). Both steady and eddyingcases display the two-gyre structure imposed by thewind, with the division between the gyres near the lat-itude of zero wind stress curl at 358S. Both cases alsohave a strong western boundary current, and this currentseparates from the coast and penetrates to about 58E. Inthe eddying case, the western boundary region is nar-


FIG. 4. A snapshot of the surface vorticity field for the high-res-olution case. The field shows small-scale eddy structures in the west-ern boundary region, with the eddies tapering off away from thisregion. The contours in the above figure range from 24 3 1025 to4 3 1025 s21.

FIG. 5. The time-averaged surface eddy kinetic energy for the eddy-permitting base case. The eddy kinetic energy is largest in the westernboundary current and near the northern domain boundary. In thefigure, the range of values is 0–0.3 m2 s22, with a contour intervalof 0.05 m2 s22.

FIG. 6. The time-averaged temperature section for the low-reso-lution (solid) and eddy-permitting (dashed) cases (8C). The temper-ature has been averaged from 68 to 88E.

rower and stronger, although the penetration into theinterior is similar.

Figure 4 shows a snapshot of the surface vorticity forthe high-resolution base case. Disturbances identifiableas mesoscale eddies are present, typically forming inthe western boundary region where the velocity shearsare greatest and near the latitude of greatest down-welling (258S). Clearly distinguishable eddies are lessvisible polewards of 358S, where the wind stress curlchanges sign and there is no longer an input of availablepotential energy from the Ekman pumping. The eddiestypically have a horizontal scale of about 100–200 km,which is somewhat larger than the radius of deformation(which is about 50 km) and which is significantly largerthan the grid size, which is about 15 km. However, theyare still much smaller than the domain size.

Figure 5 shows the eddy kinetic energy near the sur-face for the eddying base case. Again, the eddy energyis clearly largest in the western boundary region, withsmaller values spreading outward into the gyre. Thereare also moderate amounts of eddy kinetic energy at thenorthern boundary of the domain; this energy is pro-duced by eddies that form as a result of the horizontalvelocity shear caused by the wall (these latitudes areexcluded from the analysis). The ratio of the eddy tomean kinetic energy at the surface varies from littlelarger than unity (in the eastern gyre) to about 15 (justeast of the western boundary current, where much ofour analysis is carried out) with a mean value of 7. Thisis comparable to, if a little smaller than, the real oceangyre, where the ratio of the kinetic energies has beenvariously reported as approximately 25 by Wyrtki et al.(1976), 16 by Stammer (1997), and having gyre averageof 5.6 (Richardson 1983).

Figure 6 shows a depth–latitude potential temperaturesection averaged over the gyre region for the low res-olution (solid) and eddy permitting (dashed) cases. Ingeneral, the thermocline stratification is confined to theupper 400 m. This is shallower than in the real oceansince the latitudinal extent (and thus total range of tem-peratures) has been limited in this study for computa-tional reasons, although the gradient is roughly consis-tent with that in the ocean. The model thermocline canusefully be separated into various regions: (i) the ven-tilated thermocline, which is subject to Ekman pumpingand encompasses all isotherms that outcrop between thenorthern boundary and approximately 278S; (ii) the


FIG. 7. The eddy kinetic energy averaged from 68 to 88E, plottedwith a contour interval of 5 3 1023 m2 s22. Also shown with dashedcontours are the time-averaged isopycnals. The eddy kinetic energyis largest in the former mode water region near 258–308S. This is theregion that sees the largest change when eddies are added.

FIG. 8. The time-averaged vertical temperature derivative profilefor the low-resolution (solid) and eddying (dashed) cases. The tem-perature has been averaged from 68 to 88E and from 178 to 258S. Theventilated thermocline is indicated by VT, the low-resolution modewater region by MW, and the internal thermocline by IT. The plussigns indicate the depth of the ventilated thermocline, evaluated asthe depth of the maximum temperature gradient.

mode water region, which is also subject to Ekmanpumping but has regions of isothermal water in the low-resolution case and encompasses all isotherms that out-crop between 278 and 358S; and (iii), the internal ther-mocline, which encompasses all isotherms that outcropfrom 358S to the southern boundary, in regions of Ek-man suction (i.e., the subpolar gyre). We next examinehow the eddies affect both the location of individualisotherms (the details of the stratification) and the over-all regional gradient (the mean stratification) in each ofthese regions.

b. Ventilated thermocline and mode water regions

1) THE DEPTH OF THE VENTILATED THERMOCLINE

In the uppermost region of the ventilated thermocline,changes in stratification due to eddies are modest (Fig.6). However, the isotherms that outcrop in the regionof most vigorous eddy activity change position morenoticeably. The 138, 148, and 158 isotherms deepen andoutcrop at latitudes poleward of their low resolutioncounterparts. Figure 7 shows the eddy kinetic energysection, averaged from latitudes 6 to 8. Indeed, the ed-dies are strongest in the mode water region, where thechanges in the local isotherms are most noticeable. Inparticular, the warm pool of the mode water region hasdiminished and the separation between ventilated andinternal thermocline is less apparent. The change in thelocal isotherms is also apparent in profiles of the strat-ification. Figure 8 shows time-averaged profiles of thevertical temperature gradient for both the low-resolutionand eddy-permitting cases. (The profile is also spatiallyaveraged within the diagnostics region.) The entire pro-file has been smeared out by the eddies, so that the local

slope has changed and the vertical temperature gradientis more uniform through the column.

The eddies appear to play a large role in setting thesurface outcrops and local slope in the mode water re-gion. However, this does not ensure they are affectingthe overall mean slope of the ventilated thermocline. InFig. 8, the average temperature gradient in the Ekmanpumping region (listed at the bottom of the figure andaveraged from the surface to the depth of the subsurfacemaximum) is nearly the same in the two cases, with theeddying case gradient only slightly shallower. Accord-ingly, the depth of the ventilated thermocline, definedby the depth of the subsurface maximum in the tem-perature gradient and indicated by the 1 in the figure,remains nearly unchanged. Thus, we find support thatthe eddies are affecting the individual isotherms in theventilated thermocline, but the eddies do not seem toconstrain the average gradient (or slope) in this region.The scaling for this depth is still likely set by the Ekmanpumping and a mean balance in the buoyancy equation,with little contribution from the eddies.

Analysis of the terms in the buoyancy equation showsthat, indeed, the eddy terms have a small overall effect


FIG. 9. The buoyancy equation terms for the ventilated thermocline. The terms are plotted for (left) the low-resolution case away fromthe surface, (center) the eddy-permitting case away from the surface, and (right) the eddy-permitting case near the surface. The terms havebeen averaged along constant isopycnals over the diagnostics region equatorward of 258S. The terms are negative horizontal advection,2 x 2 y; negative vertical advection, 2 z; vertical diffusion, ky zz; horizontal eddy convergence, 2( )x 2 ( )y; vertical eddyub yb wb b u9b9 y9b9convergence, 2( )z; and convection, the change in the buoyancy tendency due to the convective adjustment scheme. The horizontalw9b9diffusion term, kh( xx 1 yy), is not included because it is too small to discern on the figure.b b

on the dominant balance in the ventilated thermocline.Figure 9 shows the buoyancy equation terms for thelow-resolution case away from the surface (left), theeddy-permitting case away from the surface (center),and the eddy-permitting case near the surface (right) forthe ventilated thermocline. The various terms are time-and spatially averaged in the diagnostics region, withthe spatial average performed along isopycnal surfaces.The surface region represents the mixed layer and isidentified as grid boxes where the change in temperaturedue to convection lies above a threshold value. The restof the water column constitutes the ‘‘away from thesurface’’ region. Beneath the surface, the upper ther-mocline dominant balance is between the mean advec-tion terms in both the low-resolution and eddying cases.Away from mode water regions, the horizontal eddyflux convergence is much smaller than the mean ad-vection terms, and the vertical eddy flux convergenceis close to zero. Along isotherms 12–16.5, which isthrough the mode-water region, the eddy flux conver-gence is proportionately larger and eddies do participatein the thermodynamic balance.

In the model ventilated thermocline, we find thatUe/U ø 3 and Le/L ø 1/15. The ratio of the lengthscales is found by Fourier analyzing the surface kineticenergy and identifying the maximum energy containingscale as the location of the peak of the spectrum (notshown). Thus, the ratio of the eddy and mean terms inthe ventilated thermocline is

U Le em 5 ø 0.2. (12)VT UL

An explicit average of the buoyancy equation termsalong the isotherms in the ventilated thermocline givesa similar value for this ratio, so that the overall contri-bution of the eddy terms is small, though not whollynegligible. The eddies have affected the placement ofindividual isotherms in accordance with creating a moreuniform slope, but they have not overwhelmed the dom-inant balance found in the low-resolution buoyancyequation.

The homogenization of slope in the (former) modewater region is consistent with a reduction in the avail-able potential energy (APE) of the system. The meanvolume averaged available potential energy is defined by

21 (r 2 r̃ )APE 5 dV, (13)EV dr̃/dz

where is the time mean density field and V is thervolume of the domain (Oort et al. 1989). Here, is ther̃time-averaged density averaged at constant height overthe domain,

1r̃ 5 r dx dy, (14)EEL Lx y

where Lx and Ly are the lengths of the domain in the


FIG. 10. The divergent part of the eddy buoyancy flux vectors atthe surface. The western boundary region is excluded from the figure.This plot excludes any contribution from the vertical fluxes.

longitudinal and latitudinal directions. In our model,calculating the APE by integrating over the domaingives

23low-resolution APE ø 1320 J m and23eddy-permitting APE ø 1170 J m . (15)

To approximately quantify the APE within the ventilatedthermocline, we take the integrand in (13) as an APEdensity and compute this integral over the volume with-in the 118 isotherm, near the base of the ventilated ther-mocline. However, remains the average at constantr̃depth over the whole domain. We also exclude the west-ern and northern boundary regions from the integral tofocus on the gyre portion of the thermocline. This com-putation yields

23low-resolution APE ø 648 J m and23eddy-permitting APE ø 442 J m . (16)

The simulation with eddies has a lower value of avail-able potential energy. In the low-resolution case, themode water represents a storage of APE, since a com-paratively light water mass exists at depth and forms asharp front with the heavier water at its poleward edge(identified as the steep 138 isotherm in Fig. 6). Wheneddies are added, the 138 isotherm outcrop is pushedpoleward, the slope decreases, and the mode water isreplaced with comparatively less dense water, all con-sistent with a release of the stored APE and a lowerfinal APE state.

2) SURFACE PROCESSES

The fact that the eddy-permitting isotherms outcroppoleward from their low-resolution counterparts (Fig.6) suggests the eddies may be playing an important roleat the surface. Figure 10 shows the divergent part ofthe eddy buoyancy flux vectors and the isopycnal con-tours at the surface, where any contribution from thevertical fluxes has been excluded. In this plot, the west-ern boundary region is excluded. The eddy fluxes arelargely directed across the mean isopycnals in the eddy-rich portions of the subtropical gyre (between longitudes8 and 12 and equatorward of 358S), suggesting the ed-dies have a diabatic effect at the surface.

The right panel of Fig. 9 shows the buoyancy equationterms averaged along the ventilated thermocline iso-therms near the surface, where the isotherms tend to besteeply sloping and convection is nonzero. Surface re-storing is implemented via a vertical diffusion term, sothe large negative diffusion in the figure represents thelarge removal of heat at the surface. Aside from theregion around isotherms 13–13.5, the horizontal eddyflux convergence term mixes heat poleward across iso-therms and participates in the first-order balance in thebuoyancy equation. Though the eddies are not playinga large role in setting the slope of the isotherms withinthe ventilated thermocline, they are playing a dominant

role in setting the outcrop latitudes, implying they mayaffect the subduction of mass from the mixed layer intothe thermocline.

To determine the effect of the eddy surface processeson the structure of the thermocline, we ran the low-resolution model with the same parameters as previouslydescribed, but we restored the surface temperature tothat in the eddy permitting case using a very short re-storing time scale (so that the surface temperatures areessentially prescribed to be those in the eddying case).The result captures the low-resolution response to theeddy-determined surface outcrops. Figure 11 shows thetemperature section for the original low-resolution case(solid), the eddy-permitting case (dashed), and the low-resolution case restored to the eddy-permitting surfacetemperatures (dotted, referred to as low-resolution eddy-restored case below). This plot can be compared withFig. 6, although the contour interval has been increasedto 28C. In this case, the position of the 148C isotherm


FIG. 11. The time-averaged temperature section, similar to Fig. 6.However, the dotted line shows the low-resolution case restored tothe eddy-permitting case temperature field at the surface. The contourinterval is 28C.

FIG. 12. As in Fig. 9, but averaged along the isotherms in the internal thermocline.

of the low-resolution eddy-restored case is more similarto the eddy-permitting case than the low-resolution case.The altered surface restoring diminishes the mode waterin the low-resolution case, implying the surface eddymixing and resulting change in subduction rates into themain thermocline are largely responsible for the changesseen in the ventilated thermocline structure. We willreturn to this point in section 5a.

c. The internal thermocline

The internal thermocline is the lower portion of thethermocline containing isotherms that do not outcrop inthe region of Ekman downwelling. In noneddying mod-els this is a diffusive transition zone connecting the baseof the ventilated thermocline to the abyss, and the dia-pycnal diffusion is of first order importance no matterhow small the diffusivity. We now explore how thisstructure changes in the presence of mesoscale eddies.

THE THICKNESS OF THE INTERNAL THERMOCLINE

As seen in Figs. 6 and 8, the inclusion of this eddyflux causes the internal thermocline to thicken. Thethickness is calculated as the width of the lower half-maximum of the temperature gradient profile. In ourcontrol integration, the internal thermocline thickness is110 m for the low-resolution case and 140 m for theeddying case. Although small, this increase is robustand found in all our integrations.

Figure 12 shows the buoyancy equation terms aver-aged along the internal thermocline isotherms. In theeddying case, the horizontal eddy flux convergence isas large as the diffusion term, and together they balancethe mean vertical advection. To estimate the contributionof the eddies to the buoyancy equation, we find that theratio of the eddy to mean terms averaged over the in-ternal thermocline isotherms has increased to mIT ø 0.8.This is larger than the corresponding ratio in the ven-tilated thermocline. As noted previously, this is because


FIG. 13. The terms in the TEM form of the buoyancy equation, averaged along isotherms for the (left) ventilatedthermocline and (right) internal thermocline. The along-isopycnal eddy flux convergence is v* · = , and the cross-bisopycnal flux convergence is Gz. In regions where v* · = is not zero, the eddies are inducing a cross-isopycnal massbflux.

the eddies have a strong barotropic component so thatthe amplitude of the eddies does not decay in the verticalas quickly as the amplitude of the mean flow.

The thickening of the internal thermocline by theeddy fluxes suggests that the eddies are effectively trans-ferring thickness, or buoyant water mass, from the sur-face into the interior. From the right panel of Fig. 12,the poleward eddy fluxes are converging at the surfaceand creating a source of buoyancy. In steady state, thisbuoyancy is fluxed both to the atmosphere (identifiedas a negative diffusion term) and, by way of an eddyand mean flux, along isotherms into the interior (cf.Marshall 1997; Hazeleger and Drijfhout 2000). In theinterior, the along-isopycnal eddy buoyancy fluxes areconvergent, supplying mass to and thereby thickeningthe internal thermocline. We now diagnose this from atransformed Eulerian mean (TEM) perspective.

d. Residual circulation and the transformed Eulerianmean

Without approximation, the buoyancy equation in (3)can be written in the TEM form as

]b1 (v 1 v*) · =b 5 2G 1 k b , (17)z y zz]t

where

u9b9 y9b9 u9b9 y9b9v* 5 2 , 2 , 11 2 1 2 1 2 1 2[ ]b b b bz z z zz z x y

(18)

is a three-dimensional ‘‘eddy-induced velocity’’ and

v9b9 · =bG 5 , (19)

]b/]z

and the reader can verify that

G 1 v* · =b [ = · v9b9 (20)z

(Andrews et al. 1987). This transformation separates theeddy flux convergence (nonuniquely) into the contri-butions from the along-isopycnal and cross-isopycnaleddy fluxes. The cross-isopycnal flux convergence is re-expressed as Gz, which accounts for the convergence ofthe explicitly diapycnal eddy buoyancy fluxes. Thealong-isopycnal flux convergence is reexpressed as ad-vection by an eddy-induced, volume-conserving veloc-ity, v* (that is, = · v* 5 0). When these along-isopycnaleddy fluxes have net convergence, or v* · = is nonzero,bthen v* has a component that crosses the mean isopyc-nals. Put another way, convergent along-isopycnalbuoyancy fluxes can drive a cross-isopycnal mass flux.

Figure 13 shows the TEM form of the eddying buoy-ancy equation terms averaged along isotherms in theventilated thermocline (left) and internal thermocline


FIG. 14. The depth of the 118C isotherm at 258S. The low-resolutioncase is the solid line, and the eddy-permitting case is the dashed line.In the gyre, the depths are largely similar. However, within the westernboundary region, the eddying case shows a steeper slope over anarrower region.

(right). In the ventilated thermocline away from the sur-face, the eddy flux convergence is entirely due to thealong-isopycnal fluxes and Gz is close to zero. Alongisotherms 16–18, which lie within 50 m of the surface,Gz is large, again indicating that the eddies are explicitlycross-isopycnal and diabatic at the surface.

In the internal thermocline, where the buoyancy equa-tion is a three-way balance between the mean verticaladvection, eddy flux convergence, and vertical diffu-sion, we find that Gz is again small. Given this obser-vation, the eddy convergence can be reexpressed as aneddy-induced cross-isopycnal advection,

(u9b9) 1 (y9b9) 1 (w9b9) 5 u*b 1 y*b 1 w*b .x y z x y z

(21)

An explicit diagnosis of the individual eddy advectionterms reveals that w* z dominates over u* x 1 y* yb b b(not shown), so that the eddy convergence terms can beapproximated by w* z. The balance in the buoyancybequation becomes

(w 1 w*)b ø k b .z y zz (22)

The left-hand side of the above equation is the re-sidual mean upwelling, wres, through the base of thethermocline.

What would happen if the vertical diffusion were togo to zero? In this case vres · = must be zero and therebwould be no residual mean flow crossing isopycnals.However, unlike the low-resolution models, there seemsto be a priori no requirement that the thickness of theinternal thermocline need go to zero in this limit. Theconvergent eddy buoyancy fluxes, and their associatededdy-induced mass flux, would balance the mean up-welling. However, these convergent buoyancy fluxescould continue to mix buoyant water masses from themixed layer into the internal thermocline, thickening thefront even in the absence of explicit diffusion. We revisitthis issue in section 5b.

e. Western boundary region

In contrast to the gyre portion, the eddies play a dom-inant role in setting the stratification in the westernboundary current region. In the low-resolution case, aplot of the buoyancy equation terms reveals a balancebetween the mean advection terms and the horizontaldiffusivity (not shown). If the horizontal diffusivity isreduced, the solution becomes unsteady. In the eddy-permitting case, the horizontal diffusion term is small,but it is replaced by a large-eddy convergence term. Wemust conclude that the eddies are important in settingthe stratification within the western boundary region,and that for the low-resolution case the horizontal dif-fusivity is (perhaps unrealistically) substituting for theireffects.

Figure 14 shows the depth of the 118 isotherm in thelow-resolution and eddy-permitting cases. In the west-

ern boundary region, the slope becomes steeper and theregion as a whole becomes narrower. Clearly, the eddiesare responsible for setting the slope of the isotherms inthe western boundary region, and this slope changeswhen the eddies are explicitly included rather than pa-rameterized. Further study is warranted but is beyondthe scope of this paper.

5. Varying the winds and diffusivity

We now explore how the various properties of thethermocline, in particular its depth and thickness, varywith the strength of the wind and the diffusivity. Table1 indicates the experiments performed: the base caseprovides a pivot around which we separately vary thewind and the diffusivity, integrating each experimentfor more than 100 yr to ensure that a near-equilibriumis obtained.

a. Depth of the ventilated thermocline

In a noneddying model, the balance between the meanadvection terms implies the depth of the ventilated ther-mocline should depend on the strength of the Ekmanpumping to the half power, as given in (7). As describedin Table 1, we ran the low-resolution and eddy-permit-ting experiments with a total of four different windstrengths. It is worth emphasizing that in our model, thechanged parameter is the strength of the wind; this pa-rameter then enters the scaling theory via the Ekmanpumping. Figure 15 shows the depth of the thermoclineas a function of downwelling strength on a log plot forboth the low-resolution and eddy-permitting cases. Thebase case from our earlier analysis is given a normalizeddownwelling strength of 1. The low-resolution case,


FIG. 15. A log–log plot of the depth of the ventilated thermoclineas a function of the normalized downwelling strength for the low-resolution case (circles) and eddy-permitting case when plotted as afunction of the Ekman pumping (stars) and the eddy-permitting casewhen plotted as a function of the total downwelling (squares). Thebase case analyzed in section 4 is the case with normalized down-welling of 1.

shown with circles in the figure, scales as the Ekmanpumping (or wind strength) to the half power, consistentwith an advective balance.

For the eddying cases, the stars show the depth ofthe thermocline as a function of the Ekman pumpingstrength (as inferred from the wind strength). Thesedepths do not follow a dependence on the Ekman pump-ing to the half power, as a mean advection balance wouldsuggest; the dependence is somewhat steeper for the lowwind cases, although further analysis reveals a dominantbalance between the mean advection terms in the buoy-ancy equation in all cases. This discrepancy suggeststhe difference arises from an eddy-induced change tothe mean downwelling. The squares in the figure showthe depths as a function of the explicit mean down-welling (as opposed to the Ekman pumping velocity),where we have calculated this value as the averagedownwelling within the diagnostics region at the up-permost grid box. These depths do support a dependenceon the mean downwelling to the half power, consistentwith the balance of terms in the buoyancy equation andthe resulting scaling theory predictions. However, be-cause the mean downwelling has changed, the eddiesare affecting the thermocline depth indirectly. As wesaw in section 4b(2), the eddies are diabatic at the sur-face and tend to shift the outcrop latitudes and affectthe mean subduction from the mixed layer into the in-terior. This fact enters the scaling theory via the meandownwelling, and this eddy-induced change is small andis more dramatic for lower winds.

b. Internal thermocline thickness

In a noneddying model, the internal thermoclinethickness scales as the diffusion to the half power:

1/42fL1/2d ; k . (23)y1 2DbWe

When eddies are added, there is an eddy-induced down-welling in the region beneath the main ventilated ther-mocline where the eddy fluxes are convergent. Here,the diffusion must balance the residual mean verticaladvection,

(w 1 w*)b ø k b .z y zz (24)

We again assume that the fluxes mix down the meanbuoyancy gradient with diffusivity ke, so that

y9b9 ø 2k b , (25)e y

where ke ; UeLe. Then using (18) and assuming the xvariations are smaller than the variations in y we have

byw 1 2k b 5 [w 1 (k s) ]b ø k b , (26)e z e y z y zz1 2[ ]bz y

where s 5 2 y/ z is the slope of the isopycnals.b bLet us assume that Ue ; CU, where C is an O(1)

constant and U is the horizontal velocity scale and thatW ; Ud/L, where d is the thermocline thickness. Thisgives

Ud UL ke y1 Cs ; . (27)1 2L L d

Strictly speaking, this three-way balance cannot be usedas a scaling theory, but it is instructive to examine sev-eral limits of this scaling. If the eddy fluxes (and so ke)are zero and using thermal wind balance

Dbd DbDU ; ; , (28)

fL fLi

then (27) reduces to the classical diffusive scaling forthe thermocline thickness (23). If instead ky → 0 andthe terms on the left-hand side remain finite and balanceeach other, then

d ; sL .e (29)

That is, the thickness of the internal thermocline is theproduct of its slope and the eddy scale, a seeminglysensible result. This suggests that the thickness of thethermocline might remain finite for small diffusivity,provided it has a finite slope that is independent of thevalue of the diffusivity (in order to provide baroclinicinstability), and provided the eddy scale is nonzero. Ifthe eddy scale is the deformation radius, (DbD)1/2/ f, andnoting that the slope of the isopycnals is given by D/L,where D is the depth of the ventilated thermocline (7),we obtain


FIG. 16. A log–log plot of the thickness of the internal thermoclineas a function of the normalized vertical diffusivity for the low-res-olution case (circles) and eddy-permitting case (stars). The base caseanalyzed in section 4 is the case with normalized diffusivity of 1.

1/43 2W LEd ; . (30)1 2Dbf

A third balance is conceivable if the mean flow is weakand the eddies balance the explicit diffusion. In this casethe slope of the isopycnals is determined by the natureof diabatic forcing at the surface, rather than the wind.The balance is then

UL kes ; . (31)L d

If s ; (d/L), U ; dDb/( fL) (by thermal wind) weobtain

1/32k fLfor L 5 Le1 2Db

d ; (32)1/72 4 6 1/2k f L (Dbd)for L 5 L 5 . e d31 2Db f

The first case is the classical diffusive scaling for thedepth of the thermocline, and this arises because of thechoice Le 5 L. The second choice uses the deformationscale as the eddy scale. Other possibilities exist, but wedo not explore this case further.

If eddies are present and the diffusion and mean flowremain important, as in our base case, then the modelthermocline should lie between the two limits of (23)and (29) or (30). In this case, the thickness will havesome dependence on the diffusivity, with the power lawlying between (no eddies) and diminishing depen-1/2ky

dence (no diffusion). However, we should note that thisargument is very heuristic: no attempt has been madeto actually solve the equations, and the structure of wand w* may not support such a balance.

Figure 16 shows the thickness of the internal ther-mocline as a function of the normalized diffusivity forthe low-resolution case (circles) and eddying case(stars). The low-resolution case indicates a dependenceon the diffusivity to the half power. By comparison, theeddy-permitting thickness is thicker than the low-res-olution counterpart for all diffusivities tested. This isconsistent with the idea that the eddies are adding thick-ness by subducting buoyant water from the mixed layerinto the internal thermocline. In this portion of param-eter space, the dependence on the diffusivity remainsclose to, if somewhat shallower than, . Further in-1/2ky

vestigation, probably with a model using isopycnal co-ordinates and very high vertical resolution, would benecessary to verify that the thickness remains finite asthe diffusivity goes to zero.

6. Summary and discussion

In this paper, we have explored how the inclusion ofmesoscale eddies affects the structure of the main ther-mocline. In a case without eddies, this thermocline hasan upper, advective thermocline, a weakly stratified

mode water region, and a lower diffusive region. Wehave examined how both the details of the stratificationand the overall mean stratification change with the ad-dition of eddies, and Fig. 17 summarizes our view ofthe main subtropical thermocline away from the westernboundary current and in the presence of eddies.

We find that eddies have a quantitative and in someplaces qualitative effect on thermocline structure. Ageneral conclusion is that eddies have a largely adiabaticeffect in the ocean interior, away from the surface. Thatis to say, they do not directly transform water massproperties and their effects are, in principle, parame-terizable by an advective term. However, near the sur-face their effects are clearly diabatic with decidedlyacross-isopycnal divergent buoyancy flux terms that no-ticeably affect outcrop latitudes.

Even though the model produces a vigorous meso-scale eddy field, we find that the dynamics of the modelventilated thermocline (away from the surface) remaindominated by the mean advective dynamics, with theeddy convergence terms being smaller than mean terms.Of course, this result is to some degree a function ofhow strong the eddies are in our numerical model, butthey would have to be significantly and arguably un-realistically stronger (the eddy kinetic energy wouldhave to be an order of magnitude larger) to qualitativelyalter this balance. However, the eddies do tend to mixaway the mode water region, which represents a store-house of available potential energy in the low-resolutioncase. Indeed, the mode water is almost completely erod-ed away by the effects of eddies, suggesting that sea-sonal effects, not included in our simulations, may playa role in its maintenance in the real ocean. By changingthe buoyancy equation balance in the mixed layer, ed-


FIG. 17. A schematic of the thermocline in an eddying ocean. In the ventilated thermocline,the balance remains between the mean advection terms. At the surface in the subpolar gyre,convergent eddy fluxes drive eddy subduction into the internal thermocline. The warm surfacewater mass is spread along the isopycnals by convergent eddy fluxes, which drive an eddy-induced advection. This advection is balanced by the mean upwelling through the base of thethermocline, and residual upwelling must be balanced by the vertical diffusion.

dies also change the mean subduction rate of water intothe ventilated thermocline. This enters the scaling the-ory via the mean downwelling, so that changes to theventilated thermocline depth can be traced to the dia-batic eddy effects at the surface. Still, overall, the depthof the ventilated thermocline continues to depend on themean downwelling to the half power, as in classicalnoneddying theory.

The internal thermocline is, perhaps surprisingly,more affected by the presence of eddies than is theventilated thermocline. This arises because the intensityof eddies decays more slowly with depth than does thatof the mean flow. In the absence of eddies and in simplegeometries the base of the ventilated thermocline is co-incident with the level at which vertical velocity van-ishes. This is an advective–diffusive region, which col-lapses to a front in the limit of zero diffusivity. Thelocation of the front appears to be little altered by thepresence of mesoscale eddies, but the dynamics of thefront itself does change. Typically, the internal ther-mocline thickens: thickness created in the surface layeris effectively mixed along isopycnals by eddy fluxes,separating the isotherms of the internal thermocline andthickening the front. These eddy fluxes can be associatedwith an eddy-induced downwelling that partially bal-ances the mean upwelling through the thermocline. Anynet residual mean upwelling must then be balanced bythe diffusion. In the case of vanishingly small diffusivity(which is difficult to reach in a z-coordinate model) theresidual circulation would have to go to zero, but themean thickness of the internal thermocline might remain

finite as the eddies continue to transfer buoyant watermasses from the surface to the interior of the isopycnals.In this case potential vorticity within the internal ther-mocline might become homogenized so determining thethickness of the thermocline. Other possibilities andscalings are discussed in the text, but a definitive de-termination of the thickness of the internal thermoclinein the presence of mesoscale eddies is beyond our cur-rent grasp.

Last, we remark that although eddies may not be re-sponsible for setting the depth of the ventilated ther-mocline in midgyre, equilibration of the western bound-ary region does rely on the presence of eddies or aparameterization of their effects. Thus, for example, inthe noneddying integrations the horizontal diffusivityunavoidably plays a role in the western boundary cur-rent dynamics. Such a picture is generally consistentwith that of Radko and Marshall (2003) who find thaton a b plane, the mass balance of the thermocline isdominated by eddy shedding, but this shedding occursin the western boundary layer. As b increases eddiesbecome less important in the gyre, the interior is Sver-drupian and the gyre region depth is given by the meanadvective scaling.

In general, our results are robust across the parameterspace explored in this paper, but that space is necessarilylimited. At the surface, we have used a restoring scheme,with the restoring strength such that the heat fluxes inthe western boundary region are the correct order ofmagnitude when compared with the real ocean. Basedon the ideas of Huang (1989), Drijfhout (1994) sug-


gested that the eddy-induced modification of the surfaceoutcrops will depend on the effective advection timescale of the eddies and the effective restoring time overthe depth of the mixed layer. In our model, the restoringtime is short in comparison with the eddy advectiontime, and the eddies can modify the surface outcropsand mixed-layer buoyancy balances. Thus, a change inrestoring time scale would likely change the subductionrates out of the mixed layer and therefore change theinternal thermocline thickness. This serves to emphasizethe role of interactions between the mixed layer andmesoscale eddies, a worthy topic for further investi-gation.

Acknowledgments. We thank Kirk Bryan, Isaac Held,Paul Kushner, Timour Radko, and an anonymous re-viewer for their helpful comments. We also thank Ste-phen Griffies for help with the numerical model andJohn Marshall for providing space and PC resources toCCH. This work was supported in part by the NationalScience Foundation and by a National Defense Scienceand Engineering Graduate Fellowship and a Clare LuceFellowship to CCH.

REFERENCES

Andrews, D. G., J. R. Holton, and C. B. Levy, 1987: Middle At-mosphere Dynamics. Academic Press, 489 pp.

Chelton, D. B., R. A. deSzoeke, M. G. Schlax, K. E. Naggar, and N.Siwertz, 1998: Geographical variability of the first-baroclinicRossby radius of deformation. J. Phys. Oceanogr., 28, 433–460.

Cox, M. D., 1985: An eddy resolving numerical-model of the ven-tilated thermocline. J. Phys. Oceanogr., 15, 1312–1324.

Drijfhout, S. S., 1994: Sensitivity of eddy-induced heat transport todiabatic forcing. J. Geophys. Res., 99, 18 481–18 499.

Gent, P. R., J. Willebrand, T. McDougall, and J. C. McWilliams, 1995:Parameterizing eddy-induced tracer transports in ocean circu-lation. J. Phys. Oceanogr., 25, 463–474.

Griffies, S. M., R. C. Pacanowski, and R. W. Hallberg, 2000: Spuriousdiapycnal mixing associated with advection in a z-coordinateocean model. Mon. Wea. Rev., 128, 538–564.

Hazeleger, W., and S. S. Drijfhout, 2000: Eddy subduction in a modelof the subtropical gyre. J. Phys. Oceanogr., 30, 677–695.

Huang, R. X., 1988: On boundary value problems of the ideal-fluidthermocline. J. Phys. Oceanogr., 18, 619–641.

——, 1989: Sensitivity of a multilayered oceanic general circulationmodel to the sea surface thermal boundary condition. J. Geophys.Res., 94, 18 011–18 021.

Kamenovich, I., J. Marotzke, and P. H. Stone, 2000: Factors affectingheat transport in an ocean general circulation model. J. Phys.Oceanogr., 30, 175–194.

Karsten, R., H. Jones, and J. Marshall, 2002: The rold of eddy transferin setting the stratification and transport of a circumpolar current.J. Phys. Oceanogr., 32, 39–54.

Luyten, J. R., J. Pedlosky, and H. Stommel, 1983: The ventilatedthermocline. J. Phys. Oceanogr., 13, 292–309.

Marshall, D. P., 1997: Subduction of water masses in an eddyingocean. J. Mar. Res., 55 (2), 201–222.

Marshall, J., H. Jones, R. Karsten, and R. Wardle, 2002: Can eddiesset ocean stratification? J. Phys. Oceanogr., 32, 26–38.

Nakamura, M., and Y. Chao, 2000: On the eddy isopycnal thicknessdiffusivity of the Gent–McWilliams subgrid mixing parameter-ization. J. Climate, 13, 502–510.

Oort, A. H., S. C. Ascher, S. Levitus, and J. P. Peixoto, 1989: Newestimates of the available potential-energy in the World Ocean.J. Geophys. Res., 94 (C3), 3187–3200.

Pacanowski, R. C., and S. M. Griffies, 1999: The MOM-3 Manual,Alpha Version. NOAA/Geophysical Fluid Dynamics Laboratory,580 pp.

Radko, T., and J. Marshall, 2003: Equilibration of a warm pumpedlens on a b plane. J. Phys. Oceanogr., 33, 885–899.

Rhines, P. B., 1975: Waves and turbulence on a beta-plane. J. FluidMech., 69, 417–443.

——, and W. R. Young, 1982: Homogenization of potential vorticityin planetary gyres. J. Fluid Mech., 122, 347–367.

Richardson, P. L., 1983: Eddy kinetic-energy in the North-Atlanticfrom surface drifters. J. Geophys. Res., 88 (NC7), 4355–4367.

Robinson, A. R., and H. Stommel, 1959: The oceanic thermoclineand the associated thermohaline circulation. Tellus, 11, 295–308.

Salmon, R., 1990: The thermocline as an internal boundary layer. J.Mar. Res., 44, 695–711.

Samelson, R. M., 1999: Internal boundary layer scaling in ‘‘two-layer’’ solutions of the thermocline equations. J. Phys. Ocean-ogr., 29, 2099–2102.

——, and G. K. Vallis, 1997: Large-scale circulation with small dia-pycnal diffusion: The two-thermocline limit. J. Mar. Res., 55,223–275.

Smith, K. S., and G. K. Vallis, 2002: The scales and equilibration ofmidocean eddies: Forced-dissipative flow. J. Phys. Oceanogr.,32, 1699–1721.

Stammer, D., 1997: Global characteristics of ocean variability esti-mated from regional TOPEX/Poseidon altimeter measurements.J. Phys. Oceanogr., 27, 1743–1769.

Stommel, H., and J. Webster, 1963: Some properties of the thermo-cline equations in a subtropical gyre. J. Mar. Res., 44, 695–711.

Vallis, G. K., 2000: Thermocline theories and WOCE: A mutualchallenge. International WOCE Newsletter, No. 39, WOCE In-ternational Project Office, Southampton, United Kingdom, 3–5.

——, and M. E. Maltrud, 1993: Generation of mean flows and jetson a beta plane and over topography. J. Phys. Oceanogr., 23,1346–1362.

Veronis, G., 1975: The role of models in tracer studies. NumericalModels of the Ocean Circulation, National Academy of Sciences,133–146.

Welander, P., 1959: An advective model of the ocean thermocline.Tellus, 11, 309–318.

Wyrtki, K., L. Magaard, and J. Hager, 1976: Eddy energy in oceans.J. Geophys. Res., 81, 2641–2646.

Documents

2428 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME › bibliography › related_files › cec0401.pdf · 2428 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34 q 2004 American Meteorological Society