Parameterization of Eddy Fluxes near Oceanic...

Parameterization of Eddy Fluxes near Oceanic Boundaries

RAFFAELE FERRARI

Massachusetts Institute of Technology, Cambridge, Massachusetts

JAMES C. MCWILLIAMS

University of California, Los Angeles, Los Angeles, California

VITTORIO M. CANUTO

NASA Goddard Institute for Space Studies, and Department of Applied Mathematics and Physics, Columbia University,New York, New York

MIKHAIL DUBOVIKOV

NASA Goddard Institute for Space Studies, New York, New York

(Manuscript received 24 May 2006, in final form 12 September 2007)

ABSTRACT

In the stably stratified interior of the ocean, mesoscale eddies transport materials by quasi-adiabaticisopycnal stirring. Resolving or parameterizing these effects is important for modeling the oceanic generalcirculation and climate. Near the bottom and near the surface, however, microscale boundary layer turbu-lence overcomes the adiabatic, isopycnal constraints for the mesoscale transport. In this paper a formalismis presented for representing this transition from adiabatic, isopycnally oriented mesoscale fluxes in theinterior to the diabatic, along-boundary mesoscale fluxes near the boundaries. A simple parameterizationform is proposed that illustrates its consequences in an idealized flow. The transition is not confined to theturbulent boundary layers, but extends into the partially diabatic transition layers on their interiorwardedge. A transition layer occurs because of the mesoscale variability in the boundary layer and the associatedmesoscale–microscale dynamical coupling.

1. Introduction

Eddy fluxes of momentum, buoyancy, and materialtracers exert a profound influence on the oceanic gen-eral circulation and its associated material distributions.These fluxes must be represented in modern oceanicgeneral circulation models (OGCMs) and climate mod-els where the oceanic horizontal grid resolution is usu-ally O(100) km or larger. At this resolution all the me-soscale and microscale fluxes are subgrid scale, andtheir transport effects must be parameterized. Al-though more powerful computers may soon decreasethe feasible grid scale to a marginal mesoscale eddy

resolution of O(25) km, even finer grids of O(10) km orbetter are needed to adequately resolve the fluxes pro-duced by mesoscale motions (Paiva et al. 1999; Smith etal. 2000). Furthermore, even marginal eddy resolutionrequires some parameterization of the missing eddyfluxes (Roberts and Marshall 1998). This problem haselicited a large literature on mesoscale parameteriza-tion schemes in the oceanic interior, in addition to aneven larger literature on parameterization of micro-scale turbulent fluxes in the planetary boundary layers(BLs). At present the parameterizations do not accountconsistently for interactions between mesoscale and mi-croscale turbulence. The goal of this paper is first toreview what is known about feedbacks of microscaleturbulence on mesoscale eddy fluxes and then topresent a parameterization framework that accountsfor these interactions in the top and bottom near-boundary regions.

Mesoscale parameterizations used in OGCMs repre-

Corresponding author address: Raffaele Ferrari, Department ofEarth, Atmospheric, and Planetary Sciences, Massachusetts Insti-tute of Technology, 54-1420, 77 Massachusetts Avenue, Cam-bridge, MA 02139.E-mail: [email protected]

2770 J O U R N A L O F C L I M A T E VOLUME 21

DOI: 10.1175/2007JCLI1510.1

© 2008 American Meteorological Society

JCLI1510

sent the adiabatic release of potential energy by baro-clinic instability, as suggested by Gent and McWilliams(1990, hereafter GM), the stirring and mixing of mate-rial tracers along isopycnal surfaces, and momentumtransport by lateral Reynolds stress (e.g., Smith andMcWilliams 2003). Their quasi-adiabatic, material con-servation properties have yielded significant improve-ments in OGCM solutions (Danabasoglu et al. 1994).However, this parameterization framework breaksdown close to the top and bottom boundaries wheremesoscale eddy fluxes develop a diabatic component,both because of the vigorous microscale turbulence inBLs and because geostrophic eddy motions are con-strained to follow the boundary (i.e., bottom topogra-phy or upper free surface), while the isopycnal surfacesoften intersect the boundaries. Griffies (2004) reviewsthe main deficiencies of the adiabatic formalism at theboundaries. First, adiabatic parameterizations, like inGM produce very large tracer transports in the oceanicBLs in disagreement with observations and eddy-resolving numerical experiments. Second, the adiabaticparameterization does not include any diabatic flux ofheat and salt, which are known to play an importantrole in the heat and freshwater budgets at strong oce-anic currents, like the Antarctic Circumpolar Current(Hallberg and Gnanadesikan 2006). A different para-digm is necessary to extend eddy parameterizationsinto the BLs.

Previous discussions of mesoscale eddy parameter-izations near boundaries argue that the normal compo-nent of an eddy flux must vanish at the boundary, eventhough tangential advective and diffusive componentsare allowable and even desirable (Danabasoglu andMcWilliams 1995; Large et al. 1997; Treguier et al.1997; McDougall and McIntosh 2001; Killworth 2001).However, no explicit parameterization forms are pre-sented in these works. Common practice in ocean mod-els has been to adopt ad hoc tapering functions to turnoff the GM parameterization at the boundaries. Thisapproach has the advantage of eliminating spurious BLtransports, but it does not result in parameterization formesoscale eddy fluxes within the BLs, except for thespurious effects due to tapering. This is at odds with theobservational evidence that eddy fluxes have a strongimpact both on the material composition and exchangerate between the surface BL and pycnocline and on theair–sea fluxes of heat (Robbins et al. 2000; Price 2001;Weller 2003).

More recently Greatbatch and Li (2000) and Griffies(2004) have proposed parameterizations that includeeddy transport in the BL. The parameterizations differin some details, but in both cases the basic idea is toextend the interior eddy transport into the BLs through

analytic continuation of the interior parameterizationformulas. This is essentially the same argument origi-nally proposed by Treguier et al. (1997). In this paperwe extend the approach of Treguier et al. to explicitlyaccount for the properties of the eddy fluxes within theBLs; that is, we derive formulas for the eddy-inducedtransports that depend on the local eddy fluxes. Wethen use physical arguments about eddy statistics in theupper ocean to derive a BL parameterization. A majoradvantage of our approach is that it provides expres-sions that can be checked versus high-resolution nu-merical simulations. The analytical continuation ap-proach does not provide any guidance on how to vali-date the parameterization.

Climate models are known to be sensitive to thetreatment of mesoscale eddy fluxes in the BLs (Griffies2004; Gnanadesikan et al. 2007). Consider as an illus-trative example the upper-ocean temperature differ-ence between two global numerical simulations thatboth use the GM scheme to parameterize mesoscaleeddies but adopt two different tapering functions toturn off eddy transport in the BLs (Fig. 1). The twosimulations are run with the Massachusetts Institute ofTechnology (MIT) OGCM (Marshall et al. 1997) andare identical in everything (i.e., parameters, initial con-ditions, boundary conditions, forcing, and parameter-ization schemes) except that one uses the taperingscheme proposed by Gerdes et al. (1991) and the otheruses the tapering scheme suggested in Danabasoglu andMcWilliams (1995). The two tapering schemes producesea surface temperature differences as large as a fewdegrees in regions where the models have strong heatexchange with the atmosphere (e.g., western boundarycurrents and the Antarctic Circumpolar Current).There is no obvious pattern to suggest that one solutionis superior to the other. They both suffer from biasesgenerated by an improper treatment of eddy fluxes inthe BL.

The goal of this paper is to derive a simple closurescheme that accounts for diabatic eddy transports in theBLs, but does not suffer from the arbitrariness ofpresent tapering functions or heuristic arguments basedon quasigeostrophic theory inappropriate for the BL(e.g., Treguier et al. 1997). In section 2 we discuss guid-ing principles for the structure of mesoscale and micro-scale fluxes near the oceanic boundaries. We argue thatthere must be a vertical transition layer that separatesthe quasi-adiabatic interior and the diabatic BL. Thistransition region plays an important role in the ex-change of properties between the boundaries and theinterior, and we make it an explicit part of the param-eterization scheme. Explicit parameterization formulasthat represent the eddy transports in all three regions

15 JUNE 2008 F E R R A R I E T A L . 2771

are given in section 3. In section 4 we illustrate some ofthe implications of our parameterizations in an ideal-ized circulation. Finally, in section 5 we present ourconclusions. The implementation and impact of theseparameterizations in a global oceanic model is the topicof a separate paper (Danabasoglu et al. 2008).

2. Mesoscale and microscale fluxes in the generalcirculation

The oceanic general circulation is significantly af-fected by a variety of processes occurring at space andtime scales too small to be resolved explicitly inOGCMs, and their influences need to be parameterizedwith variables that are explicitly included in the models.In terms of a Reynolds decomposition of variables intoa slowly changing “mean” and fluctuation components,we seek a representation of the eddy fluxes u�i c� forscalar variables c (including buoyancy) and u�i u�j for mo-mentum, where the prime superscript denotes a fluc-tuation and the overbar denotes an average over fluc-tuations. It is physically sensible to further separate thefluctuations in two dynamical classes: the “momentumbalanced” mesoscale eddies (McWilliams 2003) and the

microscale “unbalanced” turbulence (Joyce 1977; Davis1994; Garrett 2001). This suggests using a triple decom-position of variables into mean, mesoscale, and micro-scale components and writing c � cm � ce � ct withsubscripts m, e, and t denoting the mean, mesoscale,and microscale, respectively. Angle brackets will indi-cate large-scale averages; that is, cm � �c�. The meanbalance for any advectively conserved tracer c has thefollowing form in terms of the triple decomposition:

�tcm � um · �cm � �� · �uece� � � · �utct� � �m �1�

in which �m represents the mean sources and sinks of cexplicitly resolved in models. In large-scale OGCMsthat do not adequately resolve the eddies, parameter-izations are needed for both mesoscale and microscaletracer fluxes on the right side of (1). An analogous issuearises in the mean momentum balance where param-eterizations are needed for mesoscale and microscaleReynolds stresses.

Equations such as (1) are valid under the assumptionthat there are spectral gaps among mean, mesoscale,and microscale components, so that correlations be-tween fluctuations on different scales can be neglected.A gap plausibly exists between mesoscale, momentum-

FIG. 1. Temperature differences at 170 m between two equilibrated climate simulations runwith the MIT OGCM. The simulations are identical except for the use of different taperingfunctions applied to the GM scheme. The two tapering functions used are those proposed byDanabasoglu and McWilliams (1995) and Gerdes et al. (1991) and described in section 5. TheGM schemes use a diffusivity of GM � 1000 m2 s�1. The simulations are run for 1000 yearswith a horizontal resolution of 4°. (Simulations run by Alistair Adcroft.)


Fig 1 live 4/C

balanced currents and microscale, unbalanced turbu-lence, such as breaking internal waves, shear instability,double diffusion, and mixing in the surface and bottomBLs. It is less clear that such a gap exists between thelarge-scale circulation and the mesoscale eddies (Scottand Wang 2005). Nonetheless, a dynamical gap doesexist, and it is given an operational definition by thechoice of an OGCM’s horizontal resolution and eddydiffusivities. A coarsely resolved, diffusive OGCM hasa mean circulation but no eddies, while a finely re-solved OGCM with a more advective circulation pro-duces vigorous mesoscale eddies (though they may bedifficult to resolve accurately). This distinction allowsus to talk separately about mean and eddy motions.

In the oceanic modeling literature eddy parameter-izations are usually derived under the assumption thatmesoscale and microscale fluxes act independently. Thejustification is that mesoscale eddies represent the adia-batic rearrangement of buoyancy surfaces and tracersunder the influences of gravity and rotation, while mi-croscale turbulence controls all the irreversible, dia-batic processes that modify the buoyancy and tracerconcentrations of the water parcels. The microscale tur-bulent fluxes are typically parameterized as verticaldowngradient with Fickian laws both for momentumand for tracers (Gregg et al. 2003). The diffusivities and viscosities are set to small, albeit climatically im-portant, values in the oceanic interior, consistent withdirect measurements (Ledwell et al. 1993; Toole et al.1994). Turbulence is intensified near the oceanicboundaries by boundary fluxes (i.e., wind stress andbuoyancy fluxes at the surface and drag at the bottom)and consequent flow instabilities. This turbulence gen-erates overturning motions that often make the buoy-ancy and velocity profiles well mixed in the BL. Tradi-tional BL models (e.g., Kraus and Turner 1967; Mellorand Yamada 1974; Price et al. 1986; Large at al. 1994)represent these fluxes with vertical profiles for (z) and(z) that are strongly enhanced compared to interiorvalues.

The parameterization of mesoscale eddy fluxes is stillin its infancy. Most formulations are local in the sensethat the eddy term is calculated with local values andgradients of the resolved quantities. The few exceptionsto the rule include parameterizations that depend onvertically integrated quantities and are therefore non-local in the vertical. Parameterizations are derived foreddy tracer fluxes, such as temperature, salinity, andbiogeochemical quantities (GM; Greatbatch and Lamb1990; Visbeck et al. 1997; Treguier et al. 1997; Killworth1997), while retaining simple horizontal diffusion foreddy momentum fluxes with an eddy viscosity as smallas is consistent with numerical stability. This approach

has been shown to capture the most important eddyeffects on the mean circulation in the limit of smallRossby numbers typically found in the ocean (Treguieret al. 1997; Drijfhout and Hazeleger 2001; Wardle andMarshall 2000). These schemes can be extended to ac-count more accurately for eddy momentum fluxes(Wardle and Marshall 2000; Smith and McWilliams2003; Plumb and Ferrari 2005; Ferreira et al. 2005), butexperience with these extensions in OGCMs is still lim-ited.

Most of the existing mesoscale parameterization pro-posals (section 1) are intended for the oceanic interiorwhere the effect of eddies is predominantly adiabatic.However, near horizontal and vertical boundaries ed-dies can develop diabatic behavior, and some ad hocform of adjustment has to be made to avoid false eddytransports through the solid boundaries (Danabasogluand McWilliams 1995; Large et al. 1997; McDougalland McIntosh 2001; Killworth 2001). We formulate aparameterization scheme that expresses the essentialdiabatic nature of eddy fluxes in the BLs as a modifi-cation of the existing adiabatic parameterizations. Weborrow the basic framework of Treguier et al. (1997)but extend it beyond quasigeostrophic theory. As astarting point we divide the ocean into three differenttypes of layers according to different properties of themesoscale fluxes (Fig. 2).

a. Oceanic interior

Eddy fluxes in the oceanic interior are largely alongisopycnal surfaces with a much weaker diapycnal com-ponent. Thus it is useful to project the full flux alongand across the mean isopycnal surfaces and consider

FIG. 2. A conceptual model of eddy fluxes in the upper ocean.Mesoscale eddy fluxes (blue arrows) act to both move isopycnalsurfaces and stir materials along them in the oceanic interior, butthe fluxes become parallel to the boundary and cross density sur-faces within the BL. Microscale turbulent fluxes (red arrows) mixmaterials across isopycnal surfaces, weakly in the interior andstrongly near the boundary. The interior and the BL regions areconnected through a transition layer where the mesoscale fluxesrotate toward the boundary-parallel direction and develop a dia-batic component.


Fig 2 live 4/C

the two components separately (Andrews and McIn-tyre 1978; Ferrari and Plumb 2003),

�uebe� � ��uebe� � �bm

|�bm|2� �bm �

�uebe� · �bm

|�bm|2�bm,

�2�

where b is buoyancy. The decomposition is shown inFig. 3. After taking its divergence, the cross-gradientflux component—the so-called skew flux—is equivalentto a mean buoyancy advection,

� · ��uebe� � �bm

|�bm|2� �bm��

�uebe� � �bm

|�bm|2 � · �bm.

�3�

The nondivergent, eddy-induced velocity ume is givenby the curl of the vector streamfunction �:

ume � � � �, � � ��uebe� � �bm

|�bm|2. �4�

The velocity ume represents the rectified transport gen-erated by eddies advecting buoyancy perturbationsalong mean isopycnal surfaces.

The residual buoyancy flux, Fe{b}, is directed acrossmean isopycnal surfaces and represents eddy mixing ofwater masses with different mean buoyancy (Plumband Ferrari 2005),

Fe{b} ��uebe� · �bm

|�bm|2�bm. �5�

The buoyancy budget can then be conveniently written as

�tbm � �um � ume� · �bm � �� · Fe{b} � � · �utbt� � �m.

�6�

In the oceanic interior, mesoscale eddies satisfyquasigeostrophic scaling (Treguier et al. 1997) and the

expressions for the vector streamfunction and the dia-batic flux at leading order in Rossby number reduce to

� z � �uebe�

�zbm, Fe{b} � 0. �7�

McDougall and McIntosh (2001) point out that diapyc-nal fluxes, despite being small, appear to be of climaticimportance when estimated according to (5). However,direct measurements exclude large diapycnal fluxes inthe oceanic interior and suggest that eddy fluxes mustbe aligned with the instantaneous isopycnals. The solu-tion to the conundrum is that the diapycnal componentis a result of averaging at a fixed vertical position in-stead of averaging along fixed isopycnal surfaces. Mc-Dougall and McIntosh (2001) hence propose to formu-late eddy parameterizations in isopycnal coordinates.Adopting isopycnal averages, however, raises problemsin the unstratified BLs because isopycnal coordinatesbecome ill defined. Furthermore, Killworth (2001) findsthat parameterizations based on isopycnal coordinatesdo not have more skill than z-based parameterizationsthat ignore residual buoyancy fluxes. We therefore pro-ceed with a formulation in z coordinates and set theresidual flux in (7) to zero.

For passive tracers c � b, there is an additional along-isopycnal residual eddy flux component in Fe{c} ��uece� � � � �cm, whose effect on mean buoyancy istrivial. This residual flux appears at leading order inRossby number and must be parameterized (Solomon1971; Redi 1982).

b. Boundary layer

In the turbulent BLs the eddy flux component nor-mal to the boundaries vanishes, so the fluxes becomeparallel to the boundaries. We assume that each of thenormal components of mean, mesoscale, and micro-scale velocities vanish separately. The implication isthat mesoscale eddies mix tracers along the boundariesand are not constrained to be along isopycnal surfaces.The decomposition in (2) is not very useful to progresstoward eddy parameterizations in such a situation. Anobvious problem is that the eddy-induced velocity andthe residual flux do not individually satisfy no-normalflux boundary conditions, even though the full eddybuoyancy flux does. For a boundary with a unit normalvector n, the normal components of the eddy-inducedvelocity and the residual flux are

ume · n � � � � · n, Fe{b} · n � �� n� · �bm. �8�

Both components imply an eddy buoyancy flux acrossthe boundaries whenever the vector streamfunction hasa component along the boundaries, but they cancelwhen added together. The issue disappears in the pres-

FIG. 3. Decomposition of the buoyancy eddy flux �uebe� into twocomponents, along and across mean buoyancy surfaces bm. Thealong-isopycnal component � � �bm represents the advectiveskew flux, while the across-isopycnal component Fe{b} representsthe diapycnal residual flux.


ence of a well-mixed BL where �bm · n → 0 approach-ing the boundary. In this case � has no componentalong the boundary,1 and Fe{b} is along the boundary.However, BLs can be stratified and, in such cases, onecan retain the desirable property of vanishing cross-boundary eddy-induced advection by using an appro-priate gauge transformation in the definition of �. Adetailed derivation is presented in appendix A. Herewe write the result for the case when the boundaries arehorizontal,

� � ��webe�

|�horbm|2z � �horbm �

�uhebe� � �horbm

|�horbm|2,

�9�


|�horbm|2�horbm, �10�

where uhe and �hor are the horizontal components ofthe eddy-induced velocity and gradient operator. Asimilar decomposition is introduced by Held andSchneider (1999) for a zonal flow. These definitionsdiffer from (4) and (5), but the difference is physicallyinconsequential because the two definitions give thesame flux divergences. Most importantly, at the bound-aries the first term in (9) vanishes and the eddy-inducedvelocity and the residual flux are horizontal. Thus, byusing an appropriate gauge transformation, we haveeliminated any cross-boundary advection. A general-ization to arbitrary boundaries is given in appendix A.In this paper we restrict our analysis to surface andbottom BLs, both because side boundaries are compu-tational artifacts in OGCMs (compared to coastlines innature) and because approximate thermal wind balanceand no-slip, insulating boundary conditions usually im-ply weak near-boundary flow.

The expressions (9) and (10) confirm that there is nophysical justification to taper eddy parameterizations atthe oceanic boundaries. Observations (Weller 2003)and eddy-resolving numerical simulations (Oschlies2002) show that mesoscale fluxes penetrate into thesurface BL, and neither the eddy-induced streamfunc-tion nor the residual flux vanish. Our goal is to findguiding principles to parameterize these fluxes. As wediscuss in section 3, mixing length arguments suggestthat the horizontal fluxes of buoyancy are directeddown the mean buoyancy gradient both in the oceaninterior and at the boundaries, �uhebe� � �GM�horbm.

Substituting the downgradient closure in (9) and (10),the parameterization problem is reduced to choosingthe appropriate forms for the eddy diffusivity GM andthe vertical fluxes.

Within the BLs momentum and tracers are well ho-mogenized in the vertical as a result of all turbulentprocesses that actively mix the BLs on time scalesshorter than the mesoscale eddy turnover times. Con-tinuity then demands that the mesoscale vertical veloc-ity is linear in z. Hence, we expect the mean climato-logical gradients �Hbm and the horizontal eddy fluxes�uhebe� to be constant across the BL and the verticalflux �whebe� to be linear in z and vanish at the boundary.Both approximations are in agreement with resultsfrom the eddy-resolving numerical experiments de-scribed in Cessi et al. (2006).

Marshall (1997), Treguier et al. (1997), and Marshalland Radko (2003) assume that the eddy-induced massexchange between the BL and the interior is continu-ous. These heuristic arguments have been recently veri-fied in eddy-resolving numerical simulations (Cessi etal. 2006; Cerovecki et al. 2006, manuscript submitted toJ. Phys. Oceanogr., hereafter CPH). Continuity of �across the BL base, together with the assumption thatany along-boundary transport does not change in theacross-boundary direction, implies that the verticalcomponent of � vanishes in the BL because it is zero inthe interior as per (9). The horizontal component of �,which represents overturning circulations, must go tozero together with the vertical eddy flux. These simplerules are used in the next section to build a parameter-ization scheme.

The residual flux (10) is small in the oceanic interiorbecause of a large cancellation between �uhebe� · �horbm

and �webe��zbm. The first component does not changethrough the BL base, while the second component es-sentially vanishes because the vertical stratification be-comes much weaker. As a result, in the BL there is alarge residual flux in the horizontal that crosses densitysurfaces (Treguier et al. 1997). A schematic of the ver-tical structure of the eddy transport in the surface BL isshown in Fig. 4. There is no consistent treatment ofthese fluxes in present parameterization schemes.

In this section we argue that the eddy-induced hori-zontal velocities (given by the vertical derivative of �)and the residual fluxes are independent of depth withinthe BL. In the absence of any shear, mesoscale eddiescannot drive any restratification within the BL. Thisseems at odds with Oschlies’s (2002) result that the BLdepth is reduced in eddy-resolving numerical models asa result of increased surface velocity shears. HoweverYoung (1994), Haine and Marshall (1998), and Bocca-letti et al. (2007) show that the enhanced shears are not

1 The normal component of the eddy-induced velocity is givenby � · [� � n] � � · � � n. In a well-mixed BL, the first termvanishes because � is directed along n. The second term alsovanishes because n · � � n � 0 for any differentiable surface(Sneddon 1957).


associated with mesoscale eddies, but with submeso-scale processes like frontogenesis and frontal instabili-ties within the BL. Mesoscale shear is dominated bybaroclinic mode one (Wunsch 1997). Hence changes inmesoscale eddy velocity through the BL scale with theratio of the BL depth to the vertical scale of mode oneare typically less than 10% of the total velocity. Forpresent purposes, we can safely ignore such smallshears and assume that lateral mesoscale fluxes are, toleading order, depth independent.

Submesoscale eddies drive weak horizontal fluxescompared to mesoscale motions but dominate the ver-tical fluxes of buoyancy and tracers (Boccaletti et al.2007). The vertical fluxes drive ageostrophic circula-tions that tilt isopycnals from the vertical to the hori-zontal and achieve restratification. Fox-Kemper andFerrari (2008) show that submesoscale restratificationcan be parameterized independently of mesoscale pro-cesses with an additional eddy-induced overturningstreamfunction that acts only within the BL. A treat-ment of submesoscale dynamics goes beyond the scopeof this paper and the reader is referred to Fox-Kemperet al. (2008) for a thorough discussion. Notice, however,that the framework described in section 3 can be easilyextended to include the submesoscale parameterizationscheme described in Fox-Kemper and Ferrari (2008).

c. Transition layer

Boundary layer thickness h and tracer distributionsexhibit subsynoptic and mesoscale heterogeneity (Pol-

lard et al. 1996; Legg et al. 1998; Ferrari and Rudnick2000; Weller 2003). An average over this heterogeneityresults in a transition layer with diabatic mixing oflarge-scale tracer distributions at a rate intermediatebetween the large BL and small interior rates. We de-fine the transition layer as the layer containing allisopycnals within an averaging area and time intervalthat are intermittently exposed to strong turbulent mix-ing. This can occur either by entrainment into the BL asa result of mesoscale and internal wave heaving (i.e.,vertical isopycnal displacements) or by other processes,such as subsynoptic surface fluxes, which induce sub-grid-scale changes in h relative to isopycnal surfaces.The bottom of the transition layer is often character-ized by large stratification and vertical tracer gradientsgenerated through turbulent BL entrainment. Strongstratification plays an important role in parameteriza-tion schemes for microscale turbulent fluxes (Price etal. 1986; Large at al. 1994; Large et al. 1997) in relationto the rate that tracer and momentum anomalies passbetween the BL and the mostly geostrophic interiorflow.

From a mesoscale parameterization perspective, thetransition layer represents the region that connects theadiabatic, eddy-induced, and tracer mixing in the strati-fied interior with the boundary-parallel, diapycnalfluxes in the BL. The thickness of the transition layer Dis not well known yet, and analyses of relevant mea-surements and eddy-resolving computations are neededto provide better guidance. We present some prelimi-nary scaling analysis to estimate D.

FIG. 4. Eddy-induced circulation (light gray contours) and diabatic residual flux (dark grayarrows) in the meridional plane of a hypothetical zonally averaged channel flow. The blacklines represent the mean buoyancy surfaces that outcrop at the surface in response to diabaticsurface fluxes. The eddy-induced circulation, based on the definition in (9), goes linearly tozero at the surface within the BL in response to the strong vertical mixing that erases meso-scale vertical shears. A diabatic flux parallel to the boundary appears at the surface, as definedin (10). (right) Matching of the eddy-induced circulation between the BL and the interioroccurs through a transition layer according to the expression in (25). The streamfunction andits vertical derivative are continuous everywhere and give a continuous eddy-induced velocity.The diabatic flux acts both in the boundary and transition layers. (left) Eddy-induced circu-lation obtained setting the transition layer thickness to zero. A discontinuity develops in thederivative of � and the associated eddy-induced velocity. Such discontinuities are unphysicaland occasionally trigger convective instabilities (Griffies 2004).


The magnitude of D is set both by subsynoptic andmesoscale fluctuations of h. The first contribution is therange of instantaneous h values around its time- andarea-averaged value as diagnosed from a microscale BLparameterization applied locally. Danabasoglu et al.(2008) find that the base of the mixed layer tracks wellthe deepest values reached by the BL in the recent past.Hence, the difference between BL and mixed layerdepth is a useful proxy for the synoptic contribution toD. Johnston and Rudnick (2007, submitted to J. Phys.Oceanogr.), using CTD and ADCP data, find that thebase of the transition layer owing to subsynoptic vari-ability is typically 10% deeper than the mixed layerdepth because turbulent mixing partly penetrates intothe stratified layer below the BL. This difference can besafely ignored at the level of approximation implicit inmesoscale parameterizations.

The second contribution to D is due to eddy heaving.A fluid parcel within a mesoscale eddy undergoes avertical displacement �e as it is adiabatically advectedalong a tilted isopycnal surface; hence �e be/�zbm

when �zbe is neglected in the denominator. As a result,particles within a distance �e from the BL base are epi-sodically lifted into the BL and experience diabaticmixing. An estimate of the heaving contribution to thetransition layer is the set of vertical levels z for whichthe root-mean-square �e is larger than the distance fromthe BL base h,

�rms � ��e2�

��be2�

�zbm� |z � h|. �11�

Kuo et al. (2005) computed the transition layer due toeddy heaving in a high resolution simulation of an oce-anic jet configured to mimic the Antarctic CircumpolarCurrent system. The model was run without a BLscheme, and no mixed layer developed at the oceanicsurface. Despite the lack of a surface BL, eddy fluxeshad a diabatic component within a transition layer in-cluding all levels for which �rms was larger than thedistance from the surface. This simulation confirms thatthe diabatic component of the mesoscale eddy fluxes isnot confined to the mixed layer, contrary to what isassumed in Treguier et al. (1997) and Griffies (2004).Diabatic fluxes develop wherever isopycnals outcrop atthe surface in response to surface boundary conditions,regardless of whether there is a homogenized surfacemixed layer.

Fluctuations of BL depth on synoptic time scales arewell documented in the literature (e.g., Davis et al.1981a,b). Here we use a combination of climatologyand satellite data to estimate the eddy heaving contri-bution to the transition layer as defined in (11). TheWorld Ocean Atlas climatology (Conkright et al. 1998)

is used to compute �zbm. Buoyancy fluctuations be areestimated from altimetric measurements, assuming eq-uipartition between eddy kinetic and available poten-tial energies (Larichev and Held 1995; Eden 2007), sothat

�rms �� be2�

��zbm�2 ��|ue|2�

�zbm. �12�

The near-surface eddy kinetic energy per unit mass ona global scale is computed from altimetric observationsof sea level anomalies through the thermal wind rela-tionship (Stammer 1997). The sea level data used arefrom the final combined processing of the Ocean To-pography Experiment (TOPEX)/Poseidon and Euro-pean Remote Sensing Satellite (ERS)-1/2. Anomaliesare computed over a 5-yr time interval (October 1992–October 1997). Details of the calculation are given inLe Traon et al. (1998). Wunsch (1997) confirmed thataltimetric estimates capture the bulk of the eddy kineticenergy associated with mesoscale eddies in low baro-clinic modes needed for our calculation.

In Fig. 5a we show the transition layer depth esti-mated as the set of isopycnals that are occasionally en-trained in the surface mixed layer according to (12).The mixed layer base is defined as the depth wherepotential density is 0.1 kg m�3 larger than the surfacevalue (Fig. 5b). The calculation is done in terms of MLdepth instead of BL depth because there are no globalestimates of BL thickness. The transition layer is mostsignificant in regions of enhanced eddy activity, like theAntarctic Circumpolar Current and western boundarycurrents where it can be as deep as the mixed layerdepth. This estimate is likely to be an upper bound onD because �rms is calculated using annually averagedvertical stratification. Annual averages tend to smoothout the strong stratification that often develops at theBL base and result in an overestimate of D.

Equation (11) can be used to derive an estimate oftransition layer thickness in terms of mean climatologi-cal variables that are available in coarse resolutionmodels. Following Large et al. (1997) we express thebuoyancy fluctuations be as the product of a horizontaleddy mixing length R times the mean horizontal densitygradient |�horbm|. This scaling is supported by altimetricobservations with the eddy mixing length given by thefirst baroclinic Rossby deformation radius (Stammer1997),

R 1

�| f | ��H

�

��zbm�1�2 dz�, �13�

with H the oceanic depth and � the free surface eleva-tion. Within a few degrees of the equator, (13) must bereplaced by the equatorial deformation radius as dis-


cussed in Griffies (2004). Under these approximations(11) becomes

�rms |�horbm|

�zbmR � |z � h|. �14�

This expression further captures some of the synopticvariability of BL depth because it includes any mixedlayer below the BL where �zbm 0. Mixed layersdeeper than the BL are part of the transition layer.They are a signature of regions where deep mixing tookplace in the past but is no longer active.

Now that we have an operational definition for thedepth of the transition layer, we can proceed to deriveguiding principles to parameterize the transition fromdiabatic to adiabatic eddy transport. In the transitionlayer the mean stratification is not necessarily zero, andmean buoyancy gradients can change in the vertical.However, eddy statistics transition smoothly betweenthe interior and BL values (Cessi et al. 2006; Kuo et al.2005). From a parameterization perspective this sug-gests that the full diffusivity tensor � that relates me-soscale fluxes and mean gradients must changesmoothly though the transition layer,

�uebe� � ��bm. �15�

Using the decomposition in (9) and (10), the diffusivitytensor is composed of three terms: a symmetric termrepresenting lateral diffusion across isopycnals,

Fe{b} · �bm

|�horbm|2�

�uebe� · �bm

|�horbm|2, �16�

and two antisymmetric terms representing the eddy-induced overturning streamfunctions in the x–z and y–zplanes,

� � z � ��webe�

|�horbm|2�horbm. �17�

The eddy-induced overturning streamfunction in thex–y plane � · z is set to zero as required by continuitywith the interior. Continuity of the diffusivity tensorthen implies that both (16) and (17) are continuouswithin the transition layer and merge smoothly at theboundary and transition layer bases. Given continuityof the buoyancy field and its derivatives, this impliescontinuity of the fluxes.

3. Mesoscale eddy parameterization

We can now proceed to develop parameterizationsfor the eddy-induced velocity and the residual tracerfluxes applying the formalism developed in section 2.

FIG. 5. (a) Transition layer depth estimated as the ratio of the altimetric eddy kinetic energy and theclimatological stratification at the mixed layer base, see (11) (TOPEX/Poseidon data); (b) mixed layerdepth defined as the depth where potential density is 0.1 kg m�3 larger than the surface value (World OceanAtlas).


Fig 5 live 4/C

a. Oceanic interior

In most OGCMs mesoscale fluxes in the interior arecomputed according to the parameterization first pro-posed by GM; that is, the fluxes are prescribed to bealong mean density surfaces with the horizontal com-ponent satisfying a downgradient closure,

�uhebe� � ��GM�horbm, �webe� � �GM

|�horbm|2

�zbm.

�18�

Substituting these flux forms in the expressions for theeddy-induced streamfunction and residual flux into (5)and (7), or equivalently (9) and (10), yields

�I � ��GM

z � �horbm

�zbm, Fe{b} � 0. �19�

This form is the small-isopycnal-slope approximation inz coordinates for the thickness flux parameterization inisopycnal coordinates originally proposed by GM. Thisparameterization causes depletion of mean availablepotential energy (e.g., as in local baroclinic instability)and, implicitly, vertical diffusion of mean horizontalmomentum (e.g., as in isopycnal form stress for a geo-strophic flow). The horizontal diffusivity can be set ei-ther to a constant throughout the whole ocean or, fol-lowing OGCM modeling practices after Smagorinsky(1963), it can be made flow dependent (Visbeck et al.1997; Killworth 1997; Large at al. 2001; Smith andMcWilliams 2003; Smith and Gent 2004).

The parameterization for the residual fluxes of trac-ers other than buoyancy is represented through addi-tional isotropic diffusion along isopycnals as first sug-gested by Solomon (1971) and Redi (1982),

Fe{c} � �GM��cm � �bm

|�bm|2 � � �bm. �20�

We assume that the along-isopycnal diffusivity isequivalent to the GM diffusivity because other choiceswould lead to different mixing rates of buoyancy andtracers along the boundaries. CPH compare diffusivi-ties estimated from passive and active tracers in theupper layers of an ocean simulation and do not findsubstantial differences. However, our expressions canbe generalized to allow for different along-isopycnaland GM diffusivities if future studies suggest that theyare indeed different.

The parameterization form in (20) is often replacedby its small-slope approximation, which is numericallymore stable and less expensive (Griffies 2004),

Fe{c} � ��GM��horcm � �zcmS� � �GM�|S|2�zcm

� S · �horcm�z. �21�

Here S � ��horbm/�zbm is the isopycnal slope vector.We will use the small-slope approximation throughoutthe paper, although all expressions can be extended tofinite slopes.

Ferreira et al. (2005) recently proposed parameteriz-ing eddies in models written in a transformed Eulerianmean formulation. In this formulation mesoscale eddiescontribute as a stress �eddy in the mean momentumequation given by

�eddy � f �I, �22�

and through the residual fluxes Fe{b} and Fe{c} in thebuoyancy and tracer equations. From our perspectiveany of GM, Ferreira et al. (2005), or some other pa-rameterization for (7) are equivalent. Our purpose is tospecify how the interior prescriptions for �I, Fe{b}, andFe{c} should be modified near the boundaries.

b. Boundary and transition layers

In section 2 we argue that the symmetric and anti-symmetric components of the mesoscale diffusivity ten-sor make a smooth transition from the interior to thesurface values. We now derive a parameterizationscheme consistent with these arguments.

The horizontal eddy fluxes are down the mean buoy-ancy gradient in the oceanic interior, representing thetendency of mesoscale eddies to reduce lateral buoy-ancy gradients and release available potential energy.Consistent with the argument that eddy statistics arecontinuous throughout the boundary and transition lay-ers (section 2), we require that the horizontal eddy flux�uhebe� remains down the horizontal buoyancy gradient,as in (18), with a continuous eddy diffusivity GM. Thevertical component of the mesoscale eddy flux �webe�cannot be given by (18) because the vertical flux de-creases to zero toward the surface, both to satisfy theno-normal-flow boundary condition and because theisopycnal eddy displacements be are reduced by thegravitational stiffness of the air–sea interface. In the BLwe expect �webe� to be linear in z and �horbm to beindependent of z because strong mixing erases verticalgradients in both the horizontal mesoscale velocitiesand in the mean buoyancy gradients (section 2). A formfor the vertical flux that satisfies these requirementsand gives a continuous and differentiable eddy-inducedstreamfunction � is

�webe�

|�horbm|2� �

z � �

h � �

�webe�

|�horbm|2�z��h

for � h � z �.

�23�

The subscript z � �h indicates that the vertical fluxand horizontal gradients are evaluated at the BL base


(z � �h). The h in (23) is the large-scale boundarylayer thickness averaged in time over any rapid BLfluctuations during which the eddy field is unlikely tochange much, say a week. The time averaging is neces-sary because eddies respond only to low-frequencyvariations in BL depth, not to sudden deepening pulsesor the diurnal cycle.

An arguably simpler choice is to set �webe� to be alinear function of z and neglect the |�horbm|2 terms in(23). However, the resulting expression gives a nondif-ferentiable eddy-induced streamfunction and infinite

eddy-induced velocities whenever �horbm vanishes.This can be checked by substituting (23) into the ex-pression for the eddy-induced streamfunction in (9).

The vertical fluxes in the transition layer are not lin-ear in z because mixing is active only intermittently andshears can develop through geostrophic adjustment.Continuity of the eddy statistics requires that � is con-tinuous across the BL and transition layer boundaries.The simplest choice that allows for shear and hasenough degrees of freedom to satisfy continuity of �,and its vertical derivative is

�webe�

|�horbm|2�

�z � h�2

D2 ��h � D � �

h � �

�webe�


��webe�

|�horbm|2�z��h�D�

�z � �

h � �

�webe�


, for �h � D � z � h. �24�

The transition layer thickness D is estimated withthe algorithm described in section 2c. The subscriptz � �h � D indicates that the vertical flux and hori-zontal gradients are evaluated at the base of the tran-sition layer (z � �h � D). The expressions for thevertical eddy fluxes in (23) and (24) are continuousboth at the BL base and at the transition layer base.The choice of imposing continuity of �webe�/ |�horbm|2

instead of �webe� guarantees that � remains continuouseven where �horbm vanishes.

We now have analytical expressions for the meso-scale fluxes in the boundary and transition layers. To

complete the parameterization, we must constrain theremaining two free parameters: the vertical fluxes at theboundary and transition layer bases in (23) and (24).This is done by requiring that the vertical derivative ofthe eddy transport, �z�, is continuous at the bases ofthe boundary and transition layers. Substituting theseflux forms in the expressions for the eddy-inducedstreamfunction in (9) gives

� � ��GMG�z�z ��horbm

�zbm|z��h�D, �25�

where the vertical structure function G(z) is defined by

G�z� � ��

z � �

2�h � �� D �2 �D

�, for �h � z � �BL�

��z � h�2

�h � D � ��2 � �h � ��2 �1 �D � h � �

� �z � �

2�h � �� D �2 �D

�, for �h � D � z �h

�transition layer�.�26�

The function G(z) guarantees continuity and differen-tiability through the boundary and transition layers. Atthe base of the transition layer G � 1 and �zG � 1/�,where � is a characteristic vertical length scale for theeddy fluxes below the transition layer, defined by

� ��zbm ��zzbm|z��h�D.

Using the relationship between � and �webe� in (17),the expression in (25) can be used to write final expres-sions for the fluxes in the transition and boundary layer,

�uhebe� � ��GM�horbm,

�webe� � �GMG�z�|�horbm|2

�zbm|z��h�D. �27�

Substituting the expressions for buoyancy fluxes in thedefinition of the residual flux (10) gives

Fb{b} � ��GM�1 � G�z��zbm

�zbm|z��h�D��horbm. �28�

The derivation of the expression for the residual tracerflux is sketched in appendix B and is


Fb{c} � ��GM��horcm � G�z��zcm

�zbm|z��h�D�horbm�� GMG�z�� |�horbm|2

�zbm|z��h�D

�zcm

�zbm�

�horbm · �horcm

�zbm|z��h�D�z. �29�

This expression matches the small-slope Redi tensor inthe ocean interior and it reduces to (28) when b � c.The formula can be generalized to match the finiteslope Redi tensor in the interior, but we stick to com-mon practice and report only the small slope expres-sions. Notice that the small slope approximation ismade only in the adiabatic interior. No such assumptionis made in the boundary and transition layers whereslopes can, and usually do, become large.

Equations (25), (28), and (29) constitute the full pa-rameterization scheme in the boundary and transitionlayers. These equations follow from the eddy flux de-composition in (9) and (10) under the assumptions thatthe horizontal eddy fluxes of all tracers are down theirlocal mean horizontal gradients and the vertical eddyfluxes decrease to zero within the boundary and tran-sition layers. The decrease is linear in the BL and qua-dratic in the transition layer. These simple require-ments give a parameterization scheme that ensures thefollowing properties:

1) The residual velocities and residual fluxes vanish atthe oceanic boundaries.

2) The residual streamfunction and the residual veloc-ities are continuous everywhere.

3) The residual circulation releases available potentialenergy everywhere.

4) The shear in the residual velocities acts to restratifythe water column through the transition layer butnot within the boundary layer (as long as buoyancyis well mixed through the mixed layer).

5) The residual fluxes and their divergence are con-tinuous if the bm and cm gradients are continuous.

The first property is imposed through the vertical struc-ture functions G(z) and G�(z), which vanish at theboundaries. The second property follows from the con-tinuity of �z. The third property is guaranteed by hav-ing the residual streamfunction in the direction of thelocal horizontal buoyancy gradient so that the release

of available potential energy is nonnegative definite ev-erywhere; that is, �webe� � � � �horbm · z � 0. Thefourth property is consistent with the observation thatvertical mixing erases mesoscale shears in the BL sothat any shear in the horizontal eddy-induced velocity isconfined below the BL in the transition layer and in theoceanic interior. The fifth property guarantees that theeddy forcing in the tracer equations does not producesingularities starting from smooth tracer distributions.

c. Boundary and transition layers: Alternativeexpressions

Treguier et al. (1997) suggested that parameteriza-tion of eddy fluxes at the oceanic boundaries shouldinclude along-boundary eddy-induced velocity and re-sidual flux of buoyancy. We further argue that theeddy-induced velocity should be continuous in z withzero shear in the BL and with a constant shear in thetransition layer. There are many parameterizationforms that can satisfy these requirements. One suchform—our preference—is derived in section 3b and re-sults in the expressions (25) and (28). Alternative formshave been recently proposed in the literature. We wishto compare our approach to these alternative schemes.

Danabasoglu et al. (2008) assume that the eddy-induced streamfunction is set at the base of the transi-tion layer and decays to zero quadratically in the tran-sition layer and linearly in the BL. Continuity of theeddy-induced velocity is imposed at the transition andboundary layer bases. These simple rules fully deter-mine �:

� � �GMG1�z�z ��horbm

�zbm�

z��h�D

� �GMG2�z�z ��

�z

�horbm

�zbm�

z��h�D, �30�

where the vertical structure functions that guaranteecontinuity are

G1�z� � �2�z � ��

2�h � �� D, for �h � z � �BL�

�z � h�2

�h � D � ��2 � �h � ��2 �2�z � ��

2�h � �� D, for �h � D � z �h �transition layer�,

G2�z� � �D�z � ��

2�h � �� D, for �h � z � �BL�

�z � h�2�D � h � ��

�h � D � ��2 � �h � ��2 �D�z � ��

2�h � �� D, for �h � D � z �h �transition layer�.


The main difference between this approach and the oneproposed in section 3b is that the direction of � is hereset by the buoyancy gradients at the transition layerbase instead of being set by the local buoyancy gradientas in (9). The implication is that in this formulation �is not guaranteed to reduce the mean potential energy;that is, �webe� � � � �horbm · z is not sign definite.

However, this expression reduces to (9) if the directionof the horizontal buoyancy gradients does not veermuch in the transition and boundary layers.

Danabasoglu et al. (2008) assume that the diabaticresidual buoyancy flux grows linearly from zero in theinterior to along boundary in the BL, that is,

Fe{b} � ��GM��horbm � G3�z��horbm�,

Fe{c} � ��GM��horcm � G3�z�S�zcm� � �GMG3�z��|S|2�zcm � S · �horcm�z, �31�

where

G3�z� � �0, for �h � z � �BL�

�z � h�

D, for �h � D � z �h �transition layer�.

In the BL this approach gives the approximately sameresidual buoyancy flux as (28) because �zbm 0, whilethe residual tracer flux differs by the vertical compo-nent. In the transition layer the details of the verticalstructure of the residual fluxes are also different. Fur-thermore, (31) does not guarantee continuity of thevertical flux divergence at the transition layer base. De-spite these differences, the two expressions are likely tobe quite similar in most situations, because they bothcapture the transition from along-isopycnal residualfluxes in the interior to along-boundary fluxes in the BLwith some degree of smoothness.

The parameterization forms in (30) and (31) reduceto those of Griffies (2004) in the limit of vanishing tran-sition layer thickness D. A zero D has the effect ofintroducing discontinuities in the eddy-induced veloci-ties, and it is not consistent with high-resolution nu-merical results (Kuo et al. 2005). In the next section wediscuss why such discontinuities are undesirable andcan lead to spurious numerical instabilities. In summarywe believe that the expressions in (25), (28), and (29)constitute a superior parameterization scheme with anumber of desirable properties, but the forms in (30)and (31) may often produce similar results in practice.

4. Numerical experiment

Our ideas about eddy parameterizations at the oce-anic boundaries are now illustrated through a simplenumerical experiment. We study the spreading of awarm lens that outcrops in a mixed layer at the oceanicsurface and lies on top of a stratified ocean, as shown inFig. 6a.

The geometry, forcing, and boundary conditions do

not depend on the zonal direction x. Thus we can solvefor the zonally averaged circulation in the form

ut � u · �u � fz � u � ��p � �Fe{u} � � · ��u�,

�32�

0 � �pz � b, �33�

�y � wz � 0, �34�

bt � �u � ume� · �b � �� · Fe{b} � � · ��b� � �z.

�35�

The overbar here denotes the zonal average over thedomain length Lx:

b �1

Lx�

0

Lx

b�x, y, z, t� dx. �36�

The surface buoyancy flux is represented by �. Weimpose a rigid upper lid at z � 0 and a flat bottom atz � �H and use periodic boundary conditions in x. Thezonally averaged equations are solved with the MITOGCM (Marshall et al. 1997).

The model ocean is 2000 km wide and 2000 m deepwith a flat bottom (Table 1). The horizontal resolutionis 100 km, while the vertical resolution is variable: 20 min the upper 200 m, increasing exponentially with depthwith a stretching scale of 1.2. Boundary conditions areperiodic on the lateral walls. Heat satisfies the zero fluxboundary conditions at the surface and bottom. Formomentum we use no slip at the bottom and free slip atthe surface. The parameters used in the simulation arelisted in Table 1. At this resolution mesoscale and mi-croscale processes are subgrid scale and must be pa-rameterized.


The microscale fluxes are parameterized as downgra-dient fluxes of momentum and buoyancy: in the interiorthe microscale viscosity is set to � 2 � 10�4 m2 s�1

and the microscale diffusivity to � 2 � 10�5 m2 s�1. ABL is represented in the upper 140 m by increasing themicroscale viscosity and diffusivity to 1 � 10�3 m2 s�1.Convective adjustment is used to remove unstablestratification.

The horizontal components of the mesoscale Reyn-olds stress tensor are parameterized with a horizontalviscosity in the horizontal momentum equation Fe{u} ��A �horuhor, with A � 2 � 105 m2 s�1. The mesoscalebuoyancy fluxes are parameterized as an eddy-inducedvelocity ume � � � � and a diffusive flux Fe{b}. In theoceanic interior we use the GM scheme for � in (19)with a diffusivity GM of 1000 m2 s�1. The residualbuoyancy flux Fe{b} is set to zero under the assumptionthat eddy fluxes are adiabatic. In the transition andboundary layers, � and Fe{b} are computed from theexpressions in (25) and (28).

We start the simulations with a background stratifi-cation of N � 1 � 10�3 s�1. The warm lens is repre-sented as an anomaly with a maximum amplitude of3.5°C at the surface, confined to the upper 600 m andwith an horizontal extent of about 500 km. The zonalvelocity field is initially in geostrophic balance, whilethe meridional and vertical velocities start from zero.

TABLE 1. Parameters used in the spreading warm lensexperiment.

Boundary layer depth h 140 mOceanic depth H 2000 mVertical diffusivity in the boundary layer BL 10�3 m2 s�1

Vertical viscosity in the boundary layer BL 10�3 m2 s�1

Vertical diffusivity in the interior 2 � 10�5 m2 s�1

Vertical viscosity in the interior 2 � 10�4 m2 s�1

Horizontal viscosity A 2 � 105 m2 s�1

Eddy diffusivity GM 1000 m2 s�1

FIG. 6. Evolution of a warm lens on top of a stably stratified ocean: (left) the initial condition and (right) thesolution after one year. There is a boundary layer in the upper 140 m. The color contours represent temperatureand the lines represent the eddy-induced circulation: full lines for clockwise rotation and dashed lines for anti-clockwise rotation. (top) Solutions using the parameterization scheme described in (9) and (10) with the transitionlayer thickness estimated according to (11) and (bottom) solutions obtained using the same parameterization, buta transition layer 10 times thicker.


Fig 6 live 4/C

The initial geostrophic shear is chosen to be baro-clinically unstable (i.e., in a high-resolution, three-dimensional simulation the lens would slump and shedbaroclinic eddies). No eddies develop in our model be-cause the grid is too coarse to resolve the wavenumbersat which the instability should develop. This is wherethe mesoscale parameterization comes in. The GM pa-rameterization scheme mimics the effect of the unre-solved eddy field and acts to flatten the mean isopyc-nals.

In Fig. 6 we show the stratification and the eddy-induced streamfunction produced by the GM param-eterization scheme in the interior and the scheme in(25), (28), and (29) in the boundary and transition lay-ers. The eddy-induced streamfunction has a smoothtransition from the adiabatic interior to the BL withoutcreating spurious shears in the BL. The lens spreads ata similar rate in the interior and in the upper ocean sothat no discontinuities develop at the transition layerbase. The temperature anomaly in the BL decreaseswith time as a result of the diapycnal horizontal eddyflux.

The thickness D of the transition zone plays an im-portant role in the evolution of the lens. In Figs. 6a and6b, we show solutions with D computed according tothe formulas given in section 2c. For the parameterschosen in this example, D 30 m. One can see a slightdiscontinuity in the eddy-induced circulation across theBL base. In Figs. 6c and 6d, we show a solution wherewe increased D by 10 times: as a result, the transitionbetween the BL and the interior is more gradual. Theguiding principles used to determine D are preliminary.Direct numerical simulations of an eddying field out-cropping at the surface are necessary to settle this issue.The question of how D is determined is of crucial im-

portance because the behavior of the eddy fluxes at thebase of the BL can substantially modify the ventilationof the oceanic interior.

The parameterization proposed here differs in im-portant aspects from parameterization described in theliterature. Figure 7a shows the overturning circulationobtained by applying the GM parameterizationthroughout the whole water column. The overturningcirculation acts to spread the lens laterally and reducethe available potential energy. The overturning is larg-est at the surface where the stratification is weak. How-ever, the condition � � 0 at z � 0 forces the stream-lines to close at the surface and generates two surface-intensified overturning cells. These cells create a largeshear in the upper ocean and restratify the boundarylayer despite strong vertical mixing. Both the surfaceoverturning circulation and the strong restratificationare physically implausible: there is no numerical or ob-servational evidence for strong surface-trapped recircu-lations of this kind.

A common solution to avoid surface-intensified over-turning cells in current OGCMs is to set � to zerobefore reaching the BL through ad hoc tapering func-tions, as shown in Fig. 7b (Gerdes et al. 1991; Danaba-soglu et al. 1994). This has the awkward consequence ofcreating a strong horizontal eddy-induced mass flux im-mediately below the BL and none within it (Large et al.1997). Furthermore the tapering approach does not in-clude any diabatic horizontal flux. As a result the lensspreading proceeds faster in the oceanic interior than atthe surface because there is no parameterization ofeddy transport in the boundary layer. What Gerdes etal. (1991) and Danabasoglu et al. (1994) failed to realizeis that the restratification associated with the surface-trapped circulations can be eliminated by imposing that

FIG. 7. The eddy-induced overturning circulation produced by two different parameterization schemes for theslumping of a warm lens on top of a stably stratified ocean for the same initial condition shown in Fig. 6: (left) theGM parameterization and (right) the GM parameterization with the tapering scheme suggested by Danabasogluet al. (1994) to suppress spurious circulations in regions of weak stratification.


Fig 7 live 4/C

� has a linear gradient in z within the BL so that theeddy-induced velocity has zero shear. No spuriousboundary layer restratification occurs in this case, buteddy transport is not artificially suppressed at the oce-anic surface. Preliminary experiments with the MITgeneral circulation model show that these differencesare significant and strongly support the approach de-scribed here (Ferrari 2006).

More recently, Griffies (2004) has proposed to taper� linearly to zero using a scheme similar to the onepresented here but without a transition layer. Withouta transition layer, discontinuities in the eddy-inducedvelocity develop at the BL base (Fig. 4), and a convec-tive adjustment scheme is necessary to prevent genera-tion of unstable stratification (Griffies et al. 2005). Thegeneration of convective instabilities by mesoscale mo-tions has no physical justification and compromises theproper exchange of properties between the BL and theoceanic interior. In a sense the Griffies approach goestoo far in suppressing the excessive GM restratificationin regions of low stratification and ends up producingadditional vertical mixing.

5. Conclusions

Mesoscale eddy transport of large-scale momentumand material tracers exerts a profound influence on theoceanic general circulation and on the exchange ofheat, freshwater, and other material between the oceanand the atmosphere. The mesoscale parameterizationsused in OGCMs typically represent the adiabatic re-lease of potential energy by baroclinic instability in theinterior, as suggested by GM. However, as the surfaceis approached, eddy fluxes develop a substantial dia-batic component because density becomes verticallywell mixed in the boundary layer (BL) while eddy andmean motions are kinematically constrained to benearly parallel to the boundary (i.e., horizontal near thesurface). As a result, quasi-adiabatic schemes lead tounphysical behaviors at the boundaries. Too oftenOGCM practice has been to taper the mesoscale eddyfluxes to vanish within the boundary layers. This is notphysically correct. Mesoscale eddy fluxes play an im-portant role in the tracer transport along the oceanicboundaries and must be parameterized in coarse reso-lution models. The solution proposed in this paper is tomodify the interior parameterization to account for thedifferent eddy fluxes in the BL with a smooth transitionthrough a layer of subgrid-scale intermittent diabaticmixing.

A parameterization for the mesoscale eddy fluxesnear the oceanic boundaries is derived starting from the

physical argument that eddy fluxes are adiabatic in theinterior, diabatic in the BLs, and zero in their normalcomponent at the boundaries. We restrict the discus-sion to tracer fluxes, but our arguments are likely to begermane to momentum fluxes as well and could be ap-plied to derive matching conditions for momentumfluxes at the ocean surface. We split the mesoscaletracer flux �uece� into an advective component ume · ��c�and a residual component Fe{c}, usually represented asa diffusion process. The constraint that there is no eddyflux in and out of the ocean can be translated into theconditions that the eddy-induced velocity ume and thediffusive flux Fe{c} are oriented parallel to the bound-aries at the solid walls and at the free surface. Equa-tions (25), (28), and (29) constitute the full parameter-ization scheme and satisfy the following properties:

1) In the interior we assume that an adiabatic param-eterization scheme is known, be it the scheme ofGM or a scheme based on residual eddy stress (Fer-reira et al. 2005). This scheme is then used to deriveexpressions for ume and Fe{c} away from the bound-aries.

2) In the BLs we assume that ume has no vertical shear,as long as buoyancy is well mixed through the mixedlayer, in the spirit of well-mixed BL models. Theresidual buoyancy flux is parameterized as a down-gradient flux parallel to the boundaries; that is, thereis a diapycnal eddy flux.

3) The interior and BL parameterizations are matchedby quadratically interpolating the eddy fluxesthrough a transition layer of thickness D. Thematching condition of continuous eddy-induced ve-locity and diffusive flux divergence at the boundaryand transition layer bases are sufficient to deriveexpressions for ume and Fe{c} in the boundary layers.The transition layer represents the region throughwhich the mesoscale eddy flux develops a diabaticcomponent through intermittent BL mixing beneaththe resolution scale of the mean circulation. Consis-tent with the idea of intermittent mixing, we allowfor shear in ume so that restratification occurs in thetransition layer. We propose a simple scaling argu-ment to estimate D based on eddy heaving, but amore satisfactory parameterization of D will requirehigh-resolution numerical simulations and carefulfield experiments.

The new parameterization scheme is illustrated by anidealized zonal flow that uses a GM parameterizationscheme in the interior. The proposed BL parameteriza-tion represents a clear improvement over taperingschemes described in the literature. The new param-


eterization scheme does not to trigger any numericalinstabilities in the low-stratification regions near theboundaries (unlike a recent parameterization proposedby Griffies 2004). It is quite possible that the boundarylayer closure scheme can be applied to interior regionsof low stratification as well, thus eliminating any needto introduce artificial tapering functions to maintain nu-merical stability.

Acknowledgments. This research was supported byNSF Grants OCE02-41528 (RF), CE02-21177 (JM),and OCE03-36755 (joint). The authors wish to thank J.Marshall and A. Plumb for many useful conversationsand suggestions and G. Danabasoglu for advice aboutOGCM implementation of the parameterization. V.Canuto and M. Dubovikov were included as coauthorsduring the review process, because they contributed toresolving some inconsistencies in the original expres-sions for the eddy-induced streamfunction and the re-sidual fluxes.

APPENDIX A

Definitions of � and Fe{b} at Oceanic Boundaries

Eddy fluxes veer from along isopycnal to alongboundary as the boundary is approached. To representthis transition, it is useful to modify the definition of �in (4) so that the transition to no-normal boundary flowis achieved through a layer with finite thickness. Thetransition can be achieved by exploiting the arbitrari-ness in the choice of the vector streamfunction � todefine an eddy-induced velocity that satisfies no-normal flow boundary conditions. Following Treguieret al. (1997), we generalize the definition of � to be

� � ��uebe� � �bm

|�bm|2�

�uebe� · �bm

|�bm|2�, �A1�

where � is an arbitrary three-dimensional vector thatmay be spatially and temporally variable. The residualbuoyancy flux must be modified accordingly:


|�bm|2��bm � � � �bm�. �A2�

The residual flux remains proportional to the diapycnalflux in this generalized formulation, but it points in adirection that depends on the choice of �, not down themean buoyancy gradient. The eddy-induced circulationremains largely unaltered in the oceanic interior wherediapycnal fluxes are weak, but it is modified at theboundaries where eddy-driven diapycnal transports arelarge.

The vector � can be chosen so that the component of

the diapycnal flux that cancels the advection throughthe boundary is retained in the definition of � so thatthe no-normal flow boundary condition is satisfied.Consider a boundary surface with unit normal, n Aform for � that satisfies this boundary conditions is

� �n · �bm

|�bm|2 � |n · �bm|2n � �bm. �A3�

The corresponding vector streamfunction is

� � ��uebe� · n

|�bm|2 � |n · �bm|2n � �bm

��uebe� � �bm� · n

|�bm|2 � |n · �bm|2n. �A4�

It has two components: the first represents a circulationdirected across the boundaries that vanishes becausethe eddy velocity normal to the boundary is zero, andthe second represents a circulation along the boundary.

The associated diabatic residual flux is directed par-allel to the boundary:


|�bm|2 � |n · �bm|2��bm � ��bm · n�n�. �A5�

The definition in (A4) satisfies the no-normal flow con-dition ume · n � 0, and can be applied close to theboundaries.

The gauge invariance can now be used to defineeddy-induced circulations and residual fluxes that van-ish at all boundaries and retain useful forms in the in-terior. The choice of gauge in the interior is inconse-quential because away from boundaries there is nobuoyancy flux across isopycnals and the gauge terms in(A1) and (A2) vanish for any choice of �. Hence wecan define � as in (A3) where n is set to the vertical unitvector at the surface and smoothly transitions to theunit vector normal to topography toward the bottom.For a flat bottom ocean n � z can be used everywhere;this choice might be appropriate even in the presence oftopographic relief if eddy fluxes are weak at the oceanbottom.

APPENDIX B

Residual Fluxes of Passive Tracers

The decomposition of the eddy tracer flux into ad-vective and residual components corresponding to (9)and (10) is

�uece� � � � �cm � Fe{c}, �B1�

where � is given in (9) and the residual flux is


Fe{c} � � �webe�

|�horbm|2z � �horbm �

�wece�

|�horcm|2z � �horcm� � �cm

� ��uhebe� � �horbm

|�horbm|2�

�uhece� � �horcm

|�horcm|2 � � �cm ��uece� · �cm

|�horcm|2�horcm. �B2�

We used a downgradient closure for the horizontalflux of buoyancy. It seems consistent to assume thesame for the tracer flux, that is,

�uhece� � ��GM�horcm. �B3�

Imposing the downgradient closures for the horizontalfluxes of buoyancy and tracer, the residual flux in (B2)becomes

Fe{c} � ��GM�horcm ��webe��zcm

|�horbm|2�horbm � ��wece� � �webe�

�horbm · �horcm

|�horbm|2 �z.

Imposing continuity of the vertical fluxes and their di-vergences at the boundary and transition layer inter-faces gives the expression in (29). The expression in-volves division by the weak stratification in the mixed

layer and might be problematic for numerical imple-mentation. An alternative expression that satisfies thesame properties, but avoids division by a weak stratifi-cation, is

Fe�c� � ��GM��horcm � G�z��zcm

�zbm|z��h�D�horbm�� GMG�z�� |�horbm|2

�zbm|z��h�D2 �zcm �

�horbm � �horcm

�zbm|z��h�D2 �zbm�z.

The function G(z) has the same form of G(z) in (26)except for the definition of �. At the base of the tran-sition layer G � 1 and �zG � 1/�2, where �2 � �/2. Thisalternative expression is not our favorite choice be-cause it does not preserve the symmetric nature of thediffusivity tensor, but it might be advantageous to en-sure numerical stability.

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