Ocean Modelling 72 (2013) 92–103

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Ocean Modelling

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Using a resolution function to regulate parameterizations of oceanicmesoscale eddy effects

1463-5003/$ - see front matter Published by Elsevier Ltd.http://dx.doi.org/10.1016/j.ocemod.2013.08.007

⇑ Address: NOAA Geophysical Fluid Dynamics Laboratory, 201 Forrestal Rd.,Princeton, NJ 08540-6649, USA. Tel.: +1 609 452 6508.

E-mail address: Robert.Hallberg@noaa.gov1 This is just a simple function that goes smoothly between the appropriate

equatorial and mid-latitude definitions of the deformation radius without the needfor any arbitrary transition latitude (see, e.g. Chelton et al., 1998).

Robert Hallberg ⇑NOAA Geophysical Fluid Dynamics Laboratory, 201 Forrestal Rd., Princeton, NJ 08540, USAAtmospheric and Oceanic Sciences Program, Princeton University, Princeton, NJ 08540, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 12 April 2013Received in revised form 16 August 2013Accepted 20 August 2013Available online 5 September 2013

Keywords:Ocean modelEddy parameterizationResolution functionEddy permitting

Mesoscale eddies play a substantial role in the dynamics of the ocean, but the dominant length-scale ofthese eddies varies greatly with latitude, stratification and ocean depth. Global numerical ocean modelswith spatial resolutions ranging from 1� down to just a few kilometers include both regions where thedominant eddy scales are well resolved and regions where the model’s resolution is too coarse for theeddies to form, and hence eddy effects need to be parameterized. However, common parameterizationsof eddy effects via a Laplacian diffusion of the height of isopycnal surfaces (a Gent–McWilliams diffusiv-ity) are much more effective at suppressing resolved eddies than in replicating their effects. A variant ofthe Phillips model of baroclinic instability illustrates how eddy effects might be represented in oceanmodels. The ratio of the first baroclinic deformation radius to the horizontal grid spacing indicates wherean ocean model could explicitly simulate eddy effects; a function of this ratio can be used to specifywhere eddy effects are parameterized and where they are explicitly modeled. One viable approach isto abruptly disable all the eddy parameterizations once the deformation radius is adequately resolved;at the discontinuity where the parameterization is disabled, isopycnal heights are locally flattened onthe one side while eddies grow rapidly off of the enhanced slopes on the other side, such that the totalparameterized and eddy fluxes vary continuously at the discontinuity in the diffusivity. This approachshould work well with various specifications for the magnitude of the eddy diffusivities.

Published by Elsevier Ltd.

1. Introduction

Mesoscale eddies are ubiquitous in the ocean, and are of leadingorder importance to the dynamics of major current systems, suchas the Antarctic Circumpolar Current (e.g. Hallberg and Gnanadesi-kan, 2006), the Kuroshio (e.g. Waterman et al., 2011), and the GulfStream (e.g. Chassignet and Marshall, 2008). Credible models of theocean’s dynamics need to either explicitly resolve eddies or toparameterize their effects.

The dominant spatial scales of baroclinic ocean mesoscale ed-dies can be broadly characterized by the first baroclinic deforma-tion radius, which is the distance that a nonrotating first-modeinternal gravity wave would propagate in one inertial timescale(e.g. Gill, 1982). With an appropriate regularization at the equator,1

the first baroclinic deformation radius is given byLDef ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2

g= f 2 þ 2bcg� �q

, where cg is the first-mode internal gravity

wave speed, f is the Coriolis parameter, and b ¼ @f@y is its meridional

gradient.In idealized models of baroclinic instability, the upper and low-

er bounds of unstable wavelengths are proportional to the defor-mation radius, while the most unstable wavenumber is theinverse of the deformation radius (see, e.g., the textbook by Pedlo-sky (1987)). The observed dominant eddy length-scales in theocean vary more slowly with latitude than does the first baroclinicdeformation radius (Stammer, 1997), but this may reflect thegreater influence of higher baroclinic modes in the tropics and ofeffectively barotropic eddies in higher latitudes.

Numerical ocean models need to represent the effects of meso-scale eddies, either by explicitly resolving them or via a suitableparameterization, if they are to replicate the dynamical responseof the real ocean. As will be illustrated later, the ratio of a model’sgrid spacing to the deformation radius gives a good indication ofwhether a model will be locally capable of explicitly resolving eddyeffects. However, as both the deformation radius and an oceanmodel’s grid spacing vary in space, one should ask where, notwhether, a global ocean model can explicitly represent eddies.Fig. 1 shows the ocean model horizontal resolution required forthe baroclinic deformation radius to be twice the grid spacing,based on a nominally eddy permitting ocean model after one year

Fig. 1. The horizontal resolution needed to resolve the first baroclinic deformation radius with two grid points, based on a 1/8� model on a Mercator grid (Adcroft et al., 2010)on Jan. 1 after one year of spinup from climatology. (In the deep ocean the seasonal cycle of the deformation radius is weak, but it can be strong on continental shelves.) Thismodel uses a bipolar Arctic cap north of 65�N. The solid line shows the contour where the deformation radius is resolved with two grid points at 1� and 1/8� resolutions.

R. Hallberg / Ocean Modelling 72 (2013) 92–103 93

of spin-up from climatology. At the coarse resolution that is typicalof the ocean components of CMIP5 coupled climate models (nom-inally 1� resolution), an ocean model only resolves the deformationradius in deep water in a narrow band within a few degrees of theequator; any important extratropical eddy effects will need to beparameterized. At a much higher resolution, such as a 1/8� Merca-tor grid, the deformation radius is resolved in the deep ocean in thetropics and mid-latitudes, but even in this case eddies are not re-solved on the continental shelves or in weakly stratified polar lat-itudes. An unstructured and adaptive grid ocean model could helpto address this issue, but such models are not yet in widespreaduse for global ocean climate modeling, and even then computa-tional speed may dictate the use of models that do not resolvemesoscale eddies everywhere.

In this paper, a series of numerical simulations of a variant ofthe Phillips (1954) model of baroclinic instability are used toexamine the effects of resolution on a numerical model’s abilityto exhibit the net overturning circulation driven by mesoscale ed-dies. The effects of a commonly used parameterization of eddy ef-fect, both on the models’ explicitly resolved eddies and on the netoverturning, are examined. Based on these results, a simple pre-scription is offered for the typical situation in global ocean mod-els, where eddies are resolved in only part of the domain and inthat portion it is desired that the model be allowed to explicitlysimulate their effects, but in the remainder of the domain thateddies be entirely parameterized. Specifically, the eddy diffusivi-ties should be multiplied by a ‘‘resolution function’’, ranging from0 to 1, of the ratio of the baroclinic deformation radius to the

model’s effective grid spacing, eD ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDx2 þ Dy2ð Þ=2

p. The resolu-

tion function that works best for the cases presented here rapidlymakes a transition from 1 when this ratio is greater than a valueof about 2 (the exact value is not very important and can be cho-sen to be higher) to 0 for larger values. In the idealized case pre-sented here, this prescription is found to give a reasonablerepresentation of the net eddy-driven overturning over a widerange of resolutions.

2. The test configuration and model

Phillips (1954) analyzed the baroclinic instability that arises ina simple two-layered quasigeostrophic model of a geostrophicallysheared flow in a reentrant channel. This problem has the advan-tage that many of the properties of the eddies, including necessaryconditions for the growth of instabilities, the growth rate, energet-ics and vertical structure of the exponentially growing linearmodes can be calculated analytically, as has been documented inmany textbooks on geophysical fluid dynamics (e.g. Pedlosky,1987; Vallis, 2006).

This study examines instabilities of a stacked shallow watervariant of the Phillips problem, which is described by the momen-tum and continuity equations:

@un

@tþ f þ k̂ � r � un

� �� un ¼ �r Mn þ

12

unk k2� �

�r � T� dn2cD u2k ku2; ð1Þ

@hn

@tþr � hnunð Þ ¼ 3� 2nð Þ c g3=2

x � g3=2;Ref

� ��r � Khrg3=2

� �h i:

ð2Þ

Here un is the horizontal velocity in layer n, where n = 1 for thetop layer and n = 2 for the bottom layer. hn ¼ gn�1=2 � gnþ1=2 is thethickness of layer n, which is bounded above and below by inter-faces at heights gn�1=2 and gnþ1=2. These equations are solved in a2000 m deep channel that is 1200 km long and reentrant in thex-direction, and 1600 km wide in the y-direction with verticalwalls at the northern and southern boundaries. The Coriolis param-eter, f, varies linearly in the y-direction between 6.49 � 10�5 s�1

and 9.69 � 10�5 s�1, following the common b-plane approxima-tion. The horizontal stress tensor, T, is parameterized with a shearand resolution dependent Smagorinsky biharmonic viscosity (Grif-fies and Hallberg, 2000). The Montgomery potentials,Mn ¼ p=q0 þ gz, in the two layers are given by a vertical integrationof the hydrostatic equation, so that

Fig. 2. (Top) A side view looking westward of the interface height profile towardwhich the zonal-mean interior interface heights are damped, along with the zonalvelocities (color) and free-surface height (exaggerated by a factor of 50) that are ingeostrophic balance with this density structure and no bottom flow. (Bottom)Profiles of the deformation radius (blue) and layer potential vorticity (red) of thereference profile.

94 R. Hallberg / Ocean Modelling 72 (2013) 92–103

M1 ¼ gg1=2 and M2 ¼ M1 þ gDq=q0ð Þg3=2; ð3Þ

where Dq ¼ 0:002q0 is the difference in density between the twolayers, g is the gravitational acceleration of 9.8 m s�2, and q0 isthe mean density. These equations are solved numerically usingthe Generalized Ocean Layered Dynamics (GOLD) model (Hallbergand Adcroft, 2009), with 10 different horizontal resolutions rangingfrom 2.5 km up to 80 km; because all of the relevant physicalparameters for this case are self-scaling (e.g., by using a Smagorin-sky viscosity), the only model parameter that was changed betweenthe various resolutions was the time step.

These equations are essentially the standard nonlinear 2-layerstacked shallow water model, but with three modifications. Thefirst is the inclusion of a Laplacian diffusion of the internal inter-face height, with coefficient Kh. An eddy parameterization basedon the extraction of available potential energy via the diffusionof isopycnal heights in layered models was the original inspirationfor the Gent and McWilliams (1990) parameterization in z-coordi-nate models, and this parameterization is the equivalent in thistwo-layered system. The sensitivity of this system to the value ofKh is a large part of this study. The numerical model described heredoes not require this term for numerical stability, and in the refer-ence cases Kh is set to 0. The second modification of a standard twolayer model is the inclusion of a large bottom drag, to prevent thebarotropization of the eddies and to keep these solutions in anoceanographic regime. Mesoscale eddies in the ocean are observedto have substantial geostrophic shears, even if they are often de-scribed as ‘‘equivalent barotropic’’ because the direction of the mo-tion tends to be largely the same throughout the water column.Arbic and Flierl (2004) demonstrate that a strong quadratic bottomdrag serves the purpose of giving idealized eddies this structure,while noting that in the real ocean form drag exerted by roughbathymetry could fill an analogous role.

The third modification is the inclusion of a transfer of mass be-tween layers that acts to drive the zonal mean interior interfaceheight back toward a specified profile, shown in Fig. 2, with a rateof c = 1/ (10 days). The specified density profile has a reversal inthe potential vorticity gradient of the lower layer betweeny = 600 km and y = 1100 km, (Fig. 2)(b), and the flow is subject tobaroclinic instability. By damping the zonal mean interface heightvia a zonally uniform mass flux, this forcing avoids damping baro-clinic eddies, while energizing the flow and ensuring that the mod-el equilibrates in a state of vigorous eddy activity. The dampingrate used here is strong enough that the results do not dependstrongly on its particular value. A similar forcing strategy was em-ployed by Jackson et al. (2008) in their study of statistically equil-ibrated stratified shear instability.

As will be shown later, a key indicator of the model’s ability toresolve baroclinic instability is the ratio of the first-mode baroclin-ic deformation radius to the diagonal grid spacing. In the two-layersystem studied here, the first-mode baroclinic gravity wave speedis well approximated by

cg �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigDqh1h2

q0 h1 þ h2ð Þ

s; ð4Þ

although in the general multilayered or continuously stratified casecg is given by the second largest eigenvalue of the normal modedecomposition equation (see chapter 6 of Gill, 1982). The expres-

sion for the deformation radius used here, LDef ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2

g= f 2 þ 2bcg� �q

,

converts smoothly from an appropriate mid-latitude deformationradius to the equatorial deformation radius, and can be appliedwithout modification in a global model; for the mid-latitudeb-plane used here, the b term in the denominator reduces the defor-mation radius by between 1 and 2%. The deformation radius of thereference profile ranges between 47 km at the southern end of the

domain and 19 km at the north (Fig. 2(b)), although in the barocli-nically unstable latitudes the deformation radius is in the narrowerrange of �22 to 39 km. The actual calculations presented here usethe local instantaneous deformation radius, as determined by itera-tively solving for the eigenvalues of the modal decomposition equa-tion. The relevant grid spacing for determining whether thedynamics can be well represented is the coarsest direction, anappropriate measure of which is proportional to the diagonal grid

spacing, eD ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDx2 þ Dy2ð Þ=2

p. For the simple isotropic Cartesian

grid used here this is simply the grid spacing in each direction,but with a locally anistropic grid it picks out the least-well resolveddirection. It will be shown later that there need to be roughly 2 gridpoints per deformation radius to explicitly represent the eddytransports, so the x-direction grid spacings that are required to ade-quately resolve the deformation radius in the baroclinically unsta-ble zone range from �11 km to 20 km.

3. Results

a. Simulations without parameterized eddy effects.When there are no eddy parameterizations and at high resolu-

tion, the upper layer develops a vigorous baroclinic eddy field, asillustrated by the upper layer velocities in Fig. 3. Small perturba-tions in the initial conditions grow to finite amplitude within about150 days, and the solutions are in statistically steady states there-after; the flow fields after 10 years, shown in Fig. 3, are qualita-tively similar to other fields in the turbulent but statisticallysteady period of the run. While there is a mature field of coherentvortices up to resolutions of 10–20 km, even the coarser resolu-tions show some feeble fluctuations from the zonal mean. Recallthat the damping only applies to zonal mean properties, so even

R. Hallberg / Ocean Modelling 72 (2013) 92–103 95

slowly growing fluctuations have time to become substantial. Thenature of the forcing also ensures that the zonal mean state isstrongly unstable throughout the run, and that the eddies are notable to drive the solution into a state of marginal stability.

The upper layer relative vorticity field (Fig. 4) shows clear quali-tative differences between all of the resolutions. Even the 5 km res-olution run does not exhibit nearly as rich of a field of filaments andcoherent structures as does the 2.5 km run, and there is every reasonto expect that even finer resolution models would exhibit an evenricher small-scale structure. These runs are not converged in thatthey do not resolve a full enstrophy cascade, and thus none of themshould be characterized as fully eddy resolving by all metrics.

However, for many ocean modeling purposes, it is not the eddyproperties themselves that are of primary interest, but rather theeffect that the eddies have on the large-scale circulation. One goodmeasure of the large-scale impact of the eddies is the long-termmean overturning circulation that is driven by the eddies,

Fig. 3. Instantaneous upper layer velocity (speed in color, with directions given by the20 km, (e) 33 km, and (f) 50 km. No parameterization of the eddy effects is used in thes

�VðyÞ ¼Z

v1h1dxt

; ð5Þ

where the overbar denotes a time-average and v is the northwardvelocity. As shown in Fig. 5, the highest resolutions exhibit verysimilar overturning transports, but at resolutions coarser thanabout 10–15 km, the eddy-driven overturning progressively weak-ens. There is no abrupt cut-off to the net eddy transport; in facteven at the coarsest resolution (50 and 80 km), the weak mean-dering evident in Figs. 3 and 4 contributes much of the overturn-ing that occurs. But if the objective is to simulate an eddy fieldthat reproduces the overturning of the fully resolved flows, thisis qualitatively achieved only out to resolutions of order 15–25 km.

On timescales long enough to neglect storage of mass, the time-mean northward transport balances the integrated diapycnal massflux, so that

arrows) at day 3650 at horizontal resolutions of (a) 2.5 km, (b) 5 km, (c) 10 km, (d)e simulations. The lower layer velocities are much smaller in magnitude.

Fig. 4. Instantaneous upper layer relative vorticity normalized by the local Coriolis parameter at day 3650 at horizontal resolutions of (a) 2.5 km, (b) 5 km, (c) 10 km, (d)20 km, (e) 33 km, and (f) 50 km. These are the same simulations and times as are depicted in Fig. 3.

96 R. Hallberg / Ocean Modelling 72 (2013) 92–103

VðyÞ ¼Z

v1h1dxt

¼Z y

0

Zwdxdy

t

�Z y

0

Z@h1

@tdxdy

t

�Z y

0

Zwdxdy

t

; ð6Þ

where w is an upward diapycnal velocity. For the 15 year averagesshown in Fig. 5, this balance holds quite well. Additionally, thevelocity and thickness can be decomposed into the zonal- and tem-poral-mean and anomalies from that mean, in which case the trans-port can be written as

VðyÞ ¼Z

v1h1tdx ¼

Zv1

xth1xt

dxþZ

v 01h01tdx: ð7Þ

In all of the runs, the rectified anomalies contribute most of theoverturning; only in the lowest resolution runs is the transport bythe time-mean velocity (due to the viscous terms breaking geostro-phy) noticeable.

b. Effects of a common eddy parameterization.

It is common in coarse resolution ocean models to use an iso-pycnal height diffusivity or its advective counterpart in depth-space to parameterize the restratifying effects of eddies (e.g. Gentet al., 1995). This approach can emulate the adiabatic slumpingeffects that eddies exert in isopycnal surfaces, and by design theyextract available potential energy from the mean flow. [Such aparameterization is sometimes called a thickness diffusivity(e.g. Eden et al., 2009), but this is a misnomer derived fromflat-bottom or reduced gravity models; with variable bottomtopography a literal interpretation of the isopycnal height diffu-sivity as a thickness diffusivity leads to a substantial near-bottomsource of available potential energy, and this interpretationworks contrary to the interpretation of such a parameterizationas defining a streamfunction at the interfaces (Ferrari et al.,2010).] A parameterization of the eddy effects via an isopycnalheight diffusivity is also very effective at suppressing barocliniceddies, as is vividly illustrated with snapshots of the upper layerrelative vorticity with different values of the isopycnal height dif-fusion (Fig. 6). An isopycnal height diffusion directly extracts the

Fig. 5. Time-mean zonally integrated overturning transport for the Phillips modelas a function of horizontal resolution, Dx, averaged over years 6–20 with no eddyparameterization. (A) Shows the time-mean overturning transport as a function oflatitude for each of 10 different horizontal resolutions. (B) The heavy black line isthe peak value of the time-mean zonally integrated overturning, while the red linesshow the peak mean transport plus and minus the RMS variability; the blue dashedlines show the maximum and minimum 30-day mean overturning transport overthe same period. (For interpretation of the references to colour in this figure legend,the reader is referred to the web version of this article.)

R. Hallberg / Ocean Modelling 72 (2013) 92–103 97

available potential energy of baroclinic eddies, and even a modestisopycnal height diffusivity substantially shortens the eddy life-time and greatly reduces the structural richness of the eddy field.

Fig. 7 illustrates the effects that introducing an isopycnal-heightdiffusion has upon the overturning transport in the 10 km resolu-tion case, in which the eddies could be explicitly represented. Withsmall values of the diffusivity (here up to about 500 m2 s�1), theoverturning is similar to its value without the parameterization,but with modest values of the isopycnal-height diffusivity (here1500 m2 s�1 to 3000 m2 s�1), the total overturning transport de-clines substantially before increasing almost linearly with the larg-est diffusivities. The model is able to reproduce the overturning ofa highly resolved model, either by omitting the parameterizationaltogether or by using a sufficiently large diffusivity, here empiri-cally determined to be about 8000 m2 s�1. (The actual values ofthe diffusivities at which these changes occur are a strong functionof the specific details of the configuration, and the choice of a forc-ing that keeps the flow in a strongly baroclinically unstable state;the actual values should not be interpreted as relevant to the realocean, although it may be reasonable to expect that the qualitativebehavior of this model should be quite representative of other

systems.) Fig. 7(b) illustrates the reason behind the behavior, bydecomposing the overturning transport into its explicitly resolvedportion and that due to the parameterization:

VðyÞ ¼Z

v1xth1

xtdxþ

Zv 01h01

tdx

� �þZ

Kh

@g3=2

@y

t

dx: ð8Þ

The resolved eddy transports are strongly suppressed by even mod-est values of the diffusivity, while the parameterized transports in-crease nearly linearly with the diffusivity. The eddy length-scaleshere are smaller than the scale of the front or the mean-free-pathof the eddies, so the Laplacian isopycnal-height diffusion is muchmore effective at suppressing the resolved eddies than it is at mim-icking their effects.

c. Introducing a resolution function.No eddy parameterization perfectly captures all of the effects of

eddies, but unless the eddies are adequately resolved they cannotbe explicitly represented. As illustrated in Fig. 1, over a broad rangeof resolutions, global ocean models are able to resolve the domi-nant eddy length scales over a portion of the globe, but do not re-solve them in weakly stratified, high latitudes, or shallow waters.Traditionally this has led to a compromise in choosing whether toparameterize eddies. The results presented above suggest insteadthat it might be preferable to choose where to parameterize eddiesand where to allow the model to represent them explicitly.

Although this paper is focused on the idea of introducing a res-olution function to control where to apply eddy parameterizations,the parameterization itself of the eddy effects clearly matters too.Both the appropriate magnitude of an isopycnal height diffusivityand its spatial structure depend on the physical system that isbeing studied. Even with the simple system studied here, changingparameters can greatly alter the appropriate diffusivity. This can beillustrated by diagnosing an implied diffusivity from the eddy per-mitting cases using

K Impliedh ðyÞ ¼

Zv 01h01

tdxZ

@g3=2

@y

t

dx: ð9Þ

Fig. 8 shows strong variations in the implied diffusivity from (9)for three case – the standard case discussed previously, a casewhere the intensity of the baroclinic instability is increased bydriving the zonal mean internal interface height toward its targetvalue 4 times more strongly than in the standard case, and a casewhere the intensity of the baroclinic instability is greatly reducedby reducing the amplitude of the target interface height changesacross the jet (shown in Fig. 2) from 600 m to 250 m. Increasingthe damping rate c in (2) by a factor of 4 increases the constant dif-fusivity that best matches the explicit eddy transport to about11,000 m2 s�1 (as determined from the equivalent of Fig. 7(a)) fromabout 8000 m2 s�1 in the standard case, while decreasing the baro-clinicity of the system by reducing change in interface heightacross the jet from 600 m to 250 m reduces this value to1150 m2 s�1. Fig. 8 also shows that, in each case, the implied diffu-sivity is broadly peaked around the baroclinically unstable latituderange. As seen in Fig. 2, the meridional gradient of the lower layertarget potential vorticity is reversed (a necessary condition forbaroclinic instability) for y = 520 to 1050 km in the standard case,but the elevated diffusivities in Fig. 8 extend about 100 km furtheron either side. With the weakly unstable case the largest implieddiffusivity is substantially more localized in latitude than in thestandard case, primarily because in this case the target profile onlymeets the necessary conditions for baroclinic instability for y = 650to 950 km. As discussed later, a number of approaches have beensuggested for prescribing the spatial pattern and magnitude of dif-fusivities in eddy parameterizations, most of which are more con-sistent with the profiles in Fig. 8 than simply prescribing a spatiallyconstant diffusivity. However, to focus the discussion here on the

Fig. 6. Instantaneous upper layer relative vorticity normalized by the local Coriolis parameter at day 3650 with 10 km horizontal resolution and isopycnal height diffusivities,Kh , of (a) 0, (b) 500 m2 s�1, (c) 1000 m2 s�1, (d) 2000 m2 s�1, (e) 3000 m2 s�1, and (f) 8000 m2 s�1.

98 R. Hallberg / Ocean Modelling 72 (2013) 92–103

resolution function itself, the interface height diffusivities through-out this section are given by a constant diffusivity, as determinedfrom Fig. 7, times a function of the resolution relative to the defor-mation radius.

An appropriate measure of whether the baroclinic eddy dynam-ics are likely to be well resolved is the ratio, RH , of the horizontalgrid spacing to the first-mode deformation radius,

RH ¼LDefeD ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2

g= f 2 þ 2bcg� �q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDx2 þ Dy2ð Þ=2

p : ð10Þ

When RH is small, the baroclinic eddy dynamics are not resolved bythe grid spacing, and an eddy parameterization is required. WhenRH is large, an eddy parameterization is unnecessary and may becounterproductive. These considerations suggest that the eddy dif-fusivity should be multiplied by a resolution function, FðRHÞ, whichdecreases from 1 for small RH to 0 for large RH , making the transitionbetween the two when RH is of order 1.

One candidate resolution function that meets these criteria is

F4 RHð Þ ¼ 11þ 1

4 R4H

: ð11Þ

This function goes smoothly between the correct limits, and hasbeen used in both NOAA/GFDL’s 1� ESM2G coupled climate model(Dunne, 2012) and in 1/8� global ocean simulations (Adcroft et al.,2010). The application of this resolution function to a hierarchy ofmodel resolutions, with the diffusivity specified byKH ¼ 8000 m2 s�1 F4ðRHÞ, is shown in Fig. 9. The dimensional con-stant, 8000 m2 s�1, is empirically determined from the value inFig. 7 at which the parameterized eddies match the resolved over-turning; subsequent studies will examine how the idea of using aresolution function might fit with more sophisticated ideas fordetermining this diffusivity. As shown in Fig. 9(a), this choice of aresolution function is not particularly successful, in that there is astrong resolution dependence of the overturning, with a minimumat about 20 km resolution. Only the 10 km case is close to the

Fig. 7. Meridional peak value of the time-mean and zonally integrated overturningtransport, averaged over years 6–20 at 10 km horizontal resolution as function ofthe isopycnal height diffusivity. (A) Shows the peak mean value (heavy black line)along the mean plus and minus one standard deviation (red) along the minimumand maximum 30-day average peak transports (dashed blue). (B) Shows themeridional peak zonally integrated time-mean overturning total transport (black),the peak overturning transport due to the parameterized eddies (red) and the peakoverturning transport due to the resolved eddies (blue). The red and blue lines donot exactly add up to the black line because the peak transports can occur atdifferent latitudes, as shown later in Figs. 8 and 9. (For interpretation of thereferences to colour in this figure legend, the reader is referred to the web version ofthis article.)

Fig. 8. The implied isopycanl height diffusivities as a function of latitude ascalculated by dividing the time-mean transient eddy volume flux by the slope of thetime-and zonal-mean internal interface height times the length of the domain fromsimulations at 2.5 km resolution. The configuration used throughout this paper isshown in black, while a blue line is from a case where the damping rate c in (2) isincreased fourfold, to 1/(2.5 days), and the heavy red lines is from case where thebaroclinicity of the system is reduced by decreasing the meridional drop in thetarget interface height across the jet from 600 m to 250 m. The light dashed red lineis the same as the heavy red line but multiplied by a factor of 5 for easiercomparison with the other lines. Those latitudes where the magnitude of the slopeof the time- and zonal-mean interface is less than 2�10�5 are excluded. (Forinterpretation of the references to colour in this figure legend, the reader is referredto the web version of this article.)

R. Hallberg / Ocean Modelling 72 (2013) 92–103 99

2.5 km reference solution. The reason behind this failure can beseen in the profiles of the mean diffusivity shown in Fig. 9(b). Thischoice of resolution function varies so slowly that many of the sim-ulations are using diffusivities that are large enough to suppress theresolved variability but too weak to parameterize the true overturn-ing. This is vividly illustrated by the 20 km resolution case, in whichthe overturning is substantially weaker than the correspondingsolution shown in Fig. 5(a), which does not include any eddyparameterization. As shown in Fig. 9(c), the diffusivity is strong en-ough to greatly suppress the explicit eddy overturning, but muchtoo weak to accomplish the overturning via the parameterized dif-fusive transport.

The proposed resolution function described in (11) can be mademore abrupt by increasing the power of RH in the denominator. Pro-gressively increasing this power improves the solutions in the Phil-lips model test case presented here, which leads to the idea oftesting the limiting case of a step-function. Fig. 10 illustrates theoverturning transport with the step function resolution function

FStep RHð Þ ¼1 RH < RCrit

0 RH P RCrit; ð12Þ

with RCrit ¼ 2, and a total isopycnal-height diffusivity ofKH ¼ 8000 m2 s�1 FStepðRHÞ. The convergence of solutions across

resolutions in Fig. 10(a) is remarkable. The peak values agree closelyacross resolution. The transports on the flanks of the jet are toolarge in the cases that rely on a parameterization (e.g., 33 km reso-lution), but this reflects the use of a spatially constant diffusivitythat was chosen to match the peak transport instead of the taperedprofiles diagnosed in Fig. 8, and not a problem with the resolutionfunction (in these cases FStepðRHÞ is 1 everywhere). The time-and zo-nal-mean diffusivities (shown in Fig. 10(b)) in intermediate casesshows a smooth change in values due to the temporal and spatialfluctuations of the front at which the deformation radius is large en-ough to be resolved. The total overturning fluxes due to combinedeffects of the parameterization and the resolved eddies match andblend smoothly (Fig. 10(c)).

Similar results to those shown in Fig. 10 hold over a wide rangeof values of RCrit > 2 (for which values even resolvable eddies aresuppressed and parameterized), but degrades noticeably for smal-ler values of RCrit (when some unresolvable eddies are not param-eterized). The test case presented in Fig. 10 is stronglybaroclinically unstable, and setting RCrit ¼ 1 gives an overturningthat is only modestly degraded compared with RCrit ¼ 2, probablybecause there are broad range of unstable wavenumbers and thelonger waves are able to accomplish the eddy transport even whenthe most unstable wavelength of the continuous solution is poorlyresolved. Setting RCrit ¼ 0:7 leads to a qualitatively dissimilar over-turning, with a greatly reduced peak overturning for marginally re-solved cases (similarly to Fig. 5) and an abrupt decrease in theoverturning where the resolution function (12) disables theparameterized eddy fluxes.

The standard test case used here is strongly baroclinicallyunstable, but it can easily be modified to be much more weaklyunstable by reducing the drop in the target interface heightacross the jet from 600 m to 250 m. (Setting the drop in interfaceheight to just 150 m would have made the jet baroclinically sta-ble.) Fig. 11 is the equivalent of Fig. 10 for this weakly unstablecase with the total isopycnal height diffusivity given byKH ¼ 1150 m2 s�1 FStepðRHÞ with RCrit ¼ 2. Again, there is a strong

Fig. 9. Overturning transport averaged over years 6–20 when the isopycnal heightdiffusivity is specified as Kh ¼ 8000 m2 s�1/ [1 + 0.25ðLDef Þ=ðDxÞ4]. (A) Time meanzonally integrated overturning transport as a function of resolution; the 2.5 kmreference case (dashed) is the only one that does not use an isopycnal-heightdiffusivity. (B) The time- and zonal-mean isopycnal height diffusivity. (C) The timemean zonally integrated overturning at 20 km resolution due to the parameterizeddiffusive transport (red), the explicitly resolved flow (blue) and the total (dashedmagenta). (For interpretation of the references to colour in this figure legend, thereader is referred to the web version of this article.)

Fig. 10. Overturning transport averaged over years 6–20 when the isopycnal heightdiffusivity is specified as Kh ¼ 8000 m2 s�1 where LDef < 2Dx and 0 elsewhere. (A)Time mean zonally integrated overturning transport as a function of resolution. Thelines for the 25 km, 33 km and 40 km runs are nearly indistinguishable. (B) Thetime- and zonal-mean diffusivity. (C) The time mean zonally integrated overturningat 16 km resolution due to the parameterized diffusive transport (red), theexplicitly resolved flow (blue) and the total (dashed magenta). (For interpretationof the references to colour in this figure legend, the reader is referred to the webversion of this article.)

100 R. Hallberg / Ocean Modelling 72 (2013) 92–103

similarity in the overturning transports across a wide range ofresolutions. The 16 km solution is somewhat of an outlier inFig. 11; in this case the parameterized eddy fluxes are only activeon the northern flank of the jet but the decision to use a constantdiffusivity appropriate to the center of the jet instead of a latitu-dinally tapered diffusivity, as diagnosed in Fig. 8, acts to strength-en the resolved baroclinic instability and associated transport in

the center of the jet itself. For this weakly unstable case, interme-diate resolution solutions with values of RCrit below about 2 donot reproduce the high-resolution transport. The difference inthe more restrictive appropriate value of RCrit between thestrongly and weakly unstable cases lies in the fact that thelong-wave cut-off of baroclinic instability and the wavelengthwith the fastest growth rate both move to smaller scales for

Fig. 11. Similar to Fig. 10, but for a weakly unstable case where the interface heightdrops by only 250 m across the jet instead of 600 m. In this case, the isopycnalheight diffusivity is specified as Kh ¼ 1150 m2 s�1 where LDef < 2Dx and 0elsewhere. (A) Time mean zonally integrated overturning transport, averaged overyears 6–20, as a function of resolution. The lines for the 25 km, 33 km and 40 kmruns are nearly indistinguishable. (B) The time- and zonal-mean diffusivity. (C) Thetime mean zonally integrated overturning at 16 km resolution due to theparameterized diffusive transport (red), the explicitly resolved flow (blue) andthe total (dashed magenta). (For interpretation of the references to colour in thisfigure legend, the reader is referred to the web version of this article.)

2 The appropriate value for RCrit will depend on the numerical methods andclosures being used and how well eddies with spatial scales close to the grid spacingare described. The GOLD model configurations described here use an Arakawa C-gridspatial discretization of the dynamic core with a biharmonic Smagorinsky lateralviscosity. The tests described here would need to be repeated to determine the rightvalue of RCrit to use with other numerical discretizations and grid-scale closures.

R. Hallberg / Ocean Modelling 72 (2013) 92–103 101

weaker baroclinic shears relative to bL2Def . (See, for example, chap-

ter 6.6 of Vallis (2006) for a thorough discussion of the linearphase of baroclinic instability in the Phillips model.) Since muchof the ocean is only weakly unstable, and the critical value for

this marginally unstable case also works for the strongly unstablecase, the value of RCrit ¼ 2 should be appropriate for realisticocean modeling.2

Using a step function change in the diffusivity might seem like aterrible idea, because it introduces an artificial discontinuity to dif-fusivity in the solution. However, if the resolution is sufficientlyfine that eddies can grow, the total mass fluxes will vary continu-ously, because otherwise a growing interface height discontinuityand strong associated baroclinicity would develop. In the runsshown here, the continuous transition of the fluxes from theparameterized to the explicit is accomplished by a flattening ofthe instantaneous isopycnal slopes on the side with the strongparameterization and some steepening (but not to an extent thatis highly atypical) on the side where the parameterization is sup-pressed. Another objection to an abruptly changing diffusivitymight be that gradients in a diffusivity are analogous to addingan advective term to a spatial smoother, since

@

@xj@C@x

� �¼ @j@x

@C@xþ j

@2C@x2 ð13Þ

and an abrupt change in the diffusivity is therefore akin to addingan infinite advective speed. However, this does not impose a limita-tion on the stability of the model, because when implemented in amodel these changes actually occur over a grid spacing, meaningboth that the effective cell Reynolds number is guaranteed to be1, and that the CFL time-step limit on the apparent ‘‘advective’’ termin (13) is equivalent to the time-step limit already imposed by thediffusivity.

The simulations presented here strongly suggest that multipli-cation by an abruptly varying resolution function may be an idealway to automatically transition between using parameterizationsof eddy effects in one part of the domain, and allowing an oceanmodel to represent eddy effects explicitly in other regions.

4. Discussion and summary

Most large-scale ocean models include both regions where spa-tial resolution is adequate for eddy effects to be captured explicitlyand regions where the dominant eddy scales are unresolved andany significant effects of eddies need to be parameterized. Thecommon approach has traditionally been to make a choice be-tween parameterizing eddies or not throughout the model’s do-main. This paper instead proposes that eddies should beexplicitly represented in numerical ocean models in those portionsof the domain where the model’s resolution is sufficiently fine andparameterized where it is not. This paper uses an idealized modelof baroclinic instability to demonstrate that this objective can beaccomplished by multiplying the parameterized eddy fluxes by afunction, ranging from 1 to 0, of the ratio of the baroclinic deforma-tion radius to the model’s grid spacing.

Eddy parameterizations suppress mesoscale eddies. Real eddiestend to be self-regulating because they modify their environmentto suppress the source of their own growth, and successful eddyparameterizations do the same thing. For instance, one of the keyprinciples behind the successful Gent–McWilliams eddy parame-terization is that it deliberately extracts available potential energywithout diabatic mixing (Gent et al., 1995). Eddy parameteriza-tions also tend to be scale selective, operating much more rapidlyon smaller horizontal scales than larger scales. Because eddy

102 R. Hallberg / Ocean Modelling 72 (2013) 92–103

spatial scales in the ocean are often comparable to or smaller thanthe scales of the large-scale structures that drive the eddy instabil-ities, eddy parameterizations are typically at least as effective atsuppressing eddies as they are at reproducing their effects on thelarge-scale structure. As a result of this eddy suppression, thisstudy obtained the best results when the resolution function waschosen to change quite abruptly to fully disable the parameteriza-tions where the eddy scales are adequately resolved.

Existing eddy parameterizations introduce only an imperfectrepresentation of some of the effects of eddies. Some improve-ments in large-scale measures of eddy effects, such as the SouthernOcean overturning response to wind-stress changes, can be ob-tained by adopting parameterizations whose intensity and struc-ture are free to vary greatly with the model’s simulated state(Gent and Danabasoglu, 2011). However, other eddy effects arenot well captured by presently available parameterizations. For in-stance, in the case of the Southern Ocean, Hallberg and Gnanadesi-kan (2006) demonstrate that explicitly modeled eddies lead to acoherent transport of water over significant distances, includingsouthward transports of watermasses that are lighter than any thatexist at a given location in the mean. [The atmospheric analog ofthis effect is wintertime cold-air outbreaks (e.g. Held and Schnei-der, 1999).] This is something that no local diffusive parameteriza-tion will ever be able to capture. So even if ocean models hadgreatly improved theories for predicting the dependence of theparameterized eddy-transport strength and structure on the largescale ocean state, there would still be a compelling justificationto rely upon numerical ocean models to explicitly represent eddyeffects where ever possible.

There are many different approaches for prescribing the inten-sity and structure of eddy effects parametrically based on the mod-el’s local mean state (e.g. Visbeck et al., 1997; Held and Larichev,1996; Treguier et al., 1997; Danabasoglu and Marshall, 2007,etc.), or based on an auxiliary energy equation that allows the eddyeffects to be based on past states and to spread in space (e.g. Cessi,2008; Eden and Greatbatch, 2008; Marshall and Adcroft, 2010).Although the utility of the resolution function was demonstratedhere only for a very simple parameterization with a constant iso-pycnal-height diffusivity that was empirically determined for asingle specific configuration, the same idea will apply equally wellwith any of these other more elaborate parameterizations. Thepresent study suggests a way to apply these earlier ideas onlywhere they are needed, and should be seen as complementary tothat past work.

The GOLD simulations presented here do not require any iso-pycnal height diffusivity for numerical stability. However, thereare substantial ancillary benefits of using a Gent–McWilliams dif-fusivity in depth-coordinate B-grid ocean models: it suppressesthe well-known checkerboard null mode in the density field, andit suppresses numerical diapycnal mixing arising from advectivetruncation errors arising from grid-scale tracer variations (Griffieset al., 2000; Ilicak et al., 2012). When it is desirable to retain aneddy parameterization as a numerical closure, it might be advis-able to follow the suggestion of Roberts and Marshall (1998) touse a more highly scale-selective biharmonic interface height dif-fusion (which is much less effective at suppressing well-resolvededdies) with a strength based on numerical considerations, as asupplement to a Laplacian-based parameterization of the physicaleffects that is regulated by a resolution function.

The two-layered Phillips (1954) model of baroclinic instabilityhas been used for decades to understand the dynamics of baroclin-ic instability, but the simplicity of this model clearly imposes cer-tain limitations on what it can be used to explore. With only twolayers, there is only a single baroclinic mode. By contrast, thereare countable baroclinic modes in a continuously stratified ocean,and there is some evidence that these higher modes may be

important for shaping the properties of the eddy field and its rec-tified effects (e.g. Smith, 2007; Berloff et al., 2009). In such cases, itmay be advantageous to keep parameterizing the eddy effects untila higher-mode baroclinic deformation radius is resolved, or equiv-alently until the first mode is very well resolved. In the examplesdiscussed here, the value of ratio of the horizontal resolution tothe deformation radius at which the transition occurs does notmatter too much, provided that the transition value of RH is greaterthan about 2 and the transition is abrupt. The consideration of theeffects of the second or third baroclinic modes might argue forchoosing a transition value of order 4 or 6, since the higher modewave speeds scale approximately inversely with the mode number.Other eddy processes with smaller spatial scales that are indepen-dent of the first baroclinic deformation radius, such as submeso-scale restratification of the mixed layer (Fox-Kemper et al., 2008),should be treated separately, perhaps using a different definitionof a resolution function based on their dominant spatial scales.We have every expectation that the ideas presented here can bestraightforwardly generalized to apply to a broad range of situa-tions where marginally resolved processes sometimes need to beparameterized.

The ideas presented here have been tried in realistic globalocean models at various resolutions ranging from 1� (Dunne,2012) to 1/8� (Adcroft et al., 2010). When combined with an appro-priate closure for the parameterized eddy intensity, the resolutionfunction offers the prospect of exploiting an ocean model’s full po-tential for explicitly representing oceanic processes while behavingsensibly throughout its domain. By eliminating global decisionsabout which processes to parameterize, this approach also offersthe prospect of greatly reducing the amount of parameter tuningthat has traditionally occurred in configuring global ocean-climatemodels at various resolutions. Most global ocean models includesome regions where the mesoscale eddies are well resolved, butother regions where they are not. Moreover, these regions wherethe eddies are resolved will evolve in time with the stratificationof the ocean. (The first baroclinic deformation radius can be calcu-lated quickly enough that updating it frequently as the model’sstate evolves is a trivial part of the cost of running a realistic oceanmodel.) The application of a resolution function, as described here,is a promising avenue for creating global-scale ocean models thathave a credible representation of eddy effects throughout their do-main, via explicit resolution where possible and parameterizationwhere necessary.

Acknowledgments

I would like to thank Alistair Adcroft for numerous conversa-tions on this topic and invaluable feedback, Sonya Legg for herhelpful comments on a preliminary draft, and Matthew Harrisonfor his assistance in remaking one of the figures. I gratefullyacknowledge improvements prompted by three anonymousreviewers. This work was carried out in part while the authorwas a fellow at the Isaac Newton Institute’s 2012 programme on‘‘Multiscale Numerics for the Atmosphere and Ocean’’.

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