Ocean Modelling 48 (2012) 1–9

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Ocean Modelling

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Short communication

Grid degradation of submesoscale resolving ocean models: Benefits for offlinepassive tracer transport

M. Lévy a,⇑, L. Resplandy a,b, P. Klein c, X. Capet c, D. Iovino a, C. Ethé a

a LOCEAN-IPSL, CNRS/UPMC/IRD/MNHN, Paris, Franceb NOCS, Southampton, UKc LPO, CNRS/IFREMER/UBO, Brest, France

a r t i c l e i n f o

Article history:Received 11 October 2011Received in revised form 15 February 2012Accepted 26 February 2012Available online 12 March 2012

Keywords:Effective resolutionSubmesoscale modelsOffline transportGrid degradationPassive tracersBiogeochemical modelsVertical velocity

1463-5003/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ocemod.2012.02.004

⇑ Corresponding author. Tel.: +33 1 44 27 27 07.E-mail address: marina@locean-ipsl.upmc.fr (M. Lé

a b s t r a c t

A numerical solution for an idealized double-gyre is used to investigate the sensitivity of ocean dynamicsand passive tracer advection to horizontal resolution (Dx) in a mesoscale eddy rich regime. In agreementwith previous studies, we find that ocean dynamical solutions are strongly sensitive to grid resolution.With mesoscale resolution ðDx � Oð10Þ kmÞ, eddies are marginally resolved and their impact on tracertransport is not well represented. At submesoscale resolution ðDx � Oð1Þ kmÞ, the number of mesoscaleeddies and their energy is increased, due to the resolved submesoscales. The changes are mostly seen inthe vorticity and vertical velocity fields, and are less obvious in the temperature field. In contrast, wedemonstrate that the offline transport of passive tracer is not altered when the finest scales ðOð1Þ kmÞpresent in the dynamical solutions are discarded. We do so by showing that dynamical solutions obtainedwith Dx � Oð1Þ km can be degraded (following a flux preserving procedure) down to resolutionsDX � Oð10Þ km without significantly impacting passive tracer solutions. The reason for this stems fromthe level of dissipation/diffusion required during the integration of the dynamical model which smoothesvariance at wavelength smaller than at least 5–10 Dx. This result is used to derive a method which alle-viates data storage needs and accelerates tracer advection simulations, with a gain of the order of 103 incomputing time. The method involves three steps: (1) on-line resolution of the dynamics withDx � Oð1Þ km, (2) degradation of the 3D velocity field on a DX � Oð10Þ km grid and (3) off-line tracertransport with the degraded velocity on the DX grid. It opens promising perspectives for submesoscalebio-physical modelling at reduced numerical cost.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Mesoscale eddies (with a diameter of Oð50—300Þ km) are ubiq-uitous in all oceans and capture most of the kinetic energy of geo-strophic turbulence. This turbulence involves a scale range fromhundreds of kilometers down to a few kilometers (the submeso-scales), where a breakdown of geostrophic balance may occur (Fer-rari and Wunsch, 2009). Because of strong nonlinear interactions,resolving not only mesoscale but also submesoscale dynamics iscrucial to obtain accurate climate and biogeochemical solutions(Thomas et al., 2008; Lévy et al., 2010, 2012). In the horizontal,the narrow submesoscale vorticity structures act as transport bar-riers that strengthen mesoscale eddies and come into play in thelarge-scale transport of properties (Lévy et al., 2010). In the verti-cal, the frontogenesis (and the breakdown of geostrophic balance)associated with submesoscales leads to the development of largevertical velocities (Mahadevan and Tandon, 2006; Klein and Lap-

ll rights reserved.

vy).

eyre, 2009), which are important factors in the exchange of prop-erties between the surface mixed-layer and below. In particular,submesoscale vertical motions have important biogeochemicalconsequences because they deliver limiting nutrients to the sunlitsurface layer (Lévy et al., 2001b, 2009; Allen et al., 2005; Nagaiet al., 2008; Calil and Richards, 2010).

The resolution of oceanic submesoscales requires horizontalgrid resolution Dx � Oð1Þ km (Lévy et al., 2001b, 2010; Kleinet al., 2008; Capet et al., 2008). With present day computing capac-ities, submesoscale resolution is feasible over regional domains oflimited extension but poses a number of practical problems: first,the enormous requirement in terms of data storage, and the diffi-culties related to the treatment of this large amount of data; sec-ond, the prohibitive numerical cost required by the transport ofadditional tracers, such as biogeochemical or transient tracers(which can be numerous, up to �100 tracers for the more complexecosystem models, i.e. Follows et al. (2007)).

However, the forward integration of submesoscale resolvingdynamical models requires a certain level of dissipation and diffu-sion which smoothes the variance at wavelengths smaller than at

2 M. Lévy et al. / Ocean Modelling 48 (2012) 1–9

least 5–10 Dx (Skamarock, 2004; Marchesiello et al., 2011). Thissmoothing implies that the effective resolution of the model out-puts (DX), i.e. the smallest size of the structures which are cap-tured by the model outside the dissipative range, is less than thegrid resolution (DX > Dx) on which model equations are discret-ized and solved. The difference between the computational gridresolution Dx and the effective resolution DX is illustrated herein the context of an idealized basin scale model configuration,where the oceanic circulation is forced by atmospheric fields. Sev-eral simulations are performed, with identical configurations butdifferent model resolution Dx (and accordingly, different lateraldiffusion and dissipation); the model outputs at the highest resolu-tion are degraded onto the lower resolution grids. This is done overa resolution range Dx varying from Oð100Þ km to Oð1Þ km that cov-ers the resolution of climate models, eddy-permitting models,eddy-resolving models and submesoscale-resolving models.

The transport of tracers in such submesoscale turbulence can beperformed off-line (separately from dynamical simulations) whenthe tracers have no impact on the dynamics. In such a case, thismeans that, in practice, these simulations can be done on a coarsergrid than the submesoscale grid used to compute the dynamicalfield. Grid degradation for offline tracer models is common practiceat coarse ðOð100Þ kmÞ resolution, and is used to accelerate global,long-term simulations (Aumont et al., 1998). To take advantageof the difference between computational resolution Dx and effec-tive resolution DX at the submesoscale, this note explores the pos-sibility of grid degradation from Dx to DX for off-line tracertransport by a submesoscale velocity field.

The method is described in Section 2. Then, in Section 3, wedemonstrate that there is a factor of at least five between the com-putational resolution needed to explicitly solve submesoscaledynamics and the effective resolution of the submesoscale struc-tures that are solved (DX J 5 � Dx). To take advantage of this dif-ference, off-line simulations on a grid degraded by a factor of �5are presented and evaluated. Our results open promising perspec-tives for the computation of submesoscale off-line tracer transportat degraded resolution, which are discussed in the last section.

2. Methods

The numerical experiments are carried in a rectangular domainof dimension 3180 � 2120 � 4 km, rotated on the b-plane (Fig. 1),forced by analytical zonal wind and buoyancy forcings with the le-vel-coordinate free-surface primitive equation ocean model NEMO(Madec, 2008). The atmospheric forcing generates a strong jetwhich runs diagonally across the domain, separating a warm sub-tropical gyre from a colder subpolar gyre (Fig. 1). The jet is barocli-nically unstable and this instability is the main source of eddyenergy in the model. More detailed information about the modelconfiguration can be found in Lévy et al. (2010).

Five experiments have been performed (R1, R3, R9, R27 andR54), with different horizontal resolution (1�, 1/3�, 1/9�, 1/27�and 1/54�) and adapted lateral sub-grid scale closures (Table 1).The vertical resolution, 30 z-coordinate vertical layers, whosethickness increase with depth from 10 to 300 m, is kept identicalin all experiments. The coarse resolution experiment R1 has a hor-izontal resolution of approximately 1�. More precisely, the R1 gridis comprised of 20 � 30 regular cells in the horizontal, which havea length of 106 km in both directions. The resolution is progres-sively increased by dividing each cell equally into 2 � 2 or 3 � 3matrix as many times as needed. At coarse resolution (R1 andR3), laplacian friction dissipates momentum along horizontal sur-faces. Temperature and salinity are diffused along isopycnal sur-faces without horizontal background. At higher resolution (R9,R27 and R54), bi-harmonic friction (KM) and bi-harmonic diffusion

(KT) act along horizontal surfaces (Table 1). There is no theory to

guide the choice of the turbulent Prandtl number Pr ¼ KMKT

� �for

subgrid-scale parameterizations in ocean models. When bihar-monic closures are used (as is the case for R9, R27 and R54), someauthors have chosen Pr = 1 (e.g. Willebrand et al., 2001), others lar-ger values of Pr (Lévy et al., 2012). Here, the choice was to havePr = 1 because it led to limited noise for vertical velocity (Section3) compared to a larger Pr. Results are shown after a spin-up of100 years for all experiments.

The model primitive equations are solved on an Arakawa C-grid,i.e. with tracers (t) located at the center of grid cells and velocities(u, v and w) located at the center of the grid faces (west/east, south/north, up/down faces of the cells, respectively). Model solutionsfrom the submesoscale resolution R54 experiment are degradedfrom the parent R54 C-grid to coarser resolution C-grids (corre-sponding to the grids of R27, R9, R3 and R1, respectively). Weuse the degradation technique developed by Aumont et al.(1998) for the global ocean. More precisely, each degraded cellcomprises n2 cells of the parent R54 grid (with n = 2 for a degrada-tion on the R27 grid, n = 6 for a degradation to the R9 grid and soon). If t, u, v, and w are the values of tracers and velocities of theparent model, and T, U, V, and W are the values over the degradedgrid, the degradation is done by allocating: to T the mean over then2 values of t located inside the degraded grid cell; to U (resp. V)the mean over the n values of u (resp. v) located on the west/east(resp. south/north) faces of the degraded cell; and to W the meanover the n2 values of w located on its up/down faces (Fig. 2). Thisdegradation procedure ensures the conservation of tracers andwater fluxes, hence also preserves the non-divergence of the veloc-ities and a closed tracer budget. Thus, degrading from R54 to the R9grid implies averaging over 36 t and w points, and over 6 u and vpoints.

Off-line transport experiments are carried out by solving thetransport equation of a synthetic passive tracer, taken as an analogfor nutrients. This tracer is initially uniformly distributed in thehorizontal and, as nutrients, increases with depth: from 0 at thesurface to 1 at 100 m depth (i.e. the euphotic depth) in a hyperbolictangent manner. Three experiments are compared (Table 2). Twoof them use the 3D velocity field from the R54 model (R54OFFand R54onR9OFF) and one uses the velocity from the R9 model(R9OFF). These velocity fields have been saved at a frequency of2 days (2d-averages) and they are interpolated in time at each timestep during the off-line integration. R54OFF is performed on theparent R54 grid. R54onR9OFF is performed on the R9 grid and usesthe R54 velocity field degraded to the R9 grid, while R9OFF is alsoperformed on the R9 grid but uses the velocity field of the R9 mod-el. The time step of the offline model is 2 min on the R54 grid and20 min on the R9 grid. The same constant diffusivity is applied toall off-line experiments, Kv = 10�4 m2 s�1 in the vertical and abiharmonic Kh = �10�9 m4 s�1 in the horizontal. Note that this va-lue of Kh is the same as for the diffusivity of tracers in R54 (Table 1).In all experiments, advection is performed with a flux-correctedtransport scheme (the TVD scheme used in Lévy et al. (2001a)).

3. Results

3.1. Changing the resolution of the computational grid

An important impact of the model resolution is the emergencein the relative vorticity field of smaller and smaller eddies and fil-amentary structures resulting from non-linear interactions (Fig. 3,top panels). At 1/3� (R3), wave-like undulations start to appear, andthose become better-defined at 1/9� (R9) along the jet which flowsdiagonally across the domain. At 1/27� (R27), these instabilitiesoccasionally lead to the break-out of large eddies (200 km diame-

Fig. 1. The rotated, rectangular, idealized model domain, is representative of the western sector of the North Atlantic and characterized by a north–south gradient in sea-surface temperature (SST). The meandering of the model’s Gulf Stream is visible along �30�N.

Table 1Parameters of the model experiments.

R1 R3 R9 R27 R54

Horizontal resolution 1� 1/3� 1/9� 1/27� 1/54�Horizontal resolution 106 km 35.3 km 11.8 km 3.9 km 2.0 kmHorizontal grid points 20 � 30 60 � 90 180 � 270 540 � 810 1080 � 1620Time step 2 h 1 h 20 min 5 min 2 minEddy viscosity 105 m2 s�1 �1012 m2 s�1 �5 � 1010 m4 s�1 �5 � 109 m4 s�1 �109 m4 s�1

Eddy diffusivity 1000 m2 s�1 300 m2 s�1 �5 � 1010 m4 s�1 �5 � 109 m4 s�1 �109 m4 s�1

M. Lévy et al. / Ocean Modelling 48 (2012) 1–9 3

ter). As the resolution is increased to 1/54� (R54), more coherentaxisymmetric eddies emerge leading to a denser and well definedvortex population, covering a wide range of scales and populatingmost of the basin. The eddies have diameters between 50 and200 km. Dipole vortices and sub-mesoscale filaments are also pres-ent. The emergence of more numerous and smaller eddies with

resolution is related to the better resolution of not only the firstinternal Rossby radius of deformation but also of the Rossby radiusassociated with higher baroclinic modes, which are known to affectthe dynamics of the mesoscale turbulence (Barnier et al., 1991).The stronger frontogenesis in R54 naturally leads to more intensevertical velocities, associated with the sub-mesoscale filaments

Fig. 2. Dynamic and tracer properties of the parent and degraded grid for a grid degradation with a ratio n = 3. Small letters, dashed lines and open circles correspond to theparent model ðt;u;vÞ, plain lines, plain circles and capital letters ðT;U;VÞ denote the degraded grid. TðI; JÞ ¼

Pi�1;iþ1

Pj�1;jþ1tði; jÞ. a and A are the water fluxes along the

direction i; b and B the water fluxes along the direction j. u, v and U, V are the horizontal velocities, t and T the tracer concentrations. The vertical velocity of the parent anddegraded models (w and W, respectively), are located immediately above the tracer grid points but between two tracer boxes. Adapted from Aumont et al. (1998).

Table 2Parameters of the offline experiments.

R54OFF R54onR9OFF R9OFF

Horizontal resolution 1/54� 1/9� 1/9�Horizontal resolution 2.0 km 11.8 km 11.8 kmHorizontal grid points 1080 � 1620 180 � 270 180 � 270Time step 2 min 20 min 20 minDynamics R54 R54 R9Degradation No Yes NoEddy diffusivity �109 m4 s�1 �109 m4 s�1 �109 m4 s�1

4 M. Lévy et al. / Ocean Modelling 48 (2012) 1–9

(Fig. 4, top panels). Moreover, the manifestation of the direct cas-cade of tracers is seen as model resolution is increased. The emerg-ing mesoscale circulation stirs the large-scale SST fields intosubmesoscale structures. Stirring becomes effective in R9, whenthe first eddies are formed, and intensifies in R54 (Fig. 5, top pan-els). The differences are quantified more precisely by the examina-tion of the slope of the energy spectra of the various fields (Fig. 6a,c and e). In R54, a noticeable shallower spectrum slope (k�1 for vor-ticity, k�2 for SST, k�0 for W) is observed over the wavenumberband (k) comprised between 30 and 200 (corresponding to wave-length between 20 and 100 km). This slope gets steeper and stee-per as the model grid resolution is decreased (for instance k�3 inR9 for vorticity).

In summary, these model results illustrate that submesoscaleturbulence, which emanates from the interaction between meso-scale eddies, emerges with a computational grid resolution of 1/9�, and intensifies significantly at 1/27� and further at 1/54�. This

intensification is associated with a higher degree of horizontal stir-ring and of vertical movements. In other words, moving frommesoscale to submesoscale resolution (from R9 to R54) enables abetter definition of the velocities associated with mesoscale eddies.This is mostly seen in the vorticity and vertical velocity fields (Figs.3 and 4) and is less apparent in the temperature field (Fig. 5).

3.2. Effective resolution and degradation of the R54 model

Although a computational grid resolution of 1/54� (2 km) isfound necessary to simulate an energetic submesoscale turbulence,in R54, most of the energy is comprised at scales larger than�20 km. This is seen for instance in the spectral properties of theR54 model fields (Fig. 6a, c and e) which display a rapid steepeningof the spectral slopes for wavenumber larger than 200 (i.e. wave-lengths smaller than 20 km). This steepening reflects the dissipa-tion induced by the biharmonic operator, which selectivelydissipates energy at the smallest scales. The wavenumber at whichthe spectral slopes start to steepen (k � 200) marks the boundaryof this limited but finite dissipative range and provides a measureof the effective resolution DX of the R54 model (Skamarock, 2004;Marchesiello et al., 2011).

To illustrate this, we have degraded the R54 model outputs overthe R27, R9, R3 and R1 grids (Figs. 3–5, bottom panels). Down tothe R9 grid, hardly any change can be detected by eye: the finescale structures of the surface relative vorticity, which can onlybe simulated with a resolution of 1/54�, are preserved with excel-lent accuracy when degraded to a 1/9� grid (Fig. 3). A resolution of1/3� is sufficient to capture the main eddies and fronts but not the

Fig. 3. Model snapshots of surface relative vorticity simulated with increasing grid resolution, from 1� to 1/54� (top panels), and surface vorticity simulated with a resolutionof 1/54� and degraded on coarser resolution grids, from 1� to 1/27� (bottom panels). The color intervals are chosen to highlight the structures, but are not representative of theextremum values.

Fig. 4. Model snapshots of vertical velocity at 40 m (W) simulated with increasing grid resolution, from 1� to 1/54� (top panels), and W simulated with a resolution of 1/54�and degraded on coarser resolution grids, from 1� to 1/27� (bottom panels). The color intervals are chosen to highlight the structures, but are not representative of theextremum values. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

M. Lévy et al. / Ocean Modelling 48 (2012) 1–9 5

details of the thin filaments. At 1�, the meandering of the main jetis still detectable but eddies become hard to distinguish. Fig. 4(bottom panels) further shows that a grid degradation from 1/54�down to 1/9� does not alter the strength and the shape of the

vertical velocity field, even the finest W-structures which are clo-sely associated with thin and sharp SST fronts (Fig. 5).

The power spectra of the degraded fields provide a more quan-titative measure of the similarities between the degraded and par-

Fig. 5. Model snapshots of sea-surface temperature (SST) simulated with increasing grid resolution, from 1� to 1/54� (top panels), and SST simulated with a resolution of 1/54�and degraded on coarser resolution grids, from 1� to 1/27� (bottom panels).

6 M. Lévy et al. / Ocean Modelling 48 (2012) 1–9

ent fields (Fig. 6b, d and f). This computation confirms that theshallow part of the spectrum in the R54 simulation is capturedwith model outputs degraded on the R27 and R9 grids. There is amoderate decrease of the variance at the smallest scales whendegrading R54 on the R9 grid, but this decrease is small when com-pared to the loss of variance due to decreasing model resolutionfrom R54 to R9 (Fig. 6a, c and e). This analysis suggests that theeffective resolution of the R54 experiment is of the order of 10–20 km, i.e. 5–10 times the resolution of the actual computationalgrid. It also suggests that the use of R54 model outputs degradedon the R9 grid is sufficient to analyze processes at the smallest re-solved scales in R54, and thus that for most applications, data stor-age and data analysis on the R9 grid is sufficient for the R54 model.

3.3. Offline transport experiment

In order to take advantage of the effective resolution being lessthan the computational resolution in R54, we have tested the pos-sibility of advecting a passive tracer using the velocity field com-puted in R54 grid but degraded on the R9 grid (experimentR54onR9OFF, Table 2). This test is compared with a similar integra-tion performed on the parent R54 grid (experiment R54OFF). Interms of computing time, R54OFF is 103 times more expensivethan R54onR9OFF. The distribution of the passive tracer is initiallyhomogeneous in the horizontal and mimics the distribution ofnutrients in the vertical (0 at the surface increasing to 1 at100 m). The 3D dynamics perturbs this initial distribution bymeans of upwelling and downwelling, associated with lateral stir-ring. The results are examined after 10 days, which is a time scalelong enough for a plankton population to react to nutrient supplies.In R54OFF, the perturbed tracer field at 30 m displays strong vari-ations associated with the submesoscale filaments and the sub-mesoscale upwelling/downwelling hot spots (Fig. 7a). The resultsin R54onR9OFF are very similar (Fig. 7b), with only weak differ-ences for the highest/smallest tracer values (Fig. 7c). Close inspec-tion reveals small grid-scale noise due to the choice of the samelow horizontal diffusion value as for R54OFF.

The mean effect of the 3D dynamics, when averaged horizon-tally over the interior of the domain, is a flux of tracer from thesubsurface to the surface. This flux is seen by the removal of tracerfrom the layer 20–100 m and a supply of tracer to the upper 20 m(Fig. 7d), and has similar strength in R54OFF and R54onR9OFF. Incomparison, advection with a different 3D dynamics, issued fromthe R9 experiment (R9OFF), and thus with weaker vertical veloci-ties, leads to a noticeably reduced vertical tracer flux (Fig. 7d). Thisconfirms that the offline transport of this passive tracer by themesoscale turbulence is not well represented when velocity fieldscomputed at mesoscale resolution (i.e. resolution of R9) are used.On the other hand, with velocity fields computed at submesoscaleresolution (i.e. resolution of R54), the offline experiment on themesoscale (R9) grid presents a degree of similarity with the sameexperiment at submesoscale (R54) resolution which is extremelysatisfactory for oceanic applications, and which enables an enor-mous gain in computing time.

4. Discussion

Recent studies recognize that most of the oceanic kinetic energyis captured by mesoscale eddies (see Ferrari and Wunsch, 2009).However they point out that submesoscales, although less ener-getic, are associated with significant vertical velocities that maysignificantly impact the mesoscale eddy field (Lapeyre and Klein,2006; Thomas et al., 2008). The consequence is that an accuraterepresentation of such turbulent dynamical field requires takinginto account the whole scale range including the submesoscales(Klein et al., 2008; Capet et al., 2008; Lévy et al., 2010). In ourexperiments, the impact of submesoscales on mesoscale eddies isemphasized by the decrease of the velocity and vorticity spectralintensity with model resolution, not only at the smallest scalesbut also at scales resolved by coarser resolution models: for in-stance at wavenumber 100 (corresponding to a wavelength of20 km, i.e. about twice the grid resolution at 1/9�), the vorticityspectral intensity is noticeably reduced in R9 compared to R54

Fig. 6. (a, c, e) Power spectra of surface relative vorticity, vertical velocity at 40 m and SST shown in Figs. 3–5 simulated with different model grid resolutions. (b, d, f) Powerspectra of surface relative vorticity, vertical velocity at 40 m and SST simulated at 1/54� resolution, and degraded on lower resolution grids.

M. Lévy et al. / Ocean Modelling 48 (2012) 1–9 7

(Fig. 6a). A similar reduction is seen in the temperature spectralintensity, but with less amplitude (Fig. 6e). This illustrates theimportant role of submesoscales on mesoscale eddies even if thesesmall scales are much less energetic. To a certain extent, the under-estimation of the dynamics in the mesoscale range (i.e. above�20 km) at mesoscale resolution is due to the inhibition of the in-verse cascade of kinetic energy from submesoscales to mesoscalesbecause these submesoscales are not resolved.

Model integration requires a certain level of dissipation and dif-fusion (present for both physical and numerical reasons) that can-not be confined to the very finest scales. The consequence is thatthe effective resolution of the model is reduced; in practice, pro-cesses with wavelength N � Dx are strongly dissipated and there-fore cannot be fully resolved. Within our model setup, weevaluate that N J 5–10, and certainly this value depends on theadvective and diffusive algorithms used.

The 3D advection of a passive tracer is significantly affected bythe submesocale horizontal and vertical velocities. We see threereasons for this. First, the vertical velocities are particularly strong

in the submesocale range. Second, horizontal turbulent flows stirtracers into smaller and smaller scales (Abraham, 1998). And third,the resulting tracer field strongly depends on the scales of the ver-tical velocity field and on its phasing with the horizontal stirringfield (Klein et al., 1998; Martin et al., 2002; Smith and Ferrari,2009). Our results demonstrate that, with a model resolution ofOð1Þ km, the resolved scale range of velocities that effectively con-tribute to the submesoscale 3D advection are scales larger thanOð10Þ km. In contrast, velocity scales smaller than Oð10Þ km fallin the dissipative range; they are much less energetic and theircontribution to the 3D advection is negligible. In terms of offlinemodel integrations, degrading velocity fields from a Oð1Þ km to aOð10Þ km grid effectively comes down to neglecting the contribu-tion of the less energetic small scales, while retaining their criticalrole on larger scales.

Thus practically, the spectra shown in Fig. 6b, d and f can beused to determine the scale of the degraded grid: this scale roughlycorresponds to the limit between the dissipative range and theenergetic range, i.e. to the value of k where the slope breaks in

Fig. 7. Results of off-line transport experiments of a passive tracer (a) using the velocity fields computed at 1/54� resolution with offline tracer transport computed at 1/54�resolution (R54OFF), (b) using the velocity fields computed at 1/54� resolution, degraded on the 1/9� grid, with offline tracer transport computed at 1/9� resolution(R54onR9OFF). (a) and (b) show the tracer concentration at 30 m depth after 10 days of model integration. The initial state is uniform in the horizontal, with a verticalgradient on the vertical, increasing from 0 at the surface to 1 at 100 m and below. (c) Scatter plot of the field shown in (a) against the field shown in (b). (d) Vertical profile ofthe tracer anomaly resulting from the vertical tracer flux after 10 days of transport (i.e. profile at day 10 minus initial profile) for experiment shown in (a) (black), in (b) (red)and for an offline transport experiment using the velocity fields computed at 1/9� resolution with offline tracer transport computed at 1/9� resolution (B9OFF, blue). (Forinterpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

8 M. Lévy et al. / Ocean Modelling 48 (2012) 1–9

the submesoscale simulation R54. This value of k roughly corre-sponds to the Nyquist wavenumber of R9 (last k value of the R9spectra, corresponding to the wavelength of two grid spacings inR9); Fig. 6 clearly shows that a coarser resolution (such as the res-olution of R3) would not be suitable as it does not retain some ofthe energetic scales that are above the dissipative range (k � 50–100).

The results examined in this paper suggest that an importantgain can be made by averaging the outputs of a submesoscaleresolving ðOð1Þ kmÞ ocean dynamical model on a coarser mesoscalegrid (i.e. Oð10Þ km), without loss of information. The gain is of theorder of 102 for data storage and 103 for off-line computation. Thisis because the smallest ‘physical’ scales which are properly re-solved with a resolution of Oð1Þ km are of Oð10Þ km; the scales be-tween 1 and 10 km are less energetic and highly dissipated.Nevertheless, a dynamical model resolution of Oð1Þ km turns outto be essential to sustain an efficient inverse cascade and maintainan energetic mesoscale circulation. Regarding biogeochemicalmodels, this suggests that coarser grids could be used for the inte-gration of biogeochemical tracers when the integration of thedynamical fields is performed on a submesoscale grid.

This possibility was examined in a simple off-line experiment.Although this experiment uses a regular grid and has the sametopography before and after degradation, the degradation tech-nique that was used can be easily extended to an irregular grid

with more complex topography. In this case, the high-resolutionparent model is naturally run with a high-resolution land-seamask; the degraded offline model requires a coarse resolutionland-sea mask which is constructed the following way: each timea coarse cell comprises at least one ocean point of the parenthigh-resolution grid, it is affected an ocean value; thus the totalvolume of the ocean is larger in the coarse grid than in the high-resolution grid. Also, the velocity fields must be multiplied bythe surface of the cell to which they are attached (water fluxes aand b in Fig. 2) and by the land-sea mask prior to averaging (seeAumont et al., 1998 for more details).

The success of the proposed method relies on the fact that thedynamics is dominated by 3D advection and not by vertical mixing.The degradation of vertical mixing remains to be properlyaddressed.

Acknowledgements

The simulations were performed at IDRIS (France) from initialstates computed on the Earth Simulator (ESC, Yokohama, Japan).The post-doc of L.R. was supported by the EUR-OCEANS submeso-scale flagship. This project was supported by CNRS (INSU-LEFETANGGO project), CNES (CPUMP project) and FP7 European projectEMBRACE No. 282672.

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