Ocean Modelling 48 (2012) 1 9. Contents lists available at SciVerse ScienceDirect. Ocean Modelling. journal homepage:

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1 Ocean Modelling 48 (2012) 1 9 Contents lists available at SciVerse ScienceDirect Ocean Modelling journal homepage: Short communication Grid degradation of submesoscale resolving ocean models: Benefits for offline passive 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, France b NOCS, Southampton, UK c LPO, CNRS/IFREMER/UBO, Brest, France article info abstract Article history: Received 11 October 2011 Received in revised form 15 February 2012 Accepted 26 February 2012 Available online 12 March 2012 Keywords: Effective resolution Submesoscale models Offline transport Grid degradation Passive tracers Biogeochemical models Vertical velocity A numerical solution for an idealized double-gyre is used to investigate the sensitivity of ocean dynamics and passive tracer advection to horizontal resolution (Dx) in a mesoscale eddy rich regime. In agreement with 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 tracer transport is not well represented. At submesoscale resolution ðdx Oð1Þ kmþ, the number of mesoscale eddies and their energy is increased, due to the resolved submesoscales. The changes are mostly seen in the vorticity and vertical velocity fields, and are less obvious in the temperature field. In contrast, we demonstrate that the offline transport of passive tracer is not altered when the finest scales ðoð1þ kmþ present in the dynamical solutions are discarded. We do so by showing that dynamical solutions obtained with Dx Oð1Þ km can be degraded (following a flux preserving procedure) down to resolutions DX Oð10Þ km without significantly impacting passive tracer solutions. The reason for this stems from the level of dissipation/diffusion required during the integration of the dynamical model which smoothes variance at wavelength smaller than at least 5 10 Dx. This result is used to derive a method which alleviates data storage needs and accelerates tracer advection simulations, with a gain of the order of 10 3 in computing time. The method involves three steps: (1) on-line resolution of the dynamics with Dx Oð1Þ km, (2) degradation of the 3D velocity field on a DX Oð10Þ km grid and (3) off-line tracer transport with the degraded velocity on the DX grid. It opens promising perspectives for submesoscale bio-physical modelling at reduced numerical cost. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction Corresponding author. Tel.: address: marina@locean-ipsl.upmc.fr (M. Lévy). Mesoscale eddies (with a diameter of Oð50 300Þ km) are ubiquitous in all oceans and capture most of the kinetic energy of geostrophic turbulence. This turbulence involves a scale range from hundreds of kilometers down to a few kilometers (the submesoscales), where a breakdown of geostrophic balance may occur (Ferrari and Wunsch, 2009). Because of strong nonlinear interactions, resolving not only mesoscale but also submesoscale dynamics is crucial 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 barriers that strengthen mesoscale eddies and come into play in the large-scale transport of properties (Lévy et al., 2010). In the vertical, the frontogenesis (and the breakdown of geostrophic balance) associated with submesoscales leads to the development of large vertical velocities (Mahadevan and Tandon, 2006; Klein and Lapeyre, 2009), which are important factors in the exchange of properties between the surface mixed-layer and below. In particular, submesoscale vertical motions have important biogeochemical consequences because they deliver limiting nutrients to the sunlit surface layer (Lévy et al., 2001b, 2009; Allen et al., 2005; Nagai et al., 2008; Calil and Richards, 2010). The resolution of oceanic submesoscales requires horizontal grid resolution Dx Oð1Þ km (Lévy et al., 2001b, 2010; Klein et al., 2008; Capet et al., 2008). With present day computing capacities, submesoscale resolution is feasible over regional domains of limited extension but poses a number of practical problems: first, the enormous requirement in terms of data storage, and the difficulties related to the treatment of this large amount of data; second, the prohibitive numerical cost required by the transport of additional tracers, such as biogeochemical or transient tracers (which can be numerous, up to 100 tracers for the more complex ecosystem models, i.e. Follows et al. (2007)). However, the forward integration of submesoscale resolving dynamical models requires a certain level of dissipation and diffusion which smoothes the variance at wavelengths smaller than at /$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.

2 2 M. Lévy et al. / Ocean Modelling 48 (2012) 1 9 least 5 10 Dx (Skamarock, 2004; Marchesiello et al., 2011). This smoothing implies that the effective resolution of the model outputs (DX), i.e. the smallest size of the structures which are captured by the model outside the dissipative range, is less than the grid resolution (DX > Dx) on which model equations are discretized and solved. The difference between the computational grid resolution Dx and the effective resolution DX is illustrated here in the context of an idealized basin scale model configuration, where the oceanic circulation is forced by atmospheric fields. Several simulations are performed, with identical configurations but different model resolution Dx (and accordingly, different lateral diffusion and dissipation); the model outputs at the highest resolution are degraded onto the lower resolution grids. This is done over a resolution range Dx varying from Oð100Þ km to Oð1Þ km that covers the resolution of climate models, eddy-permitting models, eddy-resolving models and submesoscale-resolving models. The transport of tracers in such submesoscale turbulence can be performed off-line (separately from dynamical simulations) when the tracers have no impact on the dynamics. In such a case, this means that, in practice, these simulations can be done on a coarser grid than the submesoscale grid used to compute the dynamical field. Grid degradation for offline tracer models is common practice at coarse ðoð100þ kmþ resolution, and is used to accelerate global, long-term simulations (Aumont et al., 1998). To take advantage of the difference between computational resolution Dx and effective resolution DX at the submesoscale, this note explores the possibility of grid degradation from Dx to DX for off-line tracer transport by a submesoscale velocity field. The method is described in Section 2. Then, in Section 3, we demonstrate that there is a factor of at least five between the computational resolution needed to explicitly solve submesoscale dynamics and the effective resolution of the submesoscale structures that are solved (DX J 5 Dx). To take advantage of this difference, off-line simulations on a grid degraded by a factor of 5 are presented and evaluated. Our results open promising perspectives for the computation of submesoscale off-line tracer transport at degraded resolution, which are discussed in the last section. 2. Methods The numerical experiments are carried in a rectangular domain of dimension km, rotated on the b-plane (Fig. 1), forced by analytical zonal wind and buoyancy forcings with the level-coordinate free-surface primitive equation ocean model NEMO (Madec, 2008). The atmospheric forcing generates a strong jet which runs diagonally across the domain, separating a warm subtropical gyre from a colder subpolar gyre (Fig. 1). The jet is baroclinically unstable and this instability is the main source of eddy energy in the model. More detailed information about the model configuration can be found in Lévy et al. (2010). Five experiments have been performed (R1, R3, R9, R27 and R54), 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, whose thickness increase with depth from 10 to 300 m, is kept identical in all experiments. The coarse resolution experiment R1 has a horizontal resolution of approximately 1. More precisely, the R1 grid is comprised of regular cells in the horizontal, which have a length of 106 km in both directions. The resolution is progressively increased by dividing each cell equally into 2 2 or 3 3 matrix as many times as needed. At coarse resolution (R1 and R3), laplacian friction dissipates momentum along horizontal surfaces. Temperature and salinity are diffused along isopycnal surfaces without horizontal background. At higher resolution (R9, R27 and R54), bi-harmonic friction (K M ) and bi-harmonic diffusion (K T ) act along horizontal surfaces (Table 1). There is no theory to guide the choice of the turbulent Prandtl number P r ¼ K M K T for subgrid-scale parameterizations in ocean models. When biharmonic closures are used (as is the case for R9, R27 and R54), some authors have chosen P r = 1 (e.g. Willebrand et al., 2001), others larger values of P r (Lévy et al., 2012). Here, the choice was to have P r = 1 because it led to limited noise for vertical velocity (Section 3) compared to a larger P r. Results are shown after a spin-up of 100 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 solutions from the submesoscale resolution R54 experiment are degraded from the parent R54 C-grid to coarser resolution C-grids (corresponding to the grids of R27, R9, R3 and R1, respectively). We use the degradation technique developed by Aumont et al. (1998) for the global ocean. More precisely, each degraded cell comprises n 2 cells of the parent R54 grid (with n = 2 for a degradation on the R27 grid, n = 6 for a degradation to the R9 grid and so on). If t, u, v, and w are the values of tracers and velocities of the parent model, and T, U, V, and W are the values over the degraded grid, the degradation is done by allocating: to T the mean over the n 2 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 mean over the n 2 values of w located on its up/down faces (Fig. 2). This degradation procedure ensures the conservation of tracers and water fluxes, hence also preserves the non-divergence of the velocities and a closed tracer budget. Thus, degrading from R54 to the R9 grid implies averaging over 36 t and w points, and over 6 u and v points. Off-line transport experiments are carried out by solving the transport equation of a synthetic passive tracer, taken as an analog for nutrients. This tracer is initially uniformly distributed in the horizontal and, as nutrients, increases with depth: from 0 at the surface to 1 at 100 m depth (i.e. the euphotic depth) in a hyperbolic tangent manner. Three experiments are compared (Table 2). Two of them use the 3D velocity field from the R54 model (R54OFF and R54onR9OFF) and one uses the velocity from the R9 model (R9OFF). These velocity fields have been saved at a frequency of 2 days (2d-averages) and they are interpolated in time at each time step during the off-line integration. R54OFF is performed on the parent R54 grid. R54onR9OFF is performed on the R9 grid and uses the R54 velocity field degraded to the R9 grid, while R9OFF is also performed on the R9 grid but uses the velocity field of the R9 model. The time step of the offline model is 2 min on the R54 grid and 20 min on the R9 grid. The same constant diffusivity is applied to all off-line experiments, Kv =10 4 m 2 s 1 in the vertical and a biharmonic K h = 10 9 m 4 s 1 in the horizontal. Note that this value of K h is the same as for the diffusivity of tracers in R54 (Table 1). In all experiments, advection is performed with a flux-corrected transport 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 emergence in the relative vorticity field of smaller and smaller eddies and filamentary structures resulting from non-linear interactions (Fig. 3, top panels). At 1/3 (R3), wave-like undulations start to appear, and those become better-defined at 1/9 (R9) along the jet which flows diagonally across the domain. At 1/27 (R27), these instabilities occasionally lead to the break-out of large eddies (200 km diame-

3 M. Lévy et al. / Ocean Modelling 48 (2012) 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 seasurface temperature (SST). The meandering of the model s Gulf Stream is visible along 30 N. Table 1 Parameters 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 km Horizontal grid points Time step 2 h 1 h 20 min 5 min 2 min Eddy viscosity 10 5 m 2 s m 4 s m 4 s m 4 s m 4 s 1 Eddy diffusivity 1000 m 2 s m 2 s m 4 s m 4 s m 4 s 1 ter). As the resolution is increased to 1/54 (R54), more coherent axisymmetric eddies emerge leading to a denser and well defined vortex population, covering a wide range of scales and populating most of the basin. The eddies have diameters between 50 and 200 km. Dipole vortices and sub-mesoscale filaments are also present. The emergence of more numerous and smaller eddies with resolution is related to the better resolution of not only the first internal Rossby radius of deformation but also of the Rossby radius associated with higher baroclinic modes, which are known to affect the dynamics of the mesoscale turbulence (Barnier et al., 1991). The stronger frontogenesis in R54 naturally leads to more intense vertical velocities, associated with the sub-mesoscale filaments

4 4 M. Lévy et al. / Ocean Modelling 48 (2012) 1 9 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 the parent model ðt; u;vþ, plain lines, plain circles and capital letters ðt; U; VÞ denote the degraded grid. TðI; JÞ ¼ P P i 1;iþ1 j 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 and degraded 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 2 Parameters 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 km Horizontal grid points Time step 2 min 20 min 20 min Dynamics R54 R54 R9 Degradation No Yes No Eddy diffusivity 10 9 m 4 s m 4 s m 4 s 1 (Fig. 4, top panels). Moreover, the manifestation of the direct cascade of tracers is seen as model resolution is increased. The emerging mesoscale circulation stirs the large-scale SST fields into submesoscale structures. Stirring becomes effective in R9, when the first eddies are formed, and intensifies in R54 (Fig. 5, top panels). The differences are quantified more precisely by the examination 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 vorticity, k 2 for SST, k 0 for W) is observed over the wavenumber band (k) comprised between 30 and 200 (corresponding to wavelength between 20 and 100 km). This slope gets steeper and steeper as the model grid resolution is decreased (for instance k 3 in R9 for vorticity). In summary, these model results illustrate that submesoscale turbulence, which emanates from the interaction between mesoscale 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 stirring and of vertical movements. In other words, moving from mesoscale to submesoscale resolution (from R9 to R54) enables a better 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) Effective resolution and degradation of the R54 model Although a computational grid resolution of 1/54 (2 km) is found 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 the R54 model fields (Fig. 6a, c and e) which display a rapid steepening of the spectral slopes for wavenumber larger than 200 (i.e. wavelengths smaller than 20 km). This steepening reflects the dissipation induced by the biharmonic operator, which selectively dissipates energy at the smallest scales. The wavenumber at which the spectral slopes start to steepen (k 200) marks the boundary of this limited but finite dissipative range and provides a measure of the effective resolution DX of the R54 model (Skamarock, 2004; Marchesiello et al., 2011). To illustrate this, we have degraded the R54 model outputs over the R27, R9, R3 and R1 grids (Figs. 3 5, bottom panels). Down to the R9 grid, hardly any change can be detected by eye: the fine scale structures of the surface relative vorticity, which can only be simulated with a resolution of 1/54, are preserved with excellent accuracy when degraded to a 1/9 grid (Fig. 3). A resolution of 1/3 is sufficient to capture the main eddies and fronts but not the

5 M. Lévy et al. / Ocean Modelling 48 (2012) 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 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 the extremum 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 the extremum values. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) details of the thin filaments. At 1, the meandering of the main jet is 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 closely associated with thin and sharp SST fronts (Fig. 5). The power spectra of the degraded fields provide a more quantitative measure of the similarities between the degraded and par-

6 6 M. Lévy et al. / Ocean Modelling 48 (2012) 1 9 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). ent fields (Fig. 6b, d and f). This computation confirms that the shallow part of the spectrum in the R54 simulation is captured with model outputs degraded on the R27 and R9 grids. There is a moderate decrease of the variance at the smallest scales when degrading R54 on the R9 grid, but this decrease is small when compared to the loss of variance due to decreasing model resolution from R54 to R9 (Fig. 6a, c and e). This analysis suggests that the effective resolution of the R54 experiment is of the order of km, i.e times the resolution of the actual computational grid. It also suggests that the use of R54 model outputs degraded on the R9 grid is sufficient to analyze processes at the smallest resolved scales in R54, and thus that for most applications, data storage and data analysis on the R9 grid is sufficient for the R54 model Offline transport experiment In order to take advantage of the effective resolution being less than the computational resolution in R54, we have tested the possibility of advecting a passive tracer using the velocity field computed in R54 grid but degraded on the R9 grid (experiment R54onR9OFF, Table 2). This test is compared with a similar integration performed on the parent R54 grid (experiment R54OFF). In terms of computing time, R54OFF is 10 3 times more expensive than R54onR9OFF. The distribution of the passive tracer is initially homogeneous in the horizontal and mimics the distribution of nutrients in the vertical (0 at the surface increasing to 1 at 100 m). The 3D dynamics perturbs this initial distribution by means of upwelling and downwelling, associated with lateral stirring. The results are examined after 10 days, which is a time scale long enough for a plankton population to react to nutrient supplies. In R54OFF, the perturbed tracer field at 30 m displays strong variations associated with the submesoscale filaments and the submesoscale upwelling/downwelling hot spots (Fig. 7a). The results in R54onR9OFF are very similar (Fig. 7b), with only weak differences for the highest/smallest tracer values (Fig. 7c). Close inspection reveals small grid-scale noise due to the choice of the same low horizontal diffusion value as for R54OFF. The mean effect of the 3D dynamics, when averaged horizontally over the interior of the domain, is a flux of tracer from the subsurface to the surface. This flux is seen by the removal of tracer from the layer m and a supply of tracer to the upper 20 m (Fig. 7d), and has similar strength in R54OFF and R54onR9OFF. In comparison, advection with a different 3D dynamics, issued from the R9 experiment (R9OFF), and thus with weaker vertical velocities, leads to a noticeably reduced vertical tracer flux (Fig. 7d). This confirms that the offline transport of this passive tracer by the mesoscale turbulence is not well represented when velocity fields computed at mesoscale resolution (i.e. resolution of R9) are used. On the other hand, with velocity fields computed at submesoscale resolution (i.e. resolution of R54), the offline experiment on the mesoscale (R9) grid presents a degree of similarity with the same experiment at submesoscale (R54) resolution which is extremely satisfactory for oceanic applications, and which enables an enormous gain in computing time. 4. Discussion Recent studies recognize that most of the oceanic kinetic energy is captured by mesoscale eddies (see Ferrari and Wunsch, 2009). However they point out that submesoscales, although less energetic, are associated with significant vertical velocities that may significantly impact the mesoscale eddy field (Lapeyre and Klein, 2006; Thomas et al., 2008). The consequence is that an accurate representation of such turbulent dynamical field requires taking into account the whole scale range including the submesoscales (Klein et al., 2008; Capet et al., 2008; Lévy et al., 2010). In our experiments, the impact of submesoscales on mesoscale eddies is emphasized by the decrease of the velocity and vorticity spectral intensity with model resolution, not only at the smallest scales but also at scales resolved by coarser resolution models: for instance at wavenumber 100 (corresponding to a wavelength of 20 km, i.e. about twice the grid resolution at 1/9 ), the vorticity spectral intensity is noticeably reduced in R9 compared to R54

7 M. Lévy et al. / Ocean Modelling 48 (2012) 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) Power spectra of surface relative vorticity, vertical velocity at 40 m and SST simulated at 1/54 resolution, and degraded on lower resolution grids. (Fig. 6a). A similar reduction is seen in the temperature spectral intensity, but with less amplitude (Fig. 6e). This illustrates the important role of submesoscales on mesoscale eddies even if these small scales are much less energetic. To a certain extent, the underestimation of the dynamics in the mesoscale range (i.e. above 20 km) at mesoscale resolution is due to the inhibition of the inverse cascade of kinetic energy from submesoscales to mesoscales because these submesoscales are not resolved. Model integration requires a certain level of dissipation and diffusion (present for both physical and numerical reasons) that cannot be confined to the very finest scales. The consequence is that the effective resolution of the model is reduced; in practice, processes with wavelength N Dx are strongly dissipated and therefore cannot be fully resolved. Within our model setup, we evaluate that N J 5 10, and certainly this value depends on the advective and diffusive algorithms used. The 3D advection of a passive tracer is significantly affected by the submesocale horizontal and vertical velocities. We see three reasons for this. First, the vertical velocities are particularly strong in the submesocale range. Second, horizontal turbulent flows stir tracers into smaller and smaller scales (Abraham, 1998). And third, the resulting tracer field strongly depends on the scales of the vertical velocity field and on its phasing with the horizontal stirring field (Klein et al., 1998; Martin et al., 2002; Smith and Ferrari, 2009). Our results demonstrate that, with a model resolution of Oð1Þ km, the resolved scale range of velocities that effectively contribute to the submesoscale 3D advection are scales larger than Oð10Þ km. In contrast, velocity scales smaller than Oð10Þ km fall in the dissipative range; they are much less energetic and their contribution to the 3D advection is negligible. In terms of offline model integrations, degrading velocity fields from a Oð1Þ km to a Oð10Þ km grid effectively comes down to neglecting the contribution of the less energetic small scales, while retaining their critical role on larger scales. Thus practically, the spectra shown in Fig. 6b, d and f can be used to determine the scale of the degraded grid: this scale roughly corresponds to the limit between the dissipative range and the energetic range, i.e. to the value of k where the slope breaks in

8 8 M. Lévy et al. / Ocean Modelling 48 (2012) 1 9 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 vertical gradient 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 of the 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). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) the submesoscale simulation R54. This value of k roughly corresponds to the Nyquist wavenumber of R9 (last k value of the R9 spectra, corresponding to the wavelength of two grid spacings in R9); Fig. 6 clearly shows that a coarser resolution (such as the resolution of R3) would not be suitable as it does not retain some of the energetic scales that are above the dissipative range (k ). The results examined in this paper suggest that an important gain can be made by averaging the outputs of a submesoscale resolving ðoð1þ kmþ ocean dynamical model on a coarser mesoscale grid (i.e. Oð10Þ km), without loss of information. The gain is of the order of 102 for data storage and 103 for off-line computation. This is because the smallest physical scales which are properly resolved with a resolution of Oð1Þ km are of Oð10Þ km; the scales between 1 and 10 km are less energetic and highly dissipated. Nevertheless, a dynamical model resolution of Oð1Þ km turns out to be essential to sustain an efficient inverse cascade and maintain an energetic mesoscale circulation. Regarding biogeochemical models, this suggests that coarser grids could be used for the integration of biogeochemical tracers when the integration of the dynamical 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 same topography before and after degradation, the degradation technique that was used can be easily extended to an irregular grid with more complex topography. In this case, the high-resolution parent model is naturally run with a high-resolution land-sea mask; the degraded offline model requires a coarse resolution land-sea mask which is constructed the following way: each time a coarse cell comprises at least one ocean point of the parent high-resolution grid, it is affected an ocean value; thus the total volume of the ocean is larger in the coarse grid than in the highresolution grid. Also, the velocity fields must be multiplied by the surface of the cell to which they are attached (water fluxes a and b in Fig. 2) and by the land-sea mask prior to averaging (see Aumont et al., 1998 for more details). The success of the proposed method relies on the fact that the dynamics is dominated by 3D advection and not by vertical mixing. The degradation of vertical mixing remains to be properly addressed. Acknowledgements The simulations were performed at IDRIS (France) from initial states computed on the Earth Simulator (ESC, Yokohama, Japan). The post-doc of L.R. was supported by the EUR-OCEANS submesoscale flagship. This project was supported by CNRS (INSU-LEFE TANGGO project), CNES (CPUMP project) and FP7 European project EMBRACE No

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