3.1 Definition of the coarsened grid

The first step consists of defining the coordinates, the cell dimensions and the land–sea mask of the coarsened grid based on those of the finer grid. The finer grid used in the ocean dynamics and thermodynamics component is called HR hereafter and the coarser grid in the passive tracer transport component is called HRCRS hereafter. The horizontal coarsening factor used here is 3, a choice that will be discussed thereafter (Sect. 6).

As the equations are solved on an Arakawa C-type grid (Fig. 1), temperature, salinity, pressure and other scalar variables are computed at the cell's center T, while the velocities are computed at the center of each face of the cell (U and V), normal to the corresponding velocity component.

3.1.1 Coordinates

In order to build an HRCRS cell, we take a 3×3 square of HR cells (Fig. 1) and collocate the HRCRS T, U and V grid points at the same location as the corresponding HR grid points. As a result, the definition of HRCRS grid coordinates can be seen as a subsampling of HR grid coordinates.

Figure 1Schematic diagram illustrating the definition of the coarsened grid for a coarsening factor of 3 (ocean high-resolution grid is in panel a; coarsened grid for biogeochemical model is in panel b) and localization of tracer (T, blue dots), zonal velocity (U, green triangles) and meridional velocity (V, red triangles) points in both grids. Grid sections that are common to both grids are coloured in green.

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3.1.2 Horizontal dimensions

Defining the horizontal dimensions of the coarsened grid cells is necessary to compute gradient or divergence operators.

For a coarsening factor of 3, HRCRS zonal cell size is computed by summing three consecutive HR zonal cell sizes along the zonal direction, and HRCRS meridional cell size is computed by summing three consecutive HR meridional cell sizes along the meridional direction (Fig. 2).

Figure 2Horizontal dimensions of high-resolution ocean HR grid cells (blue lines, red arrows, with names following NEMO nomenclature) and the corresponding coarsened biogeochemical HRCRS grid cell (green lines).

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3.1.3 Land–sea mask

Defining the land–sea mask for HRCRS is the most strenuous task of the multi-grid algorithm, which requires ad hoc verification as it can lead to substantial errors in the final outputs. If there is at least one ocean T point in the HR 3×3 pad, then the corresponding HRCRS T point will be considered as ocean, which can be defined automatically (Fig. 3, top panel). At the interface between two coarsened grid cells, we carefully retain the shape of the HR mask by adjusting the mask at U or V points: when there is at least one ocean point in the HR interface (as in Fig. 3 middle panel), then the interface of the HRCRS corresponding cells is defined as in the ocean. On the contrary, even if the two HRCRS T points are in the ocean, the interface between them may be defined as on the land, in the event that the corresponding HR points along the interface are all on land (Fig. 3, bottom panel).

Figure 3Schematic diagram illustrating the definition of land–sea mask from high-resolution ocean grid cells (a, blue lines with land cells filled in black) to the corresponding coarsened biogeochemical grid cells (b, green lines, with numbers identifying whether the corresponding point, here T or U, is defined as land (0) or ocean (1)).

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3.1.4 Vertical dimensions

The vertical dimension of cells at T points (usually named e3t in NEMO) are used in vertical divergence and Laplacian operators throughout the biogeochemical code. For the divergence operator, it is important that the coarsening procedure preserves the HR grid volume. Hence, the vertical dimension of HRCRS grid cells e3t_HRCRS is defined as the sum of the volume of the corresponding HR grid cells divided by the HRCRS grid cell horizontal surface area:

$$\begin{array}{}\text{(4)}& \mathrm{e}\mathrm{3}{\mathrm{t}}_{\mathrm{HRCRS}}={\displaystyle \frac{\sum _{i,j\in \mathrm{HR}}\mathrm{e}\mathrm{1}\mathrm{t}\cdot \mathrm{e}\mathrm{2}\mathrm{t}\cdot \mathrm{e}\mathrm{3}\mathrm{t}}{\sum _{i,j\in \mathrm{HR}}\mathrm{e}\mathrm{1}\mathrm{t}\cdot \mathrm{e}\mathrm{2}\mathrm{t}}},\end{array}$$

as illustrated in Fig. 4c. Note that we assume that HR grid points which are land masked have their vertical dimension equal to 0.

For the vertical gradient operator, the coarsening procedure must preserve the HR grid thickness. Therefore, we define another vertical dimension for HRCRS grid cells, e3tmax, as the maximum of the vertical dimensions of the corresponding HR grid cells:

$$\begin{array}{}\text{(5)}& \mathrm{e}\mathrm{3}{\mathrm{tmax}}_{\mathrm{HRCRS}}=\underset{i,j\in \mathrm{HR}}{max}\mathrm{e}\mathrm{3}\mathrm{t},\end{array}$$

as illustrated in Fig. 4d.

Figure 4Schematic diagram illustrating the coarsening procedure for the vertical dimension (e3t) of grid cells from high-resolution ocean grid cells (blue lines, a, b) to the corresponding coarsened biogeochemical grid cells (green lines, c, d).

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3.2 Definition of coarsening operators

Once the coarser grid variables have been defined, the second step of the multi-grid algorithm consists of defining the operators used to coarsen the dynamic fields necessary to solve the passive transport equation.

3.3 Practical implementation in the NEMO ocean model

By default (i.e. without using the multi-grid algorithm), the model passive transport component uses the grid and dynamic variables computed by the dynamic component of the code (i.e. on the HR grid). It is necessary to implement the multi-grid algorithm for passive tracer transport in the NEMO ocean general circulation model (OGCM) without cancelling the default option. We describe here how the model passive transport component can switch to the HRCRS grid and dynamic variables when the multi-grid algorithm is activated.

First we present how we manage the domain loop indices. The loops in the ocean dynamic and passive transport components use the domain sizes in the three spatial directions (called $(jpi,jpj,jpk)$ in NEMO). When the multi-grid algorithm is activated, HR grid sizes $(jpi,jpj,jpk)$ are saved in other variables $(jp{i}_{\mathrm{HR}},jp{j}_{\mathrm{HR}},jp{k}_{\mathrm{HR}})$ and HRCRS coarsened grid sizes are defined as $(jp{i}_{\mathrm{HRCRS}},jp{j}_{\mathrm{HRCRS}},jp{k}_{\mathrm{HRCRS}})$. When the code is in the ocean dynamic component, the loop sizes $(jpi,jpj,jpk)$ are set to HR grid sizes $(jp{i}_{\mathrm{HR}},jp{j}_{\mathrm{HR}},jp{k}_{\mathrm{HR}})$. When the code enters the passive transport component, the loop sizes $(jpi,jpj,jpk)$ switch to HRCRS grid sizes $(jp{i}_{\mathrm{HRCRS}},jp{j}_{\mathrm{HRCRS}},jp{k}_{\mathrm{HRCRS}})$; when the code goes out of the passive transport component, the loops sizes $(jpi,jpj,jpk)$ switch back to HR grid sizes:

Secondly, we present how we manage the grid and dynamic variables used in the passive transport component. They are made available through pointers defined in an interface between the dynamical and passive transport components (called oce_trc in NEMO OGCM):

To avoid the duplication of coarsened and non coarsened grid and dynamic variables in the passive transport routine, the pointers target the coarsened grid and dynamic variables when the multi-grid algorithm is activated and the non coarsened variables when it is not activated. The choice between the two definitions is done at the pre-compilation stage of the code with a cpp key key_crs:

4.1 Model configurations

The $\mathrm{1}/\mathrm{4}{}^{\circ }$ configuration is described in Barnier et al. (2006). The $\mathrm{3}/\mathrm{4}{}^{\circ }$ configuration coordinates and bathymetry are built by applying $\mathrm{1}/\mathrm{4}{}^{\circ }$ configuration coordinates and bathymetry on the coarsening operators described above. This operation has created some fake straits in Panama and the Indonesian Throughflow, which have been manually filled. The $\mathrm{3}/\mathrm{4}{}^{\circ }$ configuration is close to the global ocean ORCA 1^∘ configuration used in various Coupled Model Intercomparison Project phase 5 (CMIP5) (Voldoire et al., 2013) and CMIP6 models (Voldoire et al., 2019).

The $\mathrm{1}/\mathrm{4}{}^{\circ }$ and $\mathrm{3}/\mathrm{4}{}^{\circ }$ resolution configurations are based on a quasi-isotropic ORCA grid (Madec and Imbard, 1996) at nominal $\mathrm{1}/\mathrm{4}{}^{\circ }$ and $\mathrm{3}/\mathrm{4}{}^{\circ }$ horizontal resolutions, respectively, at the Equator. Vertical discretization uses 75 z levels with partial cell parameterization (Barnier et al., 2006), which leads to a resolution ranging from 1 m at the surface to 450 m at depth.

The bathymetry is based on a combination of ETOPO1 (Amante and Eakins, 2009) and GEBCO08 (Becker et al., 2009). The initial state for the ocean is at rest, with temperature and salinity from WOA13 (World Ocean Atlas; https://www.nodc.noaa.gov/, last access: 10 June 2015). At the surface, the model is forced by ERA-Interim atmospheric reanalysis (Dee et al., 2011) using the CORE bulk formulae Large and Yeager (2009) to compute the turbulent air–sea fluxes. A monthly runoff climatology is built and applied to all configurations, using data for coastal runoffs and major rivers from Dai and Trenberth (2002).

A split-explicit time-splitting scheme is employed to compute the surface pressure gradient with a non-linear free surface (Levier et al., 2007). These parameterizations were already used in the Ireland–Biscay–Iberia regional configuration, described in Maraldi et al. (2013).

The momentum advection term is computed with the energy and enstrophy conserving scheme proposed by Arakawa and Lamb (1981). The advection of tracers (temperature and salinity) is computed with a total variance diminishing (TVD) advection scheme (Lévy et al., 2001; Cravatte et al., 2007).

Lateral eddy viscosity is computed with a biharmonic operator and lateral eddy diffusivity is computed with an isopycnal Laplacian operator for both active and passive tracers. As explained in Madec et al. (2017) (Sect. 9.1), lateral eddy diffusivity coefficient A^l decreases proportionally to the grid size, while the lateral eddy viscosity coefficient B^l decreases poleward as the cube of the grid cell size:

where $A_o^{\mathrm{l}}$ and $B_o^{\mathrm{l}}$ are the respective values at the Equator (see Table 1), while e_max is the maximum of e1 and e2 taken over the whole masked ocean domain located at the Equator for global configurations. The lateral eddy diffusivity coefficients shown here are used for active tracers when the multi-grid algorithm is activated or not, and used for passive tracers when the multi-grid algorithm is not activated; the choice of lateral eddy diffusivity for passive tracers when the multi-grid algorithm is activated is discussed later (Sect. 5.2).

Table 1Horizontal eddy viscosity and diffusivity values at the Equator.

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The vertical eddy viscosity and diffusivity coefficients are computed from a turbulent kinetic energy (TKE) turbulent closure model (Blanke and Delecluse, 1993), with parameters as described in Reffray et al. (2015).

4.2 Numerical experiments

To assess the multi-grid algorithm for passive tracers, three types of experiments have been set up. They differ on the passive tracer initial state or evolution. In all cases, the dynamical component is the same. The comparisons of the simulated tracer distributions are performed after 365 d of integration. At this stage, tracer fields have been transported and stirred during several mesoscale eddy turnover timescales, although tracer distributions are still far from statistical equilibrium.

In the PATCH experiments, the multi-grid algorithm is assessed on the horizontal component of the transport (Eq. 1). Here, the passive tracer vertical diffusivity component is set to 0 $(A^{\mathrm{v}}=\mathrm{0}.)$. The passive tracer C is initialized with a patch of 10^∘ radius centred on 36^∘ N, 60^∘ W (see Fig. 6a). The value of the tracer ranges from 2 at its center to 1.368 at its boundaries and is vertically uniform. Outside of the patch, the tracer initial value is set to 1. The tracer has no sources nor sinks. Hence, the passive tracer evolution in the PATCH experiment can be described by the following equations:

where R is a tensor of isoneutral slopes.

The AGE-ZDIF experiments (AGE for age tracer; ZDIF for vertical diffusion) are conceived to assess the multi-grid algorithm on the vertical eddy diffusive component of the passive tracer transport (Eq. 1). Hence, the advection and lateral diffusion are unplugged for the passive tracer. This experiment makes use of the so-called age tracer of the NEMO OGCM: its initial value is set to zero in all the ocean and its value is damped to 0 at the ocean surface, while there is a net source in the ocean interior. This tracer gives a good representation of the time evolution of a water mass after it was in contact with the ocean surface; hence, it gives some information on the ventilation of water masses. The tracer evolution in the AGE-ZDIF experiment can be described by the following equations:

Finally, the multi-grid algorithm is assessed on the full tracer transport (Eq. 1) in the AGE-FULL experiment (FULL for full equation), which also uses the age tracer but retains all terms for the tracer transport equation. Hence, the tracer evolution in the AGE-FULL experiment can be described by the following equations:

5.1 Coarsened velocities

Advection by ocean currents is one of the main drivers of the evolution of passive tracers. As a consequence, it is important that tracer transport by ocean currents on the coarsened grid HRCRS remains close to that on the HR grid. This implies that coarsened velocities share similarities with HR velocities on the spatial scales that are common. To test this, coarsened velocities are compared to HR velocities and LR velocities (Fig. 5), as done by Lévy et al. (2012) in a more idealized context. Power spectra are computed along the horizontal coordinates of the model grids. For each spectrum, the minimum wavelength corresponds to twice model grid spacing: 2⋅Δx. HRCRS velocities are the result of the coarsening of HR velocities on LR grid. As HR grid is finer than LR grid, HR velocities contains finer scales than HRCRS and LR.

In terms of kinetic energy, vertical velocities and relative vorticity, HR, HRCRS and LR have almost the same level of energy at spatial scales larger than 250 km. At smaller spatial scales, HRCRS has a level of energy comparable to HR in terms of kinetic energy and relative vorticity, while vertical velocities are less energetic than in HR. For the three quantities, at spatial scales smaller than 250 km, the level of energy of LR is significantly below HR and HRCRS levels. This suggests that ocean currents in HRCRS are, overall, more similar to those in HR than to those in LR.

Figure 5North Atlantic yearly mean surface current speed (a) and power spectrum density of (b) surface kinetic energy, (c) vertical velocities and (d) surface vorticity as functions of spatial wave number, in the HR (green lines), LR (blue lines) and HRCRS configurations (red lines). Black lines in spectra represent usual slopes to be compared with numerical spectra. All computations use daily model outputs and are averaged yearly.

5.2 Horizontal tracer transport

The evaluation of the multi-grid algorithm to resolve an advection–diffusion problem is shown here with the results of the PATCH experiment described in Sect. 4.2.

The first step is to evaluate the best value for the passive tracer horizontal diffusion coefficient in the HRCRS configuration. A comparison between HR and various HRCRS configurations differing for their horizontal diffusion coefficient (set to 300, 600, 900 and 1200 m² s⁻¹) is presented in Sect. A. This sensitivity test suggests that 900 m² s⁻¹ yields the best results for HRCRS. Using 300 m² s⁻¹ in HR and 900 m² s⁻¹ in HRCRS for the passive tracer lateral diffusion respects the ratio in spatial resolution between the two configurations (of respectively $\mathrm{1}/\mathrm{4}{}^{\circ }$ and $\mathrm{3}/\mathrm{4}{}^{\circ }$ nominal resolution), which is consistent with the linear relationship between the horizontal diffusion coefficient A^l and the cell size (see Eq. 9).

Using this value for the lateral diffusion coefficient A^l in the HRCRS configuration, and comparing outputs to those from HR and LR configurations, suggests that the HRCRS patch is closer to the HR shape than to the LR one (Fig. 6). Indeed, LR has a larger area where its concentration is higher than in HR, with a difference greater than 0.3 in its center, while the difference between HRCRS and HR does not exceed 0.2 (within the red box, after 1 year of simulation; the root mean square error (RMSE) is 0.04 between HRCRS and HR and 0.09 between LR and HR).

Figure 6Passive tracer initial condition for PATCH experiment (a), tracer concentration at 16 m depth after 1 year for the HR configuration (b) and difference with LR (c) and HRCRS configurations (d), respectively.

Horizontal velocities influence the interior EKE distribution and so the tracer distribution. In Fig. 7, we represent for a meridional section in the Gulf Stream the daily vertical distribution of zonal velocities and PATCH tracer after 1 year of simulation, for the HR, HRCRS and LR configurations. The HRCRS velocities are obtained by coarsening the HR velocities. HR and HRCRS zonal velocities are stronger than LR velocities and present more fine structures, in particular in the northern region (above 40^∘ N). HRCRS and HR tracer distribution are similar and present some differences to LR configuration. In the southern part of the 55^∘ W sections, LR configurations has a maximum of concentration, whereas HRCRS and HR have lower values. In the northern parts of the section, HR and HRCRS have some higher values of concentration in the interior and deeper ocean than in LR configuration, around 40^∘ N latitude. Despite the degraded spatial resolution operated on the velocities, the dynamics used to transport HRCRS tracer present the same pattern as the HR dynamics. The tracer using the multi-grid algorithm reproduces the HR tracer; while the dynamics is coarsened, the HRCRS tracer benefits from the higher-resolution dynamics and present a gain compared to a lower-resolution experiment.

Figure 7Vertical distribution of the zonal velocities (m s⁻¹) (a, b, c) and PATCH tracer (d, e, f) for HR (a, d), HRCRS (b, e) and LR (c, f) configurations along the 55^∘ W section in the Gulf Stream.

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5.3 Vertical diffusion

5.3.1 Choice of vertical diffusion on the coarsened grid

Vertical mixing is responsible for the ventilation of a passive tracers and thus their vertical distribution. The vertical diffusion coefficient A^v is computed by the dynamic component (so on the HR grid) but it might be coarsened on the HRCRS grid.

Here, we present the results of the AGE-ZDIF experiment in order to assess the capacity of the multi-grid algorithm to simulate the passive tracer vertical mixing. In particular, we show the sensitivity of the age tracer vertical representation to the choice of coarsening operator of the vertical mixing parameter A^v presented in Sect. 3.2.4.

Figure 8a–b present the daily vertical profiles of root mean square (rms) difference to the HR age tracer after 1 year of simulation. All the configurations have the same shape of vertical distribution, except LR and HRCRS with the max operator which have a behaviour very different to the others at the bottom of the ocean. Figure 8b presents a zoom in the first 500 m. All the configurations have some differences in the mixed layer and some HRCRS configurations are closer to HR, according to the coarsening operator used for the vertical mixing: the HRCRS with the meanlog and median operators give the best performance compared to the HR experiment, whereas the HRCRS experiments with the two extreme operators give results less comparable to LR.

Table 2 gives the RMSE with the HR solution for all HRCRS experiments and LR. The HRCRS with meanlog and median operators gives the best results. HRCRS with min and HRCRS with mean have a better skill than LR, and HRCRS with max is far from the other simulations.

Table 2RMSE with the HR solution after 1 year of simulation.

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To understand which process leads to these solutions, we present in Fig. 8c–e the divergence of the age tracer vertical diffusive fluxes averaged over the global ocean and the first year of the simulation. A positive divergence means that the net incoming fluxes are positives. Looking at the flux to the bottom (Fig. 8e) shows that the divergence is negative for HRCRS with the max operator; it is coherent with the fact that age tracer concentration is lower than the other (see Fig. 8a). It is explained by the fact that the max operator extends the zone where the A^v has a high value along the bathymetry. HRCRS with the min operator has a negative flux divergence at the bottom, but its age value is closer to HR.

Figure 8(a, b) The rms differences to HR age tracer for HRCRS and LR runs for the whole water column (a) and for the first 500 m (b). The age tracer values are instantaneous outputs after 1 year. (c–e) Horizontal mean of divergence of age tracer vertical diffusive flux mean over the first year for the whole water column (c), the top ocean (d) and the bottom ocean (e).

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5.3.2 Special case for convection

Here, we assess the multi-grid algorithm at simulating the age tracer penetration for deep convection as in the Ross and Weddell seas. We want to know if the operators selected in the previous section (meanlog and median) are appropriate for this particular situation.

Figure 9 represents the depth where the age tracer reaches the value of 10 d (Fig. 9a–b) and 100 d, along a section in the Southern Ocean, at the latitude of 58.3^∘ S. In Fig. 9a and c, we compare the HR profile (black), the LR experiment (grey) and the ensemble spread of the different solutions of HRCRS (light blue). The HR profile is included in the HRCRS ensemble, whereas the LR experiment is outside of the HRCRS spread. All the HRCRS solutions have better performance than the LR solution compared to HR. However, the spread is larger in two areas: between 140 and 80^∘ W and between 100 and 160^∘ E.

Figure 9b and d show the differences between each HRCRS solutions and HR. We can see that, outside of the two areas of deep convection, the HRCRS solutions performed with the meanlog and median operators are the closest to the HR solution, especially in the deeper case (Fig. 9d). Around 100^∘ E and 150^∘ W, the HRCRS solution with the min operator is the closest to HR solution.

In order to combine the better performance of the min operator in the convection situation and the meanlog operator in the global ocean, we add here a new operator based on the meanlog operator with a switch to the min operator in presence of convection. The HRCRS run with the meanlog operator and a switch to the min operator gives the best comparison to the HR run in the Weddell and Ross Sea areas and has good performance outside.

Figure 9Antarctic Circumpolar Current (ACC) section ($-\mathrm{58.3}{}^{\circ }$ S) of depth where age tracer value is equal to a given age: 10 d (a, b) and 100 d (c, d). Panels (a) and (c) compare HR, LR and the mean of HRCRS runs. Panels (b) and (d) show the difference between HR and all the HRCRS runs.

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Figure 10Depth where age tracer is equal to 100 d after 1 year of simulation: HR experiment (a), LR minus HR (b) and HRCRS minus HR (c).

5.4 Evaluation of the full algorithm

We have validated the multi-grid algorithm to represent the tracer advection and lateral diffusion in Sect. 5.2 and the vertical eddy diffusion in Sect. 5.3. The choice of the operator to coarsen the vertical diffusion coefficient A^v has also been discussed in Sect. 5.3. Now we present some results for the full transport (Eq. 1). We decide to continue the assessment using the meanlog operator to coarsen the vertical diffusion coefficient.

A comparison of the depth for HRCRS and LR runs to HR run is presented in Fig. 10. Figure 10a presents the HR depths where age is equal to 100 d. Figure 10b presents the depth differences between LR and HR, and Fig. 10c presents the depth differences between HRCRS and HR. Higher depths values are due to deeper age tracer penetration, caused by higher vertical mixing. Globally, there are no big differences between the three runs, except in the area where mesoscale activity is important: the Gulf Stream, the North Atlantic subpolar gyre, the Kuroshio Current and the Southern Ocean. The differences between HRCRS and HR runs are largely weaker than the differences between LR and HR runs.

In order to compare on depth the three different runs, we present in Fig. 11 the age tracer penetration along three sections. The first section starts in the Antarctic Circumpolar Current (ACC) and goes northward until the North Atlantic. For a value of 10 d, the penetration for the three experiments are close. For a value of 100 or 300 d, HRCRS is very close to HR run and LR is different from the two other experiments. The second section goes along the equatorial Pacific. For the three values of the age tracer, the corresponding depths from HR and HRCRS are very close and the depths from the LR experiment are different. The last section is inside the ACC. HRCRS and HR depths are similar for 10, 100 and 300 d. LR run depths are similar to the two other runs for 10 d, and the differences with the two other runs increase with depth.

Figure 11Atlantic (a), Pacific (b) and ACC (c) sections of age tracers after 1 year of simulation for HR (black), LR (grey) and HRCRS (blue) configurations. The age value is 10 d (dotted line), 100 d (solid line) and 300 d (dashed line).

The age tracer RMSE with the HR solution has a value of 0.91 for HRCRS and 2.30 for LR after 1 year of simulation (Table 3).

Table 3RMSE with the HR solution after 1 year of simulation.

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5.5 Computational performance

Because of the extra computational cost of the coarsening of dynamical fields, the multi-grid algorithm procedure gives better performance only with large numbers of passive tracer fields. Indeed, the expected speed-up for computing tracer advection on a 3×3 coarser-resolution grid is a priori a factor of 9. However, in practice, the time spent on defining the grid and computing dynamical fields on the coarsened grid should also be accounted for. We present in Table 4 the elapsed time spent by the HR run (without grid coarsening capacity) and HRCRS run (with grid coarsening capacity). The elapsed time spent defining the scale factors for the coarsened grid and computing the dynamic fields on the coarsened grid is actually significant. This is why, for a single tracer field, the HRCRS run (with grid coarsening capacity) uses more CPU resources than the HR run. But it should be noted that the definition of the coarsened grid and the coarsening of dynamical fields are only done once for all the tracer fields. Therefore, the net overhead cost of this extra operation is relatively small in situations with many tracer fields (as shown in the last row of Table 4).

Table 4Elapsed time (in seconds) for a 1-month run without/with the multi-grid algorithm. The values in the last row have been extrapolated.

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Overall, we estimate that using the multi-grid algorithm allows us to reduce the cost of running a full ocean model with an interactive biogeochemical component by a factor of ∼3. Table 5 provides an estimate of the breakdown of elapsed time in a global NEMO configuration coupled to PISCES biogeochemical model (Aumont et al., 2015), which simulates 24 passive tracers. Typically, running NEMO biogeochemical transport module (TOP) for 24 tracers on the same grid as the dynamics requires twice as much resources as the ocean–sea-ice component. PISCES biogeochemical component itself uses resources comparable to those used for the ocean–sea-ice component. Running a full model with a biogeochemical component based on PISCES typically requires 4 times the resources of the ocean–sea-ice component alone. Given the figures of Table 4, we estimate that the net elapsed time overhead of the biogeochemical component (transport + PISCES) would be reduced by a factor of ∼7 with the multi-grid algorithm. Therefore, a full model using our multi-grid algorithm would use 1.45 times more resources than the ocean–sea-ice component alone, therefore reducing the cost of the overall system by a factor of 3.

Table 5Typical expected speedup in elapsed time between a run without versus with multi-grid algorithm. Values have been extrapolated from the measured cost of the coarsening operation and an estimate of the net overhead of the PISCES biogeochemical (BGC) model.

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