Assuming incompressibility of the total three-dimensional flow implies: 1∇⋅u=0, where the velocity vector is written as u=(u,v,w)=(uh,w) where uh denotes the horizontal velocity. The total velocity can be written as the sum of the geostrophic (ug) and Ekman components (uEk): 2∇⋅ug+∇⋅uEk=0.

Because the Ekman circulation vanishes below the base of the Ekman layer, the vertical integration of Eq. () over the water column can be separated into the Ekman layer, extending from the surface (z=0) to its base (z=DEk), and into the ocean interior, extending from the Ekman layer base to a given depth z where we neglect ageostrophic contributions: 3∫Dek0∇⋅ugdz′+∫Dek0∇⋅uEkdz′︸Ekman Layer+∫zDek∇⋅ugdz′︸Ocean Interior=0.

The horizontal geostrophic current is calculated from the pressure gradient as 4ug=-1ρ0f∂p∂y;vg=1ρ0f∂p∂x.

Therefore, in a β-plane, the divergence of the horizontal geostrophic flow can be expressed as 5∇huh,g=-βvgf assuming f≠0 (), where f is the Coriolis parameter and β is the meridional gradient of f, without requiring the classical assumption of a linear variation of f across the domain.

The contribution of the Ekman layer in Eq. () can be expanded as ∫Dek0-βvgfdz′+∫Dek0∂wg∂zdz′+∫Dek0∇⋅uh,Ekdz′6+∫Dek0∂wEk∂zdz′.

Evaluating analytically the vertical integrals of the vertical velocity gradients gives ∫Dek0-βvgfdz′+wg(z=0)-wg(z=DEk)7+∫Dek0∇⋅uh,Ekdz′+wEk(z=0)-wEk(z=DEk).

We assume that wEk(z=DEk)=0, as all components of the three-dimensional Ekman flow are dissipated at DEk. Furthermore, the Ekman theory relates the Ekman transport, 8Uh,Ek=∫DEk0uh,Ekdz′ to the wind stress through 9∇⋅Uh,Ek=∇×τρ0f, where Uh,Ek denotes the horizontal Ekman transport in the Ekman layer, ρ0 represents the water density at the surface, and τ is the wind stress. Expanding the divergence of the Ekman transport generates ∇⋅Uh,Ek=∇⋅∫Dek0uh,Ekdz′10=∫Dek0∇⋅uh,Ekdz′-uh,Ek(z=DEk)∇⋅DEk. Consequently, ∫Dek0∇⋅uh,Ekdz′=∇×τρ0f11+uh,Ek(z=DEk)∇⋅DEk. The second term in the right-hand side corresponds to the induction term associated with variations in the Ekman layer depth. This contribution is neglected here, as Ekman velocity is assumed to vanish at DEk as mentioned above.

At the sea surface, we impose zero total vertical flow (wtot), 12wtot(z=0)=0.

This condition is exact for the stationary flow of the time-mean state, but it is only an approximation for time-varying flows. In the latter case, this approximation neglects the time derivative of sea surface height, d(η)/dt, where η represents the sea surface height relative to the mean state reference surface level. The total vertical velocity is decomposed into geostrophic and ageostrophic components (wtot=wg+wag), where the ageostrophic contribution is assumed, at first order, to be produced by surface wind stress and friction (three-dimensional Ekman flow). Under these assumptions, the surface condition implies: 13wg(z=0)=-wEk(z=0).

Substituting the Ekman transport (Eq. ) into Eq. () and imposing the surface condition of zero total vertical velocity, the contribution of the Ekman layer in Eq. () yields 14∫Dek0-βvgfdz′-wg(z=DEk)+∇×τρ0f.

In the ocean interior, only the geostrophic currents are not negligible, such that the contribution of the ocean interior to the Eq. () can be derived 15∫zDek-βvgfdz′+∫zDek∂wg∂zdz′.

Evaluating the vertical integral of the contribution of the ocean interior to Eq. () leads to 16∫zDek-βvgfdz′+wg(z=DEk)-wg(z).

Combining the Ekman layer (Eq. ) and ocean interior (Eq. ) contributions, the terms corresponding to wg(DEk) cancel each other, and the expression for the geostrophic vertical velocity in the ocean interior is obtained 17wg(z)=∇×τρ0f-∫z0βvgfdz′.

In this scenario, the horizontal geostrophic flow is divergent and generates a net geostrophic vertical velocity in the ocean interior. Equation () corresponds to the indefinite vertical integral of the LVB applied to the geostrophic flow. While the LVB is an approximation of the vorticity equation valid for large-scale flows, it provides a suitable and well-known foundation for describing the ocean interior flow, as thoroughly described in CM24. As a result of the approximations considered to derive the continuity equation (Eq. ), discrepancies between the geostrophic vertical velocity and the total vertical velocity are expected, particularly in regions where ageostrophic and nonlinear processes are non-negligible in the ocean interior.

Equation () further shows that, although wg can only be explicitly diagnosed below the Ekman layer, due to the fact that we have defined z below DEk in Eq. (), the magnitude of the vertical flow at the base of the Ekman layer differs from its surface value due to the vertically integrated divergence of the horizontal geostrophic flow within the Ekman layer. At the sea surface or if the geostrophic flow divergence is negligible, the geostrophic vertical velocity is equal to the Ekman pumping. However, CM24 reported a non-negligible contribution of the β-plane geostrophic divergence for an OGCM within the Ekman layer. This effort is consistent with that of , which, albeit different, similarly emphasises the need to revisit the classical assumption of Ekman theory. Consequently, in this study, Ekman pumping (wEk) is adopted as surface boundary condition for the computation of geostrophic vertical velocities.

To estimate the global ocean vertical velocities from the divergence of the geostrophic flow, meridional geostrophic velocities from ARMOR3D are used in combination with Ekman pumping vertical velocities derived from ERA5 wind stress as boundary condition at the ocean surface. A summary of the main characteristics of the input datasets, including physical variables, cover period, temporal frequency, and horizontal and vertical resolutions, is provided in Table .

ARMOR3D dataset () integrates satellite-derived and in situ observations. Its derivation first involves the construction of synthetic temperature and salinity fields from altimetric surface level anomaly (SLA; ), and sea surface temperature and salinity (SST and SSS) () via linear regression method and covariances calculated from historical in situ observations (EN3 dataset () and Argo floats). The synthetic and observed temperature and salinity profiles are then merged using optimal interpolation (). Finally, geostrophic velocities are computed using the thermal wind equation, referenced to the surface geostrophic velocities estimated from the altimetric absolute dynamic topography. The mixed layer depth (MLD) is obtained from the minimum of temperature and density threshold equivalent to a 0.2 °C decrease. ARMOR3D is available at 10.48670/moi-00052.

Ekman pumping vertical velocities are computed from monthly wind stress data provided by ERA5 (), the fifth-generation reanalysis of the European Centre for Medium-Range Weather Forecasts (ECMWF). The ERA5 fields are provided at 0.25° horizontal resolution and can be downloaded from the Copernicus Climate Change Service (C3S) Climate Data Store at 10.24381/cds.f17050d7.

The resulting Observation-based LInear Vorticity Vertical Velocities (OLIV3) product consists of geostrophic vertical velocities derived from ARMOR3D meridional velocities and surface Ekman pumping from ERA5 wind stress, using Eq. () over the native ARMOR3D vertical levels and then interpolated over isopycnal levels defined by the neutral density of the ARMOR3D thermohaline field. The product spans the 1993–2019 period, with a horizontal resolution of 0.25° and 71 isopycnal levels. Figure  represents the OLIV3 time-mean (1993–2019) vertical velocity at various sigma levels across the tachocline, understood as the upper ocean layer defined by a strong vertical shear in the velocity field (CM24). The product is quality-flagged based on the time-mean relative error and interannual correlation coefficient between wg and wtot in the OGCM perfect model test and available at 10.5281/zenodo.16962780 . A low-resolution version, used in the intercomparison test of this study (Sect. 3.3), is available upon request.

To evaluate OLIV3 performance, we compare the vertical velocities with two reanalyses (GLORYS12v1, ECCOv4r4) and an observation-based product (OMEGA3D). To facilitate comparison, the main attributes of the validation product sources are summarised in Table .

Following CM24, the reference OGCM simulation for the assessment of validity of the methodology is the Nucleus for European Modelling of the Ocean (NEMO) OGCM OCCITENS run from the OCCIPUT project (). This simulation is forced by the DFS5.2 forcing set, using ERA-Interim and ERA40 reanalyses (). Neutral density field and the isopycnal surfaces were computed for this study using thermohaline and sea surface height fields applied to formulation. The MLD provided by the NEMO OCCITENS simulation is computed using a density criterion of 0.01 kgm-3 of density change from the surface following the procedure defined by . Outputs from the NEMO OGCM OCCITENS simulation are available upon request (thierry.penduff@cnrs.fr). Geostrophic velocities are derived from the model pressure field (calculated using sea surface height and the hydrostatic equation) using the geostrophic equation via the codes available at https://github.com/meom-group/CDFTOOLS (last access: April 2024).

The GLobal Ocean ReanalYsis and Simulations (GLORYS; ) assimilates via Kalman filter along-track altimeter SLA, satellite SST, sea ice concentration, and in situ temperature and salinity profiles, using NEMO as the model component . The MLD is defined following the same methodology as in the NEMO OGCM simulation. This dataset is hereafter referred to as GLORYS12v1 and accessed at https://tds.mercator-ocean.fr/thredds/glorys12v1/glorys12v1_pgn_monthlymeans.html (last access: April 2024).

The Estimating the Circulation and Climate of the Ocean (ECCO) in its fourth release, version 4 () employs a 4D-VAR assimilation scheme, integrating satellite altimetry, in situ temperature and salinity profiles from Argo, satellite sea surface salinity and temperature, and ocean bottom pressure, together with the MIT general circulation model (). The MLD is defined following the procedure developed by , which find that the optimal estimated of turbulent mixing penetration is obtained with a mixed layer depth definition of ΔT=0.8°C. This reanalysis, hereafter referred to as ECCOv4r4, is available at https://www.ecco-group.org/products-ECCO-V4r4.htm (last access: April 2024).

Finally, OMEGA3D is an observation-based global estimate of vertical velocities derived from the QG omega equation (). It is based on ARMOR3D thermohaline field and geostrophic velocities, and ERA-Interim () surface air-sea fluxes. OMEGA3D is available at 10.48670/moi-00053.

Validating OLIV3 with observational data is problematic due to the lack of a ground truth for large-scale vertical velocities. Here, the performance of OLIV3 relies on the consistency across existing w estimates: GLORYS12v1, ECCOv4r4 and OMEGA3D. The intercomparison is conducted on a common spatiotemporal resolution: annual means at 5° horizontal resolution and isopycnal levels. This choice reduces vertical grid and thermohaline structure differences. It also allows to focus on large-scale dynamics, which are better resolved by the LVB framework (CM24). The isopycnal levels are defined by the neutral density () of each dataset. For the OLIV3 and OMEGA3D datasets, the thermohaline field used to interpolate w onto isopycnal levels is ARMOR3D, since the velocity field was constructed using it (). Diagnostics are computed over the overlapping 23-year period (1993–2015) and include the time-mean horizontal pattern, the time-mean vertical gradient between isopycnals, the interannual variance and correlation coefficient (R). Regions within the equator band (5° S/N) are excluded as the geostrophic equation cannot be solved at these latitudes.

Twenty-seven-year mean geostrophic vertical velocities (wg) stemming from the OLIV3 product at σ26 isopycnal surface are presented in Fig. a. This isopycnal level was chosen to assess the vertical velocity estimates across most of the extension of the global subtropical gyres, while maintaining a focus on thermocline dynamics, where the LVB framework performs best (see CM24 for the North Atlantic Ocean). The geostrophic vertical velocity field (OLIV3) within the tachocline reproduces the well-known wind-driven circulation features represented by Ekman pumping (Fig. b), generally with upwelling at tropical latitudes and downwelling at the subtropics. This emphasises the role of wind-driven divergence as the primary driver of vertical flow within the upper ocean (e.g. ). The Pacific and Atlantic eastern tropical upwelling systems that continue along the eastern coast up to subtropical latitudes are associated with maximum positive values near the coast around 0.2 md-1 (). The anticyclonic circulation of subtropical systems is characterised by negative w (downwelling) with maximum values found along western boundaries. This pattern differs from the agreement among reanalyses about upwelling in most of the western boundary current systems, in particular the Gulf Stream and the Kuroshio Current (). In the Gulf Stream case, this poor agreement with reanalyses aligns with the lack of confidence in the LVB as an estimator of the vertical flow, as we already demonstrated in the Atlantic Ocean. Outside the Northern Hemisphere subtropical band, some upwelling occurs over the extension of the Gulf Stream and Kuroshio systems, which can be explained by Ekman pumping ().

To analyse the temporal and vertical variability of the vertical velocity estimates, Fig.  shows regionally averaged wg as a function of isopycnal level and year for three distinct regions in the North Atlantic Ocean: the eastern tropical, subtropical and subpolar gyres (purple regions in Fig. a). The bottom of the layer was selected as the level where the estimates' climatology changes sign. Figure  evidences that the direction of geostrophic vertical velocity estimates maintains their sign throughout most of the layer's thickness. Before reaching this depth, the annual velocities reduce in amplitude, supporting the baroclinic nature of vertical flow found in the North Atlantic tachocline (CM24). This behaviour is consistent with the requirement of a level of no motion at depth to satisfy the Sverdrup balance ().

The velocity sign remains unchanged over time in the three regions (Fig. ), but temporal variability is evident across them. Weaker upwelling events in the North Atlantic Tropical Gyre (e.g. 1995, 2001, 2005, and 2010 in the top panel of Fig. ) seem associated with the negative North Atlantic Oscillation (NAO) phases shown by and , while positive NAO phases correspond to stronger upwelling. This suggests that the variability of the eastern tropical gyre is modulated by the phase of the NAO. Interestingly, the magnitude of the oceanic response does not reflect the NAO intensity presented in the references above. Focusing on the subtropical gyre (central panel in Fig. ), maximum downwelling events do not correlate with the NAO index as clearly as the tropical upwelling does. The atmospheric changes modify both the intensity of the downwelling and the location of the subduction maximum (). Therefore, the out-of-phase relationship between subtropical vertical flow variability and the NAO index may suggest that this region does not fully reproduce the gyre interannual variability. However, other components can influence the variability of the region as the NAO is not the sole contributor to atmospheric forcing variability (). In the subpolar gyre (bottom panel in Fig. ), oscillations in upwelling amplitude exhibit a periodicity of approximately 5 years, with particularly strong positive velocity events during 2002–2003, 2007, and 2009. These features align with observed changes in the subpolar gyre from satellite altimetry () and volume transport estimates of the East Greenland Current (). These findings suggest that OLIV3, while capturing the baroclinic nature of the vertical velocity field in the tachocline, is also capable of transmitting some of the surface interannual signal into the ocean interior.

The vertical flow computed by applying Eq. () to the ARMOR3D dataset represents the geostrophic component only. Therefore, before comparing OLIV3 with estimates of the total vertical flow, the limitations of this formulation are evaluated using an OGCM simulation, considered as a “perfect model test”. In this approach, wg is computed from the model output meridional geostrophic velocity (vg) and wind stress, following Eq. (). This global-scale analysis extends the regional assessment conducted by CM24 for the North Atlantic Ocean. To evaluate the ability of the geostrophic component to represent the spatiotemporal variability of the total vertical flow, we examine three diagnostics: the absolute relative error between the time-mean geostrophic vertical velocity (wg) and the total vertical velocity (wtot) at σ26, their interannual correlation coefficient at σ26, as well as the relative error in the time-mean vertical gradient between the bottom of the mixed layer (MLD) and σ27 computed using an OGCM simulation over the period 1993–2015 (Fig. ). The time-mean vertical gradient of vertical velocities between the base of the maximum mixed layer and σ27 has been normalised by the distance (in meters) between these isopycnal levels, as follows: 18Δzw=|wσ27|-|wσMLD||zσ27-zσMLD|.

Negative values indicate a decrease in magnitude with depth, while positive values indicate an increase. The velocity fields were smoothed with a 5° running mean to retain large-scale structures that LVB can describe (CM24).

As shown in CM24 for the North Atlantic, wg succeeds in estimating wtot over most parts of the basins. Figure a shows that values are accurate, with a relative error below 50 %, across most of the global tropical and subtropical gyres (yellow, orange and red regions), as well as the eastern part of the subpolar Pacific gyre and the Arctic Beaufort gyre. This result shows that the geostrophic vertical velocity field generally reproduces the spatial structure and amplitude of the thermocline vertical flow within the major gyres in the model simulation, suggesting that a similar behaviour may occur in the real ocean.

Relative errors exceed 50 % in several regions. These include the intergyres bands, where vertical velocities change sign, as well as larger regions of intense current systems, such as the Gulf Stream and its north-eastward extension, the Brazil–Malvinas Current, the Kuroshio Current, the Eastern Australia Current and the Algulhas Current (see locations in ). High relative errors are also found in regions dominated by strong zonal flows, like the deep tropics, such as the Equatorial Currents and Countercurrents, but also open ocean poleward extensions of western boundary currents like the North Atlantic Drift (NAD), and the Antarctic Circumpolar Current (ACC), more visible on deeper isopycnals (not shown). They are characterised by an intense zonal component, which geostrophic part does not generate divergence and geostrophic vertical velocity (CM24). Most eastern boundaries, particularly eastern boundary upwelling systems (e.g. California or Benguela), as well as the northern Indian basins, suffer from high errors in the estimation of their time-mean vertical velocities. Many of these discrepancies typically arise in regions where the LVB no longer holds (hatching in Fig. a). In such areas, nonlinear processes, friction and lateral diffusion become essential to close the time-mean vorticity budget (e.g. ).

It is interesting to note that in certain regions, the large differences are not consistent with the local validity of the LVB at the current isopycnal level. For example, in the North Atlantic intergyre region, a large relative difference between wg and wtot appears as a narrow band around 15° N, despite the LVB terms showing relative agreement within 10 % (i.e. no hatching). In contrast, within the tropical gyre (centred around 10° N), the relative error between wg and wtot is found to fall below 50 %, yet the LVB is not valid on σ26 there (hatching areas). Indeed, as for the North Atlantic application of LVB (CM24), the geostrophic vertical velocity at a given depth is computed via vertical integration of the meridional transport above that level. Therefore, local deviations from the LVB at a given depth, implying that the export of water is not fully conserved, do not necessarily lead to large errors in the integrated geostrophic velocity. This is more pronounced at deeper levels, where the amplitude of the meridional transport is reduced due to the baroclinic structure of the tachocline flow.

Considering now the reconstruction of the interannual variability, the LVB method appears to be strikingly accurate (Fig. b). Correlation between wg and wtot exceeds 0.9, indicating that the geostrophic component explains more than 80 % of the total vertical velocity variance, across most of the tropical, subtropical global ocean and northern subpolar gyres (only Pacific subpolar gyre visible), and the Beaufort polar gyre. These good results extend those previously reported for the North Atlantic. Weaker, or even negative, correlation values are found within the major western boundary current systems of the Pacific, Atlantic and Indian Oceans, where nonlinear dynamics are stronger. Similarly large differences also characterise Atlantic and Pacific deep tropics, the open ocean poleward extensions of western boundary currents, and the entire ACC, visible southeast of Africa. These regions are all dominated by flows with strong zonal components relative to the meridional flow, likely generating mostly ageostrophic vertical velocities that cannot be captured by the LVB framework. Remarkably, very high correlations persist even in regions where the time-mean geostrophic component fails to replicate the total vertical velocity pathways (Fig. a), as well as in regions where the LVB does not hold, such as the mixed layer (translucent black surfaces in Fig. a).

Figure c illustrates the ability of time-mean wg to represent the vertical structure of the velocity field shown in Fig. a, by displaying the relative error of a proxy of the time-mean vertical gradient (Eq. ) between wg and wtot. Note that the vertical gradient of the time-mean total vertical velocity is almost everywhere positive (non-dotted areas), indicating a decrease in the magnitude of the vertical velocity toward the base of the thermocline. This structure is consistent with a baroclinic velocity field, generating a tachocline, as underlined in the North Atlantic (CM24). This vertical gradient appears to be well represented by wg, where the five main subtropical gyres are characterised by a relative error smaller than 20 %. The magnitude of wtot grows with depth (dotted regions) within western boundary current systems such as the Gulf Stream, Kuroshio, and Brazil–Malvinas Currents, the North Pacific and North Atlantic deep tropics, and the intergyre regions between the major tropical and subtropical gyres. Nonetheless, in the Pacific and Atlantic tropical gyres, the errors in the vertical gradient of the total and geostrophic vertical flow are larger than in the subtropical gyres. The spatial distribution of these errors is similar to the pattern of relative errors above 10 % between wg and wtot (Fig. a). Therefore, these results suggest the limitations of the LVB framework in reproducing the time-mean value at a given depth and the vertical structure of the total vertical flow at tropical regions and western boundary current systems.

These findings evidence the relevance of the geostrophic LVB framework for capturing and explaining the dynamics of the large-scale vertical motion in an OGCM simulation. Most notably, the high and widespread synchrony at annual frequencies between wtot and wg suggests that the geostrophic component strongly dominates the interannual variability, and therefore likely in the real ocean as well. While nonlinear processes influence the mean amplitude of the vertical flow, they play a smaller role in modulating this variability. Thus, extending previous results from the North Atlantic Ocean to global scales, it shows that geostrophic vertical velocity provides a reliable estimate of the total vertical flow for studying the climatological flow structure within the interior of the major gyres and the interannual variability of vertical motion throughout much of the tropical and subtropical oceans, parts of northern subpolar gyres and polar gyres. This supports the relevance of applying the same reconstruction methodology to observation-based data. In the rest of the paper, we assess the named OLIV3, an observation-based estimate of the global thermocline vertical velocities.

Due to the lack of direct observations for large-scale vertical velocities, it is necessary to evaluate the performance of OLIV3 relative to commonly used products. In particular, acknowledging the multidimensional nature of the vertical velocity field, a comprehensive evaluation must address the mean three-dimensional structure, as well as the temporal signals relevant for future studies.

Several fundamental methodological differences among OLIV3, OMEGA3D and the two reanalyses may account for the discrepancies observed in the climatological horizontal and baroclinic structure, as well as the temporal evolution of the vertical movements. These include the reconstruction methodology, the components of vertical velocity reconstructed, the atmospheric forcing, the three-dimensional horizontal velocity inputs, and the spatiotemporal resolution of the datasets.

Atmospheric forcing does not appear to be the primary source of discrepancies, as all datasets employ variants of ERA atmospheric reanalysis products to force the oceanic surface. Similarly, both OLIV3 and OMEGA3D are based on the ARMOR3D geostrophic velocity field, while reanalyses compute vertical velocities directly from the total assimilated horizontal velocity field. Despite their common input, OLIV3 and OMEGA3D exhibit large differences. In contrast, OLIV3 aligns more closely with the reanalyses despite their distinct origin in either observation-based or assimilated horizontal velocity fields. However, the ageostrophic component of the horizontal velocity field is negligible in most of the tropical and subtropical tachocline, as discussed in CM24. Again, while these differences may induce some discrepancies, they are not dominant.

Differences in native spatial and temporal resolution may play a significant role, even when data are averaged to a common resolution. Certain phenomena may persist across scales and not be entirely removed (). For example, GLORYS12v1 has a resolution of 1/12°, capturing smaller mesoscale features compared to ECCOv4r4 at 1° resolution. Higher spatial resolutions allow GLORYS12v1 to preserve events at seasonal and sub-seasonal timescales, which could not be maintained in coarser native resolutions. Previous studies on reanalysis intercomparison (e.g. ) have shown that such differences due to resolution (in particular in temperature and salinity) are more pronounced in the tropics. In particular, shows how the coarser resolution of ECCO leads to very distinct results from other eddy-permitting datasets. The spatial resolutions in the mentioned study range from 0.25 to 1°. Therefore, the uncertainty observed here between GLORYS12v1 and ECCOv4r4 probably reproduces and magnifies the uncertainty observed in the cited studies.

OLIV3 and reanalyses estimate w through vertical integration of horizontal velocity fields, because the governing equations only contain the vertical derivative of w. In contrast, OMEGA3D employs the omega equation, which, although it also requires vertical integration, explicitly includes second-order vertical derivatives and horizontal derivatives of w. Consequently, Dirichlet (vertical velocities are set to zero) and Neumann (partial derivatives of vertical velocity are set to zero) conditions are imposed as boundary conditions (). The inclusion of the horizontal and second-order derivatives of w in the OMEGA3D framework may explain the discrepancies with the other datasets. However, a comprehensive examination of the sources of these differences would require a separate in-depth investigation, given the complex physics, constraints and assumptions underlying the omega equation.

Beyond the importance of providing an estimate of the vertical profile of thermocline vertical velocities, one might wonder how ocean interior wg compares with Ekman pumping (wEk) from Eq. (), the most commonly used observation-based reference product for vertical transfers between thermocline and nitracline, and the surface waters. Indeed, this wind-based computation is frequently employed to validate against observations, or calculate water mass fluxes (e.g. ), as well as transport of biogeochemical tracers () and marine ecosystem parameters, from fish () up to whales (). In these types of studies, Ekman pumping is generally considered as a vertical velocity proxy at a variety of levels depending on questions, time-scales and community habits. This level ranges from the bottom of the Ekman layer to that of the winter mixed layer (), an isopycnal surface close to σ26 or a fixed depth of a few tens of meters up to 200 m (). Here, the interannual synchrony of the Ekman pumping at the ocean surface and both the geostrophic and total vertical velocities is examined in isopycnal and depth coordinates (Fig. ) to highlight the effect of the geostrophic current divergence on the variability of vertical velocities in the ocean interior. Specifically, Fig. a displays the correlation between wEk at ocean surface and wtot on σ26, while Fig. b represents the correlation difference between wg and wtot shown Fig. b. Figure c and d illustrate, respectively, the correlation between wg and wtot at 100 m depth, and the correlation between wEk at the ocean surface and wtot at 100 m depth. This comparison also illustrates the extent to which Ekman pumping can serve as a reliable proxy for describing the temporal variability of the total vertical flow within the ocean interior. Again, the computation is conducted with the reference OGCM simulation, considered to be a dynamically coherent estimate of the real ocean.

Ekman pumping estimates well (R>0.7) up to very well (R>0.9) the variability of total vertical velocities along σ26 over a relatively small proportion of the thermocline, limited to certain parts of the subtropical gyres and the tropical Atlantic Ocean (Fig. a). However, comparison with the correlation between wtot and wg (Fig. b), quantified in Fig. b, shows that wg is accurate over much larger areas of the globe. In other words, wEk is only more accurate than wg in regions where both estimates are poor (R<0.7), such as western boundary currents and intense open ocean currents, such as the NAD and ACC with a strong zonal component. Since wg is derived from the geostrophic meridional velocity (vg), this suggests that, in these regions, vg reproduces an interannual variability not synchronised with wtot, thereby further reducing the low correlation of wEk. Moreover, vertical movement in most eastern boundary upwelling systems, such as Benguela or Peru, shows weak correlations with Ekman pumping. This is consistent with the importance of remote forcing from trapped waves on the coast in these regions . A marked decrease in the correlation coefficient is observed towards the western boundaries, where the isopycnal surfaces deepen, within the western boundary current systems, as well as in the deep tropics. When comparing this pattern to the correlation between wg and wtot shown in Fig. b, it is remarkable that the areas with high correlation are even more extensive in all basins.

We extended the comparison to other vertical levels, in particular to depths of 50 (not shown) and 100 m (Fig. c and d), and reached the same conclusion. Overall, these results show that the Ekman pumping alone is insufficient to account for interannual variability in vertical flow in most regions of the globe. However, the inclusion of geostrophic meridional transport divergence in Ekman pumping, i.e. wg, brings about a significant improvement. This highlights the advancement of geostrophic vertical velocities as an estimator of total vertical flow variability.