Estimates of 3D ocean circulation are needed to improve our
understanding of ocean dynamics and to assess their impact on marine
ecosystems and Earth climate. Here we present the OMEGA3D product, an
observation-based time series of (quasi-)global 3D ocean currents covering
the 1993–2018 period, developed by the Italian Consiglio Nazionale delle
Ricerche within the European Copernicus Marine Environment Monitoring
Service (CMEMS). This dataset was obtained by applying a diabatic
quasi-geostrophic (QG) diagnostic model to the data-driven CMEMS-ARMOR3D weekly
reconstruction of temperature and salinity as well as ERA Interim fluxes. Outside
the equatorial band, vertical velocities were retrieved in the upper 1500 m at

The full OMEGA3D product (released on 31 March 2020) is available
upon free registration at

The recognition of the key role played by the oceans in the Earth system led the United Nations to proclaim the Decade of Ocean Science for Sustainable Development (2021–2030). Major efforts will consequently be made in the next years to analyse state-of-the-art observations and models and provide the indispensable knowledge basis to preserve the marine environment through effective, science-informed policies. Providing accurate reconstructions of 3D ocean circulation time series is a fundamental part of this effort, aimed at better describing ocean dynamics and assessing their responses and feedbacks to natural and anthropogenic pressures. However, assessing the time-evolving lateral and vertical transport of energy, momentum, gases, nutrients, marine organisms and pollutants would require repeated synoptic observations of the 3D ocean state and surface forcings that cannot be presently achieved even with the most advanced technologies. Hence, a combination of measurements collected from in situ and remote-sensing platforms and proper modelling frameworks is needed to describe the ocean circulation both in the ocean interior and at the domain boundaries. Two main complementary approaches can be followed to this end: the assimilation of observations in global ocean circulation numerical models (Carrassi et al., 2018; Moore et al., 2019; Stammer et al., 2016) and the combination of diagnostic models and purely data-driven reconstructions. The latter is presently more widely used for surface circulation retrievals, but its extension to 3D ocean state reconstruction is generating growing interest as advanced statistical and machine learning tools are becoming more computationally efficient (Buongiorno Nardelli et al., 2012; Buongiorno Nardelli and Santoleri, 2005; Guinehut et al., 2012; Lopez-Radcenco et al., 2018; Mulet et al., 2012; Rio et al., 2016; Ubelmann et al., 2015; Yan et al., 2020). Both approaches present advantages and drawbacks: data assimilation in prognostic models may guarantee a description of the ocean state evolution that is fully consistent with the physics represented by the model, but the uncertainties in its initialization, the limits of the parameterizations of unresolved processes and the difficulties to properly represent model and observation errors and to account for their representativeness can significantly reduce models' ability to reproduce non-assimilated observations. Conversely, synergic use of satellite in situ observations and data-driven reconstruction methodologies, in combination with simpler dynamical models (often limited to zero-order balances, as in the retrieval of geostrophic currents from sea level data), can provide snapshots that better match independent observations (Mulet et al., 2012; Rio et al., 2016; Ubelmann et al., 2016).

The OMEGA3D product, developed by the Consiglio Nazionale delle Ricerche within the European Copernicus Marine
Environment Monitoring Service (CMEMS;

The accuracy of the QG velocities depends on the input data and on the theoretical limits of the model and parameterization used. Omega forcings are estimated here from the multi-year CMEMS product ARMOR3D (Guinehut et al., 2012), providing a statistical reconstruction of 3D temperature and salinity fields from a combination of in situ profiles and satellite observations of sea surface temperature, salinity and topography. ERA Interim air–sea fluxes are used to evaluate the forcing terms due to vertical mixing (Dee et al., 2011). QG approximation implies that the omega equation cannot be solved at the Equator, and increased errors are expected in the low-latitude bands.

A direct validation of the vertical velocities is not possible due to the lack of direct reference observations. As such, OMEGA3D vertical velocity mean pattern and variability have been compared here with two global model reanalyses that include vertical velocity fields as disseminated output, namely Estimating the Circulation and Climate of the Ocean (ECCO; Forget et al., 2015) and Simple Ocean Data Assimilation (SODA; Carton et al., 2018).

Total horizontal and geostrophic components are instead compared with fully independent velocity estimates obtained from drifting-buoy and Argo float displacement. For reference, a similar comparison is carried out between two reanalyses, SODA and CMEMS GLORYS (Global Ocean Reanalysis and Simulation; Drévillon et al., 2018), and drifter data.

Two datasets are taken as input for OMEGA3D processing:

ERA Interim assimilates several observations of upper-air atmospheric variables (e.g. satellite radiances, temperature, wind vectors, specific humidity and ozone) through a four-dimensional variational (4D-VAR) system, running with a 12-hourly analysis cycle. OMEGA3D diabatic forcings take in input of the mean daily fields of the zonal and meridional components of the turbulent surface stress, the surface latent and heat flux, the surface net solar and thermal radiation, and total precipitation and evaporation (needed to estimate the equivalent surface salinity flux in KPP).

The numerical tool used to retrieve the OMEGA3D product is designed to run on a non-uniform vertical grid that displays a refined mesh close to the surface. The vertical layer thickness increases with the square of depth, and the final grid includes 75 vertical levels between 2.5 and 1482.5 m. This grid was specifically designed to obtain more accurate numerical solutions within the ocean's upper boundary layer (Kalnay de Rivas, 1972; Sundqvist and Veronis, 1970). Preprocessing of input data thus includes as a first step the vertical interpolation of ARMOR3D data on OMEGA3D vertical layers (using Python class scipy.interpolate.interp1d set to cubic spline interpolation; Virtanen et al., 2019) and the mapping of ERA Interim data on OMEGA3D horizontal grid (using Python class scipy.interpolate.griddata set to fit data to a piecewise cubic, continuously differentiable, curvature-minimizing polynomial surface; Virtanen et al., 2019).

As ARMOR3D data may occasionally display density inversions along the water
column that are not compatible with the QG omega solution, vertical profiles of
potential density are adjusted to impose static stability: moving from the
surface to depth, density is set to
the upper level value plus a 0.0001 kg m

A diabatic

In Eq. (1),

In the above definitions,

In one of the analytical steps to obtain Eq. (1), the details of which are
given elsewhere (Buongiorno
Nardelli et al., 2018b; Giordani et al., 2006), the following two equations
are found:

Once vertical velocities are retrieved through the omega solution, these two
equations allow the estimation of the horizontal ageostrophic components
(

All equations used for the OMEGA3D retrieval are solved here numerically (Buongiorno Nardelli et al., 2018b). At each grid point in the interior domain (i.e. excluding the boundaries), the omega equation is rewritten substituting derivatives with central finite differences and considering a non-staggered grid. Vertical derivatives are computed considering a variable grid spacing, increasing with the square of depth (Kalnay de Rivas, 1972), and adopting a finite difference scheme of second-order accuracy (Sundqvist and Veronis, 1970).

At the surface and topographical boundaries Dirichlet conditions are
imposed (namely vertical velocities of zero), and Neumann conditions are imposed at the
bottom and lateral boundaries (namely partial derivatives of vertical velocity are set to zero). The latter are imposed through forward–backward finite
schemes and make the solution unsuitable for modelling current topography
interactions along the coasts. Grouping all factors and multiplying

Vertical velocities are finally used to integrate Eq. (3) by a simple trapezoidal rule to obtain ageostrophic horizontal velocities.

Three different ocean state reconstruction time series have been compared with OMEGA3D. All of them are based on ocean general circulation models assimilating both in situ and satellite observations, though significantly differing in terms of numerical schemes used, input data ingested and assimilation strategies.

The first dataset considered is the third release of version 4 of ECCO
(Forget et al.,
2015; Fukumori et al., 2018), hereafter ECCOv4r3, covering the 1992–2015
period and available at

The second dataset is version 3.4.2 of SODA (Carton et
al., 2018), hereafter SODAv3.4.2, which covers the 1991–2017 period and can
be downloaded from

The third product used for the comparison is the output of the first version
of the

GLORYS12v1 data can be freely downloaded at

As for OMEGA3D, ECCOv4r3, SODAv3.4.2 and GLORYS12v1, surface forcings are all taken from ERA Interim (Dee et al., 2011).

Two fully independent in situ datasets have been considered for the
validation of OMEGA3D horizontal velocities: Surface Velocity Program (SVP)
data (Lumpkin et al., 2013) from the NOAA Global
Drifter Program (covering the period 1993–2018 and freely available at

In order to minimize wind slippage, SVP drifters are drogued with a 7 m long holey sock centred at 15 m depth, and their velocity estimates are considered representative of currents at 15 m depth (Lumpkin et al., 2017). Before carrying out the validation, individual 6-hourly SVP drifters were averaged over a running time window (inversely scaled with the Coriolis parameter) to remove the signal due to inertial oscillations (Buongiorno Nardelli et al., 2018b).

YOMAHA velocities are estimated by measuring the displacement of profiling Argo floats during their submerged phase (Lebedev et al., 2007). Argo floats drift at a predefined parking pressure and emerge only for near-real-time data transmission through ARGOS–IRIDIUM satellites. Most of these instruments follow a profiling cycle of approximately 10 d, and their parking level is set to 1000 m.

Vertical velocities cannot be measured in the open ocean due to their
relatively small magnitude (of the order of 1–100 m d

OMEGA3D vertical velocities are thus compared here with the output of the only two ocean climate reanalysis systems that presently distribute vertical velocity time series of comparable length. Considering that vertical velocity fields are provided at different space–time resolutions, this comparison only describes the mean patterns and the amount of variability captured by each product.

Mean vertical velocity at 100 m

Specifically, mean vertical velocity patterns at 100 m depth and associated
variability (standard deviation) are computed here from OMEGA3D, SODAv3.4.2
and ECCOv4r3 over their 23-year overlapping period (1993–2015), focusing on
the domain covered by the OMEGA3D product and thus excluding the 5

Given its 5 d sampling, SODAv3.4.2 could be expected to reveal a stronger
variability than OMEGA3D (7 d sampling), and both are expected to display
much higher values than ECCOv4r3 (providing monthly averaged fields).
Conversely, though associated patterns display very similar features,
the maximum OMEGA3D standard deviation value exceeds SODAv3.4.2 by a factor of

For the sake of a more consistent comparison with ECCOv4r3, OMEGA3D and SODAv3.4.2 vertical velocity standard deviations have also been estimated after low-pass filtering the latter two time series (by a five-point and seven-point moving window, respectively) to only keep frequencies lower than monthly, like those provided by ECCOv4r3 (Fig. 2 should thus be compared to Fig. 1f). Even in that case, the variability observed in ECCOv4r3 is sensibly lower than those retrieved from higher-spatial-resolution products, likely revealing the limits of the mesoscale parameterization used in ECCOv4r3 in terms of vertical exchanges.

Mean monthly vertical velocity patterns and standard deviations
computed from OMEGA3D

OMEGA3D horizontal velocity accuracy has been assessed in terms of mean bias
and root mean square differences (RMSDs) with respect to space–time-co-located in situ reference observations. Estimated metrics have then been
compared to those estimated for geostrophic velocities directly obtained from
the Data Unification and Altimeter Combination System (DUACS) altimeter data (when looking at SVP
velocities; AVISO

To build our matchup databases, OMEGA3D velocities have been interpolated at the same nominal depth of drifter measurements through a weighted average of the two closest levels.

The first assessment covered surface currents as measured by SVP drifters.
As SVP drifters may occasionally lose their drogue, thus failing to correctly represent
15 m depth currents, only drogued SVP drifter data collected within

Number of matchups within

Mean and root mean square differences between OMEGA3D

Main characteristics of CMEMS OMEGA3D product.

Mean biases between SVP and OMEGA3D velocities (Fig. 4a) display similar
values and patterns to what obtained from altimeter-derived geostrophic
velocities (Fig. 4c), with a slight underestimation of the current
intensities. OMEGA3D actually appears more biased than DUACS geostrophic
velocities close to the tropical band, likely due to the fact that the omega
equation is derived from the

Directly comparing OMEGA3D and DUACS RMSD demonstrates that quasi-geostrophic velocities also improve with respect to geostrophic velocities (by a few centimetres per second), mainly along the Antarctic Circumpolar Current and in the western boundary currents (Fig. 5a).

The second assessment focused on velocities provided by the YOMAHA dataset,
which are representative of currents at 1000 m depth. In that case, in order
to increase the number of samples, a temporal window of

RMSD of OMEGA3D quasi-geostrophic
and geostrophic horizontal velocities vs. drifters in

A general overestimation of deep currents is revealed by looking at mean
biases with respect to YOMAHA observations. The mean bias attains around 5 cm s

Mean and root mean square differences between OMEGA3D

The OMEGA3D product is distributed as part of the CMEMS catalogue
(

The basic characteristics of the OMEGA3D product are summarized in Table 1.

The 1993–2018 OMEGA3D time series provides weekly observation-based estimates
of the 3D vertical and horizontal ocean currents in the upper 1500 m of the
global oceans. The product is obtained by applying a quasi-geostrophic
diagnostic model (based on the omega equation) that includes the effect of
both geostrophic advection and upper-layer turbulent mixing and delivers,
for the first time, estimates of the vertical velocities based on a
combination of satellite and in situ observations. The OMEGA3D time series is
provided over a

The model cannot be applied in the equatorial band (where QG approximation fails), and it does not include any parameterization of bottom-boundary-layer mixing. Dirichlet conditions are thus set at the bottom. As a consequence, considering that the domain is also limited to the upper 1500 m, OMEGA3D is not suited for studies of bottom-boundary dynamics or equatorial dynamics. Dirichlet conditions are also applied at coastal boundaries. Even if the effect of lateral-boundary conditions only propagates a few grid points due to the elliptical nature of the omega equation (Buongiorno Nardelli et al., 2001, 2012, 2018a), this also makes the OMEGA3D product unsuitable for studies of coastal dynamics (the product is considered reliable approximately 100 km away from masked coastal areas). As such, OMEGA3D is mostly suited to describe the role and long-term variability of open-ocean, large mesoscale dynamics and air–sea interactions (here parameterized through KPP), for example regarding the vertical exchanges and water mass transformation outside the equatorial band.

The author declares that there is no conflict of interest.

This work has been carried out as part of the Copernicus Marine Environment Monitoring Service Multi Observations Thematic Assembly Center (CMEMS MOB TAC), funded through subcontracting agreement no. CLS-SCO-18-0004 between Consiglio Nazionale delle Ricerche and Collecte Localisation Satellites (CLS), the latter of which is presently leading the CMEMS MOB TAC. The 83-CMEMS-TAC-MOB contract is funded by Mercator Ocean as part of its delegation agreement with the European Union, represented by the European Commission, to set up and manage CMEMS.

This paper was edited by Giuseppe M. R. Manzella and reviewed by Gabriela Pilo and one anonymous referee.