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Figure 5. Mean state of the SPMW layer thickness [m] (1993-2018) same as Figure 3. Superimposed with purple/green contour lines is the mean state of the horizontal 
component [Sv] (a-e) and the entrainment component [Sv] (f-j) of the subduction/obduction rate from the kinematic approach. The purple contour lines denote positive 
values and the green contour lines negative values. Bathymetry at 1,000 and 2,000 m is also displayed with gray contour lines. 
similar way, it is plausible to consider that this hypothesis can also be applied to the SPMW in the North Atlantic. 
Another study based on a high-resolution model (Xu et al., 2018) mapped the diapycnal water mass trans- 
formation in the upper North Atlantic Ocean and also agrees that surface buoyancy loss does not account for all 
he transformation. Mixing is responsible for significant transformation around the subpolar gyre (Xu 
et al., 2018), especially along the steep isopycnal slopes. A more recent study in the North Atlantic (Fröhle 
et al., 2022), based on a Lagrangian analysis on an eddy-rich ocean model, highlights the substantial role of water 
mass formation below the sea surface due to diapycnal mixing. 
We need to point out here that in order to be able to compare directly the results of the thermodynamic approach 
with the one from the kinematic approach all the data sets need to have the same time resolution. As we described 
in the method session, OMEGA3D has a sample per week centered on Wednesday. Thus, although the atmo- 
spheric data from ERA5 are on hourly resolution allowing us to actually calculate daily fluxes, we calculated the 
buoyancy fluxes using one sample per week to be consistent with the OMEGA3D and ARMOR3D data sets. 
However, when the thermodynamic calculation is done using daily data, the resulting transformation and for- 
mation estimates (Figure 4) are lower than the one calculated from the mean weekly fluxes, and they are closer to 
'he ones calculated by for example, Grist et al. (2016) and Brambilla et al. (2008). Moreover, the formation at 
59 = 27.3 kg m is close to zero when using the daily fluxes and not negative as obtained from the weekly 
averaged data (Figure 4b). 
The kinematic approach allows for a separate analysis of the individual components contributing to the net 
transport across the mixed layer, which in turn determines the subduction and obduction rates. These components 
are derived from both the vertical and horizontal velocities, as well as the temporal variations in the MLD. 
Darticularly, the entrainment component, which is influenced by the shoaling or deepening of the mixed layer 
ıtself, is not directly dependent on the velocity of water mass across the mixed layer base. Consequently, when 
there is a rapid deepening of the MLD, water that has previously left the mixed layer has the potential to re-enter it. 
Conversely, a swift shoaling of the MLD allows water to escape from the mixed layer and move toward the 
interior of the ocean. Kwon et al. (2013) suggested this as the primary mechanism driving the subduction of mode 
waters in the Southern Ocean. They argued that the detrainment of mixed layer waters into the stratified pyc- 
1ocline occurs during early spring over an expanded outcrop area. A similar hypothesis was also drawn by 
3rambilla et al. (2008) for the SPMW in the North Atlantic. In Figure 5, we compared the horizontal and 
entrainment components. We did not include a map of the vertical component because our computations showed 
that it has a negligible contribution over the entire domain (one to two orders of magnitude smaller than the other 
two terms). This is supported by the small vertical component needed to allow motion along a sloping isopycnal 
surface. Similarly to Kwon et al. (2013), we found that the entrainment component is the main mechanism driving 
ıhe subduction/obduction rates (Figures 5f-5j), with the horizontal component mostly acting as a modulator 
(Figures 5a-5e). In the two companion papers from Brambilla et al. (2008) and Brambilla and Talley (2008) it was 
argued that the net formation of SPMW obtained from the thermodynamic approach can be interpreted as the loss 
hy the entrainment to the interior layers due to vigorous mixing that strongly alters the SPMW properties. This is 
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