Review of “Global Thermocline Vertical Velocities: A Novel Observation Based Estimate” by Diego Cortés-Morales, Alban Lazar, Diana Ruiz Pino, and Juliette Mignot.

In this manuscript, Cortés-Morales et al. develop and validate a new vertical velocity estimate for the ocean. The manuscript is generally well-written, and the topic is timely. I believe that providing a vertical velocity dataset is extremely important for the oceanographic community. I have a few comments that, while not dramatically changing the manuscript and its results, will require some revisions from the authors. I hope these will help clarify key aspects of the manuscript.

For these reasons, I recommend a major revision of the manuscript.

Please see my specific comments below.

Major Comments:

1) Ekman Boundary Condition

Ekman dynamics are not geostrophic in the traditional sense, as they involve other processes (i.e., wind stress, vertical momentum transfer, and viscosity) and do not include the pressure gradient. Thus, it is confusing to refer to a vertical velocity that incorporates such dynamics as "geostrophic". In my opinion, this is more complex. A purely geostrophic vertical velocity would be represented by (2), with wg(zref) determined solely by geostrophic processes (perhaps using $\partial_t\eta$, where $\eta$ is the sea surface height). I would understand that you do no want to alter your framework, which is reasonable; however, you need to clarify this point and use more accurate terminology.

In section 2.1, if one derives the full equation, it becomes evident that this condition must be imposed by "patching" the Ekman layer and the interior ocean. Therefore, the condition exists not at the ocean surface but at the base of the Ekman layer.

The sentence on lines 106-107 contradicts what is discussed in section 3.5. If the Ekman condition governs wg, how can wg at the boundary differ from Ekman? Either there is an issue or something is unclear in your method.

In section 3.5, you begin by stating, “one might wonder how near-surface wg compares with Ekman pumping (wEk)”, yet your methodological description suggests they are the same (i.e., surface boundary conditions). You are comparing at different depths. This entire paragraph, discussion, and section are confusing; I do not understand the goal or conclusion. More importantly, how is wEk defined at different depths? Should it not be defined at the base of the Ekman layer? Are you setting it to random depths for comparison? This does not make sense.

Lines 518-526: I find this paragraph confusing. Please rewrite it based on the comments above.

2) “Overuse” of the Correlation

Even if you demonstrate a good correlation, this does not imply good representation; it only indicates good synchrony. A correlation can still be good whereas one component is significantly stronger than the other. To fully illustrate good representation, you also need to check that the variances are consistent. Hence, if you want to demonstrate good representation, it would be more effective to use the coefficient of determination (R2: https://en.wikipedia.org/wiki/Coefficient_of_determination). I suggest to use instedad of correlation for the particular purpose of your study.

Furthermore, correlation (normalized projection) and the coefficient of determination (R2; relative similarity = 1− relative error) are two distinct diagnostics. They can converge, but they do not have to. Please be specific and avoid conflating the two.

More importantly, how do you compute a local correlation with a "time mean" term (as indicated in the figure title 3b)? If w=fct(x,y,σ,t), the correlation is computed between two scalars... Projecting one scalar onto another does not make sense. The text references interannual correlation, thought, so please clarify!

3) Description of the dataset

I believe there was a missed opportunity to present the dataset effectively. Adding a single figure that illustrates vertical velocity at a range of key levels (surface, within the thermocline, and below the thermocline) would be beneficial. A single figure with six eight panels showing vertical velocity in color and the corresponding isopycnal depth in contour would serve as the main visual aid to describe the dataset.

This figure, along with a descriptive paragraph, should be included in section 2.2.

4) Two datasets (?)

Two datasets are provided and attached to the manuscript (isopycnal level and vertical level). However, only the isopycnal dataset is presented, described, and analyzed. I suggest removing the vertical-level dataset for consistency, as it seems surprising, to say the least, to include a supplementary dataset that is neither referenced nor discussed in the manuscript. Alternatively, a full description and analysis of the second dataset are required.

Minor Comments:

*) In the introduction (lines 54-58), you should also mention a strategy for reconstructing the full 3D circulation (u,v,w) based on thermal wind balance and Argo float deep displacement knowledge for the reference-level horizontal velocity. The vertical velocity derives similarly from a Sverdrup balance (equivalent to your approach): reference Colin de Verdière and Ollitrault (2016) and Colin de Verdière et al. (2023).

Ref:

Colin de Verdière, A., and M. Ollitrault, 2016. A Direct Determination of the World Ocean Barotropic Circulation. Journal of Physical Oceanography, 46, 255-273.

Colin de Verdière, A., T. Meunier and M. Ollitrault, 2019, Meridional overturning and heat transport from Argo floats displacements and the planetary geostrophic method: applications to the subpolar North Atlantic, Journal of Geophycical Research, 124, 6270-6285.

*) In several places, “Naveira Garabato” has been referred to as “Garabato.” Please revise this. *) Line 97: To the best of my knowledge, you do not need to assume the $\beta$-plane (i.e., tangent slope to sphere, $f$ varies linearly, or $\beta$ is constant). You only need to acknowledge that $\beta=\partial_yf$ (i.e., the meridional derivative of $f$ exists). *)After equation (2): Please add “assuming $f\neq0$.” This clarification is important.

*) Line 102: Please replace “geostrophic flow on a $\beta$-plane” with “horizontal geostrophic flow.” *) Line 122: This is the reference paper for the ANDRO dataset (deep displacements of Argo floats). I think that this is not be the correct reference, or I don’t understand why you are citing it in this context.

*) Sentences on lines 123-124: I do not find a reference for that. However, it is “admitted” in the community that this approach is not ideal. Have you tested your ability to execute this? How does the surface error propagate at depth? From my understanding, this is not as straightforward as it appears.

Additionally, I am concerned that the boundary conditions for the vertical velocity are Ekman-based, while for horizontal flows, they are geostrophic. There exists a subtle but fundamental difference between the two. Have you carefully checked this? For instance, is it dynamically consistent to have a horizontal flow at the surface that does not “see” to the wind, while a vertical flow derives from a horizontal flow that does?

*) Line 230, equation (4): Why is there an absolute value? This seems illogical. Please make a simple difference. Furthermore, it would be mathematically more accurate to use \Delta w / \Delta \sigma on the left; the use of $\partial$ is mathematically ambiguous (i.e., unclear) and physically inconsistent (in terms of units).

*) Lines 284-293: Please specify the frequency of the vertical velocity (i.e., w_tot from the OGCM). If it is once every 5 days or lower, I am not surprised by these results. However, if it is hourly, I would be surprised that the vertical flow energy is not dominated by inertial motion, which would weaken the correlation with the geostrophic flow.

*) Figure 3: This does not represent a vertical gradient but rather a diapycnal gradient. Additionally, if you wish to show a relative term (which is debatable, since the denominator can be 0), you should use an absolute value in the denominator (otherwise, the sign becomes ambiguous, as the denominator can be both positive and negative).

*) In several places, “x” should be replaced by “$\times$” (e.g., line 303).

*) Caption of Figure 7: Please replace “,,” with “,” and replace “and” with “, and.” *) For Figures 9 and lines 463-465: Again, R2 would provide much more informative insights!