The global ocean mixed layer depth derived from an energy approach

Moreles, Efraín; Romero, Emmanuel; Ramos-Musalem, Karina; Tenorio-Fernandez, Leonardo

doi:https://doi.org/10.5194/egusphere-2024-4079

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https://doi.org/10.5194/egusphere-2024-4079

Preprints

15 Jan 2025

| 15 Jan 2025

The global ocean mixed layer depth derived from an energy approach

Efraín Moreles, Emmanuel Romero, Karina Ramos-Musalem, and Leonardo Tenorio-Fernandez

Abstract. The mixed layer depth (MLD) is critical for understanding ocean-atmosphere interactions and internal ocean dynamics. Traditional methods for determining the MLD commonly rely on hydrographic thresholds that vary spatially and temporally with local oceanographic conditions, limiting their global consistency and applicability. To address this, we propose an energy-based methodology that defines the MLD as the depth at which the work done by buoyancy (WB) reaches 20 J m^-3. Based on the structural change in WB, the MLD criterion identifies the upper ocean's well-mixed layer in energetic terms. This approach provides a robust criterion based on physical principles, which is globally and temporally consistent and easy to implement. Our methodology aligns with turbulent boundary layer dynamics while maintaining quasi-homogeneity in density and temperature for most of the global ocean throughout the year. A global monthly MLD climatology derived from this method demonstrates its reliability across diverse oceanic conditions and its accuracy in regions and seasons where conventional methods struggle. Our study advances the development of MLD energy-based methodologies by providing a single energy value to define the MLD globally during all months. This energy-based approach could offer significant potential for advancing the study of dynamic, and thermodynamic processes, including heat content and vertical exchanges. It could also serve as a robust tool for validating ocean circulation models and to support intercomparison studies in initiatives such as the Ocean Model Intercomparison Project (OMIP) and the International Coupled Model Intercomparison Project (CMIP). Future research will explore its applicability to high-frequency processes and regional variability, further enhancing its utility for understanding and modeling oceanic phenomena.

Received: 20 Dec 2024 – Discussion started: 15 Jan 2025

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Journal article(s) based on this preprint

18 Sep 2025

The global ocean mixed layer depth derived from an energy approach based on buoyancy work

Efraín Moreles, Emmanuel Romero, Karina Ramos-Musalem, and Leonardo Tenorio-Fernandez

Ocean Sci., 21, 2019–2039, https://doi.org/10.5194/os-21-2019-2025,https://doi.org/10.5194/os-21-2019-2025, 2025

Short summary

Efraín Moreles, Emmanuel Romero, Karina Ramos-Musalem, and Leonardo Tenorio-Fernandez

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2024-4079', Brandon Reichl, 21 Feb 2025
Review of “The global ocean mixed layer depth derived from an energy approach” by Moreles et al.
Review by Brandon Reichl
Summary
In this manuscript the authors propose to define ocean surface mixed layer depth by computing the work required to move a water parcel from the mixed layer depth against gravity (buoyancy work, or WB in the manuscript). The authors argue that this WB approach is advantageous to a density threshold method because it links to physical quantities and dynamical forces in the ocean boundary layer. This work then proposes a value for the integrated buoyancy work that it argues yields accurate estimates of the MLD throughout the Pacific Ocean. The resulting MLD from WB is then compared to a subset of other existing MLD estimates in certain (“challenging”) regions, and finally a new MLD climatology is analyzed and discussed.
The MLD is not an objective quantity, it requires defining timescales and spatial scales since the ocean surface boundary layer is under consistent and variable external forcing. Furthermore, identifying a mixed layer invokes some qualitative analysis to quantify what is meant by “mixed”. This ambiguity makes the present goal of deriving a “global ocean” mixed layer depth challenging, and I acknowledge the efforts the authors have devoted to this matter. I find the proposed WB method for estimating MLD is potentially interesting, but the paper presently relies heavily on ad-hoc and empirical arguments. This limitation makes it difficult to judge the value of the WB approach versus other approaches. The study would therefore benefit significantly if it can identify quantifiable metrics to bolster its claims.
The desire for a MLD metric led to proposing PE anomaly in the Reichl et al. (2022) analysis of MLDs, as it quantifies the energetic distance of a column of sea water from being well mixed. This naturally leads to the PE anomaly as a possible basis for identifying the MLD, but it is not obvious that the same energetic distance should define the MLD in all regions or on all timescales. WB itself may provide another interesting metric for quantifying mixed layers, but it differs in important ways from the PE anomaly. WB provides an integrated measure of stratification of the column by considering the displacement of a particle from the mixed layer base. This yields some insight into the stratification, but less information than if you evaluated the buoyancy work associated with displacement of all particles within the mixed layer. It is also not obvious that a single WB quantity should define the MLD at all locations and on all timescales.
The WB method may offer some physical insights into when and why different MLD methods differ, and it is closely related to the threshold method (see my 2nd comment under "Other Comments") such that it may better ground the threshold values used in present studies. My overall opinion of this article is that there is potential to provide an interesting perspective on MLD identification in the ocean based on these methods, but presently the gaps in metrics and presentation are significant, as detailed below.
Major concerns
The WB method for estimating MLD has merit over a density threshold method since it invokes a dynamical quantity. However, I am unsure that this dynamical quantity truly addresses two main shortcomings of the threshold method. It would help if this paper better identified what it does and doesn’t improve upon other methods.
WB does not offer any physical guidance for choosing a threshold value, so the threshold remains ad-hoc. The paper argues that 20 J/m3 is a universally applicable value, but this was derived empirically based on arguments about its vertical gradient over limited regions of the ocean. There is no physical significance offered for the integrated buoyancy work of 20 J/m3, which would justify it from first principles for identifying the upper ocean mixed layer in all seasons and locations.

The new method is still sensitive to the details of the choice of the threshold value and the reference depth. Interestingly though, some of this sensitivity is reduced since WB is an integrated quantity and therefore less responsive to noise in a profile.

I did not agree with the claim in the text: “There is a correspondence between our methodology and that of Reichl et al. (2022), suggesting that our energy-based methodology is consistent with the turbulence approach of the mixed layer formation”. I do not find the correspondence between the WB and PE anomaly to be obvious, other than both using energy based criteria. PE anomaly is a column integrated energy value that quantifies the distance of the column from being perfectly homogenous, hence it has units of J/m2. WB only considers the energetics of a single parcel advected through the column, hence it has units of J/m3. One could construct different idealized profiles that give the same WB MLD yet have different PE anomalies, which was identified as a shortcoming of other MLD methods in Reichl et al. 2022. Perhaps the WB method has less PE anomaly sensitivity than other methods, but an evaluation of WB in terms of PE anomaly was not attempted in this paper. These important differences from the PE anomaly do not support the statement that this method connects to boundary layer turbulence in the same way as argued for PE anomaly.

The article claims the WB MLD with an energy value of 20 J/m3 is “accurate, robust, and of global applicability”. This claim is based on looking at “challenging regions”, which are defined where a suite of existing methods disagree with each other in MLD magnitude. One issue is that this approach appears to weight the analysis to deep MLD regions (usually convective regions), but this appears to deemphasize shallow MLD regions (e.g., the Arctic Ocean (more on this below) and summer time mixed layers in biologically productive regions) that are just as valuable. Furthermore, I did not find a convincing and quantitative argument in section 3.2.2 for why the WB method estimates a better and/or more consistent MLD than other methods in these regions; it mostly relies on ad-hoc arguments.

I found the analysis of the climatology lacked a significant new/novel result that advances beyond previous work on mixed layer climatologies. It would be useful if issues in previous climatologies could be identified and to discuss why the WB method was able to provide a more meaningful estimate of a MLD climatology (or for what contexts it may be more meaningful).

The article states that it offers globally applicable mixed layer depths and provides a mixed layer depth climatology. Constructing a gridded mixed layer depth climatology requires many decisions, and little technical detail on how this is done was provided. Nor is the resulting climatology data made available, which would severely limit any potential impacts of this work. The climatology is produced only using Argo data, which neglects a lot of additional oceanic profile data that exists. This also leads to coastal oceans and the entire Arctic Ocean being absent from this work.

Other Comments
What reference pressure is used to define the potential density in equation 3? Can a fixed reference pressure be used as a suitable approximation?

One can rewrite the WB criteria (equation 3, by rewriting the 2nd terms RHS w/ H defined as the MLD and _Hmean defined as the depth integral mean of potential density over the MLD): WB = g*H*[_Hmean - rho_ref], which is now expressed similarly to the threshold method: delta = [rho(z) - rho_ref]. This makes two notable differences of WB from the threshold method more apparent: (1) the density difference is now from the average of the density over the column instead of at the MLD, and (2) the density difference is effectively weighted by the MLD (and gravity). These have some interesting implications: (1) integrating the potential density makes the MLD less sensitive to noise in the profile, and (2) weighting the threshold by the MLD reduces the depth of deeper MLDs with the same energy value. The first effect seems advantageous for practical use. The second effect makes sense from the perspective that it takes more energy to raise a parcel from a deeper depth, but since deeper MLDs often experience more turbulence and mixing, it is not straightforward to me that the second effect is obviously “better” for MLD identification. The ocean does not experience uniform levels of energy inputs to turbulence. (This is a main reason I am not convinced there should be a single universal energy value for either the WB method or the PE anomaly method.) Furthermore, the 2nd point above about MLD weighting also offers an interpretation of why shallower depths are associated with larger density differences, e.g. Figure 7.

L336: “It is also easy to implement numerically.” < Easy compared to what?
Citation: https://doi.org/10.5194/egusphere-2024-4079-RC1
- AC1: 'Reply on RC1', Efrain Moreles, 28 Apr 2025
  
  Dear Brandon Reichl,
  Thank you very much for your constructive comments and feedback on our manuscript! Please refer to the author's response document in the attached file for the revision. We also prepared a marked-up manuscript version highlighting the changes made to the earlier version of the manuscript.
  
  Citation: https://doi.org/10.5194/egusphere-2024-4079-AC1
RC2:
'Comment on egusphere-2024-4079', H. Giordani, 24 Feb 2025

The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2024-4079/egusphere-2024-4079-RC2-supplement.pdf

Citation: https://doi.org/10.5194/egusphere-2024-4079-RC2
- AC3: 'Reply on RC2', Efrain Moreles, 28 Apr 2025
  
  Dear H. Giordani,
  Thank you very much for your constructive comments and feedback on our manuscript! Please refer to the author's response document in the attached file for the revision. We also prepared a marked-up manuscript version highlighting the changes made to the earlier version of the manuscript.
  
  Citation: https://doi.org/10.5194/egusphere-2024-4079-AC3
RC3:
'Comment on egusphere-2024-4079', Anonymous Referee #3, 24 Mar 2025
Review of “The global ocean mixed layer depth derived from an energy approach” by Moreles et al.
The mixed layer is an important oceanographic parameter, playing a significant role in understanding upper-ocean dynamics and air-sea interactions. Therefore, accurately calculating the mixed layer depth (MLD) is crucial. Over time, many researchers have proposed various methods to estimate the MLD, which can generally be categorized into three types: threshold methods, energy-based methods, and geometric shape methods. However, due to the very nature of the mixed layer, there is no perfect method. How can we quantitatively define "mixing"? It always requires some reference values, and the use of such reference values inherently leads to spatial or temporal dependence in the applicability of different methods.
This work approaches the problem from an energy perspective, proposing a new method for calculating MLD based on the amount of buoyancy work. The authors demonstrate through practical applications that this method performs reasonably well. Overall, I find this to be a very interesting study. It not only helps us better understand the mixed layer, but also provides an additional option for calculating its depth. However, I have several comments as follows:
I believe the authors may have overstated the performance of their proposed method. In the abstract, they claim that this method provides “a robust criterion based on physical principles.” However, it should be noted that: 1) The authors still use a threshold value (20 J/m³) as the criterion for defining the mixed layer. This value is derived only from a few observational sections, and whether it is applicable globally remains questionable. 2) The authors’ evaluation of “good” or “bad” performance seems to focus on regions where traditional methods do not perform well. Is this method better across all seasons and global ocean regions? In areas where density profiles change gradually, the MLD itself is inherently ambiguous—how should a “good” standard be defined in such cases? 3) The authors mention in the introduction that “in regions with vertically compensated layers, the density threshold may overestimate the MLD.” Can the proposed method in this paper avoid this issue? Considering that the method is still based on density, it seems unlikely that this problem can be completely avoided.

Although the authors express the buoyancy work integral in the form of Equation 3, in essence, this equation is the depth integration of density anomalies. This means the method still heavily depends on the choice of threshold value—especially in summer with shallow mixed layers when density increases slowly with depth. In such cases, different thresholds could lead to significantly different results. I recommend the authors conduct a deeper discussion and analysis of the limitations of their method, rather than focusing solely on its advantages.

Although there is no universally accepted standard method for determining MLD, there is a parameter that can reflect the effectiveness of a given method to some extent—the Quality Index (https://doi.org/10.1029/2003JC002157). I suggest the authors consider using this parameter to evaluate the performance of their method.
Citation: https://doi.org/10.5194/egusphere-2024-4079-RC3
- AC2: 'Reply on RC3', Efrain Moreles, 28 Apr 2025
  
  Dear Anonymous Referee,
  Thank you very much for your constructive comments and feedback on our manuscript! Please refer to the author's response document in the attached file for the revision. We also prepared a marked-up manuscript version highlighting the changes made to the earlier version of the manuscript.
  
  Citation: https://doi.org/10.5194/egusphere-2024-4079-AC2

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2024-4079', Brandon Reichl, 21 Feb 2025
Review of “The global ocean mixed layer depth derived from an energy approach” by Moreles et al.
Review by Brandon Reichl
Summary
In this manuscript the authors propose to define ocean surface mixed layer depth by computing the work required to move a water parcel from the mixed layer depth against gravity (buoyancy work, or WB in the manuscript). The authors argue that this WB approach is advantageous to a density threshold method because it links to physical quantities and dynamical forces in the ocean boundary layer. This work then proposes a value for the integrated buoyancy work that it argues yields accurate estimates of the MLD throughout the Pacific Ocean. The resulting MLD from WB is then compared to a subset of other existing MLD estimates in certain (“challenging”) regions, and finally a new MLD climatology is analyzed and discussed.
The MLD is not an objective quantity, it requires defining timescales and spatial scales since the ocean surface boundary layer is under consistent and variable external forcing. Furthermore, identifying a mixed layer invokes some qualitative analysis to quantify what is meant by “mixed”. This ambiguity makes the present goal of deriving a “global ocean” mixed layer depth challenging, and I acknowledge the efforts the authors have devoted to this matter. I find the proposed WB method for estimating MLD is potentially interesting, but the paper presently relies heavily on ad-hoc and empirical arguments. This limitation makes it difficult to judge the value of the WB approach versus other approaches. The study would therefore benefit significantly if it can identify quantifiable metrics to bolster its claims.
The desire for a MLD metric led to proposing PE anomaly in the Reichl et al. (2022) analysis of MLDs, as it quantifies the energetic distance of a column of sea water from being well mixed. This naturally leads to the PE anomaly as a possible basis for identifying the MLD, but it is not obvious that the same energetic distance should define the MLD in all regions or on all timescales. WB itself may provide another interesting metric for quantifying mixed layers, but it differs in important ways from the PE anomaly. WB provides an integrated measure of stratification of the column by considering the displacement of a particle from the mixed layer base. This yields some insight into the stratification, but less information than if you evaluated the buoyancy work associated with displacement of all particles within the mixed layer. It is also not obvious that a single WB quantity should define the MLD at all locations and on all timescales.
The WB method may offer some physical insights into when and why different MLD methods differ, and it is closely related to the threshold method (see my 2nd comment under "Other Comments") such that it may better ground the threshold values used in present studies. My overall opinion of this article is that there is potential to provide an interesting perspective on MLD identification in the ocean based on these methods, but presently the gaps in metrics and presentation are significant, as detailed below.
Major concerns
The WB method for estimating MLD has merit over a density threshold method since it invokes a dynamical quantity. However, I am unsure that this dynamical quantity truly addresses two main shortcomings of the threshold method. It would help if this paper better identified what it does and doesn’t improve upon other methods.
WB does not offer any physical guidance for choosing a threshold value, so the threshold remains ad-hoc. The paper argues that 20 J/m3 is a universally applicable value, but this was derived empirically based on arguments about its vertical gradient over limited regions of the ocean. There is no physical significance offered for the integrated buoyancy work of 20 J/m3, which would justify it from first principles for identifying the upper ocean mixed layer in all seasons and locations.

The new method is still sensitive to the details of the choice of the threshold value and the reference depth. Interestingly though, some of this sensitivity is reduced since WB is an integrated quantity and therefore less responsive to noise in a profile.

I did not agree with the claim in the text: “There is a correspondence between our methodology and that of Reichl et al. (2022), suggesting that our energy-based methodology is consistent with the turbulence approach of the mixed layer formation”. I do not find the correspondence between the WB and PE anomaly to be obvious, other than both using energy based criteria. PE anomaly is a column integrated energy value that quantifies the distance of the column from being perfectly homogenous, hence it has units of J/m2. WB only considers the energetics of a single parcel advected through the column, hence it has units of J/m3. One could construct different idealized profiles that give the same WB MLD yet have different PE anomalies, which was identified as a shortcoming of other MLD methods in Reichl et al. 2022. Perhaps the WB method has less PE anomaly sensitivity than other methods, but an evaluation of WB in terms of PE anomaly was not attempted in this paper. These important differences from the PE anomaly do not support the statement that this method connects to boundary layer turbulence in the same way as argued for PE anomaly.

The article claims the WB MLD with an energy value of 20 J/m3 is “accurate, robust, and of global applicability”. This claim is based on looking at “challenging regions”, which are defined where a suite of existing methods disagree with each other in MLD magnitude. One issue is that this approach appears to weight the analysis to deep MLD regions (usually convective regions), but this appears to deemphasize shallow MLD regions (e.g., the Arctic Ocean (more on this below) and summer time mixed layers in biologically productive regions) that are just as valuable. Furthermore, I did not find a convincing and quantitative argument in section 3.2.2 for why the WB method estimates a better and/or more consistent MLD than other methods in these regions; it mostly relies on ad-hoc arguments.

I found the analysis of the climatology lacked a significant new/novel result that advances beyond previous work on mixed layer climatologies. It would be useful if issues in previous climatologies could be identified and to discuss why the WB method was able to provide a more meaningful estimate of a MLD climatology (or for what contexts it may be more meaningful).

The article states that it offers globally applicable mixed layer depths and provides a mixed layer depth climatology. Constructing a gridded mixed layer depth climatology requires many decisions, and little technical detail on how this is done was provided. Nor is the resulting climatology data made available, which would severely limit any potential impacts of this work. The climatology is produced only using Argo data, which neglects a lot of additional oceanic profile data that exists. This also leads to coastal oceans and the entire Arctic Ocean being absent from this work.

Other Comments
What reference pressure is used to define the potential density in equation 3? Can a fixed reference pressure be used as a suitable approximation?

One can rewrite the WB criteria (equation 3, by rewriting the 2nd terms RHS w/ H defined as the MLD and _Hmean defined as the depth integral mean of potential density over the MLD): WB = g*H*[_Hmean - rho_ref], which is now expressed similarly to the threshold method: delta = [rho(z) - rho_ref]. This makes two notable differences of WB from the threshold method more apparent: (1) the density difference is now from the average of the density over the column instead of at the MLD, and (2) the density difference is effectively weighted by the MLD (and gravity). These have some interesting implications: (1) integrating the potential density makes the MLD less sensitive to noise in the profile, and (2) weighting the threshold by the MLD reduces the depth of deeper MLDs with the same energy value. The first effect seems advantageous for practical use. The second effect makes sense from the perspective that it takes more energy to raise a parcel from a deeper depth, but since deeper MLDs often experience more turbulence and mixing, it is not straightforward to me that the second effect is obviously “better” for MLD identification. The ocean does not experience uniform levels of energy inputs to turbulence. (This is a main reason I am not convinced there should be a single universal energy value for either the WB method or the PE anomaly method.) Furthermore, the 2nd point above about MLD weighting also offers an interpretation of why shallower depths are associated with larger density differences, e.g. Figure 7.

L336: “It is also easy to implement numerically.” < Easy compared to what?
Citation: https://doi.org/10.5194/egusphere-2024-4079-RC1
- AC1: 'Reply on RC1', Efrain Moreles, 28 Apr 2025
  
  Dear Brandon Reichl,
  Thank you very much for your constructive comments and feedback on our manuscript! Please refer to the author's response document in the attached file for the revision. We also prepared a marked-up manuscript version highlighting the changes made to the earlier version of the manuscript.
  
  Citation: https://doi.org/10.5194/egusphere-2024-4079-AC1
RC2:
'Comment on egusphere-2024-4079', H. Giordani, 24 Feb 2025

The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2024-4079/egusphere-2024-4079-RC2-supplement.pdf

Citation: https://doi.org/10.5194/egusphere-2024-4079-RC2
- AC3: 'Reply on RC2', Efrain Moreles, 28 Apr 2025
  
  Dear H. Giordani,
  Thank you very much for your constructive comments and feedback on our manuscript! Please refer to the author's response document in the attached file for the revision. We also prepared a marked-up manuscript version highlighting the changes made to the earlier version of the manuscript.
  
  Citation: https://doi.org/10.5194/egusphere-2024-4079-AC3
RC3:
'Comment on egusphere-2024-4079', Anonymous Referee #3, 24 Mar 2025
Review of “The global ocean mixed layer depth derived from an energy approach” by Moreles et al.
The mixed layer is an important oceanographic parameter, playing a significant role in understanding upper-ocean dynamics and air-sea interactions. Therefore, accurately calculating the mixed layer depth (MLD) is crucial. Over time, many researchers have proposed various methods to estimate the MLD, which can generally be categorized into three types: threshold methods, energy-based methods, and geometric shape methods. However, due to the very nature of the mixed layer, there is no perfect method. How can we quantitatively define "mixing"? It always requires some reference values, and the use of such reference values inherently leads to spatial or temporal dependence in the applicability of different methods.
This work approaches the problem from an energy perspective, proposing a new method for calculating MLD based on the amount of buoyancy work. The authors demonstrate through practical applications that this method performs reasonably well. Overall, I find this to be a very interesting study. It not only helps us better understand the mixed layer, but also provides an additional option for calculating its depth. However, I have several comments as follows:
I believe the authors may have overstated the performance of their proposed method. In the abstract, they claim that this method provides “a robust criterion based on physical principles.” However, it should be noted that: 1) The authors still use a threshold value (20 J/m³) as the criterion for defining the mixed layer. This value is derived only from a few observational sections, and whether it is applicable globally remains questionable. 2) The authors’ evaluation of “good” or “bad” performance seems to focus on regions where traditional methods do not perform well. Is this method better across all seasons and global ocean regions? In areas where density profiles change gradually, the MLD itself is inherently ambiguous—how should a “good” standard be defined in such cases? 3) The authors mention in the introduction that “in regions with vertically compensated layers, the density threshold may overestimate the MLD.” Can the proposed method in this paper avoid this issue? Considering that the method is still based on density, it seems unlikely that this problem can be completely avoided.

Although the authors express the buoyancy work integral in the form of Equation 3, in essence, this equation is the depth integration of density anomalies. This means the method still heavily depends on the choice of threshold value—especially in summer with shallow mixed layers when density increases slowly with depth. In such cases, different thresholds could lead to significantly different results. I recommend the authors conduct a deeper discussion and analysis of the limitations of their method, rather than focusing solely on its advantages.

Although there is no universally accepted standard method for determining MLD, there is a parameter that can reflect the effectiveness of a given method to some extent—the Quality Index (https://doi.org/10.1029/2003JC002157). I suggest the authors consider using this parameter to evaluate the performance of their method.
Citation: https://doi.org/10.5194/egusphere-2024-4079-RC3
- AC2: 'Reply on RC3', Efrain Moreles, 28 Apr 2025
  
  Dear Anonymous Referee,
  Thank you very much for your constructive comments and feedback on our manuscript! Please refer to the author's response document in the attached file for the revision. We also prepared a marked-up manuscript version highlighting the changes made to the earlier version of the manuscript.
  
  Citation: https://doi.org/10.5194/egusphere-2024-4079-AC2

Peer review completion

AR: Author's response | RR: Referee report | ED: Editor decision | EF: Editorial file upload

AR by Efrain Moreles on behalf of the Authors (28 Apr 2025) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (29 Apr 2025) by Anne Marie Treguier

RR by Hervé Giordani (19 May 2025)

RR by Brandon Reichl (23 May 2025)

Suggestions for revision or reasons for rejection

This is my second time reviewing this manuscript, which proposes an alternative definition of the ocean surface mixed layer depth (MLD) based on a buoyancy work diagnostic. I found the revision partially satisfied my previous concerns, but presented several new concerns detailed below.

As stated from the previous version, I find the subject matter interesting, and the alternative buoyancy work “WB” metric MLD definition provides a useful contextualization of differences between traditional simple threshold MLD methods and methods like potential energy (PE) anomaly. However, I think the presentation still needs to better characterize differences between threshold methods, WB, and PE anomaly. I found the paper addresses how WB differs from threshold methods (Section 2.2 is a nice addition in that regard). But it doesn’t really clarify how the WB metric differs from PE anomaly (and how they offer different perspectives).

I found that the tone of this paper implies that WB has provided a new superior metric compared to PE anomaly, threshold methods, and algorithm methods. I personally didn’t find that to be objectively proven, and would suggest adjusting the narrative to more objectively present WB in the context of the other metrics. I therefore still suggest major revisions to the text, with more specific comments below.

General comments

1. As is already noted in the text, equation 1 is valid for a small perturbation with a locally referenced potential density. It is probably true that the error in using a surface reference pressure is usually small, but I did not find the citation provided in response to my previous comment was sufficient to justify why it is adequate to use surface referenced potential density for the purpose of computing buoyancy work over large vertical displacements. There is an error associated with this assumption, it would be prudent to quantify if/when that error matters for these calculations. I would guess that the deepest winter convective mixed layers, e.g., in deep water production regions such as the Labrador Sea, are situations where the error could be significant. (It may be useful to consider the approach we took in Reichl et al., 2022, where we evaluated the conceptually similar error associated with assuming that surface referenced potential density was the homogenized quantity for PE, rather than conservative temperature and absolute salinity.)

2. I spent much time trying to understand the arguments on the connection of WB to boundary layer turbulence that surrounds equations 4-7. I find the text to be vague with cursory arguments.
-Equation 4 (similar to eq 1) is only appropriate for infinitesimal vertical displacements with a locally referenced potential density. There is an error associated with the approximation applying it to large vertical displacements. It should be explained.
-The manipulation to go from equation 6 to equation 7 is not explained/justified.
-Equation 7 is dependent on the definition of the vertical coordinate “z”, one could add an arbitrary constant “delta” to “z” and it changes the answer. Equation 3 is independent of the definition of the vertical coordinate, so this indicates that there is a problem with equation 7 and its interpretation.
-Equation 7 is compared to a 2008 paper by Herrmann et al., where a similar looking equation is presented. However, Herrmann et al. appear to consider bulk mixed layer dynamics in the derivation and cited work. Thus, their N^2(h) is the buoyancy frequency evaluated at the mixed layer depth (“h”) under a bulk mixed layer, and it is not the same as N^2(z) in this paper. This difference is crucial because (a) by using “h” instead of “z” you should drop any dependence on the vertical coordinate, and (b) N^2(h) in Herrmann et al. should be a function of time, the stratification at the MLD base changes as the bulk mixed layer deepens and is not equal to the stratification N^2(z=h) in the initial condition (after some time/entrainment, the uniform density of the fluid within the mixed layer will be lighter due to mixing with the surface relative to the initial density at the depth of the MLD base, hence the density jump across the mixed layer base, N^2(h) in Herrmann et al., is greater than N^2(z=h) in equation 7. Maybe I missed something here (if so it could be explained better in the text), but for this reason I find it problematic to invoke the interpretation of equation 7 having the same meaning as Herrmann et al.’s equation. While the equations look similar, these distinctions seem to be critical for the connection to boundary layer turbulence energetics and, most importantly, mixing energy (that is most relevant for the TKE sink term) vs displacement energy.
-Equation 7 is interpreted to mean the “columnar buoyancy”, or the buoyancy flux required to mix a water column to depth “h”. I think what is important here about columnar buoyancy is that it is a buoyancy flux equivalent for the energy required to mix a water column over depth “h” (with some assumptions about TKE production and dissipation fraction). The relevant part here is not the columnar buoyancy itself, but that it yields an estimate of the energy requirement to homogenize the column.

3. The text claims WB “advances the development of MLD energy-based methodologies”, which might give the impression that the WB metric improves some inadequacy over the PE anomaly metric. I did not find a claim that WB advances any aspect of PE anomaly supported by the presented results. Another impression given by the text is that WB is aligned with boundary layer turbulence dynamics in a similar way to PE anomaly. I think the distinctions between WB and PE anomaly in that regard needs better clarified. While WB does use an energetically formulated criteria and it allows one to connect the threshold family of metrics to column integral aspects of the PE anomaly metric (this seems clear from section 2.2), it is otherwise distinct from PE anomaly in important ways (it measures displacement, rather than homogenization). This is not a criticism of the utility of WB, I think this concept nicely connects the threshold method to more fundamental fluid mechanics concepts via the work associated with particle displacement. But I wish to reemphasize a point from my previous review, that WB is not obviously aligned with turbulent boundary layer physics in the same way as PE anomaly, and thus does not obviously “advance” the PE anomaly method.

We discussed the energy requirement to homogenize a column of seawater (without a bulk mixed layer assumption) in detail in Reichl et al., 2022 (e.g., building to our equation 11, this is discussed by other work before ours, see references on PE anomaly within). If both energy-based approaches give the energetic distance from well-mixed (as indicated at L12, L57, L113, L131, L357, L417, L424, L432, etc.) then PE anomaly and WB should be equivalent metrics, but they are not. I therefore personally think PE anomaly is the energy metric more directly aligned with concepts mixing, at least these differences should be discussed in the text.

4. I found section 2.2 a very useful addition to understand how WB and the threshold methods compare to one another. I found some of the emphasis on “energy” to be confusing though. “Energetically homogenous” is confusing to me, it sounds like the kinetic or potential energy of the column is homogenous? I think this is not the point, it is meant that a particle displaced in the vertical will not feel any net restoring forces along its path? This concept comes up throughout the text so it should be more clearly conveyed.

Specific comments

L173: I didn’t understand the justification for saying it works for “different regions and ocean conditions, such as polar seas, intermediate and deep water formation regions, and barrier and compensated layers”, the remainder of the paragraph seems to indicate why it can struggle in certain scenarios? And makes the point why MLD computations that are globally applicable are difficult (a useful narrative!).

L205: Clarification: Does “ocean variables” mean that potential density, conservative temperature, and absolute salinity are each interpolated individually? So that it is not potential density computed from the interpolation of conservative temperature and absolute salinity?

L355-359: What is meant by physically realistic is unclear here. The threshold method still gives insight into the depth where the density changes by some value. It is a perfectly valid metric in that regard. It is important to clarify instead what the metrics measure relative to each other. I personally find the PE anomaly metric a better indicator of a mixing depth based on the arguments here than WB since the PE anomaly metric directly measures homogenization and WB measures displacement. Why is the concept of PE anomaly not discussed in this context?

L435: Note that you could also compute PE anomaly from rho_0. These arguments about simple formulas and short scripts are entirely subjective. I personally don’t think computing PE directly involved any complex formula, either. In fact, I would argue the calculation of rho_0 itself is much more complex than computing PE, particularly if using one of the high-order fits for the equation of state.

L444: The energy required to homogenize a column of seawater is by definition where the PE anomaly concept comes from. If part of the utility of WB is as a “proxy” for this metric, why not use PE anomaly directly?

L450: The discussions surrounding PE anomaly (based on mixing rho_0) vs “delta PE” (based on mixing conservative temperature and absolute salinity) metrics in Reichl et al. (2022) seem extremely relevant to this conversation.

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RR by Anonymous Referee #3 (24 May 2025)

ED: Reconsider after major revisions (24 May 2025) by Anne Marie Treguier

AR by Efrain Moreles on behalf of the Authors (19 Jun 2025) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (23 Jun 2025) by Anne Marie Treguier

RR by Brandon Reichl (15 Jul 2025)

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The revisions satisfied many of my comments from the previous version.

I have a few remaining comments, I would prefer they are addressed before recommending the manuscript for publication.

General Comments:

1. More precise language could be used throughout the text where WB is referred to only as “energy”, since there can be different energy approaches and considerations (e.g., potential energy anomaly). E.g., a more appropriate title could be “The global ocean mixed layer depth derived from buoyancy work”.

2. I don’t have a general issue stating that the details of the connection to boundary layer turbulence are beyond the scope of this work. However, that language is not always consistently reflected by the text. E.g., in the abstract it says this approach “aligns with turbulent boundary layer dynamics” yet later the claim is only that it is plausible to establish a relationship with the buoyancy flux, and hence to connect with the turbulent kinetic energy budget of the upper ocean. Since it is said the link between WB and turbulence boundary layer dynamics will be explored in future work I think it is more appropriate to leave it at that and to not overemphasize a connection that is only claimed as plausible in the text.

3. I remain skeptical that this work establishes a precise, global and seasonally applicable range for WB. As I’ve mentioned throughout the review process, this is why we were reluctant to propose a global value of PE anomaly in Reichl et al. (2022). But ultimately a range of 12.5-20 J/m3 for WB is given in this work based on most of the ocean which has been profiled by Argo. This is ultimately not that different from the range of 10-25 J/m2 given as a plausible range in Reichl et al. 2022 (one I do not necessarily emphasize, I did not intend our work to come up with a PE anomaly value that reproduced Holte and Talley). So, (A) I personally don’t think there is a significant difference between a range of 10-25 and 12.5-20 (yes it is a factor of 2, but neither range was formally derived). (B) Regardless of point A, the region where the values seem to differ the most from this mean value are the deep water formation regions. This is also where the quality index is lowest. This is one of the most important regions globally for interior water mass formation. I would have been interested for this study to comment more on the values of WB in those regions and its implications in the pursuit of a globally universal value. This might also emphasize regions where WB could provide additional insight beyond the delta threshold established in Section 3.1.

Specific comments:
L47: I suggest to replace gravitational potential energy with potential energy anomaly
Figure 3: Winter MLDs in convection regions approach 500-1000m by most MLD metrics (including WB as commented in the text). I suggest revisiting the colorbar to not saturate out the deepest values, a nonlinear/logarithmic increment may help. Similar for WB in Figure 4 and the saturated regions. These regions are geographically small, but have significant roles in interior ventilation.
L112: “buoyancy losses required to mix” implies causation, but I think this is more of a statement of consistency and not causation? I suggest “buoyancy deficit compared to that of a homogeneous column in potential density”. There seems to be some approximation here in the details, by assuming the potential density of the mixed column is the mean potential density of the unmixed column.
L123: I suggest clarifying it is “potential” density here.
L133: A connection to turbulence and boundary layers to me implies a connection to the mixing layer (by definition). Here WB is explicitly disconnected from the timescales of active mixing. Maybe I’m being too picky, but there seems to be some contradiction here (see also general comment “2” above).
L169: WB_ref defined? I think I understand it, but I suggest writing the equation here to make it clear for the reader.
S3.3: This ignores the deep convection regions, which happen to also be the regions where the quality index figure shows the worst performance (see also general comment “3” above).

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ED: Reconsider after major revisions (15 Jul 2025) by Anne Marie Treguier

AR by Efrain Moreles on behalf of the Authors (25 Jul 2025) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (28 Jul 2025) by Anne Marie Treguier

RR by Brandon Reichl (05 Aug 2025)

ED: Publish as is (11 Aug 2025) by Anne Marie Treguier

AR by Efrain Moreles on behalf of the Authors (13 Aug 2025) Manuscript

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18 Sep 2025

The global ocean mixed layer depth derived from an energy approach based on buoyancy work

Efraín Moreles, Emmanuel Romero, Karina Ramos-Musalem, and Leonardo Tenorio-Fernandez

Ocean Sci., 21, 2019–2039, https://doi.org/10.5194/os-21-2019-2025,https://doi.org/10.5194/os-21-2019-2025, 2025

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Efraín Moreles, Emmanuel Romero, Karina Ramos-Musalem, and Leonardo Tenorio-Fernandez

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Efraín Moreles, Emmanuel Romero, Karina Ramos-Musalem, and Leonardo Tenorio-Fernandez

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The surface mixed layer depth (MLD), where ocean properties are uniform, is key to ocean-atmosphere interactions and ocean dynamics. We propose an energy-based method that defines the MLD using a constant value of buoyancy work that accurately captures the upper ocean's well-mixed layer. This approach is globally and temporally consistent and reliable, improving MLD estimates, aiding ocean model validation, and advancing studies of ocean heat content and vertical exchanges.


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