Determining the depth and pumping speed of the equatorial Ekman layer from surface drifter trajectories

Paldor, Nathan; De-Leon, Yair

doi:https://doi.org/10.5194/egusphere-2025-89

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https://doi.org/10.5194/egusphere-2025-89

Preprints

24 Jan 2025

| 24 Jan 2025

Determining the depth and pumping speed of the equatorial Ekman layer from surface drifter trajectories

Nathan Paldor and Yair De-Leon

Abstract. Trajectories of more than 500 drogued surface drifters launched since 1979 in the equatorial ocean are analyzed by employing the results of a new Lagrangian theory of wind-driven transport along the equator forced by the prevailing Trade winds. The analysis yields robust estimates of 45 meters for the Ekman layer’s depth and 1.0 meters/day for the upwelling speed of deep water into the layer.

Received: 09 Jan 2025 – Discussion started: 24 Jan 2025

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Journal article(s) based on this preprint

22 Jul 2025

Determining the depth and upwelling speed of the equatorial Ekman layer from surface drifter trajectories

Nathan Paldor and Yair De-Leon

Ocean Sci., 21, 1461–1468, https://doi.org/10.5194/os-21-1461-2025,https://doi.org/10.5194/os-21-1461-2025, 2025

Short summary

Nathan Paldor and Yair De-Leon

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2025-89', Anonymous Referee #1, 20 Feb 2025

The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-89/egusphere-2025-89-RC1-supplement.pdf

Citation: https://doi.org/10.5194/egusphere-2025-89-RC1
- AC1: 'Reply on RC1', Nathan Paldor, 26 Mar 2025
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-89/egusphere-2025-89-AC1-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2025-89-AC1
RC2:
'Comment on egusphere-2025-89', Anonymous Referee #2, 21 Feb 2025

Review of Paldor and De-Leon “Determining the depth and pumping speed of the equatorial Ekman layer from surface drifter trajectories”.
Paldor and De-Leon investigate equatorial upwelling velocity by applying a recently developed Ekman theory for the equatorial beta plane to surface drifter data. By fitting the theory to Lagrangian trajectories, they derive an Ekman layer depth and an upwelling vertical velocity. This is a well-conceived and straightforward study that highlights new aspects of equatorial Ekman dynamics.
However, the authors should elaborate further on the underlying theory, the assumptions made, and, in particular, the applicability of drifters with drogues representing velocity at 15 m depth. Additionally, they should discuss the role of stratification. In the equatorial cold tongue regions of the Atlantic and Pacific, mixed layer depths are often extremely shallow (e.g., ~10 m). How would this affect the Ekman layer depth?
A primary concern is that the selection criteria for drifters could introduce a significant bias in the estimated Ekman layer depth and vertical velocity. The authors should also expand their literature review, acknowledging that different studies have reported varying vertical velocities depending on the meridional scale used in their analyses. I will provide a few examples, including Poulain (1993), which reports very high values for small meridional scales.
Overall, the manuscript is well written and suitable for publication after revision. Below are my specific comments:
L54: Minimum potential. More information would be helpful here to make this understandable without referring directly to Paldor (2024).
L101: Did you check for drifters that lost their drouges? Should be noted.
L103: Typically, there is substantial meridional wind, particularly in the equatorial Atlantic. How do you account for this effect? With the selection criteria used, a significant bias could be introduced, as only drifters associated with specific non-Ekman dynamics—such as tropical instability waves, Yanai waves, or meridional wind forcing—are considered. A way to test for such a bias would be to calculate the mean meridional drifter velocity as a function of latitude. It would be interesting to see how this quantity compares to the meridional velocities derived from Ekman theory.
L111: I do not understand this sentence. Please clarify. What does it mean? L is 2° or is L approaching y(0)? Or do you mean if y(0) approches L=2°? Is the erratic behavior a consequence of meridional wind forcing?
L113: Criterion for selection of drifters. Would it be a better criterion to consider the strength of the meridional wind, i.e., to only use drifters in cases of weak meridional wind? Additionally, what is the initial velocity of the drifters—for example, is it influenced by meridional wind forcing or, more importantly, by tropical instability waves, Yanai waves, etc.? Drifters could be deployed under different conditions.
L140: Result of the Ekman layer depth. There must be a strong bias in the mean since you specifically neglect drifters that do not flow in the expected direction or that cross the equator. As a result, you include all drifters that may be driven by other motions away from the equator but exclude those drifting toward it.
What would be the equatorial divergence if calculated from all drifters, i.e., the mean meridional velocity averaged along the equator at different latitudes (3°, 4°, 5°)? If there is a difference, how could it be explained?
L141: Can this be called oscillation free. How do you subtract the oscillation from the drifter velocity? To my understanding this would require the knowledge of the initial velocity.
L142: The derived velocity corresponds to the velocity at the depth of the drifters' drogue. What assumptions do you make about the vertical structure of the Ekman velocity? Would it be useful to also consider Argo drift velocities, which represent surface velocity, to calculate mean meridional velocities? This could provide an estimate of the vertical structure of the Ekman flow near the equator.
L147: I think one of the main results is the dependence of the derived vertical velocity on the meridional scale, which could also explain many previous estimates (see references below). This aspect should be given more emphasis.
References:
Bubnov, V. A., 1987: Vertical motions in the central equatorial Pacific. Oceanol. Acta, Spec.Vol. 6, 15–17.
Gouriou, Y., and G. Reverdin, 1992: Isopycnal and diapycnal circulation of the upper equatorial Atlantic Ocean in 1983–1984. J. Geophys. Res., 97(C3), 3543–3572.
Halpern, D., and H. P. Freitag, 1987: Vertical motion in the upper ocean of the equatorial eastern Pacific. Oceanol. Acta, Spec. Vol. 6, 19–26.
Halpern, D., R. A. Knox, D. S. Luther, S. G. H. Philander, 1989: Estimates of Equatorial Upwelling Between 140° and 110°W During 1984. J. Geophys. Res., 94, 8018-8020.
Hansen, D. V., and C. A. Paul, 1984: Genesis and effects of long waves in the equatorial Pacific, J. Geophys. Res., 89, 10,431–10,440.
Hansen, D. V., and C. A. Paul, 1987: Vertical motion in the eastern equatorial Pacific inferred from drifting buoys. Oceanol. Acta, Spec. Vol. 6, 27–32.
Poulain, P.-M., 1993: Estimates of Horizontal Divergence and Vertical Velocity in the Equatorial Pacific. J. Phys. Oceanogr., 23, 601–607.
Quay, P. D., M. Stuiver, and W. S. Broecker, 1983: Upwelling rates for the equatorial Pacific Ocean derived from the bomb 14C distribution. J. Mar. Res., 41, 769–792.

Citation: https://doi.org/10.5194/egusphere-2025-89-RC2
- AC2: 'Reply on RC2', Nathan Paldor, 26 Mar 2025
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-89/egusphere-2025-89-AC2-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2025-89-AC2

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2025-89', Anonymous Referee #1, 20 Feb 2025

The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-89/egusphere-2025-89-RC1-supplement.pdf

Citation: https://doi.org/10.5194/egusphere-2025-89-RC1
- AC1: 'Reply on RC1', Nathan Paldor, 26 Mar 2025
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-89/egusphere-2025-89-AC1-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2025-89-AC1
RC2:
'Comment on egusphere-2025-89', Anonymous Referee #2, 21 Feb 2025

Review of Paldor and De-Leon “Determining the depth and pumping speed of the equatorial Ekman layer from surface drifter trajectories”.
Paldor and De-Leon investigate equatorial upwelling velocity by applying a recently developed Ekman theory for the equatorial beta plane to surface drifter data. By fitting the theory to Lagrangian trajectories, they derive an Ekman layer depth and an upwelling vertical velocity. This is a well-conceived and straightforward study that highlights new aspects of equatorial Ekman dynamics.
However, the authors should elaborate further on the underlying theory, the assumptions made, and, in particular, the applicability of drifters with drogues representing velocity at 15 m depth. Additionally, they should discuss the role of stratification. In the equatorial cold tongue regions of the Atlantic and Pacific, mixed layer depths are often extremely shallow (e.g., ~10 m). How would this affect the Ekman layer depth?
A primary concern is that the selection criteria for drifters could introduce a significant bias in the estimated Ekman layer depth and vertical velocity. The authors should also expand their literature review, acknowledging that different studies have reported varying vertical velocities depending on the meridional scale used in their analyses. I will provide a few examples, including Poulain (1993), which reports very high values for small meridional scales.
Overall, the manuscript is well written and suitable for publication after revision. Below are my specific comments:
L54: Minimum potential. More information would be helpful here to make this understandable without referring directly to Paldor (2024).
L101: Did you check for drifters that lost their drouges? Should be noted.
L103: Typically, there is substantial meridional wind, particularly in the equatorial Atlantic. How do you account for this effect? With the selection criteria used, a significant bias could be introduced, as only drifters associated with specific non-Ekman dynamics—such as tropical instability waves, Yanai waves, or meridional wind forcing—are considered. A way to test for such a bias would be to calculate the mean meridional drifter velocity as a function of latitude. It would be interesting to see how this quantity compares to the meridional velocities derived from Ekman theory.
L111: I do not understand this sentence. Please clarify. What does it mean? L is 2° or is L approaching y(0)? Or do you mean if y(0) approches L=2°? Is the erratic behavior a consequence of meridional wind forcing?
L113: Criterion for selection of drifters. Would it be a better criterion to consider the strength of the meridional wind, i.e., to only use drifters in cases of weak meridional wind? Additionally, what is the initial velocity of the drifters—for example, is it influenced by meridional wind forcing or, more importantly, by tropical instability waves, Yanai waves, etc.? Drifters could be deployed under different conditions.
L140: Result of the Ekman layer depth. There must be a strong bias in the mean since you specifically neglect drifters that do not flow in the expected direction or that cross the equator. As a result, you include all drifters that may be driven by other motions away from the equator but exclude those drifting toward it.
What would be the equatorial divergence if calculated from all drifters, i.e., the mean meridional velocity averaged along the equator at different latitudes (3°, 4°, 5°)? If there is a difference, how could it be explained?
L141: Can this be called oscillation free. How do you subtract the oscillation from the drifter velocity? To my understanding this would require the knowledge of the initial velocity.
L142: The derived velocity corresponds to the velocity at the depth of the drifters' drogue. What assumptions do you make about the vertical structure of the Ekman velocity? Would it be useful to also consider Argo drift velocities, which represent surface velocity, to calculate mean meridional velocities? This could provide an estimate of the vertical structure of the Ekman flow near the equator.
L147: I think one of the main results is the dependence of the derived vertical velocity on the meridional scale, which could also explain many previous estimates (see references below). This aspect should be given more emphasis.
References:
Bubnov, V. A., 1987: Vertical motions in the central equatorial Pacific. Oceanol. Acta, Spec.Vol. 6, 15–17.
Gouriou, Y., and G. Reverdin, 1992: Isopycnal and diapycnal circulation of the upper equatorial Atlantic Ocean in 1983–1984. J. Geophys. Res., 97(C3), 3543–3572.
Halpern, D., and H. P. Freitag, 1987: Vertical motion in the upper ocean of the equatorial eastern Pacific. Oceanol. Acta, Spec. Vol. 6, 19–26.
Halpern, D., R. A. Knox, D. S. Luther, S. G. H. Philander, 1989: Estimates of Equatorial Upwelling Between 140° and 110°W During 1984. J. Geophys. Res., 94, 8018-8020.
Hansen, D. V., and C. A. Paul, 1984: Genesis and effects of long waves in the equatorial Pacific, J. Geophys. Res., 89, 10,431–10,440.
Hansen, D. V., and C. A. Paul, 1987: Vertical motion in the eastern equatorial Pacific inferred from drifting buoys. Oceanol. Acta, Spec. Vol. 6, 27–32.
Poulain, P.-M., 1993: Estimates of Horizontal Divergence and Vertical Velocity in the Equatorial Pacific. J. Phys. Oceanogr., 23, 601–607.
Quay, P. D., M. Stuiver, and W. S. Broecker, 1983: Upwelling rates for the equatorial Pacific Ocean derived from the bomb 14C distribution. J. Mar. Res., 41, 769–792.

Citation: https://doi.org/10.5194/egusphere-2025-89-RC2
- AC2: 'Reply on RC2', Nathan Paldor, 26 Mar 2025
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-89/egusphere-2025-89-AC2-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2025-89-AC2

Peer review completion

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

AR by Nathan Paldor on behalf of the Authors (27 Mar 2025) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (04 Apr 2025) by Benjamin Rabe

RR by Anonymous Referee #1 (14 Apr 2025)

RR by Anonymous Referee #2 (19 Apr 2025)

Suggestions for revision or reasons for rejection

Second review of Paldor and De-Leon “Determining the depth and upwelling speed of the equatorial Ekman layer from surface drifter trajectories”.

The authors take into account most of the points raised in the reviews. I have still some points that should be taken into account to enhance the presentation of this interesting topic. I think the manuscript can be accepted after minor revision.

Minor points:
L19: 500-600: more than 500? In the text you wrote ~700, in table 1 531-610.

L27: deep water: deep water is a terminus used for water masses of the deep ocean. Better would be here "thermocline water".

L30: Hadley: Hadley circulation refers to meridional winds, while the Walker circulation refers to the zonal winds at the equator.

L56: avoid repeating “filters out”: method () to separate fast …

L58: meaning of “general” unclear, can be removed

L63: Why is there an underscore below U? What is the meaning?

L69: difficult to quantify instead of hard to quantify

L71: A similar result, but from shipboard measurements, was derived by Johnson et al. (2001), https://doi.org/10.1175/1520-0485(2001)031<0839:EPOHVD>2.0.CO;2

L100: estimations … estimated, better estimations … derived

L118: initial westward directed velocity: Should one note that "unbalanced flow" is meant here to differentiate from westward flow associated with equatorial Kelvin and Rossby waves?

L124: comma missing after (water density)

L132: I would suggest removing “when the inertial oscillations shown Fig. 1a are filtered out” as it is a repetition.

L152f: I still do not fully understand this criterion. What do you do with drifters deployed at the equator, such as in Hans et al. (2024, https://doi.org/10.1029/2023JC020870), their Fig. 1? Are such drifters, deployed at the equator, included or not? Would it make sense to count all drifters that reach a position near the equator and subsequently move poleward, regardless of their prior trajectory, including whether they may have crossed the equator earlier or not?

L155: launch position: Here, I would suggest to define a starting position for the calculation instead of a launch position. Anyway, taken the launch position does not yield zero meridional velocity at the trajectory start.

L158f: No information …: It might not be used here, but the Global Drifter Program (GDP) flags the presence or absence of drogues in its dataset and information is available. Might not be the case for older drifter. See, e.g. https://doi.org/10.5194/essd-13-645-2021

L171: If I understand correctly only mean tau_x is used instead of tau_x calculated for each drifter separately (i.e. averaged over the period of drift and longitude range) and then averaged for all drifters. This should be stated clearly right from the beginning. Which longitude ranges and periods are used for averaging?

L195: “these values” are unclear: individual Ekman layer depths?

L197: How is H estimated, just as an arithmetic mean? As the distribution is skewed, one could use other statistical expression such as the median. What is the difference between mean and median?

L223: Tau_x particularly varies with time

L229: whereas …: This is not true for inertial oscillations on a beta-plane, see e.g., Ripa (1997, https://doi.org/10.1175/1520-0485(1997)027<0633:IOATPA>2.0.CO;2)

L237: I don’t know how the given equation for the period fit to Figure 1a showing 3 oscillations in 80 days for zero wind stress.

L245f: shallower with …: Is the Ekman layer indicated in Fig. 4 (black line in lower panel)? If this black line indicates the Ekman layer depth, the Ekman layer becomes deeper with distance from the equator. Otherwise please include the Ekman layer depth in the lower panel of Figure 4.

L256: Here the assumption should be mentioned that the drifter velocity at 15 m is representative for the whole Ekman layer.

L258: spatial and temporal changes of zonal wind stress

Hide

ED: Publish subject to minor revisions (review by editor) (22 Apr 2025) by Benjamin Rabe

AR by Nathan Paldor on behalf of the Authors (23 Apr 2025) Author's response Author's tracked changes Manuscript

ED: Publish as is (29 Apr 2025) by Benjamin Rabe

AR by Nathan Paldor on behalf of the Authors (29 Apr 2025) Manuscript

Journal article(s) based on this preprint

22 Jul 2025

Determining the depth and upwelling speed of the equatorial Ekman layer from surface drifter trajectories

Nathan Paldor and Yair De-Leon

Ocean Sci., 21, 1461–1468, https://doi.org/10.5194/os-21-1461-2025,https://doi.org/10.5194/os-21-1461-2025, 2025

Short summary

Nathan Paldor and Yair De-Leon

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