Extension of Ekman (1905) wind-driven transport theory to the &beta;-plane

Paldor, Nathan; Friedland, Lazar

doi:https://doi.org/10.5194/egusphere-2022-831

Preprints

https://doi.org/10.5194/egusphere-2022-831

Preprints

31 Aug 2022

| 31 Aug 2022

Extension of Ekman (1905) wind-driven transport theory to the β-plane

Nathan Paldor and Lazar Friedland

Abstract. The seminal, Ekman (1905)’s, f-plane theory of wind driven transport at the ocean surface is extended to the β-plane by substituting the pseudo angular momentum for the zonal velocity in the Lagrangian equation. The addition of the β term implies that equations become nonlinear, which greatly complicates the analysis. Though rotation relates the momentum equations in the zonal and the meridional directions, the transformation to pseudo angular momentum greatly simplifies the longitudinal dynamics, which yields a clear description of the meridional dynamics in terms of a slow drift compounded by fast oscillations, which can then be applied to describe the motion in the zonal direction. Both analytical expressions and numerical calculations underscore the critical role of the equator in determining the trajectories of water columns forced by eastward directed wind stress even when the water columns are far from the equator. Our results demonstrate that the averaged motion in the zonal direction is highly dependent on the meridional oscillations and for some initial conditions can be as large as the meridional mean motion.

Received: 24 Aug 2022 – Discussion started: 31 Aug 2022

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Journal article(s) based on this preprint

24 Jan 2023

Extension of Ekman (1905) wind-driven transport theory to the β plane

Nathan Paldor and Lazar Friedland

Ocean Sci., 19, 93–100, https://doi.org/10.5194/os-19-93-2023,https://doi.org/10.5194/os-19-93-2023, 2023

Short summary

Nathan Paldor and Lazar Friedland

Interactive discussion

Status: closed

CC1:
'Comment on egusphere-2022-831', Adrian Constantin, 15 Sep 2022

On page 1, it should be pointed out that effects beyond the equatorial f-plane approximation were also recently considered in the paper
https://aip.scitation.org/doi/full/10.1063/1.5083088
This enforces actually the importance of the present study, since while the above paper is more general as it
deals with Ekman flows in spherical coordinates, the approach pursued in this preprint takes advantage of the
simpler structure of the beta-plane approximation to obtain more detailed information about the dynamics.

Citation: https://doi.org/10.5194/egusphere-2022-831-CC1
- RC1:
  'Reply on CC1', Anonymous Referee #1, 17 Sep 2022
  
  The authors discuss the extension of Ekman's classical wind-drift solution to the beta-plane.
  
  Of particular interest is the equatorial case. The authors use a mixture of the analytical and numerical
  
  approaches to show, among other results, that the averaged motion in the zonal direction is highly
  
  dependent on the meridional oscillations and for some initial conditions can be as large as the
  
  meridional mean motion. The paper is well-written and definitely of great interest. I strongly recommend
  
  acceptance after a minor revision. The author should also point out that extensions of the
  
  classical Ekman approach are also available in the papers A. Constantin and R. S. Johnson,
  
  Ekman-type solutions for shallow-water flows on a rotating sphere: A new perspective
  
  on a classical problem, Phys. Fluids 31, 021401 (2019); doi: 10.1063/1.5083088;
  
  M. F. Cronin and W. S. Kessler, Near-Surface Shear Flow in the Tropical Pacific Cold Tongue Front,
  
  J. Phys. Oceanogr. 39 (2009), 1200-1215.
  
  Citation: https://doi.org/10.5194/egusphere-2022-831-RC1
  - AC4: 'Reply on RC1', Nathan Paldor, 14 Nov 2022
    
    We thank the reviewer for the accolates. In the revised version we will reference works in which the transport of the Ekman layer was extended numericaly to sphrical coordinates.
    
    Citation: https://doi.org/10.5194/egusphere-2022-831-AC4
- AC1: 'Reply to RC1 (also CC1)', Nathan Paldor, 07 Nov 2022
  
  We thank the reviewer for the accolates. In the revised version we will reference works in which the transport of the Ekman layer was extended numericaly to sphrical coordinates.
  
  Citation: https://doi.org/10.5194/egusphere-2022-831-AC1
- AC2: 'Reply on CC1', Nathan Paldor, 07 Nov 2022
  
  See our response to RC1
  
  Citation: https://doi.org/10.5194/egusphere-2022-831-AC2
RC2:
'Comment on egusphere-2022-831', Nicolas Grisouard, 25 Oct 2022

Review of `Extension of Ekman (1905) wind-driven transport theory to the beta-plane’ by Paldor and Friedland

This manuscript provides solutions to a vertically-integrated version of the equations of motion on a beta-plane, subjected to constant eastward (or westward) wind stress. I was a bit surprised to find that these solutions had never been derived, but not shocked. Indeed, I have come to realize that Ekman’s theory has not been extended as much as one would think, in light of its fundamental influence on our representation of how the ocean moves. Of course, the model is crude: it is a beta-plane that claims to describe motions as far away as 30º around the equator at least, and the structure of the dominant east- and westerlies that gives the ocean its gyre structure is completely absent. But to tackle these problems, one would have to start with the present analysis.

The derivations are not technically difficult. Rather, the difficulty seems to have lied in casting the equations in a useful form, and to have the appropriate strategy to solve them. Therefore, I enjoyed reading the mathematical developments in section 3, ‘Analysis’.

My main comments are about the presentation: figures could be improved, and the order of the presentation got me confused at times.

A. I was confused by why the authors presented their numerical simulations in figure 2 before solving the equations in section 3. For example, I could not wrap my head around why the red curves in figure 2 oscillated and the blue curves did not. In section 3, I finally understood the idea behind the potential well, and how starting at the bottom of it (y=0) vs. slightly off of it (y=0.05) yielded oscillations in the latter and not the former. Maybe deriving first, and showing solutions later, might help.

B. A point that is probably closely related to my previous one, is that I still do not understand what y=0 represents. It seems easy enough at first: it is the point, around which you start the expansion, and it is located 1/b away from the equator. But when I start digging, I don’t understand how this point was chosen: it is not the initial latitude, and I do not understand why starting from y≠0 yields oscillations and not y=0. If I started at y=0.05, why couldn’t I re-define y=0 and b to be located at the new starting point, and still see oscillations? I believe some magic happens when introducing the pseudo angular momentum (l. 69) and the phrase ‘We note that (…) to the latitude’ (ll. 70-71), but to expand on it could help the reader understand faster.

Some specific comments:

1. Ll. 32-33: the first sentence of point 4 should be rewritten, I understand the general meaning, but I suspect that command that the word ‘that’ are being misused.

2. Figure 1 does not need to be so big, but font sizes would need to be increased.

3. Ll. 42-45: my first reaction was to think that there is no such thing as in the ocean, since the dominant zonal winds alternate latitudinally with the atmospheric circulation cells. You mention this in the discussion, but a word about what this is meant to simplify (or, equivalently, how far away from the equator are winds more or less in the same direction) would be welcome here.

4. Ll. 53-55: I believe that only Gill (1982) covers the time-dependent problem.

5. L. 83: ‘f_0=10^{-4}1/s’ should be ‘f_0=10^{-4}\ s^{-1}’ (unit)

6. L. 84: ‘to b of (1)’ should be ‘to b of O(1)’. Besides, I am not sure what this sentence really brings. At the risk of unfairly exaggerating, I read ‘b=O(1) and Gamma<<1 so the theory should be applicable to b=O(1) and Gamma<<1’. If it is a condition, it is never assessed later.

7. Figure 2 is a bit confusing: two curves have the same colour (green), and I believe that on the left panel, the two green curves superpose. Furthermore, I believe that those green curves are <x> and y_m, which should be specified in the caption or legend. Also, I would flip the order of the panels: placing y before x is counter to standard usage.

8. Like in figure 2, panels in figure 3 should be flipped. Having later times on front of earlier times is also counter to standard usage.

9. L. 166: ‘In the 1’ should be ‘In section 1’ (I think)

10. L. 187: ‘beta’ should be ‘b’ (I think); delete ‘clearly’ at the end of the line (no offence; my personal opinion is that it is for the reader to decide if it is clear).

11. L. 188: ‘illustrates’ should be ‘illustrated’

12. L. 195 and 198: I don’t really see the point of the bold fonts.

Citation: https://doi.org/10.5194/egusphere-2022-831-RC2
- AC3: 'Reply on RC2', Nathan Paldor, 14 Nov 2022
  
  We agree with the reviewer’s general remarks and were happy to read that the reviewer “… enjoyed reading the mathematical development in section 3, ‘Analysis.”.
  As for the two main comments:
  A. We accepted this point. In the revised version the numerical solutions will follow the analysis. In accordance with this change, old Fig. 3 is split into two figures: 3 for the y-dynamics and 4 for the x-dynamics.
  B. We agree – the meaning of y₀ is much subtler (and intriguing) on the b-plane than on the f-plane. In the revised version the numerical calculations will be initiated at the fixed point y₀=0 (and V₀ ≠ 0). As expected – our preliminary calculations show that the change of initial conditions does not affect the general features of the results.
  Specific comments:
  Thank you for your careful reading. All typos and miss-worded phrases will be corrected in the revised version.
  
  Citation: https://doi.org/10.5194/egusphere-2022-831-AC3

Interactive discussion

Status: closed

CC1:
'Comment on egusphere-2022-831', Adrian Constantin, 15 Sep 2022

On page 1, it should be pointed out that effects beyond the equatorial f-plane approximation were also recently considered in the paper
https://aip.scitation.org/doi/full/10.1063/1.5083088
This enforces actually the importance of the present study, since while the above paper is more general as it
deals with Ekman flows in spherical coordinates, the approach pursued in this preprint takes advantage of the
simpler structure of the beta-plane approximation to obtain more detailed information about the dynamics.

Citation: https://doi.org/10.5194/egusphere-2022-831-CC1
- RC1:
  'Reply on CC1', Anonymous Referee #1, 17 Sep 2022
  
  The authors discuss the extension of Ekman's classical wind-drift solution to the beta-plane.
  
  Of particular interest is the equatorial case. The authors use a mixture of the analytical and numerical
  
  approaches to show, among other results, that the averaged motion in the zonal direction is highly
  
  dependent on the meridional oscillations and for some initial conditions can be as large as the
  
  meridional mean motion. The paper is well-written and definitely of great interest. I strongly recommend
  
  acceptance after a minor revision. The author should also point out that extensions of the
  
  classical Ekman approach are also available in the papers A. Constantin and R. S. Johnson,
  
  Ekman-type solutions for shallow-water flows on a rotating sphere: A new perspective
  
  on a classical problem, Phys. Fluids 31, 021401 (2019); doi: 10.1063/1.5083088;
  
  M. F. Cronin and W. S. Kessler, Near-Surface Shear Flow in the Tropical Pacific Cold Tongue Front,
  
  J. Phys. Oceanogr. 39 (2009), 1200-1215.
  
  Citation: https://doi.org/10.5194/egusphere-2022-831-RC1
  - AC4: 'Reply on RC1', Nathan Paldor, 14 Nov 2022
    
    We thank the reviewer for the accolates. In the revised version we will reference works in which the transport of the Ekman layer was extended numericaly to sphrical coordinates.
    
    Citation: https://doi.org/10.5194/egusphere-2022-831-AC4
- AC1: 'Reply to RC1 (also CC1)', Nathan Paldor, 07 Nov 2022
  
  We thank the reviewer for the accolates. In the revised version we will reference works in which the transport of the Ekman layer was extended numericaly to sphrical coordinates.
  
  Citation: https://doi.org/10.5194/egusphere-2022-831-AC1
- AC2: 'Reply on CC1', Nathan Paldor, 07 Nov 2022
  
  See our response to RC1
  
  Citation: https://doi.org/10.5194/egusphere-2022-831-AC2
RC2:
'Comment on egusphere-2022-831', Nicolas Grisouard, 25 Oct 2022

Review of `Extension of Ekman (1905) wind-driven transport theory to the beta-plane’ by Paldor and Friedland

This manuscript provides solutions to a vertically-integrated version of the equations of motion on a beta-plane, subjected to constant eastward (or westward) wind stress. I was a bit surprised to find that these solutions had never been derived, but not shocked. Indeed, I have come to realize that Ekman’s theory has not been extended as much as one would think, in light of its fundamental influence on our representation of how the ocean moves. Of course, the model is crude: it is a beta-plane that claims to describe motions as far away as 30º around the equator at least, and the structure of the dominant east- and westerlies that gives the ocean its gyre structure is completely absent. But to tackle these problems, one would have to start with the present analysis.

The derivations are not technically difficult. Rather, the difficulty seems to have lied in casting the equations in a useful form, and to have the appropriate strategy to solve them. Therefore, I enjoyed reading the mathematical developments in section 3, ‘Analysis’.

My main comments are about the presentation: figures could be improved, and the order of the presentation got me confused at times.

A. I was confused by why the authors presented their numerical simulations in figure 2 before solving the equations in section 3. For example, I could not wrap my head around why the red curves in figure 2 oscillated and the blue curves did not. In section 3, I finally understood the idea behind the potential well, and how starting at the bottom of it (y=0) vs. slightly off of it (y=0.05) yielded oscillations in the latter and not the former. Maybe deriving first, and showing solutions later, might help.

B. A point that is probably closely related to my previous one, is that I still do not understand what y=0 represents. It seems easy enough at first: it is the point, around which you start the expansion, and it is located 1/b away from the equator. But when I start digging, I don’t understand how this point was chosen: it is not the initial latitude, and I do not understand why starting from y≠0 yields oscillations and not y=0. If I started at y=0.05, why couldn’t I re-define y=0 and b to be located at the new starting point, and still see oscillations? I believe some magic happens when introducing the pseudo angular momentum (l. 69) and the phrase ‘We note that (…) to the latitude’ (ll. 70-71), but to expand on it could help the reader understand faster.

Some specific comments:

1. Ll. 32-33: the first sentence of point 4 should be rewritten, I understand the general meaning, but I suspect that command that the word ‘that’ are being misused.

2. Figure 1 does not need to be so big, but font sizes would need to be increased.

3. Ll. 42-45: my first reaction was to think that there is no such thing as in the ocean, since the dominant zonal winds alternate latitudinally with the atmospheric circulation cells. You mention this in the discussion, but a word about what this is meant to simplify (or, equivalently, how far away from the equator are winds more or less in the same direction) would be welcome here.

4. Ll. 53-55: I believe that only Gill (1982) covers the time-dependent problem.

5. L. 83: ‘f_0=10^{-4}1/s’ should be ‘f_0=10^{-4}\ s^{-1}’ (unit)

6. L. 84: ‘to b of (1)’ should be ‘to b of O(1)’. Besides, I am not sure what this sentence really brings. At the risk of unfairly exaggerating, I read ‘b=O(1) and Gamma<<1 so the theory should be applicable to b=O(1) and Gamma<<1’. If it is a condition, it is never assessed later.

7. Figure 2 is a bit confusing: two curves have the same colour (green), and I believe that on the left panel, the two green curves superpose. Furthermore, I believe that those green curves are <x> and y_m, which should be specified in the caption or legend. Also, I would flip the order of the panels: placing y before x is counter to standard usage.

8. Like in figure 2, panels in figure 3 should be flipped. Having later times on front of earlier times is also counter to standard usage.

9. L. 166: ‘In the 1’ should be ‘In section 1’ (I think)

10. L. 187: ‘beta’ should be ‘b’ (I think); delete ‘clearly’ at the end of the line (no offence; my personal opinion is that it is for the reader to decide if it is clear).

11. L. 188: ‘illustrates’ should be ‘illustrated’

12. L. 195 and 198: I don’t really see the point of the bold fonts.

Citation: https://doi.org/10.5194/egusphere-2022-831-RC2
- AC3: 'Reply on RC2', Nathan Paldor, 14 Nov 2022
  
  We agree with the reviewer’s general remarks and were happy to read that the reviewer “… enjoyed reading the mathematical development in section 3, ‘Analysis.”.
  As for the two main comments:
  A. We accepted this point. In the revised version the numerical solutions will follow the analysis. In accordance with this change, old Fig. 3 is split into two figures: 3 for the y-dynamics and 4 for the x-dynamics.
  B. We agree – the meaning of y₀ is much subtler (and intriguing) on the b-plane than on the f-plane. In the revised version the numerical calculations will be initiated at the fixed point y₀=0 (and V₀ ≠ 0). As expected – our preliminary calculations show that the change of initial conditions does not affect the general features of the results.
  Specific comments:
  Thank you for your careful reading. All typos and miss-worded phrases will be corrected in the revised version.
  
  Citation: https://doi.org/10.5194/egusphere-2022-831-AC3

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 (15 Nov 2022) Author's response

EF by Polina Shvedko (16 Nov 2022) Manuscript

EF by Polina Shvedko (16 Nov 2022) Author's tracked changes

ED: Referee Nomination & Report Request started (20 Nov 2022) by Anne Marie Treguier

RR by Anonymous Referee #1 (21 Nov 2022)

RR by Nicolas Grisouard (02 Dec 2022)

RR by Anonymous Referee #3 (02 Dec 2022)

Suggestions for revision or reasons for rejection

While this work has merit as a mathematical exercise, its value in understanding the ocean is less clear.

The problem of inertial oscillations on the rotating earth has already been solved cleanly and exactly by Early (2012), following Ripa (1997), see eqn. (16) therein. These equations are simpler than those presented in the paper under review, as well as being free from questionable approximations. Moreover, unlike the equations in this paper which require numerical integration, the solution on the rotating earth have exact analytic solutions, see Appendix A of Early (2012)—which in fact extended earlier work by the first author of this paper. Therefore, I am unclear in principle what a beta plane study might add.

The virtue of using the beta plane equations for inertial oscillations was challenged by Ripa (1997), who showed that they are inconsistent in that they do not improve the representation of angular momentum from the f-plane case, and give an incorrect zonal drift velocity. An improved alternative was suggested therein. In other words, the starting equations are potentially flawed. These issues are not mentioned by the authors.

On a more intuitive level, inertial oscillations are known to arise from the interaction of the departure of the earth from sphericity as a result of its rotation, leading to a component of true gravity that is tangent to the surface of the earth. This has been clearly articulated by Durran (1993) and Early (2012). Thus, one would not expect that simply adding the beta term to f would lead to more realistic representation of inertial oscillations, as they involve curvature and the already-neglected centrifugal force. The use of the beta plane equations for the problem at hand therefore requires rigorous justification.

Finally, I had questions about the parameters regimes being explored. To be of oceanographic interest, gamma and other parameters should be chosen to be physically meaningful. In other words, if the solutions presented have particles moving at 1000 km / hour, that would not be very interesting. Therefore it is essential to connect the parameter space of solutions back to physically relevant values.

The most interesting part of the paper to me was the articulation of the equator as a kind of trapping point via the adiabatic analysis. Despite my concerns, if the use of the beta plane could be justified, either globally or in the vicinity of the equator, and the connection to the spherical solution made clear, I could see it becoming a useful contribution.

Hide

ED: Reconsider after major revisions (09 Dec 2022) by Anne Marie Treguier

AR by Nathan Paldor on behalf of the Authors (19 Dec 2022) Author's response Author's tracked changes Manuscript

ED: Publish as is (02 Jan 2023) by Anne Marie Treguier

AR by Nathan Paldor on behalf of the Authors (02 Jan 2023) Manuscript

Journal article(s) based on this preprint

24 Jan 2023

Extension of Ekman (1905) wind-driven transport theory to the β plane

Nathan Paldor and Lazar Friedland

Ocean Sci., 19, 93–100, https://doi.org/10.5194/os-19-93-2023,https://doi.org/10.5194/os-19-93-2023, 2023

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Nathan Paldor and Lazar Friedland

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When the wind blows over an ocean the resulting mean current can assume many directions relative to the wind direction and does not have to perpendicular to the direction of the wind. This in contrast to a simpler, 120 year-old, theory that completely ignored Earth's sphericity and predicted that the mean ocean current should always be perpendicular to the direction of the overlying wind. In the new theory the direction of the mean current is determined by the values of several parameters.


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