the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Extension of Ekman (1905) winddriven transport theory to the βplane
Nathan Paldor
Lazar Friedland
Abstract. The seminal, Ekman (1905)’s, fplane theory of wind driven transport at the ocean surface is extended to the βplane 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.

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The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.

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(891 KB)

The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.
 Preprint
(891 KB)  BibTeX
 EndNote
 Final revised paper
Journal article(s) based on this preprint
Nathan Paldor and Lazar Friedland
Interactive discussion
Status: closed

CC1: 'Comment on egusphere2022831', Adrian Constantin, 15 Sep 2022
On page 1, it should be pointed out that effects beyond the equatorial fplane 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 betaplane approximation to obtain more detailed information about the dynamics.
Citation: https://doi.org/10.5194/egusphere2022831CC1 
RC1: 'Reply on CC1', Anonymous Referee #1, 17 Sep 2022
The authors discuss the extension of Ekman's classical winddrift solution to the betaplane.
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 wellwritten 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,
Ekmantype solutions for shallowwater 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, NearSurface Shear Flow in the Tropical Pacific Cold Tongue Front,
J. Phys. Oceanogr. 39 (2009), 12001215.
Citation: https://doi.org/10.5194/egusphere2022831RC1 
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/egusphere2022831AC4

AC4: 'Reply on RC1', Nathan Paldor, 14 Nov 2022

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/egusphere2022831AC1 
AC2: 'Reply on CC1', Nathan Paldor, 07 Nov 2022
See our response to RC1
Citation: https://doi.org/10.5194/egusphere2022831AC2

RC1: 'Reply on CC1', Anonymous Referee #1, 17 Sep 2022

RC2: 'Comment on egusphere2022831', Nicolas Grisouard, 25 Oct 2022
Review of `Extension of Ekman (1905) winddriven transport theory to the betaplane’ by Paldor and Friedland
This manuscript provides solutions to a verticallyintegrated version of the equations of motion on a betaplane, 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 betaplane 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 redefine 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. 7071), but to expand on it could help the reader understand faster.
Some specific comments:
1. Ll. 3233: 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. 4245: 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. 5355: I believe that only Gill (1982) covers the timedependent 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/egusphere2022831RC2 
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 ydynamics and 4 for the xdynamics.
B. We agree – the meaning of y_{0} is much subtler (and intriguing) on the bplane than on the fplane. In the revised version the numerical calculations will be initiated at the fixed point y_{0}=0 (and V_{0} ≠ 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 missworded phrases will be corrected in the revised version.
Citation: https://doi.org/10.5194/egusphere2022831AC3

AC3: 'Reply on RC2', Nathan Paldor, 14 Nov 2022
Interactive discussion
Status: closed

CC1: 'Comment on egusphere2022831', Adrian Constantin, 15 Sep 2022
On page 1, it should be pointed out that effects beyond the equatorial fplane 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 betaplane approximation to obtain more detailed information about the dynamics.
Citation: https://doi.org/10.5194/egusphere2022831CC1 
RC1: 'Reply on CC1', Anonymous Referee #1, 17 Sep 2022
The authors discuss the extension of Ekman's classical winddrift solution to the betaplane.
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 wellwritten 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,
Ekmantype solutions for shallowwater 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, NearSurface Shear Flow in the Tropical Pacific Cold Tongue Front,
J. Phys. Oceanogr. 39 (2009), 12001215.
Citation: https://doi.org/10.5194/egusphere2022831RC1 
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/egusphere2022831AC4

AC4: 'Reply on RC1', Nathan Paldor, 14 Nov 2022

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/egusphere2022831AC1 
AC2: 'Reply on CC1', Nathan Paldor, 07 Nov 2022
See our response to RC1
Citation: https://doi.org/10.5194/egusphere2022831AC2

RC1: 'Reply on CC1', Anonymous Referee #1, 17 Sep 2022

RC2: 'Comment on egusphere2022831', Nicolas Grisouard, 25 Oct 2022
Review of `Extension of Ekman (1905) winddriven transport theory to the betaplane’ by Paldor and Friedland
This manuscript provides solutions to a verticallyintegrated version of the equations of motion on a betaplane, 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 betaplane 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 redefine 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. 7071), but to expand on it could help the reader understand faster.
Some specific comments:
1. Ll. 3233: 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. 4245: 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. 5355: I believe that only Gill (1982) covers the timedependent 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/egusphere2022831RC2 
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 ydynamics and 4 for the xdynamics.
B. We agree – the meaning of y_{0} is much subtler (and intriguing) on the bplane than on the fplane. In the revised version the numerical calculations will be initiated at the fixed point y_{0}=0 (and V_{0} ≠ 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 missworded phrases will be corrected in the revised version.
Citation: https://doi.org/10.5194/egusphere2022831AC3

AC3: 'Reply on RC2', Nathan Paldor, 14 Nov 2022
Peer review completion
Journal article(s) based on this preprint
Nathan Paldor and Lazar Friedland
Nathan Paldor and Lazar Friedland
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The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.
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