the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 30 Oct 2025
Horizontal transport on the continental shelf driven by periodic rotary wind stress
Abstract. Wind driven circulation on a linearly sloping continental shelf is studied by employing the Lagrangian equations of motion forced by periodic rotary wind stress. The analysis yields explicit approximate expressions for the water column trajectories in the longshore and cross-shore directions, and these expressions are verified by numerical integration of the governing nonlinear equations. The periodic rotary wind stress generates a steady longshore drift directed with land to its left when the wind rotates counterclockwise at sub-inertial frequencies and with land to its right in all other frequencies. Counterclockwise rotation of the wind at the local inertial frequency results in a strong resonance manifested in very fast longshore drift.
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Notice on discussion status
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Preprint
(861 KB)
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
(861 KB) - Metadata XML
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- Final revised paper
Journal article(s) based on this preprint
Interactive discussion
Status: closed
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RC1: 'Comment on egusphere-2025-5187', Robert Weller, 14 Nov 2025
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AC1: 'Reply on RC1', Nathan Paldor, 08 Jan 2026
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-5187/egusphere-2025-5187-AC1-supplement.pdf
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AC1: 'Reply on RC1', Nathan Paldor, 08 Jan 2026
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RC2: 'Comment on egusphere-2025-5187', Anonymous Referee #2, 03 Jan 2026
I read the paper with great interest. Through a theoretical approach, the authors found a mean drift driven by a variable wind over a sloping continental shelf. I have the following questions for the authors’ reference.
The authors assumed that the bottom stress of a wind-driven current can be neglected, stated in Lines 92-93. This should be justified carefully. Near the coast (but beyond the gray region shown in Figure 1), the convergence/divergence of alongshore wind-driven Ekman transport induces a sea-surface slope, which in turn generates a barotropic current. Near sea bed, this barotropic current induces the bottom shear stress. It is exactly the bottom Ekman tranpsort that drives the compensating shoreward corrent shown in Figure 1. Hence, it is highly questionable to exclude the bottom shear stress. Certainly, the authors can state that this study only consider regions not very close to the coast, thus bottom shear is not that important. However, this requires the water depth being at least three times greater than the Ekman frictional thickness. In this case, the wind-driven Ekman current actually cannot feel the sea bed. Unfortunately, the authors treated the water depth H as the Ekman thickness (Eq. 5), thus the Ekman transport feels the topography even in deep water, which is not ture.
Other issues are as follows.
- This paper actually considers the movement of centroid of water column, instead of the surface water that is often focused in Ekman dynamics. Please state it clearly.
- The overall mathematics was unclear to me (and perhaps to most readers). Some key steps were missing. Some examples will be given.
- I don’t see the necessity of introducing the variable D (=U+y). Can you explain in which way it simplifies the mathematics or makes the physics more transparent ?
- Why can the solution of (12-13) be written as (14-15)Â ? Equation (16) was split into (18) and the equation in Line 160, why? It excludes the possibility that G(t) could be associated with sin(omega*t). I can see that the authors intentionally split the solution into an oscillatory (at frequency omega) term and an inertial (at frequency f) term, then verified that they can obtain such a solution that satisfies the equations. It is necessary to prove that the solution of the equations is unique.
- The derivation of (25) is unclear.
- It is unclear how one can know that D is oscillatory based on (23). It seems to me the second term on R.H.S. has a non-zero periodic mean.
- It is unclear how one can get the relationship in Line 183 based on (24). If delta_y uses the oscillatory solution in (19) and delta_D is also oscillatory, the time-average of all terms should be zero.
- The authors compared the numerical and theoretical solutions to the equations in Section 4. It is unsurprising that they are consistent. What’s more reasonable is to compare the theoretical results with the simulation of a hydrodynamic model (either 2D or 3D).
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Citation: https://doi.org/10.5194/egusphere-2025-5187-RC2 -
AC2: 'Reply on RC2', Nathan Paldor, 08 Jan 2026
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-5187/egusphere-2025-5187-AC2-supplement.pdf
Interactive discussion
Status: closed
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RC1: 'Comment on egusphere-2025-5187', Robert Weller, 14 Nov 2025
Well written.  It would be good to note that the change in direction of forced wind components at the inertial frequency was observationally verified in Weller (1981)(JGR, vol 86 C3 pages 1969-1977). A suggestion is to make it clear perhaps in lined 10-15 that the fluid is not stratified. Perhaps the abstract should include words noting northern hemisphere and homogenous fluid.  For a coastal oceanographer the normal thinking might be of a surface wind-driven layer overlaying and a bottom boundary layer and the merging of the two as the water shoals.  Any idea how stratification would change the solutions?  and would a bottom boundary layer have a rectified current as well?
Citation: https://doi.org/10.5194/egusphere-2025-5187-RC1 -
AC1: 'Reply on RC1', Nathan Paldor, 08 Jan 2026
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-5187/egusphere-2025-5187-AC1-supplement.pdf
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AC1: 'Reply on RC1', Nathan Paldor, 08 Jan 2026
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RC2: 'Comment on egusphere-2025-5187', Anonymous Referee #2, 03 Jan 2026
I read the paper with great interest. Through a theoretical approach, the authors found a mean drift driven by a variable wind over a sloping continental shelf. I have the following questions for the authors’ reference.
The authors assumed that the bottom stress of a wind-driven current can be neglected, stated in Lines 92-93. This should be justified carefully. Near the coast (but beyond the gray region shown in Figure 1), the convergence/divergence of alongshore wind-driven Ekman transport induces a sea-surface slope, which in turn generates a barotropic current. Near sea bed, this barotropic current induces the bottom shear stress. It is exactly the bottom Ekman tranpsort that drives the compensating shoreward corrent shown in Figure 1. Hence, it is highly questionable to exclude the bottom shear stress. Certainly, the authors can state that this study only consider regions not very close to the coast, thus bottom shear is not that important. However, this requires the water depth being at least three times greater than the Ekman frictional thickness. In this case, the wind-driven Ekman current actually cannot feel the sea bed. Unfortunately, the authors treated the water depth H as the Ekman thickness (Eq. 5), thus the Ekman transport feels the topography even in deep water, which is not ture.
Other issues are as follows.
- This paper actually considers the movement of centroid of water column, instead of the surface water that is often focused in Ekman dynamics. Please state it clearly.
- The overall mathematics was unclear to me (and perhaps to most readers). Some key steps were missing. Some examples will be given.
- I don’t see the necessity of introducing the variable D (=U+y). Can you explain in which way it simplifies the mathematics or makes the physics more transparent ?
- Why can the solution of (12-13) be written as (14-15)Â ? Equation (16) was split into (18) and the equation in Line 160, why? It excludes the possibility that G(t) could be associated with sin(omega*t). I can see that the authors intentionally split the solution into an oscillatory (at frequency omega) term and an inertial (at frequency f) term, then verified that they can obtain such a solution that satisfies the equations. It is necessary to prove that the solution of the equations is unique.
- The derivation of (25) is unclear.
- It is unclear how one can know that D is oscillatory based on (23). It seems to me the second term on R.H.S. has a non-zero periodic mean.
- It is unclear how one can get the relationship in Line 183 based on (24). If delta_y uses the oscillatory solution in (19) and delta_D is also oscillatory, the time-average of all terms should be zero.
- The authors compared the numerical and theoretical solutions to the equations in Section 4. It is unsurprising that they are consistent. What’s more reasonable is to compare the theoretical results with the simulation of a hydrodynamic model (either 2D or 3D).
Â
Â
Â
Citation: https://doi.org/10.5194/egusphere-2025-5187-RC2 -
AC2: 'Reply on RC2', Nathan Paldor, 08 Jan 2026
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-5187/egusphere-2025-5187-AC2-supplement.pdf
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Nathan Paldor
Lazar Friedland
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
(861 KB) - Metadata XML
Well written.  It would be good to note that the change in direction of forced wind components at the inertial frequency was observationally verified in Weller (1981)(JGR, vol 86 C3 pages 1969-1977). A suggestion is to make it clear perhaps in lined 10-15 that the fluid is not stratified. Perhaps the abstract should include words noting northern hemisphere and homogenous fluid.  For a coastal oceanographer the normal thinking might be of a surface wind-driven layer overlaying and a bottom boundary layer and the merging of the two as the water shoals.  Any idea how stratification would change the solutions?  and would a bottom boundary layer have a rectified current as well?