the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 16 Jun 2025
On the applicability of linear wave theories to simulations on the mid-latitude β-plane
Abstract. The applicability of one-dimensional (zonally invariant) harmonic and trapped wave theories for Inertia-Gravity waves to simulations on the mid-latitude β-plane is examined by comparing the analytical estimates in the geostrophic adjustment and Ekman adjustment problems with numerical simulations of the linearized rotating shallow water equations. The spatial average of the absolute differences between the theoretical solutions and the simulations, ε(t), is calculated for values of the domain's north-south extent, L, ranging from L = 4 to L = 60 (where L is measured in units of the deformation radius). The comparisons show that: (i) Though ε oscillates with time, its low-pass filter, εLP(t), increases with time. (ii) In small domains, εLP(t) in harmonic theory is significantly smaller than in trapped wave theory, while the opposite occurs in large domains. (iii) The accuracy of the harmonic wave theory decreases with L for 0 < L < 20, while for L > 20 the trend changes with time. (iv) The accuracy of the trapped wave theory increases with L in the geostrophic adjustment problem, while in the Ekman adjustment problem, its best accuracy is obtained when L ≈ 30. (v) There is a range of L and t values for which no theory provides reasonable approximations, and this range is wider in the Ekman adjustment problem than in the geostrophic adjustment problem. Non-linear simulations of a multilayered stratified ocean show that internal inertia-gravity waves exhibit the same characteristics as the wave solutions of the linearized rotating shallow water equations in a single layer ocean.
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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
(1525 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
(1525 KB) - Metadata XML
- BibTeX
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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-2025-2529', Anonymous Referee #1, 30 Jul 2025
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RC2: 'Reply on RC1', Anonymous Referee #2, 26 Aug 2025
This manuscript explores two one-dimensonal wave theories for inertia-gravity waves on a mid-latitude beta-plane and consider both geostrophic and Ekman adjustments. They focus mainly on linear theories but also present a few nonlinear simulations. They define a metric to quantify the differences between linear and nonlinear theories and show how this varies in different regimes.
This research is of interesting to the community and well written but I strongly recommend that the following concerns are addressed before it is published.
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Please give citations for the particular scalings you are using in the two cases.
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The authors work very hard to have one set of equations that allows for the two types of adjustments. This makes things mathematically complicated and I’m not sure how much is gained. If there is a big benefit to this, please emphasize this as it’s not entirely clear to me.
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In section 2.5, going from equation 14 to 15, the quadratic term is ignored. This is because the authors want to get the equation that has a known special function a solution. However, this is an approximation, and introduces more error into the equation. Since the eigenvalue problem is solved numerically, you can easily keep in this term and that should yield a more accurate theory. Please work on doing this or give a very good reason why not.
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Section 3 discusses harmonic, trapped and semi-infinite domains. These have all been previously studied in the context of the shallow water model. In reality, waves will not be purely harmonic or trapped, but some hybrid of the two. If that’s the case, then why spend so much time focusing on each of these limits? Again, with numerical solutions you can study any of these waves and you need not restrict yourselves to these relatively simple cases.
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Solving the inhomogeneous eigenvalue problem is interesting. But when you do this you are essentially decomposing the inhomogeneous part of the equation in terms of your eigenfunctions. These don’t change at all and don’t change the eigenfunctions. This is not something that we see very much in the literature and is worthy of further discussion.
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In section 5, when introducing the MITgcm, be explicit as to what equations you are solving. It does not solve the shallow water equations you have focused on up to this point.
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In section 5, you now start to consider averages of the numerical solutions. But the theory does not mention temporal averages at all. If you want to have a good comparison, you should consider temporal averages in the linear theory.
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Page 11, you say the number of modes summed is 10^4, then later its reduced to 500. How sensitive is your result to this number?
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Lots of emphasis is put on this epsilon parameter, the so called error estimator, for many different cases. I’m not convinced this is physically interesting. If you feel it is, please give some better justification for why you are considering it so intently.
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Section 6 redoes everything for two layers. Why not do it all together? It was reduce what is a rather long manuscript.
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Section 6.2 is called numerical simulations of a multilayered ocean and the authors use the MITgcm. This is not a layered model. Please be vary careful in your descriptions.
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In general, this paper seems to be a lot of different things thrown together and they don’t necessarily flow very cleanly. Please work harder to make this a coherent story, not just a series of interesting results.
- AC2: 'Reply on RC2', Nathan Paldor, 18 Sep 2025
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AC1: 'Reply on RC1', Nathan Paldor, 18 Sep 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-2529/egusphere-2025-2529-AC1-supplement.pdf
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RC2: 'Reply on RC1', Anonymous Referee #2, 26 Aug 2025
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2025-2529', Anonymous Referee #1, 30 Jul 2025
-
RC2: 'Reply on RC1', Anonymous Referee #2, 26 Aug 2025
This manuscript explores two one-dimensonal wave theories for inertia-gravity waves on a mid-latitude beta-plane and consider both geostrophic and Ekman adjustments. They focus mainly on linear theories but also present a few nonlinear simulations. They define a metric to quantify the differences between linear and nonlinear theories and show how this varies in different regimes.
This research is of interesting to the community and well written but I strongly recommend that the following concerns are addressed before it is published.
-
Please give citations for the particular scalings you are using in the two cases.
-
The authors work very hard to have one set of equations that allows for the two types of adjustments. This makes things mathematically complicated and I’m not sure how much is gained. If there is a big benefit to this, please emphasize this as it’s not entirely clear to me.
-
In section 2.5, going from equation 14 to 15, the quadratic term is ignored. This is because the authors want to get the equation that has a known special function a solution. However, this is an approximation, and introduces more error into the equation. Since the eigenvalue problem is solved numerically, you can easily keep in this term and that should yield a more accurate theory. Please work on doing this or give a very good reason why not.
-
Section 3 discusses harmonic, trapped and semi-infinite domains. These have all been previously studied in the context of the shallow water model. In reality, waves will not be purely harmonic or trapped, but some hybrid of the two. If that’s the case, then why spend so much time focusing on each of these limits? Again, with numerical solutions you can study any of these waves and you need not restrict yourselves to these relatively simple cases.
-
Solving the inhomogeneous eigenvalue problem is interesting. But when you do this you are essentially decomposing the inhomogeneous part of the equation in terms of your eigenfunctions. These don’t change at all and don’t change the eigenfunctions. This is not something that we see very much in the literature and is worthy of further discussion.
-
In section 5, when introducing the MITgcm, be explicit as to what equations you are solving. It does not solve the shallow water equations you have focused on up to this point.
-
In section 5, you now start to consider averages of the numerical solutions. But the theory does not mention temporal averages at all. If you want to have a good comparison, you should consider temporal averages in the linear theory.
-
Page 11, you say the number of modes summed is 10^4, then later its reduced to 500. How sensitive is your result to this number?
-
Lots of emphasis is put on this epsilon parameter, the so called error estimator, for many different cases. I’m not convinced this is physically interesting. If you feel it is, please give some better justification for why you are considering it so intently.
-
Section 6 redoes everything for two layers. Why not do it all together? It was reduce what is a rather long manuscript.
-
Section 6.2 is called numerical simulations of a multilayered ocean and the authors use the MITgcm. This is not a layered model. Please be vary careful in your descriptions.
-
In general, this paper seems to be a lot of different things thrown together and they don’t necessarily flow very cleanly. Please work harder to make this a coherent story, not just a series of interesting results.
- AC2: 'Reply on RC2', Nathan Paldor, 18 Sep 2025
-
-
AC1: 'Reply on RC1', Nathan Paldor, 18 Sep 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-2529/egusphere-2025-2529-AC1-supplement.pdf
-
RC2: 'Reply on RC1', Anonymous Referee #2, 26 Aug 2025
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Itamar Yacoby
Hezi Gildor
Nathan Paldor
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
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Written in a separate PDF-file.