the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 08 Feb 2022
Analytical solution of the ray equations of Hamilton for Rossby waves on stationary shear flows
Abstract. The asymptotic behavior of Rossby waves in the ocean interacting with a shear stationary flow is considered. It is shown that there is a qualitative difference between the problems for the zonal and non-zonal background flow. Whereas only one critical layer arises for a zonal flow, then several critical layers can exist for a non-zonal flow. It is established that the integrated ray equations of Hamilton are equivalent to the asymptotic behavior of the Cauchy problem solution. Explicit analytical solutions are obtained for the tracks of Rossby waves as a function of time and initial parameters of the wave disturbance, as well as the magnitude of the shear and angle of inclination of the flow to the zonal direction. On the example of Rossby waves on a shear flow, the ray equations of Hamilton are analytically integrated. The obtained explicit expressions make it possible to calculate in real-time the Rossby wave tracks for any initial wave direction and any shear current inclination angle. It is shown qualitatively that these tracks for a non-zonal flow are strongly anisotropic.
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Interactive discussion
Status: closed
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RC1: 'Comment on os-2022-5', Anonymous Referee #1, 03 Mar 2022
The comment was uploaded in the form of a supplement: https://os.copernicus.org/preprints/os-2022-5/os-2022-5-RC1-supplement.pdf
-
AC1: 'Reply on RC1', Tatyana Belonenko, 10 Mar 2022
Dear Reviewer,
We thank the anonymous reviewers for carefully reading the manuscript. Please find below a detailed point-by-point response to all comments.
- The main issue I have with this work is that it teaches us nothing new about “Ocean Science”. The authors make no attempt to relate the solutions they compute to an oceanographic phenomenon. I’m not sure whether this work warrants publication as a contribution in applied mathematics or fluid mechanics but it is definitely not a contribution in oceanography. The authors do not attempt to contribute anything to our understanding of phenomena observed in physical oceanography.
The next step in the study is to find a specific application to ocean processes using satellite, ground and model data. At this stage, we show that there is a qualitative difference between the problems for zonal and nonzonal background flow and confirm this with examples.
- From a mathematical standpoint, it is rare for nonlinear, 4th order, systems to poses analytic solutions. However, for the most part, numerical solutions are both easy to compute and very accurate (unless there is a singularity in the problem which is not the case here). The ray equation is no more than the calculation of trajectories in space when the (group) velocity is spatially variable (including the spatial variability of the wavenumber). It is unclear what has been gained from the few trajectories calculated in this manuscript. The main message of this paper is too thin mathematically. As highlighted by the authors, the attraction of all trajectories to the critical point (e.g. latitude) is well known and was highlighted in previous papers by the same authors (Gnevyshev et al., 2020a; 2020b). Solutions of such basic mathematical systems are publishable only when they constitute significant contributions to our understanding of oceanographic processes.
You caught the main physical idea - this is the extreme sensitivity of Rossby wave tracks. But we wanted to say that such sensitivity exists only for non-zonal currents. Tracks on the zonal flow have a continuous dependence on the initial conditions and in this sense, the zonal task is extremely predictable.
When you discuss about “…unless there is a singularity in the problem which is not the case here etc.” you're mistaken. The fact that the Rossby baroclinic waves have an extremely unpleasant singularity on the hyperbolic Booth lemniscate is obtained in the works of Longuet-Higgins. The intersection point of the lemniscate with the abscissa axis was also highlighted by Pedlosky [13]. Apparently, we need to write a separate paper on this topic to explain the specifics of Rossby waves as a system consisting of two types of waves: waves with positive and waves with negative dispersion. The hyperbolic lemniscate is the boundary of these types of waves. On the lemniscate there is a singularity of Rossby waves, as at the boundary of the change of the sign of the dispersion of Rossby waves.
- The description of the underlying assumptions and the methodology employed is very poor. The extremely weak pedagogical presentation typifies both the English style and the mathematical analysis. If the authors wish to publish this work in a different journal they should tend much more seriously to both of these aspects of communication with their readers. A few examples are listed below as an aid to the authors if they intend to submit the manuscript to another journal.
We took into account all your comments and improved the text. Once again, we want to thank you for your help.
Citation: https://doi.org/10.5194/os-2022-5-AC1
-
AC1: 'Reply on RC1', Tatyana Belonenko, 10 Mar 2022
-
RC2: 'Comment on os-2022-5', Anonymous Referee #2, 05 Mar 2022
I read this manuscript and doubted my eyes. I felt like I was back in the world half a century ago. In general, theory is meaningful when it can explain the actual phenomenon concisely or when it is effective for interpreting observations and experimental data from a new perspective. This paper is not of that sort. It deals with the behavior of Rossby waves in a linear shear flow, showing the sameness between the ray theory based on the simple dispersion relation of Rossby waves considering the shear frequency and the solution of the Cauchy problem. This problem was calculated almost half a century ago using a coordinate system that follows the shear flow, and several papers were already published even with pointing out the limitations of introducing the wave packet, as cited in the references. Therefore, in modern times, it is just an exercise for college students studying wave packets and ray theory for Rossby waves in a shear flow.
Unfortunately, this paper is not worth publishing in Ocean Science.
Citation: https://doi.org/10.5194/os-2022-5-RC2 -
CC1: 'Reply on RC2', Tatyana Belonenko, 10 Mar 2022
Dear reviewer,
We are not sure you are right. The ray equations of Hamilton-Jacobi have not been integrated yet for any types of waves. We mean the integration analytically in an explicit form and in inhomogeneous media. If you have another information, you must have provided the appropriate links in your review.
However, you did not give any references to the fact that such work has already been done by someone. Thus, your negative statements are not supported by any facts.
The very idea of looking for an analytical solution in the problem where there is an explicit transverse inhomogeneity and a plane wave in this variable at the same time is not entirely trivial. We got the right solution and sure that it can have the interest to specialists.
Citation: https://doi.org/10.5194/os-2022-5-CC1 -
AC4: 'Reply on RC2', Tatyana Belonenko, 06 Apr 2022
Dear reviewer,
We are not sure you are right. The ray equations of Hamilton-Jacobi have not been integrated yet for any types of waves. We mean the integration analytically in an explicit form and in inhomogeneous media. If you have another information, you must have provided the appropriate links in your review.
However, you did not give any references to the fact that such work has already been done by someone. Thus, your negative statements are not supported by any facts.
The very idea of looking for an analytical solution in the problem where there is an explicit transverse inhomogeneity and a plane wave in this variable at the same time is not entirely trivial. We got the right solution and sure that it can have the interest to specialists.
Citation: https://doi.org/10.5194/os-2022-5-AC4
-
CC1: 'Reply on RC2', Tatyana Belonenko, 10 Mar 2022
-
RC3: 'Comment on os-2022-5', Anonymous Referee #3, 12 Mar 2022
The manuscript is concerned with theoretical investigation of Rossby wave dynamics on currents. Its radical novelty is in considering Rossby wave dynamics on non-zonal currents, which, as the authors have demonstrated, changes the wave kinematics qualitatively. The authors were able to solve analytically the Hamiltonian ray equations. The solutions are quite elegant. The significance of the findings is in the very different overall picture of Rossby wave dynamics, which allows to predict new phenomena and, in particular, re-interpret the old simulations of Rossby wave dynamics carried out by Peter Killworth and his group.
The results are mathematically sound. The presentation is clear, easy to follow, but needs an extra polishing. The English needs some help from a native speaker.
The main weakness of the work is that the authors made no serious attempt to link their results with observations.
I see two options:
(i) To revise the paper by adding a discussion on how the results can be applied for interpreting observations. What of the newly found phenomena could be observed, at least in principle, and how? Discuss the gap between the theory and reality of altimeter observations of Rossby waves. Is it possible to obtain useful information on baroclinic Rossby waves from ADCP arrays? When the SKIM system of measuring surface currents from satellites becomes functioning would it be possible to use it for Rossby wave observations? The diminant tendency nowadays is to employ a variety of multiple sensors distribbuted somehow in the ocean, what are perspective of this approach for monitoring Rossby wave dynamics?
(ii) The manuscript can be re-submitted to a more mathematically oriented journal (e.g. JFM, Phys Fluids, Proc. Roy Soc, Wave Motion, etc).
In any case, the English needs some extra polishing, a copy editor help is needed.
Citation: https://doi.org/10.5194/os-2022-5-RC3 -
AC2: 'Reply on RC3', Tatyana Belonenko, 12 Mar 2022
Dear reviewer,
We are very grateful to you for the review. We really appreciate that you are able to understand the results presented in the article.
Our results have two aspects. The first is that asymptotic analysis works even where it doesn't seem to work. They are extremely simple to analyze so they can be tested in different ways. That's why we submitted the paper to OS. However, there is a second aspect: the conclusions are extremely unpleasant for the applied analysis. And this is very clearly seen in our solution which show the extreme sensitivity of the results for non-trivial solutions.
There are not any appropriate in situ observations that would allow us to verify the conclusions. Altimetric observations characterize the sea surface and have insufficient spatial resolution of 0.25°. Buoys and gliders describing the variability of oceanological parameters at different horizons are also not suitable for describing the asymptotic behavior of Rossby waves. Nowadays the only method for describing the time-spatial variability of Rossby waves in the ocean is the construction of Hovmöller diagrams. However, this simple method cannot describe the asymptotic behavior of Rossby waves on non-zonal flows, in particular the situations with adhering and overshooting of the waves (see please
Gnevyshev, V.G., Badulin, S.I., Belonenko, T.V., 2020. Rossby waves on non-zonal currents: structural stability of critical layer effects. Pure Appl. Geophys., 177: 5585–5598. https://doi.org/10.1007/s00024-020-02567-0;
Gnevyshev V.G., Badulin S.I., Koldunov A.V., and Belonenko T.V. 2020. Rossby Waves on Non-zonal Flows: Vertical Focusing and Effect of the Current Stratification. Pure Appl. Geophys. 178(8), 3247 – 3261. https://doi.org/10.1007/s00024-021-02799-8).
Going further, we have come to the conclusion, that the concept of searching for continuously differentiable solutions for the World Ocean is suitable more or less for the open ocean, and here we agree with the results of Kilworth with his students. However, we have taken a step to expand this approach. We assumed that it is necessary to expand the class of solutions in the analysis of interaction of the Rossby waves with an inhomogeneous flow. We added the reflected from the zonal or non-zonal flow waves to the analysis. Thus, we have moved from the class of continuously differentiable, to simply continuous solutions. Moreover, we used the so-called crosslinking conditions in the Miles-Ribner problem:
see please Gnevyshev, V.G. and Belonenko, T.V., 2021. Vortex Layer on the β-Plane in the Miles – Ribner Formulation. Pole on the Real Axis. Physical Oceanography, [e-journal] 28(5), pp. 486-498. doi:10.22449/1573-160X-2021-5-486-498.
Here we considered the solution for the vortex layer and established that the previous researchers who tried to solved this problem made a silly mistake. They missed an extremely important solution, which may be useful for interpreting and analyzing the role of non-zonality in the interaction of the Rossby waves with the flow. And we could see such examples in altimetry (DOI: 10.21046/2070-7401-2021-18-5-242-251).
These results are extremely revolutionary and have no analogues.
Citation: https://doi.org/10.5194/os-2022-5-AC2 -
RC4: 'Reply on AC2', Anonymous Referee #3, 12 Mar 2022
I appreciate the difficulties of observing directly the found peculiarities of Rossby wave dynamics. I am
aware of the limitations of satellite altimetry. Having said that, I still encourage the authors (i) to discuss
the gap between the theory and the observations; (ii) I also strongly encourage the authors to think how
the new phenomena they found might manifest in the dynamics of larger scales observable from satellites.
For example, we cannot see motion of molecules by an unaided eye, but we can predict its macroscopic results.
What are the "macroscopic" implications of the new dynamics?
Large scale ocean circulation models are routinely run in many institutions. It would have been illuminating to see
manifestations of the new Rossby dynamics in one such simulations. What would be the requirements for the model reslution?
Citation: https://doi.org/10.5194/os-2022-5-RC4 -
AC3: 'Reply on RC4', Tatyana Belonenko, 14 Mar 2022
Nonlinear effects of Rossby waves can be observed in the World Ocean. You can see our work on the Agulhas vortices (Gnevyshev, V. G., A. A. Malysheva, T. V. Belonenko, and A. V. Koldunov (2021), On Agulhas eddies and Rossby waves travelling by forcing effects, Russ. J. Earth. Sci., 21, ES5003, doi:10.2205/2021ES000773). We show the clear tracks there and they are well identified. Identification of linear Rossby waves can be found in the works of La Casce (https://doi.org/10.1002/2017GL075430). He says that it is the long-wave limit of Rossby waves that we observe in the ocean. And the short-wave limit is practically not visible from the field data. Our paper supports this result.
Our result is qualitative and characterizes the asymptotic behavior of the waves in the absence of other factors that in reality can affect their propagation. However, the asymptotic conditions for the wave propagation are cannot be observed in the nature. There are two main reasons for it. The first reason is that Rossby waves in the ocean do not exist in a vacuum, but interact with other processes - currents, vortices and other waves. Another reason is that the very kinematics of Rossby waves is unstable and the wave tracks can vary significantly. The Green function of Rossby waves demonstrates that only long Rossby waves have a clear almost zonal direction and the waves always go west (Lighthill, 1967). On the contrary, the short Rossby waves do not have clearly defined directions (we also illustrated this in the paper). The short Rossby waves can move in any direction. Perhaps, that is why the short Rossby waves are not being observed in the ocean. It is difficult to choose the only direction of the waves, the only track that the wave will follow.
Our paper is devoted to the Hamilton-Jacobi equations i.e. the ray equations. This is the short-wave limit where the wavelength is less than 50 km. How can they be identified in the ocean? There is no way because a) the altimetry space resolution 0.25°; b) we obtained the result that there are no stable tracks for Rossby waves (in the short-wave linear approximation). Once again, the task is extremely sensitive to the initial data.
Dear reviewer, the is no sense for us to prove our point of view since two other reviewers have already written the negative reviews. One of them didn't understand at all what the article was about. If you want to continue the discussion, you can email us if you are not afraid to interact with the Russians. We are always open for collaboration.
Citation: https://doi.org/10.5194/os-2022-5-AC3
-
AC3: 'Reply on RC4', Tatyana Belonenko, 14 Mar 2022
-
RC4: 'Reply on AC2', Anonymous Referee #3, 12 Mar 2022
-
AC2: 'Reply on RC3', Tatyana Belonenko, 12 Mar 2022
Interactive discussion
Status: closed
-
RC1: 'Comment on os-2022-5', Anonymous Referee #1, 03 Mar 2022
The comment was uploaded in the form of a supplement: https://os.copernicus.org/preprints/os-2022-5/os-2022-5-RC1-supplement.pdf
-
AC1: 'Reply on RC1', Tatyana Belonenko, 10 Mar 2022
Dear Reviewer,
We thank the anonymous reviewers for carefully reading the manuscript. Please find below a detailed point-by-point response to all comments.
- The main issue I have with this work is that it teaches us nothing new about “Ocean Science”. The authors make no attempt to relate the solutions they compute to an oceanographic phenomenon. I’m not sure whether this work warrants publication as a contribution in applied mathematics or fluid mechanics but it is definitely not a contribution in oceanography. The authors do not attempt to contribute anything to our understanding of phenomena observed in physical oceanography.
The next step in the study is to find a specific application to ocean processes using satellite, ground and model data. At this stage, we show that there is a qualitative difference between the problems for zonal and nonzonal background flow and confirm this with examples.
- From a mathematical standpoint, it is rare for nonlinear, 4th order, systems to poses analytic solutions. However, for the most part, numerical solutions are both easy to compute and very accurate (unless there is a singularity in the problem which is not the case here). The ray equation is no more than the calculation of trajectories in space when the (group) velocity is spatially variable (including the spatial variability of the wavenumber). It is unclear what has been gained from the few trajectories calculated in this manuscript. The main message of this paper is too thin mathematically. As highlighted by the authors, the attraction of all trajectories to the critical point (e.g. latitude) is well known and was highlighted in previous papers by the same authors (Gnevyshev et al., 2020a; 2020b). Solutions of such basic mathematical systems are publishable only when they constitute significant contributions to our understanding of oceanographic processes.
You caught the main physical idea - this is the extreme sensitivity of Rossby wave tracks. But we wanted to say that such sensitivity exists only for non-zonal currents. Tracks on the zonal flow have a continuous dependence on the initial conditions and in this sense, the zonal task is extremely predictable.
When you discuss about “…unless there is a singularity in the problem which is not the case here etc.” you're mistaken. The fact that the Rossby baroclinic waves have an extremely unpleasant singularity on the hyperbolic Booth lemniscate is obtained in the works of Longuet-Higgins. The intersection point of the lemniscate with the abscissa axis was also highlighted by Pedlosky [13]. Apparently, we need to write a separate paper on this topic to explain the specifics of Rossby waves as a system consisting of two types of waves: waves with positive and waves with negative dispersion. The hyperbolic lemniscate is the boundary of these types of waves. On the lemniscate there is a singularity of Rossby waves, as at the boundary of the change of the sign of the dispersion of Rossby waves.
- The description of the underlying assumptions and the methodology employed is very poor. The extremely weak pedagogical presentation typifies both the English style and the mathematical analysis. If the authors wish to publish this work in a different journal they should tend much more seriously to both of these aspects of communication with their readers. A few examples are listed below as an aid to the authors if they intend to submit the manuscript to another journal.
We took into account all your comments and improved the text. Once again, we want to thank you for your help.
Citation: https://doi.org/10.5194/os-2022-5-AC1
-
AC1: 'Reply on RC1', Tatyana Belonenko, 10 Mar 2022
-
RC2: 'Comment on os-2022-5', Anonymous Referee #2, 05 Mar 2022
I read this manuscript and doubted my eyes. I felt like I was back in the world half a century ago. In general, theory is meaningful when it can explain the actual phenomenon concisely or when it is effective for interpreting observations and experimental data from a new perspective. This paper is not of that sort. It deals with the behavior of Rossby waves in a linear shear flow, showing the sameness between the ray theory based on the simple dispersion relation of Rossby waves considering the shear frequency and the solution of the Cauchy problem. This problem was calculated almost half a century ago using a coordinate system that follows the shear flow, and several papers were already published even with pointing out the limitations of introducing the wave packet, as cited in the references. Therefore, in modern times, it is just an exercise for college students studying wave packets and ray theory for Rossby waves in a shear flow.
Unfortunately, this paper is not worth publishing in Ocean Science.
Citation: https://doi.org/10.5194/os-2022-5-RC2 -
CC1: 'Reply on RC2', Tatyana Belonenko, 10 Mar 2022
Dear reviewer,
We are not sure you are right. The ray equations of Hamilton-Jacobi have not been integrated yet for any types of waves. We mean the integration analytically in an explicit form and in inhomogeneous media. If you have another information, you must have provided the appropriate links in your review.
However, you did not give any references to the fact that such work has already been done by someone. Thus, your negative statements are not supported by any facts.
The very idea of looking for an analytical solution in the problem where there is an explicit transverse inhomogeneity and a plane wave in this variable at the same time is not entirely trivial. We got the right solution and sure that it can have the interest to specialists.
Citation: https://doi.org/10.5194/os-2022-5-CC1 -
AC4: 'Reply on RC2', Tatyana Belonenko, 06 Apr 2022
Dear reviewer,
We are not sure you are right. The ray equations of Hamilton-Jacobi have not been integrated yet for any types of waves. We mean the integration analytically in an explicit form and in inhomogeneous media. If you have another information, you must have provided the appropriate links in your review.
However, you did not give any references to the fact that such work has already been done by someone. Thus, your negative statements are not supported by any facts.
The very idea of looking for an analytical solution in the problem where there is an explicit transverse inhomogeneity and a plane wave in this variable at the same time is not entirely trivial. We got the right solution and sure that it can have the interest to specialists.
Citation: https://doi.org/10.5194/os-2022-5-AC4
-
CC1: 'Reply on RC2', Tatyana Belonenko, 10 Mar 2022
-
RC3: 'Comment on os-2022-5', Anonymous Referee #3, 12 Mar 2022
The manuscript is concerned with theoretical investigation of Rossby wave dynamics on currents. Its radical novelty is in considering Rossby wave dynamics on non-zonal currents, which, as the authors have demonstrated, changes the wave kinematics qualitatively. The authors were able to solve analytically the Hamiltonian ray equations. The solutions are quite elegant. The significance of the findings is in the very different overall picture of Rossby wave dynamics, which allows to predict new phenomena and, in particular, re-interpret the old simulations of Rossby wave dynamics carried out by Peter Killworth and his group.
The results are mathematically sound. The presentation is clear, easy to follow, but needs an extra polishing. The English needs some help from a native speaker.
The main weakness of the work is that the authors made no serious attempt to link their results with observations.
I see two options:
(i) To revise the paper by adding a discussion on how the results can be applied for interpreting observations. What of the newly found phenomena could be observed, at least in principle, and how? Discuss the gap between the theory and reality of altimeter observations of Rossby waves. Is it possible to obtain useful information on baroclinic Rossby waves from ADCP arrays? When the SKIM system of measuring surface currents from satellites becomes functioning would it be possible to use it for Rossby wave observations? The diminant tendency nowadays is to employ a variety of multiple sensors distribbuted somehow in the ocean, what are perspective of this approach for monitoring Rossby wave dynamics?
(ii) The manuscript can be re-submitted to a more mathematically oriented journal (e.g. JFM, Phys Fluids, Proc. Roy Soc, Wave Motion, etc).
In any case, the English needs some extra polishing, a copy editor help is needed.
Citation: https://doi.org/10.5194/os-2022-5-RC3 -
AC2: 'Reply on RC3', Tatyana Belonenko, 12 Mar 2022
Dear reviewer,
We are very grateful to you for the review. We really appreciate that you are able to understand the results presented in the article.
Our results have two aspects. The first is that asymptotic analysis works even where it doesn't seem to work. They are extremely simple to analyze so they can be tested in different ways. That's why we submitted the paper to OS. However, there is a second aspect: the conclusions are extremely unpleasant for the applied analysis. And this is very clearly seen in our solution which show the extreme sensitivity of the results for non-trivial solutions.
There are not any appropriate in situ observations that would allow us to verify the conclusions. Altimetric observations characterize the sea surface and have insufficient spatial resolution of 0.25°. Buoys and gliders describing the variability of oceanological parameters at different horizons are also not suitable for describing the asymptotic behavior of Rossby waves. Nowadays the only method for describing the time-spatial variability of Rossby waves in the ocean is the construction of Hovmöller diagrams. However, this simple method cannot describe the asymptotic behavior of Rossby waves on non-zonal flows, in particular the situations with adhering and overshooting of the waves (see please
Gnevyshev, V.G., Badulin, S.I., Belonenko, T.V., 2020. Rossby waves on non-zonal currents: structural stability of critical layer effects. Pure Appl. Geophys., 177: 5585–5598. https://doi.org/10.1007/s00024-020-02567-0;
Gnevyshev V.G., Badulin S.I., Koldunov A.V., and Belonenko T.V. 2020. Rossby Waves on Non-zonal Flows: Vertical Focusing and Effect of the Current Stratification. Pure Appl. Geophys. 178(8), 3247 – 3261. https://doi.org/10.1007/s00024-021-02799-8).
Going further, we have come to the conclusion, that the concept of searching for continuously differentiable solutions for the World Ocean is suitable more or less for the open ocean, and here we agree with the results of Kilworth with his students. However, we have taken a step to expand this approach. We assumed that it is necessary to expand the class of solutions in the analysis of interaction of the Rossby waves with an inhomogeneous flow. We added the reflected from the zonal or non-zonal flow waves to the analysis. Thus, we have moved from the class of continuously differentiable, to simply continuous solutions. Moreover, we used the so-called crosslinking conditions in the Miles-Ribner problem:
see please Gnevyshev, V.G. and Belonenko, T.V., 2021. Vortex Layer on the β-Plane in the Miles – Ribner Formulation. Pole on the Real Axis. Physical Oceanography, [e-journal] 28(5), pp. 486-498. doi:10.22449/1573-160X-2021-5-486-498.
Here we considered the solution for the vortex layer and established that the previous researchers who tried to solved this problem made a silly mistake. They missed an extremely important solution, which may be useful for interpreting and analyzing the role of non-zonality in the interaction of the Rossby waves with the flow. And we could see such examples in altimetry (DOI: 10.21046/2070-7401-2021-18-5-242-251).
These results are extremely revolutionary and have no analogues.
Citation: https://doi.org/10.5194/os-2022-5-AC2 -
RC4: 'Reply on AC2', Anonymous Referee #3, 12 Mar 2022
I appreciate the difficulties of observing directly the found peculiarities of Rossby wave dynamics. I am
aware of the limitations of satellite altimetry. Having said that, I still encourage the authors (i) to discuss
the gap between the theory and the observations; (ii) I also strongly encourage the authors to think how
the new phenomena they found might manifest in the dynamics of larger scales observable from satellites.
For example, we cannot see motion of molecules by an unaided eye, but we can predict its macroscopic results.
What are the "macroscopic" implications of the new dynamics?
Large scale ocean circulation models are routinely run in many institutions. It would have been illuminating to see
manifestations of the new Rossby dynamics in one such simulations. What would be the requirements for the model reslution?
Citation: https://doi.org/10.5194/os-2022-5-RC4 -
AC3: 'Reply on RC4', Tatyana Belonenko, 14 Mar 2022
Nonlinear effects of Rossby waves can be observed in the World Ocean. You can see our work on the Agulhas vortices (Gnevyshev, V. G., A. A. Malysheva, T. V. Belonenko, and A. V. Koldunov (2021), On Agulhas eddies and Rossby waves travelling by forcing effects, Russ. J. Earth. Sci., 21, ES5003, doi:10.2205/2021ES000773). We show the clear tracks there and they are well identified. Identification of linear Rossby waves can be found in the works of La Casce (https://doi.org/10.1002/2017GL075430). He says that it is the long-wave limit of Rossby waves that we observe in the ocean. And the short-wave limit is practically not visible from the field data. Our paper supports this result.
Our result is qualitative and characterizes the asymptotic behavior of the waves in the absence of other factors that in reality can affect their propagation. However, the asymptotic conditions for the wave propagation are cannot be observed in the nature. There are two main reasons for it. The first reason is that Rossby waves in the ocean do not exist in a vacuum, but interact with other processes - currents, vortices and other waves. Another reason is that the very kinematics of Rossby waves is unstable and the wave tracks can vary significantly. The Green function of Rossby waves demonstrates that only long Rossby waves have a clear almost zonal direction and the waves always go west (Lighthill, 1967). On the contrary, the short Rossby waves do not have clearly defined directions (we also illustrated this in the paper). The short Rossby waves can move in any direction. Perhaps, that is why the short Rossby waves are not being observed in the ocean. It is difficult to choose the only direction of the waves, the only track that the wave will follow.
Our paper is devoted to the Hamilton-Jacobi equations i.e. the ray equations. This is the short-wave limit where the wavelength is less than 50 km. How can they be identified in the ocean? There is no way because a) the altimetry space resolution 0.25°; b) we obtained the result that there are no stable tracks for Rossby waves (in the short-wave linear approximation). Once again, the task is extremely sensitive to the initial data.
Dear reviewer, the is no sense for us to prove our point of view since two other reviewers have already written the negative reviews. One of them didn't understand at all what the article was about. If you want to continue the discussion, you can email us if you are not afraid to interact with the Russians. We are always open for collaboration.
Citation: https://doi.org/10.5194/os-2022-5-AC3
-
AC3: 'Reply on RC4', Tatyana Belonenko, 14 Mar 2022
-
RC4: 'Reply on AC2', Anonymous Referee #3, 12 Mar 2022
-
AC2: 'Reply on RC3', Tatyana Belonenko, 12 Mar 2022
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