Effects of Rotation and Topography on Internal Solitary Waves Governed by the Rotating-Gardner Equation

Helfrich, Karl; Ostrovsky, Lev

doi:https://doi.org/10.5194/npg-2022-3

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https://doi.org/10.5194/npg-2022-3

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21 Jan 2022

Status: this preprint is currently under review for the journal NPG.

Effects of Rotation and Topography on Internal Solitary Waves Governed by the Rotating-Gardner Equation

Karl Helfrich^1, and Lev Ostrovsky^2, Karl Helfrich and Lev Ostrovsky Karl Helfrich^1, and Lev Ostrovsky^2,

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¹Department of Physical Oceanography, Woods Hole Oceanographic Institution, Woods Hole, MA USA
²University of Colorado Boulder, Boulder, Colorado, and University of North Carolina at Chapel Hill, Chapel Hill, NC USA
The authors contributed equally to this work.

Received: 18 Jan 2022 – Discussion started: 21 Jan 2022

Abstract. Nonlinear oceanic internal solitary waves are considered under the influence of the combined effects of saturating nonlinearity, the earth's rotation, and horizontal depth inhomogeneity. Here the basic model is the extended Korteweg-de Vries equation that includes both quadratic and cubic nonlinearity (the Gardner equation) with additional terms incorporating slowly varying depth and weak rotation. The complicated interplay between these different factors is explored using an approximate adiabatic approach and then through numerical solutions of the governing variable-depth, rotating Gardner model. These results are also compared to analysis in the Korteweg-de Vries limit to highlight the effect of the cubic nonlinearity. The study explores several particular cases considered in the literature that included some of these factors to illustrate limitations. Solutions are made to illustrate the relevance of this extended Gardner model for realistic oceanic conditions.

Karl Helfrich and Lev Ostrovsky

Status: final response (author comments only)

RC1:
'Comment on npg-2022-3', Anonymous Referee #1, 10 Mar 2022

The review of the manuscript

"Effects of Rotation and Topography on Internal Solitary Waves Governed by the Rotating-Gardner Equation"

by Karl R. Helfrich and Lev Ostrovsky

This referee was asked to join the discussion at the late stage, when the previous comments were no longer visible on the system. Hence, my comments are complitely independent of the previous discussion which I have not seen.

The paper continues the line of research on internal waves within the scope of the KdV-type models. This study is based on the use of the variable-depth, rotating Gardner model, i.e. it aims to highlight the effects due to the common action of nonlinearity (including cubic nonlinearity), dispersion, rotation and depth inhomogeneity. Analysis is based on the previously developed adiabatic theory (with some corrections), and is compared to the results of direct numerical simulations of this model equation using a pseudospectral method. The results presented in a number of high-quality Figures look instructive and convincing, showing good agreement for the case of weak nonlinearity, and clarifying the limitations of the adiabatic theory when the amplitude parameter is increased.

In the view of this referee, the following points need to be clarified in order to put the research in the context of other studies:

1. Could you, please, provide a reference to the derivation of this model equation from the primitive equations. Can it be derived in a systematic way? Does this model include all terms appearing at the relevant order of the amplitude parameter?

If not, please, explain why and when we can disregard some terms appearing in the systematic asymptotic derivation (e.g., the 5th derivative and nonlinear-dispersive terms in the extended KdV equation). I suspect that there could be some additional terms in the case of variable topography and rotation.

Also, please, compare to the results in this paper and discuss in that context:

A. Karczewska, P. Rozmej, Can simple KdV-type equations be derived for shallow water problem with bottom bathymetry? Comm. Nonlin. Sci. & Numer. Sim. 82 (2020) 105073.

2. The results of the study need to be compared and contrasted to the two recent papers by Yury Stepanyants:

- Yu. A. Stepanyants, The effects of interplay between the rotation and shoaling for a solitary wave on variable topography, Stud. Appl. Math. 142 (2019) 465-486.

- Y.A. Stepanyants, Nonlinear waves in a rotating ocean (the Ostrovsky equation and its generalizations and applications), Izvestiya, Atmospheric and Oceanic Physics, 56 (2020) 20-42.

In particular, in these studies it was shown that for solitary waves moving towards shallower waters, the terminal decay caused by rotation can be suppressed by the shoaling effect. Is this result confirmed in your studies?

3. This sentence in the Introduction is only partially correct:

"It is also known that rotation destroys internal solitons due to resonant radiation of inertia-gravity waves (terminal damping; see Grimshaw at al., 1998a)."

This is true in the absense of currents. However, the sign of the rotation coefficient in the Ostrovsky equation may be changed by the underlying shear flow, and then the equation supports solitons, see the first examples in

A. Alias, R.H.J. Grimshaw, and K.R. Khusnutdinova, Coupled Ostrovsky equations for internal waves in a shear flow, Phys. Fluids 26 (2014) 126603.

4. Lines 19-20, should it be 'specific case'?

To summarise, assuming that the model equation can be justified, the study makes perfect sense, and the results are convincing. The validity of the model equation is not entirely clear and needs to be clarified by putting the research in the context of some other studies in the field, as suggested above.

Citation: https://doi.org/10.5194/npg-2022-3-RC1
- AC1: 'Reply on RC1', Lev Ostrovsky, 05 Apr 2022
  
  See attachment please.
  
  Citation: https://doi.org/10.5194/npg-2022-3-AC1
- AC2: 'Reply on RC1', Lev Ostrovsky, 05 Apr 2022
  
  The answer is placed as a Supplement for RC1. Sorry, we are a bit confused.
  
  Citation: https://doi.org/10.5194/npg-2022-3-AC2
RC2:
'Comment on npg-2022-3', Anonymous Referee #2, 21 Mar 2022

The comment was uploaded in the form of a supplement: https://npg.copernicus.org/preprints/npg-2022-3/npg-2022-3-RC2-supplement.pdf

Citation: https://doi.org/10.5194/npg-2022-3-RC2
- AC3: 'Reply on RC2', Lev Ostrovsky, 05 Apr 2022
  
  The comment was uploaded in the form of a supplement: https://npg.copernicus.org/preprints/npg-2022-3/npg-2022-3-AC3-supplement.pdf
  
  Citation: https://doi.org/10.5194/npg-2022-3-AC3

Karl Helfrich and Lev Ostrovsky

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Short summary

This paper studies internal solitary waves (ISWs) in the ocean under simultaneous action of the main defining factors: nonlinearity (quadratic and cubic), earth rotation, and depth variation. Both analytical (adiabatic) theory and numerical simulation are used in the study. The results show a complex interplay of the factors mentioned above that can result in amplification of the ISW or, on the contrary, its destruction. The calculations are made for realistic oceanic parameters.


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