Limiting amplitudes of fully nonlinear interfacial tides and solitons

Aguiar-González, Borja; Gerkema, Theo

doi:https://doi.org/10.5194/npg-23-285-2016

Articles | Volume 23, issue 4

https://doi.org/10.5194/npg-23-285-2016

© Author(s) 2016. This work is distributed under
the Creative Commons Attribution 3.0 License.

https://doi.org/10.5194/npg-23-285-2016

© Author(s) 2016. This work is distributed under
the Creative Commons Attribution 3.0 License.

Articles | Volume 23, issue 4

Research article

|

18 Aug 2016

Research article |

| 18 Aug 2016

Limiting amplitudes of fully nonlinear interfacial tides and solitons

Borja Aguiar-González and Theo Gerkema

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Final revised paper (published on 18 Aug 2016)
Preprint (discussion started on 19 Jan 2016)

Interactive discussion

Status: closed

AC: Author comment | RC: Referee comment | SC: Short comment | EC: Editor comment

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RC1: 'Subject', Anonymous Referee #1, 16 Feb 2016
- AC1: 'Response to Referee 1', Borja Aguiar-González, 15 Mar 2016
RC2: 'On the manuscript “Limiting amplitudes of fully nonlinear interfacial tides and solitons”', Anonymous Referee #2, 01 Mar 2016
- AC2: 'Response to Referee 2', Borja Aguiar-González, 15 Mar 2016
RC3: 'referee report', Anonymous Referee #3, 17 Mar 2016
- AC3: 'Response to Referee 3', Borja Aguiar-González, 11 Apr 2016

Peer-review completion

AR: Author's response | RR: Referee report | ED: Editor decision

AR by Borja Aguiar-González on behalf of the Authors (14 Apr 2016) Author's response Manuscript

ED: Reconsider after major revisions (further review by Editor and Referees) (19 Apr 2016) by Victor Shrira

AR by Borja Aguiar-González on behalf of the Authors (25 Apr 2016) Author's response Manuscript

ED: Reconsider after major revisions (further review by Editor and Referees) (25 Apr 2016) by Victor Shrira

ED: Reconsider after major revisions (further review by Editor and Referees) (27 Apr 2016) by Victor Shrira

ED: Referee Nomination & Report Request started (01 May 2016) by Victor Shrira

RR by Vasiliy I. Vlasenko (10 May 2016)

RR by Anonymous Referee #3 (23 May 2016)

Suggestions for revision or reasons for rejection

This is a substantially revised and improved version of the manuscript (NPG-2016-1). The authors have made a thorough attempt to address my previous concerns. However, I still have two remaining issues that are serious and must be addressed.

The authors’ response to my first comment regarding inclusion of the barotropic forcing is incorrect. Their model employs a rigid lid. As a consequence, x-integration of the continuity equation (45) (in the new manuscript) does give (48) with the undetermined function of time C(t), as they show. In the case when h_t = 0, this function is clearly the depth-integrated, barotropic, transport. It must be independently specified (because of the rigid lid). However, it is not necessarily equal to zero for all time simply because u_i = 0 at t=0 as the authors claim (line 273). The authors are making a choice to set C(t) = 0 and oscillate the topography. There is no mathematical or physical problem with making the opposite choice to take h_t = 0 and specify C(t). Both are approximations to the real ocean. However, the latter is less of an idealization than choosing the oscillating topography. Specifying the barotropic transport C(t) does not allow for spatial variations of that transport as would occur with a free surface (in which case an additional dynamical equation for the barotropic mode would be required). Specification of C(t) is common in fully-nonlinear models of this type as as, for example, in the work of Lamb (1994, JGR 99) and the book by Vlasenko et al (2005) (their equation 1.85b is the equivalent of setting C(t) = Q_0 sin(\omega t)) and related papers. The fact that the authors have reduced the stratification to two layers and assumed weak non-hydrostatic dispersion does not change the situation.

Again, I suggest that the authors consider reformulating the problem for an imposed barotropic flow rather then a moving topography. However, at a minimum they need to revise their discussion of the issue to take the comments above into account.

The second concern has to do with the quasi-linear model and the interpretation of those calculations. The revised version now makes clear to me what has been done and it then raises new issues. They use this quasi-linear model to argue that the radiated internal tide amplitude reaches a limiting amplitude because of the forcing of the first harmonic. However, from Fig. 2 the departure of the quasi-linear result from the linear calculation occurs when Fr \approx 1 and larger. This is just the regime when nonlinearity in the baroclinic mode is O(1). So while there is generation of the first harmonic by the quasi-linear terms (in B11), it is not at all clear that this is a significant factor in the response in the fully-nonlinear model at Fr \approx 1. Said another way, I am not convinced that this second harmonic from the quasi-nonlinear terms is more important than other nonlinear effects (that will also result in higher harmonics) that occur when Fr = O(1). Lastly, while it may be a fine point considering the preceding discussion, I do not see a clear limiting amplitude in the quasi-linear results in Fig. 5. Neither a local maximum nor a well-define asymptote is obvious. If it does occur is must be for Fr > 1.6, even further into the regime where the quasi-linear model and interpretation is suspect.

I also suggest strongly that they remove the material on the quasi-linear calculations. The results are suspect and potentially misleading regarding the source of any nonlinearity in radiated baroclinic mode when Fr = O(1).

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ED: Reconsider after major revisions (further review by Editor and Referees) (23 May 2016) by Victor Shrira

AR by Borja Aguiar-González on behalf of the Authors (27 May 2016) Author's response Manuscript

ED: Referee Nomination & Report Request started (30 May 2016) by Victor Shrira

RR by Anonymous Referee #3 (06 Jun 2016)

Suggestions for revision or reasons for rejection

The authors are wrong about the barotropic flow issue. If the assumption of a rigid lid is removed, then the barotropic flow must be solved for as part of the problem. But with a rigid lid the barotropic flow must be specified. Their equation 48 with the topographic motion set to zero (U=0, as my most recent review suggested that they do, but still retaining a stationary topography) is still correct and C(t) is the barotropic mass flux. For the moment consider the shallow water limit (the non-hydrostatic dispersion terms are zero) then their equations (41), (42) and (48) (with U=0) are equivalent to (2.1)-(2.3) in Sandstrom and Quon (1993), where the barotropic flux is imposed (q(t) in that paper). The addition of the non-hydrostatic terms does not change this (see my earlier review comment on barotropic flux in Lamb (1994, JGR 99) and the book by Vlasenko et al (2005)). The authors seem to implicitly understand this as they impose a purely baroclinic flow in the far field (lines 273-275). They are specifying the barotropic flux! Setting it to zero is simply a specific choice of C(t), but not the only one. As before, I think the authors just need to discuss briefly that they choose to oscillate the topography instead of imposing the barotropic flux through C(t) and that with a rigid lid there is no need to solve independently for the barotropic flow.

Regarding the second point: They say in the abstract “we use the model equations to investigate the role of the initial stages of the internal tide on limiting the amplitudes of solitons under fully nonlinear conditions.” My point was that for Fr = O(1) the initial stages (i.e. the internal tide generated by the topography) is a fully-nonlinear problem. The Sandstrom and Quon papers do break the problem up, and I agree that a unified approach is preferred. However, Sandstrom and Quon use the shallow water model (fully nonlinear) to study the generation, and then the weakly non-hydrostatic model for the far field. The authors should at least show the radiated internal tide from the fully-nonlinear and the quasi-linear models agree with each other in the immediate neighborhood of the topography for the Fr = 0.5-1.6 range. If they do not agree, then the quasi-linear results are not directly relevant.

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ED: Publish subject to minor revisions (further review by Editor) (06 Jun 2016) by Victor Shrira

AR by Borja Aguiar-González on behalf of the Authors (22 Jun 2016) Author's response Manuscript

ED: Publish subject to technical corrections (07 Jul 2016) by Victor Shrira

AR by Borja Aguiar-González on behalf of the Authors (18 Jul 2016) Manuscript

Short summary

We derive a new two-fluid layer model consisting of forced rotation-modified Boussinesq equations for studying the limiting amplitudes of tidally generated fully nonlinear, weakly nonhydrostatic dispersive interfacial tides and solitons. Numerical solutions show that solitons attain in some cases a limiting table-shaped form, but may also be limited well below that state by saturation of the underlying quasi-linear internal tide under increasing barotropic forcing.