Quasi-Linear Model of Tsunami Run-Up on a Beach with a Seafloor Described by the Piecewise Continuous Function

Round 1

Reviewer 1 Report

In this paper, an approximate method for calculating the run-up of small-amplitude waves on the shore is developed. In essence, it is based on a linear description of the wave motion in the seaward part (up to the shoreline) and the geometric continuation of the wave field to the land area. To some extent, this approach has been used many times in the literature, although usually the solution of the linear wave equation in the seaward part is matched with the well-known analytical solution of the nonlinear problem of wave run-up on a plane slope. A purely linear continuation of the wave field on a dry coast, as I remember, was done in (Mazova R.Kh., and Pelinovskiy Ye.N. Linear theory of tsunami wave run-up on a beach, Izvestiya. Atmospheric and Oceanic Physics, 1982, Vol. 18, No. 2, 124-128). In essence, it corresponds to the quasi-linear continuation of the wave field in the work under review.

Basically, I have three comments:

1. The authors use only one small parameter - the ratio of wave height to depth (nonlinearity parameter). However, in Section 3 (formula 3.5) they obtain a condition for the smallness of the nonlinearity, which guarantees that the waves will not break. This condition exactly coincides with the breaking criterion, which follows from the exact solution of the nonlinear problem of wave run-up on a plane slope. The authors do not refer here to this exact criterion, which is derived in a more elegant form in the papers cited in the article (and in several others). Such a coincidence is obvious to me, since the authors use the constant slope approximation in the vicinity of a fixed shoreline. If they had considered a more complicated case, when the slope angle changes abruptly on a fixed edge (and this is a typical situation in geophysics, when, for example, the Dean profile x^2/3 is stitched with an almost flat bank), then the breaking would be possible on the bank ( and not at the point of a fixed shoreline), and the author's formulas will not be suitable (as well as the exact solution of a nonlinear problem, which is not available in analytical form).

2. The authors, for reasons not clear to me, require the disappearance of the wave field at infinity, where the depth is constant (formulas 1.7, 1.8, 1.21, 2.10). Obviously, this is not the case, since the solution of the initial problem with zero initial velocity gives two waves, one of which goes to infinity with a constant amplitude (since the depth is constant). The same applies to the wave reflected from the coast, its amplitude decreases in the zone of increasing depth, but then becomes constant on the shelf of constant depth.

3.English must be polished.

It seems to me that the article, in principle, can be published after a major revision.

Author Response

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Author Response File: Author Response.docx

Reviewer 2 Report

The work presents the quasi-linear theory of tsunami run-up and run-down on a beach with complex bottom topography. The presented method shows potentialities that can be implemented in  estimating tsunami run up. I see however some points that need to be addressed.

     1.  The authors replaced the moving boundary with a stationary boundary by applying a transformation. Authors need explain it clearly and justify the same.

2. Fix grammatical errors like “… the effect the different beach profiles and initial wave locations are considered”.

3. Linearizing the problem using the method of perturbations requires detailed explanation.

Author Response

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Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

My comments are accounted and I recommend publishing as it is