|This manuscript considers internal solitary waves with multiple scales in the horizontal (waves with two or more humps). This version of the manuscript has been polished up from the previous version I have read, and I am essentially treating it as a new manuscript. The English is OK though no better than OK, and I give a list of corrections that is most certainly not exhaustive below The technical graphics range from good (e.g. Figure 5) to somewhat grainy (e.g. Figure 1). The figures captions are very sparse and the discussion of the Figures in the paper is similarly sparse. In some cases the fonts are quite small at the sizes of the panels shown. In the final analysis, none of this really matters to my assessment of the paper, but it would matter to a casual reader and certainly to students. I have been intrigued by multi-scaled solitary waves for many years, and have tried to calculate them by various means using a solver for the DJL equation. While I have been successful in constructing multi-scaled solitary waves using two separate computations that are stitched together, I have never been able to generate them spontaneously just by setting a value for the APE and letting the variational algorithm do its thing. I worked quite hard to try to make this happen with the parameters the author provides (and is to be commended for taking care to ensure this is the case), and was not successful. It is possible that the free surface which the author accounts for in his asymptotic procedure, but which is absent in the formulation of the DJL equation I use, is essential for this, but I cannot see a physical reason why. I am thus left wondering about whether the waves shown are a mathematical curiosity, or whether they could actually be observed. But that is my opinion and should not stand in the way of publication of the manuscript. I do note it, since science should be replicated and in this case, I was unable to replicate the science reported.|
The manuscript, in my opinion, starts to drift when the topic of trapped cores is brought up. . Trapped cores either invalidate the assumptions used to derive the DJL equation, or are a dynamical feature that naturally evolves, for example when a wave shoals. They have been extensively discussed in the literature, and none of this discussion appears in either the text or the references. I understand the author wants to present his ideas about wave stability and hence is not obliged to fully review the literature, but some commentary would help the reader orient the present study.
I thus hope the author will adopt a subset of the suggestions below, and after this I think the manuscript can be published.
1) The title seems excessively general. At the very least “in a stratified fluid” should be added
2) “then” refers to a comparison in time as in “I ate lunch and then I ate supper”. “than” is the correct word when the author states “the family of solutions is richer than two-humped structures”. A similar error occurs at other points in the article.
3) The description of what we did in Dunphy et al isn’t quite right. It might make sense to describe Lamb and Wan’s work first since their result is what allowed us to construct the multi-scaled solitary wave solution. The part about nearly identical profiles isn’t really relevant to that aspect of the study. I suppose the author feels it is important to mention since he argues that very small differences in density profiles can make the difference in whether multiscaled waves do or do not exist in his formulation. In the two pycnocline example from Lamb and Wan we were following, small differences in stratification made no difference to the calculation.
4) On page 3 the discussion of the Weierstrass approximation theorem has been expanded but I still think it’s a bit unclear. At the very least the author should state that f(z) is now taken to be a polynomial (may be before equation (11)). It would be helpful to tell the reader whether going from the general expression (11) to the specific result (17) is algorithmic, or whether it just kind of worked out for this choice of f(z). I also think the section heading “Multiscaling” occurs in an odd place, since the first sentence ties in very nicely to the last pargraph of the previous section. Finally, it would be very useful to have a table with f(z), P_N(z) for the various special cases discussed, possible with a column for relevant figures.
5) Is phase velocity the correct term? I am not aware of a group velocity for solitary waves, so wouldn’t propagation speed be easier to understand?
6) Both the figure captions, and the discussion of the figures in the text are very brief. Figure 1 is very useful. I would tighten the axes to show how special the region required really is, and I would add a vertical line at the value of alpha used on page 4. Then I would describe this in the text.
7) On page 7 the author states that waves “evolve”. This is misleading, since there is no temporal evolution, merely a tracing of the form that the wave takes in parameter space.
English: “In a Russian journal”, “dissimilated” should be changed to “disseminated” or some similar word, “Assumption of small, albeit finite”, “a priori” not “a priory”, “Multisclaed” not “Multscaled” , “fourth order” not “forth order”