Prof. David Ketcheson | Dispersive effects of bathymetry on water waves

Okay. Quite a quiet, gentle person. Thanks.

Yep.

Okay, thanks.

So let's resume our session. Our next speaker is David Ketcheson, and he will talk about dispersive effect of bathymetry on water waves.

Thank you, and I want to say thanks also to the organizers of this whole program. I was able to be here for 2 months earlier in the program, and that and this week have really been great for me as someone who doesn't primarily work in this community to get to know people and really learn a lot. So I will say first and foremost, I am not an expert on water waves, especially get that clear since there are so many experts on water waves here in the room. So I'm looking forward to lots of criticism. And helpful comments about what I'm doing. Let me just mention it's not going to the next slide. On my screen, it's showing the next slide. So I'm still sharing screen. It's just not moved on. I don't know. Oh, maybe it went to sleep or something. It says resume share. So maybe because I shared it like half an hour ago. Yeah, that's a new thing to me. OK, let me stop and restart. It's a new update. I didn't know that. So sorry.

No problem.

I updated everything this morning. All right, and now I can't find the play button.

There we go.

OK, so yeah, let me just mention— Do you want me to make that disappear? Yeah, if you can.

Thanks.

Oh, but now it's still not updating there. This is really strange.

That is strange.

I've got absolutely no clue why that's doing this.

Okay. Okay. Okay.

Let me just drag that out of the way. Great. So for purposes of credit, you can consider these people as co-authors. For purposes of blame, just me. So the first part of this talk is joint work with Giovanni Russo, actually done just in the past few weeks. And the second part is work done a few years ago with Manuel Quesada de Luna when he was at KAUST. So, that first part will be 1D depicted here, flow where the wave is flowing to the right or left over this periodic bathymetry. And then, in the second part, we'll turn that sideways. So, the flow on the top would then be into the screen, or there's an isometric view of what it'll be like here at the bottom.

Yeah. Okay.

So, I'm interested in what happens to water waves when they're flowing over a periodic bottom are periodic bathymetry. And here's just— actually, this is a correct depiction of the kind of scales that I'm interested in, as well as establishing notation. OK. So delta is the length of the period of the bathymetry. A is the surface. A to 0 is some reference unperturbed surface height. B is the bottom. This capital H, that bar seems to have gotten moved. It should be the distance between the reference surface height and the bathymetry. So it is a function of x. And here, I'm going to start— probably a dumb thing to do, come to a conference on dispersive waves and say I'm starting with a non-dispersive model. But I hope you'll humor me by sort of throwing the dispersion out at the beginning. The other sources of dispersion, we'll be able to more easily see The source of dispersion that I'm interested in. So, I'm starting with the shallow water equations, first-order hyperbolic system, right? No dispersion. And we're going to do some analysis in a moment where we'll make this formal assumption, although we'll see that it's not really strongly satisfied by the waves that come out of it. But that is that the waves are long compared to the period. of the bathymetry. And here, I've also shown another kind of periodic bathymetric structure, just to emphasize that the kind of results that you'll see here, and also the analysis, is valid for arbitrary periodic bathymetry. It doesn't matter if it's piecewise constant, sinusoidal, or something in between. So first, let me just show some solutions. This is really just to remind you how first-order hyperbolic PDEs behave. So, this is a wave in 1-meter deep water. The initial wave is very small, because that's what we're going to end up finding out that we need to be interested in here, just 1.5 centimeters, and it's propagating 500 meters. Obviously, there are concerns about the physicality of this, but let's just look at the math for now. And, as we always see with hyperbolic PDEs, we have a shock that forms here. This sort of N-wave. And this thing will gradually decay over time. It takes quite a long distance because it's such a small wave in this case. OK, on the other hand, here's what happens if you have that same wave, but now we have these sort of alternating deep and shallow regions, as depicted on the side here, where the unperturbed depth ranges from 20 to 100 centimeters. And this kind of behavior is familiar to everyone in the room here. We've seen lots of this kind of thing. But what's novel here is that I'm solving a first-order hyperbolic PDE. There are no dispersive terms in the equations. So I won't say much about the numerical methods here, although that's my primary area of work. So here, I'm using ClawPack. It's actually quite a challenge to solve this problem because Typical numerical discretizations, even if you start with a flat surface, will generate lots of waves because you have to have this balance between the bathymetric term in the momentum equation and the flux term. Yeah. OK, so my goal, then, is to explain this behavior from a mathematical perspective using multiple-scale perturbation theory to derive some constant coefficient PDEs that effectively describe the behavior of these waves. Obviously, we expect that they'll have dispersive terms in them. And the approach that I'm using was developed by Darryl Young in his PhD thesis about 20 years ago. OK, so we start from the shallow water equations. But h and u are not good variables to do this analysis with because they are rapidly varying with the bathymetry. So instead, we'll switch switch to the surface height eta and the momentum q. And here are what the equations look like in those variables. And then, we introduce a fast variable— I'll call it y— that is related to our original coordinate x by this small parameter delta. And we formally consider x and y now to be independent variables. So we replace all of our x derivatives with This d by dx plus 1 over delta d by dy. We assume some perturbation expansion in delta for each of our dependent variables. The leading terms always end up not to depend on the fast variable. So I've already assumed that here. And additionally, everything is assumed to be periodic in y, because of the periodic structure of the bathymetry. So now, yeah, our goal is to find evolution equations for the averages of eta and q, where we've averaged over each period of the medium. All right. So now we just make these 2 substitutions, that rule and those series expansions, into our original governing equations. And we collect terms at each order of delta. And then average them. So, at order 1 over delta, we get only this one term, which means that it must vanish. There's nothing to balance it. And the only way for this to vanish is that q itself, the momentum, has to be order delta. And if we follow this through, this means that the perturbation of the surface height is also order delta. And the surface height itself is just some— that leading term is just some constant independent of x and t. At the next order, we get these equations, just these y derivatives of what are now the leading terms in q and eta, and that their derivatives have to be equal to 0. So, we can also just sort of remove the y dependence of these 2 terms.

Right?

OK.

So, now we have just this form for the expansion. So far, nothing very interesting has happened. But now, things get more complicated, and we start finding out what these equations will be like. So, at each order, what we do is collect terms of order delta to the n, solve for the y derivatives of the highest order terms, average that, set it equal to 0, because everything has to be periodic in y. So, when we integrate over one period, we should get 0. Integrate the y derivative over one period. And unfortunately, the number of terms that appear here, because of the nature of calculus, And of actually just algebraic products too, grows exponentially. So this becomes painful. You need to use Mathematica already at the next order. So at order delta to the 1, we get this system. And when we take— so these are y derivatives on the left-hand side. So when we average the right-hand side, we should get 0. And this term actually— Because h is just a function of b, and this is the derivative of b, when we integrate this over one period, we get 0. And this starts happening a lot. A lot of terms that appear will average to 0. So we just get this equation, which is actually just the linear wave equation written in first-order form with this sound speed. Looks about right. It has at least the right sort of structure, square root of g times h. Oh, so the— The bracket notation here means the mean over one period. At the next order, I've omitted all the details because they wouldn't fit on the slide now. There's a number of other terms that appear, but all of their coefficients actually average to 0. And we get this pair of equations. And we have— oh, sorry. This should have a delta multiplying it to properly match here.

OK.

But we get the first-order wave equation. Then we get these first nonlinear terms. And they come in at the next order. So this is a weakly nonlinear system, because we assumed that our— or because we found that our waves had to be small.

Yeah? Can you go back to where you had qy squared? So is it q squared 1?

Sorry, you're asking about this term? Oh, no, no.

It's not squared.

Sorry. Yeah, yeah, yeah, of course.

Yeah, yeah.

Probably I should have put the superscript indices in parentheses or something.

But yeah.

OK. OK. At the next order, what happens? Finally, the thing that we were looking for. So, we have all the terms we had before, the linear part. This equation always is just this simple one, right? At all orders. But here, we get the sort of transport, the quadratic nonlinearities. And then, we get— this actually comes out, I mentioned this to Sergey last night, as a 3rd derivative in time. But I've used the equations themselves to exchange time derivatives for space derivatives, because that makes more sense to me. And then, we have some additional nonlinear terms.

Right.

But most interestingly, this dispersive term appears here. These coefficients, and you'll notice they're not 1, 2, 3, 4, 'cause there were many others, but they turned out to be 0. There are these complicated averages. So this double bracket has a meaning that's not so important for understanding what's going on here. But these get to be more and more complicated averages of the variation of the depth.

OK. Yeah.

So there we have dispersion. OK. And so here again is the solution over that periodic bottom. This is exactly what I showed you before. And now, here's using pseudospectral methods rather than finite volume methods, the solution of this homogenized constant coefficient system.

OK.

For exactly the same scenario. And here's the comparison between the two. So I think actually, like I said, this is work I just— I mean, I've still been working on it this week. So it's quite possible, and I think likely, that there may still be some source of error. We don't expect these to match perfectly because we've truncated. We haven't included the higher-order terms. So, there will be disagreement at long times. I think it's actually a little bit worse than it ought to be, and that there's still probably something wrong here.

Which is the right one?

So, sorry, the blue one is the solution of the first-order hyperbolic system, and that one is right. So, it's incredibly over-resolved. It is the true solution. correctness of pixels on the screen, let's say. OK. So what causes this? If you listened to my talk 2 months ago, then I talked about this quite a bit. I'm going to relate it to some things that have been discussed this week. So in Daryl Young's work 20 years ago, and this very nice paper with Randy Levesque, they looked at waves in an elastic medium with a periodic structure, the simplest case being just alternating between 2 materials, as depicted here. And you can think of these materials as having different densities and different stiffnesses. And if you consider weakly nonlinear elastic waves in this medium, then you get the same kind of behavior that I've been showing you here for water waves. And there, they noted that if you, for instance, take these blue layers to have, let's say, order 1 density and very, very high stiffness, k, and you take the red layers to have very low stiffness, so they're very spongy and very low density, then this is like a system of mass and springs. In a certain asymptotic limit, it becomes equivalent to a lattice. And we've had some discussion of lattices already this week. And we know that those lead to dispersion. So in some way, you can relate this to a discrete structure like a lattice. But you get these solitary waves even when you're very far from those asymptotic regimes in which the material is like a lattice. So, you could also think of here, the deep regions with a lot of water as being somewhat like the masses, and the ones here with a little bit of water as being like the springs. Although, the analogy is rougher there, and there's not a formal limit in which this becomes equivalent to a lattice. At least, I think some intuition can be gotten from that. There's also some really nice work from all the way back 30 years ago by Santosa and Symes, where they did this for linear waves and showed that linear waves in a periodic structure always exhibit effective dispersion. OK, so now I'm going to go into the 2D setting, where we'll turn everything sideways. Here's the depiction. Again, I'll show you examples that involve piecewise constant bathymetry, but the same kind of behavior is observed with other kinds of periodic bathymetry. Here's the shallow water equations in 2D. Again, no dispersion, first order system. And actually, let me go back and make one more comment here. So, in— this work, they showed that the dispersive terms that appear when you apply this multiple scales analysis, they all have coefficients that vanish if the impedance between these 2 materials is matched, in which case there would be no reflections. So you can think of reflection as being responsible in some way for this effective dispersion. Sort of the large-scale effect of all these small-scale reflections is a dispersed Is the dispersion. Now, in this setting, we're not going to have reflection, at least not as a sort of primary effect, because I'm thinking of roughly a plane wave propagating along these sort of troughs and ridges, right? Instead, what turns out to be responsible for dispersion is the fact that there's a difference in the speed of propagation in these 2 regions, right? So the characteristic speed of these gravity waves, of course, goes like square root of gh. The waves in the deep regions want to move faster. So here's a top-down view of the system. You can think, if you have a plane wave moving along this way, then the parts of it that are over the deep water want to go faster. And the parts over the shallower water want to go slower. And this ends up leading to energy propagation in the transverse direction and somehow leads to a dispersive effect. And if you go back to that Santos and Symes paper, They actually did this in 3D, or maybe an arbitrary number of dimensions, I don't recall. And you can see, actually, that this effect is there, although they don't explicitly call it out in their paper. It's there in the analysis. And Manuel and I wrote a paper on linear waves exploring this kind of dispersion directly more recently. OK, so unsurprisingly, Now, if you run water waves over this, since they're nonlinear shallow water waves, you get the combination of nonlinearity and dispersion leads to solitary waves. So this is a channel. These waves have propagated almost 800 meters. The channel is only 1 meter wide, to give you an idea of the scales involved. And the sort of— What's the wavelength? top or left half of the channel is shallow, and the right or bottom half here is deep. And this is just a couple of cross-sections through this solution along the middle of the deep and shallow parts of the channel.

Do you have periodic boundary conditions?

Yes. Thank you, Mark. You know the answer to that question, but thanks for reminding me to say it. So, yeah. So, we use periodic boundary conditions. And why? Although, I'll come back to that in just a moment. Yeah, actually, right here. So perfect question. So we have this— we imagine infinite periodic structure here in the transverse direction. But if you think about it carefully, you realize that the solution of that problem, when this is a plane wave, is actually identical to flow in this channel where we've just taken half of one period repeated anti-symmetrically. Infinitely many times, of course, where we also have to assume that there's no friction at the walls. But because of the symmetry of the structure, these 2 have the same solution. So you can think of this as really a property of flow in a channel where the cross-section is not rectangular, rather than about flow over periodic bathymetry. And once we realize that— We found that there's very nice work that's been done on that already, a very long time ago, by Peregrine. He analyzed this starting from the Euler equations, so including inherent water wave dispersion as well. And the result, to a certain order at least, is this sort of KDV equation with a modified dispersion coefficient. So, if kappa was 1 here, then this would be The regular KDV model for flow in a channel. But when you have a non-rectangular cross-section, you get some other value of kappa here, where to find it, you have to solve some elliptic PDE. And actually, I didn't mention it, but if you do the homogenization analysis for this problem in 2D, like I did in 1D, then you find that the coefficients of the terms that appear are solutions of elliptic PDEs.

Yes?

So, Sorry, just to understand, where do we get the situation when the resonant dispersion dominates? Yes, very good question. I'll try to say something about that in a moment. Yeah, thanks. So this is, I think, actually the first natural question you would have. You say, there's some other source of dispersion in water waves. Well, water waves are naturally dispersive. So would you actually see this effect? Or is it always smaller? than the natural dispersion from water waves. So this is what we got by homogenizing the shallow water equations, again, a KDV-type equation to this order. And this is just comparing the homogenized model with direct solutions of the shallow water equations and then averaging that across the channel. And you can see, similar to the comparison that I showed earlier, the agreement gets worse over time. Although, this is what I expected to see in the 1D case too, that when they're starting to break up into solitons, that there's still reasonable agreement. So, I think there's some improvements still to be made there in the 1D setting. Okay. So, this is to answer your question. And here I've plotted for a particular choice of type of channel cross-section, but for a range of So this is a piecewise constant bathymetry with 2 different depths in the channel. And along this axis, we're altering the deeper part of it. So we're changing the ratio of those 2 depths. And this is the dispersion relation for the shallow water equations homogenized for the KDV equation, which—

What's KDV?

is just over a flat bottom of the average depth, right? And for Peregrine's model, which takes everything into account. And you can see, as it should, Peregrine's model agrees with the KDV equation in the limit where the channel is almost rectangular, right? Well, here it's a rectangular channel. And eventually, it gets close to the homogenized shallow water model, when the bathymetric dispersion becomes extremely strong. And in between, it sort of combines these 2 effects. So once again, b0 is the ratio of what? So let's see. I'm trying to remember. Oh, yeah. OK. So we have a channel. And one side has bathymetry at 0. And the other side has bathymetry at b0. And each of those occupies half of the channel.

Yeah.

Thanks for asking. And the surface, the unperturbed surface height is 75 centimeters. So at this point, you would get a dry state on the other side. So this is sort of the full range of possibilities. And so you can see there's a regime where each of these is dominant. And this is just to show you an example of actual flow in a channel rather than periodic bathymetry and with a different kind of You know, cross-section there, you get similar behavior. Yeah. And then, okay, so to come back to the question of physicality of this all, I think these 1-dimensional waves, there's no hope of realizing them. I mean, even as a non-expert, right? They're 1 centimeter tall. They have to propagate, you know, hundreds of meters, right? There's other physical effects that are going to dominate and I think disrupt them. But in 2D, There are regimes where these waves should be experimentally realizable. This is a simulation of the full 2-fluid free boundary problem. So here we're simulating actually the air and the water using the Navier-Stokes equations. We're including viscosity, surface tension, friction, the boundaries, et cetera. What I'm not showing you here— here, let me go back. Because I want you to be able to see the waves. Oh, I must have cut that slide.

OK.

So the bottom is approximately here. I can find the slide later for someone who's interested. This is not— the bottom is not here. That's what I want to emphasize, to give you an idea of the scale that's necessary to get this kind of domination. If you had a flat bottom, no matter whether it was up here, somewhere in between, or all the way down here, would get wave breaking very quickly. These waves are large enough amplitude, they would have broken long before when this result appears. But in this situation, you get not the perfect solitary waves that we saw before. And there's obviously some 3-dimensional structure back here. But the leading wave actually looks pretty uniform and has a fairly traditional solitary wave-like structure to it.

Okay.

So, I hope that in the future sometime we can actually see these things in practice. Yeah, some interesting points maybe for people who work in nonlinear dispersive waves is that in these systems, you can choose parameters that will change the dispersion coefficient without needing to change your fluid or something like that. You're just changing the geometry of the problem, which may be easier in a lot of settings like with water waves. Yeah. And there's also maybe a lot of interesting work to be done, I think, looking at the homogenized equations that come out of these. And yeah, what remains an open problem is really developing understanding directly of the variable coefficient first order systems. I think in general, variable coefficient PDEs are much less understood. Right, than their constant coefficient counterparts. And this is an instance of that.

Yeah.

OK, there's obviously a huge number of references on water waves over bathymetry. I'm not gonna try to mention those. I'm actually just focusing here on references of waves in periodic structures, nonlinear waves in periodic structures. And I'm sure that I'm still leaving out lots of things. probably by some people in the room, so forgive me for that. And please let me know what I'm leaving out that I should know about. But here are a few interesting related references and some of the papers related to things that I showed here. So thank you.

We have time for one more question. But then in your approximation, do you still get the number of solitons right even if the distribution is not correct?

Yeah. It seems. Yeah. At least in the examples that I showed and all the examples I've seen.

But kind of how I'm grading, I think the failure in the averaging is that the averaging of the collisions is not equal to the inherent collision. So you're missing faces. Perhaps where you are weak is the average of collisions.

Yeah.

The breaking into the past distance. That's, I think, the average you're taking.

OK. Yeah. In the paper of Yang and Levesque, they actually carry out this process to significantly higher order, 2 or 3 more orders than what I've done here. And what comes out are higher and higher order dispersive terms. Which I guess would be most important when you have interaction of the solitary waves.

So in your homogenized equation, you said that you swapped x and t derivatives to get the 3rd derivative, 3 derivatives in space. But just like others, like Roberto this morning showed, you could take other choices.

Right. We could write that as xxt or— yeah.

And you can choose it in order to try to get a better fit to the dispersion of the system.

Oh, so, OK, I didn't realize that making different choices there will—

A fitting of the dispersion to the high wave numbers.

OK.

That might be helpful.

Oh, yeah, because I guess these are only equivalent really in the limit of small delta.

Yeah.

Oh, and one thing I forgot to mention, so these waves end up being just on the order of, like, 4 or 5 times delta periods long, which reminded me of— there was a talk about lattices. I think Panos Kevrakidis talked about the solitary waves that come out were like 4 or 5 sites wide. And I think the reason is the same here. So this small parameter isn't so small in practice.

Yeah.

Just a little comment. There is one natural situation in fluids Perhaps more natural than what you're considering at the moment, where all this could be relevant. So imagine a pipe, yeah, and the fluid flow in a pipe, and quite often you get gas, air bubbles inside of the pipe, yeah, which kind of— so you can get a configuration where you have a 2-phase mixture, and it's fluid, gas, fluid, gas, and it's similar kind of to your masses and springs, yeah?

Yes.

Gas is like your spring, and fluid is your mass.

Yes.

And so—

So there's actually a very recent nice paper by Sergei Gavrylyuk and Giovanni Russo, one of my collaborators here, on shockwaves, actually, in alternating layer gas mixtures. And I have some unpublished simulations. Actually, I have a skipped slide here, but—

Yeah, sorry. Just don't know.

Showing that you can get solitary waves, the same kind, in solutions of the compressible Euler equations of gas dynamics. So in this setting. Yeah.

Can you do nearly periodic configuration rather than exactly periodic?

Sorry?

Yeah. Can you kind of remove this strict periodicity condition? Oh.

If you— you mean if you add some randomness to it?

Or— Yes. Some randomness.

So yes, but you lose the really nice structure of the solitary waves. And the more you move away from a perfectly periodic structure, the more disordered the resulting waves become.

Thank you. So let's thank the speaker again and move to the next one. The next speaker is Alberto Villois from Torino, and he will speak about dissipative solitons and breathers in quadratic microresonators. No worries.

All right. So, well, this work has been done in collaboration with Dimitris Kriabin when I was in Bath, when I was a postdoc in Bath a few years ago. I started working on dissipative solitons in quadratic microsonators. And now I am extending this work, working with Miguel at University of Turin. So I guess most of the audience should be familiar with what solitons and breeders are, so I'm not going to describe