HYDW08 | Prof. David Ketcheson | Solitary Water Waves Created by Variations in Bathymetry

So I can leave the floor to David Ketcheson, who's going to tell us about solitary water— sorry, solitary water waves created by variations in bathymetry.

Thank you, thank you, Constanza.

Thank you to the organizers for inviting me to speak and to all the speakers who've made this a very interesting meeting.

Yeah, so before I start, I just want to emphasize this is Everything I'll present today is joint work with Manuel Quesada de Luna, who's also at KAUST.

And I'm going to focus on maybe qualitative aspects and leave out most of the technical details, but I'll provide references at the end for those who want to delve into those. So the starting place is is a funny one maybe for this meeting since it's a non-dispersive system of equations, but I'm starting from the shallow water equations, I think familiar to everyone in the audience, and they've been mentioned many times already during the conference.

Here they are in 2 dimensions.

Of course, they're depth averaged, right? h here is the depth, u and v are the velocity components, and b is the bathymetry.

So this is a first-order nonlinear hyperbolic system, and of course, generically, solutions of the system exhibit shock formation after a finite time.

And so we look for the vanishing viscosity weak solutions. And with a flat bottom, right, if b is constant, at least this model has no dispersion in it, no traveling wave solutions. Right? And so it's quite different from most of the wave models that have been discussed here. But I'm going to look at not a flat bottom but a particular kind of bathymetry depicted here.

So I'm thinking about initially something that's a plane wave or close to a plane wave traveling along this direction, but in the transverse direction to that, there's this variation consisting of like sort of deep channels and then shallower ridges.

Here's a top-down view. And imagine that this goes on forever in the y direction, right, or along this axis. And I'm interested in when waves propagate over long distances denoted here by L, much longer than this period of the bathymetry, which I'll denote by omega.

So first, my interest is really in the solutions of the nonlinear problem.

But let's go and look at what happens for small amplitude waves to understand the effects of bathymetry.

And so I'm going to assume, like I said before—

well, no, this is a new assumption— so that the wavelength of the waves I'm interested in is much longer than than the period of the bathymetry. And the small parameter there we'll denote by delta. That will be the parameter used to homogenize in a moment. And then roughly speaking, we'll assume that the waves are small compared to the overall depth in order to linearize the equations. So we introduce a fast scale, which is related to our original scale, but only in the y direction. The fast scale in the y direction.

And we assume that the bathymetry now formally just depends on this fast scale. And if we linearize the system with these assumptions, then we end up with the system at the bottom here, where eta now is the surface height, and mu and nu are the momenta.

So h times the velocity, right?

And eta star here is the unperturbed surface height about which we've linearized.

OK.

And then what I'm going to do now is expand all these dependent variables as power series in my small parameter delta and derive a constant coefficient system.

Right here we still have variable coefficients because b depends On space. So we introduce these formal expansions, right, insert them into the equations, and after, you know, several pages of derivations, which I'll leave out here, we, we obtain a new system of PDEs with constant coefficients, but extending to, you know, whatever order we care to go to in delta.

Right, and it turns out that only even powers of delta will appear in this, and here's just the sort of first corrections to that lowest order system. So at lowest order, we get essentially the original system but linearized, which appears on the left-hand side here.

This M denotes the arithmetic average, and H denotes a harmonic average of these quantities over one period of the medium.

And then the terms appearing on the right here, as you might guess, and it turns out to be correct, these are dispersive terms, these third derivative spatial terms, and the coefficients modifying them are given by solving certain elliptic PDEs that depend on the bathymetry and taking averages of those solutions.

This is what usually happens in this kind of homogenization procedure.

And an important fact here, not surprising, shouldn't be surprising, is that if the bottom is flat, then all of these coefficients, and indeed all the, all the higher-order terms, will vanish, right?

And we'll just end up with the linearized first-order system. Okay.

Yeah, so going back to this piecewise constant diffimetry, and I should say I'll talk about this later in the talk, but There's nothing essential about this piecewise constant structure to any of the results that I'll show you. Similar things happen if you have a, you know, smoothly varying periodic bottom.

Okay, but this one's easy— easier for us to compute with for a few different reasons. Okay, so here's the structure of the, the bottom, right?

And here I'm just showing you for an example of one of these dispersive term coefficients, right?

And I want to emphasize here that the dispersion is strong when the bathymetry variation, which here is given by this b0, right, is large relative to the overall depth, where eta star here is that average overall depth, right?

Yeah, the smaller that is, the less important this source of dispersion will be.

Okay, and here's just a comparison between, in red, the, the solution of the fully nonlinear shallow water equations, and in black, that homogenized linear system with the dispersive terms that I just derived. And this is for a very small amplitude wave so that they agree for, you know, relatively large times. Here the x-axis is given in units of omega. So here the wave has traveled almost 50— of course, it's traveling in the perpendicular direction, but it's traveled 50 of those units. You see that eventually at later times, even for this small amplitude wave, there are differences. And this is because of the nonlinearities in the shallow water model. And in fact, I'm showing you in red here a slice along, I think it's the A layer, and in dotted blue, a slice along the B layer. So I'll show a lot of 1-dimensional plots even though the problem that I'm dealing with here is 2-dimensional. Usually I'll be either averaging in the y direction or just taking slices through the solution.

OK.

So yeah, so the first-order hyperbolic system, of course, has these characteristic speeds given by u plus or minus the square root of gh.

So this means that waves in deep regions move faster. So from a sort of physical intuition point of view, we can imagine that if we have a plane wave, right, moving in this direction— now I'm looking sort of top-down at the bathymetry—

then the parts of that plane wave that are in deep water want to sort of move faster and get ahead. And what this leads to then is propagation of energy in the transverse or in the y direction. And this is what leads to that dispersive effect that we see in the homogenized equations. And I'll refer to this in the rest of the talk as bathymetric dispersion because it has, of course, nothing to do with the dispersion that's in, say, the KDV or Boussinesq equations. That's an inherent property of water waves and that actually was ignored in the derivation of the shallow water equations. Right? So this is a totally separate effect.

And yeah, if you go back to this really great SIAP paper from 1991 of Santos and Symes, actually you can work out that this dispersion is predicted there.

And there's a more recent paper where we looked at this kind of dispersion very specifically and worked out more details of it for linear waves. Those are both references for the linear wave problem.

So let's go back to the nonlinear problem. If you start with a plane wave here, I've just started with Gaussian. So what I'm showing here, notice that the, the, in the y direction, the length of the domain is just 1. So I have periodic boundary conditions at the top and bottom.

So the— and I have a deep region on the bottom and a shallow region on the top. with a discontinuity where this dotted line is in the bathymetry. And I start with a Gaussian plane wave propagating to the right, and I get this train of waves which, if we look at slices along the y direction through the solution, look like this familiar train of solitary waves.

And in fact, like it says here, generic positive initial data leads to the formation of solitary waves, even though we're solving here first-order hyperbolic PDE, right?

And generically you would get wave breaking here, but because of the variation in the bathymetry, we don't see wave breaking, and instead we see these solitary waves. And the 2 curves in this figure just correspond to slices along, you know, the middle of, of the deep region and the middle of the shallow region. And you see that there is some structure also in the y-direction to these solitary waves, right? The amplitude varies with y.

So let's look at that structure a little more. So here I've averaged each solitary wave in y and plotted its amplitude as a function of x. And then, under the same rescaling that you would use for KDV solitons, you can see that these solitary waves all have the same shape.

And in fact, it's very, very close to a hyperbolic secant squared shape.

Here's the structure in y. So here I'm just plotting at the peak in x for each wave its amplitude as a function of y.

Okay, and you can see that for the smaller amplitude waves, they're, they're nearly uniform in y, but for larger amplitude solitary waves, they have a very non-uniform structure.

And in fact, on the right here, we plot on the horizontal axis the mean of the amplitude of the wave, and on the vertical axis the difference between the highest point and the lowest point of the peak, sort of sliced through at y. And you can see that it's highly nonlinear, right?

I think it's also clear from this figure. This dashed line is just a quadratic fit to these values. So, and I didn't show it here, but If you look at the y component of the velocity or momentum of these solitary waves, and if you look at it as a function of y, at the peak in x, it has the structure of the first derivative of a hyperbolic secant squared.

Okay, and so you might be wondering also about the relationship between the speed and amplitude of these things. Of course, for some equations like KDV, it's linear.

Here the blue dots correspond to the solitary waves in the results that I just showed you, and the purple line is a linear fit.

Actually, it's—

no, it's not a fit. It's a linear function based on an equation that I'll derive a little bit later in the talk.

But you can see that there seems to be some nonlinearity in this relationship.

Actually, the dotted line is, is a quadratic fit. And I forgot to mention, the simulations that I'm showing you that produced all of this data are done with finite volume Godunov-type methods, the typical methods that we use for solving first-order hyperbolic PDEs. In particular, here I'm using ClawPack. And I can talk more about that if there are questions.

Okay, so yeah, here are some interactions of these solitary waves. Again, I'm just showing the average in the y direction, right?

Of course, these are actually 2-dimensional interactions. This is an overtaking collision, and maybe the thing to emphasize here is that after they've interacted and separated, there seems to be no radiation. Whereas for a head-on collision, when we zoom in and look at the area between them after they've interacted, there seems to be—

and these are very, very highly refined simulations.

This structure seems to be truly mathematical, not numerical.

OK, so after thinking about this, we realized that there's a relationship between this and flow in a channel, at least a particularly idealized kind of channel. So here what I'm thinking about is sort of taking a vertical slice through the middle of this A layer and through the middle of this B layer and, and just chopping the simulation off at those points. And because of the symmetry of the domain here, right, there's a sort of reflective symmetry in the flow. And so the solution of this problem over periodic bathymetry, truncated in the way I just described, is also the solution to the flow in a channel problem where you have a finite domain here, but you have no friction at the walls.

You have slip boundary conditions at the walls.

Yeah, so, and that's true also if your initial condition is a plane wave, right?

Then this symmetry exists. You have to have that same symmetry— actually, yeah, yeah, but you have to have that same symmetry in the initial data, essentially.

And so once you start looking at that, you realize that this is connected to very old work—

well, at least from almost 50 years ago— on flow in non-rectangular channels. In particular, there's a very nice 1950s '68 paper by Peregrine, where he looked at this and he derived the KDV equation shown here. Actually, it's in a little bit of a different form in that paper, but it's effectively this equation. And this is for waves in a non-rectangular channel.

He doesn't start from the shallow water equations.

Of course, he started from the more fundamental Euler system and derives this one-way KDV-type model. So of course it's different from what I've been looking at for both of these reasons, right?

And in particular, because he starts from the Euler equations, the resulting model includes not only this dispersion due to bathymetry, but the sort of more inherent water wave dispersion that occurs even over a flat bottom. And so this kappa here, which is sometimes referred to as the shape factor, It's—

well, first, it's also given by the average of the solution of an elliptic PDE, just like the coefficients of the dispersive terms in the model that I showed you earlier. And it somehow accounts for both this bathymetric dispersion and the natural dispersion of water waves over a flat bottom.

So we can compare these 2 models.

Here I'm showing you in green one-dimensional you know, solutions of Peregrine's equation, right, using pseudo-spectral methods.

And in black, the solution of the 2-dimensional shallow water equations, but I've averaged it in the y direction. And you can— what I want to emphasize is that there are regimes where the 2 agree very well, right?

And that's in the case where this bathymetric dispersion is really dominant and that the frequency dispersion, the other source of dispersion in water waves, is much weaker, relatively speaking. And there are also regimes where the two are quite different because the dispersion that appears in the KDV equation classically is significant, right? It is important relative to the bathymetric dispersion.

Yeah.

Yeah. So we attempted also to homogenize the fully nonlinear shallow water model.

And the techniques that we used for the linear setting at least formally fail.

They break down in the nonlinear setting with large bathymetry. If you assume small bathymetry or if you assume small amplitude waves, you can do it. When both are large, we weren't able to fully get the homogenized result. So what we did instead was, based on the similarity of the solitary waves with KDV solitary waves, at least when you average in the y direction, we use the sort of variation of coefficients approach, hypothesizing a KDV-type equation and then trying to work out what the coefficient of the dispersive term should be. And the result is, is this system, which looks like Peregrine's system and looks like KDV, right?

But the factor gamma here, which I won't say too much about due to time, depends on the amplitude of the, of the solution, right? But so I'm referring to this model as shallow water H for homogenized, and you can see that at least in a certain regime, it agrees very well with the fully nonlinear variable coefficient shallow water equations. But in other regimes—

and this is predicted by the assumptions that you make when you derive the model— these 2 can behave differently. Or if you look at long enough times, at least, these will separate.

So another way to look at these models and compare them is to look at the dispersion relation. And so here I'm looking at the dispersive coefficient— yeah, not the dispersion relation, but the coefficient of that 3rd derivative term in each of these models as a function of how much the bathymetry varies. And here I have a fixed unperturbed depth Of 0.75, right?

The channel width is one, or the bathymetry period is one.

And in blue is the the original KDV equation, right? Which doesn't where which doesn't account for variation in the bottom. And here we've used the average of the two depths, or the depth of the the KDV system. In green is Peregrine's model, right?

And in red is the shallow water homogenized model.

And remember that there are 2 sources of dispersion here.

Whoops.

And in the limit on the left, we have no dispersion from bathymetry because the bottom is in fact flat.

And you can see there that Peregrine's model and KDV coincide just as you would expect.

On the right, we have extremely strong bathymetric dispersion where the bathymetry almost the high points of it almost come up to the surface and there Peregrine's and the shallow water homogenized model become close to each other.

And Peregrine's model, which is the only one that incorporates both of these effects, sort of interpolates between these 2.

And this is the scenario corresponding to this simulation which I showed you earlier.

where both of these bathymetric and frequency dispersion are important. Here's the other scenario that I looked at where I had very good agreement between Peregrine's model and the shallow water model. And you can see that in this case, only the bathymetric dispersion is ever really significant. So I mentioned at the beginning that for other bathymetric profiles, you know, as long as it's periodic, you get the same thing. So that allows you to think about other kinds of channels, like channels with sloping sides. So here's an example of that, and here's a— the result of a simulation. Again, I'm just showing averages in y, but you get these similar kinds of solitary waves appearing. We've also done simulations with sinusoidally varying bottom And the behavior is similar.

I'm interested in the question of whether you could observe these experimentally. So we've also conducted simulations of the full, you know, 3D Navier-Stokes equations, and including, you know, some of these competing effects that might interfere with formation of these solitary waves, like for friction at the walls, surface tension, and so forth.

And this is just an example of one of those simulations. As you can see, it's not as clean and beautiful as what you get with the simple model. But this is just starting from a sort of Gaussian hump. And it appears that generic initial data again leads to a train of what are roughly solitary waves. So Just to summarize, variations in bathymetry are a source of dispersion from water waves, and this is totally distinct, right, from the dispersion appearing in most water wave models. That is, you can start from a model that has neglected the second kind of dispersion, right, and still find the first one.

And in the presence of periodic bathymetry or in a channel, this dispersion can lead to formation of solitary waves on its own, right, without And so one thing that this means is that you can get formation of solitary waves on time scales that are—

that wouldn't be predicted, for instance, by KDV, you know, or by any model that assumes a flat bottom. And in regimes where, you know, other models might predict wave breaking. Yeah, either of these dispersive sources can be dominant depending on the flow and the domain.

And here are the references.

So most of what I talked about in this talk is in this first paper just published this year in JFM. But I think people here might be interested in some of this earlier work, these 2 papers on dispersion of linear and nonlinear waves in periodic media more generally.

So this isn't unique to the shallow water equations.

Any first-order hyperbolic system with variable coefficients will lead to this kind of behavior.

You can get the same kind of thing, including solitary wave formation, for instance, in the Euler equations for compressible gas dynamics or equations of nonlinear elasticity. And we've also looked at—

so for first-order hyperbolic systems, generically, you think about viscous shock formation, or the vanishing viscosity limit, at least shock formation. And it's interesting to look at how those waves behave in the presence of this, you know, these periodically varying materials. What happens when you, you know, have a shock wave in the presence of this effective dispersion that I've described? And we've explored that in these 2 papers. All of these are also, you know, up on arXiv. So that's all. Thank you very much for your attention.

Thank you, David. Um, are there any questions? You can either unmute yourself or pop them in the chat. I see there's a question from Germain.

Yes, hello David. I was wondering why the bathymetric dispersion is winning against transverse instability. I would expect that because of the variation in bathymetry, as you have explained, the speed— when the water depth is higher, the speed is higher. But so I would expect that the soliton would be unstable transversely, and apparently it is not the case in your numerics. Have you looked to the stability of your solution?

So yeah, this is something I'm interested in.

We haven't done an analysis of the stability.

Certainly, if you look at the solutions, you can see that there's a lot of energy transport in the y direction, right?

That is what keeps the solitary wave from sort of undergoing a snaking instability, presumably. But yeah, we haven't done a mathematical analysis of it, and that would be interesting to do.

I think it's a bit challenging.

So it would be straightforward to do that, I think, for the You know, homogenized equation, but I don't know. Maybe maybe people here know. I don't know how to do it for the you know variable coefficient first order system.

German, is your hand still up or?

Yes, I have a second question if I may. The green curve when you compare with the peregrine solution, is it not an airy function? Of x divided by t to the power one-third, the linear solution of the KDV when you suppress the nonlinear term.

This this curve you're talking about? Yes.

I think so, because it looks like an airy function of x.

It does look like an airy function.

I agree.

Yeah.

I mean, this is just the solution of the KDV equation, right?

So yes, but it's a linear yes, but it is a linear one.

No.

That's why I'm asking.

That's why I'm asking.

Oh, okay. Yeah, no, here we're solving the nonlinear KDV equation.

Yeah.

Okay, but you can be in a limit where it—

This is— yeah, I think here we're close to that regime, right?

Because the amplitude of the wave is relatively small compared to the overall—

I think in this case, the nonlinear term is no more Applying, so I think you are switching to the linear regime.

Yeah, approximately. Yeah.

Okay, thanks. I agree.

Any other questions for David? I don't see any, so thanks, David, for your talk.

Thank you.