Summer 2010 Undergrad Research Program

It is widely known that most of the natural laws of physics can be stated in terms of partial differential equations, since derivatives occur naturally in the form of velocity, flux, etc.

Our research focuses on waves that arise from non-linear PDE’s (i.e. partial differential equations). The first part of the internship was focused on soliton waves and the Inverse Scattering Transform technique of solving PDE’s and then we moved on to the tidal bore formation.

A word about solitons…

These waves marked a revolution in the field of nonlinear physics as it was discovered that numerous PDE models have soliton solutions. Mathematicians Korteweg and de Vries were the first ones to formulate the equation (KdV equation) governing weakly nonlinear shallow water waves:

This equation has a sech2 solitary wave solution:

Soon after that soliton solutions started appearing in other contexts, and their study is to this day an active research field.

So, what is a soliton wave and how is it different?

In 1965 mathematicians Kruskal and Zabusky coined the word soliton to describe a self-reinforcing solitary wave, which retains its shape while it moves at a constant speed. Unlike other waves, it only has a single peak and no trough and does not disperse as it travels. Overall, it is a very stable structure.

Soliton discovery is dated back to the 19th century when a Scottish naval engineer, John Russell described a phenomenon he called the wave of translation:

The unique qualities of this phenomenon are used in many disciplines: for example, the non-linear Schrodinger equation (NLS), modeling deep water rogue waves, has a soliton solution. Other areas affiliated with solitons include telecommunications, atmospheric sciences, fluid dynamics, optics and neurological sciences.

Our next objective was to understand the formation of tidal bores, as we believed that solitons were present at some stages of the process. Hubert Chanson in his The Rumble Sound Generated by a Tidal Bore Event in Baie du Mont Saint Michel work defines a tidal bore as follows:

There are very few places in the world that have this phenomenon and, conveniently enough, the Turnagain Armof Cook Inlet, Alaska, is one of them.

There exist two types of tidal bores:

• A turbulent bore is characterized by a turbulent wave front with a steady stream behind it.
• An undular bore has a smooth wave front followed by an oscillatory wave train.

For a bore to exist, the velocity of the flow approaching the bore must exceed the velocity of a long wave of small amplitude in water of the same depth. We looked at the bores in the context of the shallow water wave theory and used the shallow water wave equations:

where u(x,t) is the horizontal velocity and h(x,t) is the vertical depth.

Applying the method of characteristics and the necessary jump conditions, we were able to come up with equations for the bore’s path and speed.

Bore speed:

Bore's path:

where V is the speed of the incoming tide wall, h1 is the height of the wall and h0 is the initial water depth.

These formulas will be used for our Girdwood trip: after observing the bore wave, we will calculate its height and speed.

So far in the research, our group came to the conclusion that all modeling of the turbulent tidal bores are found numerically by solving the shallow water equations (or some variant of ) with certain boundary conditions.

There are more models for undular bores, but they are not as accurate as one would wish. The only link with solitons we found so far was the KdV model with the Burgers-type term:

where v is a small dissipation coefficient and u is the free surface height. This model, however, has some major disadvantages:

• It does not reveal transition conditions
• It describes only the established regime and not the bore formation

We will keep looking for solutions and, hopefully, the Girdwood trip, brings some clarification to this matter.

Our research paper can be found here.

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