Here you will find information on my PhD research on numerical methods for internal / inertial waves in enclosed geometries. A short introduction is given below, for more extensive information go to the reports page.

A Short Introduction

Internal waves come in two varieties, firstly there are internal gravity waves driven by density gradients and restored by the gravitational force. Secondly there are the inertial waves, driven by rotation. These waves are of physical relevance in, for example, geophysics where inertial waves occur in the liquid core of the earth. Also internal waves occur in the oceans (or other natural fluid basins) where the waves transport energy, or generate mixing where they hit the boundaries. Biologists might be interested in the potential for transport of nutrients.

Under simplifying asumptions the governing equations reduce to a second order hyperbolic equation for the streamfunction where the constant . As common in oceanography the z-axis points upward, rotation is along this axis. This equation was first studied by Henry Poincaré and is for this reason often called the Poincaré equation. It is also (less widely) known as the Sobolev equation

The nature of this equation reveals itself when characteristic coordinates are introduced

These characteristics form straight lines in the xz-plane with a fixed angle with the z-axis. It turns out that the general solution can be written in terms of two arbitrary functions

which may look familiar for those aquanted with the wave equation. Indeed it is nothing more than the old wave equation, but in spatial coordinates. The next step is specifying the boundary conditions, no-flow through the (solid!) boundaries seems reasonable. This is expressed by which translates to

at the boundary.

This has far-reaching consequences, namely: it must be the case that at the boundary. Together with the invariance of the arbitrary functions along the characteristic coordinates we conclude that boundary conditions are caried over from boundary point to boundary point along the characteristics. This makes prescription of (Dirichlet) boundary conditions a precarious matter... It is easy to overspecify or underspecify the solution. Also, small changes to the boundary can have tremendous impact on the solution. Some authors proclaim the problem of obtaining solutions ill-posed for this reason.

Below is a picture of a pressure distribution. Here the sloping wall has a tremendous impact on the solution, it has a focusing effect on the characteristics.This follows from the fixed angle of the characteristics with the vectical. Two characteristics will get closer to each other upon reflection, this gives rise to a fractal structure repeating itself in increasingly smaller scales with an attractor as limitting orbit for this reproduction.

It is worthwile to note that these solutions were overlooked for a long time. It was widely believed that the Poincaré equation yields modal solutions, and indeed these are the solutions that are found in regular domains like rectangles or ellipses. Unfortunately these domains were mostly used as prototypes, they do however reveal the complete picture, symetry-breaking is needed for the special solutions to show up. Still, the wave attractors are seem by some as merely a mathematical curiosity. More laboratory experiments and numerical modeling seems justified.

Numerical results

Below is shown the solution in a circular domain. The resolution is 200x200, the matrices involved are only 400x400. This kind of cellular pattern is often called a modal solution. (Only the part inside the circle is relevant.)

The solution is however far from unique, below are two images of higer modes (more cells).

The real test is to try obtaining the fractal solutions. The nice thing is that our curent numerical method does not only find this solution, but also yields the two independent solutions that correspond to the two fundamental intervals.

Adding those two together nicely reproduces the fractal pattern. Note that this picture is in characteristic coordinates.

Experiments

One might question the validity of these solutions. Can this fracal structure, or even such a high energy square attractor as shown above ever be physical reality ? Laboratory experiments have been performed that show the attractor, allthough fractal structure is yet to be discovered. Below are shown the trapezoid basin (front left basin) which is expected to inhibit the above structure, and to the right the (significantly larger...) rotating turntable of the Coriolis laboratory of Grenoble.

This project (NUMWAT) is funded by NWO.