Wave attractors

Internal waves: Wave attractors

In a uniformly stratified fluid monochromatic (single frequency) internal gravity waves propagate under a fixed angle with respect to the vertical. In 2D, the ray paths, along which the energy propagates, are identical to the characteristics of the governing (canonical) hyperbolic equation.

Because the frequency is conserved upon reflection at a sloping boundary, so is the angle with the vertical, and hence reflection is nonspecular.

The energy propagation path is traced out by the web of connected characteristics.

It can be seen that from an arbitrary starting point X, say (the blue dot in the top figure) the energy piles up on a single periodic orbit (red), which is therefore referred to as a wave attractor. The whole plane is filled with such webs. Each web has its own invariant 'partial pressure'. Each point in the plane (with the exception of the periodic orbit) is at the intersection of two webs and the streamfunction value at such a point is simply the difference of their partial pressures. (The pressure being the sum of these). Note that the attractor still shows up in the streamfunction field (bottom figure), as a location where gradients in the streamfunction (and thus velocities) increase.

Changing the frequency of the wave for a fixed-depth geometry is, upon stretching the geometry, identical to retaining the wave propagation angle with the vertical to 45 degrees while changing the (scaled) depth. Hence, the change in shape of the attractor to either a change in frequency or in depth (in fact, to the single lumped parameter containing all of the descriptive parameters) is shown over some range of "Depths".

In this figure, for each depth just the surface reflections of the asymptotic (attractor) state are indicated.

For the previous example, taken at a depth of 0.9, just two surface reflections arise. While the problem is linear, the figure shows an intricate change of regimes, reminiscent of figures obtained for iterative maps. The final figure shows a "dressed" version of this, where the color associates to any initial surface point X from which webs are launched the summation of this web's surface positions.

Publications:

Maas, L.R.M., Benielli, D., Sommeria, J., Lam, F.-P. A. (1997)

Observation of an internal wave attractor in a confined stably-stratified fluid. Nature, 388: 557-561. ( Download PDF-file )

Maas, L.R.M. (1995)

Focusing of internal waves and the absence of eigenmodes. 'Aha Huliko' a 1995 conference proceedings, eds: P.M. Müller and D. Henderson. ( Download PDF-file )

Maas, L.R.M., Lam, F.-P. A. (1995)

Geometric focusing of internal waves. Journal of Fluid Mechanics, 300: 1-41. ( Download PDF-file )