Focusing on wave attractors

Internal waves: Focusing on wave attractors

Internal gravity waves in continuously stratified fluids
A continuously stratified fluid supports internal gravity waves. These waves propagate obliquely through a fluid. Upon reflection they conserve their propagation angle with respect to the vertical. As a consequence in a wedge, whose slope is less than that of the internal wave characteristics, waves get focused into the apex. During reflection from the sloping bottom they get focused and intensify.
When the sidewalls of a container slope steeper than the characteristics, these waves are focused upon reflection, but are reflected back into the interior. However, in enclosed domains focusing reflections dominate over defocusing ones, so that internal waves tend to be steered towards certain periodic orbits (wave attractors), where viscous and nonlinear effects act to absorb these. An example of the streamfunction field of standing internal waves in such a configuration is shown at the right. The following two movies present experimental results (see one of accompanying papers) obtained in the fluid dynamics Laboratory of Dr. J. Sommeria, who was then at the Ecole Normale et Superieure de Lyon. It presents a side-view of a uniformly-stratified fluid, which is visualized by means of fluorescent dye, injected in alternating horizontal layers, upon using a vertical laser sheet. The fluid has a sloping wall at the right, and the table on which the tank sits is oscillated vertically. By the parametric excitation mechanism internal waves are generated, that become visible 5 minutes after the start of the oscillation. See the internal wave experiment. These waves appear to be localized on a limit cycle: a wave attractor (here: the obliquely oriented, rectangular shaped object). Two distinct phases can be discerned, which is perhaps better appreciated by subtracting the initial, horizontal dye lines, shown here. 1: An initial phase, in which the localized oscillations are all in phase. In this growing phase, the wave thus appears to be standing. (This can be appreciated by noting that there are certain instances at which in the latter movie the whole picture is black). And, 2 a quasi-stationary final state, in which there appears to be a continuous propagation of phase (and, hence, energy) and the waves thus, paradoxically are of propagating type. This phase propagation is seen from the (black) nodal lines that 'cross over' in a direction perpendicular to the edges of the rectangular box-shaped attractor. Given that internal wave energy propagation is perpendicular to its phase propagation direction (having opposite vertical components) the sense in which each of these phase lines propagates is consistent with a energy propagating around (and into) the attractor in a clockwise direction, consistent with the sense dictated by the focusing at the sloping side. Similar wave properties are found in inertial waves, arising in homogeneous rotating fluids.
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 ) Unpublished webscript: Leo Maas, Uwe Harlander, Astrid Manders & Arno Swart (2003) Stratified fluid experiments in an annulus with sloping inner wall under slight modulation of the rotation speed. ( Download PDF-file )
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Thierry Dauxois Bruce Sutherland