Internal wave attractors in 3D geometries : A dynamical systems approach

Abstract

We study the propagation in three dimensions of internal waves using ray tracing methods and traditional dynamical systems theory. The wave propagation on a cone that generalizes the Saint Andrew's cross justifies the introduction of an angle of propagation that allows to describe the position of the wave ray in the horizontal plane. Considering the evolution of this reflection angle for waves that repeatedly reflect off an inclined slope, a new trapping mechanism emerges that displays the tendency to align this angle with the upslope gradient.In the rather simple geometry of a translationally invariant canal, we show first that this configuration leads to trapezium-shaped attractors, very similar to what has been extensively studied in two-dimensions. However, we also establish a direct link between the trapping and the existence of two-dimensional attractors.In a second stage, considering a geometry that is not translationally invariant, closer to realistic configurations, we prove that although there are no two-dimensional attractors, one can find a structure in three-dimensional space with properties similar to internal wave attractors: a one-dimensional attracting manifold. Moreover, as this structure is unique, it should be easy to visualize in laboratory experiments since energy injected in the domain would eventually be confined to a very thin region in three-dimensional space, for which reason it is called a super-attractor.

Reflections and focusing of inertial waves in a librating cube with the rotation axis oblique to its faces

“…All six polar edges where beams focus correspond to large wavevector and velocity magnitudes |k| and |a|. The sensitive dependence on the initial abscissa z 0 for the determination of which edge the beams focus on is reminiscent of the chaos reported in Maas (2005) and Pillet et al (2019) for wavebeam reflections in some geometries. Beams emitted from the polar edge at p 0 (−0.5, y 0 , −0.5) in direction a − focus, in a similar way, to a point on one of the three polar edges connected to the north pole.…”

Stably stratified square cavity subjected to horizontal oscillations: responses to small amplitude forcing

“…Maas & Lam 1995; Wotherspoon 1995), there is still much to learn (e.g. Bajars et al 2013; Pillet et al 2018; Pillet, Maas & Dauxois 2019), and approaches from the study of how inertial waves reflect in the obliquely rotating cube (Wu et al 2020) can be adapted for the internal wave problem.…”

Boundary-confined waves in a librating cube

“…As well as experiments, this librating geometry has also been studied theoretically using linear inviscid ray tracing (e.g. Manders & Maas 2003; Pillet, Maas & Dauxois 2019). Those studies focused primarily on the wave attractors in the bulk flow, but also considered the critical slope attractor, where for a particular forcing frequency the slope of the inclined wall is critical and a branch of the wave attractor becomes trapped in the wall, while other branches of the attractor continue to reside in the bulk.…”