Aspect ratio effects, multistability and quantisation in wave attractors.

Internal and inertial waves play a substantial role in ocean dynamics. They can transport a considerable amount of kinetic energy over long distances, and their amplitude in the abyssal ocean can reach gigantic vertical scales of several hundreds of meters. At the same time, packets of internal and inertial waves conserve a fixed angle with respect to gravity or the rotation axis upon reflection, which makes both their linear and nonlinear dynamics rather peculiar. Most hydrodynamical systems in closed domains can be described in terms of modes. In this framework, one usually assumes eigenfunctions satisfying the boundary conditions, for example Fourier standing modes in rectangular domains. These modes oscillate in time at every point in space but do not propagate in a specific spatial direction. Internal and inertial waves constitute a remarkable exception to this approach. It has been shown that, in a general geometry, wave beams of travelling waves converge toward a limiting path, known as a wave attractor, while global modes form a set of zero measure. Rectangular tanks aligned with gravity and/or rotation, actually represent an exceptional but very important case. Our work focuses on two aspects of internal waves in this context: first, the influence of the aspect ratio on the transition to turbulence and mixing for structurally stable wave attractors; second, the interplay between wave-attractor regimes and modal structures in the vicinity of rectangular geometries. Surprisingly, a conventional rectangular geometry may exhibit much more complex and strongly multistable regimes than those observed for simple wave attractors. We demonstrate competition between different triadic instability pairs, leading to multistability and a nearly uniform picket-fence spectrum, which is markedly different from the spectrum resulting from cascades of triadic instabilities driven by large-aspect-ratio wave attractors.