Nonspecular mechanism for chaotic ray scattering of internal waves in three-dimensional anisotropic stadiums

Fluids, subject to symmetry breaking by stratification, support propagation of anisotropic internal waves (IWs). In the vertical plane, rays representing energy paths obey a nonspecular reflection law, as their inclination is solely dictated by their frequency. Although satisfying the linear Poincaré equation, in basins having sloping walls, ray dynamics exhibits nonlinear effects such as convergence onto wave attractors. In contrast, in the horizontal plane of a basin with vertical walls, IWs reflect specularly and follow chaotic ray paths. Here we present an analysis of these competing effects in a 3D IW ray billiard of a stadium having sloping walls. We show and explain how varying the wall's slope shifts the ray dynamics between regimes of near ergodicity, chaotic scattering, and nonchaotic scattering with self-similar patterns, despite the basin being closed. The rich results stemming from the interplay between elliptical ergodicity and hyperbolic focusing relate to a broader context of physical phenomena.