From wave attractors to quantized mode multistability: internal-wave dynamics in bounded domains

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Exceptionally, the seminar will start at 10.45 instead of 11 AM

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Internal and inertial waves are key components of geophysical fluid dynamics. They can transport energy over long distances, and in the abyssal ocean their vertical scales may reach hundreds of meters. Unlike many other hydrodynamic systems, they obey a geometric propagation constraint: upon reflection, wave beams preserve a fixed angle with respect to gravity (or the rotation axis). This property makes both linear focusing and nonlinear energy transfer highly nontrivial in bounded domains.

In generic non-orthogonal containers, repeated reflections produce wave attractors, i.e. limiting ray cycles that are structurally stable and concentrate energy. In such regimes and comparable vertical and horizontal scales, transition to turbulence is described by cascades of triadic resonances. We first discuss this attractor framework in large-aspect-ratio domains, where the forcing-amplitude dependence reveals distinct nonlinear scenarios: below a threshold, energy transfer is dominated by superharmonic pathways. As the forcing increases, parametric subharmonic instability appears, which propagate to multiples of half the forcing frequency. Only after they saturate to the same energy level they give rise to triadic resonance cascades. Spectral and time-frequency diagnostics indicate that geometry and aspect ratio strongly affect the pathway to turbulence, the kinetic-energy evolution, and the organization of coherent structures.

We then approach a geometrically exceptional but physically important limit: rectangular domains aligned with gravity and/or rotation. In this geometry, internal waves are quantized by boundary conditions, and only discrete mode families are admissible. One might expect this setting to be simpler than attractor dynamics; surprisingly, we observe the opposite. Combining laboratory measurements (PIV), targeted modal diagnostics, and high-resolution DNS with moving-boundary forcing, we show that rectangular domains can exhibit strong multistability and spectral complexity. A primary triadic resonance generates a dense, nearly uniform picket-fence frequency comb through nonlinear frequency mixing, while the dominant spatial content remains concentrated in a small set of global modes. Near-degenerate mode pairs and weak detuning provide a natural reduced-model interpretation of the slow envelopes. This behavior is qualitatively different from the broadband-like spectra associated with cascading attractor turbulence.

Overall, there are two complementary messages. First, aspect ratio is a true control parameter for nonlinear internal-wave dynamics in attractor geometries, including the route to turbulent states. Second, near-rectangular quantized systems are not a trivial limit of attractor physics: they can be even more multistable and spectrally intricate, with competition between triadic pathways and robust discrete comb formation. These findings help to give a unified perspective on how geometry selects nonlinear energy-transfer mechanisms in bounded internal-wave systems.