Instabilities and Attractors in Large Aspect Ratio Stratified/Rotating Fluids

Abstract

Internal and inertial waves are ubiquitous in the world’s oceans, and the energy that they transfer from the global tide to the smaller scales may impact the vertical mixing and global currents. The dynamics of internal and inertial waves in closed domains possesses a remarkable property of self-focusing on geometrical limit cycles called wave attractors. The significant growth of wave amplitude at the wave attractors results in instabilities for viscous fluids. The scenarios of the transition to turbulence and the description of the fully turbulent regimes differ substantially from the cases observed in closed domains in the absence of wave attractors. Our previous studies demonstrated the key role of a cascade of triadic resonances as the route to fully developed wave turbulence, either with overturning events or not. Presently we consider a shallow trapezoid and the wave attractors that form after multiple reflections from the horizontal boundaries. The saturation time of (n,1) wave attractors in laminar viscous fluids grows linearly with n if the wavemaker is located at the sidewall. Next we show that in a shallow elongated domain with small aspect (depth-to-length) ratio, the frequency spectrum of (1,1) wave attractor motion may exhibit significant peaks at integer and halfinteger multiples of the forcing frequency. For an aspect ratio of about one tenth the temporal average of the total kinetic energy grows monotonically with the amplitude and exhibits a bend at a particular amplitude. Below this amplitude the cascade transferring energy to integer superharmonic components prevails, while above it the amplitudes of waves at integer and half-integer multiples of the forcing frequency are comparable. The spatial spectra of the waves are compared for domains of aspect ratio varying from small values to values close to unity. It is shown that in the former case (i.e. for elongated shallow domains) the spectrum exhibits two zones at small and high wave numbers characterized by different slopes. The fully turbulent regimes show a trend toward a long-term evolution leading to new regimes with a complex resonant dynamics of large-scale coherent structures.