Аннотация:Internal and inertial waves play an important role in the ocean dynamics trough energy
cascades from global tidal forcing to small-scale internal waves and resulting mixing. Inertial
waves in closed domains possess a remarkable property of focusing on the limit cycles called
wave attractors. The growth of amplitude at wave attractors results in instabilities in case of
viscous fluids. The scenarios of transition to turbulence and description of fully turbulent
regimes differ substantially from the cases observed in closed domains in 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 [1]. In
present report we consider a shallow trapezium and the wave attractors with multiple
reflections from the horizontal boundaries. The exact ray theory solution for the coordinates of
such (n,1) wave attractors was obtained, as well as the expression for the Lyapunov exponents
[2]. Next we show that in a shallow domain with small aspect (depth-to-length) ratio, the
frequency spectrum of (1,1) wave attractor motion may exhibit significant peaks at integer and
half-integer multiples of the forcing frequency. For the aspect ratio of about one tenth the
temporal average of total kinetic energy grows monotonically with amplitude and have a bend
at a particular amplitude. Below this amplitude the cascade transferring energy to
superharmonic components prevails, while above this value the amplitudes of subharmonic and
superharmonic waves are comparable. The spatial spectra of waves in the domains of the aspect
ratio varying from small values to the values close to unity are compared. It is shown that in
the former case (i.e. for shallow domains) the spectrum has two zones at small and high wave
numbers characterized by different slopes. The fully turbulent regimes show the trend toward
long-term evolution leading to new regimes with complex resonant dynamics of large-scale
coherent structures.
[1] Brouzet, C., Sibgatullin, I. N., Scolan, H., Ermanyuk, E. V., and Dauxois, T. (2016).
Internal wave attractors examined using laboratory experiments and 3d numerical
simulations. Journal of Fluid Mechanics, 793:109–131.
[2] Sibgatullin, I., Petrov, A., Xu, X., and Maas, L. (2022). On (n,1) wave attractors:
Coordinates and saturation time. Symmetry, 14(2):319