Abstract
—The hydrodynamic system that admits the development of internal wave attractors under biharmonic forcing is investigated. It is shown that in the case of low amplitude of external forcing the wave pattern consists of two attractors that interact between themselves only slightly: the total energy of the system is equal to the sum of energies of the components with high accuracy. In the nonlinear case the attractors interact in the more complex way which leads to the development of a cascade of triad interactions generating a rich set of time scales. In the case of closely adjacent frequencies of the components of a biharmonic perturbation, the nonlinear “beating” regime develops, namely, the mean energy of the system of coupled attractors performs oscillations at a large time scale that corresponds to the beating period. It is found that the high-frequency energy fluctuations corresponding to the same mean energy can differ by an order of magnitude depending on whether the envelope of the mean value increases or decreases.
Highlights
In a fluid with uniform density stratification the internal waves obey a specific dispersion relation [1] which connects the wave frequency and the angle of inclination of the wave beam with respect to the vector of gravity but does not contain the length scale
The interaction of the surface tides with the bottom topography leads to generation of internal tides; in this case, under certain conditions, the internal tidal motion can exist in form of attractors [9,10,11,12]
It should be noted that almost all the literature in the field of simulation of internal tides and wave attractors is devoted to the investigation of monochromatic external forcing [3, 6]
Summary
We will consider a tank of trapezoidal form with the greater base at top filled with a uniformly stratified liquid in the gravity field (Fig. 1) This geometric structure is fairly well studied in literature, namely, there is a detailed classification of the geometric configurations of attractors within the framework of the ray path theory [5], the stream functions are constructed within the framework of the inviscid liquid model [22], the linear [18] and nonlinear [23] scaling for the width of attractor’s wave beams is investigated, and the energy cascade in the nonlinear regime is described [24]. All the following results were obtained from the numerical simulation of the nonlinear system of the Navier–Stokes equations in the Boussinesq approximation
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