Numerical Simulation of Three-Dimensional Wave Attractors

Abstract

Internal and inertial waves obey very specific dispersion relation, which defines the direction of wave propagation with respect to the gravity or axis of rotation, but which does no define the wavelength. Particular cases of scale effects are described in (Brouzet et al., 2017). In closed geometrics the presence of a monochromatic wavemaker can produce wave beams, which after multiple reflections from the boundaries may approach a closed loop – the wave attractor. In ideal fluids the concentration of energy on the wave attractor can grow without any limits. In viscous stratified or rotating fluids the concentration will be balanced with dissipation due to viscosity, which results in appearance of wave attractors of finite width. The characterization of the flow with the Reynolds number based on the boundary conditions is questionable in this case, since on the attractor the velocity can be several times magnified. When the wave beam is reflected from an inclined plane, the horizontal component of velocity rotates, as was first described by O. Phillips, while preserving the angle with the gravity or axis of rotation. With the help of ray tracing it can be shown that due to this effect the three-dimensional accumulation of wave energy can occur. First qualitative and quantitative correspondence of laboratory and numerical simulation of wave attractors in the pseudo-2D laboratory tank with trapezoidal section was described in (Brouzet et al., 2016), and importance of dissipation on the lateral boundaries was shown (F. Beckebanze et al., 2018). For high Schdmidt number there appear the folded structures, which can interact with the background wave motion (Sibgatullin, Kalugin 2016). In (Brouzet et al., 2016b), (Dauxois et al., 2018) cascade of triadic resonances in (1,1) produced by a wave attractor was demonstrated. Three-dimensional accumulation of wave energy in trapezoidal frustum with a localized wavemaker was investigated in (Pillet et al., 2018). Numerical simulations of the present work had showed the importance of phase shift in transversal direction. An attractor can have the same form as the 2D attractor in any given longitudinal cut, but the phase of oscillation can change up to counter-phase. Interplay of 3D concentration of waves beams, dissipation and phase shifting impact the final energy distribution in transversal direction. First three-dimensional simulations (Sibgatullin et al., 2017) of tidal and symmetric forcing on the rotating layer of fluid with inclined walls showed three-dimensional twisted structure of waves attractors for precession of one boundary of the layer in opposite direction to the rotation of the layer. With growth of the amplitude of the external forcing the the instability of triadic resonance appears, but in contrast to the internal wave attractors, triadic resonances take place in azimuthal (transversal to the trapeze) section. The turbulent regimes generated by the background wave attractors are studied, with analysis of full power income and total dissipation. The research was supported by the Program of Fundamental Research of the Presidium of the Russian Academy of Sciences No. 26