Application of statistical and spectral methods to computational modeling of internal wave attractors

Direct numerical simulation of internal gravity waves propagation and formation of internal waves attractors in trapezoidal tank, which was filled with stably stratified salt solution of constant buoyancy frequency. The left vertical boundary oscillates with constant time frequency and has the form of semi-cosine of the tank height. The right wall is inclined to the vertical and performs focusing of waves, the other two boundaries are horizontal. The upper wall has stress free boundary conditions, on the other boundaries no-slip condition is imposed. Navier-Stokes equations in Boussinesq approximation and diffusive transport of salt are used as mathematical model. Direct numerical simulation is performed with the help of spectral element approach of 8-th order and modified code nek5000. Hilbert transforms and time-frequency diagrams were applied to the results of direct numerical simulation of internal wave attractors. In particular, with the help of phase reconstruction we obtained wave vectors, corresponding to time frequencies from time-frequency diagrams. To obtain phase images Hilbert transforms with filtration over narrow intervals of frequencies were used. Time-frequency diagrams for moderate forcing amplitudes show appearance of daughter waves of frequencies, corresponding to triadic resonance, which is also demonstrated with the help of bispectra: product of amplitudes on the background of cross product of frequency intervals. The results are close to data of experiments, being carried out at ENS de Lyon.