On the kinetics of stratified flows : wave dynamics and mean flow formation
Waves are ubiquitous in geophysical and industrial flows. Their study is necessary because they impact the dynamics at many spatial and temporal scales, shaping the large-scale flows and influencing the small-scale mixing.
When the interactions between waves are weakly nonlinear, the weak wave turbulence theory (WWT) gives a theoretical framework to study such waves. Namely, it allows us to derive a kinetic equation describing the evolution of the wave energy spectrum. This equation is of interest because it gives access to the wave dynamics on kinetic (long) time scales due to wave-wave interactions, that are not affordable with direct numerical simulations. The theory predicts steady solution(s) to the kinetic equation for many systems. For others, like internal gravity waves, controversy exists about the predictions for three reasons. Firstly, almost all predictions are obtained in the hydrostatic limit. Secondly, the presence of slow, large-scale flows (shear modes or vortical modes) that do not satisfy the WWT assumptions. Thirdly, the mathematical validity of the predictions is still subject to debate.
In this talk, I present some answers to these problems. On the one hand, we compare non-hydrostatic WWT predictions of 2D internal gravity waves to direct numerical simulations without shear modes. In the weakly nonlinear regime, we observe the formation of an energy spectrum compatible with the theoretical prediction for high wave frequencies. The size and typical velocity of the small frequency, large scale flow is predicted using WWT and phenomenological arguments, even beyond the weakly nonlinear regime. On the other hand, we performed forced dissipated simulations of the kinetic equation for internal gravity waves in the hydrostatic limit. The simulations allow us to observe the self-similar development of the energy spectrum and reveal the importance of non-local interactions (with waves of very different wave vectors modulus and frequency) in the early stage of the simulation. Then, local interactions are responsible for smoothing the energy spectrum on longer time scales. The structure and scalings of the steady spectrum are discussed and compared to theoretical predictions.
Contact Juan Ignacio Polanco for more information or to schedule a discussion with the seminar speaker.