Interactions between Turbulence and Waves: Kinetic Approach

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The dynamics of the ocean is a fascinating and complex subject. There are many types of waves and flows in the ocean: surface gravity waves are the familiar waves that can be observed breaking rhythmically near the shore; internal waves are hidden beneath the surface and take the form of modulations between layers of different water density; ocean eddies are circular currents of water, and finally, turbulence includes myriad types of unsteady movements of water. These different flows span a wide range of length and time scales, ranging from centimeters to thousands of kilometers, and from a few seconds to several years. Crucially, all these flows interact with each other, creating a system of immense complexity whose study remains a major challenge. This project attempts to build a rigorous mathematical theory to characterize the interactions between internal waves and large scale, two-dimensional eddies. The motivation is recent data from physical oceanography, indicating that processes operating on small scales can significantly affect oceanic energy flows on the large scale. The project will contribute to the study of this phenomenon by analyzing how energy is transferred in the reverse direction, from large scale horizontal motions in eddies to small scale isotropic turbulence where the energy of the eddy is dissipated. Additionally, the project will provide research opportunities for postdoctoral scholars.The interaction of waves and turbulence within a rotating stratified fluid is of crucial importance for the understanding of the ocean and atmosphere dynamics. The project will examine this interaction by employing the hydrostatic rotating Boussinesq equations. The project aims to utilize ideas from the wave turbulence theory on a novel system of partial differential equations that is equivalent to the hydrostatic rotating Boussinesq equations but has the advantage of explicitly separating wave motions from vorticity effects. This PDE system will be used to derive kinetic-like equations that describe the flow of spectral energy density between large scale vortices and internal waves. The mathematical predictions will be compared to oceanographic data of internal waves as well as to high resolution direct numerical simulations of rotating stratified turbulence.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.