Interactions between Turbulence and Waves: Kinetic Approach

海洋动力学是一门迷人而复杂的学科。海洋中有许多类型的波浪和流：表面重力波是我们熟悉的波浪，可以在海岸附近观察到有节奏的破裂；内波隐藏在表面之下，以不同水密度层之间调制的形式存在；海洋漩涡是水的循环流，最后，湍流包括无数种不稳定的水运动。这些不同的水流跨越了很长的长度和时间尺度，从几厘米到几千公里，从几秒钟到几年。至关重要的是，所有这些流动相互作用，创造了一个极其复杂的系统，其研究仍然是一个主要挑战。这个项目试图建立一个严格的数学理论来描述内波和大尺度二维涡流之间的相互作用。其动机是来自物理海洋学的最新数据，这些数据表明，在小尺度上运作的过程可以显著影响大尺度上的海洋能量流动。该项目将有助于研究这一现象，通过分析能量是如何在相反的方向上转移的，从涡旋中的大规模水平运动到涡旋能量耗散的小规模各向同性湍流。此外，该项目将为博士后学者提供研究机会。旋转层状流体中波浪和湍流的相互作用对于理解海洋和大气动力学具有至关重要的意义。该项目将通过采用流体静力旋转布辛尼斯克方程来检验这种相互作用。该项目旨在利用波浪湍流理论的思想建立一个新的偏微分方程系统，该系统相当于流体静力旋转的Boussinesq方程，但具有明确地将波浪运动与涡量效应分开的优点。该PDE系统将用于导出描述大尺度涡旋和内波之间谱能量密度流动的类动力学方程。数学预测将与海洋内波数据以及旋转分层湍流的高分辨率直接数值模拟进行比较。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.