Dynamics of Nonlinear Internal Wave Beams in Stratified Flows

本文对密度分层流体中非线性内波束的传播和相互作用进行了理论研究。这些是源自内部重力波运动的非各向同性性质的基本时谐扰动。重点关注以下具体问题：(i)波束在短尺度次谐波扰动下的不稳定性，根据最近的实验和数值研究，这是击穿和湍流产生的一个突出机制；（ii）非线性光束从倾斜边界的三维反射，包括近临界反射的情况，当入射到水平面的角度几乎与边界斜率相匹配时；（三）非线性光束的三维碰撞和相关的能量散射。这些物理现象是通过制定数学模型来解决的，这些数学模型是用渐近和数值方法分析的。将理论预测与相关的实验室实验和现场观测进行了比较。内波束在自然流体中起着重要作用，如海洋、湖泊和大气，其中密度分层主要由温度变化引起；在大气中由于雷暴而产生的内部波束，以及在海洋中由于潮汐流与大陆架和海底地形的相互作用而产生的内部波束，有许多观测结果。这个项目的动机是需要了解和定量模拟从潮汐产生的内波束到深海混合的能量转移机制，而深海混合反过来又影响海洋环流、气候变化、污染物扩散和营养循环。研究结果在地球物理流体力学和应用数学中具有重要的应用价值。

AkylasDMS-1107335 A theoretical investigation is made of the propagation and interactions of nonlinear internal wave beams in density-stratified fluids. These are fundamental time-harmonic disturbances that derive from the non-isotropic nature of internal gravity-wave motion. Attention is focused on the following specific problems: (i) the instability of wave beams to short-scale subharmonic disturbances, a prominent mechanism of breakdown and turbulence generation according to recent experimental and numerical studies; (ii) three-dimensional reflections of nonlinear beams from a sloping boundary, including the case of near-critical reflection, that occurs when the angle of incidence to the horizontal nearly matches the boundary slope; (iii) three-dimensional collisions of nonlinear beams and the associated scattering of energy. These physical phenomena are tackled by formulating mathematical models that are analyzed by asymptotic and numerical methods. The theoretical predictions are compared against related laboratory experiments and field observations. Internal wave beams play an important part in natural fluid bodies, like the oceans, lakes, and the atmosphere, where density stratification is mainly caused by temperature variations; there are numerous observations of internal wave beams in the atmosphere due to thunderstorms, as well as in oceans due to the interaction of tidal currents with the continental shelf and sea-floor topography. This project is motivated by the need to understand and model quantitatively the mechanisms responsible for the transfer of energy from tide-generated internal wave beams to deep-ocean mixing, which in turn affects ocean circulation, climate change, pollutant dispersal, and nutrient cycles. Moreover, the results of this study would be of basic value in geophysical fluid dynamics and applied mathematics.