Nonlinear Wave Dynamics in Stratified Flows

本文将研究重力波在分层流中的两个传播问题：内部波束的传播问题和非线性调制波与诱导平均流的共振相互作用问题。由于重力提供了一个优选方向，由振荡源产生的内波扰动是各向异性的；它们不是圆柱形的正面，而是呈放射状向外伸展的直梁。用渐近模型和数值模型研究这种光束在不同流动条件下的传播特性。特别强调将放在三维和非轴对称梁的非线性效应是最明显的。其次，对调制波束的谐振自加速进行理论研究；波列包络和诱导的水平平均流之间的这种强烈的非线性相互作用会导致几乎垂直传播的波列不稳定和破裂。虽然在日常经验中不像海洋表面波那样熟悉，但实际上，内部重力波在海洋、湖泊和大气中无处不在，它们在传递动量和能量方面发挥着重要作用。特别是内部重力波束常常是由雷暴触发的，从数值模拟和实地观测中有证据表明，大气中重力波活动的一个重要组成部分是以波束状结构的形式出现的。因此，了解内部重力波束的传播特性将有助于天气预报。该奖项由数学科学部和工程局联合资助，作为数学科学优先领域的一部分。

Two problems of gravity wave propagation in stratified flows will be studied: the propagation of internal wave beams and the resonant interaction of nonlinear modulated wavetrains with the induced mean flow. Since gravity provides a preferred direction, internal wave disturbances generated by an oscillatory source are anisotropic; rather than cylindrical fronts, they form straight beams stretching radially outwards. The propagation characteristics of such beams will be studied under various flow conditions using asymptotic and numerical models. Particular emphasis will be placed on three-dimensional and non-axisymmetric beams for which nonlinear effects are most pronounced. Secondly, a theoretical study will be made of the so-called resonant self-acceleration of modulated wavetrains; this strong nonlinear interaction between the wavetrain envelope and the induced horizontal mean flow can cause instability and breakdown of nearly vertically propagating wavetrains. Although not as familiar from everyday experience as ocean surface waves, internal gravity waves are in fact ubiquitous in oceans, lakes and in the atmosphere, where they play an important part in transferring momentum and energy. Internal gravity wave beams in particular are often triggered by thunderstorms, and there is evidence from numerical simulations and field observations that a significant component of gravity wave activity in the atmosphere is in the form of beam-like structures. Understanding the propagation characteristics of internal gravity wave beams will thus prove useful in weather forecasting. This award is being jointly funded by the Division of Mathematical Sciences and the Directorate for Engineering as part of the Mathematical Sciences Priority Area.