Collaborative Research: Water Waves--Nonlinearity, dissipation, and forcing

海洋表面的波浪在天气预报和气候模拟、沿海社区和近海工业的安全以及海外航运方面发挥着重要作用。在这个项目中，研究人员将重点放在经常被近似或完全忽略的物理效应上，但这是准确预测海浪观测行为所必需的。例子包括海浪在传播过程中穿过深海并随后到达海岸线的耗散；波浪产生过程中风的时间依赖性；和浅水中波浪的分散。将耗散包括在内将有助于更好地理解波能是如何演变的。在产生波浪的风中包含时间依赖性将使我们更好地理解能量从空气转移到水的初始阶段。在浅水波模式中加入耗散和弥散将有助于提高沿海地区波浪的预测能力。研究者的研究工具包括建模、分析、计算机模拟和实验室实验。虽然重点是水波，他们可以进行实验室实验，但数学分析更广泛地适用于其他物理系统，并对偏微分方程的研究感兴趣。研究人员建议对下列问题进行分析、数值和实验研究：(A)深水波。他们考虑了自由传播波的频率降频和风对波浪产生的影响。他们正在考虑两种频率降频的模型，这两种模型在如何模拟气流的旋转部分方面有所不同。为了模拟风力产生的波浪，他们考虑了空气和水中随时间变化的剪切流。由此产生的波的稳定性问题是非标准的，理解如何解决它是一个核心的数学问题。(B)浅水波。他们寻求由于底部、壁面和表面边界层引起的色散和耗散的精确模型。他们将从惠瑟姆方程开始，并将其推广到包括表面张力效应、耗散效应、非水平测深和双向波。他们将从解析和数值两方面寻找小振幅和大振幅的解，并研究它们的稳定性。他们将通过比较数值模拟和实验进一步研究如何最好地包括由于底部边界层引起的耗散。(C)三波偏微分方程。在许多物理应用中出现的三波偏微分方程，描述了色散波列之间最简单的非线性相互作用，没有耗散。研究人员提出了一种求解任意边界条件下的一般解的方法。偏微分方程通解的例子很少，所以他们再增加一个例子将是一个数学上的突破。

Waves on the ocean's surface play important roles in weather forecasting and climate modeling, in the safety of coastal communities and offshore industries, and in overseas shipping. In this project, the investigators focus on physical effects that are often approximated or neglected altogether, but that are needed to predict accurately the observed behavior of ocean waves. Examples include the dissipation of ocean swell during propagation across the deep ocean and subsequently onto the shoreline; the time-dependence of wind in the wave-generation process; and dispersion of waves in shallow water. The inclusion of dissipation will lead to a better understanding of how wave energy evolves. The inclusion of time-dependence in the wind that generates waves will allow for a better understanding of the initial period during which energy is transferred from air to water. The inclusion of dissipation and dispersion in models for shallow-water waves will allow for better predictive capabilities of waves in coastal areas. The research tools of the investigators include modeling, analysis, computer simulations, and laboratory experiments. While the emphasis is on water waves, for which they can conduct laboratory experiments, the mathematical analysis is more broadly applicable to other physical systems and is of interest in the study of partial differential equations.The investigators propose analytic, numerical, and experimental investigations of the following: (A) Deep-water waves. They consider the frequency downshifting of freely propagating waves and wave generation due to wind. They are considering two models of frequency downshifting that differ in how the rotational part of the flow is modeled. To model wind-generated waves, they are allowing for time-dependent shear flows in both the air and water. The resulting stability problem for waves is non-standard, and understanding how to address it is a central mathematical question. (B) Shallow-water waves. They seek accurate models of dispersion and of dissipation due to the bottom, wall, and surface boundary layers. They will start with a Whitham equation and generalize it to include surface tension effects, dissipative effects, nonhorizontal bathymetry, and bidirectional waves. They will look, both analytically and numerically, for small- and large-amplitude solutions, and study their stability. They will further investigate how best to include dissipation that is due to the bottom boundary layer by comparing numerical simulations and experiments. (C) Three-wave partial differential equations. The three-wave partial differential equations, which arise in many physical applications, describe the simplest possible nonlinear interactions among dispersive wave trains, without dissipation. The investigators propose a solution method using a Painleve-analysis to obtain the general solution for arbitrary boundary conditions. There are few examples of general solutions of partial differential equations, so their adding one more example would be a mathematical breakthrough.