Nonlinear Analysis of Three-Dimensional Water-Wave Patterns via Exponential Asymptotics

由于移动的水面和水下源（如船舶和潜艇）或由于流过海山和水下山脊的水流而产生的三维水波模式，从日常经验中是熟悉的。 除了它们的美学吸引力，这些波现象也是各种科学领域应用的基础，包括地球物理流体动力学，船舶流体动力学，海洋学和气象学。 本研究计画将发展一种新的数学方法来分析三维水波型态。 与以前的所有分析处理相比，新的方法将不限于小陡度的波，其目的是从理论上解释最近的数值模拟不均匀的海底地形的电流引起的水波揭示了几个显着的功能。该项目还将邀请研究生参与研究。通过移动表面和水下源生成稳定的三维（3D）水波模式是流体动力学的一个基本课题，可应用于地球物理流体建模、海洋学和船舶流体动力学。 解决这些问题的分析通常的方法是基于线性化的水波方程，假设无限小的干扰。 从这种线性分析出发，本项目将设计一种分析（渐近）技术，使稳定的三维自由表面重力波模式，由于各种类型的波源的非线性处理。 动机来自最近发现（由PI的小组）的一个非线性机制，该机制控制了广泛的流动条件下的2D强迫重力波的振幅，严重限制了线性分析的有效性。 这种机制取决于所有阶的扰动幅度（非线性参数），并可以捕获的扰动扩展，超越所有阶的非线性参数。 本项目将开发适用于3D波型的超越所有阶次（指数渐近）技术。 这种新的方法将用于推进对两种不同流态下三维强迫重力水波非线性效应的理解：（i）由于完全局部地形上的水流或由于自由表面上移动的局部压力片而引起的有限深度水面上的波;以及（ii）由于流过从自由表面淹没在有限深度处的点源或偶极子，或由于在自由表面上移动的压力片而产生的深水波。 这些问题模拟（i）海山和水下山脊上的海洋流动以及浅水中的船舶尾流;和（ii）深水中的船舶波型。 渐近分析的目的是从理论上解释这些数学模型最近的数值模拟所揭示的几个引人注目的非线性特征，并阐明尚未被数值捕获的新的非线性方面。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

Three-dimensional water-wave patterns due to moving surface and submerged sources, such as ships and submarines, or due to currents flowing over seamounts and underwater ridges, are familiar from everyday experience. Apart from their aesthetic appeal, these wave phenomena also are fundamental to applications in various scientific fields, including geophysical fluid dynamics, ship hydrodynamics, oceanography, and meteorology. This research project will develop a novel mathematical methodology for analyzing three-dimensional water-wave patterns. In contrast to all previous analytical treatments, the new approach will not be restricted to waves of small steepness and aims to explain theoretically several striking features revealed by recent numerical simulations of water waves induced by currents over uneven ocean bottom topography. The project will also involve graduate students in the research.The generation of steady three-dimensional (3D) water-wave patterns by moving surface and submerged sources is a fundamental topic of fluid dynamics with applications to geophysical fluid modeling, oceanography, and ship hydrodynamics. The usual way of tackling these problems analytically is based on the linearized water-wave equations, assuming infinitesimally small disturbances. Departing from such linear analysis, this project will devise an analytical (asymptotic) technique that will enable nonlinear treatment of steady 3D free-surface gravity wave patterns due to various types of wave sources. Motivation comes from the recent discovery (by the PI's group) of a nonlinear mechanism that controls the amplitude of 2D forced gravity waves for a broad range of flow conditions, limiting severely the validity of linear analysis. This mechanism depends on all orders of the disturbance amplitude (nonlinearity parameter) and can be captured by perturbation expansions that go beyond all orders in the nonlinearity parameter. This project will develop such a beyond-all-orders (exponential asymptotics) technique suitable for 3D wave patterns. This new methodology will be used to advance understanding of nonlinear effects in 3D forced gravity water waves in two distinct flow regimes: (i) waves on water of finite depth due to a stream over fully localized topography or due to a localized pressure patch moving on the free surface; and (ii) deep-water waves due to flow past a point source or doublet submerged at finite depth from the free surface or due to a pressure patch moving on the free surface. These problems model (i) oceanic flows over seamounts and underwater ridges as well as ship wakes in shallow water; and (ii) ship wave patterns on deep water. The asymptotic analysis will aim to explain theoretically several striking nonlinear features revealed by recent numerical simulations of these mathematical models and to shed light on novel nonlinear aspects that as yet have not been captured numerically.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.