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

由于移动的水面和水下源，如船只和潜艇，或由于在海山和水下山脊上流动的水流，造成的三维水波模式，从日常经验中是熟悉的。除了美观之外，这些波浪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。本研究计划将发展一种新的数学方法来分析三维水波模式。与以前所有的分析方法相比，新方法将不局限于小陡度的波浪，其目的是从理论上解释最近由不均匀海底地形上的水流引起的水波的数值模拟所揭示的几个显著特征。该项目还将涉及研究生的研究。通过移动的水面和水下源产生稳定的三维水波模式是流体动力学的一个基本主题，它应用于地球物理流体建模、海洋学和船舶流体动力学。解析处理这些问题的通常方法是基于线性化的水波方程，假设有无穷小的扰动。从这种线性分析出发，本项目将设计一种解析（渐近）技术，该技术将能够非线性处理由各种类型的波源引起的稳定的三维自由表面重力波模式。动机来自于最近（由PI的小组）的一项非线性机制的发现，该机制可以在广泛的流动条件下控制二维强迫重力波的振幅，这严重限制了线性分析的有效性。这种机制取决于扰动幅度（非线性参数）的所有阶数，并且可以通过超越非线性参数中所有阶数的扰动展开来捕获。本项目将开发一种适用于三维波形的超全阶（指数渐近）技术。这种新方法将用于促进对两种不同流动状态下三维强迫重力水波非线性效应的理解：(i)由于在完全局部地形上的水流或由于在自由表面上移动的局部压力斑块而产生的有限深度水上的波；（ii）由于水流经过一个点源或从自由表面在有限深度上淹没的双重波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(1)模拟海底山和海底脊上的海洋流动以及浅水中的船舶尾迹；（ii）深水船舶波浪模式。渐近分析将旨在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显著的非线性特征，并阐明尚未在数值上捕获的新的非线性方面。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。

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.