Motion of Complex Singularities and Integrability in Surface Dynamics

该研究项目致力于研究不同运动流体之间的界面或流体表面的表面动力学。表面动力学包括水波破碎和白浪，这是海洋和大气之间能量交换的主要机制，使其成为全球气候动力学的一个关键组成部分。这些都是强非线性现象，需要求解完全非线性的流体力学方程。流氓波是另一个强非线性大表面波的例子，它在海洋中自发发生。 流体的相对运动（风在水面上）引起它们共同界面的不稳定性，例如开尔文-亥姆霍兹不稳定性（KHI）。这种不稳定性最近成为实验科学家在不同成分超流氦之间的界面背景下关注的焦点，也将在该项目中得到解决。本研究将致力于发展一种新型的保角映射和新的工具，以有效地描述自由表面和界面的强非线性表面动力学。正是斯托克斯在世纪首次使用保角变换作为研究具有自由表面的流体定常流动的工具。在这种方法中，由流体占据的域被共形地映射到更简单的域，如较低的复杂半平面。曲面的动力学被简化为共形映射的动力学。它提供了一个巨大的优势，表面动力学的分析和高精度模拟，允许恢复流体动力学通过复杂的分支削减和极点在复杂的域中的运动。该项目旨在推进地表动力学和可积性领域的研究，并开发实用工具，以确定影响全球气候动态的地表重力波消散简化模型。该研究将包括流氓和破碎波动力学分析通过运动的分支削减，以及在不同的实验情况下的超流体的界面动力学的可积性的探索。 为了解决高振幅水波的统计问题，将采用分析方法以及开发高性能计算工具。该奖项反映了NSF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。

This research project is devoted to a study of surface dynamics that arise either at the interface between different moving fluids or at the fluid's surface. Surface dynamics include breaking of water waves and whitecapping, which are the primary mechanisms for the exchange of energy between the ocean and atmosphere making them a crucial ingredient of the global climate dynamics. These are strongly nonlinear phenomena which require to solve fully nonlinear hydrodynamics equations. Rogue waves are another example of strongly nonlinear large surface waves, which occur spontaneously in the ocean. Relative motion of fluids (wind over the water) induces instability of their common interface such as Kelvin-Helmholtz instability (KHI). This instability that recently became a focus of attention of experimental scientists in the context of the interface between different components superfluid Helium, will also be addressed in this project. This research will focus on development a new type of conformal map and new tools for the efficient description of the strongly nonlinear surface dynamics both for free surface and interfaces. It was Stokes who in the 19th century first used conformal mapping as a tool for studying of the steady flow of the fluid with a free surface. In this approach domains occupied by fluids are conformally mapped into simpler domains such as a lower complex half plane. The dynamics of the surface is then reduced to the dynamics of the conformal map. It gives an enormous advantage for both the analysis and high precision simulations of surface dynamics by allowing to recover the fluid dynamics through the motion of the complex branch cuts and poles in the complex domain. This project is aimed towards advancing the fields of surface dynamics and integrability as well as developing practical tools to identify the reduced models for dissipation of surface gravity waves, affecting global climate dynamics. The research will include an analysis of rogue and breaking waves dynamics through the motion of branch cuts as well as an exploration of integrability of interface dynamics of superfluids in different experimental situations. To address statistics of high amplitude water waves, analytical methods will be employed as well as development of high-performance computing tools.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.