New Directions in Water Waves

波在流体运动中无处不在。数学为研究和理解水波传播提供了一个形式化的框架。对水波的数学描述的第一个重大突破通常归功于斯托克斯（发表于1848年的一篇论文）。尽管如此，即使是最简单的水波描述，也会以丰富的数学结构继续让研究人员感到惊讶。控制流体波动的方程常常难以解析求解。在这种情况下，可以通过简化方程来构造近似的“渐近”解，但代价是对解的行为施加额外的假设。另一种近似解的方法是使用计算机模拟（数值分析）。该奖学金的目标是在三种不同的物理环境中找到复杂水波行为的渐近和数值近似。第一种是内波，即发生在海洋等层状流体内部的波。与在海面上看到的波浪不同，它们的大小非常巨大，可以达到100米高，长达数公里。由于内波在世界海洋和大气中分布能量、热量、污染物和生物物质方面的重要性，人们对其进行了研究。然而，大多数研究关注的是所谓的“1型”波，这是观测最多的内波类型，具有最简单的垂直结构。最近的野外观测表明，“2型”波比以前认为的更为普遍。此外，对这些波的新颖实验和数学研究揭示了这些波的许多有趣的特征，这些特征尚未得到充分的探索。它们有一个复杂的垂直结构，可以有一个再循环的被困区域。在研究期间，将构建一个复杂的数值算法来计算模式2解，并使用代码来探索它们的波特性。这些代码将是开源的，为研究人员提供一个定制的工具来研究这些波浪。第二个项目是关于旋转表面波的。当研究波浪在水面上传播时，大多数文献假设流体的涡度为零（即粒子不“自旋”）。这种假设并不总是有效的，例如当水面上的强风引起非恒定的切变流时。最近，人们对具有非零涡度的波产生了兴趣，这种波被称为旋转波。现有文献大多关注定涡量，发现水面和内部流动具有悬垂波和内部滞止点等奇异结构。对于非恒定、非零涡度，大多数使用的公式不考虑这些有趣的特征，严重限制了波的形式。推导出了一种新的公式，克服了它的缺点，并用于严格地证明波浪的特征。波浪还没有使用这个公式进行数值恢复，这是该奖学金的第二个研究目标。第三个主题涉及一种被称为“奇粘度”的现象。它已经在流体或类流体系统中被观察到，其中粒子在旋转，例如由于外部磁场而绕轴旋转的磁性粒子组成的流体。我想探讨奇粘度在著名的高原-瑞利不稳定性中的作用。这种不稳定性是由于一种被称为表面张力的力造成的，它会导致一柱流体破裂成液滴（就像从水龙头流出的液体一样）。必须恢复方程的正确形式，在此基础上，我将对系统进行渐近分析。奇粘度在流体流动中的作用是一个不断发展的研究领域，它将有助于理解“活性物质”，无论是自然存在的（细菌）还是合成产生的（磁性颗粒）。

Waves are ubiquitous in fluid motion. Mathematics provides a formal framework to study and understand water wave propagation. The first major breakthrough in the mathematical description of water waves is typically attributed to Stokes (a paper published in 1848). Despite this, even the simplest descriptions of water waves continue to surprise researchers with rich mathematical structure. The equations governing wave motion in fluids are oftentimes too difficult to solve analytically. In such cases, approximate "asymptotic" solutions can be constructed by simplifying the equations, at the cost of imposing additional assumptions about the behaviour of the solution. Another way to approximate solutions is through the use of computer simulation (numerical analysis). The goal of the fellowship is to find both asymptotic and numerical approximations of complex water wave behaviour in three different physical settings.The first concerns internal waves, which are waves which occur inside stratified fluids such as the ocean. Unlike waves seen on the ocean surface, they can be of ginormous size, reaching heights of 100's of metres and being kilometres long. Internal waves have been studied due to their importance in distributing energy, heat, pollutants and biological matter in the worlds' oceans and atmosphere. However, the majority of research concerns so-called "mode-1" waves, the most observed type of internal wave with the simplest vertical structure. Recent field observations have demonstrated that "mode-2" waves are more common than previously believed. Furthermore, novel experimental and mathematical research of these waves has uncovered a plethora of interesting features of these waves that remain underexplored. They have a complex vertical structure and can have a trapped region of recirculation. During the fellowship, a sophisticated numerical algorithm will be constructed to compute mode-2 solutions, and the code will be utilised to explore their wave properties. This code will be made open-source, to provide a bespoke tool for researchers to use in their studies of these waves.The second project is on rotational surface waves. When studying waves propagating on the surface of water, most literature assumes the vorticity of the fluid is zero (i.e. particles don't 'spin'). This assumption is not always valid, such as when strong winds at the water surface induce non-constant shear currents. Recently, there has been a spell of interest concerning waves with non-zero vorticity, known as rotational waves. Most existing literature concerns constant vorticity, where it is found the water surface and interior flow have exotic structures such as overhanging waves and internal stagnation points. For non-constant, non-zero vorticity, most formulations used do not allow for these interesting features, severely restricting the form of the wave. A recent formulation of the equations was derived which overcomes and shortcomings, and was used to rigorously prove features of the waves. Waves are yet to be recovered numerically using this formulation, which is the second research objective of the fellowship. The third topic concerns a phenomenon known as "odd viscosity". It has been observed in fluid or fluid-like systems where particles are rotating, such as a fluid composed of magnetic particles spinning on an axis due to an external magnetic field. I wish to explore the role of odd-viscosity on the famous Plateau-Rayleigh instability. This instability is due to a force known as surface tension, and causes a column of fluid to break into droplets (as can be observed of fluid coming from a tap). The correct form of the equations must be recovered, upon which I will perform asymptotic analysis on the system. The role of odd viscosity on fluid flows in a growing field of research which will be helpful in understanding 'active matter', both occurring naturally (bacteria) and being synthetically produced (magnetic particles).