New Directions in Water Waves

在流体运动中，波无处不在。数学为研究和理解水波传播提供了一个正式的框架。水波的数学描述的第一个重大突破通常归功于斯托克斯(1848年发表的一篇论文)。尽管如此，即使是对水波的最简单描述也会以丰富的数学结构继续让研究人员感到惊讶。控制流体中波动运动的方程常常太难解析求解。在这种情况下，可以通过简化方程来构造近似的“渐近”解，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)。该研究会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。第一种是关于内波，即在海洋等分层流体中发生的波。与海洋表面的波浪不同，它们可以是巨大的，高达百米高的S，有几公里长。由于内波在世界海洋和大气中分配能量、热量、污染物和生物物质的重要性，人们对内波进行了研究。然而，大多数研究涉及所谓的“1模”波，这是一种观察最多的内波类型，具有最简单的垂直结构。最近的野外观测表明，“模式2”波比之前认为的更为常见。此外，对这些波进行了新颖的实验和数学研究，发现了这些波的许多有趣的特征，但仍未得到充分的探索。它们有一个复杂的垂直结构，并可能有一个被困的再循环区域。在联谊会期间，将构建一个复杂的数值算法来计算模式2的解，并将利用该代码来探索它们的波动特性。这个代码将是开源的，为研究人员提供一个定制的工具来研究这些波。第二个项目是关于旋转表面波的。在研究波在水面上的传播时，大多数文献都假定流体的涡量为零(即颗粒不“自旋”)。这一假设并不总是有效的，例如当水面上的强风引起非恒定切变流时。最近，人们对涡度非零的波--旋转波--产生了浓厚的兴趣。现有的文献大多涉及恒定涡度，发现水面和内部流动具有奇异的结构，如悬挑波和内部停滞点。对于非常量、非零涡度，使用的大多数公式都不考虑这些有趣的特征，从而严重限制了波的形式。导出了一种新的公式，克服了它的缺点，并被用来严格证明波的特征。波浪还没有用这个公式进行数值恢复，这是该研究基金的第二个研究目标。第三个话题与一种被称为“奇数粘度”的现象有关。在粒子旋转的流体或类流体系统中已经观察到了这种现象，例如由于外部磁场而使磁性粒子在轴上旋转的流体。我想探讨奇粘性在著名的高原-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，并导致一柱液体破裂成液滴(从水龙头流出的液体可以观察到)。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。在一个不断发展的研究领域中，奇数粘度对流体流动的作用将有助于理解“活性物质”，既有自然产生的(细菌)，也有合成产生的(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).