Wave Models for Atmospheric Fluid Dynamics

通过旋转、分层和潮湿气团的数学流体力学，可以理想化对我们在北美经历的天气的理论理解。大气的气象学显示出两种截然不同的动力风场：控制每日大陆天气的大尺度Rossby波和具有更快和更小尺度影响的重力波。这项研究计划的目标是了解这些大气波的强烈非线性行为的数学工具。与大气科学家合作，这项工作旨在识别和量化理想化流体模型中的波传播机制-并将物理原因与观测和预测模型输出联系起来。虽然这项研究的重点是地球物理波传播的渐近方法技术，但同样重要的是设计和实现说明科学意义的计算演示。**以前的工作已经发展了理论和计算工具，用于研究二维旋转浅水方程中地球球面上的Rossby波的传播。虽然Rossby波的线性理论可以追溯到拉普拉斯，但事实证明，急流和赤道信风的包含并不完全理解。2013年，SFU的一篇博士论文首次完整地确定了中纬度和赤道Rossby波集的全球结构。特别是，关键的发现是存在一个无限的离散赤道波模集合(以前假设是有限的)。将这种新的光谱理解扩展到大气的3D分层方程仍然是一个开放的挑战。从科学上讲，Rossby波结构的这种二分法影响了中纬度和热带天气之间的通信；并为为什么在气候数据中只发现了最长波长的Rossby波信号的问题提供了一种潜在的解决方案。**与NCAR科学家的合作揭示了云边缘运动的新机制。与大多数以不稳定方式演化的云不同，这一新过程涉及一种潮湿的重力波，它作为稳定的激波在水平方向传播。云物理的数学模型出人意料地简单，对非对流云的演化具有广泛的适用性。我们认为，最近对“洞穴”云的模拟是一个偶然的机会，可以为观测到的天气事件验证这一冲击波理论。这团奇特的云团的预报动力学将直接受到冲击波理论的检验。我们乐观地认为，这个例子只是观察这种云边缘动态的几个真实世界场景中的第一个。**在科学研究计划的过程中，通常情况下，专门的技术被开发出来，保证单独传播到更广泛的数学界。针对非线性常微分方程组的近分离线轨迹设计了一种匹配的渐近分析。第一个应用于正弦强迫的非线性摆得到了描述扰动哈密顿系统混沌层庞加莱截面的“晶须映射”。在包含弱阻尼的情况下，我们发现了一条不稳定的路径，其中包括一条意外的近分离线“奇怪吸引子”，在通往熟悉的耗散混沌的道路上。这一新的混沌特征背后的数学要求将渐近分析进一步扩展到更高的阶数。

A theoretical understanding of the weather we experience in North America can be idealized through the mathematical fluid mechanics of a rotating, stratified and moist airmass. The meteorology of the atmosphere displays two distinct dynamical wind regimes: large-scale Rossby waves, which govern the daily continental weather; and gravity waves, which have faster and smaller-scale influences. The objectives of this program of research are mathematical tools for understanding the strongly nonlinear behaviour of these atmospheric waves. In collaboration with atmospheric scientists, this work aims to identify and quantify mechanisms for wave propagation within idealized fluid models - and connect physical cause with observations and forecast model outputs. While the focus of this research is the technology of asymptotic methods for geophysical wave propagation, an equally important aspect is the design and implementation of computational demonstrations that illustrate the scientific implications.**Previous work has developed theoretical and computational tools for studying the propagation of Rossby waves on the spherical geometry of the Earth within the 2D rotating shallow water equations. While the linear theory of Rossby waves dates back to Laplace, the inclusion of the jetstream and equatorial tradewinds proved incompletely understood. A 2013 SFU Doctoral thesis gave the first complete determination of the global structure of the midlatitude and equatorial Rossby waves sets. In particular, the key discovery was the existence of an infinite set (previously hypothesized to be finite) of discrete equatorial wave modes. Extending this new spectral understanding to the 3D stratified equations for the atmosphere remains an open challenge. Scientifically, this dichotomy of Rossby wave structure impacts the communication between midlatitude and tropical weather; and offers one potential resolution to the question of why only the longest wavelength Rossby wave signals have been identified in climatological data.**A collaboration with NCAR scientists has uncovered a new mechanism for the motion of cloud edges. Unlike most clouds which evolve in an unstable fashion, this new process involves a moist gravity wave that propagates horizontally as a stable shock. The mathematical model for the cloud physics is unexpectedly simple and potentially has broad applicability to the evolution of non-convective clouds. We believe that a recent simulation of a "holepunch" cloud is a fortuitous opportunity to verify this shock wave theory for an observed weather event. The forecast dynamics of this curious cloud will be tested directly against the shock wave theory. We are optimistic that this example is just a first of several real-world scenarios for the observation of this cloud edge dynamics.**During the course of a scientific program of research, it is often the case that specialized techniques are developed that warrant separate dissemination to the broader mathematical community. A matched asymptotic analysis has been designed specific to near-separatrix trajectories of nonlinear ODE systems. A first application to the sinusoidally-forced nonlinear pendulum obtains the "whisker map" that describes the Poincare section of the chaotic layer for a perturbed Hamiltonian system. With the inclusion of weak damping, we have discovered a pathway of instability that includes an unexpected near-separatrix "strange attractor" en route to the familiar dissipative chaos. The mathematics behind this new chaotic feature requires further extension of the asymptotic analysis to a higher-order.