Wave Models for Atmospheric Fluid Dynamics

通过旋转、分层和潮湿气团的数学流体力学，我们可以理想化地理解我们在北美经历的天气。大气的气象学显示出两种不同的动力学风态：大尺度的罗斯贝波，它控制着每日的大陆天气;重力波，它具有更快和更小的尺度影响。该研究计划的目标是理解这些大气波的强非线性行为的数学工具。与大气科学家合作，这项工作旨在确定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的技术，一个同样重要的方面是设计和实施的计算演示，说明了科学的影响。以前的工作已经开发了理论和计算工具，用于研究Rossby波在地球的球形几何形状内的2D旋转浅水方程的传播。虽然罗斯贝波的线性理论可以追溯到拉普拉斯，但急流和赤道信风的包含被证明是不完全理解的。2013年SFU博士论文给出了中纬度和赤道Rossby波集的全球结构的第一个完整的确定。特别是，关键的发现是存在一个无限集（以前假设是有限的）离散赤道波模式。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。科学上，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;并为为什么只有最长波长的Rossby波信号在气候数据中被识别的问题提供了一个潜在的解决方案。与NCAR科学家的合作发现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，这个新过程涉及作为稳定冲击水平传播的潮湿重力波。云物理学的数学模型出乎意料地简单，并可能对非对流云的演变具有广泛的适用性。我们相信，最近的一个模拟的“洞穿”云是一个偶然的机会，以验证这一冲击波理论的观测天气事件。这片奇特云团的预测动力学将直接与冲击波理论进行对比。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场景中的第一个。在科学研究计划的过程中，通常会开发出专门的技术，以保证单独传播给更广泛的数学界。针对非线性常微分方程系统的近分界线轨迹，设计了一种匹配渐近分析方法。第一个应用正弦强迫非线性摆获得的“晶须地图”，描述了庞加莱节的扰动哈密顿系统的混沌层。随着弱阻尼的加入，我们发现了一个不稳定的路径，其中包括一个意想不到的近分界线的“奇怪的吸引子”途中熟悉的耗散混沌。这个新的混沌特征背后的数学需要进一步扩展到高阶的渐近分析。

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.