Wave Models for Atmospheric Fluid Dynamics

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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.