December  7, 2023

Fields Institute Graduate School Information Day
Saturday, November 5, 2005

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Lucy Campbell, Carleton University
Some mathematical problems in geophysical fluid dynamics
In the past few decades, theoretical studies of "geophysical fluid dynamics" (GFD) have greatly advanced our understanding of the dynamics of the atmosphere and the oceans. The evolution of fluids like water and air are modeled using partial differential equations. In GFD we try to make these models more realistic by adding in effects like the earth's rotation and density stratification (layering) of the atmosphere and oceans. In general, we can't find exact analytic solutions of the equations unless many simplifying assumptions are made. But approximate solutions can sometimes be found, which are valid in certain regimes of time and space. They can be derived using techniques such as the method of "matched asymptotic expansions", "multiple scaling", and "perturbation" methods.

In my presentation, I will describe some of these methods and explain how they are applied to some problems in GFD that I have examined in recent years. These problems involve "wave-mean-flow interactions". The waves are treated as sinusoidal perturbations superimposed on a steady basic flow. The evolution of the perturbations is then governed by PDEs that are generally nonlinear and can, under certain conditions, contain singularities. I will also present some results obtained using numerical techniques (finite differences and spectral methods) and talk about some of the geophysical phenomena that can be explained by the results of these mathematical studies.

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