**Lucy Campbell, Carleton University**

*Some mathematical problems in geophysical fluid dynamic*s

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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