## Fast-slow interactions in fluids

Many physical systems
involve the interaction of phenomena with widely separated time
scales. Motivated by applications to geophysical fluids, I am
interested in the situation where the fast degrees of freedom have
weak amplitudes. In this case, the dynamics be can reduced
to evolution on a so-called "slow manifold" where the fast motion is
filtered out. An important problem concerns the limitations of this
approach: the asymptotic expansions defining the slow manifold
diverge, and fast motion is always generated. My recent results show how this fast motion can be captured in simple models using exponential asymptotics (with I Yavneh, Technion, E I Olafsdottir and A Olde Daalhuis, Edinburgh).

## Instabilities of balanced flows

The existence of fast modes (inertia-gravity waves) in large-scale geophysical flows leads to a number of instabilities which are not captured in balanced models describing the dynamics on a slow manifold (such as the quasi-geostrophic model). I study these instabilities in simple shear flows using asymptotic methods (with I Yavneh, Technion, and D G Dritschel, St Andrews).

## Passive-scalar mixing

The evolution of the concentration of
scalars (such as chemicals) in complex flows raises a number of
issues, many of which with industrial or environmental applications.
My research is this area focusses on flows dominated by their
large-scale components which can be modelled stochastically. With P H
Haynes (Cambridge), we have recently elucidated the mechanism
that sets the decay rate of the concentration of a scalar that is
released suddenly in such flows.

## Fluids as dynamical systems

Regarding fluids, or more
generally continuous media, as infinite-dimensional dynamical
systems often proves fruitful. My interest in this area lies in the
analysis of the relationship between the
properties of perfect fluids and those of finite-dimensional
Hamiltonian systems.
In a recent project with D Wirosoetisno (Durham), we examine the persistence
and stability of steady fluid flows under perturbations of their
domain.
A natural extension concerns the evolution of fluid flows in slowly changing domains; this naturally brings about fluid-dynamical versions of the theory of adiabatic invariant and Hannay-Berry angles.