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.