The investigator undertakes a theoretical and numerical study of nonlinear interactions between waves and vortices in problems spanning several subject areas, namely atmosphere and ocean fluid dynamics, small-scale engineering applied to particle mixing, and the dynamics of rotating superfluids in quantum mechanics. Project topics include energy exchanges between random waves and vortices in the ocean, the role of wave-induced vortex structures for particle dispersal in the deep ocean, the design and dynamics of small-scale engineering devices to model the acoustic mixing of immersed particles, and the peculiar fluid dynamics that attends the quantum-mechanical interactions between waves and point vortices in rotating superfluids. A common theme of these problems is that they involve huge separations of scale between different dynamical components of the flows. Such separations present mathematical difficulties and make straightforward numerical simulations impractical.
The investigator takes up several problems that involve analysis of the interactions between waves and vortices in fluid flows: ocean fluid dynamics relevant to climate dynamics, small-scale fluid engineering applied to acoustic mixing, and rotating superfluids in condensed matter physics. The ocean is a vast environment in which a multitude of dynamical components such as waves and currents interact, and these interactions eventually determine the present ocean state and also its future development. Our ability to predict ocean dynamics hinges delicately on our ability to understand and to model these interactions, because a direct computer simulation far exceeds our computational resources and will do so for decades to come. This is so because different dynamical components of the flows occur at vastly different scales of length or time. The investigator mathematically and computationally studies the interactions between waves and vortices, using modern mathematical theory to simplify and thereby to reduce the complexity of the interactions that must be modelled on a computer. Similar scale-separated interactions among dynamical components of flows are studied in other complex systems of technological relevance, such as non-invasive particle mixing techniques using wave engineering, and the detailed study of superfluid behaviour that has recently become possible using breakthroughs in laboratory technology.
Interactions between waves and vortices are familiar to everyone who has ever sat in a bathtub and watched surface waves spinning into the vortex over the plughole. Similar interactions are in fact taking place in the atmosphere and ocean, and those interactions are well known to be significant for the long term dynamics of these systems. However, in many cases of practical interest the waves that are involved in these interactions are too small in scale to be resolved by direct numerical simulations. In other words, the effects due to these waves need to be put in "by hand" in such models, combining theory and empirical data to get the best results. It is in the theoretical study of such wave-vortex interactions that the present project made various contributions. For example, one important practical question is whether unresolved small scale waves can contribute to mixing and stirring in the deep ocean, say more than a mile deep in the water column. The ocean is dark and slow there, and very little is known by direct observation or simulation. Our study showed for the first time how internal gravity waves, which owe their restoring mechanism to a combination of the Earth rotation and the stable density stratification of the ocean, can lead to the irreversible diffusion of particles in the horizontal at such depth. Other results of our project concern the peculiar fluid mechanical behaviour of both waves and vortices in superfluid helium and the potential for very fast internal waves in the atmosphere to signal the formation and propgation of tsunamis at sea level by an observable oscillation in the ionosphere, which is at 100km altitude, within an hour or so of the tsunami generation.
