Fluid dynamics and nonlinear waves

Developing a predictive theory of wave breaking that can be incorporated into operational wave models is a major challenge in water-wave dynamics. Such a theory would enable accurate estimates of the forces exerted by breaking waves on coastal protection structures—informing robust, long-lasting designs—and support the optimal placement of wave-energy converter farms that harvest renewable energy from the ocean.

In cutting-edge research led by Matt Turner and Tom Bridges, breaking waves are being studied through the lens of the superharmonic instability of steep waves. Working with collaborators at ENS Paris-Saclay, the team has shown that an unstable mode with a dipole-shaped eigenfunction provides a robust mechanism for wave breaking in nature, even when realistic effects such as air density, weak viscosity, and surface tension are included.

Another active direction focuses on modelling the extraction of green energy using offshore wave-energy converters. Current work by Matt Turner and Tom Bridges develops simplified models that capture the coupled dynamics of vessel motion and fluid motion. The vessel dynamics are described using geometric mechanics for rigid bodies, while the fluid is represented through a hierarchy of models ranging from the shallow-water equations, to irrotational two- and three-dimensional flow, and ultimately to the full inviscid Euler equations with exact vorticity.

A related research avenue uses complex-analysis techniques—particularly conformal mapping—to design efficient numerical schemes for coupled sloshing problems. By employing the Schottky–Klein prime function, the team can solve problems involving multiply connected domains, such as vessels containing internal baffles.

Vortex dynamics plays a central role in understanding transport phenomena in geophysical flows, including the movement of heat, tracers, and chemicals in the ocean and atmosphere. Current research led by Matt Turner investigates how coherent vortices respond to nonlinear perturbations, with particular emphasis on identifying regions in parameter space where these perturbations persist over long times.

Another active direction concerns the prediction of leap-frogging behaviour in systems of point vortices near solid boundaries. Recent work has established numerical criteria—based on vortex strengths and initial separations—for determining when two vortices exterior to a circular cylinder will propagate in a leap-frogging motion. Ongoing research extends these ideas to more general doubly and multiply connected domains, revealing how domain geometry influences vortex interaction patterns.

Research in hydrodynamic stability, led by Matt Turner, focuses on the absolute and convective instability properties of spatially periodic flows. A particular emphasis is placed on a mixing layer bounded by a spatially periodic wall. The approach combines analytical and numerical techniques: Floquet theory is applied in the spatial direction to construct modal solutions and derive dispersion relations for wave-like disturbances.

To characterize the global stability of these flows, the team identifies absolute instabilities by locating saddle points in the complex dispersion relation. This framework allows for the systematic study of how flow parameters and boundary geometry influence the onset and growth of instabilities. Current work extends these methods to more complex geometries and flow configurations, aiming to uncover universal patterns in the transition from laminar to unstable or turbulent behaviour in spatially periodic systems.