Staircase Formation in Fluid Dynamical Systems

One of the most interesting features of certain fluid dynamical systems is their tendency to form layers or "staircases", in which a key physical quantity, such as the density, exhibits a staircase structure with depth, rather than being more smoothly distributed as one might expect. This phenomenon occurs in systems that appear to be rather different physically, and so a research question of some interest is whether, at heart, the underlying physics of layering is the same in these different systems. Understanding staircase formation is not only of intrinsic scientific interest but is vital to understanding turbulent transport. Layered and unlayered systems have very different transport properties; thus it is essential to understand the process in order that turbulent transport can be realistically parameterised in large oceanographic or atmospheric models.
In the context of atmospheres and oceans there are two particular systems of interest that are known to be susceptible to layering. One is double-diffusive convection; in the oceans the key quantities are heat and salt, which diffuse at very different rates - the process is then known as thermohaline or thermosolutal convection. Convection can then be driven either by an unstable solutal gradient with a stable thermal gradient (which occurs in warmer oceans) or, alternatively, by an unstable thermal gradient with a stable solutal gradient (the situation in colder oceans). In both these cases, the density is observed to take on a staircase structure, which appears to be remarkably resilient. Interestingly, a similar process is believed to be of importance in stellar cores, where the competing elements are heat and a compositional gradient. The other layering process at work in the atmosphere is that of the formation of layers (staircases) in potential vorticity, which is manifested by the appearance of strong jets. This process is relevant not only for our own atmosphere, but also for the outer layers of Jupiter's atmosphere, which is characterised by a strongly banded structure.
This project will explore, through simplified models, the entire nature of staircase formation - the physical ingredients necessary for their formation, the scale at which they initially form, and how subsequent staircase mergers occur. The model equations will, initially at least, be nonlinear partial differential equations in one spatial direction (height, for example, for the case of thermohaline convection) and time. These will be simpler than the full equations of three-dimensional fluid dynamics, but will nonetheless allow a detailed analysis and understanding of what exactly is needed to provide layering. The project will involve a combination of analytical and asymptotic approaches, together with numerical solutions of the model equations.