Staircase structures in fluid dynamical systems

Stratified turbulent fluids exhibit a wide variety of fascinating behaviours. One of the most interesting is the phenomenon of staircase formation, where the density field spontaneously evolves into a series of well-mixed layers separated by sharp interfaces with high gradients. Staircases appear in a wide range of contexts, from the geophysical examples of oceanic thermohaline staircases and atmospheric potential vorticity staircases, to the E×B staircase of plasma physics. In this thesis we present models for staircases in stirred stratified convection and double diffusive convection. We derive a one-dimensional horizontally-averaged mixing-length model from the Boussinesq equations, which we apply first to layering in stirred stratified convection, and then to double-diffusive layering. In stirred stratified convection, the model consists of equations for the buoyancy and kinetic energy, closed via a length scale parameterised in terms of the variables. We investigate the linear stability of the system, determining the effects of varying the viscous and molecular diffusivities. A novel choice of boundary conditions allows us to investigate the behaviour of numerical solutions to very late times. Staircase solutions undergo layer mergers, which we demonstrate occur on a logarithmic timescale, providing a link with other models of layering. We also present an experimental study to test predictions of the theory. For double-diffusive convection, the model consists of three equations, for temperature, salinity and energy. We present a linear stability analysis for a general three-component flux-law system, which we apply to our specific model. A suitable parameterisation for the length scale allows the model to produce staircases in salt fingering convection without the need for any external forcing. In diffusive convection, some energy source is required for layering. Temperature and salinity fluxes through staircases increase during layer mergers, accounting for heightened fluxes in observed staircases in comparison with non-layered states.