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.