Localised states in natural doubly diffusive convection

Fluids subject to both thermal and compositional variations can undergo doubly diffusive convection when these properties both affect the fluid density and diffuse at different rates. This phenomenon can lead to the formation of a variety of patterns, including salt fingers and thermohaline staircases, which have been identified throughout the world’s oceans. In this thesis, we consider natural doubly diffusive convection driven by opposing thermal and solutal gradients in the horizontal direction and aim to determine how states in this system are affected by the physical parameters that characterise the strength of the thermal gradients, the balance between thermal and solutal
gradients, and ratios between thermal, solutal and viscous diffusivities. In the particular case when the imposed thermal and solutal gradients balance, a motionless conduction state exists but destabilises when the gradients are sufficiently large. We determine the nature of the associated primary bifurcation using a weakly nonlinear analysis and extend the resulting primary convection branches using numerical continuation to find that large-amplitude steady convection states can coexist with the stable conduction state for both sub- and supercritical bifurcations. We proceed by considering vertically extended domains where spatially localised states, known as convectons, have been found to lie on a pair of secondary branches that intertwine when the onset of convection is subcritical. This process is known as homoclinic snaking and is usually associated with bistability. Here, we show that convectons persist into parameter regimes where the primary bifurcation is supercritical and there is no bistability. We finally consider how the system changes when the imposed thermal and solutal gradients do not balance and the motionless conduction state does not exist. We focus on how the form of convectons change with increasing imbalance and how these localised states cease to exist in sufficiently thermally dominated flows.