Double-diffusive convection arises in fluids comprising two diffusive elements that compete to influence the density of the fluid, such as heat and salt in the ocean. Instabilities in such systems have previously been shown to lead to the formation of layered structures, known as ‘density staircases’. We are interested in the modal interactions that lead to the formation of such layers. Specifically, we begin by constructing highly truncated models of double-diffusive convection comprising horizontally uniform ‘elevator modes’, which are thought to be critical in the formation of layers. We study the interactions of modes in the truncated models, and show that we can obtain layered structures in systems having as few as nine modes.
In view of the importance of elevator modes in layer
formation, we then proceed to study the stability of the elevator modes themselves. That is, we apply perturbations to an initial state comprising fully developed elevator modes to determine the fastest growing secondary modes. We first study the stability of steady elevator modes, before extending our analysis to elevator modes that are oscillatory in time. We discover that the form of the dominant secondary mode in each case is highly dependent on the amplitude of the elevator mode. Furthermore, this dependency is influenced by the parameters governing the background basic state and the fluid itself.
Interestingly, the secondary modes arising from oscillatory elevator modes are similar in structure to those arising from steady elevator modes, except that the former modes are oscillatory in time. The fastest-growing secondary modes in the oscillatory case are found to be similar to those that lead to layer formation in our truncated models. This suggests that the elevator modes may first become unstable to secondary modes, which then interact with the elevator modes to generate layering modes. The layering modes then grow in amplitude to eventually dominate and form a layered state. This constitutes a relatively simple mechanism describing the initial formation of a layered state from a stably stratified background density gradient.
We compare our results with those of a purely hydrodynamic system, to show that large amplitude elevator modes appear to become unstable to a hydrodynamic, rather than a diffusive instability. We conclude that, although diffusive effects are vital for the formation of the primary instability, the system may eventually become driven, at least in part, by viscous shear effects, providing the elevator modes grow to a sufficient amplitude.
