Salt fingering staircases and the three-component Phillips effect

Understanding the dynamics of staircases in salt fingering convection presents a long-standing theoretical challenge to fluid dynamicists. Although there has been significant progress, particularly through numerical simulations, there are a number of conflicting theoretical explanations as to the driving mechanism underlying staircase formation. The Phillips effect proposes that layering in stirred stratified flow is due to an antidiffusive process, and it has been suggested that this mechanism may also be responsible for salt fingering staircases. However, the details of this process, as well as mathematical models to predict the evolution and merger dynamics of staircases, have yet to be established. We generalise the theory of the Phillips effect to a three-component system (e.g. temperature, salinity, energy) and demonstrate a regularised nonlinear model of layering based on mixing length parametrisations. The model predicts both the inception of layering and its long-term evolution through mergers, while generalising, and remaining consistent with, previous results for double-diffusive layering based on flux ratios. Our model of salt fingering is formulated using spatial averaging processes, and closed by a mixing length parametrised in terms of the kinetic energy and the ratio of the temperature and salt gradients. The model predicts a layering instability for a bounded range of parameter values in the salt fingering regime. Nonlinear solutions show that an initially unstable linear buoyancy gradient develops into layers, which proceed to merge through a process of stronger interfaces growing at the expense of weaker ones. Our results indicate that these mergers are responsible for the characteristic increase of buoyancy flux through thermohaline staircases.