A one-dimensional model of staircase formation in diffusive convection

An important feature of the dynamics of double-diffusive fluids is the spontaneous formation of thermohaline staircases, where wide regions of well-mixed fluid are separated by sharp density interfaces. Recent developments have produced a number of one-dimensional reduced models to describe the evolution of such staircases in the salt fingering regime relevant to mid-latitude oceans; however, there has been significantly less work done on layer formation in the diffusive convection regime. We aim to fill this gap by presenting a new model for staircases in diffusive convection based on a regularisation of the $\gamma$ -instability (Radko 2003 J. Fluid Mech. vol. 805, 147–170), with a range of parameter values relevant to both polar oceans and astrophysical contexts. We use the results of numerical simulations to inform turbulence-closure parametrisations as a function of the horizontally averaged kinetic energy $e$ , and ratio of the haline to thermal gradients $R_0^*$ . These parametrisations result in a one-dimensional model that reproduces the critical value of $R_0^*$ for the layering instability, and the spatial scale of layers, for a wide range of parameter values, although there is a mismatch between the range of $R_0^*$ for layer formation in the model and observational values from polar oceans. Staircases form in the one-dimensional model, evolving gradually through layer merger events that closely resemble simulations.