A Model for Staircase Formation in Fingering Convection

Fingering convection is a convective instability that occurs in fluids where two buoyancy-changing scalars with different diffusivities have a competing effect on density. The peculiarity of this form of convection is that, although the transport of each individual scalar occurs down-gradient, the net density transport is up-gradient. In a suitable range of non-dimensional parameters, solutions characterized by constant vertical gradients of the horizontally averaged fields may undergo a further instability, which results in the alternation of layers where density is roughly homogeneous with layers where there are steep vertical density gradients, a pattern known as "doubly-diffusive staircases". This instability has been interpreted in terms of an effective negative diffusivity, but simplistic parameterizations based on this idea, obviously, lead to ill-posed equations. Here we propose a mathematical model that describes the dynamics of the horizontally-averaged scalar fields and the staircase-forming instability. The model allows for unstable constant-gradient solutions, but it is free from the ultraviolet catastrophe that characterizes diffusive processes with a negative diffusivity.