Development and long-term evolution of layers in turbulent stratified fluids

A fascinating aspect of stratified turbulence is the spontaneous formation of density staircases, consisting of layers with nearly uniform density separated by narrow interfaces with large density gradients. For example, double diffusive staircases appear in regions of the ocean where the overall stratification is stable, and layers can also be induced experimentally by stirring a fluid with a stable salt gradient. One leading theory for layering is the Phillips Effect: layering occurs due to the dependence of the turbulent density flux on the density gradient. If the flux is a decreasing function of the gradient for a finite range of gradients, then negative diffusion causes perturbations to grow into systems of layers and interfaces. We present a model for stirred stratified layering derived from the Boussinesq equations encapsulating the Phillips effect. Solutions to the model produce layered regions which evolve indefinitely through layer mergers. We include molecular and viscous diffusion, both of which act to stabilise the system. A novel investigation into the long-term dynamics of solutions reveals a simple scaling law for the time dependence of the number of layers, suggesting a self-similar structure to merger dynamics, and a link to Cahn-Hilliard models of layering.