Development and long-term evolution of density staircases in stirred stratified turbulence

We formulate and analyze a model describing the development and evolution of density staircases in stratified turbulence, using it to investigate the long-term evolution and merger of layers and demonstrating an inverse logarithmic relationship for their decay. Starting from the Boussinesq equations, including viscous stresses and density diffusion, we use spatial averaging and simple closure assumptions to derive a system of equations describing the evolution of the density profile of a turbulent stratified fluid in terms of the horizontally averaged buoyancy and turbulent kinetic energy. Subject to critical conditions on the buoyancy gradient, the model predicts the development of a system of well-mixed layers separated by sharp interfaces. A linear stability analysis determines critical conditions for layer formation and demonstrates both a minimum and maximum initial density gradient necessary for layering. Increasing the viscosity decreases the maximum unstable wave number, thereby increasing the vertical length scale of layers. Increasing the diffusivity has a similar effect, but can also suppress the instability entirely by decreasing the unstable range of gradients. The long-term nonlinear evolution shows that the layers undergo successive mergers, with each merger increasing the magnitude of steps in the density staircase. In particular, by applying boundary conditions of fixed buoyancy, instead of the previously adopted condition of zero buoyancy flux, we reduce the influence of the boundaries at late times, allowing us to investigate the long-term evolution of layer mergers in stratified turbulence in detail. For long times

t

, we infer a general law describing the evolution of the number of layers

N

as

1

/

N

∼

ln

t

, suggesting a self-similar structure to merger dynamics and a link to Cahn-Hilliard models of layering.