Turbulent mixing in stratified fluids: layer formation and energetics

Abstract

Water with constant initial salt stratification was mixed with a horizontally moving vertical rod. The initially linear density profile turned into a series of steps when mixing was weak, in agreement with instability theory by Phillips (1972) and Posmentier (1977). For stronger mixing no steps formed. However, in all cases mixed layers formed next to the top and bottom boundaries and expanded into the interior due to the no-flux condition at the horizontal boundaries. The critical Richardson number Rie, dividing experiments with steps and ones without, increases with Reynolds number Re as Rie ≈ exp(Re/900). Steps evolved over time, with small ones forming first and larger ones appearing later. The interior seemed to reach an equilibrium state with a collection of stationary steps. The boundary mixed layers continued to penetrate into the interior. They finally formed two mixed layers separated by a step, and ultimately acquired the same densities so the fluid became homogeneous. The length scale of the equilibrium steps, ls, is a linear function of U/Ni, where U is the speed of the stirring rod and Ni is the buoyancy frequency of the initial stratification. The mixing efficiency Rf also evolved in relation to the evolution of the density structure. During the initiation of the steps, Rf showed two completely different modes of evolution depending on the overall Richardson number of the initial state, Rio. For Rio [Gt ] Rie, Rf increased initially. However for Rio near Rie, Rf decreased. Then the steps reached an equilibrium state where Rf was constant at a value that depended on the initial stratification. The density flux was measured to be uniform in the layered interior irrespective of the interior density gradient during the equilibrium state. Thus, the density (salt) was transported from the bottom boundary mixed layer through the layered interior to the top boundary mixed layer without changing the interior density structure. The relationship between Ril and Rf was found for Ril > 1, where Ril is the Richardson number based on the thickness of the interface between the mixed layers. Rf decreases as Ril increases, consistent with the most crucial assumption of the instability theory of Phillips/Posmentier.