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Turbulent mixing in stratified fluids: layer formation and energetics
Young-Gyu Park, John A. Whitehead and Anand Gnanadesikan
Journal of Fluid Mechanics 279, 279-311, 1994.
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 a greement with the 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 dur to the no flux condition at the horizontal boundaries. The critical Richardson numer Ri_c , dividing experiments with steps and ones without, increases with Reynolds number Re as Ri_c ~ exp(Re/900). Steps eevolved over time, with small ones forming first and large ones appearing later. The interior seem 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 that the fluid became homogeneous. The length scale of the equilibrium steps l_s, is a linear function of U/N where U is the speed of the stirring rod and N is the buoyancy frequency of the initial stratification. The mixing efficiency R_f also evolved in relation to the evolution of the density structure. During the initiation of the steps, R_f showed two completely different modes of evolution depending on the overall Richardson number of the initial state Ri_0. For Ri_0 > Ri_c, R_f increased initially. However, for Ri_0 near Ri_c, R_f decreased. Then the steps reached an equilibrium state where R_f 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 without changing the interior density structure. The relationship between Ri_l and R_f was found for Ri_l>1, where Ri_l is the Richardson number based on the thickness of the interface between the mixed layers. R_f decreases as Ri_l increases, consistent with the most crucial assumption of the instability theory of Phillips/Posmentier.