The spread of mixing in a stratified fluid and its analogue in rotating fluids

Abstract

The asymptotic consequences due to mixing by an oscillating bottom mixer in a stably stratified fluid in a container are analyzed theoretically. This mixed region is found to move laterally away from the mixer along its neutral buoyancy level in the form of a horizontal wedge. With the beginning of a rather arbitrary initial configuration. the two-dimensional wedge is found always to approach a parabolic profile associated with which the mixed region flow is horizontally columnar. Horizontal pressure gradient generated from density differences across the interface pushes the unmixed fluid towards the wall of the container. causing a return flow in free shear layers along levels marked by the top and bottom of the mixer. The lateral speed of the tip of the mixed fluid is found to be proportional to the square of the Brunt-Väisäla frequency. and the fourth power of the mixed region and inversely proportional to the vertical eddy viscosity of the mixed fluid. Qualitative verification of the theory is achieved with a simple laboratory model experiment.

A perfect analogue to the above phenomenon is found in the vertical spread of angular momentum mixing in a homogeneous rotating fluid. The differential in Coriolis force between the mixed and unmixed region generates the necessary pressure gradient in this case. The rate of vertical advance of the angular momentum front now becomes directly proportional to the square of the Coriolis frequency and the fourth power of the radial extent of the ring of mixing, and inversely proportional to the lateral eddy viscosity due to mixing. Some evidence of the vertically moving wedge formation is found in an experiment by Turner (1966).