MATHEMATICAL CONDITIONS FOR QUASI-ISOLATION OF TWO WATER MASSES OVER A FINITE TIME INTERVAL

Abstract

We consider the problem of the long-term preservation of differences between two adjoining water masses in a stratified rotating fluid. Absolute non-mixing is impossible in a real oceanic environment owing to molecular and turbulent diffusion, front deformation, internal waves, and local hydrodynamic instabilities. We therefore introduce the more physically accurate notion of quasi-isolation: two water masses are said to be quasi-isolated over a time interval if the total transport of marker substance across the moving surface separating them remains small compared with the amount of substance in each of the regions. The underlying model is the system of Boussinesq equations in a rotating frame, augmented by transport equations for temperature, salinity, and a passive water-mass indicator. An exact expression is obtained for the relative flux through the unsteady interface. A theorem on finite-time quasi-isolation is stated and proved under the assumption that the normal relative velocity, the effective diffusivity, and the area of the deforming interface are all small. Estimates are derived for the time over which the contrast between the water masses is preserved, and a dimensionless quasi-isolation criterion is formulated. The roles of stable stratification, the Richardson number, the Earth's rotation, and tangential transport along the front are discussed.