Analogous behavior of homogeneous, rotating fluids and stratified, non-rotating

Abstract

It has been pointed out in the past that analyses of homogeneous, rotating fluids (system S) and stratified, non-rotating fluids (system S) lead to the same mathematical problem in certain circumstances. The analogy is explored here for the case of steady, linear two-dimensional flows where the temperature variations imposed along side boundaries drive the stably stratified flow and swirl or zonal velocity imposed at the top and bottom boundaries drive the rotating flow. In the fluid S buoyancy boundary layers exist near the side boundaries and these serve the same function in adjusting the temperature between boundary and interior values that the Ekman layer serves to adjust the zonal velocity in the rotating case. The mathematical properties of the two layers are identical. In system S any changes which take place in the interior and tend to alter the stratification are resisted just as changes in the interior which tend to alter the vertical vorticity are resisted in system ?. Processes which give rise to changes in the interior are concentrated in the boundary layers. Simple problems are worked out which exhibit the analogous behavior for two-dimensional systems driven by symmetric and by anti-symmetric conditions imposed at the boundaries. The temperature in the interior of the stratified fluid is the average of the imposed boudary values. This means that with symmetric driving temperatures the fluid is simply restratified by the imposed conditions and with anti-symmetric boundary values the temperature of the fluid interior is unchanged. Buoyancy boundary layers are necessary to adjust interior to boundary temperatures in the latter case. The rotating system has an analogous behavior. In three dimensions the analogy breaks down. The reason is that the third dimension in system S does not involve the constraint and therefore is literally an added degree of freedom whereas in system ? the third dimension also involves the constraint. The difference between the two systems is exhibited by an analysis of the three-dimensional stratified problem with symmetric and anti-symmetric boundary conditions. The symmetric problem in system S differs qualitatively from both the two-dimensional case and the rotating fluid problem. The three-dimensional problem in system ? is qualitatively similar to the two-dimensional one.