It has long been known that turbulent channel flows tend to have a zone of mean shear with nearly zero absolute vorticity in homogeneous rotating fluids. This feature was observed either in a numerical experiment for the Couette flow or in a tank experiment for the Poiseuille flow. This paper is intended to give a plausible account of how this curious property of turbulent channel flows is maintained in a rotating system, on the basis of an analogy between rotating and stratified fluids. The analogy itself has been well-known since Veronis. In reality, some arguments along this line have already been attempted for the presence of the shear zone of nearly zero absolute vorticity by such as Bradshaw and Tanaka et al. Nevertheless convincing explanation does not seem to have been proposed so far. Here the argument is improved first by showing that the zonally averaged flow in the rotating fluids is governed by the same equations as those for the zonally averaged flow in stratified fluids, even though the second moment may differ in the two systems. Then, as is usual, the shear zone of zero absolute vorticity in the rotating fluids is identified with nearly homogeneous layers in stratified fluids. The Couette flow in the rotating system corresponds to the thermal convection driven by cooling at the top and heating at the bottom. Likewise and remarkably the Poiseullie flow in the rotating system can be compared to the stratified fluid that suffers from steady body cooling along with the stabilizing heating from above.
In order to confirm the validity of this analogy numerical experiments were carried out for the zonally averaged field in the stratified system to model the turbulent channel flows of Couette and Poiseullie types in the rotating system. In both cases, thermal convection yielded nearly homogeneous layers. The profile of the mean density profile in stratified fluids is translated to the mean zonal flow in rotating fluids including the linear shear due to the Coriolis parameter. The results showed a good agreement between the picture based on the analogy with stratified fluids and previous numerical or tank experiments in rotating fluids. Additional experiments were carried out, where fluctuating forms of body cooling were superposed to the inherent regular one in the stratified system corresponding either to the Couette or the Poiseuille flow in the rotating system. Even under the fluctuating body forcing, we observed the robust formation of nearly homogeneous layers that corresponds to the shear zone of zero absolute vorticity. The results suggest that the basic mechanism for maintaining the shear zone of nearly zero absolute vorticity is the inertial instability analogous to the gravitational instability in stratified fluids and that the maintenance of the shear zone of zero absolute vorticity does not depend on the details of turbulent motion. |
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