On non-linear hydrodynamic instability and enhanced transport in differentially rotating flows

In this paper we argue that differential rotation can possibly sustain hydrodynamic turbulence in the absence of a magnetic field. We explain why the non-linearities of the hydrodynamic equations (i.e. turbulent diffusion) should not be neglected, either as a simplifying approximation or based on boundary counditions (Hawley et al. 1999). The consequences of lifting this hypothesis are studied for the flow stability and the enhanced turbulent transport. We develop a simple general model for the energetics of turbulent fluctuations in differentially rotating flows. By taking into account the non-linearities of the equations of motions, we give constraints on the mean flow properties for the possible development of shear instability. The results from recent laboratory experiments on rotating flows show - in agreement with the model - that the pertinent parameter for stability appears to be the Rossby number Ro. The laboratory experiments seem to be compatible with Ro<-1/2 and Ro > 1 in the inviscid or high rotation rates limit. Our results, taken in the inviscid limit, are coherent with the classical linear stability analysis, in the sense that the critical perturbation equals zero on the marginal linear stability curve. We also propose a prescription for turbulent viscosity which generalize the (beta)-prescription derived in Richard & Zahn (1999).