  We generalize an analogy between rotating and stratified shear flows. This
analogy is summarized in Table 1. We use this analogy in the unstable case
(centrifugally unstable flow v.s. convection) to compute the torque in
Taylor-Couette configuration, as a function of the Reynolds number. At low
Reynolds numbers, when most of the dissipation comes from the mean flow, we
predict that the non-dimensional torque $G=T/\nu^2L$, where $L$ is the cylinder
length, scales with Reynolds number $R$ and gap width $\eta$, $G=1.46
\eta^{3/2} (1-\eta)^{-7/4}R^{3/2}$. At larger Reynolds number, velocity
fluctuations become non-negligible in the dissipation. In these regimes, there
is no exact power law dependence the torque versus Reynolds. Instead, we obtain
logarithmic corrections to the classical ultra-hard (exponent 2) regimes: $$
G=0.50\frac{\eta^{2}}{(1-\eta)^{3/2}}\frac{R^{2}}{\ln[\eta^2(1-\eta)R^
2/10^4]^{3/2}}.$$ These predictions are found to be in excellent agreement with
available experimental data. Predictions for scaling of velocity fluctuations
are also provided.
