Transitions from three- to two-dimensional turbulence in a rotating system
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Motivated by the variation of Coriolis effects on planetary scale flows, we explore rotating turbulent flows in a 1 m diameter tank, as the rotation rate is varied. For fast rotation (Rossby number Ro ' 0.1), the flow becomes quasi-two-dimensional (2D) and leads to an inverse cascade of energy from the injection scale to the scale of the system. In the low-rotation case (Ro ' 1), the flow is three-dimensional (3D), and only small vortices are observed. A gradual transition is found in the intermediate cases, where structures of increasing size are formed for faster rotation. The statistics of the velocity increments are compared for the different rotation rates. We observe a transition from typical intermittent behavior in the case of the 3D flow to scale-independent (self-similar) statistics for fast rotation. The self-similar 2D statistics match the predictions for 2D turbulence when using the relative (Sp vs. S3) scaling, but the scaling of the pth order structure functions (Sp) with distance (`) display an anomalous slope Sp ∼ ` p/2 . This scaling is further confirmed by the slope of the energy spectrum, where E(k) ∼ k −2 . The β- and γ- tests of the hierarchical symmetry model [She and L´evˆeque, Phys. Rev. Lett., 72 p.336, (1994)] are also applied. β remains constant at β ' 0.75 for low and high rotation rates, indicating flows that are highly intermittent in both cases. The value for γ changes from γ3D = 0.18 to γ2D = 0.34 which is the expected value for self-similar turbulence. The combination of these statistics with quantitative visualization shows that the coherent structures which populate the flow produce intermittent statistics in all the cases above, but that the intermittency is scale-independent in the 2D case. Finally, we apply the Beck-Tsallis nonextensive entropy [C. Beck, Physica, 277A p.115 (2000)]. The model is slightly modified and used to fir the velocity difference histograms, yielding a value for the nonextensivity parameter q. The value of q is found to agree with other 3D flows for the low rotation rate. In the case of 2D flows, we find a value which is nearly constant at q ' 1.32 ± 0.03, thus quantifying the departure from Gaussian scaling.