## Transitions from three- to two-dimensional turbulence in a rotating system

##### Abstract

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.