Scaling in three-dimensional and quasi-two-dimensional rotating turbulent flows

Abstract

We have made velocity time series measurements (using hot film probes) and velocity field measurements (using particle image velocimetry) on turbulent flow in a rotating annulus. For low annulus rotation rates the Rossby number was of order unity and the flow was three-dimensional (3D), but at high rotation rates the Rossby number was only about 0.1, comparable to the value for oceans and the atmosphere on large length scales. The low Rossby number (quasi-geostrophic) flow was nearly two-dimensional (2D), as expected from the Taylor–Proudman theorem. For the 3D flow we found that the probability distribution function (PDF) for velocity differences along the direction of the flow, δv(d)=v(x0+d)−v(x0), was Gaussian for large separations d and non-Gaussian (with exponential tails) for small d, as has been found for nonrotating turbulent flows. However, for low Rossby number flow, the PDF was self-similar (independent of d) and non-Gaussian. The exponents characterizing the structure functions, Sp=〈(δv)p〉∼dζp were obtained by the extended self-similarity method. For 3D flow the exponents departed from p/3 with increasing p, as has been found for turbulence in nonrotating flows, while for the quasi-2D turbulent flow, the exponents increased linearly with p, as expected for a self-similar flow. We applied the β-test of the hierarchical structure model [She and Lévêque, Phys. Rev. Lett. 72, 336 (1994)] and found that β remained constant at β≃0.75 as the rotation was increased from the 3D to the 2D regime; this indicates that both the quasi-2D and 3D flows are highly intermittent. The PIV images provided another indication of the intermittency—both the quasi-2D and 3D flows had coherent vortices which could be distinguished from the background flow. We also applied the γ-test of the hierarchical structure model and found that γ increased from 0.18 for the 3D flow to 0.34 for the quasi-2D flow; the latter value is in accord with expectation for self-similar turbulence. We conclude that our rotating 3D flow is similar to nonrotating turbulent flows, while the rotating quasi-2D turbulence is different from both the 3D rotating turbulence and from nonrotating 2D turbulence studied in other experiments.