Fluctuating hydrodynamics and turbulence in a rotating fluid: Universal properties

Abstract

We analyze the statistical properties of three-dimensional ($3d$) turbulence in a rotating fluid. To this end we introduce a generating functional to study the statistical properties of the velocity field $\mathbf{v}$. We obtain the master equation from the Navier-Stokes equation in a rotating frame and thence a set of exact hierarchical equations for the velocity structure functions for arbitrary angular velocity $\mathbf{\Omega }$. In particular we obtain the {\em differential forms} for the analogs of the well-known von Karman-Howarth relation for $3d$ fluid turbulence. We examine their behavior in the limit of large rotation. Our results clearly suggest dissimilar statistical behavior and scaling along directions parallel and perpendicular to $\mathbf{\Omega }$. The hierarchical relations yield strong evidence that the nature of the flows for large rotation is not identical to pure two-dimensional flows. To complement these results, by using an effective model in the small-$\Omega$ limit, within a one-loop approximation, we show that the equal-time correlation of the velocity components parallel to $\mathbf{\Omega }$ displays Kolmogorov scaling $q^{-5/3}$, where as for all other components, the equal-time correlators scale as $q^{-3}$ in the inertial range where $\mathbf{q}$ is a wavevector in $3d$. Our results are generally testable in experiments and/or direct numerical simulations of the Navier-Stokes equation in a rotating frame.

24 pages in preprint format; accepted for publication in Phys. Rev. E (2011)

References in corpus (9)

• Intermittency in two-dimensional Ekman-Navier-Stokes turbulence
• Scaling and energy transfer in rotating turbulence
• Intermittency in Two-Dimensional Turbulence with Drag
• The effects of violating detailed balance on critical dynamics
• Weak and strong dynamic scaling in a one-dimensional driven coupled-field model: Effects of kinematic waves
• Critical behaviour of a fluid in a random shear flow: Renormalization group analysis of a simplified model
• Universality classes in anisotropic non-equilibrium growth models
• Signatures of two-dimensionalisation of 3D turbulence in presence of rotation
• On two-dimensionalization of three-dimensional turbulence in shell models