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Register a new userMechanism of mean flow generation in rotating turbulence through inhomogeneous helicity
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Recent numerical simulations showed that the mean flow is generated in inhomogeneous turbulence of an incompressible fluid accompanied with helicity and system rotation. In order to investigate the mechanism of this phenomenon, we carry out a numerical simulation of inhomogeneous turbulence in a rotating system. In the simulation, an external force is applied to inject inhomogeneous turbulent helicity and the rotation axis is taken to be perpendicular to the inhomogeneous direction. No mean velocity is set in the initial condition of the simulation. The simulation results show that only in the case with both the helical forcing and the system rotation, the mean flow directed to the rotation axis is generated and sustained. We investigate the physical origin of this flow-generation phenomenon by considering the budget of the Reynolds-stress transport equation. It is found that the pressure diffusion term has a large contribution in the Reynolds stress equation and supports the generated mean flow. It is shown that a model expression for the pressure diffusion can be expressed by the turbulent helicity gradient coupled with the angular velocity of the system rotation. This implies that inhomogeneous helicity can play a significant role for the generation of the large-scale velocity distribution in incompressible turbulent flows.
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We discuss a mean-field theory of generation of large-scale vorticity in a rotating density stratified developed turbulence with inhomogeneous kinetic helicity. We show that the large-scale nonuniform flow is produced due to ether a combined action of a density stratified rotating turbulence and uniform kinetic helicity or a combined effect of a rotating incompressible turbulence and inhomogeneous kinetic helicity. These effects result in the formation of a large-scale shear, and in turn its interaction with the small-scale turbulence causes an excitation of the large-scale instability (known as a vorticity dynamo) due to a combined effect of the large-scale shear and Reynolds stress-induced generation of the mean vorticity. The latter is due to the effect of large-scale shear on the Reynolds stress. A fast rotation suppresses this large-scale instability.
The large-scale structures in the ocean and the atmosphere are in geostrophic balance, and a conduit must be found to channel the energy to the small scales where it can be dissipated. In turbulence this takes the form of an energy cascade, whereas one possible mechanism in a balanced flow at large scales is through the formation of fronts, a common occurrence in geophysical dynamics. We show in this paper that an iconic configuration in laboratory and numerical experiments for the study of turbulence, that of the Taylor-Green or von Karman swirling flow, can be suitably adapted to the case of fluids with large aspect ratios, leading to the creation of an imposed large-scale vertical shear. To this effect we use direct numerical simulations of the Boussinesq equations without net rotation and with no small-scale modeling, and with this idealized Taylor-Green set-up. Various grid spacings are used, up to $2048^2times 256$ spatial points. The grids are always isotropic, with box aspect ratios of either $1:4$ or $1:8$. We find that when shear and stratification are comparable, the imposed shear layer resulting from the forcing leads to the formation of multiple fronts and filaments which destabilize and further evolve into a turbulent flow in the bulk, with a sizable amount of dissipation and mixing, and with a cycle of front creation, instability, and development of turbulence. The results depend on the vertical length scales for shear and for stratification, with stronger large-scale gradients being generated when the two length scales are comparable.
We present a model describing evolution of the small-scale Navier-Stokes turbulence due to its stochastic distortions by much larger turbulent scales. This study is motivated by numerical findings (laval, 2001) that such interactions of separated scales play important role in turbulence intermittency. We introduce description of turbulence in terms of the moments of the k-space quantities using a method previously developed for the kinematic dynamo problem (Nazarenko, 2003). Working with the $k$-space moments allows to introduce new useful measures of intermittency such as the mean polarization and the spectral flatness. Our study of the 2D turbulence shows that the energy cascade is scale invariant and Gaussian whereas the enstrophy cascade is intermittent. In 3D, we show that the statistics of turbulence wavepackets deviates from gaussianity toward dominance of the plane polarizations. Such turbulence is formed by ellipsoids in the $k$-space centered at its origin and having one large, one neutral and one small axes with the velocity field pointing parallel to the smallest axis.
Kraichnan seminal ideas on inverse cascades yielded new tools to study common phenomena in geophysical turbulent flows. In the atmosphere and the oceans, rotation and stratification result in a flow that can be approximated as two-dimensional at very large scales, but which requires considering three-dimensional effects to fully describe turbulent transport processes and non-linear phenomena. Motions can thus be classified into two classes: fast modes consisting of inertia-gravity waves, and slow quasi-geostrophic modes for which the Coriolis force and horizontal pressure gradients are close to balance. In this paper we review previous results on the strength of the inverse cascade in rotating and stratified flows, and then present new results on the effect of varying the strength of rotation and stratification (measured by the ratio $N/f$ of the Brunt-Vaisala frequency to the Coriolis frequency) on the amplitude of the waves and on the flow quasi-geostrophic behavior. We show that the inverse cascade is more efficient in the range of $N/f$ for which resonant triads do not exist, $1/2 le N/f le 2$. We then use the spatio-temporal spectrum, and characterization of the flow temporal and spatial scales, to show that in this range slow modes dominate the dynamics, while the strength of the waves (and their relevance in the flow dynamics) is weaker.
We analyse the nonlinear dynamics of the large scale flow in Rayleigh-Benard convection in a two-dimensional, rectangular geometry of aspect ratio $Gamma$. We impose periodic and free-slip boundary conditions in the streamwise and spanwise directions, respectively. As Rayleigh number Ra increases, a large scale zonal flow dominates the dynamics of a moderate Prandtl number fluid. At high Ra, in the turbulent regime, transitions are seen in the probability density function (PDF) of the largest scale mode. For $Gamma = 2$, the PDF first transitions from a Gaussian to a trimodal behaviour, signifying the emergence of reversals of the zonal flow where the flow fluctuates between three distinct turbulent states: two states in which the zonal flow travels in opposite directions and one state with no zonal mean flow. Further increase in Ra leads to a transition from a trimodal to a unimodal PDF which demonstrates the disappearance of the zonal flow reversals. On the other hand, for $Gamma = 1$ the zonal flow reversals are characterised by a bimodal PDF of the largest scale mode, where the flow fluctuates only between two distinct turbulent states with zonal flow travelling in opposite directions.
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