Intermittency in the enstrophy cascade of two-dimensional fully developed turbulence: Universal features

Intermittency (externally induced) in the two-dimensional (2D) enstrophy cascade is shown to be able to maintain a finite enstrophy along with a vorticity conservation anomaly. Intermittency mechanisms of three-dimensional (3D) energy cascade and 2D enstrophy cascade in fully developed turbulence (FDT) seem to have some universal features. The parabolic-profile approximation (PPA) for the singularity spectrum f(α) in multi-fractal model is used and extended to the appropriate microscale regimes to exhibit these features. The PPA is also shown to afford, unlike the generic multi-fractal model, an analytical calculation of probability distribution functions (PDF) of flow-variable gradients in these FDT cases and to describe intermittency corrections that complement those provided by the homogeneous-fractal model.     

Physical mechanism of the two-dimensional enstrophy cascade

In two-dimensional turbulence, irreversible forward transfer of enstrophy requires anticorrelation of the turbulent vorticity transport vector and the inertial-range vorticity gradient. We investigate the basic mechanism by numerical simulation of the forced Navier-Stokes equation. In particular, we obtain the probability distributions of the local enstrophy flux and of the alignment angle between vorticity gradient and transport vector. These are surprisingly symmetric and cannot be explained by a local eddy-viscosity approximation. The vorticity transport tends to be directed along streamlines of the flow and only weakly aligned down the fluctuating vorticity gradient. All these features are well explained by a local nonlinear model. The physical origin of the cascade lies in steepening of inertial-range vorticity gradients due to compression of vorticity level sets by the large-scale strain field.     

Energy and enstrophy transfer in decaying two-dimensional turbulence

We present experimental data on the direct enstrophy cascade in decaying two-dimensional turbulence. Velocity and vorticity fields are obtained using particle tracking velocimetry. From those fields we directly compute the enstrophy and energy flux by using a filtering technique inspired by large-eddy simulations. This allows considerable insight into the physical processes of turbulence when compared with structure-function or spectral analysis. The direct cascade of enstrophy is weakly forward, with almost as much backscatter as down-scale enstrophy transfer, whereas the inverse energy cascade is strongly upscale with a modest amount of backscatter.     

Crossover in the enstrophy decay in two-dimensional turbulence in a finite box

The numerical simulation of two-dimensional decaying turbulence in a large but finite box presented in this Letter uncovered two physically different regimes of enstrophy decay. During the initial stage, the enstrophy Omega, generated by a random Gaussian initial condition, decays as Omega(t) proportional variant t(-gamma) with gamma approximately 0.7-0.8. After that, the flow undergoes a transition to a gas or fluid composed of distinct vortices. Simultaneously, the magnitude of the decay exponent crosses over to gamma approximately 0.4. A theory predicting N(t) proportional variant t(-xi) and the magnitudes of exponents gamma=2/5 and xi=4/5 is presented.     

Isotropization of two-dimensional hydrodynamic turbulence in the direct cascade

We present results of numerical simulation of the direct cascade in two-dimensional hydrodynamic turbulence (with spatial resolution up to ). If at the earlier stage (at the time of order of the inverse pumping growth rate τ-Γ_max ^?1), the turbulence develops according to the same scenario as in the case of a freely decaying turbulence [1, 2]: quasi-singular distribution of di-vorticity are formed, which in k-space correspond to jets, leading to a strong turbulence anisotropy, then for times of the order of 10τ turbulence becomes almost isotropic. In particular, at these times any significant anisotropy in the angular fluctuations for the energy spectrum (for a fixed k) is not visible, while the probability distribution function of vorticity for large arguments has the exponential tail with the exponent linearly dependent on vorticity, in the agreement with the theoretical prediction [3].     

Inverse cascade regime in shell models of two-dimensional turbulence

We consider shell models that display an inverse energy cascade similar to two-dimensional turbulence (together with a direct cascade of an enstrophylike invariant). Previous attempts to construct such models ended negatively, stating that shell models give rise to a "quasiequilibrium" situation with equipartition of the energy among the shells. We show analytically that the quasiequilibrium state predicts its own disappearance upon changing the model parameters in favor of the establishment of an inverse cascade regime with Kolmogorov scaling. The latter regime is found where predicted, offering a useful model to study inverse cascades.     

Anisotropic characteristics of the kraichnan direct cascade in two-dimensional hydrodynamic turbulence

The statistical characteristics of the Kraichnan direct cascade for two-dimensional hydrodynamic turbulence are numerically studied (with spatial resolution 8192 × 8192) in the presence of pumping and viscous-like damping. It is shown that quasi-shocks of vorticity and their Fourier partnerships in the form of jets introduce an essential influence in turbulence leading to strong angular dependencies for correlation functions. The energy distribution as a function of modulus k for each angle in the inertial interval has the Kraichnan behavior, ~k ^–4, and simultaneously a strong dependence on angles. However, angle average provides with a high accuracy the Kraichnan turbulence spectrum E _k = C _Kη^2/3k^–3, where η is the enstrophy flux and the Kraichnan constant C _K ? 1.3, in correspondence with the previous simulations. Familiar situation takes place for third-order velocity structure function S ₃ ^L which, as for the isotropic turbulence, gives the same scaling with respect to the separation length R and η, S ₃ ^L = C ₃ηR ³, but the average over the angles and time differs from its isotropic value.     

Kolmogorov-Kraichnan scaling in the inverse energy cascade of two-dimensional plasma turbulence

Turbulence in plasmas that are magnetically confined, such as tokamaks or linear devices, is two dimensional or at least quasi two dimensional due to the strong magnetic field, which leads to extreme elongation of the fluctuations, if any, in the direction parallel to the magnetic field. These plasmas are also compressible fluid flows obeying the compressible Navier-Stokes equations. This Letter presents the first comprehensive scaling of the structure functions of the density and velocity fields up to 10th order in the PISCES linear plasma device and up to 6th order in the Mega-Ampère Spherical Tokamak (MAST). In the two devices, it is found that the scaling of the turbulent fields is in good agreement with the prediction of the Kolmogorov-Kraichnan theory for two-dimensional turbulence in the energy cascade subrange.     

The effect of Lagrangian chaos on locking bifurcations in shear flows

The effect of an externally imposed perturbation on an unstable or weakly stable shear flow is investigated, with a focus on the role of Lagrangian chaos in the bifurcations that occur. The external perturbation is at rest in the laboratory frame and can form a chain of resonances or cat's eyes where the initial velocity v(x0)(y) vanishes. If in addition the shear profile is unstable or weakly stable to a Kelvin-Helmholtz instability, for a certain amplitude of the external perturbation there can be an unlocking bifurcation to a nonlinear wave resonant around a different value of y, with nonzero phase velocity. The interaction of the propagating nonlinear wave with the external perturbation leads to Lagrangian chaos. We discuss results based on numerical simulations for different amplitudes of the external perturbation. The response to the external perturbation is strong, apparently because of non-normality of the linear operator, and the unlocking bifurcation is hysteretic. The results indicate that the observed Lagrangian chaos is responsible for a second bifurcation occurring at larger external perturbation, locking the wave to the wall. This bifurcation is nonhysteretic. The mechanism by which the chaos leads to locking in this second bifurcation is by means of chaotic advective transport of momentum from one chain of resonances to the other (Reynolds stress) and momentum transport to the vicinity of the wall via chaotic scattering. These results suggest that locking of waves in rotating tank experiments in the presence of two unstable modes is due to a similar process. (c) 2002 American Institute of Physics.     

Inverse energy cascade in two-dimensional turbulence: deviations from gaussian behavior

High-resolution numerical simulations of stationary inverse energy cascade in two-dimensional turbulence are presented. Deviations from Gaussian behavior of velocity differences statistics are quantitatively investigated. The level of statistical convergence is pushed enough to permit reliable measurement of the asymmetries in the probability distribution functions of longitudinal increments and odd-order moments, which bring the signature of the inverse energy flux. No measurable intermittency corrections could be found in their scaling laws. The seventh order skewness increases by almost two orders of magnitude with respect to the third, thus becoming of order unity.     

Phonon drag effect in three- and two-dimensional electron systems

The nonequilibrium phonon flow drags the electrons, and depending upon experimental conditions manifests itself in the acoustoelectric current, acoustomagnetic field or acoustoelectric field. The results of these phenomena in Sn, Al, Ga, Ag measured with SQUID technique are discussed. In the two-dimensional (2D) case the phonon drag is studied on the interface of bicrystals and on the cleavage (111) surface of Ge and on the inversion layer on (111) (100) planes of Si. In all these cases the phonon drag is about two orders of magnitude larger than in metals with the same charge density. This is due to the drag of surface electrons by nonequilibrium phonon of the whole specimen. The Kohn resonance of phonons with Fermi surface and topological transitions on Fermi surface of 2D electrons produced sharp singularities of phonon drag effect in 2D cases.     

Lagrangian chaos and Eulerian chaos in shear flow dynamics

Shear flow dynamics described by the two-dimensional incompressible Navier-Stokes equations is studied for a one-dimensional equilibrium vorticity profile having two minima. These lead to two linear Kelvin-Helmholtz instabilities; the resulting nonlinear waves corresponding to the two minima have different phase velocities. The nonlinear behavior is studied as a function of two parameters, the Reynolds number and a parameter lambda specifying the width of the minima in the vorticity profile. For parameters such that the instabilities grow to a sufficient level, there is Lagrangian chaos, leading to mixing of vorticity, i.e., momentum transport, between the chains of vortices or cat's eyes. Lagrangian chaos is quantified by plotting the finite time Lyapunov exponents on a grid of initial points, and by the probability distribution of these exponents. For moderate values of lambda, there is Lagrangian chaos everywhere except near the centers of the vortices and near the boundaries, and there are competing effects of homogenization of vorticity and formation of structures associated with secondary resonances. For smaller values of lambda Lagrangian chaos occurs in the regions in the centers of the vortices, and the Eulerian behavior of the flow undergoes bifurcations leading to Eulerian chaos, as measured by the time series of several Galilean invariant quantities. A discussion of Lagrangian chaos and its relation to Eulerian chaos is given.(c) 2001 American Institute of Physics.     

Probability density functions of the enstrophy flux in two dimensional grid turbulence

Probability density functions of the enstrophy flux in two dimensional grid turbulence are found to be strongly non-Gaussian and can be mimicked by stretched exponential functions. Evidence of this behavior is found in experiments using turbulent soap films and numerical simulations. The enstrophy flux itself is found to be constant for a range of scales corresponding to the enstrophy cascade.