Inverse energy cascade in three-dimensional isotropic turbulence

We study the statistical properties of homogeneous and isotropic three-dimensional (3D) turbulent flows. By introducing a novel way to make numerical investigations of Navier-Stokes equations, we show that all 3D flows in nature possess a subset of nonlinear evolution leading to a reverse energy transfer: from small to large scales. Up to now, such an inverse cascade was only observed in flows under strong rotation and in quasi-two-dimensional geometries under strong confinement. We show here that energy flux is always reversed when mirror symmetry is broken, leading to a distribution of helicity in the system with a well-defined sign at all wave numbers. Our findings broaden the range of flows where the inverse energy cascade may be detected and rationalize the role played by helicity in the energy transfer process, showing that both 2D and 3D properties naturally coexist in all flows in nature. The unconventional numerical methodology here proposed, based on a Galerkin decimation of helical Fourier modes, paves the road for future studies on the influence of helicity on small-scale intermittency and the nature of the nonlinear interaction in magnetohydrodynamics.     

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.     

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.     

Free energy cascade in gyrokinetic turbulence

In gyrokinetic theory, the quadratic nonlinearity is known to play an important role in the dynamics by redistributing (in a conservative fashion) the free energy between the various active scales. In the present study, the free energy transfer is analyzed for the case of ion temperature gradient driven turbulence. It is shown that it shares many properties with the energy transfer in fluid turbulence. In particular, one finds a (strongly) local, forward (from large to small scales) cascade of free energy in the plane perpendicular to the background magnetic field. These findings shed light on some fundamental properties of plasma turbulence, and encourage the development of large-eddy-simulation techniques for gyrokinetics.     

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].     

Dynamics of energy condensation in two-dimensional turbulence

We report a numerical study, supplemented by phenomenological explanations, of "energy condensation" in forced 2D turbulence in a biperiodic box. Condensation is a finite size effect which occurs after the standard inverse cascade reaches the size of the system. It leads to the emergence of a coherent vortex dipole. We show that the time growth of the dipole is self-similar, and it contains most of the injected energy, thus resulting in an energy spectrum which is markedly steeper than the standard k{-5/3} one. Once the coherent component is subtracted, however, the remaining fluctuations have a spectrum close to k{-1}. The fluctuations decay slowly as the coherent part grows.     

Analysis of energy cascade models of turbulence

The exponent tau(p) that describes the scaling of the order-p moment of the energy dissipation field in fully developed turbulence is studied in a range which extends to negative values of p. The curve tau(p) vs p and its derivative tau(')(p) are estimated in various ways, including a two-scale method. Predictions of recent cascade models agree with our findings, within the statistical errors, for moderately large, positive p. Evident discrepancies already appear, however, in the interval pin[0,1], and sometimes become dramatic for p<0. In the discussion of a class of cascade models, we present a scaling law which relates different moments to one another.     

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.     

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.     

Free shear turbulence intensity anisotropies and the energy cascade

An extended energy cascade theory is presented, which correctly predicts the Reynolds number dependence of centerline turbulence intensities in the self-preserving region of fully turbulent free shear flows, shown clearly by the experimental measurements. The model indicates that below Re = 10⁶ too few stages exist to completely erase the cascade's “memory” of initial conditions.     

Inverse energy cascade correlated with turbulent-structure generation in toroidal plasma

We report the first experimental observation of the inverse energy cascade correlated with the generation of large turbulent structures. Spectral energy is nonlinearly transferred from the unstable region of the spectrum into large coherent structures and into broadband turbulence in agreement with theoretical expectations. These results are obtained by producing plasma in the H-1 heliac whose parameters allow a single-field, Hasegawa-Mima-type model to be used for the spectral energy transfer analysis.     

Physical mechanism of the two-dimensional inverse energy cascade

We study the physical mechanisms of the two-dimensional inverse energy cascade using theory, numerics, and experiment. Kraichnan's prediction of a -5/3 spectrum with constant, negative energy flux is verified in our simulations of 2D Navier-Stokes equations. We observe a similar but shorter range of inverse cascade in laboratory experiments. Our theory predicts, and the data confirm, that inverse cascade results mainly from turbulent stress proportional to small-scale strain rotated by 45 degrees. This "skew-Newtonian" stress is explained by the elongation and thinning of small-scale vortices by large-scale strain which weakens their velocity and transfers their energy upscale.     

Field theoretic calculation of energy cascade rates in non-helical magnetohydrodynamic turbulence

Energy cascade rates and Kolmogorov’s constant for non-helical steady magnetohydrodynamic turbulence have been calculated by solving the flux equations to the first order in perturbation. For zero cross helicity and space dimensiond = 3, magnetic energy cascades from large length-scales to small length-scales (forward cascade). In addition, there are energy fluxes from large-scale magnetic field to small-scale velocity field, large-scale velocity field to small-scale magnetic field, and large-scale velocity field to large-scale magnetic field. Kolmogorov’s constant for magnetohydrodynamics is approximately equal to that for fluid turbulence (≈ 1.6) for Alfvén ratio 05 ≤r _A ≤ ∞. For higher space-dimensions, the energy fluxes are qualitatively similar, and Kolmogorov’s constant varies asd ^1/3. For the normalized cross helicity σ_c →1, the cascade rates are proportional to (1 − σ_c)/(1 + σ_c , and the Kolmogorov’s constants vary significantly with σ_cc.