Shell Model Of Turbulence Research Articles

Shell models of MHD turbulence and the problem of superequipartition

Within the framework of the shell model of turbulence, we consider the long-time evolution of the magnetic field in a turbulent flow of conducting fluid. A weak initial magnetic field rapidly increases up to magnitudes corresponding to the equipartition with kinetic energy, and then the stage of slow evolution begins. It is shown that in some cases, the magnetic energy can significantly exceed the kinetic energy of the turbulence at this stage. We discuss the possible observational evidences for such a superequipartition of the magnetic field. It is shown that the superequipartition is realized if the initial energy of the magnetic field exceeds the kinetic energy, although even in such cases the equipartition is reached in many realizations.

Instanton calculus in shell models of turbulence

It has been shown recently that the intermittency of the Gledzer-Ohkitani-Yamada (GOY) shell model of turbulence has to be related to singular structures whose dynamics in the inertial range includes interactions with a background of fluctuations. In this paper we propose a statistical theory of these objects by modeling the incoherent background as a Gaussian white-noise forcing of small strength Gamma. A general scheme is developed for constructing instantons in spatially discrete dynamical systems and the Cramer function governing the probability distribution of effective singularities of exponent z is computed up to first order in a semiclassical expansion in powers of Gamma. The resulting predictions are compared with the statistics of coherent structures deduced from full simulations of the GOY model at very high Reynolds numbers.

Symmetries, invariants, and cascades in a shell model of turbulence

Reduced wave number models of turbulence, namely shell models, show cascade processes and anomalous scaling of correlators which might be analogous to what is observed in Navier-Stokes (NS) turbulence. The scaling properties of the shell models depend on the specific symmetries and invariants of the models. A shell model is investigated. It is argued that this model might have a closer resemblance than the standard Gledzer-Ohkitani-Yamada model to the NS turbulence. The shell model investigated here coincides with the Sabra model proposed by L'vov et al. [Phys. Rev. E. 58, 1811 (1998)] for a specific choice of the free parameters of their model. For this choice of parameters, besides the energy and the "helicity," the model has a cubic inviscid invariant.

Anomalous scaling from controlled closure in a shell model of turbulence

We present a model of hydrodynamic turbulence for which the program of computing the scaling exponents from first principles can be developed in a controlled fashion. The model consists of N suitably coupled copies of the “Sabra” shell model of turbulence. The couplings are chosen to include two components: random and deterministic, with a relative importance that is characterized by a parameter called ε. It is demonstrated, using numerical simulations of up to 25 copies and 28 shells that in the N→∞ limit but for 0<ε⩽1 this model exhibits correlation functions whose scaling exponents are anomalous. The theoretical calculation of the scaling exponents follows verbatim the closure procedure suggested recently for the Navier–Stokes problem, with the additional advantage that in the N→∞ limit the parameter ε can be used to regularize the closure procedure. The main result of this paper is a finite and closed set of scale-invariant equations for the 2nd and 3rd order statistical objects of the theory. This set of equations takes into account terms up to order ε4 and neglects terms of order ε6. Preliminary analysis of this set of equations indicates a K41 normal scaling at ε=0, with a birth of anomalous exponents at larger values of ε, in agreement with the numerical simulations.

Energy dissipation statistics in a shell model of turbulence

The Reynolds number dependence of the statistics of energy dissipation is investigated in a shell model of fully developed turbulence. The results are in agreement with a model which accounts for fluctuations of the dissipative scale with the intensity of energy dissipation. It is shown that the assumption of a fixed dissipative scale leads to a different scaling with Reynolds which is not compatible with numerical results.

Large deviation statistics of the energy-flux fluctuation in the shell model of turbulence

The energy-flux fluctuation in the shell model of turbulence is numerically analyzed from the large deviation statistical point of view. We first observe that the rate function defined in the inertial range is independent of the Reynolds number. The rate function derived by the cascade model of the log-Poisson statistics turns out to be in good agreement with the present numerical result in the region where strong singularity of fluctuation exits. This fact may imply the universality as well as the robustness of the large deviation statistical quantities in turbulence.

Multi-time, multi-scale correlation functions in turbulence and in turbulent models

A multifractal-like representation for multi-time, multi-scale velocity correlation in turbulence and dynamical turbulent models is proposed. The importance of subleading contributions to time correlations is highlighted. The fulfillment of the dynamical constraints due to the equations of motion is thoroughly discussed. The predictions stemming from this representation are tested within the framework of shell models for turbulence.

Helicity advection in turbulent models

Helicity transfer in a shell model of turbulence is investigated. In particular, we study the scaling behavior of helicity transfer in a dynamical model of turbulence lacking inversion symmetry. We present some phenomenological and numerical support to the idea that Helicity becomes -at scale small enough- a passively-advected quantity.

Improved shell model of turbulence

We introduce a shell model of turbulence that exhibits improved properties in comparison to the standard (and very popular) Gledzer, Ohkitani, and Yamada (GOY) model. The nonlinear coupling is chosen to minimize correlations between different shells. In particular, the second-order correlation function is diagonal in the shell index and the third-order correlation exists only between three consecutive shells. Spurious oscillations in the scaling regime, which are an annoying feature of the GOY model, are eliminated by our choice of nonlinear coupling. We demonstrate that the model exhibits multiscaling similar to the GOY model. The scaling exponents are shown to be independent of the viscous mechanism as is expected for Navier-Stokes turbulence and other shell models. These properties of the model make it optimal for further attempts to achieve understanding of multiscaling in nonlinear dynamics.

Universal Scaling Exponents in Shell Models of Turbulence: Viscous Effects Are Finite-Sized Corrections to Scaling

In a series of recent works it was proposed that shell models of turbulence exhibit inertial range scaling exponents that depend on the nature of the dissipative mechanism. If true, and if one could imply a similar phenomenon to Navier-Stokes turbulence, this finding would cast strong doubts on the universality of scaling in turbulence. In this Letter we propose that these ``nonuniversalities'' are just corrections to scaling that disappear when the Reynolds number goes to infinity.

Helicity transfer in turbulent models

Helicity transfer in a shell model of turbulence is investigated. We show that a Reynolds-independent helicity flux is present in the model when the large scale forcing breaks inversion symmetry. The equivalent in shell models of the ``$2/15$ law,'' obtained from helicity conservation in Navier-Stokes equations, is derived and tested. The odd part of the helicity flux statistics is found to be dominated by a few very intense events. In a particular model, we calculate analytically leading and subleading contributions to the scaling of triple velocity correlation.

Intermittency, chaos and singular fluctuations in the mixed Obukhov-Novikov shell model of turbulence

The multiscaling properties of the mixed Obukhov-Novikov (ON) shell model of turbulence are investigated numerically and compared with those of the complex Gledzer-Okhitani-Yamada (GOY) model, mostly studied in recent years. Two types of generic singular fluctuations are identified: first, self-similar solutions propagating from large to small scales and building up intermittency, second, complex-time singularities which are argued to encode “blockings” in the cascade. A robust dynamic rescaling method is developed to characterize these objects. It is shown that the scaling exponent of self-similar solutions selected by the dynamics is compatible with large-order statistics, only when it departs enough from the Kolmogorov value. Complex-time singularities on the other hand suffer a “depinning” transition which, in a remarkable way, is found to occur in the phase diagrams of both the ON and GOY models at a place such that the scaling exponent of corresponding self-similar solutions take the same value ≈ 0.855.

Towards a Two-Fluid Picture of Intermittency in Shell Models of Turbulence

Intermittency in the Gledzer-Okhitani-Yamada model of turbulence is explained in terms of collisions of coherent solitonlike structures with a random background issuing from the disintegration of their predecessors. This two-fluid picture is substantiated by the elucidation of local dynamical mechanisms leading to anomalous growth of coherent structures, their detection in true signals involving forcing and dissipation, and an investigation of their statistics.

(1+1)-dimensional turbulence

A class of dynamical models of turbulence living on a one-dimensional dyadic-tree structure is introduced and studied. The models are obtained as a natural generalization of the popular GOY shell model of turbulence. These models are found to be chaotic and intermittent. They represent the first example of (1+1)-dimensional dynamical systems possessing non trivial multifractal properties. The dyadic structure allows us to study spatial and temporal fluctuations. Energy dissipation statistics and its scaling properties are studied. The refined Kolmogorov hypothesis is found to hold.

Cascades of energy and helicity in the GOY shell model of turbulence

The effect of extreme hyperviscous damping, νknp,p=∞ is studied numerically in the GOY shell model of turbulence. It has resently been demonstrated [Leveque and She, Phys. Rev. Lett. 75, 2690 (1995)] that the inertial range scaling in the GOY model is non-universal and dependent on the viscous damping. In the present study it is shown that the deviation from Kolmogorov scaling is due to the cascade of the second inviscid invariant. This invariant is non-positive definite and in this sense analogous to the helicity of 3D turbulent flow.

Inertial- and Dissipation-Range Asymptotics in Fluid Turbulence

We propose and verify a wave-vector-space version of generalized extended self-similarity [R. Benzi et al., Europhys. Lett. 32, 709 (1995)] and broaden its applicability to uncover intriguing, universal scaling in the far dissipation range by computing high-order ( $\ensuremath{\le}20$) structure functions numerically for (1) the three-dimensional, incompressible Navier-Stokes equation (with and without hyperviscosity) and (2) the Gledzer-Ohkitani-Yamada shell model for turbulence. Also, in case (2), with Taylor-microscale Reynolds numbers $4\ifmmode\times\else\texttimes\fi{}{10}^{4}\ensuremath{\le}{\mathrm{Re}}_{\ensuremath{\lambda}}\ensuremath{\le}3\ifmmode\times\else\texttimes\fi{}{10}^{6}$, we find that the inertial-range exponents ( ${\ensuremath{\zeta}}_{p}$) of the order- $p$ structure functions do not approach their Kolmogorov value $p/3$ as ${\mathrm{Re}}_{\ensuremath{\lambda}}$ increases.

Helicity Causes Chaos in a Shell Model of Turbulence

We consider numerical solutions to a shell model of decaying homogeneous isotropic turbulence with helicity. In phase space, the stable manifold of the Komolgorov equilibrium has zero helicity in all shells. Trajectories starting near the stable manifold miss the equilibrium and enter a chaotic regime. Exponential growth of the helicity characterizes the divergence from the equilibrium. Erratic fluctuations of helicity and dissipation follow. As a result the energy decays intermittently. At high ${\mathrm{Re}}_{\ensuremath{\lambda}}$ the energy decay is an irregular staircase descending roughly as $E\ensuremath{\propto}{t}^{\ensuremath{-}1}$.

Some recent advances in the theory of homogeneous isotropic turbulence

We review some advances in the theory of homogeneous, isotropic turbulence. Our emphasis is on the new insights that have been gained from recent numerical studies of the three-dimensional Navier Stokes equation and simpler shell models for turbulence. In particular, we examine the status of multiscaling corrections to Kolmogorov scaling, extended self similarity, generalized extended self similarity, and non-Gaussian probability distributions for velocity differences and related quantities. We recount our recent proposal of a wave-vector-space version of generalized extended self similarity and show how it allows us to explore an intriguing and apparently universal crossover from inertial- to dissipation-range asymptotics.

Temporal intermittency and cascades in shell models of turbulence.

The two-dimensional- (2D) and three-dimensional- (3D) like Gletzer, Okhitani, and Yamada shell models are examined. The 2D-like model shows a transition from statistical quasiequilibrium to cascade of enstrophy as a function of the spectral ratio of energy to enstrophy. The transition is related to the ratio of time scales, corresponding to eddy turnover times, between shells. The anomalous scaling, giving rise to nonlinear scaling functions, is also connected to the ratio of eddy turnover times. This is illustrated in a simple stochastic model, where the structure function \ensuremath{\zeta}(q) becomes independent of q. In the 3D-like model the multiscaling is also influenced by the existence of a second nonpositive-definite inviscid invariant, the helicity. \textcopyright{} 1996 The American Physical Society.

Scaling properties of the velocity increments correlation function

An experimental analysis of the scaling properties of the velocity increments correlation function is presented. It is found that the scaling behavior predicted by a shell model of turbulence, called the Gledzer-Ohkitani-Yamada (GOY) model, is in substantial agreement with present experimental results.