Anomalous Scaling Exponents Research Articles

Time evolution of wave-packets in quasi-1D disordered media

We have investigated numerically the quantum evolution of a wave-packet in a quenched disordered medium described by a tight-binding Hamiltonian with long-range hopping (band random matrix approach). We have obtained clean data for the scaling properties in time and in the bandwidth b of the packet width and its fluctuations with respect to disorder realizations. We confirm that the fluctuations of the packet width in the steady-state show an anomalous scaling and we give a new estimate of the anomalous scaling exponent. This anomalous behaviour is related to the presence of non-Gaussian tails in the distribution of the packet width. Finally, we have analysed the steady state probability profile and we have found finite band corrections of order 1/b with respect to the theoretical formula derived by Zhirov in the limit of infinite bandwidth. In a neighbourhood of the origin, however, the corrections are $O(1/\sqrt{b})$.

Anomalous scaling exponents and coherent structures in high Re fluid turbulence

The scaling exponents of the velocity structure functions are predicted by a simple model based on the probability density function (PDF) of the local turbulent kinetic energy fluctuations with no hypotheses on the shape of the most intermittent coherent structures. The energy fluctuations are presumed to be induced by statistically independent coherent structures and, therefore, are characterized by exponential-like PDFs. Accounting also for the scaling of the energy with the scale, a recursive formula is obtained which permits the calculation of the scaling exponents of the velocity structure functions. The predicted exponents are checked to be in good agreement with previous theoretical and experimental results. The main hypotheses behind the present approach are also validated by analyzing experimental velocity time series acquired in a jet flow at moderately high Reλ. It is also shown that present approach allows the calculation of the scaling exponents also in cases of nonhomogeneous turbulence.

Nonperturbative spectrum of anomalous scaling exponents in the anisotropic sectors of passively advected magnetic fields

We address the scaling behavior of the covariance of the magnetic field in the three-dimensional kinematic dynamo problem when the boundary conditions and/or the external forcing are not isotropic. The velocity field is Gaussian, space homogeneous, and $\ensuremath{\delta}$ correlated in time, and its structure function scales with a positive exponent $\ensuremath{\xi}.$ The covariance of the magnetic field is naturally computed as a sum of contributions proportional to the irreducible representations of the SO(3) symmetry group. The amplitudes are nonuniversal, determined by boundary conditions. The scaling exponents are universal, forming a discrete, strictly increasing, spectrum indexed by the sectors of the symmetry group. When the initial mean magnetic field is zero, no dynamo effect is found, irrespective of the anisotropy of the forcing. The rate of isotropization with decreasing scales is fully understood from these results.

Dynamical correlations of two-dimensional vortex-like defects

High-order dynamical correlations of defects (quantum vortices, disclinations, etc.) in thin films are examined by starting from the Langevin equation for the defect motion. It is demonstrated that the dynamical correlation functions F 2n of the vorticity or disclinicity behave as F 2n ∼y 2/r 4n , where r is the characteristic scale and y is the renormalized fugacity. Therefore below the Berezinskii-Kosterlitz-Thouless transition temperature the F 2n are characterized by anomalous scaling exponents. The behavior differs strongly from the normal law F 2n ∼F 2 n obeyed by equal-time correlation functions; the unequal-time correlation functions appear to be much larger. The phenomenon resembles intermittency in turbulence.

Anisotropic nonperturbative zero modes for passively advected magnetic fields.

An analytic assessment of the role of anisotropic corrections to the isotropic anomalous scaling exponents is given for the d-dimensional kinematic magnetohydrodynamics problem in the presence of a mean magnetic field. The velocity advecting the magnetic field changes very rapidly in time and scales with a positive exponent xi. Inertial-range anisotropic contributions to the scaling exponents, zeta(j), of second-order magnetic correlations are associated with zero modes and have been calculated nonperturbatively. For d=3, the limit xi-->0 yields zeta(j)=j-2+xi(2j(3)+j(2)-5j-4)/[2(4j(2)-1)], where j (j>or=2) is the order in the Legendre polynomial decomposition applied to correlation functions. Conjectures on the fact that anisotropic components cannot change the isotropic threshold to the dynamo effect are also made.

Enstrophy and dissipation must have the same scaling exponent in the high Reynolds number limit of fluid turbulence

Writing the Poisson equation for the pressure in the vorticity-strain form, it is shown that the pressure has a finite inertial range spectrum in the high Reynolds number limit of isotropic turbulence only if the anomalous scaling exponents μ and μω for the dissipation and enstrophy (squared vorticity) are equal. Since a finite inertial range pressure spectrum requires only very weak assumptions about high Reynolds number turbulence, it is concluded that the inference from experiment and direct numerical simulation that these exponents are different must be a finite-range scaling result which will not survive taking the high Reynolds number limit.

Diffusion in one‐dimensional multifractal porous media

We examine the scaling properties of one‐dimensional random walks on media with multifractal diffusivities, which is a simple model for transport in scaling porous media. We find both theoretically and numerically that the anomalous scaling exponent of the walk is dw=2 + K(− 1) where K(− 1) is the scaling exponent of the reciprocal spatially averaged (“dressed”) resistance to diffusion. Since K(− 1) > 0, the walk is subdiffusive; the walkers are effectively trapped in a hierarchy of barriers. The trapping is dominated by contributions from a specific order of singularity associated with a phase transition between anomalous and normal diffusion. We discuss the implications for transport in porous media.

Anomalous Scaling in Passive Scalar Advection: Monte Carlo Lagrangian Trajectories

We present a numerical method which is used to calculate anomalous scaling exponents of structure functions in the Kraichnan passive scalar advection model [R. H. Kraichnan, Phys. Fluids 11, 945 (1968)]. This Monte Carlo method, which is applicable in any space dimension, is based on the Lagrangian path interpretation of passive scalar dynamics, and uses the recently discovered equivalence between scaling exponents of structure functions and relaxation rates in the stochastic shape dynamics of groups of Lagrangian particles. We calculate third and fourth order anomalous exponents for several dimensions, comparing with the predictions of perturbative calculations in large dimensions. We find that Kraichnan's closure appears to give results in close agreement with the numerics. The third order exponents are compatible with our own previous nonperturbative calculations.

Structure of the three-point correlation function of a passive scalar in the presence of a mean gradient

The three-point correlation function of a passive scalar advected by a random, incompressible velocity field in the presence of a mean gradient is investigated by means of phenomenological Hopf equations. Numerical solutions are provided in the case where the velocity is Gaussian, white in time, and with a power law in space, $〈[v(\stackrel{\ensuremath{\rightarrow}}{r})\ensuremath{-}v(\stackrel{\ensuremath{\rightarrow}}{0}){]}^{2}〉\ensuremath{\sim}{r}^{2\ensuremath{-}\ensuremath{\epsilon}},$ and in the model introduced by Shraiman and Siggia [C. R. Acad. Sci. 321, 279 (1995)]. Anomalous scaling exponents are found in both models. The numerics agrees with all the available analytic, perturbative results. In addition, the angular dependence of the correlation function is explicitly determined. In the Batchelor limit of random advection by a smooth velocity field, the exponent is found to remain very close to 1, as found experimentally. In this limit, the three-point correlation function is found to be very well represented, away from collinearity, by an explicit integral representation.

Temporal multiscaling in hydrodynamic turbulence

On the basis of the Navier-Stokes equations, we develop the high Reynolds number statistical theory of different-time, many-point spatial correlation functions of velocity differences. We find that their time dependence is not scale invariant: n-order correlation functions exhibit infinitely many distinct decorrelation times that are characterized by anomalous dynamical scaling exponents. We derive exact scaling relations that bridge all these dynamical exponents to the static anomalous exponents ${\mathrm{\ensuremath{\zeta}}}_{\mathrm{q}}$ of the standard structure functions. We propose a representation of the time dependence using the Legendre-transform formalism of multifractals that automatically reproduces all the newly found bridge relationships.

Instanton for random advection

A path integral over trajectories of 2n fluid particles is identified with a 2nth order correlation function of a passive scalar convected by d-dimensional short-correlated multiscale incompressible random velocity flow. Strong intermittency of the scalar is described by means of an instanton calculus (saddle point plus fluctuations about it) in the path integral at n\ensuremath{\gg}:d. The anomalous scaling exponent of the 2nth scalar's structural function is found analytically.

THE PHENOMENOLOGY OF SMALL-SCALE TURBULENCE

I have sometimes thought that what makes a man's work classic is often just this multiplicity [of interpretations], which invites and at the same time resists our craving for a clear understanding. Wright (1982, p. 34) , on Wittgenstein's philosophy ▪ Abstract Small-scale turbulence has been an area of especially active research in the recent past, and several useful research directions have been pursued. Here, we selectively review this work. The emphasis is on scaling phenomenology and kinematics of small-scale structure. After providing a brief introduction to the classical notions of universality due to Kolmogorov and others, we survey the existing work on intermittency, refined similarity hypotheses, anomalous scaling exponents, derivative statistics, intermittency models, and the structure and kinematics of small-scale structure—the latter aspect coming largely from the direct numerical simulation of homogeneous turbulence in a periodic box.

Anomalous scaling in random shell models for passive scalars.

A shell-model version of Kraichnan's [Phys. Rev. Lett. 72, 1016 (1994)] passive scalar problem is introduced which is inspired by the model of Jensen, Paladin, and Vulpiani [Phys. Rev. A 45, 7214 (1992)]. As in the original problem, the prescribed random velocity field is Gaussian and \ensuremath{\delta} correlated in time, and has a power-law spectrum \ensuremath{\propto}${\mathit{k}}_{\mathit{m}}^{\mathrm{\ensuremath{-}}\ensuremath{\xi}}$, where ${\mathit{k}}_{\mathit{m}}$ is the wave number. Deterministic differential equations for second- and fourth-order moments are obtained and then solved numerically. The second-order structure function of the passive scalar has normal scaling, while the fourth-order structure function has anomalous scaling. For \ensuremath{\xi}=2/3 the anomalous scaling exponents ${\mathrm{\ensuremath{\zeta}}}_{\mathit{p}}$ are determined for structure functions up to p=16 by Monte Carlo simulations of the random shell model, using a stochastic differential equation scheme, validated by comparison with the results obtained for the second- and fourth-order structure functions. \textcopyright{} 1996 The American Physical Society.

Normal and anomalous scaling in a problem of a passively advected magnetic field.

Recently M. Vergassola [Phys. Rev. E 53, R3021 (1996)] considered the possibility of anomalous scaling in the three-dimensional dynamo problem. It has been shown that the two-point correlation function of magnetic field, advected by a white-in-time random velocity field with zero helicity has anomalous inertial-range scaling exponent in a statistically steady state. In this work we demonstrate that for the same problem the scaling of covariance of magnetic helicity is normal. The difference in scalings comes from the fact that unlike magnetic energy magnetic helicity in this problem is conserved. It is also conjectured that even small violation of parity invariance of the velocity field makes existence of the steady-state solution impossible.

Scaling behavior in turbulence is doubly anomalous.

It is shown that the description of anomalous scaling in turbulent systems requires the simultaneous use of two normalization scales. This phenomenon stems from the existence of two independent (infinite) sets of anomalous scaling exponents that appear in leading order, one set due to infrared anomalies and the other due to ultraviolet anomalies. To expose this clearly we introduce here a set of local fields whose correlation functions depend simultaneously on the two sets of exponents. Thus the Kolmogorov picture of ``inertial range'' scaling is shown to fail because of anomalies that are sensitive to the two ends of this range.

Nonuniversality of the scaling exponents of a passive scalar convected by a random flow.

We consider a passive scalar convected by a multiscale random velocity field with short yet finite temporal correlations. Taking Kraichnan's limit of a white Gaussian velocity as a zero approximation we develop the perturbation theory with respect to a small correlation time and small non-Gaussianity of the velocity. We derive the renormalization (due to temporal correlations and non-Gaussianity) of the operator of turbulent diffusion. That allows us to calculate the respective corrections to the anomalous scaling exponents of the scalar field and show that they continuously depend on velocity correlation time and the degree of non-Gaussianity. The scalar exponents are thus nonuniversal as was predicted by Shraiman and Siggia on a phenomenological ground.

Anomalous scaling exponents of a white-advected passive scalar.

For Kraichnan's problem of passive scalar advection by a velocity field delta correlated in time, the limit of large space dimensionality $d\ensuremath{\gg}1$ is considered. Scaling exponents of the scalar field are analytically found to be ${\ensuremath{\zeta}}_{2n}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}n{\ensuremath{\zeta}}_{2}\ensuremath{-}2(2\ensuremath{-}{\ensuremath{\zeta}}_{2})n(n\ensuremath{-}1)/d$, while those of the dissipation field are ${\ensuremath{\mu}}_{n}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}2(2\ensuremath{-}{\ensuremath{\zeta}}_{2})n(n\ensuremath{-}1)/d$ for orders $n\ensuremath{\ll}d$. The refined similarity hypothesis ${\ensuremath{\zeta}}_{2n}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}n{\ensuremath{\zeta}}_{2}+{\ensuremath{\mu}}_{n}$ is thus established by a straightforward calculation for the case considered.

Vortex Tubes, Local Anisotropy and Anomalous Scaling

It has recently been proposed [1] that in highly developed hydrodynamic turbulence the dissipation field has an anomalous scaling behaviour independent of the velocity fields. The anomalous scaling exponent is ``subcritical" so that the Kolmogorov scaling behaviour of the velocity fields is retained in the limit of Re\to \infty. On the other hand, the formation of vortex tubes leading to local isotropy-breaking has been shown to give rise to anomalous scaling behaviour of the moments of the derivatives of the velocity field, \partial_\alpha u_\beta, in terms of which one is able to define an "anisotropy factor" [2]. We argue that the formation of the locally two-dimensional structures, namely the vortex tubes, is also what gives rise to the anomalous behaviour of the dissipation field, since the dissipation field itself is defined in terms of the velocity gradients as \epsilon \equiv \langle \nu \vert \nabla u\vert ^2\rangle, so that the ``anisotropy factor" [2] and \epsilon can be related by operator algebra. We also examine the lowest order term in the expansion of \epsilon in terms of the nonlinearity in the Navier Stokes equations. It is the logarithmic singularity in this term that later sums, via ladder diagrams, up to a correction to the scaling behaviour. We show how vortex tubes and sheets may contribute.

Probability distribution functions for Navier–Stokes turbulence

The probability distribution function (PDF), P(Δur), of a velocity difference, Δur, over a distance r in incompressible fluid turbulence is derived from the Navier–Stokes equations using the underlying functional probability distribution. Two different types of approximation are used to evaluate the resulting functional integral for P(Δur). The first is based on the saddle-point technique. It is used to examine the non-Gaussian features of P(Δur) and to demonstrate, in particular, that its tail has the characteristic exponential form associated with intermittency. The second approximation is developed for the purpose of deriving the anomalous scaling exponents of the structure functions from P(Δur). It represents P(Δur) as the integral with respect to the spatially averaged dissipation rate, εr, of the product of the PDF of Δur conditioned to a particular εr, and the PDF of εr. These are coarse-grained PDFs, obtained using the renormalization group. The former is approximately Gaussian, whereas the latter is given by a constrained Gaussian functional integral, which is evaluated approximately using a transformation that enables it to be represented in terms of a Poisson process. This approach yields the scaling exponents in the form discussed by She and Lévêque [Phys. Rev. Lett. 72, 336 (1994)], and also gives the complete third-order structure function exactly.

Scaling relations for a randomly advected passive scalar field.

A recent ansatz for dissipation terms gave anomalous inertial-range scaling exponents ({proportional_to}{ital n}{sup 1/2},{ital n}{r_arrow}{infinity}) for the {ital n}th-order structure functions of a passive scalar field advected by a random velocity field. Analysis of a series expansion for the conditional mean of a dissipation term suggests that the ansatz gives the only possible anomalous scaling. Anomaly of inertial-range scaling is supported by realizability inequalities on the dissipation field. Predictions for conditional means and structure functions are compared with simulations.