Publications

The Eshelby problem refers to the response of a two-dimensional elastic sheet to cutting away a circle, deforming it into an ellipse, and pushing it back. The resulting response is dominated by the so-called Eshelby kernel, which was derived for purely elastic (infinite) material, but has been employed extensively to model the redistribution of stress after plastic events in amorphous solids with finite boundaries. Here, we discuss and solve the Eshelby problem directly for amorphous solids, taking into account possible screening effects and realistic boundary conditions. We find major modifications compared to the classical Eshelby solution. These modifications are needed for modeling correctly the spatial responses to plastic events in amorphous solids.

The response of amorphous solids to mechanical loads is accompanied by plasticity that is generically associated with \u201cnon-affine\u201d quadrupolar events seen in the resulting displacement field. To develop a continuum theory, one needs to assess when these quadrupolar events have a finite density, allowing the development of a field theory. Is there a transition, as a function of the material parameters and the nature of the loads, from isolated plastic events whose density is zero to a regime governed by a finite density? And if so, what is the nature of this transition? The aim of the paper is to explore this issue. The motivation for the present study stems from recent research in which it was shown that gradients of the quadrupolar fields act as dipole charges that can screen elasticity. Analytically soluble examples of mechanical loading that lead to screening and emergent length scales (that are absent in classical elasticity) have been analyzed and tested. However, \u201cgradients of quadrupolar fields\u201d make sense only when the density of quadrupoles is finite, and hence, the issue is central to this article. The article introduces a notion of polarizability under the strain of Eshelby quadrupoles and concludes that the onset of a density of such quadrupoles with random orientations can only appear when the polarizability is finite.

A significant amount of attention was dedicated in recent years to the phenomenon of jamming of athermal amorphous solids by increasing the volume fraction of the microscopic constituents. At a critical value of the volume fraction, pressure shoots up from zero to finite values with a host of critical exponents discovered and discussed. In this paper, we advance evidence for the existence of a second transition, within the jammed state of two-dimensional granular systems, that separates two regimes of different mechanical responses. Explicitly, highly packed systems are quasielastic with quadrupole screening, and more loosely jammed systems exhibit anomalous mechanics with dipole screening. Evidence is given for a clear transition between these two regimes, reminiscent of the intermediate hexatic phase of crystal melting in two-dimensional crystals. Theoretical estimates of the screening parameters and the pressure where transition takes place are provided.

Applying very small purely radial strains on amorphous solids in radial geometry one observes elastic responses that break the radial symmetry. Without any plasticity involved, the responses indicate mode coupling contributions even for minute strains. We show that these symmetry-breaking responses are due to disorder, typical to amorphous configurations. The symmetry-breaking responses are quantitatively explained using the classical Michell solutions which are excited by mode coupling.

Applying constant tensile stress to a piece of amorphous solid results in a slow extension, followed by an eventual rapid mechanical collapse. This "creep"process is of paramount engineering concern, and as such was the subject of study in a variety of materials, for more than a century. Predictive theories for τw, the expected time of collapse, are incomplete, mainly due to its dependence on a bewildering variety of parameters, including temperature, system size, tensile force, but also the detailed microscopic interactions between constituents. The complex dependence of the collapse time on all the parameters is discussed below, using simulations of strip of amorphous material. Different scenarios are observed for ductile and brittle materials, resulting in serious difficulties in creating an all-encompassing theory that could offer safety measures for given conditions. A central aim of this paper is to employ scaling concepts, to achieve data collapse for the probability distribution function (pdf) of lnτw. The scaling ideas result in a universal function which provides a prediction of the pdf of lnτw for out-of-sample systems, from measurements at other values of these parameters. The predictive power of the scaling theory is demonstrated for both ductile and brittle systems. Finally, we present a derivation of universal scaling function for brittle materials. The ductile case appears to be due to a plastic necking instability and is left for future research.

The theoretical understanding of the low-frequency modes in amorphous solids at finite temperature is still incomplete. The study of the relevant modes is obscured by the dressing of interparticle forces by collision-induced momentum transfer that is unavoidable at finite temperatures. Recently, it was proposed that low-frequency modes of vibrations around the thermally averaged configurations deserve special attention. In simple model glasses with bare binary interactions, these included quasilocalized modes whose density of states appears to be universal, depending on the frequencies as D(ω)∼ω4, in agreement with the similar law that is obtained with bare forces at zero temperature. In this paper, we report investigations of a model of silica glass at finite temperature; here the bare forces include binary and ternary interactions. Nevertheless, we can establish the validity of the universal law of the density of quasilocalized modes also in this richer and more realistic model glass.

Fatigue caused by cyclic bending of a piece of material, resulting in its mechanical failure, is a phenomenon that had been studied for ages by engineers and physicists alike. In this Letter we study such fatigue in a strip of athermal amorphous solid. On the basis of atomistic simulations we conclude that the crucial quantity to focus on is the {\em accumulated damage}. Although this quantity exhibits large sample-to-sample fluctuations, its dependence on the loading determines the statistics of the number of cycles to failure. Thus we can provide a scaling theory for the Wöhler plots of mean number of cycles for failure as a function of the loading amplitude.

A careful examination of the Quasi-Localized Modes (QLMs) that exist in a generic atomistic model of a glass former reveals at least two types of them, each exhibiting a different density of states, one depending on the frequency as and the other as . The properties of the glassy energy landscape that is responsible for the two types of modes is examined here, explaining the analytic feature responsible for the creations of (at least) two families of QLMs. It is argued that the QLMs that are revealed by diagonalizing the Hessian matrix are not related to possibly existing Two-Level Systems (TLS).

It has been established that the low frequency quasilocalized modes of amorphous solids at zero temperature exhibit universal density of states, depending on the frequencies as D(omega) similar to omega(4). It remains an open question whether this universal law extends to finite temperatures. In this Letter we show that well quenched model glasses at temperatures as high as T-g/3 possess the same universal density of states. The only condition required is that average particle positions stabilize before thermal diffusion destroys the cage structure of the material. The universal density of quasilocalized low frequency modes refers then to vibrations around the thermally averaged configuration of the material.

Theoretical treatments of frictional granular matter often assume that it is legitimate to invoke classical elastic theory to describe its coarse-grained mechanical properties. Here, we show, based on experiments and numerical simulations, that this is generically not the case since stress autocorrelation functions decay more slowly than what is expected from elasticity theory. It was theoretically shown that standard elastic decay demands pressure and torque density fluctuations to be normal, with possibly one of them being hyperuniform. However, generic compressed frictional assemblies exhibit abnormal pressure fluctuations, failing to conform with the central limit theorem. The physics of this failure is linked to correlations built in the material during compression from a dilute configuration prior to jamming. By changing the protocol of compression, one can observe different pressure fluctuations, and stress autocorrelations decay at large scales.

Low-frequency quasilocalized modes of amorphous glass appear to exhibit universal density of states, depending on the frequencies as D(omega) similar to omega(4). To date, various models of glass formers with short-range binary interaction and network glass with both binary and ternary interactions were shown to conform with this law. In this paper, we examine granular amorphous solids with long-range electrostatic interactions and find that they exhibit the same law. To rationalize this wide universality class, we return to a model proposed by Gurevich, Parshin, and Schober (GPS) [Phys. Rev. B 67, 094203 (2003)] and analyze its predictions for interaction laws with varying spatial decay, exploring this wider-than-expected universality class. Numerical and analytic results are provided for both the actual system with long-range interaction and for the GPS model.

Catastrophic events in nature can be often triggered by small perturbations, with "remote triggering" of earthquakes being an important example. Here we present a mechanism for the giant amplification of small perturbations that is expected to be generic in systems whose dynamics is not derivable from a Hamiltonian. We offer a general discussion of the typical instabilities involved (being oscillatory with an exponential increase of noise) and examine in detail the normal forms that determine the relevant dynamics. The high sensitivity to external perturbations is explained for systems with and without dissipation. Numerical examples are provided using the dynamics of frictional granular matter. Finally, we point out the relationship of the presently discussed phenomenon to the highly topical issue of "exceptional points" in quantum models with non-Hermitian Hamiltonians.

Electrostatic theory preserves charges, but allows dipolar excitations. Elasticity theory preserves dipoles, but allows quadrupolar (Eshelby-like) plastic events. Charged amorphous granular systems are interesting in their own right; here we focus on their plastic instabilities and examine their mechanical response to external strain and to an external electric field, to expose the competition between elasticity and electrostatics. In this paper a generic model is offered, its mechanical instabilities are examined, and a theoretical analysis is presented. Plastic instabilities are discussed as saddle-node bifurcations that can be fully understood in terms of eigenvalues and eigenfunctions of the relevant Hessian matrix. This system exhibits moduli that describe how electric polarization and stress are influenced by strain and the electric field. Theoretical expression for these moduli are offered and compared to the measurements in numerical simulations.

In amorphous solids at finite temperatures the particles follow chaotic trajectories which, at temperatures sufficiently lower than the glass transition, are trapped in "cages." Averaging their positions for times shorter than the diffusion time, one can define a time-averaged configuration. Under strain or stress, these average configurations undergo sharp plastic instabilities. In athermal glasses the understanding of plastic instabilities is furnished by the Hessian matrix and its eigenvalues and eigenfunctions. Here we propose an uplifting of Hessian methods to thermal glasses, with the aim of understanding the plastic responses in the time-averaged configuration. We discuss a number of potential definitions of Hessians and identify which of these can provide eigenvalues and eigenfunctions which can explain and predict the instabilities of the time-averaged configurations. The conclusion is that the nonaffine changes in the average configurations during an instability is accurately predicted by the eigenfunctions of the low-lying eigenvalues of the chosen Hessian.

It was discovered recently that frictional granular materials can exhibit an important mechanism for instabilities, i.e., the appearance of pairs of complex eigenvalues in their stability matrix. The consequence is an oscillatory exponential growth of small perturbations which are tamed by dynamical nonlinearities. The amplification can be giant, many orders of magnitude, and it ends with a catastrophic system-spanning plastic event. Here we follow up on this discovery, explore the scaling laws characterizing the onset of the instability, the scenarios of the development of the instability with and without damping, and the nature of the eventual system-spanning events. The possible relevance to earthquake physics and to the transition from static to dynamic friction is discussed.

We report on a combined theoretical and numerical study of counterflow turbulence in superfluid He-4 in a wide range of parameters. The energy spectra of the velocity fluctuations of both the normal-fluid and superfluid components are strongly anisotropic. The angular dependence of the correlation between velocity fluctuations of the two components plays the key role. A selective energy dissipation intensifies as scales decrease, with the streamwise velocity fluctuations becoming dominant. Most of the flow energy is concentrated in a wave-vector plane which is orthogonal to the direction of the counterflow. The phenomenon becomes more prominent at higher temperatures as the coupling between the components depends on the temperature and the direction with respect to the counterflow velocity.

Frictional granular matter is shown to be fundamentally different in its plastic responses to external strains from generic glasses and amorphous solids without friction. While regular glasses exhibit plastic instabilities due to the vanishing of a real eigenvalue of the Hessian matrix, frictional granular materials can exhibit a previously unnoticed additional mechanism for instabilities, i.e., the appearance of a pair of complex eigenvalues leading to oscillatory exponential growth of perturbations that are tamed by dynamical nonlinearities. This fundamental difference appears crucial for the understanding of plasticity and failure in frictional granular materials. The possible relevance to earthquake physics is discussed.

Three-dimensional anisotropic turbulence in classical fluids tends towards isotropy and homogeneity with decreasing scales, allowing-eventually-the abstract model of homogeneous and isotropic turbulence to be relevant. We show here that the opposite is true for superfluid He-4 turbulence in three-dimensional counterflow channel geometry. This flow becomes less isotropic upon decreasing scales, becoming eventually quasi-two-dimensional. The physical reason for this unusual phenomenon is elucidated and supported by theory and simulations.

Numerical Simulations are employed to create amorphous nano-films of a chosen thickness on a crystalline substrate which induces strain on the film. The films are grown by a vapor deposition technique. Using the exact relations between the Hessian matrix and the shear and bulk moduli we explore the mechanical properties of the nano-films as a function of the density of the substrate and the film thickness. The existence of the substrate dominates the mechanical properties of the combined substrate-film system. Scaling concepts are then employed to achieve data collapse in a wide range of densities and film thicknesses.

We report a joint experimental and theoretical investigation of the probability distribution functions (PDFs) of the normal and tangential (frictional) forces in amorphous frictional media. We consider both the joint PDF of normal and tangential forces together, and the marginal PDFs of normal forces separately and tangential forces separately. A maximum entropy formalism is utilized for all these cases after identifying the appropriate constraints. Excellent agreements with both experimental and simulation data are reported. The proposed joint PDF predicts giant slip events at low pressures, again in agreement with observations.

Standard approaches to magnetomechanical interactions in thermal magnetic crystalline solids involve Landau functionals in which the lattice anisotropy and the resulting magnetization easy axes are taken explicitly into account. In glassy systems one needs to develop a theory in which the amorphous structure precludes the existence of an easy axis, and in which the constituent particles are free to respond to their local amorphous surroundings and the resulting forces. We present a theory of all the mixed responses of an amorphous solid to mechanical strains and magnetic fields. Atomistic models are proposed in which we test the predictions of magnetostriction for both bulk and nanofilm amorphous samples in the paramagnetic phase. The application to nanofilms with emergent self-affine free interfaces requires a careful definition of the film "width" and its change due to the magnetostriction effect.

The large-scale turbulent statistics of mechanically driven superfluid He4 was shown experimentally to follow the classical counterpart. In this paper, we use direct numerical simulations to study the whole range of scales in a range of temperatures T [1.3,2.1] K. The numerics employ self-consistent and nonlinearly coupled normal and superfluid components. The main results are that (i) the velocity fluctuations of normal and super components are well correlated in the inertial range of scales, but decorrelate at small scales. (ii) The energy transfer by mutual friction between components is particulary efficient in the temperature range between 1.8 and 2 K, leading to enhancement of small-scale intermittency for these temperatures. (iii) At low T and close to Tλ, the scaling properties of the energy spectra and structure functions of the two components are approaching those of classical hydrodynamic turbulence.

We revisit the problem of the stress distribution in a frictional sandpile with both normal and tangential (frictional) inter-granular forces, under gravity, equipped with a new numerical method of generating such assemblies. Numerical simulations allow a determination of the spatial dependence of all the components of the stress field, principle stress axis, angle of repose, as a function of systems size, the coefficient of static friction and the frictional interaction with the bottom surface. We compare these results with the predictions of a theory based on continuum equilibrium mechanics. Basic to the theory of sandpiles are assumptions about the form of scaling solutions and constitutive relations for cohesiveless hard grains for which no typical scale is available. We find that these constitutive relations must be modified; moreover for smaller friction coefficients and smaller piles these scaling assumptions break down in the bulk of the sandpile due to the presence of length scales that must be carefully identified. Fortunately, for larger friction coefficient and for larger piles the breaking of scaling is weak in the bulk, allowing an approximate analytic theory which agrees well with the observations. After identifying the crucial scale, triggering the breaking of scaling, we provide a predictive theory to when scaling solutions are expected to break down. At the bottom of the pile the scaling assumption breaks always, due to the different interactions with the bottom surface. The consequences for measurable quantities like the pressure distribution and shear stress at the bottom of the pile are discussed. For example one can have a transition from no dip in the base-pressure to a dip at the center of the pile as friction increases.

Below the phase transition temperature Tc≃10-3K He3-B has a mixture of normal and superfluid components. Turbulence in this material is carried predominantly by the superfluid component. We explore the statistical properties of this quantum turbulence, stressing the differences from the better known classical counterpart. To this aim we study the time-honored Hall-Vinen-Bekarevich-Khalatnikov coarse-grained equations of superfluid turbulence. We combine pseudospectral direct numerical simulations with analytic considerations based on an integral closure for the energy flux. We avoid the assumption of locality of the energy transfer which was used previously in both analytic and numerical studies of the superfluid He3-B turbulence. For T

We consider the problem of how to determine the force laws in an amorphous system of interacting particles. Given the positions of the centers of mass of the constituent particles we propose an algorithm to determine the interparticle force laws. Having n different types of constituents we determine the coefficients in the Laurent polynomials for the n(n+1)/2 possibly different force laws. A visual providing the particle positions in addition to a measurement of the pressure is all that is required. The algorithm proposed includes a part that can correct for experimental errors in the positions of the particles. Such a correction of unavoidable measurement errors is expected to benefit many experiments in the field.

Amorphous solids yield in strain-controlled protocols at a critical value of the strain. For larger strains the stress and energy display a generic complex serrated signal with elastic segments punctuated by sharp energy and stress plastic drops having a wide range of magnitudes. Here we provide a theory of the scaling properties of such serrated signals taking into account the system-size dependence. We show that the statistics are not homogeneous: they separate sharply to a regime of "small" and "large" drops, each endowed with its own scaling properties. A scaling theory is first derived solely by data analysis, showing a somewhat complex picture. But after considering the physical interpretation one discovers that the scaling behavior and the scaling exponents are in fact very simple and universal.

The mechanical properties of amorphous solids like metallic glasses can be dramatically changed by adding small concentrations (as low as 0.1%) of foreign elements. The glass-forming-ability, the ductility, the yield stress and the elastic moduli can all be greatly effected. This paper presents theoretical considerations with the aim of explaining the magnitude of these changes in light of the small concentrations involved. The theory is built around the experimental evidence that the microalloying elements organise around them a neighbourhood that differs from both the crystalline and the glassy phases of the material in the absence of the additional elements. These regions act as isotropic defects that in unstressed systems modify the shear moduli. When strained, these defects interact with the incipient plastic responses which are quadrupolar in nature. It will be shown that this interaction interferes with the creation of system-spanning shear bands and increases the yield strain. We offer experimentally testable estimates of the lengths of nano-shear bands in the presence of the additional elements.

The determination of the normal and transverse (frictional) interparticle forces within a granular medium is a long-standing, daunting, and yet unresolved problem. We present a new formalism that employs the knowledge of the external forces and the orientations of contacts between particles (of any given size), to compute all the interparticle forces. Having solved this problem, we exemplify the efficacy of the formalism showing that the force chains in such systems are determined by an expansion in the eigenfunctions of a newly defined operator.

Amorphous solids yield at a critical value of the strain (in strain-controlled experiments); for larger strains, the average stress can no longer increase - the system displays an elastoplastic steady state. A long-standing riddle in the materials community is what the difference is between the microscopic states of the material before and after yield. Explanations in the literature are material specific, but the universality of the phenomenon begs a universal answer. We argue here that there is no fundamental difference in the states of matter before and after yield, but the yield is a bona fide first-order phase transition between a highly restricted set of possible configurations residing in a small region of phase space to a vastly rich set of configurations which include many marginally stable ones. To show this, we employ an order parameter of universal applicability, independent of the microscopic interactions, that is successful in quantifying the transition in an unambiguous manner.

Long-ranged dipole-dipole interactions in magnetic glasses give rise to magnetic domains having labyrinthine patterns on the scale of about 1 micron. Barkhausen Noise then results from the movement of domain boundaries which is modeled by the motion of elastic membranes with random pinning. Here we propose that on the nanoscale new sources of Barkhausen Noise can arise. We propose an atomistic model of magnetic glasses in which we measure the Barkhausen Noise which results from the creation of new domains and the movement of domain boundaries on the nanoscale. The statistics of the Barkhausen Noise found in our simulations is in striking disagreement with the expectations in the literature. In fact we find exponential statistics without any power law, stressing the fact that Barkhausen Noise can belong to very different universality classes. In the present model the essence of the phenomenon is the fact that the spin response Green's function is decaying too rapidly for having sufficiently large magnetic jumps. A theory is offered in excellent agreement with the measured data without any free parameter.

The existence of a static lengthscale that grows in accordance with the dramatic slowing-down observed at the glass transition is a subject of intense interest. Due to limitations on the relaxation times reachable by standard molecular-dynamics techniques (i.e. a range of about 4-5 orders of magnitude) it was until now impossible to demonstrate a significant enough increase in any proposed lengthscale. In this letter we explore the typical scale at unprecedented lower temperatures. A Swap Monte Carlo approach allows us to reach a lengthscale growth by more than 500%. We conclude by discussing the relationship between the observed lengthscale and various models of the relaxation time, proposing that the associated increase in relaxation time approaches experimental values.

The quantization of vortex lines in superfluids requires the introduction of their density L(r,t) in the description of quantum turbulence. The space homogeneous balance equation for L(t), proposed by Vinen on the basis of dimensional and physical considerations, allows a number of competing forms for the production term P. Attempts to choose the correct one on the basis of time-dependent homogeneous experiments ended inconclusively. To overcome this difficulty we announce here an approach that employs an inhomogeneous channel flow which is very suitable to distinguish the implications of the various possible forms of the desired equation. We demonstrate that the originally selected form which was extensively used in the literature is in strong contradiction with our data. We therefore present a new form of an inhomogeneous equation for L(r,t) that is in agreement with our data and propose that it should be considered for further studies of superfluid turbulence.

In this paper we focus on the mechanical properties of oligomeric glasses (waxes), employing a microscopic model that provides, via numerical simulations, information about the shear modulus of such materials, the failure mechanism via plastic instabilities, and the geometric responses of the oligomers themselves to a mechanical load. We present a microscopic theory that explains the numerically observed phenomena, including an exact theory of the shear modulus and of the plastic instabilities, both local and system spanning. In addition we present a model to explain the geometric changes in the oligomeric chains under increasing strains.

The study of metallic glasses which exhibit magnetic interactions was mainly restricted to the amorphous solid regime and its properties. Here we study the glass transition in fluids where particles are endowed with spins, such that magnetic and positional degrees of freedom are coupled. Results for slowing down in the spin time-correlation functions are described, and the effects of magnetic fields on the glass transition are studied. Aging effects in such systems and the corresponding data collapse are presented and discussed.

A nuclear capture reaction of a single neutron by ultracold superfluid He3 results in a rapid overheating followed by the expansion and subsequent cooling of the hot subregion, in a certain analogy with the big bang of the early universe. It was shown in a Grenoble experiment that a significant part of the energy released during the nuclear reaction was not converted into heat even after several seconds. It was thought that the missing energy was stored in a tangle of quantized vortex lines. This explanation, however, contradicts the expected lifetime of a bulk vortex tangle, 10-5-10-4 s, which is much shorter than the observed time delay of seconds. In this paper we propose a scenario that resolves the contradiction: the vortex tangle, created by the hot spot, emits isolated vortex loops that take with them a significant part of the tangle's energy. These loops quickly reach the container walls. The dilute ensemble of vortex loops attached to the walls can survive for a long time, while the remaining bulk vortex tangle decays quickly.

"Microalloying", which refers to the addition of small concentrations of a foreign metal to a given metallic glass, has been used extensively in recent years in attempts to improve the mechanical properties of these glasses. The results are haphazard and nonsystematic. In this paper we provide a microscopic theory of the effect of microalloying, exposing the delicate consequences of this procedure and the large parameter space which needs to be controlled. In particular we consider two very similar models which exhibit opposite trends for the change of the shear modulus, and explain the origins of this difference as displayed in the various microscopic structures and properties.

Experimental and simulational studies of the dynamics of vortex reconnections in quantum fluids showed that the distance d between the reconnecting vortices is close to a universal time dependence d=D[κ|t ₀-t|]α with α fluctuating around 1/2 and κ=h/m is the quantum of circulation. Dimensional analysis, based on the assumption that the quantum of circulation κ=h/m is the only relevant parameter in the problem, predicts α=1/2. The theoretical calculation of the dimensionless coefficient D in this formula remained an open problem. In this Letter we present an analytic calculation of D in terms of the given geometry of the reconnecting vortices. We start from the numerically observed generic geometry on the way to vortex reconnection and demonstrate that the dynamics is well described by a self-similar analytic solution which provides the wanted information.

In this paper we extend the recent theory of shear localization in two-dimensional (2D) amorphous solids to three dimensions. In two dimensions the fundamental instability of shear localization is related to the appearance of a line of displacement quadrupoles that makes an angle of 45 ^â̂̃ with the principal stress axis. In three dimensions the fundamental plastic instability is also explained by the formation of a lattice of anisotropic elastic inclusions. In the case of pure external shear stress, we demonstrate that this is a 2D triangular lattice of similar elementary events. It is shown that this lattice is arranged on a plane that, similarly to the 2D case, makes an angle of 45 ^â̂̃ with the principal stress axis. This solution is energetically favorable only if the external strain exceeds a yield-strain value that is determined by the strain parameters of the elementary events and the Poisson ratio. The predictions of the theory are compared to numerical simulations and very good agreement is observed.

The usefulness of glasses, and particularly of metallic glasses, in technological applications is often limited by their toughness, (which is defined as the area under the stress vs. strain curve before plastic yielding). Recently, toughness was found to increase significantly by the addition of small concentrations of foreign atoms that act as pinning centers. We model this phenomenon at zero temperature and quasi-static straining with randomly positioned particles that participate in the elastic deformation, but are pinned in the non-affine return to mechanical equilibrium. We find a very strong effect on toughness via the increase of both the shear modulus and the yield stress as a function of the density of pinned particles. Understanding the results calls for analyzing separately the elastic, or Born term, and the contributions of the excess modes that result from glassy disorder. Finally, we present a scaling theory that collapses the data on one universal curve as a function of rescaled variables.

In recent research it was found that the fundamental shear-localizing instability of amorphous solids under external strain, which eventually results in a shear band and failure, consists of a highly correlated array of Eshelby quadrupoles all having the same orientation and some density rho. In this paper we calculate analytically the energy E(rho, gamma) associated with such highly correlated structures as a function of the density rho and the external strain gamma. We show that for strains smaller than a characteristic strain gamma(Y) the total strain energy initially increases as the quadrupole density increases, but that for strains larger than gamma(Y) the energy monotonically decreases with quadrupole density. We identify gamma(Y) as the yield strain. Its value, derived from values of the qudrupole strength based on the atomistic model, agrees with that from the computed stress-strain curves and broadly with experimental results. DOI: 10.1103/PhysRevE.87.022810

We consider the intermittent behavior of superfluid turbulence in He4. Because of the similarity in the nonlinear structure of the two-fluid model of superfluidity and the Euler and Navier-Stokes equations, one expects the scaling exponents of the structure functions to be the same as in classical turbulence for temperatures close to the superfluid transition T_λ and also for T≪T_λ. This is not the case when the densities of normal and superfluid components are comparable to each other and mutual friction becomes important. Using shell model simulations, we propose that in this situation there exists a range of scales in which the effective exponents indicate stronger intermittency. We offer a bridge relation between these effective and the classical scaling exponents. Since this effect occurs at accessible temperatures and Reynolds numbers, we propose that experiments should be conducted to further assess the validity and implications of this prediction.

We present a short review of the theory of shear localization which results in shear bands in amorphous solids. As this is the main mechanism for the failure of metallic glasses, understanding the instability is invaluable in finding how to stabilize such materials against the tendency to shear localize. We explain the mechanism for shear localization under external shear-strain, which in 2-dimensions is the appearance of highly correlated lines of Eshelby-like quadrupolar singularities which organize the non-affine plastic flow of the amorphous solid into a shear band. We prove analytically that such highly correlated solutions in which N equi-distant quadrupoles are aligned with equal orientations are minimum energy states when the strain is high enough. The line lies at 45 degrees to the compressive stress. We use the theory to first predict the yield strain at zero temperature and quasi-static conditions, but later generalize to the case of finite temperature and finite shear rates, deriving the Johnson-Samwer T-2/3 law.

Over the past decade, computer simulations have had an increasing role in shedding light on difficult statistical physical phenomena, and in particular on the ubiquitous problem of the glass transition. Here in a wide variety of materials the viscosity of a supercooled liquid increases by many orders of magnitude upon decreasing the temperature over a modest range. A natural concern in these computer simulations is the very small size of the simulated systems compared to experimental ones, raising the issue of how to assess the thermodynamic limit. Here we turn this limitation to our advantage by performing finite size scaling on the system size dependence of the relaxation time for supercooled liquids to emphasize the importance of a growing static length scale in the theory of glass transition. We demonstrate that the static length scale that was discovered by us in Physica A0378-437110.1016/j.physa.2011.11.020 391, 1001 (2012) fits the bill extremely well, allowing us to provide a finite-size scaling theory for the α-relaxation time of the glass transition, including predictions for the thermodynamic limit based on simulations in small systems.

Kelvin waves propagating on quantum vortices play a crucial role in the phenomenology of energy dissipation of superfluid turbulence. Previous theoretical studies have consistently focused on the zero-temperature limit of the statistical physics of Kelvin-wave turbulence. In this letter, we go beyond this athermal limit by introducing a small but finite temperature in the form of non-zero mutual friction dissipative force; A situation regularly encountered in actual experiments of superfluid turbulence. In this case we show that there exists a new typical length scale separating a quasi-inertial range of Kelvin-wave turbulence from a far-dissipation range. The letter culminates with analytical predictions for the energy spectrum of the Kelvin-wave turbulence in both of these regimes.

We address the cross-effects between mechanical strains and magnetic fields on the plastic response of magnetoelastic amorphous solids. It is well known that plasticity in non-magnetic amorphous solids under external strain γ is dominated by the codimension-1 saddle-node bifurcation in which an eigenvalue of the Hessian matrix vanishes at γ _P like √ γ _P-γ. This square-root singularity determines much of the statistical physics of elasto-plasticity, and in particular that of the stress-strain curves under athermal-quasistatic conditions. In this letter we discuss the much richer physics that can be expected in magnetic amorphous solids. Firstly, magnetic amorphous solids exhibit codimension-2 plastic instabilities, when an external strain and an external magnetic field are applied simultaneously. Secondly, the phase diagrams promise a rich array of new effects that have been barely studied; this opens up a novel and extremely rich research program for magnetoplastic materials.

In superfluid 3He-B, turbulence is carried predominantly by the superfluid component. To explore the statistical properties of this quantum turbulence and its differences from the classical counterpart, we adopt the time-honored approach of shell models. Using this approach, we provide numerical simulations of a Sabra shell model that allows us to uncover the nature of the energy spectrum in the relevant hydrodynamic regimes. These results are in qualitative agreement with analytical expressions for the superfluid turbulent energy spectra that were found using a differential approximation for the energy flux.

Tetrahedral liquids such as water and silica-melt show unusual thermodynamic behavior such as a density maximum and an increase in specific heat when cooled to low temperatures. Previous work had shown that Monte Carlo and mean-field solutions of a lattice model can exhibit these anomalous properties with or without a phase transition, depending on the values of the different terms in the Hamiltonian. Here we use a somewhat different approach, where we start from a very popular empirical model of tetrahedral liquids - the Stillinger-Weber model - and construct a coarse-grained theory which directly quantifies the local structure of the liquid as a function of volume and temperature. We compare the theory to molecular-dynamics simulations and show that the theory can rationalize the simulation results and the anomalous behavior.

Fractal decimation reduces the effective dimensionality D of a flow by keeping only a (randomly chosen) set of Fourier modes whose number in a ball of radius k is proportional to kD for large k. At the critical dimension D _c=4/3 there is an equilibrium Gibbs state with a k ^-5 ^/3 spectrum, as in V. L'vov et al., Phys. Rev. Lett. 89, 064501 (2002)PRLTAO0031-900710.1103/PhysRevLett.89.064501. Spectral simulations of fractally decimated two-dimensional turbulence show that the inverse cascade persists below D=2 with a rapidly rising Kolmogorov constant, likely to diverge as (D-4/3) ^-2 ^/3.

Brittle materials exhibit sharp dynamical fractures when meeting Griffith's criterion, whereas ductile materials blunt a sharp crack by plastic responses. Upon continuous pulling, ductile materials exhibit a necking instability that is dominated by a plastic flow. Usually one discusses the brittle to ductile transition as a function of increasing temperature. We introduce an athermal brittle to ductile transition as a function of the cutoff length of the interparticle potential. On the basis of extensive numerical simulations of the response to pulling the material boundaries at a constant speed we offer an explanation of the onset of ductility via the increase in the density of plastic modes as a function of the potential cutoff length. Finally we can resolve an old riddle: In experiments brittle materials can be strained under grip boundary conditions and exhibit a dynamic crack when cut with a sufficiently long initial slot. Mysteriously, in molecular dynamics simulations it appeared that cracks refused to propagate dynamically under grip boundary conditions, and continuous pulling was necessary to achieve fracture. We argue that this mystery is removed when one understands the distinction between brittle and ductile athermal amorphous materials.

We study the elastic theory of amorphous solids made of particles with finite range interactions in the thermodynamic limit. For the elastic theory to exist, one requires all the elastic coefficients, linear and nonlinear, to attain a finite thermodynamic limit. We show that for such systems the existence of nonaffine mechanical responses results in anomalous fluctuations of all the nonlinear coefficients of the elastic theory. While the shear modulus exists, the first nonlinear coefficient B₂ has anomalous fluctuations and the second nonlinear coefficient B₃ and all the higher order coefficients (which are nonzero by symmetry) diverge in the thermodynamic limit. These results call into question the existence of elasticity (or solidity) of amorphous solids at finite strains, even at zero temperature. We discuss the physical meaning of these results and propose that in these systems elasticity can never be decoupled from plasticity: the nonlinear response must be very substantially plastic.

The effect of finite temperature T and finite strain rate γ on the statistical physics of plastic deformations in amorphous solids made of N particles is investigated. We recognize three regimes of temperature where the statistics are qualitatively different. In the first regime the temperature is very low, T

Amorphous solids that underwent a strain in one direction such that they responded in a plastic manner "remember" that direction also when relaxed back to a state with zero mean stress. We address the question "what is the order parameter that is responsible for this memory?" and is therefore the reason for the different subsequent responses of the material to strains in different directions. We identify such an order parameter which is readily measurable, we discuss its trajectory along the stress-strain curve, and propose that it and its probability distribution function must form a necessary component of a theory of elastoplasticity.

We propose a method to predict the value of the external strain where a generic amorphous solid will fail by a plastic response (i.e., an irreversible deformation), solely on the basis of measurements of the nonlinear elastic moduli. While usually considered fundamentally different, with the elastic properties describing reversible phenomena and plastic failure epitomizing irreversible behavior, we show that the knowledge of some nonlinear elastic moduli is enough to predict where plasticity sets in.

An effective temperature Teff which differs from the bath temperature is believed to play an essential role in the theory of elastoplasticity of amorphous solids. Here, we introduce a natural definition of Teff appearing naturally in a Boltzmann-like distribution of measurable structural features without recourse to any questionable assumption. The value of Teff is connected, using theory and scaling concepts, to the flow stress and the mean energy that characterize the elastoplastic flow.

We address the now classical problem of a diffusion process that crosses over from a ballistic behavior at short times to a fractional diffusion (sub- or super-diffusion) at longer times. Using the standard non-Markovian diffusion equation we demonstrate how to choose the memory kernel to exactly respect the two different asymptotics of the diffusion process. Having done so we solve for the probability distribution function as a continuous function which evolves inside a ballistically expanding domain. This general solution agrees for long times with the probability distribution function obtained within the continuous random walk approach but it is much superior to this solution at shorter times where the effect of the ballistic regime is crucial.

Using two extremely different models of glass formers in two and three dimensions we demonstrate how to encode the subtle changes in the geometric rearrangement of particles during the scenario of the glass transition. We construct a statistical mechanical description that is able to explain and predict the geometric rearrangement, the temperature-dependent thermodynamic functions and the alpha-relaxation time within the measured temperature range and beyond. The theory is based on an up-scaling to proper variables (quasi-species) which is validated using a simple criterion. Once constructed, the theory provides an accurate predictive tool for quantities like the specific heat or the entropy at temperatures that cannot be reached by measurements. In addition, the theory identifies a rapidly increasing typical length scale xi as the temperature decreases. This growing spatial length scale determines the alpha-relaxation time as tau(alpha) similar to exp(mu xi/T), where mu is a typical chemical potential per unit length.

Strongly correlated amorphous solids are a class of glass formers whose interparticle potential admits an approximate inverse power-law form in a relevant range of interparticle distances. We study the steady-state plastic flow of such systems, first in the athermal quasistatic limit and second at finite temperatures and strain rates. In all cases we demonstrate the usefulness of scaling concepts to reduce the data to universal scaling functions where the scaling exponents are determined a priori from the interparticle potential. In particular we show that the steady plastic flow at finite temperatures with efficient heat extraction is uniquely characterized by two scaled variables; equivalently, the steady-state displays an equation of state that relates one scaled variable to the other two. We discuss the range of applicability of the scaling theory, and the connection to density scaling in supercooled liquid dynamics. We explain that the description of transient states calls for additional state variables whose identity is still far from obvious.

The shear modulus and yield stress of amorphous solids are important material parameters, with the former determining the rate of increase in stress under external strain and the latter being the stress value at which the material flows in a plastic manner. It is therefore important to understand how these parameters can be related to the interparticle potential. Here a scaling theory is presented such that given the interparticle potential, the dependence of the yield stress, and the shear modulus on the density of the solid can be predicted in the athermal limit. It is explained when such prediction is possible at all densities and when it is only applicable at high densities.

In light of some recent experiments on quasi two-dimensional (2D) turbulent channel flow we provide here a model of the ideal case, for the sake of comparison. The ideal 2D channel flow differs from its three-dimensional (3D) counterpart by having a second quadratic conserved variable in addition to the energy and the latter has an inverse rather than a direct cascade. The resulting qualitative differences in profiles of velocity V and energy K as a function of the distance from the wall are highlighted and explained. The most glaring difference is that the 2D channel is much more energetic, with K in wall units increasing logarithmically with the Reynolds number Reτ instead of being Reτ independent in 3D channels.

The Shintani-Tanaka model is a glass-forming system whose constituents interact via an anisotropic potential depending on the angle of a unit vector carried by each particle. The decay of time-correlation functions of the unit vectors exhibits the characteristics of generic relaxation functions during glass transitions. In particular it exhibits a stretched exponential form, with the stretching index β depending strongly on the temperature. We construct a quantitative theory of this correlation function by analyzing all the physical processes that contribute to it, separating a rotational from a translational decay channel. These channels exhibit different relaxation times, each with its own temperature dependence. Interestingly, the separate decay function of each of these processes is a temperature-independent function, and is shown to scale (exhibit data collapse) at different temperatures. Taken together with temperature-dependent weights determined a priori by statistical mechanics this allows one to generate the observed correlation function in quantitative agreement with simulations at different temperatures. This underlines the danger of concluding anything about glassy relaxation functions without detailed physical scrutiny.

Experimental measurements of the specific heat in glass-forming systems are obtained from the linear response to either slow cooling (or heating) or to oscillatory perturbations with a given frequency about a constant temperature. The latter method gives rise to a complex specific heat with the constraint that the zero frequency limit of the real part should be identified with thermodynamic measurements. Such measurements reveal anomalies in the temperature dependence of the specific heat, including the so called "specific heat peak" in the vicinity of the glass transition. The aim of this paper is to provide theoretical explanations of these anomalies in general and a quantitative theory in the case of a simple model of glass formation. We first present interesting simulation results for the specific heat in a classical model of a binary mixture glass former. We show that in addition to the formerly observed specific heat peak there is a second peak at lower temperatures which was not observable in earlier simulations. Second, we present a general relation between the specific heat, a caloric quantity, and the bulk modulus of the material, a mechanical quantity, and thus offer a smooth connection between the liquid and amorphous solid states. The central result of this paper is a connection between the micromelting of clusters in the system and the appearance of specific heat peaks; we explain the appearance of two peaks by the micromelting of two types of clusters. We relate the two peaks to changes in the bulk and shear moduli. We propose that the phenomenon of glass formation is accompanied by a fast change in the bulk and the shear moduli, but these fast changes occur in different ranges of the temperature. Last, we demonstrate how to construct a theory of the frequency dependent complex specific heat, expected from heterogeneous clustering in the liquid state of glass formers. A specific example is provided in the context of our model for the dynamics of glycerol. We show that the frequency dependence is determined by the same α -relaxation mechanism that operates when measuring the viscosity or the dielectric relaxation spectrum. The theoretical frequency dependent specific heat agrees well with experimental measurements on glycerol. We conclude the paper by stating that there is nothing universal about the temperature dependence of the specific heat in glass formers-unfortunately, one needs to understand each case by itself.

We present a personal view of the state of the art in turbulence research. We summarize first the main achievements of the recent past, and then point ahead to the main challenges that remain for experimental and theoretical efforts.

Recently, the existence and properties of unbounded cavity modes, resulting in extensive plastic deformation failure of two-dimensional sheets of amorphous media, were discussed in the context of the athermal shear-transformation-zones (STZ) theory. These modes pertain to perfect circular symmetry of the cavity and the stress conditions. In this paper we study the shape stability of the expanding circular cavity against perturbations, in both the unbounded and the bounded growth regimes (for the latter the unperturbed theory predicts no catastrophic failure). Since the unperturbed reference state is time dependent, the linear stability theory cannot be cast into standard time-independent eigenvalue analysis. The main results of our study are as follows: (i) sufficiently small perturbations are stable; (ii) larger perturbations within the formal linear decomposition may lead to an instability; this dependence on the magnitude of the perturbations in the linear analysis is a result of the nonstationarity of the growth; and (iii) the stability of the circular cavity is particularly sensitive to perturbations in the effective disorder temperature; in this context we highlight the role of the rate sensitivity of the limiting value of this effective temperature. Finally we point to the consequences of the form of the stress dependence of the rate of STZ transitions. The present analysis indicates the importance of nonlinear effects that were not taken into account yet. Furthermore, the analysis suggests that details of the constitutive relations appearing in the theory can be constrained by the modes of macroscopic failure in these amorphous systems.

The aim of this paper is to discuss some basic notions regarding generic glass-forming systems composed of particles interacting via soft potentials. Excluding explicitly hard-core interaction, we discuss the so-called glass transition in which a supercooled amorphous state is formed, accompanied by a spectacular slowing down of relaxation to equilibrium, when the temperature is changed over a relatively small interval. Using the classical example of a 50-50 binary liquid of N particles with different interaction length scales, we show the following. (i) The system remains ergodic at all temperatures. (ii) The number of topologically distinct configurations can be computed, is temperature independent, and is exponential in N. (iii) Any two configurations in phase space can be connected using elementary moves whose number is polynomially bounded in N, showing that the graph of configurations has the small world property. (iv) The entropy of the system can be estimated at any temperature (or energy), and there is no Kauzmann crisis at any positive temperature. (v) The mechanism for the super-Arrhenius temperature dependence of the relaxation time is explained, connecting it to an entropic squeeze at the glass transition. (vi) There is no Vogel-Fulcher crisis at any finite temperature T>0.

We present experimental and theoretical results addressing the Reynolds number (Re) dependence of drag reduction by sufficiently large concentrations of rodlike polymers in turbulent wall-bounded flows. It is shown that when Re is small the drag is enhanced. On the other hand, when Re increases, the drag is reduced and eventually, the maximal drag reduction asymptote is attained. The theory is shown to be in agreement with experiments, explaining the universal and rationalizing some of the the nonuniversal aspects of drag reduction by rodlike polymers.

We address the relaxation dynamics in hydrogen-bonded supercooled liquids near (but above) the glass transition, measured via broadband dielectric spectroscopy (BDS). We propose a theory based on decomposing the relaxation of the macroscopic dipole moment into contributions from hydrogen-bonded clusters of s molecules, with smin ≤s≤ smax. The existence of smax is translated into a sum rule on the concentrations of clusters of size s. We construct the statistical mechanics of the supercooled liquid subject to this sum rule as a constraint, to estimate the temperature-dependent density of clusters of size s. With a theoretical estimate of the relaxation time of each cluster, we provide predictions for the real and imaginary parts of the frequency-dependent dielectric response. The predicted spectra and their temperature dependence are in accord with measurements, explaining a host of phenomenological fits like the Vogel-Fulcher fit and the stretched exponential fit. Using glycerol as a particular example, we demonstrate quantitative correspondence between theory and experiments. The theory also demonstrates that the α peak and the "excess wing" stem from the same physics in this material. The theory also shows that in other hydrogen-bonded glass formers the excess wing can develop into a β peak, depending on the molecular material parameters (predominantly the surface energy of the clusters). We thus argue that α and β peaks can stem from the same physics. We address the BDS in constrained geometries (pores) and explain why recent experiments on glycerol did not show a deviation from bulk spectra. Finally, we discuss the dc part of the BDS spectrum and argue why it scales with the frequency of the α peak, providing an explanation for the remarkable data collapse observed in experiments.

A classical problem in elasticity theory involves an inhomogeneity embedded in a material of given stress and shear moduli. The inhomogeneity is a region of arbitrary shape whose stress and shear moduli differ from those of the surrounding medium. In this paper we present a semianalytic method for finding the stress tensor for an infinite plate with such an inhomogeneity. The solution involves two conformal maps, one from the inside and the second from the outside of the unit circle to the inside, and respectively outside, of the inhomogeneity. The method provides a solution by matching the conformal maps on the boundary between the inhomogeneity and the surrounding material. This matching converges well only for relatively mild distortions of the unit circle due to reasons which will be discussed in the article. We provide a comparison of the present result to known previous results.

We present here an extended version of an invited talk we gave at the international conference 'Turbulent Mixing and Beyond'. The dynamical and statistical description of stably stratified turbulent boundary layers with the important example of the stable atmospheric boundary layer in mind is addressed. Traditional approaches to this problem, based on the profiles of mean quantities, velocity second-order correlations and dimensional estimates of the turbulent thermal flux, run into a well-known difficulty, predicting the suppression of turbulence at a small critical value of the Richardson number, in contradiction to observations. Phenomenological attempts to overcome this problem suffer from various theoretical inconsistencies. Here, we present an approach taking into full account all the second-order statistics, which allows us to respect the conservation of total mechanical energy. The analysis culminates in an analytic solution of the profiles of all mean quantities and all second-order correlations, removing the unphysical predictions of previous theories. We propose that the approach taken here is sufficient to describe the lower parts of the atmospheric boundary layer, as long as the Richardson number does not exceed an order of unity. For much higher Richardson numbers, the physics may change qualitatively, requiring careful consideration of the potential Kelvin-Helmoholtz waves and their interaction with the vortical turbulence.

We study a recently introduced model of one-component glass-forming liquids whose constituents interact with an anisotropic potential. This system is interesting per se and as a model of liquids such as glycerol (interacting via hydrogen bonds) which are excellent glass formers. We work out the statistical mechanics of this system, encoding the liquid and glass disorder using appropriate quasiparticles (36 of them). The theory provides a full explanation of the glass transition phenomenology, including the identification of a diverging length scale and a relation between the structural changes and the diverging relaxation times.

The long-range elastic model, which is believed to describe the evolution of a self-affine rough crack front, is analyzed to linear and nonlinear orders. It is shown that the nonlinear terms, while important in changing the front dynamics, do not change the scaling exponent which characterizes the roughness of the front. The scaling exponent thus predicted by the model is much smaller than the one observed experimentally. The inevitable conclusion is that the gap between the results of experiments and the model that is supposed to describe them is too large and some new physics has to be invoked for another model.

The statistical mechanics of simple glass forming systems in two dimensions is worked out. The glass disorder is encoded via a Voronoi tesselation, and the statistical mechanics is performed directly in this encoding. The theory provides, without free parameters, an explanation of the glass transition phenomenology, including the identification of two different temperatures, Tg and Tc, the first associated with jamming and the second associated with crystallization at very low temperatures.

We develop an athermal version of the shear-transformation-zone (STZ) theory of amorphous plasticity in materials where thermal activation of irreversible molecular rearrangements is negligible or nonexistent. In many respects, this theory has broader applicability and yet is simpler than its thermal predecessors. For example, it needs no special effort to assure consistency with the laws of thermodynamics, and the interpretation of yielding as an exchange of dynamic stability between jammed and flowing states is clearer than before. The athermal theory presented here incorporates an explicit distribution of STZ transition thresholds. Although this theory contains no conventional thermal fluctuations, the concept of an effective temperature is essential for understanding how the STZ density is related to the state of disorder of the system.

The stability of a rapid dynamic crack in a two-dimensional infinite strip is studied in the framework of linear elastic fracture mechanics supplemented with a modified principle of local symmetry. It is predicted that a single crack becomes unstable by a finite wavelength oscillatory mode at a velocity vc, 0.8cR

We investigate whether fractal viscous fingering and diffusion-limited aggregates are in the same scaling universality class. We bring together the largest available viscous fingering patterns and a novel technique for obtaining the conformal map from the unit circle to an arbitrary singly connected domain in two dimensions. These two Laplacian fractals appear different to the eye; in addition, viscous fingering is grown in parallel and the aggregates by a serial algorithm. Nevertheless, the data strongly indicate that these two fractal growth patterns are in the same universality class.

We employ the full FENE-P model of the hydrodynamics of a dilute polymer solution to derive a theoretical approach to drag reduction in wall-bounded turbulence. We recapture the results of a recent simplified theory which derived the universal maximum drag reduction (MDR) asymptote, and complement that theory with a discussion of the cross-over from the MDR to the Newtonian plug when the drag reduction saturates. The FENE-P model gives rise to a rather complex theory due to the interaction of the velocity field with the polymeric conformation tensor, making analytic estimates quite taxing. To overcome this we develop the theory in a computer-assisted manner, checking at each point the analytic estimates by direct numerical simulations (DNS) of viscoelastic turbulence in a channel.

Drag reduction by polymers is bounded between two universal asymptotes, the von Kármán log law of the law and the maximum drag reduction (MDR) asymptote. It is theoretically understood why the MDR asymptote is universal, independent of whether the polymers are flexible or rodlike. The crossover behavior from the Newtonian von Kármán log law to the MDR is, however, not universal, showing different characteristics for flexible and rodlike polymers. In this paper we provide a theory for this crossover phenomenology.

We propose a new approach to the old-standing problem of the anomaly of the scaling exponents of nonlinear models of turbulence. We construct, for any given nonlinear model, a linear model of passive advection of an auxiliary field whose anomalous scaling exponents are the same as the scaling exponents of the nonlinear problem. The statistics of the auxiliary linear model are dominated by "statistically preserved structures" which are associated with exact conservation laws. The latter can be used, for example, to determine the value of the anomalous scaling exponent of the second order structure function. The approach is equally applicable to shell models and to the Navier-Stokes equations.

Drag reduction by bubbles in stationary turbulent flows is sensitive to the compressibility of the bubbles. Without this dynamical effect the bubbles only renormalize the fluid density and viscosity, an effect that by itself can only lead to a small percentage of drag reduction. We show in this paper that the dynamics of bubbles and their effect on the compressibility of the mixture can lead to a much higher drag reduction.

Structure functions of rough fracture surfaces in isotropic materials exhibit complicated scaling properties due to the broken isotropy in the fracture plane generated by a preferred propagation direction. Decomposing the structure functions into the even order irreducible representations of the SO(2) symmetry group indexed by (m=0,2,4,...) results in a lucid and quickly convergent description. The scaling exponent of the isotropic sector (m=0) dominates at small length scales. One can reconstruct the anisotropic structure functions using only the isotropic and the first nonvanishing anisotropic sector (m=2) [or at most the next one (m=4)]. The scaling exponent of the isotropic sector should be observed in a proposed, yet unperformed, experiment.

A celebrated universal aspect of wall-bounded turbulent flows is the von Kármán log-law-of-the-wall, describing how the mean velocity in the stream-wise direction depends on the distance from the wall. Although the log-law is known for more than 75 years, the von Kármán constant governing the slope of the log-law was not determined theoretically. In this letter we show that the von Kármán constant can be estimated from homogeneous turbulent data, i. e, without information from wall-bounded flows.

The problem of anisotropy and its effects on the statistical theory of high Reynolds number (Re) turbulence (and turbulent transport) is intimately related and intermingled with the problem of the universality of the (anomalous) scaling exponents of structure functions. Both problems had seen tremendous progress in the last 5 years. In this review we present a detailed description of the new tools that allow effective data analysis and systematic theoretical studies such as to separate isotropic from anisotropic aspects of turbulent statistical fluctuations. Employing the invariance of the equations of fluid mechanics to all rotations, we show how to decompose the (tensorial) statistical objects in terms of the irreducible representation of the SO (d) symmetry group (with d being the dimension, d = 2 or 3). This device allows a discussion of the scaling properties of the statistical objects in well-defined sectors of the symmetry group, each of which is determined by the "angular momenta" sector numbers (j, m). For the case of turbulent advection of passive scalar or vector fields, this decomposition allows rigorous statements to be made: (i) the scaling exponents are universal, (ii) the isotropic scaling exponents are always leading, (iii) the anisotropic scaling exponents form a discrete spectrum which is strictly increasing as a function of j. This emerging picture offers a complete understanding of the decay of anisotropy upon going to smaller and smaller scales. Next, we explain how to apply the SO (3) decomposition to the statistical Navier-Stokes theory. We show how to extract information about the scaling behavior in the isotropic sector. Doing so furnishes a systematic way to assess the universality of the scaling exponents in this sector, clarifying the anisotropic origin of the many measurements that claimed the opposite. A systematic analysis of direct numerical simulations (DNS) of the Navier-Stokes equations and of experiments provides a strong support to the proposition that also for the non-linear problem there exists foliation of the statistical theory into sectors of the symmetry group. The exponents appear universal in each sector, and again strictly increasing as a function of j. An approximate calculation of the anisotropic exponents based on a closure theory is reviewed. The conflicting experimental measurements on the rate of decay of anisotropy upon reducing the scales are explained and systematized, showing that isotropy is eventually recovered at small scales.

Slow crack propagation in ductile, and in certain brittle materials, appears to take place via the nucleation of voids ahead of the crack tip due to plastic yields, followed by the coalescence of these voids. Postmortem analysis of the resulting fracture surfaces of ductile and brittle materials on the Îm-mm and the nm scales, respectively, reveals self-affine cracks with anomalous scaling exponent ζâ0.8 in 3 dimensions and ζâ 0.65 in 2 dimensions. In this paper we present an analytic theory based on the method of iterated conformal maps aimed at modelling the void formation and the fracture growth, culminating in estimates of the roughening exponents in 2 dimensions. In the simplest realization of the model we allow one void ahead of the crack, and address the robustness of the roughening exponent. Next we develop the theory further, to include two voids ahead of the crack. This development necessitates generalizing the method of iterated conformal maps to include doubly connected regions (maps from the annulus rather than the unit circle). While mathematically and numerically feasible, we find that the employment of the stress field as computed from elasticity theory becomes questionable when more than one void is explicitly inserted into the material. Thus further progress in this line of research calls for improved treatment of the plastic dynamics.

We propose a theoretical model for branching instabilities in 2-dimensional fracture, offering predictions for when crack branching occurs, how multiple cracks develop, and what is the geometry of multiple branches. The model is based on equations of motion for crack tips which depend only on the time dependent stress intensity factors. The latter are obtained by invoking an approximate relation between static and dynamic stress intensity factors, together with an essentially exact calculation of the static ones. The results of this model are in qualitative agreement with a number of experiments in the literature.

Drag reduction by polymers in turbulent flows raises an apparent contradiction: the stretching of the polymers must increase the viscosity, so why is the drag reduced? A recent theory proposed that drag reduction, in agreement with experiments, is consistent with the effective viscosity growing linearly with the distance from the wall. With this self-consistent solution the reduction in the Reynolds stress overwhelms the increase in viscous drag. In this Rapid Communication we show, using direct numerical simulations, that a linear viscosity profile indeed reduces the drag in agreement with the theory and in close correspondence with direct simulations of the FENE-P model at the same flow conditions.

The connection between velocity fluctuations and the putative existence of microcracks ahead of the crack tip was studied. The dynamics of one macrocrack and one microcrack were analyzed. It was demonstrated that the interaction results in a large and rapid velocity fluctuation. An approximate methodology was proposed that will allow one to write ordinary differential equations for the positions of the tips of both macrocrack and microcrack.

We demonstrate, by using suitable shell models, that drag reduction in homogeneous turbulence is usefully discussed in terms of a scale-dependent effective viscosity. The essence of the phenomenon of drag reduction found in models that couple the velocity field to the polymers can be recaptured by an \u201cequivalent\u201d equation of motion for the velocity field alone, with a judiciously chosen scale-dependent effective viscosity that succinctly summarizes the important aspects of the interaction between the velocity and the polymer fields. Finally, we clarify the differences between drag reduction in homogeneous and in wall bounded flows.

The mechanism of drag reduction by polymers in turbulent wall-bounded flows was investigated. A new logarithmic law for the mean velocity was derived with a slope that fit existing data. The phenomenonology of the maximum drag reduction asymptote which is the maximum drag reduction attained by polymers was developed on the basis of the equations of fluid mechanics. It was observed that the mechanism of drag reduction is the suppression of the Reynolds stress in the elastic sublayer.

The stress intensity factors for arbitrarily shaped cracks in an infinite two-dimensional elastic medium were investigated. The conformal map was constructed from the exterior of the unit circle to exterior of the arbitrary crack using the method of iterated conformal maps. The method, when compared with the Schwartz-Cristoffel transformation, was found to be better. The conformal method allowed an accurate estimate of the stress intensity factors or of the subleading terms such as the T-stress and simplified the calculation of the full stress field.

An exact integro-differential equation for the conformal map from the unit circle to the boundary of an evolving cavity in a stressed 2-dimensional solid is derived. This equation provides an accurate description of the dynamics of precursors to fracture when surface diffusion is important. The solution predicts the creation of sharp grooves that eventually lead to material failure via rapid fracture. Solutions of the new equation are demonstrated for the dynamics of an elliptical cavity and the stability of a circular cavity under biaxial stress, including the effects of surface stress.

Recent direct numerical simulations of the finite-extensibility nonlinear elastic dumbbell model with the Peterlin approximation of non-Newtonian hydrodynamics revealed that the phenomenon of drag reduction by polymer additives exists (albeit in reduced form) also in homogeneous turbulence. We use here a simple shell model for homogeneous viscoelastic flows, which recaptures the essential observations of the full simulations. The simplicity of the shell model allows us to offer a transparent explanation of the main observations. It is shown that the mechanism for drag reduction operates mainly on large scales. Understanding the mechanism allows us to predict how the amount of drag reduction depends on the various parameters in the model. The main conclusion is that drag reduction is not a universal phenomenon; it peaks in a window of parameters such as the Reynolds number and the relaxation rate of the polymer.

Motivated by the large effect of turbulent drag reduction by minute concentrations of polymers, we study the effects of a weakly space-dependent viscosity on the stability of hydrodynamic flows. In a recent paper [Phys. Rev. Lett. 87, 174501, (2001)], we exposed the crucial role played by a localized region where the energy of fluctuations is produced by interactions with the mean flow (the \u201ccritical layer\u201d). We showed that a layer of a weakly space-dependent viscosity placed near the critical layer can have a very large stabilizing effect on hydrodynamic fluctuations, retarding significantly the onset of turbulence. In this paper we extend these observations in two directions: first we show that the strong stabilization of the primary instability is also obtained when the viscosity profile is realistic (inferred from simulations of turbulent flows with a small concentration of polymers). Second, we analyze the secondary instability (around the time-dependent primary instability) and find similar strong stabilization. Since the secondary instability develops around a time-dependent solution and is three dimensional, this brings us closer to the turbulent case. We reiterate that the large effect is not due to a modified dissipation (as is assumed in some theories of drag reduction), but due to reduced energy intake from the mean flow to the fluctuations. We propose that similar physics act in turbulent drag reduction.

We address the theory of quasistatic crack propagation in a strip of glass that is pulled from a hot oven towards a cold bath. This problem had been carefully studied in a number of experiments that offer a wealth of data to challenge the theory. We improve upon previous theoretical treatments in a number of ways. First, we offer a technical improvement of the discussion of the instability towards the creation of a straight crack. This improvement consists in employing Padé approximants to solve the relevant Wiener-Hopf factorization problem that is associated with this transition. Next we improve the discussion of the onset of oscillatory instability towards an undulating crack. We offer a way of considering the problem as a sum of solutions of a finite strip without a crack and an infinite medium with a crack. This allows us to present a closed form solution of the stress intensity factors in the vicinity of the oscillatory instability. Most importantly we develop a dynamical description of the actual trajectory in the regime of oscillatory crack. This theory is based on the dynamical law for crack propagation proposed by Hodgdon and Sethna. We show that this dynamical law results in a solution of the actual crack trajectory in post-critical conditions; we can compute from first principles the critical value of the control parameters, and the characteristics of the solution such as the wavelength of the oscillations. We present detailed comparison with experimental measurements without any free parameters. The comparison appears quite excellent. Finally we show that the dynamical law can be translated to an equation for the amplitude of the oscillatory crack; this equation predicts correctly the scaling exponents observed in experiments.

We address the phenomenon of drag reduction by a dilute polymeric additive to turbulent flows, using direct numerical simulations (DNS) of the FENE-P model of viscoelastic flows. It had been amply demonstrated that these model equations reproduce the phenomenon, but the results of DNS were not analyzed so far with the goal of interpreting the phenomenon. In order to construct a useful framework for the understanding of drag reduction we initiate in this paper an investigation of the most important modes that are sustained in the viscoelastic and Newtonian turbulent flows, respectively. The modes are obtained empirically using the Karhunen-Loéve decomposition, allowing us to compare the most energetic modes in the viscoelastic and Newtonian flows. The main finding of the present study is that the spatial profile of the most energetic modes is hardly changed between the two flows. What changes is the energy associated with these modes, and their relative ordering in the decreasing order from the most energetic to the least. Modes that are highly excited in one flow can be strongly suppressed in the other, and vice versa. This dramatic energy redistribution is an important clue to the mechanism of drag reduction as is proposed in this paper. In particular, there is an enhancement of the energy containing modes in the viscoelastic flow compared to the Newtonian one; drag reduction is seen in the energy containing modes rather than the dissipative modes, as proposed in some previous theories.

The anomalous scaling of correlation functions in the turbulent statistics of active scalars (like temperature in turbulent convection) is understood in terms of an auxiliary passive scalar which is advected by the same turbulent velocity field. The even-order correlation functions of the two fields are the same to leading order (up to a trivial multiplicative factor). The leading correlation functions are statistically preserved structures of the passive scalar decaying problem; thus the universality of the scaling exponents of the even-order correlations of the active scalar is demonstrated.

The statistics of two-dimensional turbulence exhibit a riddle: the scaling exponents in the regime of inverse energy cascade agree with the K41 theory of turbulence far from equilibrium, but the probability distribution functions are close to Gaussian-like in equilibrium. The skewness Sequivalent toS(3)(R)/S-2(3/2)(R) was measured as S(exp)approximate to0.03. This contradiction is lifted by understanding that two-dimensional turbulence is not far from a situation with equipartition of enstrophy, which exists as true thermodynamic equilibrium with K41 exponents in space dimension of d=4/3. We evaluate the skewness S(d) for 4/3 less than or equal todless than or equal to2, showing that S(d)=0 at d=4/3, and that it remains as small as S-exp in two dimensions.

We study the geometrical characteristic of quasistatic fractures in brittle media, using iterated conformal maps to determine the evolution of the fracture pattern. This method allows an efficient and accurate solution of the Lamé equations without resorting to lattice models. Typical fracture patterns exhibit increased ramification due to the increase of the stress at the tips. We find the roughness exponent of the experimentally relevant backbone of the fracture pattern, it crosses over from about 0.5 for small scales to about 0.75 for large scales. We propose that this crossover reflects the increased ramification of the fracture pattern.

The method of iterated conformal maps allows one to study the harmonic measure of diffusion-limited aggregates with unprecedented accuracy. We employ this method to explore the multifractal properties of the measure, including the scaling of the measure in the deepest fjords that were hitherto screened away from any numerical probing. We resolve probabilities as small as 10 ^-35, and present an accurate determination of the generalized dimensions and the spectrum of singularities. We show that the generalized dimensions D_q are infinite for q

The method of iterated conformal maps for the study of diffusion limited aggregates (DLA) is generalized to the study of Laplacian growth patterns and related processes. We emphasize the fundamental difference between these processes: DLA is grown serially with constant size particles, while Laplacian patterns are grown by advancing each boundary point in parallel, proportional to the gradient of the Laplacian field. We introduce a two-parameter family of growth patterns that interpolates between DLA and a discrete version of Laplacian growth. The ultraviolet putative finite-time singularities are regularized here by a minimal tip size, equivalently for all the models in this family. With this we stress that the difference between DLA and Laplacian growth is not in the manner of ultraviolet regularization, but rather in their deeply different growth rules. The fractal dimensions of the asymptotic patterns depend continuously on the two parameters of the family, giving rise to a \u201cphase diagram\u201d in which DLA and discretized Laplacian growth are at the extreme ends. In particular, we show that the fractal dimension of Laplacian growth patterns is higher than the fractal dimension of DLA, with the possibility of dimension 2 for the former not excluded.

We study the nature of the phase transition in the multifractal formalism of the harmonic measure of diffusion limited aggregates. Contrary to previous work that relied on random walk simulations or ad hoc models to estimate the low probability events of deep fjord penetration, we employ the method of iterated conformal maps to obtain an accurate computation of the probability of the rarest events. We resolve probabilities as small as 10⁻³⁵. We show that the generalized dimensions D_q are infinite for q

We study the dynamics of "finger" formation in Laplacian growth without surface tension in a channel geometry (the Saffman Taylor problem). We present a pedagogical derivation of the dynamics of the conformal map from a strip in the complex plane to the physical channel. In doing so we pay attention to the boundary conditions (no flux rather than periodic) and derive a field equation of motion for the conformal map. We first consider an explicit analytic class of conformal maps that form a basis for solutions in infinitely long channels, characterized by meromorphic derivatives. The great bulk of these solutions can lose conformality due to finite time singularities. By considerations of the nature of the analyticity of these solutions, we show that those solutions which are free of such singularities inevitably result in a single asymptotic "finger" whose width is determined by initial conditions. This is in contradiction with the experimental results that indicate selection of a finger of width 1/2. In the last part of this paper we show that such a solution might be determined by the boundary conditions of a finite body of fluid, e.g. finiteness can lead to pattern selection.

Traditional devices for measuring turbulence have been unable to keep up with the latest developments in theory. But detectors derived from high-energy physics may narrow the gap between experiment and theory.

The problem of anomalous scaling in passive scalar advection, especially with delta -correlated velocity field (the Kraichnan model) has attracted a lot of interest since the exponents can be computed analytically in certain limiting cases. In this paper we focus, rather than on the evaluation of the exponents, on elucidating the physical mechanism responsible for the anomaly. We show that the anomalous exponents zeta (n) stem from the Lagrangian dynamics of shapes which characterize configurations of n points in space. Using the shape-to-shape transition probability, we define an operator whose eigenvalues determine the anomalous exponents for all n, in all the sectors of the SO(3) symmetry group.

A conformal theory of fractal growth processes in two dimensions was considered, including diffusion limited aggregation (DLA) as a special case. The computation of the dimension from early stage dynamics was demonstrated. Further, a novel notion was introduced that the fractal dimension appears as a dynamical exponent already at early stages of the growth in which the cluster has not yet developed a fractal geometry.

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 ∈. It is demonstrated, using numerical simulations of up to 25 copies and 28 shells that in the N→∞ limit but for 0

The main difficulty of statistical theories of fluid turbulence is the lack of an obvious small parameter. In this paper we show that the formerly established fusion rules can be employed to develop a theory in which Kolmogorovs statistics of 1941 (K41) acts as the zero order, or background statistics, and the anomalous corrections to the K41 scaling exponents [Formula Presented] of the [Formula Presented]-order structure functions can be computed analytically. The crux of the method consists of renormalizing a four-point interaction amplitude on the basis of the fusion rules. This amplitude includes a small dimensionless parameter, which is shown to be of the order of the anomaly of [Formula Presented] [Formula Presented] Higher-order interaction amplitudes are shown to be even smaller. The corrections to K41 to [Formula Presented] result from standard logarithmically divergent ladder diagrams in which the four-point interaction acts as a \u201crung.\u201d The theory allows a calculation of the anomalous exponents [Formula Presented] in powers of the small parameter [Formula Presented] The n dependence of the scaling exponents [Formula Presented] stems from pure combinatorics of the ladder diagrams. In this paper we calculate the exponents [Formula Presented] up to [Formula Presented] Previously derived bridge relations allow a calculation of the anomalous exponents of correlations of the dissipation field and of dynamical correlations in terms of the same parameter [Formula Presented] The actual evaluation of the small parameter [Formula Presented] from first principles requires additional developments that are outside the scope of this paper.

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 [Formula Presented] correlated in time, and its structure function scales with a positive exponent [Formula Presented] 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.

The Taylor hypothesis, which allows surrogating spatial measurements requiring many experimental probes by time series from one or two probes, is examined on the basis of a simple analytic model of turbulent statistics. The main points are as follows: (i) The Taylor hypothesis introduces systematic errors in the evaluation of scaling exponents. (ii) When the mean wind V̄₀ is not infinitely larger than the root-mean-square longitudinal turbulent fluctuations v_T, the effective Taylor advection velocity V_ad should take the latter into account. (iii) When two or more probes are employed the application of the Taylor hypothesis and the optimal choice of the effective advecting wind V_ad need extra care. We present practical considerations for minimizing the errors incurred in experiments using one or two probes. (iv) Analysis of the Taylor hypothesis when different probes experience different mean winds is offered.

We show that the Sabra shell model of turbulence, which was introduced recently, displays a Hamiltonian structure for given values of the parameters. The requirement of scale independence of the flux of this Hamiltonian allows us to compute exactly a one-parameter family of anomalous scaling exponents associated with 4th-order correlation functions.

We consider flame front propagation in channel geometries. The steady-state solution in this problem is space dependent and therefore the linear stability analysis is described by a partial integro-differential equation with a space-dependent coefficient. Accordingly, it involves complicated eigenfunctions. We show that the analysis can be performed using a finite-order dynamical system in terms of the dynamics of singularities in the complex plane, yielding a detailed understanding of the physics of the eigenfunctions and eigenvalues.

A recent theoretical development in the understanding of the small-scale structure of Navier-Stokes turbulence has been the proposition that the scales η_n(R) that separate inertial from viscous behavior of many-point correlation functions depend on the order n and on the typical separations R of points in the correlation. This is of fundamental significance in itself but it also has implications for the scaling behaviour of various correlation functions. This dependence has never been observed directly in laboratory experiments. In order to observe it, turbulence data which both display a well-developed scaling range with clean scaling behaviour and are well-resolved in the small scales to well within the viscous range is required. The data of the experiments performed in the laboratory of P. Tabeling of Navier-Stokes turbulence in a helium cell with counter-rotating disks approach these criteria, and provide supporting evidence for the existence of the predicted scaling of the viscous scale.

In this short paper we describe the essential ideas behind a new consistent closure procedure for the calculation of the scaling exponents ζ_n of the nth order correlation functions in fully developed hydrodynamic turbulence, starting from first principles. The closure procedure is constructed to respect the fundamental rescaling symmetry of the Euler equation. The starting point of the procedure is an infinite hierarchy of coupled equations that are obeyed identically with respect to scaling for any set of scaling exponents ζ_n. This hierarchy was discussed in detail in a recent publication [V.S. L'vov and I. Procaccia, Physica A (1998), in press, chao-dyn/9707015]. The scaling exponents in this set of equations cannot be found from power counting. In this short paper we discuss in detail low order non-trivial closures of this infinite set of equations, and prove that these closures lead to the determination of the scaling exponents from solvability conditions. The equations under consideration after this closure are nonlinear integro-differential equations, reflecting the nonlinearity of the original Navier-Stokes equations. Nevertheless, they have a very special structure such that the determination of the scaling exponents requires a procedure that is very similar to the solution of linear homogeneous equations, in which amplitudes are determined by fitting to the boundary conditions in the space of scales. The renormalization scale that is necessary for any anomalous scaling appears at this point. The Hölder inequalities on the scaling exponents select the renormalization scale as the outer scale of turbulence L.

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.

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.

We analyze turbulent velocity signals in the atmospheric surface layer, obtained by pairs of probes separated by inertial-range distances parallel to the ground and (nominally) orthogonal to the mean wind. The Taylor microscale Reynolds number ranges up to 20 000. Choosing a suitable coordinate system with respect to the mean wind, we derive theoretical forms for second order structure functions and fit them to experimental data. The effect of flow anisotropy is small for the longitudinal component but significant for the transverse component. The data provide an estimate for a universal exponent from among a hierarchy that governs the decay of flow anisotropy with the scale size.

The phenomenology of the scaling behavior of higher order structure functions of velocity differences across a scale R in turbulence should be built around the irreducible representations of the rotation symmetry group. Every irreducible representation is associated with a scalar function of R which may exhibit different scaling exponents. The common practice of using moments of longitudinal and transverse fluctuations mixes different scalar functions and therefore may mix different scaling exponents. It is shown explicitly how to extract pure scaling exponents for correlation functions of arbitrary orders.

The theoretical calculation of the scaling exponents that characterize the statistics of fully developed turbulence is one of the major open problems of statistical physics. We review the subject, explain some of the recent developments, and point out the road ahead.

We present the first experimental tests of the recently derived fusion rules for Navier-Stokes turbulence. The fusion rules address the asymptotic properties of many-point correlation functions as some of the coordinates coalesce, and form an important ingredient of the nonperturbative statistical theory of turbulence. Here we test the fusion rules when the spatial separations lie within the inertial range, and find good agreement between experiment and theory. For inertial-range separations and for velocity increments which are not too large, a simple linear relation appears to exist for the Laplacian of the velocity fluctuation conditioned on velocity increments.

The anomalous scaling behavior of the nth-order correlation functions F_n of the Kraichnan model of turbulent passive scalar advection is believed to be dominated by the homogeneous solutions (zero modes) of the Kraichnan equation B_nF_n = 0. In this paper we present an extensive analysis of the simplest (nontrivial) case of n = 3 in the isotropic sector. The main parameter of the model, denoted as ζ_h, characterizes the eddy diffusivity and can take values in the interval 0≤ζ_h≤2. After choosing appropriate variables we can present nonperturbative numerical calculations of the zero modes in a projective two dimensional circle. In this presentation it is also very easy to perform perturbative calculations of the scaling exponent ζ₃ of the zero modes in the limit ζ_h→0, and we display quantitative agreement with the nonperturbative calculations in this limit. Another interesting limit is ζ_h→2. This second limit is singular, and calls for a study of a boundary layer using techniques of singular perturbation theory. Our analysis of this limit shows that the scaling exponent ζ₃ vanishes as √ζ₂/|Inζ₂, where ζ₂ is the scaling exponent of the second-order correlation function. In this limit as well, perturbative calculations are consistent with the nonperturbative calculations.

Elements of the analytic structure of anomalous scaling and intermittency in fully developed hydrodynamic turbulence are described. We focus here on the structure functions of velocity differences that satisfy inertial range scaling laws S-n(R)similar to R(zeta n), and the correlation of energy dissipation K-epsilon epsilon(R)similar to R(-mu). The goal is to understand from first principles what is the mechanism that is responsible for changing the exponents zeta(n) and mu from their classical Kolmogorov values. In paper II of this series [V. S. L'vov and I. Procaccia, Phys. Rev. E 52, 3858 (1995)] it was shown that the existence of an ultraviolet scale (the dissipation scale eta) is associated with a spectrum of anomalous exponents that characterize the ultraviolet divergences of correlations of gradient fields. The leading scaling exponent in this family was denoted Delta. The exact resummation of ladder diagrams resulted in a ''bridging relation,'' which determined Delta in terms of zeta(2): Delta = 2-zeta(2). In this paper we continue our analysis and show that nonperturbative effects may introduce multiscaling (i.e., zeta(n) not linear in n) with the renormalization scale being the infrared outer scale of turbulence L. It is shown that deviations from the classical Kolmogorov 1941 theory scaling of S-n(R) (zeta(n) not equal n/3) must appear if the correlation of dissipation is mixing (i.e., mu>0). We suggest possible scenarios for multiscaling, and discuss the implication of these scenarios on the values of the scaling exponents zeta(n) and their ''bridge'' with mu.

We examine the dynamic interplay between vorticity magnitude and vortex line geometry, and its relevance for curbing potential finite-time singularities in incompressible Navier-Stokes flows. We present direct numeri- cal simulations of flows with various low and mid-range Reynolds numbers and different types of forcing. The central conclusion is that the vortex lines in regions of high vorticity tend to be straight and well aligned. Such an organization indicates the existence of a self-correcting mechanism that cancels the quadratic nonlinearity inherent in the vorticity equation. We consider several relevant effects, including the observation of straight ening of vortex lines by stretching.

Fusion rules in turbulence specify the analytic structure of many-point correlation functisons of the turbulent field when a group of coordinates coalesce. We show that the existence of universal flux equilibrium in fully developed turbulent systems combined with a direct cascade induces universal fusion rules. In certain examples these fusion rules suffice to compute the multiscaling exponents exactly, and in other examples they give rise to an infinite number of scaling relations that constrain enormously the structure of the allowed theory.

In this short note we present a brief overview of our recent progress in understanding the universal statistics of fully developed turbulence, with a stress on anomalous scaling.

We consider turbulent advection of a scalar field [Formula Presented], passive or active, and focus on the statistics of gradient fields conditioned on scalar differences [Formula Presented] across a scale [Formula Presented]. In particular we focus on two conditional averages [Formula Presented] and [Formula Presented]. We find exact relations between these averages, and with the help of the fusion rules we propose a general representation for these objects in terms of the probability density function [Formula Presented] of [Formula Presented]. These results offer a way to analyze experimental data that is presented in this paper. The main question that we ask is whether the conditional average [Formula Presented] is linear in [Formula Presented]. We show that there exists a dimensionless parameter which governs the deviation from linearity. The data analysis indicates that this parameter is very small for passive scalar advection, and is generally a decreasing function of the Rayleigh number for the convection data.

We present a theoretical framework for the discussion of the scaling properties of interfaces advancing in systems with quenched disorder. In all such systems there are critical conditions at which the interface gains scale invariance for sufficiently slow growth. There are two fundamental concepts, the ''blocking surfaces'' and the ''associated processes,'' whose nature determines the scaling properties of the advancing interfaces at criticality. The associated processes define a network whose scaling properties determine all the exponents (static and dynamic) that characterize the critical growing interface via universal scaling relations. We point out in this paper that most of the physical rules that can be used to advance the interface also incorporate noncritical elements; as a result, the roughness exponent of the growing interface may deviate from that of the critical interface in a rule-dependent way. We illustrate the wide applicability of the universal scaling relations with diverse models, such as the Edwards-Wilkinson (EW) model with quenched noise, the random-field Ising model, and the Kardar-Parisi-Zhang (KPZ) model with quenched noise. It is shown that the last model is characterized by bounded slopes, whereas in the EW model the slopes are unbounded. This fact makes the KPZ model equivalent to the self-organized interface depinning model of Buldyrev and Sneppen.

We examine the possibility of a blowup of the vorticity due to self-stretching and mechanisms for its prevention. We first estimate directly from the Navier-Stokes equations the length scale of coherence in the direction of the vortex lines to be of the order of the Kolmogorov length. Alignment of vortex lines is seen to be a viscous phenomenon and may prevent some scenarios for blowup. Next we derive equations for the curvature and torsion of vortex lines. We show that the same stretching that amplifies the vorticity also tends to straighten out the vortex lines. Then we show that in well-aligned vortex tubes, the self-stretching rate of the vorticity is proportional to the ratio of the vorticity and the radius of curvature. Thus blowup of the vorticity in such tubes can be prevented by the growth of the vorticity being balanced by the straightening of the vortex lines. Implications for vorticity-strain alignment and the scaling theory of turbulence are noted. Finally, we examine the effects of viscous diffusion on the vorticity field and see how viscosity can lead to organization and alignment of vortex lines.

This paper is the first in a series of papers that aim at understanding the scaling behavior of hydrodynamic turbulence. We present in this paper a perturbative theory for the structure functions and the response functions of the hydrodynamic velocity field in real space and time. Starting from the Navier-Stokes equations (at high Reynolds number Re) we show that the standard perturbative expansions that suffer from infrared divergences can be exactly resummed using the Belinicher-L'vov transformation. After this exact (partial) resummation it is proven that the resulting perturbation theory is free of divergences, both in large and in small spatial separations. The hydrodynamic response and the correlations have contributions that arise from mediated interactions which take place at some space-time coordinates. It is shown that the main contribution arises when these coordinates lie within a shell of a ''ball of locality'' that is defined and discussed. We argue that the real space-time formalism that is developed here offers a clear and intuitive understanding of every diagram in the theory, and of every element in the diagrams. One major consequence of this theory is that none of the familiar perturbative mechanisms may ruin the classical 1941 Kolmogorov (K41) scaling solution for the structure functions. Accordingly, corrections to the K41 solutions should be sought in nonperturbative effects. These effects are the subjects of paper II (the following paper) and a future paper in this series that will propose a mechanism for anomalous scaling in turbulence, which in particular allows a multiscaling of the structure functions.

A Comment on the Letter by C. Jayaprakash, F. Hayot, and R. Pandit, Phys. Rev. Lett. 71, 12 (1993).

We study analytically and numerically the corrections to scaling in turbulence which arise due to finite size effects as anisotropic forcing or boundary conditions at large scales. We find that the deviations δζm from the classical Kolmogorov scaling ζm=m/3 of the velocity moments u(k)mk-ζm decrease like δζm(Re)=cmRe-3/10. If, on the contrary, anomalous scaling in the inertial subrange can experimentally be verified in the large Re limit, this will support the suggestion that small scale structures should be responsible, originating form viscous effects either in the bulk (vortex tubes or sheets) or from the boundary layers (plumes or swirls), as both are underestimated in our reduced wave vector set approximation of the Navier-Stokes dynamics.

Recent experiments on the dispersal of a dye by chaotic surface waves indicate that the iso-concentration curves exhibit fractal geometry on a significant range of scales. In this letter we offer a rigorous theory in which the dimension of the iso-concentration curves is calculated. It is shown that the results are universal for all types of surface waves without mean advection, and are in good agreement with the experimental measurements.

We develop a theory that is nonperturbative and free of uncontrolled approximations to understand scaling behavior in turbulence. The main tool is a connection between the dimension of the graphs of the hydrodynamic fields and the scaling exponents of their structure functions. The connection is developed in some generality for both scalar and vector fields, in terms of the geometric invariants of the gradient tensor. We show that fluid mechanics is consistent with fractal graphs for both the scalar and the vector fields, and explain how this leads to the scaling behavior of the structure functions. We derive scaling relations between various scaling exponents, and show that in the case of "strong scaling" (which is defined below) the Kolmogorov solution is unique. Our theory allows additional solutions in which a weaker version of scaling results in a spectrum of scaling exponents. In particular, we identify the dimensionless (but Reynolds-number-dependent) contributions which can lead to deviations from the Kolmogorov exponents (which are derived using dimensional analysis). Results for the dimensions of fractal level sets in hydrodynamic turbulence which are measured in experiments and simulations follow immediately from this theory.

The multifractal model of turbulence assumes a concentration of the field of vorticity magnitude on a set of dimension smaller than 3. We examine the mechanism for the exponential growth of the vorticity magnitude, and estimate the fractal dimenison of its level sets on the basis of the equations of fluid mechanics. We propose a model of multifractal turbulence in which there is a connection between the fractal dimension of the level sets and the information dimension D1 of the carrier of the field of vorticity magnitude. Under the stated conditions we can estimate the dimension D1 and evaluate, within the framework of the multifractal model of turbulence, the values of the scaling exponents of velocity and temperature structure functions. We find exponents that are consistent with the available experimental information and which differ from the ones obtained by the Kolmogorov dimensional analysis.

The fractal nature of level sets and the multifractal nature of various scalar and vector fields in hydromagnetic and hydrodynamic turbulence are investigated using data of direct simulations. It turns out that fields whose evolution is governed by stretching terms (vortex stretching, magnetic-field line stretching) exhibit ''near singularities'' that result in a multifractal scaling. Such stretching terms can lead to a rapid increase in the local value of the field. Fields without rapid local increase have no multifractal scaling. Furthermore, the simulations support recent theoretical suggestions that the fractal properties of the level sets of various fields are quite insensitive to the existence of stretching. Indeed, all the fields under study (temperature, vorticity magnitude, magnetic-field magnitude) show rather universal behavior in the geometry of their level sets, consistent with a two-dimensional geometry at small scales, with a crossover to a universal fractal geometry at large scales. The dimension at large scales is compatible with the theoretical prediction of about 2.7. The most surprising result of the simulations is that it appears that the ''near singularities'' are not efficiently eliminated by viscous dissipation, but rather seem to be strongest at the Kolmogorov cutoff. The effects of the singularities do not quite penetrate into the inertial range. We offer a simple analytic model to account for this behavior. We conclude that our findings may be due to the relatively small Reynolds numbers, but may also be indicative of generic behavior at larger Reynolds numbers. We offer some thoughts about the expected scaling behavior in the inertial range in light of our findings.

We present a Lagrangian calculation of the area of isothermal (or isoconcentration) surfaces in a medium of fluid turbulence. We argue that such surfaces appear fractal above some inner scale which depends on the Reynolds number. The fractal dimension is estimated theoretically. The theory is compared to experiments in which a dye is used as a passive scalar in various turbulent flows. We find excellent agreement in many details.

Combining methods and ideas of dynamical systems theory with the usual stability analysis for extended hydrodynamic systems we show that defect-mediated turbulence is a generic consequence of a set of physical properties which are shared by many systems. We show how the interplay between broken continuous symmetries and the dynamics of patterns leads to a universal scenario for the onset of this type of turbulence.

Chaotic dynamical systems can be organized around an underlying strange set, which is comprised of all the unstable periodic orbits. In this paper, we quantify the complexity of such an organization; this complexity addresses the difficulty of predicting the structure of the strange set from low-order data and is independent of the entropy and the algorithmic complexity. We refer to the new measure as the grammatical complexity. The notion is introduced, discussed, and illustrated in the context of simple dynamical systems. In addition, the grammatical complexity is generalized to include metric properties arising due to the nonuniform distribution of the invariant measure on the strange set.

The calculation of the scaling properties of multifractal sets is presented as an eigenvalue problem. The eigenvalue equation unifies the treatment of sets that can be organized on regular treesbe they complete or incomplete (pruned) trees. In particular, this approach unifies the multifractal analyses of sets at the borderline of chaos with those of chaotic sets. Phase transitions in the thermodynamic formalism of multifractals are identified as a crossing of the largest discrete eigenvalue with the continuous part of the spectrum of the relevant operator. An analysis of the eigenfunctions is presented and several examples are solved in detail. Of particular interest is the analysis of intermittent maps, which shows the existence of an infinite-order phase transitions, and of (Smale) incomplete maps of the interval with finite and infinite rules of pruning of the multifractal trees.

A dynamical theory for the structure functions in convective turbulence at high Rayleigh numbers is presented. A range of scales is identified in which buoyancy forces dominate and the scaling exponents differ significantly from isotropic turbulence. A crossover to Kolmogorov-type scaling is predicted as a function of the Rayleigh number.

Strange attractors in dynamical systems that go to chaos via quasiperiodicity are considered. It is shown that there exists an infinite number of points in parameter space where the topology of the strange attractors is universal. At such points the periodic points belonging to unstable periodic orbits can be organised on ternary trees which are pruned by local rules. The grammar is universal, and thus the topological entropy is universal at each of these points in parameter space. The complete understanding of the topology is used to calculate systematically the metric properties of the attractors. The spectrum of scaling indices f(α) is computed. It is found that there is no metric universality, although some aspects of the metric properties are universal. Experiments to test some of the predictions of this theory are suggested.

The spectrum of scaling exponents of multifractal, hyperbolic strange attractors is quantitatively understood by realizing that each unstable periodic point contributes one scaling exponent to the spectrum. The calculation of the f(±) function can thus be reduced to counting periodic orbits.

The application of standard dimension algorithms is not always successful in deciding whether a given data set represents deterministic chaotic motion. We propose to supplement such algorithms with a search of the underlying unstable periodic orbits. These appear to give important clues to the organization of strange attractors and to their scaling properties. It is shown how to extract these orbits from experimental data sets, and how to use to obtain information on dimensions, entropies and the spectrum of scaling exponents of the ergodic invariant measure. Finally they are used to obtain approximants for the invariant measure itself. These approximants can be used to calculate a variety of interesting invariants that characterize the chaotic motion.

Nonanalyticities in the generalized dimensions of multifractal sets of physical interest are interpreted as phase transitions. The problem is mapped onto thermodynamics of one-dimensional spin models. The spin Hamiltonians are explicitly constructed and their phase transitions discussed. This mapping can provide insight in both directions.

It is shown that the system of parametrically excited surface waves falls into the class of critical flows whose dynamics and transition to chaos can be understood from first-return maps derived in the vicinity of one saddle point in phase space. The onset of chaos is via gluing bifurcations, which are also common to Lorenz-like flows, but these are intermingled here with usual period-doubling bifurcations. Similar parametrically excited systems might show the full array of routes to chaos which appear in critical flows.

It is shown that in simple three-dimensional flows there are uncountably many different codimension-2 scenarios for the onset of chaos. We analyze these scenarios via one-dimensional Poincaré maps, and show how to construct renormalization groups to obtain new universal results. Comparisons with numerical studies are given.

Due to fluctuations around the metric entropy of chaotic dynamical systems there exists a spectrum of invariant dynamical scaling indices, complementary to the range of invariant static scaling indices which have been discovered and applied recently. The basic scaling behavior in both cases is shown to be rooted in the thermodynamic formalism of dynamical systems. An explicit simple example of these ideas is given, stressing their use in characterizing complex behavior.

We study an experimental orbit on a two-torus with a golden-mean winding number obtained from a forced Rayleigh-Bénard system at the onset of chaos. This experimental orbit is compared with the orbit generated by a simple theoretical model, the circle map, at its golden-mean winding number at the onset of chaos. The "spectrum of singularities" of the two orbits are compared. Within error, these are identical. Since the spectrum characterizes the metric properties of the entire orbit, this result confirms theoretical speculations that these orbits, taken as a whole, enjoy a kind of universality.