Publications

Soft amorphous materials are viscoelastic solids ubiquitously found around us, from clays and cementitious pastes to emulsions and physical gels encountered in food or biomedical engineering. Under an external deformation, these materials undergo a noteworthy transition from a solid to a liquid state that reshapes the material microstructure. This yielding transition was the main theme of a workshop held from January 9 to 13, 2023 at the Lorentz Center in Leiden. The manuscript presented here offers a critical perspective on the subject, synthesizing insights from the various brainstorming sessions and informal discussions that unfolded during this week of vibrant exchange of ideas. The result of these exchanges takes the form of a series of open questions that represent outstanding experimental, numerical, and theoretical challenges to be tackled in the near future.

In recent work it was shown that elasticity theory can break down in amorphous solids subjected to nonuniform static loads. The elastic fields are screened by geometric dipoles; these stem from gradients of the quadrupole field associated with plastic responses. Here we study the dynamical responses induced by oscillatory loads. The required modification to classical elasticity is described. Exact solutions for the displacement field in circular geometry are presented, demonstrating that dipole screening results in essential departures from the expected predictions of classical elasticity theory. Numerical simulations are conducted to validate the theoretical predictions and to delineate their range of validity.

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.

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.

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.

Amorphous solids appear to react elastically to small external strains, but in contrast to ideal elastic media, plastic responses abound immediately at any value of the strain. Such plastic responses are quasilocalized in nature, with the "cheapest"one being a quadrupolar source. The existence of such plastic responses results in screened elasticity in which strains and stresses can either quantitatively or qualitatively differ from the unscreened theory, depending on the specific screening mechanism. Here we offer a theory of such screening effects by plastic quadrupoles, dipoles, and monopoles, explain their natural appearance, and point out the analogy to electrostatic screening by electric charges and dipoles. For low density of quadrupoles the effect is to normalize the elastic moduli without a qualitative change compared to pure elasticity theory; for higher density of quadrupoles the screening effects result in qualitative changes. Predictions for the spatial dependence of displacement fields caused by local sources of strains are provided and compared to numerical simulations. We find that anomalous elasticity is richer than electrostatics in having a screening mode that does not appear in the electrostatic analog.

The rupture of a polymer chain maintained at temperature T under fixed tension is prototypical to a wide array of systems failing under constant external stress and random perturbations. Past research focused on analytic and numerical studies of the mean rate of collapse of such a chain. Surprisingly, an analytic calculation of the probability distribution function (PDF) of collapse rates appears to be lacking. Since rare events of rapid collapse can be important and even catastrophic, we present here a theory of this distribution, with a stress on its tail of fast rates. We show that the tail of the PDF is a power law with a universal exponent that is theoretically determined. Extensive numerics validate the offered theory. Lessons pertaining to other problems of the same type are drawn.

Diffusion-limited aggregation (DLA) has served for 40 years as a paradigmatic example for the creation of fractal growth patterns. In spite of thousands of references, no exact result for the fractal dimension D of DLA is known. In this Letter we announce an exact result for off-lattice DLA grown on a line embedded in the plane D = 3/2. The result relies on representing DLA with iterated conformal maps, allowing one to prove self-affinity, a proper scaling limit, and a well-defined fractal dimension. Mathematical proofs of the main results are available in N. Berger, E. B. Procaccia, and A. Turner, Growth of stationary Hastings-Levitov, arXiv:2008.05792.

The extreme slowing down associated with glass formation in experiments and in simulations results in serious difficulties to prepare deeply quenched, well annealed, glassy material. Recently, methods to achieve such deep quenching were proposed, including vapor deposition on the experimental side and "swap Monte Carlo" and oscillatory shearing on the simulation side. The relation between the resulting glasses under different protocols remains unclear. Here we show that oscillatory shear and swap Monte Carlo result in thermodynamically equivalent glasses sharing the same statistical mechanics and similar mechanical responses under external strain.

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.

Plastic instabilities in amorphous materials are often studied using idealized models of binary mixtures that do not capture accurately molecular interactions and bonding present in real glasses. Here we study atomic-scale plastic instabilities in a three-dimensional molecular dynamics model of silica glass under quasistatic shear. We identify two distinct types of elementary plastic events, one is a standard quasilocalized atomic rearrangement while the second is a bond-breaking event that is absent in simplified models of fragile glass formers. Our results show that both plastic events can be predicted by a drop of the lowest nonzero eigenvalue of the Hessian matrix that vanishes at a critical strain. Remarkably, we find very high correlation between the associated eigenvectors and the nonaffine displacement fields accompanying the bond-breaking event, predicting the locus of structural failure. Both eigenvectors and nonaffine displacement fields display an Eshelby-like quadrupolar structure for both failure modes, rearrangement, and bond breaking Our results thus clarify the nature of atomic-scale plastic instabilities in silica glasses, providing useful information for the development of mesoscale models of amorphous plasticity.

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

It is known [H. G. E. Hentschel et al., Phys. Rev. E 83, 061101 (2011)PLEEE81539-375510.1103/PhysRevE.83.061101] that amorphous solids at zero temperature do not possess a nonlinear elasticity theory: besides the shear modulus, which exists, none of the higher order coefficients exist in the thermodynamic limit. Here we show that the same phenomenon persists up to temperatures comparable to that of the glass transition. The zero-temperature mechanism due to the prevalence of dangerous plastic modes of the Hessian matrix is replaced by anomalous stress fluctuations that lead to the divergence of the variances of the higher order elastic coefficients. The conclusion is that in amorphous solids elasticity can never be decoupled from plasticity: the nonlinear response is very substantially plastic.

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.

In classical turbulence the kinematic viscosity ν is involved in two phenomena. The first is the energy dissipation and the second is the mechanical momentum flux toward the wall. In superfluid turbulence the mechanism of energy dissipation is different, and it is determined by an effective viscosity which was introduced by Vinen and is denoted as ν'. In this paper we show that in superfluid turbulence the transfer of mechanical momentum to the wall is caused by the presence of a quantum vortex tangle, giving rise to another effective "momentum" viscosity that we denote as νm(T). The temperature dependence of the second effective viscosity is markedly different from Vinen's effective viscosity ν'(T). We show that the notion of vortex-tension force, playing an important role in the theory of quantum turbulence, can be understood as the gradient of the Reynolds-stress tensor, which is, in fact, determined by the second newly defined kinematic viscosity νm(T).

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.

Quasistatic strain-controlled measurements of stress versus strain curves in macroscopic amorphous solids result in a nonlinear-looking curve that ends up either in mechanical collapse or in a steady state with fluctuations around a mean stress that remains constant with increasing strain. It is therefore very tempting to fit a nonlinear expansion of the stress in powers of the strain. We argue here that at low temperatures the meaning of such an expansion needs to be reconsidered. We point out the enormous difference between quenched and annealed averages of the stress versus strain curves and propose that a useful description of the mechanical response is given by a stress (or strain) -dependent shear modulus for which a theoretical evaluation exists. The elastic response is piecewise linear rather than nonlinear.

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.

We focus on the observed reduction in shear modulus when the stress on an amorphous solid is increased beyond the initial linear region. Careful numerical quasi-static simulations reveal an intimate relation between plastic failure and the reduction in shear modulus. The attainment of the smallest value of the shear modulus is identified with spreading of the regions that underwent a plastic event. We present an elementary "two-state" model that interpolates between failed and virgin regions and provides a simple and effective characterization of the phenomenon.

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.

We present a comprehensive statistical study of free decay of the quantized vortex tangle in superfluid He4 at low and ultralow temperatures 0≤T≤1.1 K. Using high-resolution vortex filament simulations with full Biot-Savart vortex dynamics, we show that for ultralow temperatures T≲0.5 K, when the mutual friction parameters α≃α

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. (C) 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

The dramatic dynamic slowing down associated with the glass transition is considered by many to be related to the existence of a static length scale that grows when temperature decreases. Defining, identifying, and measuring such a length is a subtle problem. Recently, two proposals, based on very different insights regarding the relevant physics, were put forward. One approach is based on the point-to-set correlation technique and the other on the scale where the lowest eigenvalue of the Hessian matrix becomes sensitive to disorder. Here we present numerical evidence that the two approaches might result in the same identical length scale. This provides mutual support for their relevance and, at the same time, raises interesting theoretical questions, discussed in the conclusion.

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 employ a recently developed model that allows the study of two-dimensional brittle crack propagation under fixed grip boundary conditions. The crack development highlights the importance of voids which appear ahead of the crack as observed in experiments on the nanoscale. The appearance of these voids is responsible for roughening the crack path on small scales, in agreement with theoretical expectations. With increasing speed of propagation one observes the branching instabilities that were reported in experiments. The simulations allow understanding the phenomena by analyzing the elastic stress field that accompanies the crack dynamics.

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.

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.

This letter is motivated by some recent experiments on pan-cake-shaped nano-samples of metallic glass that indicate a decline in the measured shear modulus upon decreasing the sample radius. Similar measurements on crystalline samples of the same dimensions showed a much more modest change. In this letter we offer a theory of this phenomenon; we argue that such results are generically expected for any amorphous solid, with the main effect being related to the increased contribution of surfaces with respect to the bulk when the samples get smaller. We employ exact relations between the shear modulus and the eigenvalues of the system's Hessian matrix to explore the role of surface modes in affecting the elastic moduli.

We developed a model for the enhancement of the heat flux by spherical and elongated nanoparticles in sheared laminar flows of nanofluids. Besides the heat flux carried by the nanoparticles, the model accounts for the contribution of their rotation to the heat flux inside and outside the particles. The rotation of the nanoparticles has a twofold effect: it induces a fluid advection around the particle and it strongly influences the statistical distribution of particle orientations. These dynamical effects, which were not included in existing thermal models, are responsible for changing the thermal properties of flowing fluids as compared to quiescent fluids. The proposed model is strongly supported by extensive numerical simulations, demonstrating a potential increase of the heat flux far beyond the Maxwell-Garnett limit for the spherical nanoparticles. The road ahead, which should lead toward robust predictive models of heat flux enhancement, is discussed.

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.

By comparing the response to external strains in metallic glasses and in Lennard-Jones glasses we find a quantitative universality of the fundamental plastic instabilities in the athermal, quasistatic limit. Microscopically these two types of glasses are as different as one can imagine, the latter being determined by binary interactions, whereas the former is determined by multiple interactions due to the effect of the electron gas that cannot be disregarded. In spite of this enormous difference the plastic instability is the same saddle-node bifurcation. As a result, the statistics of stress and energy drops in the elastoplastic steady state are universal, sharing the same system-size exponents.

Glasses are liquids whose viscosity has increased so much that they cannot flow. Accordingly, there have been many attempts to define a static length-scale associated with the dramatic slowing down of supercooled liquid with decreasing temperature. Here we present a simple method to extract the desired length-scale which is highly accessible both for experiments and for numerical simulations. The fundamental new idea is that low lying vibrational frequencies come in two types, those related to elastic response and those determined by plastic instabilities. The minimal observed frequency is determined by one or the other, crossing at a typical length-scale which is growing with the approach of the glass transition. This length-scale characterizes the correlated disorder in the system: on longer length-scales the details of the disorder become irrelevant, dominated by the Debye model of elastic modes. To connect the newly defined length-scale to relaxation dynamics near the glass transition, we show that supercooled liquids in which there exist random pinning sites of density ^ρim∼1ξsd exhibit complete jamming of all dynamics. This is a direct demonstration that the proposed length scale is indeed the static length that was long sought-after.

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.

A free material surface which supports surface diffusion becomes unstable when put under external nonhydrostatic stress. Since the chemical potential on a stressed surface is larger inside an indentation, small shape fluctuations develop because material preferentially diffuses out of indentations. When the bulk of the material is purely elastic one expects this instability to run into a finite-time cusp singularity. It is shown here that this singularity is cured by plastic effects in the material, turning the singular solution to a regular crack.

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.

We develop an athermal shear-transformation-zone (STZ) theory of plastic deformation in spatially inhomogeneous, amorphous solids. Our ultimate goal is to describe the dynamics of the boundaries of voids or cracks in such systems when they are subjected to remote, time-dependent tractions. The theory is illustrated here for the case of a circular hole in an infinite two-dimensional plate, a highly symmetric situation that allows us to solve much of the problem analytically. In spite of its special symmetry, this example contains many general features of systems in which stress is concentrated near free boundaries and deforms them irreversibly. We depart from conventional treatments of such problems in two ways. First, the STZ analysis allows us to keep track of spatially heterogeneous, internal state variables such as the effective disorder temperature, which determines plastic response to subsequent loading. Second, we subject the system to stress pulses of finite duration, and therefore are able to observe elastoplastic response during both loading and unloading. We compute the final deformations and residual stresses produced by these stress pulses. Looking toward more general applications of these results, we examine the possibility of constructing a boundary-layer theory that might be useful in less symmetric situations.

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.

We address the role of the nature of material disorder in determining the roughness of cracks, which grow by damage nucleation and coalescence ahead of the crack tip. We highlight the role of quenched and annealed disorders in relation to the length scales d and c associated with the disorder and the damage nucleation, respectively. In two related models, one with quenched disorder in which dâ c, the other with annealed disorder in which dâ ¡ c, we find qualitatively different roughening properties for the resulting cracks in two dimensions. The first model results in random cracks with an asymptotic roughening exponent ζâ 0.5. The second model shows correlated roughening with ζâ 0.66. The reasons for the qualitative difference are rationalized and explained.

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 polymers in wall turbulence is bounded from above by a universal maximal drag reduction (MDR) velocity profile that is a log law, estimated experimentally by Virk as V+(y+) 11.7log y+-17. Here V+(y) and y+ are the mean streamwise velocity and the distance from the wall in "wall" units. In this Letter we propose that this MDR profile is an edge solution of the Navier-Stokes equations (with an effective viscosity profile) beyond which no turbulent solutions exist. This insight rationalizes the universality of the MDR and provides a maximum principle which allows an ab initio calculation of the parameters in this law without any viscoelastic experimental input.

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.

The drag reduction by polymers in turbulent flows was analyzed. The drag reduction, in a recent theory, was found to be consistent with the effective viscocity growing linearly with the distance from the wall. the reduction in the Reynolds stress overwhelmed the increase in the viscous drag by the self-consistent solution. By using the direct numerical simulations it was shown that a linear viscosity profile reduced the drag in agreement with the theory and in close correspondence with direct simulations of the FENE-P model at the same flow conditions.

We address the effect of polymer additives on two-dimensional turbulence, an issue that was studied recently in experiments and direct numerical simulations. We show that the same simple shell model that reproduced drag reduction in three-dimensional turbulence reproduces all the reported effects in the two-dimensional case. The simplicity of the model offers a straightforward simulation of all the major effects under consideration.

Post mortem analysis of fracture surfaces of ductile and brittle materials on the mum-mm and the nm scales, respectively, reveal self-affine cracks with anomalous scaling exponent zetaapproximate to0.8 in three dimensions and zetaapproximate to0.65 in two dimensions. Attempts to use elasticity theory to explain this result failed, yielding exponent zetaapproximate to0.5 up to logarithms. We show that when the cracks propagate via plastic void formations in front of the tip, followed by void coalescence, the void positions are positively correlated to yield exponents higher than 0.5.

We report an algorithm to generate Laplacian growth patterns using iterated conformal maps. The difficulty of growing a complete layer with local width proportional to the gradient of the Laplacian field is overcome. The resulting growth patterns are compared to those obtained by the best algorithms of direct numerical solutions. The fractal dimension of the patterns is discussed.

The Eulerian statistically preserved structures in passive scalar advection were presented. The time-dependent linear operators that govern the dynamics of Eulerian correlation functions was analyzed numerically. The ways to naturally discuss the dynamics in terms of effective compact operators that display Eulerian statistically preserved structures which determine the anomalous scaling of the correlation functions were shown.

A simple model of the effect of polymeric additives to Newtonian fluids, is introduced, with the aim of understanding in simple mathematical terms some of the prominent features associated with the phenomenon of drag reduction. The model reveals that arguments concerning the turbulent cascade process do not appear essential. Drag reduction is a phenomenon that appears on the scales of the system size, involving energy containing modes. Furthermore, the main point appears to be the space dependence of the stretching of the polymer.

A shell model for drag reduction with polymer additives in homogeneous turbulence was presented. The model was used for homogeneous viscoelastic flows, which recaptures the essential observations of the full simulations. It was shown that the simplicity of the shell model allowed to offer a transparent explanation of the main observations.

Scaling exponent of the maximum growth probability in diffusion-limited aggregation (DLA) was studied. It was found that the position of maximal probability fluctuates wildly and the fluctuations didn't appear to go down with the increase in the cluster size. The loss of conjecture meant that there was no clear connection between the spectrum of singularities and the fractal dimension of DLA.

The stabilization of hydrodynamic flows was studied by small viscosity variations. The primary and secondary instability of simple channel flows was addressed. It was reiterated that the large effect was not due to a modified dissipation but due to reduced energy intake from the mean flow to the fluctuations.

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.

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.

Experiments in quasi-two-dimensional geometry (Hele-Shaw cells) in which a fluid is injected into a viscoelastic medium (foam, clay, or associating polymers) show patterns akin to fracture in brittle materials, very different from standard Laplacian growth patterns of viscous fingering. An analytic theory is lacking since a prerequisite to describing the fracture of elastic material is the solution of the bi-Laplace rather than the Laplace equation. In this Letter we close this gap, offering a theory of bi-Laplacian growth patterns based on the method of iterated conformal maps.

We address the statistical theory of fields that are transported by a turbulent velocity field, both in forced and in unforced (decaying) experiments. With very few provisos on the transporting velocity field, correlation functions of the transported field in the forced case are dominated by statistically preserved structures. In decaying experiments we identify infinitely many statistical constants of the motion, which are obtained by projecting the decaying correlation functions on the statistically preserved functions. We exemplify these ideas and provide numerical evidence using a simple model of turbulent transport. This example is chosen for its lack of Lagrangian structure, to stress the generality of the ideas.

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.

We discuss the scaling exponents characterizing the power-law behavior of the anisotropic components of correlation functions in turbulent systems with pressure. The anisotropic components are conveniently labeled by the angular momentum index (Formula presented) of the irreducible representation of the SO(3) symmetry group. Such exponents govern the rate of decay of anisotropy with decreasing scales. It is a fundamental question whether they ever increase as (Formula presented) increases, or they are bounded from above. The equations of motion in systems with pressure contain nonlocal integrals over all space. One could argue that the requirement of convergence of these integrals bounds the exponents from above. It is shown here on the basis of a solvable model (the \u201clinear pressure model\u201d) that this is not necessarily the case. The model introduced here is of a passive vector advection by a rapidly varying velocity field. The advected vector field is divergent free and the equation contains a pressure term that maintains this condition. The zero modes of the second-order correlation function are found in all the sectors of the symmetry group. We show that the spectrum of scaling exponents can increase with (Formula presented) without bounds while preserving finite integrals. The conclusion is that contributions from higher and higher anisotropic sectors can disappear faster and faster upon decreasing the scales also in systems with pressure.

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 statistical objects characterizing turbulence in real turbulent flows differ from those of the ideal homogeneous isotropic model. They contain contributions from various two- and three-dimensional aspects, and from the superposition of inhomogeneous and anisotropic contributions. We employ the recently introduced decomposition of statistical tensor objects into irreducible representations of the SO(3) symmetry group (characterized by j and m indices, where [Formula Presented] to disentangle some of these contributions, separating the universal and the asymptotic from the specific aspects of the flow. The different j contributions transform differently under rotations, and so form a complete basis in which to represent the tensor objects under study. The experimental data are recorded with hot-wire probes placed at various heights in the atmospheric surface layer. Time series data from single probes and from pairs of probes are analyzed to compute the amplitudes and exponents of different contributions to the second order statistical objects characterized by [Formula Presented] 1, and 2. The analysis shows the need to make a careful distinction between long-lived quasi-two-dimensional turbulent motions (close to the ground) and relatively short-lived three-dimensional motions. We demonstrate that the leading scaling exponents in the three leading sectors [Formula Presented] 1, and 2) appear to be different but universal, independent of the positions of the probe, the tensorial component considered, and the large scale properties. The measured values of the scaling exponent are [Formula Presented] [Formula Presented] and [Formula Presented] We present theoretical arguments for the values of these exponents using the Clebsch representation of the Euler equations; neglecting anomalous corrections, the values obtained are 2/3, 1, and 4/3, respectively. Some enigmas and questions for the future are sketched.

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.

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 address scaling in inhomogeneous and anisotropic turbulent flows by decomposing structure functions into their irreducible representation of the SO(3) symmetry group which are designated by j, m indices. Employing simulations of channel flows with Re-lambda approximate to 70 we demonstrate that different components characterized by different j display different scaling exponents, but for a given j these remain the same at different distances ham the wall. The j = 0 exponent agrees extremely well with high Re measurements of the scaling exponents, demonstrating the vitality of the SO(3) decomposition.

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.

The onset of space-time chaos is studied on the basis of a Galilean invariant model that exhibits the essential characteristics of the phenomenon. By keeping the linear part of the model extremely simple, one has better than usual control of the classes of available stationary solutions. These stationary solutions include not only spatially periodic but also a large set of spatially chaotic solutions that can be characterized by words of a symbolic language. The main proposition of this paper is that space-time chaos in Galilean invariant models can be understood in a qualitative fashion as an orbit in the space of functions that visits words in this language in a random fashion. The appearance of topological defects and other \u201csignatures\u201d of space-time chaos are a natural consequence of this dynamics. Finally, we construct a simple demonstration of this scenario.

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 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.

All statistical models of turbulence take into account Kolmogorov's exact result known as the "4/5 law" which stems from energy conservation. This law states that the energy flux expressed as a spatial derivative of the 3rd order velocity correlator equals the rate of energy dissipation. We have found an additional exact result which stems from the conservation of helicity in turbulence without inversion symmetry. It equates the flux of helicity expressed as a second spatial derivative of the 3rd order velocity correlator with the rate of helicity dissipation. This exact result must be incorporated to all statistical theories of turbulence with helicity. After submitting this paper for publication we learned that the main result was independently found by Otto Chkhetiani in JETP Lett . 63, 808 (1996).

The anomalous scaling behavior of the nth order correlation functions [formula presented] 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-|[formula presented][formula presented]=0. Previous analysis found zero modes in perturbation theory with respect to a small parameter. We present a computer-assisted analysis of the simplest nontrivial case of n=3: we demonstrate nonperturbatively the existence of anomalous scaling, and compare the results with the perturbative predictions.

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.

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.

In this paper we address nonperturbative aspects of the analytic theory of hydrodynamic turbulence. Of paramount importance for this theory are the "fusion rules" that describe the asymptotic properties of [Formula Presented]-point correlation functions when some of the coordinates tend toward one other. We first derive here, on the basis of two fundamental assumptions, a set of fusion rules for correlations of velocity differences when all the separations are in the inertial interval. Using this set of fusion rules we consider the standard hierarchy of equations relating the [Formula Presented]-order correlations (originating from the viscous term in the Navier-Stokes equations) to [Formula Presented] order (originating from the nonlinear term) and demonstrate that for fully unfused correlations the viscous term is negligible. Consequently the hierarchic chain of equations is decoupled in the sense that the correlations of [Formula Presented] order satisfy a homogeneous equation that may exhibit anomalous scaling solutions. Using the same hierarchy of equations when some separations go to zero we derive, on the basis of the Navier-Stokes equations, a second set of fusion rules for correlations with differences in the viscous range. The latter includes gradient fields. We demonstrate that every [Formula Presented]-order correlation function of velocity differences [Formula Presented] exhibits its own crossover length [Formula Presented] to dissipative behavior as a function of, say, [Formula Presented]. This length depends on [Formula Presented] and on the remaining separations [Formula Presented] When all these separations are of the same order [Formula Presented] this length scales as [Formula Presented] with [Formula Presented], with [Formula Presented] being the scaling exponent of the [Formula Presented]-order structure function. We derive a class of exact scaling relations bridging the exponents of correlations of gradient fields to the exponents [Formula Presented] of the [Formula Presented]-order structure functions. One of these relations is the well known "bridge relation" for the scaling exponent of dissipation fluctuations [Formula Presented].

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.

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.

Fusion rules in turbulence address the asymptotic properties of many-point correlation functions when some of the coordinates are very close to each other. Here we put to the experimental test some nontrivial consequences of the fusion rules for scalar correlations in turbulence. To this aim we examine passive turbulent advection as well as convective turbulence. Adding one assumption to the fusion rules, one obtains a prediction for universal conditional statistics of gradient fields. We examine the conditional average of the scalar dissipation field [Formula Presented] for [Formula Presented] in the inertial range and find that it is linear in [Formula Presented] with a fully determined proportionality constant. The implications of these findings for the general scaling theory of scalar turbulence are discussed.

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.

A physical interpretation of a recent Navier-Stokes based theory for scaling in developed hydrodynamic turbulence is presented. It is proposed that corrections to the normal Kolmogorov scaling behavior of the nth order velocity structure functions are finite Reynolds number effects which disappear when the inertial interval exceeds 5-6 decades. These corrections originate from the correlation between the velocity differences and energy dissipation which are characterized by an anomalous (subcritical) exponent. The values of the experimentally observed scaling indices for the nth order structure functions for n between 4 and 14 are in agreement with our findings.

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.

The structures in magnetohydrodynamic (MHD) flow, flux tubes in particular, are investigated with respect to coherence in the direction of the magnetic field. A length scale, which is interpreted as the diameter of the tubes, is derived from the MHD equations. This scale implies that the tendency towards alignment of flux lines in tubes is a diffusion driven phenomenon. The dynamics of the tubes is also investigated; the major conclusion is that stronger tubes are expected to be straighter. These ideas are tested out on data from numerical simulations of turbulent MHD convection. It is also seen that alignment of flux lines increases with the strength of the tube. Possible reasons for this effect are discussed.

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

Lower bounds on the fractal dimension of level sets of advecting passive scalars in turbulent fields are derived, in the limit that the scalar diffusivity kappa goes to zero. The main result is as follows: denote the Holder exponent of the velocity field u by zeta (u), with 0

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.

Under the assumption that the Kardar-Parisi-Zhang model (KPZ) possesses scale invariant solutions, there exists an exact calculation of the dynamic scaling exponent z=3/2. The authors prove that both KPZ and the related Kuramoto-Sivashinsky model indeed possess scale invariant solutions in 1+1 dimensions which are in fact the same for both models. The proof entails an examination of the higher order diagrams in the perturbation theory in terms of the dressed Green function and the correlator. Although each higher order diagram contains logarithmic divergences, endangering the existence of the scale invariant solution, the authors show that these divergences cancel in each order. The proof uses a fluctuation-dissipation theorem (FDT), which is an exact result for KPZ in 1+1 dimensions.

The long-wavelength properties of the Kuramoto-Sivashinsky equation are studied in 2+1 dimensions using numerical and analytic techniques. It is shown that this equation is not in the universality class of the Kardar-Parisi-Zhang model. Its roughening exponents are (up to logarithmic corrections) like those of the free-field theory, with dimension 2 being the marginal dimension for roughening. Assuming that the solution has logarithmic corrections, we derive a scaling relation for the exponents of the logarithmic terms. This solution is consistent order by order with the Dyson-Wyld diagrams. We explain why previous renormalization-group treatments failed.

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 an experimental study of the dynamics and interactions of laminar plumes emitted from a localized heat source. The observations are explained by a simple model of the flow structure around a plume. Using sources and sinks in a uniform flow, we reproduce the experimental shapes and extract the scaling behavior of the size of the plume. The model describes the initial stage of the interaction between plumes.

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.

The scaling exponents of two-point correlation functions in nonisotropic convective turbulence are evaluated on the basis of the Boussinesq equations. The scaling indices differ significantly from Kolmogorovs scaling for isotropic turbulence. The physical reason for this difference is the length dependent buoyancy forcing in this case, in contrast to forcing on the largest scale only in Kolmogorovs picture. The calculation appears to be in agreement with preliminary experimental findings in convective turbulence with helium as the working fluid.

The dynamics of a single topological defect and the interaction between pairs, including the process of annihilation of defects of opposite topological charge, are studied experimentally and theoretically near the onset of weak turbulence in Williams domains of electroconvecting nematics. The existence of topological defects requires a gauge field theoretical treatment, enriching the commonly used amplitude equations. The gauge field carries the interaction which is found to be of finite range in agreement with detailed observations.

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.

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 fractal invariant measure of chaotic strange attractors can be approximated systematically by the set of unstable n-periodic orbits of increasing n. Algorithms for extracting the periodic orbits from a chaotic time series and for calculating their stabilities are presented. With this information alone, important properties like the topological entropy and the Hausdorff dimension can be calculated.

We propose to look at first-return maps into a specified region of phase space as a basis for a unified renormalization scheme for dynamical systems. The choice of the region for first return is dictated by the symbolic dynamics (e.g., kneading sequence) of the relevant trajectories. The renormalization group can be formulated on the symbolic level, but once translated to maps it yields the said renormalization scheme. We show how the well-studied examples of the onset of chaos via period doubling and quasiperiodicity fit into this approach, and argue that these problems get in fact unified. The unification leads also to a generalization that allows us to study the onset of chaos in maps that belong to larger spaces of functions than those usually considered. In these maps we discover a host of new scenarios for the onset of chaos. These scenarios are physically relevant since the maps considered are reductions of simple flows. We present a theoretical analysis of some of these new scenarios, and report universal results. Finally we show that all the available renormalization groups can be found using symbolic manipulations only.

The experiment on chaos in surface waves reported by Ciliberto and Gollub is used as a case model for an understanding of the appearance of few-dimensional chaos in systems with an infinite number of degrees of freedom. Center-manifold and normal-form theories are utilized to derive the pertinent dynamical system and to provide an approximate solution of the partial differential equations. Comparisons of theory and experiment are discussed.

The transition from a mode-locked torus to a chaotic fractal attractor via the generic saddle-node bifurcation is considered. Although superficially the transition seems to be one of the known types of intermittency, it is shown to be in fact discontinuous. All the invariants that characterize chaos show a distinct jump. It is also argued that the power spectra are not sensitive to the discontinuous nature of the transition.

We discuss apparent and real 1/f divergencies in the small-frequency limit of power spectra of dynamical systems. Apparent 1/f2 divergencies appear in type-I intermittency and whenever there is infrequent random hopping between regions with chaotic motion. Real 1/f divergencies appear in types-II and -III intermittency. The universal properties of these divergencies are discussed in the context of one- and two-dimensional maps.

A generalization of the Euclidean diffusion equation is proposed for diffusion on fractals on the basis of a scaling argument, a renormalization-group theory for the Green's function, and numerical tests. Conjectures on the applicability to natural fractals (such as porous media) are presented.

Turbulent diffusion is anomalously large and considered as universal, i.e. independent of molecular properties. A recent report in Science raised doubts on this, based on measurements of pesticide mixture dispersion in air. This touches the very basis of current understanding of turbulence. We offer an explanation why and how molecular properties affect dispersion despite universal eddy transport.