closure approximations for passive scalar turbulence a comparative study on an exactly solvable model with complex features some standard closure approximations used in turbulence theory are analyzed by examining systematically the predictions these approximations produce for a passive scalar advection model consisting of a shear flow with a fluctuating cross sweep this model has a general geometric structure of a jet flow with transverse disturbances which occur in a number of contexts and it encompasses a wide variety of possible spatio temporal statistical structures for the velocity field including strong long range correlations even though the eulerian and lagrangian velocity statistics are not equal and the passive scalar statistics exhibit broader than gaussian intermittency this model is nevertheless simple enough so that many passive scalar statistics can be computed exactly and compared systematically with the predictions of the closure approximations our comparative study illustrates the strength and weaknesses of the closure approximations and points out the physical phenomena that these approximations are able or not able to describe properly in particular it is shown that the direct interaction approximation dia one of the most sophisticated closure approximations available fails to reproduce adequately the statistical features of the scalar and may even lead to absdurd predictions even though the equations it produces are rather complicated and difficult to analyze two alternative closure approximations the modified dia mdia and the renormalized lagrangian approximation rla with different levels of sophistication both are simpler to use than the dia and perform better in particular it is shown that both closure approximations always reproduce exactly the second order statistics for the scalar and that the mdia is even able to capture intermittency effects

optimal bounds on the effective bulk modulus of polycrystals the classical problem of bounding the effective bulk modulus of isotropic polycrystalline aggregates built from a single anisotropic phase is studied for three dimensional composites we find that the voigt and reuss bounds are each attained by multi layered laminate geometries and hence are optimal for two dimensional composites the reuss lower bound is proved to be optimal while a new optimal upper bound is derived

mathematical models with exact renormalization for turbulent transport ii fractal interfaces non gaussian statistics and the sweeping effect this paper continues the study of a model for turbulent transport with an exact renormalization theory which has recently been proposed and developed by the authors three important topics are analyzed with complete mathematical rigor for this model 1 renormalized higher order statistics of a passively advected scalar such as the pair distance distribution and the fractal dimension of interfaces 2 the effect of non gaussian turbulent velocity statistics on renormalization theory 3 the sweeping effect of additional large scale mean velocities a special emphasis is placed on renormalization theory in the vicinity of the value of the analogue of the kolmogorov spectrum in the model in the authors earlier paper it was established that the kolmogorov value is at a phase transition boundary in the exact renormalization theory it is found here that the qualitative model despite its simplicity contains in the vicinity of the kolmogorov value a remarkable amount of the qualitative behavior of turbulent transport which has been uncovered in recent experiments and proposed in phenomenological theories in particular the richardson 4 3 law for pair dispersion and interfaces with fractal dimension defect of 2 3 occur in the model rigorously as limits when the kolmogorov spectrum is approached as a limit from one side of the phase transition boundary alternative corrections to the richardson law with the same form as those proposed heuristically in the recent literature and interfaces with fractal dimension defect 1 3 occur in the model when the kolmogorov spectrum is approached from the other side of the phase transition it is very interesting that fractal dimension defects of roughly the value either 1 3 or 2 3 for level sets and interfaces of passive scalars have been ubiquitous in recent turbulence experiments as regards non gaussian the asymptotic normality of normalized integrals b 56 corresponding to compactly supported blobs with mean zero the proof of this latter fact is done in the same way as step 2 proposition b 3 using the fact that the corresponding random processes tilde v delta s have finite domain of dependence this concludes the proof of proposition b 4

on singular singularly perturbed initial value problems singularly perturbed systems of ordinary differential equations are considered for which the reduced problem has an infinite family of solutions recent approaches to such problems are compared an

taylor dispersion in curved channels the taylor dispersion problem is to determine the long time behavior of the concentration density of a passive scalar contaminant diffusing in an incompressible channel flow given an initial injection of contaminant which is slowly varying order epsilon in the longitudinal direction taylor showed that the cross channel mean of the concentration experiences an enhanced diffusion relative to the mean flow this is due to mechanical rather than molecular events mercer and roberts have treated this problem by a formal application of infinite dimensional center manifold theory and extended the analysis to the case of a curved channel which is slowly varying in the longitudinal direction it is shown that after the introduction of appropriate multiple scales their center manifold technique can be used to carry out the taylor dispersion calculations for a channel whose cross section has large order 1 variation in the longitudinal directions also more general underlying flows can be allowed than was heretofore possible a heuristic argument is given to show the existence of some channel shape and flow field for which effective diffusion is reduced rather than enhanced a similar advection diffusion problem but with no channel boundary has been rigorously analyzed by homogenization theory and discussed in a recent paper of majda andmore mclaughlin the two problems are briefly compared less

heat equilibrium distribution in a turbulent flow we consider a shear flow of a scale invariant gaussian random velocity field that does not depend on the coordinates in the direction of the flow we investigate a heat advection coming from a gaussian random homogeneous source we discuss a relaxation at large time of a temperature distribution determined by the forced advection diffusion equation we represent the temperature correlation functions by means of the feynman kac formula jensen inequalities are applied for lower and upper bounds on the correlation functions we show that at finite time there is no velocity dependence of long range temperature correlations low momentum asymptotics in the direction of the flow but the equilibrium heat distribution has large distance correlations low momentum behavior with an index depending on the scaling index of the random flow and of the index of the random forcing if the velocity has correlations growing with the distance a turbulent flow then the large distance correlations depend in a crucial wa

on the superdiffusive behavior of passive tracer with a gaussian drift in the present article we consider a motion of a passive tracer particle whose trajectory satisfies the ito stochastic differential equation d x t v t x t dt sqrt 2 kappa d w t where w is a brownian motion v is a stationary gaussian random field with incompressible realizations independent of w and 0 we prove the superdiffusive character of the motion under certain conditions on the energy spectrum of the velocity field the result is shown both for steady time independent and time dependent and markovian velocity fields in addition we provide explicit upper and lower bounds for the hurst exponent of the trajectory all previous rigorous results concerned explicitely solvable shear flows cases

on strong anomalous diffusion superdiffusive behavior i e x2 t t2 with 1 2 is in general not completely characterized by a unique exponent we study some systems exhibiting strong anomalous diffusion i e x t q tq q where 2 1 2 and q q is not a linear function of q this feature is different from the weak superdiffusive regime i e q const 1 2 occurring in random shear flows strong anomalous diffusion can be generated by nontrivial chaotic dynamics e g lagrangian motion in 2d time dependent incompressible velocity fields 2d symplectic maps and 1d intermittent maps typically the function q q is piecewise linear this is due to two mechanisms a weak anomalous diffusion for the typical events and a ballistic transport for the rare excursions in order to have strong anomalous diffusion one needs a violation of the hypothesis of the central limit theorem this happens only in a very narrow region of the control parameters space in the presence of strong anomalous diffusion one does not have a unique exponent and therefore one has the failure of the usual scaling p x t t f x t of the probability density this implies that the effective equation at large scale and long time for p x t obeys neither the usual fick equation nor other linear equations involving temporal and or spatial fractional derivatives

more on asymptotic approximation for matrix differential equations an asymptotic approximation is obtained for solutions of a matrix differential equation with symmetric matrix coefficient analytic on the real line our contribution consists of an asymptotic approximation that is valid even for various singularities near infinity

maximal effective diffusivity for time periodic incompressible fluid flows in this paper we establish conditions for the maximal pe 2 behavior of the effective diffusivity in time periodic incompressible velocity fields for both the pe to infty and pe to 0 limits using ergodic theory these conditions can be interpreted in terms of the lagrangian time averages of the velocity we reinterpret the maximal effective diffusivity conditions in terms of a poincare map of the velocity field the connection between the pe 2 asymptotic behavior of the effective diffusivity and t 2 asymptotic dispersion of a nondiffusive tracer is established several examples are analyzed we relate the existence of accelerator modes in a flow with pe 2 effective diffusivity and show how maximal effective diffusivity can appear as a result of a time dependent perturbation of a steady cellular velocity field also three dimensional symmetric time dependent duct velocity fields are analyzed and the mechanism for an effective diffusivity with peclet number dependence other than p

superdiffusion in nearly stratified flows in classical work matheron and the marsilly showed that superdiffusive scaling of mean square displacements occurs in transport diffusion for stratified flows with steady simple shear layers and long range spatial correlations more recently the authors have calculated a formula for the non gaussian large scale long time renormalized green function for these problems here the scaling laws and renormalized green functions for diffusion in nearly stratified flows are studied in such flows the simple shear layer with long range correlations is perturbed by incompressible flows with short range correlations here it is established that these flows belong to the same universality class as the simple shear layers with a renormalized green function with a similar structure but reflecting homogenization by the transverse displacements the tools in the analysis involve a modification of homogenization theory and also rigorous diagrammatic perturbation theory

improved rigorous upper bounds for transport due to passive advection described by simple models of bounded systems the work of j a krommes and r a smith on rigorous upper bounds for the turbulent transport of a passively advected scalar is extended in two directions 1 for their reference model improved upper bounds are obtained by utilizing more sophisticated two time constraints which include the effects of cross correlations up to fourth order numerical solutions of the model stochastic differential equation are also obtained they show that the new bounds compare quite favorably with the exact results even at large reynolds and kubo numbers 2 the theory is extended to take account of afinite spatial autocorrelation lengthlc as a reasonably generic example the problem of particle transport due to statistically specified stochastic magnetic fields in a collisionless turbulent plasma is revisited a bound is obtained which reduces for smalllc to the quasilinear limit and for largelc to the strong turbulence limit and which provides a reasonable and rigorous interpolation for intermediate values oflc

large deviations for brownian motion in a random scenery we prove large deviations principles in large time for the brownian occupation time in random scenery frac 1 t int 0 t xi b s ds the random field is constant on the elements of a partition of d into unit cubes these random constants say left lbrace xi j j in mathbb z d right rbrace consist of i i d bounded variables independent of the brownian motion bs s 0 this model is a time continuous version of kesten and spitzer s random walk in random scenery we prove large deviations principles in quenched and annealed settings

macrodispersivity for transport in arbitrary nonuniform flow fields asymptotic and preasymptotic results 1 we use homogenization theory to investigate the asymptotic macrodispersion in arbitrary nonuniform velocity fields which show small scale fluctuations in the first part of the paper a multiple scale expansion analysis is performed to study transport phenomena in the asymptotic limit e 1 where e represents the ratio between typical lengths of the small and large scale in this limit the effects of small scale velocity fluctuations on the transport behavior are described by a macrodispersive term and our analysis provides an additional local equation that allows calculating the macrodispersive tensor for darcian flow fields we show that the macrodispersivity is a fourth rank tensor if dispersion diffusion can be neglected it depends only on the direction of the mean flow with respect to the principal axes of anisotropy of the medium hence the macrodispersivity represents a medium property in the second part of the paper we heuristically extend the theory to finite e effects our results differ from those obtained in the common probabilistic approach employing ensemble averages this demonstrates that standard ensemble averaging does not consistently account for finite scale effects it tends to overestimate the dispersion coefficient in the single realization

on superdiffusive behavior of a passive tracer in a random flow abstract in this note we consider a passive tracer model describing particle dispersion in a turbulent flow the trajectory of the particle is given by the solution of an ordinary differential equation x t f x t x 0 x 0 where f x is a divergence free random vector field that is spatially homogeneous and isotropic we show that trajectories of the tracer display superdiffusive behavior when the random velocity f x decorrelates at large distances but does it not rapidly but rather at some moderate rate the main tools used in the proofs are variational principles and tauberian type theorems

stieltjes integral representation and effective diffusivity bounds for turbulent transport a stieltjes integral representation for the effective diffusivity in turbulent transport is developed this formula is valid for all peclet numbers and yields a rigorous resummation of the divergent perturbation series in peclet number provided that all diagrams are computed exactly another consequence of the integral representation is convergent upper and lower bounds on effective diffusivity for all peclet numbers utilizing a prescribed finite number of terms in their perturbation series

diffusion advection in cellular flows with large peclet numbers for periodic two dimensional cellular flows in incompressible fluids we provide explicit upper and lower estimates of the effective diffusivity which have the correct scaling behavior for large peclet numbers we demonstrate that all allowed scaling laws can occur the bounds prove that there is no residual diffusion in the infinite peclet number limit for holder continuous flows answering a problem posed by kozlov

estimation of pure qubits on circles gisin and popescu gisin n and popescu s 1999 phys rev lett 83 432 have shown that more information about their direction can be obtained from a pair of anti parallel spins compared to a pair of parallel spins where the first member of the pair which we call the pointer member can point equally along any direction in the bloch sphere they argued that this was due to the difference in dimensionality spanned by these two alphabets of states here we consider similar alphabets but with the first spin restricted to a fixed small circle of the bloch sphere in this case the dimensionality spanned by the anti parallel versus parallel alphabet is now equal however the anti parallel alphabet is found to still contain more information in general we generalize this to having n parallel spins and m anti parallel spins when the pointer member is restricted to a small circle these alphabets again span spaces of equal dimension yet in general more directional information can be found for sets with smaller n m for any fixed total number of spins we find that the optimal povms for extracting directional information in these cases can always be expressed in terms of the fourier basis our results show that dimensionality alone cannot explain the greater information content in antiparallel combinations of spins compared to parallel combinations in addition we describe an locc protocol which extracts optimal directional information when the pointer member is restricted to a small circle and a pair of parallel spins are supplied

homogenization of a diffusion process in a divergence free random field on etudie le probleme d homogeneisation pour un processus de diffusion dans un champ de derive aleatoire independant de la divergence dans le cas particulier d un petit champ gaussien on derive un developpement asymptotique de la matrice de diffusion efficace en termes de sa densite spectrale

convection enhanced diffusion for random flows we analyze the effective diffusivity of a passive scalar in a two dimensional steady incompressible random flow that has mean zero and a stationary stream function we show that in the limit of small diffusivity or large peclet number with convection dominating there is substantial enhancement of the effective diffusivity our analysis is based on some new variational principles for convection diffusion problems and on some facts from continuum percolation theory some of which are widely believed to be correct but have not been proved yet we show in detail how the variational principles convert information about the geometry of the level lines of the random stream function into properties of the effective diffusivity and substantiate the result of isichenko and kalda that the effective diffusivity behaves like 3 13 when the molecular diffusivity is small assuming some percolation theoretic facts we also analyze the effective diffusivity for a special class of convective flows random cellular flows where the facts from percolation theory are well established and their use in the variational principles is more direct than for general random flows

diffusion in layered random flows polymers electrons in random potentials and spin depolarization in random fields several related models are studied in a common framework we first reconsider the model of matheron and de marsilly for anomalous tracer dispersion in a stratified porous medium in each horizontal layer the flow velocity is constant parallel to the layer and depends randomly on the vertical coordinate z this model is mapped onto ad 1 localization problem in a random potential and equivalently onto ad 1 polymer at larget theaveraged distribution of horizontal displacementsx takes the scaling form p x t z 0 at 5 4q bxt 3 4 whereq y is independent of the details of the model q y a andb are obtained exactly for a large class of models from the lifschitz tails of the localization problem we find in the regionx t3 4 i e y thatq y y exp c y 4 3 we also obtain exactly ind 1 the scaling functions for the local and total average magnetization of spins diffusing in a random magnetic field by mapping onto a polymer problem as well as the average local concentration for diffusion in the presence of random sources and sinks these mappings are then used to study higher dimensional extensions of these models

front propagation in reactive systems with anomalous diffusion abstract we study front propagation of reactive fields in systems whose diffusive behavior is anomalous both superdiffusive and subdiffusive the features of the front propagation depend not only on the scaling exponent x t 2 t 2 but also on the detailed shape of the probability distribution of the diffusive process from the analysis of different systems we have three possible behavior of front propagation the usual fisher kolmogorov like scenario i e the field has a spatial exponential tail moving with constant speed v f and thickness the field has a spatial exponential tail but v f and change in time as a power law and finally the field has a spatial power law tail and v f increases exponentially in time a linear analysis of the front tail is in quantitative agreement with the numerical simulations it is remarkable the fact that anomalous diffusion is neither necessary nor sufficient condition for the linear front propagation moreover if the probability distribution of the transport process follows the scaling relation given by the flory argument the front propagation is standard fisher kolmogorov like even in presence of super or sub diffusion

spectral analysis and computation for homogenization of advection diffusion processes in steady flows advective diffusion plays a key role in the transport of salt heat buoys and markers in geophysical flows in the dispersion of pollutants and trace gases in the atmosphere and even in the dynamics of sea ice floes influenced by winds and ocean currents the long time large scale behavior of such systems is equivalent to an enhanced diffusion process with an effective diffusivity matrix d three decades ago a stieltjes integral representation for the homogenized matrix involving a spectral measure of a self adjoint operator was developed however analytical calculations of d have been obtained for only a few simple flows and numerical computations of the effective behavior based on this spectral representation have apparently not been attempted we overcome these limitations by providing a mathematical foundation for the computation of stieltjes integral representations of d we explore two different approaches and for each approach we derive new stieltjes integral representations and rigorous bounds for the symmetric and antisymmetric parts of d involving the molecular diffusivity and a spectral measure of a self adjoint operator that depends on the characteristics of a randomly perturbed periodic flow in discrete formulations of each approach we express explicitly in terms of standard or generalized eigenvalues and eigenvectors of hermitian matrices we develop and implement an efficient numerical algorithm that combines beneficial numerical attributes of each approach we use this method to compute the effective behavior for model flows and relate spectral characteristics to flow geometry and transport properties advective diffusion plays a key role in the transport of salt heat buoys and markers in geophysical flows in the dispersion of pollutants and trace gases in the atmosphere and even in the dynamics of sea ice floes influenced by winds and ocean currents the long time large scale behavior of such systems is equivalent to an enhanced diffusion process with an effective diffusivity matrix d three decades ago a stieltjes integral representation for the homogenized matrix involving a spectral measure of a self adjoint operator was developed however analytical calculations of d have been obtained for only a few simple flows and numerical computations of the effective behavior based on this spectral representation have apparently not been attempted we overcome these limitations by providing a mathematical foundation for the computation of stieltjes integral representations of d we explore two different approaches and for each approach we derive new stieltjes integral representations and rigorou

macrodispersion in a radially diverging flow field with finite peclet numbers 1 perturbation theory approach we study the transport behavior of a tracer in a radially diverging heterogeneous velocity field making use of homogenization theory we derive effective transport equations these effective transport equations are very similar to those defined on the local scale however the local transport parameters such as local dispersion coefficients are replaced by effective dispersion coefficients for smoothly varying heterogeneous media explicit results for effective radial dispersion coefficients are derived starting with the purely advective transport behavior infinite peclet numbers we extend our calculations to transport with finite peclet numbers we find that the impact of molecular diffusion on the effective dispersivity differs from the impact of local dispersion including local dispersion leads to effective dispersivities which are constant and equivalent to the effective dispersivities found in uniform flow configurations in contrast effective dispersivities including diffusion are not constant but depend on the radial distance we compare the results found by homogenization theory with those derived by neuweiler et al this issue by standard method of moments

boundary value problem with unbounded fast oscillatory random flows abstract we use tartar s weak convergence method in conjunction with a variational principle to prove a sharp homogenization theorem for diffusion in steady random flows the flow has a stationary and square integrable stream matrix the key of our approach is introducing approximate correctors by means of a saddle point variational principle we also obtain the two term asymptotics

convection of microstructure and related problems we study how two flows that vary spatially in two widely separated scales evolve under the dynamics of euler s equations we use a multiple scale approach which we motivate by studying some simpler model problems

an integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows precise necessary and sufficient conditions on the velocity statistics for mean field behavior in advection diffusion by a steady incompressible velocity field are developed here under these conditions a rigorous stieltjes integral representation for effective diffusivity in turbulent transport is derived this representation is valid for all peclet numbers and provides a rigorous resummation of the divergent perturbation expansion in powers of the peclet number one consequence of this representation is that convergent upper and lower bounds on effective diffusivity for all peclet numbers can be obtained utilizing a prescribed finite number of terms in the perturbation series explicit rigorous examples of steady incompressible velocity fields are constructed which have effective diffusivities realizing the simplest upper or lower bounds for all peclet numbers a nonlocal variational principle for effective diffusivity is developed along with applications to advection diffusion by random arrays of vortices a new class of rigorous examples is introduced these examples have an explicit stieltjes measure for the effective diffusivity furthermore the effective diffusivity behaves likek0 pe 1 2 in the limit of large peclet numbers wherek0 is the molecular diffusivity formal analogies with the theory of composite materials are exploited systematically

diffusion approximation for the convection diffusion equation with random drift we consider the asymptotic behavior of the solutions ofscaled convection diffusion equations t u t x x t x 1 v t 2 x x u t x with the initial condition u 0 x u 0 x as the parameter 0 under the assumptions that 0 and v t x t x r d is a d dimensional stationary zero mean incompressible gaussian random field markovian and mixing in t we show that the laws of u t t 0 in an appropriate functional space converge weakly as 0 to a type measureconcentrated on a solution of a certain constant coefficient heat equation

mathematical models with exact renormalization for turbulent transport the advection diffusion of a passive scalar by incompressible velocity fields which admit a statistical description and involve a continuous range of excited spatial and or temporal scales is very important in applications ranging from fully developed turbulence to the diffusion of tracers in heterogeneous porous media a variety of renormalization theories which typically utilize partial resummation of divergent perturbation series according to various recipes have been applied to this problem in various contexts in this paper a simple model problem for the advection diffusion of a passive scalar is introduced and the complete renormalization theory is developed with full mathematical rigor explicit formulas for the anomalous time scaling in various regimes as well as the green s function for the large scale long time ensemble average are developed here formulas for the renormalized higher order statistics are also developed the simple form of the model problem is deceptive the renormalization theory for this problem exhibits a remarkable range of different renormalization phenomena as parameters in the velocity statistics are varied these phenomena include the existence of several distinct anomalous scaling regimes as the spectral parameter tilde varepsilon is varied as well as explicit regimes in tilde varepsilon where the effective equation for the ensemble average is not a simple diffusion equation but instead involves an explicit random nonlocal eddy diffusivity we use fourier analysis and the feynman kac formula as main tools in the explicit exact renormalization theory developed here

diffusion coefficients as function of kubo number in random fields particle standard diffusion d k in 2d homogeneous isotropic stationary random velocity fields is studied numerically as a function of the dimensionless kubo number k defined as the ratio between the correlation time of the velocity field and its sweeping time the three cases corresponding to the energy spectra proportional in the inertial range respectively to k 3 k 5 3 and k 2 are considered and two different power law regimes are found for large but finite k

bounds for effective parameters of heterogeneous media by analytic continuation we give a mathematical formulation of a method for obtaining bounds on effective parameters developed by d bergman and g w milton this method in contrast to others used before does not rely on a variational principle but exploits the properties of the effective parameter as an analytic function of the component parameters the method is at present restricted to two component media