semi analytical solution of multilayer diffusion problems with time varying boundary conditions and general interface conditions we develop a new semi analytical method for solving multilayer diffusion problems with time varying external boundary conditions and general internal boundary conditions at the interfaces between adjacent layers the convergence rate of the semi analytical method relative to the number of eigenvalues is investigated and the effect of varying the interface conditions on the solution behaviour is explored numerical experiments demonstrate that solutions can be computed using the new semi analytical method that are more accurate and more efficient than the unified transform method of sheils appl math model 46 450 464 2017 furthermore unlike classical analytical solutions and the unified transform method only the new semi analytical method is able to correctly treat problems with both time varying external boundary conditions and a large number of layers the paper is concluded by replicating solutions to several important industrial environmental and biological applications previously reported in the literature demonstrating the wide applicability of the work

anomalous scaling of a passive scalar advected by the synthetic compressible flow abstract the field theoretic renormalization group and operator product expansion are applied to the problem of a passive scalar advected by a gaussian nonsolenoidal velocity field with finite correlation time in the presence of large scale anisotropy the energy spectrum of the velocity in the inertial range has the form e k k1 e and the correlation time at the wave number k scales as k 2 it is shown that depending on the values of the exponents e and the model exhibits various types of inertial range scaling regimes with nontrivial anomalous exponents explicit asymptotic expressions for the structure functions and other correlation functions are obtained they are represented by superpositions of power laws with nonuniversal amplitudes and universal independent of the anisotropy anomalous exponents calculated to the first order in e and in any space dimension these anomalous exponents are determined by the critical dimensions of tensor composite operators built of the scalar gradients and exhibit a kind of hierarchy related to the degree of anisotropy the less is the rank the less is the dimension and consequently the more important is the contribution to the inertial range behavior the leading terms of the even odd structure functions are given by the scalar vector operators the anomalous exponents depend explicitly on the degree of compressibility

correlation functions in isotropic and anisotropic turbulence the role of the symmetry group the theory of fully developed turbulence is usually considered in an idealized homogeneous and isotropic state real turbulent flows exhibit the effects of anisotropic forcing the analysis of correlation functions and structure functions in isotropic and anisotropic situations is facilitated and made rational when performed in terms of the irreducible representations of the relevant symmetry group which is the group of all rotations so 3 in this paper we first consider the needed general theory and explain why we expect different universal scaling exponents in the different sectors of the symmetry group we exemplify the theory context of isotropic turbulence for third order tensorial structure functions and in weakly anisotropic turbulence for the second order structure function the utility of the resulting expressions for the analysis of experimental data is demonstrated in the context of high reynolds number measurements of turbulence in the atmosphere

mean field approximation and a small parameter in turbulence theory numerical and physical experiments on two dimensional 2d turbulence show that the differences of transverse components of velocity field are well described by gaussian statistics and kolmogorov scaling exponents in this case the dissipation fluctuations are irrelevant in the limit of small viscosity in general one can assume the existence of a critical space dimensionality d d c at which the energy flux and all odd order moments of velocity difference change sign and the dissipation fluctuations become dynamically unimportant at dl d c the flow can be described by the mean field theory leading to the observed gaussian statistics and kolmogorov scaling of transverse velocity differences it is shown that in the vicinity of d d c the ratio of the relaxation and translation characteristic times decreases to zero thus giving rise to a small parameter of the theory the expressions for pressure and dissipation contributions to the exact equation for the generating function of transverse velocity differences are derived in the vicinity of d d c the resulting equation describes experimental data on two dimensional turbulence and demonstrates the onset of intermittency as d ensuremath d c g0 and r stackrel ensuremath rightarrow l 0 in three dimensional flows in close agreement with experimental data in addition some exact relations between correlation functions of velocity differences are derived it is also predicted that the single point probability density function of transverse velocity components in developing as well as in the large scale stabilized two dimensional turbulence is a gaussian

logarithmic violation of scaling in strongly anisotropic turbulent transfer of a passive vector field inertial range asymptotic behavior of a vector e g magnetic field passively advected by a strongly anisotropic turbulent flow is studied by means of the field theoretic renormalization group and the operator product expansion the advecting velocity field is gaussian not correlated in time with the pair correlation function of the form t t k d 1 where k k and k is the component of the wave vector perpendicular to the distinguished direction direction of the flow the d dimensional generalization of the ensemble introduced by avellaneda and majda commun math phys 131 381 1990 the stochastic advection diffusion equation for the transverse divergence free vector field includes as special cases the kinematic dynamo model for magnetohydrodynamic turbulence and the linearized navier stokes equation in contrast to the well known isotropic kraichnan s model where various correlation functions exhibit anomalous scaling behavior with infinite sets of anomalous exponents here the dependence on the integral turbulence scale l has a logarithmic behavior instead of powerlike corrections to ordinary scaling determined by naive canonical dimensions the anomalies manifest themselves as polynomials of logarithms of l the key point is that the matrices of scaling dimensions of the relevant families of composite operators appear nilpotent and cannot be diagonalized the detailed proof of this fact is given for the correlation functions of arbitrary order

stability and asymptotic behavior of a numerical solution corresponding to a diffusion reaction equation solved by a finite difference scheme crank nicolson abstract in this paper some partial differential equations arising in biochemistry are studied from the numerical point of view these nonlinear equations based on diffusion reaction processes are solved by using a crank nicolson technique and convergence is proved by using lower and upper solutions stability and asymptotic behavior are also examined a numerical algorithm has been deduced and is presently used by chemists

statistical properties of four dimensional turbulence the energy transfer and small scale intermittency in decaying turbulence in four dimensions 4d are studied by direct numerical simulation and by spectral theory in comparison with three dimensions 3d the energy transfer is more efficient in 4d than in 3d hence the exponent of energy decay is larger the kolmogorov constant is 1 31 which is smaller than 1 72 in 3d the longitudinal third order structure function is confirmed to be governed by a 1 2 law 1 2 epsilonr instead of a 4 5 law in 3d the intermittency is weaker in 4d for the total dissipation rate nu summation operator i j partial differentialu j partial differentialx i 2 and the associated velocity difference spherical velocity difference on scale r than in 3d while it is slightly stronger for the surrogated dissipation rate nu partial differentialu 1 partial differentialx 1 2 and the associated longitudinal velocity difference the scaling exponents of the spherical and longitudinal velocity differences are also evaluated indicating that the spherical velocity difference is less intermittent in 4d while the longitudinal difference is more intermittent in 4d the distribution of the eigenvalues of the strain tensor is also examined it was also found that the normalized variance of the pressure gradient epsilon 3 2 nu 1 2 in 4d is smaller than in 3d the roles of the incompressibility condition the pressure gradient and the intermittency in d dimensional turbulence are examined and the importance of the longitudinal component of turbulent velocity field in the energy transfer toward small scales are discussed burgers turbulence as the asymptote of turbulence in large dimension is suggested

self similar decay in the kraichnan model of a passive scalar we study the two point correlation function of a freely decaying scalar in kraichnan s model of advection by a gaussian random velocity field that is stationary and white noise in time but fractional brownian in space with roughness exponent 0 2 appropriate to the inertial convective range of the scalar we find all self similar solutions by transforming the scaling equation to kummer s equation it is shown that only those scaling solutions with scalar energy decay exponent a d 1 are statistically realizable where d is space dimension and 2 an infinite sequence of invariants jp p 0 1 2 is pointed out where j0 is corrsin s integral invariant but the higher invariants appear to be new we show that at least one of the invariants j0 or j1 must be nonzero possibly infinite for realizable initial data initial datum with a finite nonzero invariant the first being jp converges at long times to a scaling solution p with a d p p 0 1 the latter belongs to an exceptional series of self similar solutions with stretched exponential decay in space however the domain of attraction includes many initial data with power law decay when the initial datum has all invariants zero or infinite and also it exhibits power law decay then the solution converges at long times to a nonexceptional scaling solution with the same power law decay these results support a picture of a two scale decay with breakdown of self similarity for a range of exponents d a d 2 analogous to what has recently been found in the decay of burgers turbulence

riemann integrability versus weak continuity abstract in this paper we focus on the relation between riemann integrability and weak continuity a banach space x is said to have the weak lebesgue property if every riemann integrable function from 0 1 into x is weakly continuous almost everywhere we prove that the weak lebesgue property is stable under l 1 sums and obtain new examples of banach spaces with and without this property furthermore we characterize dunford pettis operators in terms of riemann integrability and provide a quantitative result about the size of the set of continuous nonriemann integrable functions with a locally convex topology weaker than the norm topology

instanton for random advection a path integral over trajectories of 2n fluid particles is identified with a 2nth order correlation function of a passive scalar convected by d dimensional short correlated multiscale incompressible random velocity flow strong intermittency of the scalar is described by means of an instanton calculus saddle point plus fluctuations about it in the path integral at n ensuremath gg d the anomalous scaling exponent of the 2nth scalar s structural function is found analytically

scalings of scalar structure functions in a velocity field with coherent vortical structures in planar turbulence modeled as an isotropic and homogeneous collection of two dimensional noninteracting compact vortices the structure functions s p r of a statistically stationary passive scalar field have the following scaling behavior in the limit where the peclet number pe s n r const in r l pe 1 3 for l pe 1 3 r l s p r r l pe 1 3 6 1 d for l pe 1 2 r l pe 1 3 where l is a large scale and d is the fractal codimension of the spiral scalar structures generated by the vortices d 2 3 note that l pe 1 2 is the scalar taylor microscale that stems naturally from our analytical treatment of the advection diffusion equation the essential ingredients of our theory are the locality of interscale transfer and lundgren s time average assumption a phenomenological theory explicitly based only on these two ingredients reproduces our results and a generalization of this phenomenology to spatially smooth chaotic flows yields k ln k 1 generalized power spectra for the advected scalar fields

practical stability analysis of finite difference equations by the matrix method it is shown how the various norms of the coefficient matrix of a set of finite difference equations can in many cases be employed in an easy straightforward fashion to find sufficient conditions for stability in situations involving non periodic boundary conditions and variable coefficients with the aid of the easily computed norms particularly the infinite norm one does not have to rely on the necessary condition provided by the spectral radius which has been the main basis of criticism of the matrix method in the past

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

renormalization group operator product expansion and anomalous scaling in models of turbulent advection recent progress on the anomalous scaling in models of turbulent heat and mass transport is reviewed with the emphasis on the approach based on the field theoretic renormalization group rg and operator product expansion ope in that approach the anomalous scaling is established as a consequence of the existence in the corresponding field theoretic models of an infinite number of dangerous composite fields operators with negative critical dimensions which are identified with the anomalous exponents this allows one to calculate the exponents in a systematic perturbation expansion similar to the e expansion in the theory of critical phenomena the rg and ope approach is presented in a self contained way for the example of a passive scalar field temperature concentration of an impurity etc advected by a self similar gaussian velocity ensemble with vanishing correlation time the so called kraichnan s rapid change model where the anomalous exponents are known up to order o e3 effects of anisotropy compressibility and the correlation time of the velocity field are discussed passive advection by non gaussian velocity field governed by the stochastic navier stokes equation and passively advected vector e g magnetic fields are considered

anomalous scaling in the n point functions of a passive scalar a recent analysis of the four point correlation function of the passive scalar advected by a time decorrelated random flow is extended to the n point case it is shown that all stationary state inertial range correlations are dominated by homogeneous zero modes of singular operators describing their evolution we compute analytically the zero modes governing the n point structure functions and the anomalous dimensions corresponding to them to the linear order in the scaling exponent of the two point function of the advecting velocity field the implications of these calculations for the dissipation correlations are discussed textcopyright 1996 the american physical society

scaling structure of the velocity statistics in atmospheric boundary layers 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 j5 0 2 j m j 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 j 50 1 and 2 the analysis shows the need to make a careful distinction between long lived quasi twodimensional 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 j50 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 z 2 j50 50 6860 01 z 2 j51 51 0 60 15 and z 2 j52 51 3860 10 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

structure of the three point correlation function of a passive scalar in the presence of a mean gradient the three point correlation function of a passive scalar advected by a random incompressible velocity field in the presence of a mean gradient is investigated by means of phenomenological hopf equations numerical solutions are provided in the case where the velocity is gaussian white in time and with a power law in space v stackrel ensuremath rightarrow r ensuremath v stackrel ensuremath rightarrow 0 2 ensuremath sim r 2 ensuremath ensuremath epsilon and in the model introduced by shraiman and siggia c r acad sci 321 279 1995 anomalous scaling exponents are found in both models the numerics agrees with all the available analytic perturbative results in addition the angular dependence of the correlation function is explicitly determined in the batchelor limit of random advection by a smooth velocity field the exponent is found to remain very close to 1 as found experimentally in this limit the three point correlation function is found to be very well represented away from collinearity by an explicit integral representation

anomalous scaling nonlocality and anisotropy in a model of the passively advected vector field a model of the passive vector quantity advected by the gaussian velocity field with the covariance ensuremath propto ensuremath delta t ensuremath t ensuremath mathbf x ensuremath mathbf x ensuremath ensuremath varepsilon is studied the effects of pressure and large scale anisotropy are discussed the inertial range behavior of the pair correlation function is described by an infinite family of scaling exponents which satisfy exact transcendental equations derived explicitly in d dimensions by means of the functional techniques the exponents are organized in a hierarchical order according to their degree of anisotropy with the spectrum unbounded from above and the leading minimal exponent coming from the isotropic sector this picture extends to higher order correlation functions like in the scalar model the second order structure function appears nonanomalous and is described by the simple dimensional exponent s 2 ensuremath propto r 2 ensuremath ensuremath varepsilon for the higher order structure functions s 2n ensuremath propto r n 2 ensuremath ensuremath varepsilon ensuremath delta n the anomalous scaling behavior is established as a consequence of the existence in the corresponding operator product expansions of dangerous composite operators whose negative critical dimensions determine the anomalous exponents ensuremath delta n l0 a close formal resemblance of the model with the stirred navier stokes equation reveals itself in the mixing of relevant operators and is the main motivation of the paper using the renormalization group the anomalous exponents are calculated in the o ensuremath varepsilon approximation in large d dimensions for the even structure functions up to the twelfth order

particle dispersion in a multidimensional random flow with arbitrary temporal correlations abstract we study the statistics of relative distances r t between fluid particles in a spatially smooth random flow with arbitrary temporal correlations using the space dimensionality d as a large parameter we develop an effective description of lagrangian dispersion we describe the exponential growth of relative distances r 2 t exp 2 t at different values of the ratio between the correlation and turnover times we find the stretching correlation time which determines the dependence of r 1 r 2 on the difference t1 t2 the calculation of the next cumulant of r2 shows that statistics of r2 is nearly gaussian at small times as long as d 1 and becomes log normal at large times when large d approach fails for high order moments the crossover time between the regimes is the stretching correlation time which surprisingly appears to depend on the details of the velocity statistics at t we establish the dispersion of the ln r2 in the log normal statistics

calculation of the anomalous exponents in the rapid change model of passive scalar advection to order epsilon 3 the field theoretic renormalization group and operator product expansion are applied to the model of a passive scalar advected by the gaussian velocity field with zero mean and correlation function approximately equal to delta t t k d epsilon inertial range anomalous exponents identified with the critical dimensions of various scalar and tensor composite operators constructed of the scalar gradients are calculated within the epsilon expansion to order epsilon 3 three loop approximation including the exponents in anisotropic sectors the main goal of the paper is to give the complete derivation of this third order result and to present and explain in detail the corresponding calculational techniques the character and convergence properties of the epsilon expansion are discussed the improved inverse epsilon expansion is proposed and the comparison with the existing nonperturbative results is given

exact results for conditional means of a passive scalar in certain statistically homogeneous flows for a passive scalar t r t randomly advected by a statistically homogeneous flow the probability density function pdf of its fluctuation can in general be expressed in terms of two conditional means 2 t t and t 2 t we find that in some special cases either one of the two conditional means can be obtained explicitly from the equation of motion in the cases when there is no external source and that the normalized fluctuation reaches a steady state or when a steady state results from a negative damping 2 t t t 2 t 2 t the linearity of the conditional mean in these cases follows directly from the fact that the advection equation of a passive scalar is linear on the other hand when the scalar is supported by a homogeneous white in time external source t 2 t t 2

two different rapid decorrelation in time limits for turbulent diffusion a turbulent diffusion model in which the velocity field is gaussian and rapidly decorrelating in time grdt has been widely used recently in an endeavor to understand the emergence of anomalous scaling behavior of physical fields in fluid mechanics from the underlying stochastic partial differential equations the utility of the grdt model is the fact that correlation functions of the passive scalar field solve closed partial differential equations the usual moment closure obstacle is averted we study here the sense in which the grdt model describes turbulent diffusion by a general non gaussian velocity field with nontrivial temporal structure in the limit in which the correlation time of the velocity field is taken to zero when the velocity field is rescaled in a particular manner in this rapid decorrelation limit then a limit theorem of khas minskii indeed shows that the passive scalar statistics are described asymptotically by the grdt model for a broad class of velocity field models we provide however an explicit example of a poisson blob model velocity field which has two different well defined rapid decorrelation in time limits in one the passive scalar correlation functions converge to those of the grdt model and in the other they converge to a distinct nontrivial limit in which the correlation functions do not solve closed pde s we provide both mathematical and heuristic explanations for the differences between these two limits the conclusion is that the grdt model provides a universal description of the rapid decorrelation in time limit of general non gaussian velocity field models only when the velocity field is rescaled in a particular manner during the limit process

anomalous scaling of passive scalar fields advected by the navier stokes velocity ensemble effects of strong compressibility and large scale anisotropy the field theoretic renormalization group and the operator product expansion are applied to two models of passive scalar quantities the density and the tracer fields advected by a random turbulent velocity field the latter is governed by the navier stokes equation for compressible fluid subject to external random force with the covariance t t k 4 d y where d is the dimension of space and y is an arbitrary exponent the original stochastic problems are reformulated as multiplicatively renormalizable field theoretic models the corresponding renormalization group equations possess infrared attractive fixed points it is shown that various correlation functions of the scalar field its powers and gradients demonstrate anomalous scaling behavior in the inertial convective range already for small values of y the corresponding anomalous exponents identified with scaling critical dimensions of certain composite fields operators in the quantum field terminology can be systematically calculated as series in y the practical calculation is performed in the leading one loop approximation including exponents in anisotropic contributions it should be emphasized that in contrast to gaussian ensembles with finite correlation time the model and the perturbation theory presented here are manifestly galilean covariant the validity of the one loop approximation and comparison with gaussian models are briefly discussed

anomalous scaling of a passive scalar in the presence of strong anisotropy field theoretic renormalization group and the operator product expansion are applied to a model of a passive scalar quantity straight theta t x advected by the gaussian strongly anisotropic velocity field with the covariance infinity delta t t x x epsilon inertial range anomalous scaling behavior is established and explicit asymptotic expressions for the structure functions s n r identical with are obtained they are represented by superpositions of power laws the corresponding anomalous exponents which depend explicitly on the anisotropy parameters are calculated to the first order in epsilon in any space dimension d in the limit of vanishing anisotropy the exponents are associated with tensor composite operators built of the scalar gradients and exhibit a kind of hierarchy related to the degree of anisotropy the less is the rank the less is the dimension and consequently the more important is the contribution to the inertial range behavior the leading terms of the even odd structure functions are given by the scalar vector operators for the finite anisotropy the exponents cannot be associated with individual operators which are essentially mixed in renormalization but the aforementioned hierarchy survives for all the cases studied the second order structure function s2 is studied in more detail using the renormalization group and zero mode techniques the corresponding exponents and amplitudes are calculated within the perturbation theories in epsilon 1 d and in the anisotropy parameters if the anisotropy of the velocity is strong enough the skewness factor s 3 s 3 2 2 increases going down towards the depth of the inertial range the higher order odd ratios increase even if the anisotropy is weak

statistics of fourier modes in a turbulent flow fourier series are often used to discuss the properties of a homogeneous turbulent field we investigate the statistics of fourier modes of the turbulent velocity field and of a passive scalar the statistics of individual fourier modes is known to be gaussian when the size of the system l is much larger that the integral correlation size l 0 the case where the integral size is of the order of the system size l approximately l 0 is studied by direct numerical simulations in the range 20 or approximately rlambda or approximately 80 at a given rlambda we find that the probabilities of large fluctuations become larger when the wave number increases in qualitative agreement with the notion of intermittency as the reynolds number increases however the probability density functions become closer to gaussian in sharp contrast with the behavior of velocity increments we also show that in a simple model of cascade the fourier series decomposition is not appropriate to capture intermittency effects last we discuss other issues related to our results

anomalous scaling in two and three dimensions for a passive vector field advected by a turbulent flow a model of the passive vector field advected by the uncorrelated in time gaussian velocity with power like covariance is studied by means of the renormalization group and the operator product expansion the structure functions of the admixture demonstrate essential power like dependence on the external scale in the inertial range the case of an anomalous scaling the method of finding independent tensor invariants in the cases of two and three dimensions is proposed to eliminate linear dependencies between the operators entering into the operator product expansions of the structure functions the constructed operator bases which include the powers of the dissipation operator and the enstrophy operator provide the possibility of calculating the exponents of the anomalous scaling

nonperturbative spectrum of anomalous scaling exponents in the anisotropic sectors of passively advected magnetic fields we address the scaling behavior of the covariance of the magnetic field in the three dimensional kinematic dynamo problem when the boundary conditions and or the external forcing are not isotropic the velocity field is gaussian and delta correlated in time and its structure function scales with a positive exponent xi the covariance of the magnetic field is naturally computed as a sum of contributions proportional to the irreducible representations of the so 3 symmetry group the amplitudes are non universal determined by boundary conditions the scaling exponents are universal forming a discrete strictly increasing spectrum indexed by the sectors of the symmetry group when the initial mean magnetic field is zero no dynamo effect is found irrespective of the anisotropy of the forcing the rate of isotropization with decreasing scales is fully understood from these results

anomalous scaling in the anisotropic sectors of the kraichnan model of passive scalar advection kraichnan s model of passive scalar advection in which the driving gaussian velocity field has fast temporal decorrelation is studied as a case model for understanding the anomalous scaling behavior in the anisotropic sectors of turbulent fields we show here that the solutions of the kraichnan equation for the n order correlation functions foliate into sectors that are classified by the irreducible representations of the so d symmetry group we find a discrete spectrum of universal anomalous exponents with a different exponent characterizing the scaling behavior in every sector generically the correlation functions and structure functions appear as sums over all these contributions with nonuniversal amplitudes that are determined by the anisotropic boundary conditions the isotropic sector is always characterized by the smallest exponent and therefore for sufficiently small scales local isotropy is always restored the calculation of the anomalous exponents is done in two complementary ways in the first they are obtained from the analysis of the correlation functions of gradient fields the theory of these functions involves the control of logarithmic divergences that translate into anomalous scaling with the ratio of the inner and the outer scales appearing in the final result in the second method we compute the exponents from the zero modes of the kraichnan equation for the correlation functions of the scalar field itself in this case the renormalization scale is the outer scale the two approaches lead to the same scaling exponents for the same statistical objects illuminating the relative role of the outer and inner scales as renormalization scales in addition we derive exact fusion rules which govern the small scale asymptotics of the correlation functions in all the sectors of the symmetry group and in all dimensions

sailing the deep blue sea of decaying burgers turbulence we study lagrangian trajectories and scalar transport statistics in decaying burgers turbulence we choose velocity fields solutions of the inviscid burgers equation whose probability distributions are specified by kida s statistics they are time correlated and neither time reversal invariant nor gaussian we discuss in some detail the effect of shocks on trajectories and transport equations we derive the inviscid limit of these equations using a formalism of operators localized on shocks we compute the probability distribution functions of the trajectories although they do not define markov processes as physically expected these trajectories are statistically well defined but collapse with probability one at infinite time we point out that the advected scalars enjoy inverse energy cascades we also make a few comments on the connection between our computations and persistence problems

on measures on rosenthal compacta abstract we show that if k is rosenthal compact which can be represented by functions with countably many discontinuities then every radon measure on k is countably determined we also present an alternative proof of the result stating that every radon measure on an arbitrary rosenthal compactum is of countable type our approach is based on some caliber type properties of measures parameterized by separable metrizable spaces

anomalous scaling in random shell models for passive scalars a shell model version of kraichnan s 1994 it phys rev lett bf 72 1016 passive scalar problem is introduced which is inspired from the model of jensen paladin and vulpiani 1992 it phys rev a bf 45 7214 as in the original problem the prescribed random velocity field is gaussian delta correlated in time and has a power law spectrum propto k m xi where k m is the wavenumber deterministic differential equations for second and fourth order moments are obtained and then solved numerically the second order structure function of the passive scalar has normal scaling while the fourth order structure function has anomalous scaling for xi 2 3 the anomalous scaling exponents zeta p are determined for structure functions up to p 16 by monte carlo simulations of the random shell model using a stochastic differential equation scheme validated by comparison with the results obtained for the second and fourth order structure functions

fully developed isotropic turbulence nonperturbative renormalization group formalism and fixed point solution we investigate the regime of fully developed homogeneous and isotropic turbulence of the navier stokes ns equation in the presence of a stochastic forcing using the nonperturbative functional renormalization group nprg within a simple approximation based on symmetries we obtain the fixed point solution of the nprg flow equations that corresponds to fully developed turbulence both in d 2 and 3 dimensions deviations to the dimensional scalings kolmogorov in d 3 or kraichnan batchelor in d 2 are found for the two point functions to further analyze these deviations we derive exact flow equations in the large wave number limit and show that the fixed point does not entail the usual scale invariance thereby identifying the mechanism for the emergence of intermittency within the nprg framework the purpose of this work is to provide a detailed basis for nprg studies of ns turbulence the determination of the ensuing intermittency exponents is left for future work

spatial properties of velocity structure functions in turbulent wake flows in this paper we present experimental evidence that the scaling laws for the velocity structure functions s n r v x r ensuremath v x n n 2 4 6 8 hold in various parts of the flow domain the exponents that characterize the scaling are however a function of the position in the wake that is the local strength and ubiquity of coherent structures this variation is shown to be systematic and considerably exceeds the inaccuracy involved in the determination of the exponents this is an objective indication of the influence that the organized flow structures and inhomogeneity may have on intermittency in the analysis we invoke the concept of the extended self similarity ess

on the scaling of three dimensional homogeneous and isotropic turbulence abstract in this paper we investigate the scaling properties of three dimensional isotropic and homogeneous turbulence we analyze a new form of scaling extended self similarity recently introduced in the literature we found that anomalous scaling of the velocity structure functions is clearly detectable even at a moderate and low reynolds number and it extends over a much wider range of scales with respect to the inertial range

anisotropic nonperturbative zero modes for passively advected magnetic fields an analytic assessment of the role of anisotropic corrections to the isotropic anomalous scaling exponents is given for the d dimensional kinematic magnetohydrodynamics problem in the presence of a mean magnetic field the velocity advecting the magnetic field changes very rapidly in time and scales with a positive exponent xi inertial range anisotropic contributions to the scaling exponents zeta j of second order magnetic correlations are associated with zero modes and have been calculated nonperturbatively for d 3 the limit xi 0 yields zeta j j 2 xi 2j 3 j 2 5j 4 2 4j 2 1 where j j or 2 is the order in the legendre polynomial decomposition applied to correlation functions conjectures on the fact that anisotropic components cannot change the isotropic threshold to the dynamo effect are also made

anomalous scaling of a passive vector advected by the navier stokes velocity field using the field theoretic renormalization group and the operator product expansion the model of a passive vector field a weak magnetic field in the framework of the kinematic mhd advected by the velocity field which is governed by the stochastic navier stokes equation with the gaussian random stirring force correlated in time and with the correlator proportional to k4 d 2e is investigated to the first order in e one loop approximation it is shown that the single time correlation functions of the advected vector field have anomalous scaling behavior and the corresponding exponents are calculated in the isotropic case as well as in the case with the presence of large scale anisotropy the hierarchy of the anisotropic critical dimensions is briefly discussed and the persistence of the anisotropy inside the inertial range is demonstrated on the behavior of the skewness and hyperskewness dimensionless ratios of correlation functions as functions of the reynolds number re it is shown that even though the present model of a passive vector field advected by the realistic velocity field is mathematically more complicated than on one hand the corresponding models of a passive vector field advected by synthetic gaussian velocity fields and on the other hand than the corresponding model of a passive scalar quantity advected by the velocity field driven by the stochastic navier stokes equation the final one loop approximate asymptotic scaling behavior of the single time correlation or structure functions of the advected fields of all models are defined by the same anomalous dimensions up to normalization

turbulent compressible fluid renormalization group analysis scaling regimes and anomalous scaling of advected scalar fields we study a model of fully developed turbulence of a compressible fluid based on the stochastic navier stokes equation by means of the field theoretic renormalization group in this approach scaling properties are related to the fixed points of the renormalization group equations previous analysis of this model near the real world space dimension 3 identified a scaling regime n v antonov et al theor math phys 110 305 1997 tmphah0040 577910 1007 bf02630456 the aim of the present paper is to explore the existence of additional regimes which could not be found using the direct perturbative approach of the previous work and to analyze the crossover between different regimes it seems possible to determine them near the special value of space dimension 4 in the framework of double y and expansion where y is the exponent associated with the random force and 4 d is the deviation from the space dimension 4 our calculations show that there exists an additional fixed point that governs scaling behavior turbulent advection of a passive scalar density field by this velocity ensemble is considered as well we demonstrate that various correlation functions of the scalar field exhibit anomalous scaling behavior in the inertial convective range the corresponding anomalous exponents identified as scaling dimensions of certain composite fields can be systematically calculated as a series in y and all calculations are performed in the leading one loop approximation

extended self similarity in turbulent flows we report on the existence of a hitherto undetected form of self similarity which we call extended self similarity ess ess holds at high as well as at low reynolds number and it is characterized by the same scaling exponents of the velocity differences of fully developed turbulence

anomalous scaling in anisotropic turbulence abstract we present a short review of the work conducted by our group on the subject of anomalous scaling in anisotropic turbulence the basic idea that unifies all the applications discussed here is that the equations of motion for correlation functions are always linear and invariant to rotations and therefore the solutions foliate into sectors of the symmetry group of all rotations so 3 we have considered models of passive scalar and passive vector advections by a rapidly changing turbulent velocity field kraichnan type models for which we find a discrete spectrum of universal anomalous exponents with a different exponent characterizing the scaling behavior in every sector generically the correlation functions and structure functions appear as sums over all these contributions with nonuniversal amplitudes which are determined by the anisotropic boundary conditions in addition we considered navier stokes turbulence by analyzing simulations and experiments and reached some interesting conclusions regarding the scaling exponents in the anisotropic sectors the theory presented here clarifies questions like the restoration of local isotropy upon decreasing scales we explain when the local isotropy is fully restored and when the lingering effects of the anisotropic forcing appear for arbitrarily small scales

scaling properties of the streamwise component of velocity in a turbulent boundary layer abstract experimental results concerning the statistics of the streamwise component of velocity in a turbulent boundary layer near a solid wall are presented energy spectra probability density functions and velocity structure functions are investigated as a function of the distance to the wall energy spectra and structure functions display power law scalings which depend on this distance on the other hand the relative behavior of structure functions remains constant in the logarithmic sublayer this behavior exhibits two distinct power laws at small scales relative scaling exponents are relatively close to those usually reported in isotropic and homogeneous turbulence at larger scales where shear effects are dominant relative exponents display larger intermittency corrections

finite volume schemes for multilayer diffusion this paper focusses on finite volume schemes for solving multilayer diffusion problems we develop a finite volume method that addresses a deficiency of recently proposed finite volume difference methods which consider only a limited number of interface conditions and do not carry out stability or convergence analysis our method also retains second order accuracy in space while preserving the tridiagonal matrix structure of the classical single layer discretisation stability and convergence analysis of the new finite volume method is presented for each of the three classical time discretisation methods forward euler backward euler and crank nicolson we prove that both the backward euler and crank nicolson schemes are always unconditionally stable the key contribution of the work is the presentation of a set of sufficient stability conditions for the forward euler scheme here we find that to ensure stability of the forward euler scheme it is not sufficient that the time step satisfies the classical constraint of h i 2 2 d i in each layer where d i is the diffusivity and h i is the grid spacing in the i th layer as more restrictive conditions can arise due to the interface conditions the paper concludes with some numerical examples that demonstrate application of the new finite volume method with the results presented in excellent agreement with the theoretical analysis

scaling properties of passive scalars in one dimension this paper is a set of notes about work in progress since the work is directed toward scaling it seems quite appropriate to report this in a context devoted to ben widom s scientific contributions model equations for the motion of a passive scalar in one dimension are set up and some scaling properties of their solutions are derived all these models follow the approach set up by kraichnan in which the driving velocities have gaussian correlations with scaling properties in space but zero range correlations in time the simplest and most natural versions of the model fail to satisfy an incompressibility condition in another version the model does have an incompressible flow to permit the incompressibility there are two pipes each of which is described by temperature and velocity fields which are functions of a single coordinate x and time flow from one pipe to the other transfers both mass and heat this model has two conserved quantities in the limit of zero dissipation in still another version fluid elements are interchanged in position so that there are an infinite number of conservation laws in the no dissipation limit scaling properties of the resulting two point functions are derived

multiscaling in passive scalar advection as stochastic shape dynamics the study of intermittency in fluid turbulence is hampered by the lack of understanding of its basic mechanism 1 the kraichnan model of passive scalar advection by a rapidly varying velocity 2 is an example of a simplified model which displays intermittency which is believed to be related to intermittency in turbulence a mechanism for the generation of multiscaling in the kraichnan model has been identified and multiscaling exponents have been calculated by several methods 3 4 5 6 7

invariance principle for inertial scale behavior of scalar fields in kolmogorov type turbulence abstract we prove limit theorems for small scale pair dispersion in synthetic velocity fields with power law spatial spectra and wavenumber dependent correlation times these limit theorems are related to a family of generalized richardson s laws with a limiting case corresponding to richardson s t3 and 4 3 laws we also characterize a regime of positive dissipation of passive scalars

small scale intermittency and bursting in a turbulent channel flow the statistical properties of the streamwise velocity fluctuations in a fully developed turbulent channel flow are studied experimentally by means of single hot wire measurements the intermittency features studied through the scaling of the moments of the velocity structure function computed using the extended selfsimilarity and through the probability density function of the wavelet coefficients are found to be dependent on the distance from the wall the maximum intermittency effects are observed in the region between the buffer layer and the inner part of the logarithmic region where it is known that the bursting phenomenon related to coherent structures such as low speed streaks and streamwise vortices is the dominant dynamical feature an eduction technique based on wavelet transform for identification of organized motion is developed and used to analyze the turbulent signals streamwise velocity conditional averages computed on events educed with the proposed method are reported events responsible for intermittency are found to consist of regions of high velocity gradients and are directly correlated with the observed increase of intermittency close to the wall pacs number s 47 27 nz 01 50 pa i introduction a background

anomalous scaling of a passive scalar advected by the navier stokes velocity field two loop approximation the field theoretic renormalization group and operator product expansion are applied to the model of a passive scalar quantity advected by a non gaussian velocity field with finite correlation time the velocity is governed by the navier stokes equation subject to an external random stirring force with the correlation function proportional to ensuremath delta t ensuremath t ensuremath k 4 ensuremath d ensuremath 2 ensuremath epsilon it is shown that the scalar field is intermittent already for small ensuremath epsilon its structure functions display anomalous scaling behavior and the corresponding exponents can be systematically calculated as series in ensuremath epsilon the practical calculation is accomplished to order ensuremath epsilon 2 two loop approximation including anisotropic sectors as for the well known kraichnan rapid change model the anomalous scaling results from the existence in the model of composite fields operators with negative scaling dimensions identified with the anomalous exponents thus the mechanism of the origin of anomalous scaling appears similar for the gaussian model with zero correlation time and the non gaussian model with finite correlation time it should be emphasized that in contrast to gaussian velocity ensembles with finite correlation time the model and the perturbation theory discussed here are manifestly galilean covariant the relevance of these results for real passive advection and comparison with the gaussian models and experiments are briefly discussed

anomalous scaling for passively advected magnetic fields the kraichnan model 1 for passive scalar advection is by now considered a paradigm for intermittency problems a first systematic understanding of the scalar field scaling properties has been attained the crucial property of the model is that equal time correlation functions obey closed equations of motion this stems from the small correlation time of the velocity field advecting the scalar it has been recognized that a general mechanism for intermittency and anomalous scaling is the presence of nontrivial zero modes of the closed equations satisfied by the correlation functions 2 3 4 such mechanism can be illustrated in its simplest form by considering the following linear differential equation frac d 2 y r d r 2 frac a r frac dy r dr b frac y r r 2 f frac r l 1 where r varies on the positive axis y r is the unknown function a and b are two constants and f r l varies over distances of order l and is almost constant for r l in the physical situations l is the integral scale and the mechanisms maintaining the turbulence are supposed to act at this scale as for intermittency in navier stokes turbulence we are interested in the scaling behavior in the inertial range i e for r l note that the l h s of 1 is invariant under scale transformations reflecting the scale invariance of the general solution of 1 is given by a non homogeneous and a linear combination of two homogeneous solutions the combination is fixed by the conditions that y r being a correlation function should be regular and vanish at infinity

stirring up trouble multi scale mixing measures for steady scalar sources abstract the mixing efficiency of a flow advecting a passive scalar sustained by steady sources and sinks is naturally defined in terms of the suppression of bulk scalar variance in the presence of stirring relative to the variance in the absence of stirring these variances can be weighted at various spatial scales leading to a family of multi scale mixing measures and efficiencies we derive a priori estimates on these efficiencies from the advection diffusion partial differential equation focusing on a broad class of statistically homogeneous and isotropic incompressible flows the analysis produces bounds on the mixing efficiencies in terms of the peclet number a measure of the strength of the stirring relative to molecular diffusion we show by example that the estimates are sharp for particular source sink and flow combinations in general the high peclet number behavior of the bounds scaling exponents as well as prefactors depends on the structure and smoothness properties of and length scales in the scalar source and sink distribution the fundamental model of the stirring of a monochromatic source sink combination by the random sine flow is investigated in detail via direct numerical simulation and analysis the large scale mixing efficiency follows the upper bound scaling within a logarithm at high peclet number but the intermediate and small scale efficiencies are qualitatively less than optimal the peclet number scaling exponents of the efficiencies observed in the simulations are deduced theoretically from the asymptotic solution of an internal layer problem arising in a quasi static model

renormalization group operator product expansion and anomalous scaling in a model of advected passive scalar field theoretical renormalization group methods are applied to the obukhov kraichnan model of a passive scalar advected by the gaussian velocity field with the covariance mathbf v t mathbf x mathbf v t ensuremath mathbf x ensuremath mathbf v t mathbf x mathbf v t ensuremath mathbf x ensuremath ensuremath propto ensuremath delta t ensuremath t ensuremath mathbf x ensuremath mathbf x ensuremath ensuremath varepsilon inertial range anomalous scaling for the structure functions and various pair correlators is established as a consequence of the existence in the corresponding operator product expansions of certain essential or dangerous composite operators powers of the local dissipation rate whose negative critical dimensions determine anomalous exponents the main technical result is the calculation of the anomalous exponents in the order ensuremath varepsilon 2 of the ensuremath varepsilon expansion generalization of the results obtained to the case of a slow velocity field is also presented

solution of multilayer diffusion problems via the laplace transform abstract we consider a one dimensional multilayer diffusion problem subject to nonhomogeneous boundary conditions unlike previous results that used a separation of variables technique to solve such problems with homogeneous boundary conditions here we use a laplace transform approach we reformulate the multilayer diffusion problem as a sequence of one layer diffusion problems with arbitrary time dependent functions solve a general one layer diffusion problem using the laplace transform and then use the interface conditions to determine a system of renewal type equations for the time dependent functions finally these renewal equations are solved explicitly using the laplace transform

distribution of domain sizes in the zero temperature glauber dynamics of the one dimensional potts model for the zero temperature glauber dynamics of the q state potts model we calculate the exact distribution of domain sizes by mapping the problem on an exactly soluble one species coagulation model a1a a in the long time limit this distribution is universal and from its complicated exact expression we extract its behavior in various regimes our results are tested in a simulation and compared to the predictions of a simple approximation proposed recently considering the dynamics of domain walls as a reaction diffusion model a1a a with probability q22 q21 and a1a f with probability 1 q21 we calculate the pair correlation function in the long time regime s1063 651x 96 12809 9

towards a geometrical classification of statistical conservation laws in turbulent advection abstract the paper revisits the compressible kraichnan model of turbulent advection in order to derive explicit quantitative relations between scaling exponents and lagrangian particle configuration geometry

intermittency and anomalous scaling of passive scalars in any space dimension we establish exact inequalities for the structure function scaling exponents of a passively advected scalar in both the inertial convective and viscous convective ranges these inequalities involve the scaling exponents of the velocity structure functions and in a refined form an intermittency exponent of the convective range scalar flux they are valid for three dimensional navier stokes turbulence and satisfied within errors by present experimental data the inequalities also hold for any synthetic turbulent velocity statistics with a finite correlation in time we show that for time correlation exponents of the velocity smaller than the local turnover exponent the scalar spectral exponent is strictly less than that in kraichnan s phys rev lett 72 1016 1994 phys fluids 11 945 1968 j fluid mech 64 737 1974 62 305 1974 soluble rapid change model with velocity ensuremath delta correlated in time our results include as a special case an exponent inequality derived previously by constantin and procaccia nonlinearity 7 1045 1994 but with a more direct proof the inequalities in their simplest form follow from a kolmogorov type relation for the turbulent passive scalar valid in each space dimension d our improved inequalities are based upon a rigorous version of the refined similarity hypothesis for passive scalars these are compared with the relations implied by fusion rules hypothesized for scalar gradients textcopyright 1996 the american physical society

computing the scaling exponents in fluid turbulence from first principles demonstration of multiscaling we develop a consistent closure procedure for the calculation of the scaling exponents n of the nth order correlation functions in fully developed hydro dynamic 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 n this hierarchy was discussed in detail in a recent publication by v s l vov and i procaccia the scaling exponents in this set of equations cannot be found from power counting in this paper we present in detail the lowest non trivial closure of this infinite set of equations and prove that this closure leads 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 holder inequalities on the scaling exponents select the renormalization scale as the outer scale of turbulence l we demonstrate that the solvability condition of our equations leads to non kolmogorov values of the scaling exponents n finally we show that this solutions is a first approximation in a systematic series of improving approximations for the calculation of the anomalous exponents in turbulence

normal and anomalous scaling of the fourth order correlation function of a randomly advected passive scalar for a delta correlated velocity field simultaneous correlation functions of a passive scalar satisfy closed equations we analyze the equation for the four point function to describe a solution completely one has to solve the matching problems at the scale of the source and at the diffusion scale we solve both the matching problems and thus find the dependence of the four point correlation function on the diffusion and pumping scale for large space dimensionality d it is shown that anomalous scaling appears in the first order of 1 d perturbation theory anomalous dimensions are found analytically both for the scalar field and for it s derivatives in particular for the dissipation field

anomalous scaling for a passive scalar near the batchelor limit a class of phenomenological hopf equations describing mixing of a passive scalar by random flow close to the batchelor limit i e advection by random strain and vorticity is analyzed in the batchelor limit multipoint correlators of the scalar are constructed explicitly by exploiting the sl n r symmetry of the hopf operator hopf equations close to this integrable limit are solved via singular perturbation theory based on matched asymptotic expansions the solution for the three point correlator exhibits anomalous scaling indicating persistence of the small scale anisotropy for the scalar in addition to the exponent the full configuration dependence of the correlator is obtained s1063 651x 98 08703 0 pacs number s 47 27 2i

some results and open questions on spaceability in function spaces a subset m of a topological vector space x is called lineable respectively spaceable in x if there exists an infinite dimensional linear space respectively an infinite dimensional closed linear space y subset of m boolean or 0 in this article we prove that for every infinite dimensional closed subspace x of c 0 1 the set of functions in x having infinitely many zeros in 0 1 is spaceable in x we discuss problems related to these concepts for certain subsets of some important classes of banach spaces such as c 0 1 or muntz spaces we also propose several open questions in the field and study the properties of a new concept that we call the oscillating spectrum of subspaces of c 0 1 as well as oscillating and annulling properties of subspaces of c 0 1