Statistical Estimates for the Navier–Stokes Equations and the Kraichnan Theory of 2-D Fully Developed Turbulence

A mathematical formulation of the Kraichnan theory for 2-D fully developed turbulence is given in terms of ensemble averages of solutions to the Navier–Stokes equations. A simple condition is given for the enstrophy cascade to hold for wavenumbers just beyond the highest wavenumber of the force up to a fixed fraction of the dissipation wavenumber, up to a logarithmic correction. This is followed by partial rigorous support for Kraichnan's eddy breakup mechanism. A rigorous estimate for the total energy is found to be consistent with Kraichnan's theory. Finally, it is shown that under our conditions for fully developed turbulence the fractal dimension of the attractor obeys a sharper upper bound than in the general case.     

Scaling,Renormalization and Statistical Conservation Laws in the Kraichnan Model of Turbulent Advection

We present a systematic way to compute the scaling exponents of the structure functions of the Kraichnan model of turbulent advection in a series of powers of ξ, adimensional coupling constant measuring the degree of roughness of the advecting velocity field. We also investigate the relation between standard and renormalization group improved perturbation theory. The aim is to shed light on the relation between renormalization group methods and the statistical conservation laws of the Kraichnan model, also known as zero modes.     

Bifractality of the devil’s staircase appearing in the burgers equation with brownian initial velocity

It is shown that the inverse Lagrangian map for the solution of the Burgers equation (in the inviscid limit) with Brownian initial velocity presents a bifractality (phase transition) similar to that of the Devil’s staircase for the standard triadic Cantor set. Both heuristic and rigorous derivations are given. It is explained why artifacts can easily mask this phenomenon in numerical simulations.     

Towards a geometrical classification of statistical conservation laws in turbulent advection

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.     

非线性发展方程的Wronskian解及Young图证明

给出一般非线性发展方程构造Wronskian解的间接法. 根据Young图运算的性质给出了文中命题的证明, 并讨论了置换群特征标与Young图表达式系数间的关系. 关键词： 非线性发展方程 Wronskian解 Young图 特征标     

Colligative Properties of Solutions: I. Fixed Concentrations

Using the formalism of rigorous statistical mechanics, we study the phenomena of phase separation and freezing-point depression upon freezing of solutions. Specifically, we devise an Ising-based model of a solvent--solute system and show that, in the ensemble with a fixed amount of solute, a macroscopic phase separation occurs in an interval of values of the chemical potential of the solvent. The boundaries of the phase separation domain in the phase diagram are characterized and shown to asymptotically agree with the formulas used in heuristic analyses of freezing-point depression. The limit of infinitesimal concentrations is described in a subsequent paper.     

Regularized and renormalized Bethe-Salpeter equations: Some aspects of irreducibility and asymptotic completeness in renormalizable theories

Results on the links between 2-particle irreducibility and asymptotic completeness are presented in the framework of a renormalized Bethe-Salpeter formalism, introduced recently by J. Bros from an axiomatic viewpoint, for the most simple class of renormalizable theories. These results, which involve therenormalized 2-particle irreducible kernelG (i.e. from the perturbative viewpoint the sum of renormalized Feynman amplitudes of 2-particle irreducible graphs in the channel considered), complement the general quasi-equivalence previously established by Bros forregularized (non-renormalized) Bethe-Salpeter kernels. On the one hand, a formal derivation of (2-particle) asymptotic completeness from the irreducibility ofG is given. On the other hand, the links between regularized and renormalized kernels are investigated. This analysis provides in particular a converse derivation (up to some assumptions) of the 2-particle irreducibility ofG from asymptotic completeness. As a byproduct, it also provides a more explicit justification of previous heuristic derivations by K. Symanzik of integral equations betweenF and various differences of values ofG, and a simple alternative derivation of the recently proposed renormalized Bethe-Salpeter equation.     

On the Landau–Lifschitz Degrees of Freedom in 2-D Turbulence

We show that if the Kraichnan theory of fully developed turbulence holds, then the Landau–Lifschitz degrees of freedom is bounded (up to a logarithmic term) by G ^1/2, where G is the Grashof number.     

A space-time functional formalism for classical field equations

The equations derived by R.L. Levis and R.H. Kraichnan for a space-time functional for turbulence are further investigated in the general case of classical field equations. The compact form of the equations obtained allows for their different transformations and enables to discuss the effect of initial conditions on the final results.     

Height-dependence of spatio-temporal spectra of wall-bounded turbulence – LES results and model predictions

Wavenumber–frequency spectra of the streamwise velocity component obtained from large-eddy simulations (LES) are presented. Following recent work we show that the main features, a Doppler shift and a Doppler broadening of frequencies, are captured by an advection model based on the Tennekes–Kraichnan random-sweeping hypothesis with additional mean flow. In this paper, we focus on the height-dependence of the spectra within the logarithmic layer of the flow. We furthermore benchmark an analytical model spectrum that takes the predictions of the random-sweeping model as a starting point and find good agreement with the LES data. We also quantify the influence of the LES grid resolution on the wavenumber–frequency spectra.     

Mean-field theory of the three-state chiral clock model

The phase diagram of the three-state chiral clock model, which is known to exhibit commensurate and incommensurate ordered modulated structures, is investigated in the mean-field approximation. First a numerical analysis of the mean-field equations is presented. It is based in the main on the observation that these equations define a non-linear mapping in a four dimensional space. This method of analyzing the mean-field theory proves particularly useful in the determination of the pinning transition of the incommensurate structures. Next the phase diagram is investigated analytically by means of a Landau expansion modified such as to include domain walls. It is found that in the vicinity of the order-disorder transition most features of the phase diagram can be explained quantitatively by this expansion. Finally we present a systematic lowtemperature expansion of the mean-field theory, showing that the low-temperature phase diagram obtained in the mean-field approximation is different from that of the full model.Dedicated to B. Mühlschlegel on the occasion of his 60th birthday     

Impurity spin dynamics in the Kondo problem

We derive a system of coupled linear integral equations to determine the dynamical impurity spin susceptibility in the Kondo problem. It is shown that a high temperature solution of these equations is possible by an asymptotic expansion. This yields a Korringa width as one would expect from heuristic arguments.     

The structure of theW ∞ algebra

A new definition of spectral data of a monopole is given for any compact Lie or Kac-Moody group. It is shown that the spectral data determines the irreducible monopole. In the case of maximal symmetry breaking the spectral data is shown to reduce to an earlier definition in terms of algebraic curves indexed by the nodes of the Dynkin diagram of the group. The structure of solutions to Nahm's equations corresponding to the monopole is discussed.Research supported in part by NSER C grant A8361 and FCAR grant EQ3518     

Influence of helicity on scaling regimes in model of passive scalar advected by the turbulent velocity field with finite correlation time

The advection of a passive scalar quantity by incompressible helical turbulent flow has been investigated in the frame of an extended Kraichnan model. Statistical fluctuations of the velocity field are assumed to have the Gaussian distribution with zero mean and defined noise with finite time-correlation. Actual calculations have been done up to two-loop approximation in the frame of the field-theoretic renormalization group approach. It turned out that the space parity violation (helicity) of a stochastic environment does not affect anomalous scaling which is the peculiar attribute of corresponding model without helicity. However, stability of asymptotic regimes, where anomalous scaling takes place, and the effective diffusivity strongly depend on the amount of helicity.     

Convergence of perturbation series for renormalization constants in Kraichnan model with “frozen” velocity field

An instanton for the Kraichnan model with a ‘frozen’ velocity field is found. High-order asymptotics of the quantum-field perturbation expansion for the renormalization constant Z _ν are investigated. It is shown that this expansion is convergent, and the radius of convergence is calculated.     

Properties of the functional formalism and their application to the theory of classical fluids

A general formalism is derived relating any generating functional of a hierarchy of functions to some other functionals yieldingUrsell, Husimi, and similar expansions of the original hierarchy and vice versa. There are two expansions starting with an equation of the O.-Z. type. This formalism is applied to the grand partition function with an external potential which is a generating functional for the molecular distribution functions. When the external potential is induced by adding particles to the system we obtain several hierarchies of integral equations related to each other in a simple fashion. As the Kirkwood-Salsburg, Mayer-Montroll, Green equations, the P. Y., HNC and a HNC similar approximation with their extensions are special cases of these hierarchies the relations between them become transparent. At the same time the heuristic feature in the choice of functionals and independent functions in earlier derivations of some of these equations is removed.     

Lagrangian Dispersion in Gaussian Self-Similar Velocity Ensembles

We analyze the Lagrangian flow in a family of simple Gaussian scale-invariant velocity ensembles that exhibit both spatial roughness and temporal correlations. We argue that the behavior of the Lagrangian dispersion of pairs of fluid particles in such models is determined by the scale dependence of the ratio between the correlation time of velocity differences and the eddy turnover time. For a non-trivial scale dependence, the asymptotic regimes of the dispersion at small and large scales are described by the models with either rapidly decorrelating or frozen velocities. In contrast to the decorrelated case, known as the Kraichnan model and exhibiting Lagrangian flows with deterministic or stochastic trajectories, fast separating or trapped together, the frozen model is poorly understood. We examine the pair dispersion behavior in its simplest, one-dimensional version, reinforcing analytic arguments by numerical analysis. The collected information about the pair dispersion statistics in the limiting models allows to partially predict the extent of different phases of the Lagrangian flow in the model with time-correlated velocities.     

Thermodynamics of the Farey Fraction Spin Chain

We consider the Farey fraction spin chain, a one-dimensional model defined on (the matrices generating) the Farey fractions. We extend previous work on the thermodynamics of this model by introducing an external field h. From rigorous and more heuristic arguments, we determine the phase diagram and phase transition behavior of the extended model. Our results are fully consistent with scaling theory (for the case when a “marginal” field is present) despite the unusual nature of the transition for h = 0, and the presence of long-range forces.     

Reassessment of the classical closures for scalar turbulence

In deducing the consequences of the Direct Interaction Approximation, Kraichnan was sometimes led to consider the properties of special classes of nonlinear interactions in degenerate triads in which one wavevector is very small. Such interactions can be described by simplified models closely related to elementary closures for homogeneous isotropic turbulence such as the Heisenberg and Leith models. These connections can be exploited to derive considerably improved versions of the Heisenberg and Leith models that are only slightly more complicated analytically. This paper applies this approach to derive some new simplified closure models for passive scalar advection and investigates the consistency of these models with fundamental properties of scalar turbulence. Whereas some properties, such as the existence of the Kolmogorov–Obukhov range and the existence of thermal equilibrium ensembles, follow the velocity case closely, phenomena special to the scalar case arise when the diffusive and viscous effects become important at different scales of motion. These include the Batchelor and Batchelor–Howells–Townsend ranges pertaining, respectively, to high and low molecular Schmidt number. We also consider the spectrum in the diffusive range that follows the Batchelor range. We conclude that improved elementary models can be made consistent with many nontrivial properties of scalar turbulence, but that such models have unavoidable limitations.     

A formula related to irreducible diagram expansions

A formula is presented which can be regarded as the analytic basis for irreducible diagram expansions. It expresses the off-diagonal elements of the inverse of a matrix of operators by the off-diagonal elements and by diagonal elements of various inverses of the original operator. The formula can be obtained by purely analytic means without reference to statistical considerations. No infinite processes are involved if one deals with a finite matrix of operators.