Kraichnan Turbulence via Finite Time Averages

Relations central to Kraichnan’s theory of fully developed two-dimensional turbulence are rigorously established for finite time averages. In particular, we prove that if the ratio of the averages of palinstrophy to enstrophy is large, then a large inertial range displaying an enstrophy cascade exists. Moreover, if this ratio is comparable (up to a logarithm) to the dissipation wave number (a necessary condition for turbulence), then the power law for the energy spectrum, until now derived only heuristically, is rigorously shown to provide (up to a logarithm) an upper bound for the energy spectrum. Finally we show that, deep in the dissipation range, the palinstrophy contributed by eddies smaller than a specified length decays exponentially in the corresponding wave number. The averaging times needed for these relations are bounded in terms of the generalized Grashof number, independent of the solution for which the time averages are taken. Solutions are not assumed to be on the global attractor, merely within the absorbing ball, an easily verified condition. This work was partially supported by NSF grant number DMS-0139874. This work, though completed after O.P.M. passed away, was initiated by his insistence that the physical content of our paper [FJMR] be made independent of the esoteric Hahn-Banach extension of the classical concept of a limit. An erratum to this article is available at .     

Dynamo Effect in the Kraichnan Magnetohydrodynamic Turbulence

The existence of a dynamo effect in a simplified magnetohydrodynamic model of turbulence is considered when the magnetic Prandtl number approaches zero or infinity. The magnetic field is interacting with an incompressible Kraichnan-Kazantsev model velocity field which incorporates also a viscous cutoff scale. An approximate system of equations in the different scaling ranges can be formulated and solved, so that the solution tends to the exact one when the viscous and magnetic-diffusive cutoffs approach zero. In this approximation we are able to determine analytically the conditions for the existence of a dynamo effect and give an estimate of the dynamo growth rate. Among other things we show that in the large magnetic Prandtl number case the dynamo effect is always present. Our analytical estimates are in good agreement with previous numerical studies of the Kraichnan-Kazantsev dynamo by Vincenzi (J. Stat. Phys. 106:1073–1091, 2002).     

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.     

On the Distribution of Long-Term Time Averages on Symbolic Space

The pressure was studied in a rather abstract theory as an important notion of the thermodynamic formalism. The present paper gives a more concrete account in the case of symbolic spaces, including subshifts of finite type. We relate the pressure of an interaction function to its long-term time averages through the Hausdorff and packing dimensions of the subsets on which has prescribed long-term time-average values. Functions with values in ^d are considered. For those depending only on finitely many symbols, we get complete results, unifying and completing many partial results.     

The Kraichnan–Kazantsev Dynamo

We investigate the dynamo effect generated by an incompressible, helicity-free flow drawn from the Kraichnan statistical ensemble. The quantum formalism introduced by Kazantsev [A. P. Kazantsev, Sov. Phys. JETP 26, 1031–1034 (1968)] is shown to yield the growth rate and the spatial structure of the magnetic field. Their dependences on the magnetic Reynolds number and the Prandtl number are analyzed. The growth rate is found to be controlled by the largest between the diffusive and the viscous characteristic times. The same holds for the magnetic field correlation length and the corresponding spatial scales.     

Lagrangian instanton for the Kraichnan model

We consider high-order correlation functions of the passive scalar in the Kraichnan model. Using the instanton formalism, we find the scaling exponents ζ_n of the structure functions S _n for n≫1 under the additional condition dζ₂≫1 (where d is the dimensionality of space).At n<n _c (where n _c =dζ₂/[2(2−ζ₂)]) the exponents are ζ_n=(ζ ₂/4)(2n−n ²/n _c ), while at n>n _c they are n-independent: ζ _n=ζ₂ n _c /4. We also estimate the n-dependent factors in S _n . Pis’ma Zh. éksp. Teor. Fiz. 68, No. 7, 588–593 (10 October 1998) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Probabilistic Averages of Jacobi Operators

I study the Lyapunov exponent and the integrated density of states for general Jacobi operators. The main result is that questions about these can be reduced to questions about ergodic Jacobi operators. I use this to show that for finite gap Jacobi operators, regularity implies that they are in the Cesàro–Nevai class, proving a conjecture of Barry Simon. Furthermore, I use this to study Jacobi operators with coefficients a(n) = 1 and b(n) = f(n ^ρ (mod 1)) for ρ > 0 not an integer.     

Intermittency in Turbulence: Multiplicative Random Process in Space and Time

We present a simple stochastic algorithm for generating multiplicative processes with multiscaling both in space and in time. With this algorithm we are able to reproduce a synthetic signal with the same space and time correlation as the one coming from shell models for turbulence and the one coming from a turbulent velocity field in a quasi-Lagrangian reference frame.     

Sticky Behavior of Fluid Particles in the Compressible Kraichnan Model

We consider the compressible Kraichnan model of turbulent advection with small molecular diffusivity and velocity field regularized at short scales to mimic the effects of viscosity. As noted in ref 5, removing those two regularizations in two opposite orders for intermediate values of compressibility gives Lagrangian flows with quite different properties. Removing the viscous regularization before diffusivity leads to the explosive separation of trajectories of fluid particles whereas turning the regularizations off in the opposite order results in coalescence of Lagrangian trajectories. In the present paper we re-examine the situation first addressed in ref 6 in which the Prandtl number is varied when the regularizations are removed. We show that an appropriate fine-tuning leads to a sticky behavior of trajectories which hit each other on and off spending a positive amount of time together. We examine the effect of such a trajectory behavior on the passive transport showing that it induces anomalous scaling of the stationary 2-point structure function of an advected tracer and influences the rate of condensation of tracer energy in the zero wavenumber mode.     

Finite Time Corrections in KPZ Growth Models

We consider some models in the Kardar-Parisi-Zhang universality class, namely the polynuclear growth model and the totally/partially asymmetric simple exclusion process. For these models, in the limit of large time t, universality of fluctuations has been previously obtained. In this paper we consider the convergence to the limiting distributions and determine the (non-universal) first order corrections, which turn out to be a non-random shift of order t ^−1/3 (of order 1 in microscopic units). Subtracting this deterministic correction, the convergence is then of order t ^−2/3. We also determine the strength of asymmetry in the exclusion process for which the shift is zero. Finally, we discuss to what extend the discreteness of the model has an effect on the fitting functions.     

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 J _p, p=0, 1, 2,..., is pointed out, where J ₀ is Corrsin's integral invariant but the higher invariants appear to be new. We show that at least one of the invariants J ₀ or J ₁ must be nonzero (possibly infinite) for realizable initial data. Initial datum with a finite, nonzero invariant—the first being J _p—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.     

Derivation of Maximum Entropy Principles in Two-Dimensional Turbulence via Large Deviations

The continuum limit of lattice models arising in two-dimensional turbulence is analyzed by means of the theory of large deviations. In particular, the Miller–Robert continuum model of equilibrium states in an ideal fluid and a modification of that model due to Turkington are examined in a unified framework, and the maximum entropy principles that govern these models are rigorously derived by a new method. In this method, a doubly indexed, measure-valued random process is introduced to represent the coarse-grained vorticity field. The natural large deviation principle for this process is established and is then used to derive the equilibrium conditions satisfied by the most probable macrostates in the continuum models. The physical implications of these results are discussed, and some modeling issues of importance to the theory of long-lived, large-scale coherent vortices in turbulent flows are clarified.     

Kraichnan Flow in a Square: An Example of Integrable Chaos

The Kraichnan flow provides an example of a random dynamical system accessible to an exact analysis. We study the evolution of the infinitesimal separation between two Lagrangian trajectories of the flow. Its long-time asymptotics is reflected in the large deviation regime of the statistics of stretching exponents. Whereas in the flow that is isotropic at small scales the distribution of such multiplicative large deviations is Gaussian, this does not have to be the case in the presence of an anisotropy. We analyze in detail the flow in a two-dimensional periodic square where the anisotropy generally persists at small scales. The calculation of the large deviation rate function of the stretching exponents reduces in this case to the study of the ground state energy of an integrable periodic Schrödinger operator of the Lamé type. The underlying integrability permits to explicitly exhibit the non-Gaussianity of the multiplicative large deviations and to analyze the time-scales at which the large deviation regime sets in. In particular, we indicate how the divergence of some of those time scales when the two Lyapunov exponents become close allows a discontinuity of the large deviation rate function in the parameters of the flow. The analysis of the two-dimensional anisotropic flow permits to identify the general scenario for the appearance of multiplicative large deviations together with the restrictions on its applicability.     

Mixing Time Bounds via Bottleneck Sequences

We provide new upper bounds for mixing times of general finite Markov chains. We use these bounds to show that the total variation mixing time is robust under rough isometry for bounded degree graphs that are roughly isometric to trees.     

有限外尺度对大气湍流统计特征测量的影响

有限外尺度影响下的相位结构函数和孔径平均的斜率相关函数的表达式,结果表明有限的大气外尺度对大气流流统计特征的测量有很大的影响,尤其对大尺寸的望远镜和子孔径更是如此。基于科尔莫戈罗夫模型的大气相位结构函数和相干长度仅仅是本文推导结果的近似。对结果的分析表明,测量得到的对大气湍流科尔莫戈罗夫模型的偏离有可能是大气外尺度的影响,而不完全是真正的偏离。     

Acquisition Time for Ground-to-Satellite Optical Communication System in Weak Atmospheric Turbulence with Spatial Diversity

Abstract

In this article, various issues involved in a ground-to-satellite optical communication link (i.e., acquisition time, uncertainty area, and channel noise) are discussed. Acquisition time of a free-space optical link is evaluated for coherent (sub-carrier BPSK and QPSK) and non-coherent (OOK and -PPM) modulation schemes over weak turbulent channel with transmit diversity. In the analysis, both uncorrelated and correlated beams are considered. It is seen that an increase in transmit diversity order helps to improve the acquisition time, irrespective of turbulence strength in the atmosphere. At high correlation, a marginal change in acquisition time is observed with the increase in diversity order.     

Asymptotic Behaviour of Extremal Averages of Laplacian Eigenvalues

We study the convergence of extrema of averages of eigenvalues of the Dirichlet Laplacian on domains in \(\mathbb {R}^{n}\) under both measure and surface measure restrictions. In the former case we prove that the sequence of averages to the power n / 2 is sub-additive and determine the first term in its asymptotics in the high-frequency limit. In the latter case, we show that the sequence of minimisers converges to the ball as the frequency goes to infinity. Similar results hold for Neumann boundary conditions.     

Turbulence noise

We show that the large-eddy motions in turbulent fluid flow obey a modified hydrodynamic equation with a stochastic turbulent stress whose distribution is a causal functional of the large-scale velocity field itself. We do so by means of an exact procedure of statistical filtering of the Navier-Stokes equations, which formally solves the closure problem, and we discuss the relation of our analysis with the decimation theory of Kraichnan. We show that the statistical filtering procedure can be formulated using field-theoretic path-integral methods within the Martin-Siggia-Rose (MSR) formalism for classical statistical dynamics. We also establish within the MSR formalism a least-effective-action principle for mean turbulent velocity profiles, which generalizes Onsager's principle of least dissipation. This minimum principle is a consequence of a simple realizability inequality and therefore holds also in any realizable closure. Symanzik's theorem in field theory—which characterizes the static effective action as the minimum expected value of the quantum Hamiltonian over all state vectors with prescribed expectations of fields—is extended to MSR theory with non-Hermitian Hamiltonian. This allows stationary mean velocity profiles and other turbulence statistics to be calculated variationally by a Rayleigh-Ritz procedure. Finally, we develop approximations of the exact Langevin equations for large eddies, e.g., a random-coupling DIA model, which yield new stochastic LES models. These are compared with stochastic subgrid modeling schemes proposed by Rose, Chasnov, Leith, and others, and various applications are discussed.