V. V. Lebedev

Normal and anomalous scaling of the fourth-order correlation function of a randomly advected passive scalar.

Advection of a passive scalar θ in d = 2 by a large-scale velocity field rapidly changing in time is considered. The Gaussian feature of the passive scalar statistics in the convective interval was discovered in [1]. Here we examine deviations from the Gaussianity: we obtain analytically the simultaneous fourth-order correlation function of θ. Explicit expressions for fourth-order objects, like 〈(θ1 − θ2) 〉 are derived.

Instantons and Intermittency

We describe the method for finding the non-Gaussian tails of the probability distribution function (PDF) for solutions of a stochastic differential equation, such as the convection equation for a passive scalar, the random driven Navier-Stokes equation, etc. The existence of such tails is generally regarded as a manifestation of the intermittency phenomenon. Our formalism is based on the WKB approximation in the functional integral for the conditional probability of large fluctuation. We argue that the main contribution to the functional integral is given by a coupled field-force configuration\char22{}the instanton. As an example, we examine the correlation functions of the passive scalar u advected by a large-scale velocity field \ensuremath{\delta} correlated in time. We find the instanton determining the tails of the generating functional, and show that it is different from the instanton that determines the probability distribution function of high powers of u. We discuss the simplest instantons for the Navier-Stokes equation. \textcopyright{} 1996 The American Physical Society.

Statistics of a Passive Scalar Advected by a Large-Scale 2D Velocity Field: Analytic Solution

Steady statistics of a passive scalar advected by a random two-dimensional flow of an incompressible fluid is described in the range of scales between the correlation length of the flow and the diffusion scale. That corresponds to the so-called Batchelor regime where the velocity is replaced by its large-scale gradient. The probability distribution of the scalar in the locally comoving reference frame is expressed via the probability distribution of the line stretching rate. The description of line stretching can be reduced to a classical problem of the product of many random matrices with a unit determinant. We have found the change of variables that allows one to map the matrix problem onto a scalar one and to thereby prove the central limit theorem for the stretching rate statistics. The proof is valid for any finite correlation time of the velocity field. Whatever the statistics of the velocity field, the statistics of the passive scalar (averaged over time locally in space) is shown to approach gaussianity with increase in the Peclet number

Intermittency of Burgers' Turbulence

Pe

Dynamics of Nearly Spherical Vesicles in an External Flow

(the pumping-to-diffusion scale ratio). The first

Spectra of turbulence in dilute polymer solutions

n<\ln Pe

Weak crystallization theory

simultaneous correlation functions are expressed via the flux of the square of the scalar and only one factor depending on the velocity field: the mean stretching rate, which can be calculated analytically in limiting cases. Non-Gaussian tails of the probability distributions at finite

Spectrum of quark-gluon plasma

Pe

Wave kinetics of random fibre lasers

are found to be exponential.

Nearly spherical vesicles in an external flow

We consider the tails of probability density function (PDF) for the velocity that satisfies Burgers equation driven by a Gaussian large-scale force. The saddle-point approximation is employed in the path integral so that the calculation of the PDF tails boils down to finding the special field-force configuration (instanton) that realizes the extremum of probability. For the PDFs of velocity and its derivatives