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## The fourth-order correlation function of a randomly advected passive scalar

JETP Letters. 1995. Vol. 61. No. 12. P. 1049-1054.

Advection of a passive scalar $\theta$ 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
\cite{95CFKLa}. Here we examine deviations from the Gaussianity: we obtain
analytically the simultaneous fourth-order correlation function of $\theta$.
Explicit expressions for fourth-order objects,
like $\langle(\theta_1-\theta_2)^4\rangle$ are derived.

Kolokolov I., Lebedev V., Chertkov M. et al., International Journal of Modern Physics B 1996 Vol. 10 No. 18-19 P. 2273-2309

The steady statistics of a passive scalar advected by a random two-dimensional flow of an incompressible fluid is described at scales less than the correlation length of the flow and larger than the diffusion scale. The probability distribution of the scalar is expressed via the probability distribution of the line stretching rate. The description of ...

Added: March 28, 2017

Kolokolov I., Chertkov M., Vergassola M., Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 1997 Vol. 56 No. 5 P. 5483-5499

Random advection of a Lagrangian tracer scalar field u (t,x) by a one-dimensional, spatially smooth and
short-correlated in time velocity field is considered. Scalar fluctuations are maintained by a source concentrated
at the integral scale L. The statistical properties of both scalar differences and the dissipation field are analytically
determined, exploiting the dynamical formulation of the model. The ...

Added: March 6, 2017

Kolokolov I., Chertkov M., Gamba A., Physics Letters A 1994 Vol. 192 No. 5-6 P. 435-443

We describe a new functional integral method for the computation of averages containing chronological exponentials of random matrices of arbitrary dimension. We apply these results to the rigorous study of the statistics of a passive scalar advected by a large-scale N-dimensional flow. In the delta-correlated case the statistics of the rate of line stretching appears to ...

Added: March 28, 2017

Kolokolov I., Lebedev V., Chertkov M. et al., Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 1995 Vol. 51 No. 6 P. 5609-5627

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. This corresponds to the so-called Batchelor regime where the velocity is replaced by its large-scale gradient. The probability distribution of the ...

Added: March 28, 2017

Kolokolov I., Chertkov M., Vergassola M., Physical Review Letters 1998 Vol. 80 P. 512

A model of scalar turbulent advection in compressible flow is analytically investigated. It is shown that, depending on the dimensionality d of space and the degree of compressibility of the smooth advecting velocity field, the cascade of the scalar is direct or inverse. If d>4 the cascade is always direct. For a small enough degree of compressibility, the cascade ...

Added: March 5, 2017

Kolokolov I., JETP Letters 2010 Vol. 92 No. 2 P. 107-109

The joint distribution function of two distances between three Lagrangian particles has been calculated in the
problem of chaotic twodimensional transport ...

Added: February 2, 2017

Kolokolov I., Nguyen Thanh T., Physics Letters A 2012 Vol. 376 P. 1836-1838

We study mixing of passive scalar by a chaotic velocity field with a relatively strong regular shear
component. We show that the tail of partition distribution function (PDF) of coarse-grained passive scalar
field differs qualitatively from the corresponding asymptotics in the case of isotropic flow statistics. ...

Added: December 28, 2016

Kolokolov I., Chertkov M., Falkovich G., Physical Review Letters 1998 Vol. 80 No. 10 P. 2121-2124

The probability density function (PDF) of passive scalar dissipation P sed is found analytically in
the limit of large Peclet and Prandtl numbers (Batchelor-Kraichnan regime) in two dimensions. The
tail of PDF is shown to be stretched exponent. ...

Added: March 5, 2017

Kolokolov I., Lebedev V., Chertkov M. et al., Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 1995 Vol. 52 No. 5 P. 4924-4941

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 ...

Added: March 27, 2017

Kolokolov I., Journal of Physics A: Mathematical and Theoretical 2017 Vol. 50 No. 155501 P. 1-12

The two-point correlation tensor of small-scale fluctuations of magnetic field B in a two-dimensional chaotic flow is studied. The analytic approach is developed in the framework of the Kraichnan–Kazantsev model. It is shown that the growth of the field fluctuations takes place in an essentially resistive regime and stops at large times in accordance with ...

Added: March 28, 2017

Kolokolov I., Lebedev V., Chertkov M., Physics of Fluids 2007 Vol. 19 P. 101703-1-101703-4

Passive scalar turbulence forced steadily is characterized by the velocity correlation scale L,
injection scale l, and diffusive scale rd. The scales are well separated if the diffusivity is small,
rdl ,L, and one normally says that effects of diffusion are confined to smaller scales, rrd.
However, if the velocity is single scale, one finds that a weak ...

Added: February 10, 2017

Kolokolov I., Gamba A., Journal of Statistical Physics 1996 Vol. 85 No. 3 P. 489-499

We present a functional integration method for the averaging of continuous productsPt ofN×N random matrices. As an application, we compute exactly the statistics of the Lyapunov spectrum ofPt. This problem is relevant to the study of the statistical properties of various disordered physical systems, and specifically to the computation of the multipoint correlators of a passive scalar ...

Added: March 8, 2017

Kolokolov I., Lebedev V., Falkovich G. et al., Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 1996 Vol. 54 No. 5 P. 4896

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 ...

Added: March 8, 2017

Kolokolov I., Annals of Physics 1994 Vol. 231 No. 2 P. 234-255

Starting from the Abrikosov-Ryzhkin formulation of the 1D random potential problem I find closed functional representations for various physical quantities. These functional integrals are calculated exactly without the use of any perturbative expansions. The expressions for the multipoint densities correlators are obtained. Then I evaluate the mean square dispersion of the size of localized wave ...

Added: March 28, 2017

Kolokolov I., Gamba A., Journal of Statistical Physics 1999 Vol. 94 No. 5/6 P. 759-777

We compute analytically the probability distribution function PP(ε) of the dissipation field ε=(∇θ)2 of a passive scalar θ advected by a d-dimensional random flow, in the limit of large Peclet and Prandtl numbers (Batchelor–Kraichnan regime). The tail of the distribution is a stretched exponential: for ε→∞, ln PP(ε)∼−(d2ε)1/3. ...

Added: March 5, 2017

Kolokolov I., Physics Letters A 2017 Vol. 381 P. 1036-1040

The correlation tensors of magnetic field in a two-dimensional chaotic flow of conducting fluid are studied. It is shown that there is a stage of resistive evolution where the field correlators grow exponentially with time. The two-and four-point field correlation tensors are computed explicitly in this stage in the framework of Batchelor–Kraichnan–Kazantsev model. They demonstrate ...

Added: February 14, 2017

Kolokolov I., Falkovich G., Chertkov M. et al., Physical Review Letters 1999 Vol. 83 No. 20 P. 4065-4068

Kinematic dynamo theory is presented here for turbulent conductive fluids. We describe how inhomogeneous magnetic fluctuations are generated below the viscous scale of turbulence where the spatial smoothness of the velocity permits a systematic analysis of the Lagrangian path dynamics. We find analytically the moments and multipoint correlation functions of the magnetic field at small ...

Added: March 5, 2017

Kolokolov I., Lebedev V., Sizov G. A., Journal of Experimental and Theoretical Physics 2011 Vol. 140 No. 2 P. 387-400

We analyze magnetic kinematic dynamo in a conducting fluid where the stationary shear flow is accompanied by relatively weak random velocity fluctuations. The diffusionless and diffusion regimes are described. The growth rates of the magnetic field moments are related to the statistical characteristics of the flow describing divergence of the Lagrangian trajectories. The magnetic field ...

Added: February 2, 2017

Kolokolov I., Lebedev V., Chertkov M. et al., Journal of Fluid Mechanics 2005 P. 251-260

We consider the dynamics of a polymer with finite extensibility placed in a chaotic flow
with large mean shear, to explain how the polymer statistics changes with Weissenberg
number, Wi, the product of the polymer relaxation time and the Lyapunov exponent
of the flow, ¯ λ. The probability distribution function (PDF) of the polymer orientation
is peaked around a ...

Added: February 12, 2017

Kolokolov I., Lebedev V., Kogan V. R., Journal of Physics A: Mathematical and Theoretical 2010 Vol. 43 P. 182001

We analyze the kinematic dynamo in a conducting fluid where the stationary shear flow is accompanied by relatively weak random velocity fluctuations. The diffusionless and diffusion regimes are described. The growth rates of the magnetic field moments are related to the statistical characteristics of the flow describing the divergence of Lagrangian trajectories. A degree of ...

Added: February 5, 2017

Окубо Ю. undefined., Journal of Physics: Conference Series 2017 Vol. 804 No. 012036 P. 1-8

We investigate the existence and the orthogonality of the generalized Jack symmetric functions which play an important role in the AGT relations. We show their orthogonality by deforming them to the generalized Macdonald symmetric functions. ...

Added: October 26, 2017

Andrew G. Semenov, Zaikin A., Physical Review B: Condensed Matter and Materials Physics 2013 Vol. 88 No. 5 P. 054505-1-054505-10

We investigate the effect of interacting quantum phase slips on persistent current and its fluctuations in ultrathin superconducting nanowires and nanorings pierced by the external magnetic flux. We derive the effective action for these systems and map the original problem onto an effective sine-Gordon theory on torus. We evaluate both the flux dependent persistent current ...

Added: February 9, 2015

Slunyaev A., Кокорина А. В., Water Waves, Springer 2019 P. 1-20

The issue of accounting of the wave breaking phenomenon in direct numerical simulations of oceanic waves is discussed. It is emphasized that this problem is crucial for the deterministic description of waves, and also for the dynamical calculation of extreme wave statistical characteristics, such as rogue wave height probability, asymmetry, etc. The conditions for accurate ...

Added: October 13, 2019

Marshakov A., Миронов А. Д., Морозов А. Ю., Journal of Geometry and Physics 2011 Vol. 61 P. 1203-1222

We present a summary of current knowledge about the AGT relations for conformal blocks with additional insertion of the simplest degenerate operator, and a special choice of the corresponding intermediate dimension, when the conformal blocks satisfy hypergeometric-type differential equations in position of the degenerate operator. A special attention is devoted to representation of conformal block ...

Added: February 28, 2013