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.

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We propose the new method for finding the non-Gaussian tails of probability distribution function (PDF) for solutions of a stochastic differential equation, such as convection equation for a passive scalar, random driven Navier-Stokes equation etc. Existence of such tails is generally regarded as a manifestation of 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 -instanton. As an example, we examine the correlation functions of the passive scalar u advected by a large-scale velocity field δ-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.PACS numbers 47.10.+g, 47.27.-i, 05.40.+j

We study properties of dilute polymer solutions. The probability density function (PDF) of polymer end-to-end extensions R in turbulent flows is examined. We show that if the value of the Lyapunov exponent lambda is smaller than the inverse molecular relaxation time 1/tau then the PDF has a strong peak at the equilibrium size R0 and a power tail at R>>R0. This confirms and extends the results of J. L. Lumley [Symp. Math. 9, 315 (1972)]. There is no essential influence of polymers on the flow in this regime. At lambdatau>1 the majority of molecules is stretched to the linear size R(op)>>R0, which can be much smaller than the maximal length of the molecules due to their back reaction.