### Article

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

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*)∼−(*d*2*ε*)1/3.

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—the *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. © *1996 The American Physical Society.*

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 is direct again. Otherwise it is inverse; i.e., very large scales are excited. The dynamical hint for the direction of the cascade is the sign of the Lyapunov exponent for particles separation. Positive Lyapunov exponents are associated to direct cascade and Gaussianity at small scales. Negative Lyapunov exponents lead to inverse cascade, Gaussianity at large scales, and strong intermittency at small scales.

The dynamics of a two-component Davydov-Scott (DS) soliton with a small mismatch of the initial location or velocity of the high-frequency (HF) component was investigated within the framework of the Zakharov-type system of two coupled equations for the HF and low-frequency (LF) fields. In this system, the HF field is described by the linear Schrödinger equation with the potential generated by the LF component varying in time and space. The LF component in this system is described by the Korteweg-de Vries equation with a term of quadratic influence of the HF field on the LF field. The frequency of the DS soliton`s component oscillation was found analytically using the balance equation. The perturbed DS soliton was shown to be stable. The analytical results were confirmed by numerical simulations.

Radiation conditions are described for various space regions, radiation-induced effects in spacecraft materials and equipment components are considered and information on theoretical, computational, and experimental methods for studying radiation effects are presented. The peculiarities of radiation effects on nanostructures and some problems related to modeling and radiation testing of such structures are considered.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.