### Article

## Dissipation Statistics of a Passive Scalar in a Multidimensional Smooth Flow

We compute analytically the probability distribution function P$P$(*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 P$P$(*ε*)∼−(*d*2*ε*)1/3.

We introduce a new asymptotic invariant of magnetic fields, namely, the quadratic (and polynomial) helicity. We construct a higher asymptotic invariant of a magnetic field. We also discuss various problems that can be solved by using the magnetic helicity invariant.

This classic survey considers passive scalar and vector transport processes in a random nonstationary medium, which are described by linear parabolic equations. Integration over random paths is used, along with the asymptotic behavior of the product of a large number of independent identically distributed random matrices. The most interesting effect is the appearance of concentrated structures (intermittency) of a smooth initial distribution of the transported quantity. The occurrence of intermittent distributions in the linear problem is due to the fact that the coefficients of the transport equation are stochastic. The intermittency shows itself in the rates of exponential growth of the successive moments (Lyapunov exponents) as the moment number increases. Moment equations are obtained for the scalar and vector, and are used to study temperature evolution and magnetic-field generation in a random fluid flow. These equations are differential in a medium with short time correlations and integral in the general case. The range of application of the diffusion description is analyzed. The behavior of the diffusion coefficients in the case of time reversal is examined. The properties of an individual realization of a scalar and vector are also explained, and a dynamo theorem is given on the exponential growth of the magnetic field in a random flow with renewal.

The generalized Wiedemann-Franz law for a nonisothermal quasi-neutral plasma with developedion-acoustic turbulence and Coulomb collisions has been proven. The results obtained are used to explain the anomalously low thermal conductivity in the chromosphere-corona transition region of the solar atmosphere. Model temperature distributions in the lower corona and the transition region that correspond to well-known experimental data have been determined. The results obtained are useful for explaining the abrupt change in turbulent-plasma temperature at distances smaller than the particle mean free path.

The system of equations for average velocity and Reynolds stresses are examined supposing the smallness of diffusive, relaxation and viscous processes. Such turbulent state is named ideal. It is shown that the spectrum of turbulence has the form of spectrum of absolutely black body.

Within the framework of model calculations the possibility of occurrence of the ion-acoustic oscillation instability in a plasma without current and particle fluxes, but with an anisotropic distribution function, which corresponds to heat flux is shown. The model distribution function was selected taking into account the medium conditions. The increment of ion-acoustic oscillation is investigated as functional of the distribution function parameters. The threshold condition for the anisotropic part of the distribution function, under which the build-up of ion-acoustic oscillation with the wave vector opposite to the heat flux begins is studied. The critical heat flux, which corresponds to the threshold of ion-acoustic instability, is determined. For the solar conditions, the critical heat flux proved to be close to the heat flux from the corona into the chromosphere on the boundary of the transition region. The estimations show that outside of active regions and even in active regions with weaker magnetic fields ion-acoustic turbulence can be responsible for the formation of the sharp temperature jump. The generalized Wiedemann-Franz law for a non-isothermic quasi-neutral plasma with developed ion-acoustic turbulence is discussed. This law determines the relationship between electrical and thermal conductivities in a plasma with well-developed ion-acoustic turbulence. The anomalously low thermal conductivity responsible to the formation of high temperature gradients in the zone of the temperature jump is explained. The results are used to explain some properties of stellar atmosphere transition regions.

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.