Band 25, Heft 5

A joint three-dimensional numerical model of wet convection forming convective clouds in the atmosphere is developed for systems with a two-phase dispersive medium. Based on kinetic equations of condensation/coagulation, a numerical model of the formation of water drops and ice particles in mixed clouds is developed. Actual rates of condensation and coagulation growth are used, which take into account the processes of nucleation and spontaneous freezing of drops. A series of numerical experiments was performed for determination of the effect of input parameters on the formation intensity of the size spectrum for cloud drops and ice particles. The results of numerical experiments are analyzed.

The reproduction of the current state of the Earth climate system with the help of a mathematical model is considered. The model has been developed at the Institute of Numerical Mathematics of the Russian Academy of Sciences (INM RAS). This model describes the following processes: general circulation of the atmosphere, general circulation of the ocean including the sea ice evolution, the transfer and chemical transformations of several tens of small gaseous components, and the carbonic and methane cycles.

Rayleigh–Bénard thermal convection in a doubly periodic channel with rigid walls is studied using the LES-model as an analogue of multiscale atmospheric turbulence from the viewpoint of reproduction of spectral properties. A large ratio of the horizontal size to the vertical one ensures the existence of quasi-two-dimensional large-scale flow components, and a computational grid of several tens of millions nodes allows us to reproduce explicitly the dynamics of the small-scale three-dimensional turbulent component. The decomposition of the turbulent flow into the barotropic and baroclinic components allows us to propose the transformation scheme for kinetic energy in the studied system, which explains some spectral properties of the observed atmospheric turbulence.

The paper discusses two constructive approaches to the solution of problems of polynomial approximation in the uniform (Chebyshev) norm and also attainable estimates of the solution norms for initial value problems for Hermitian systems of differential and algebraic equations.

The results of research directed to the construction of a model for four-dimensional analysis of the ocean with a high spatial resolution are presented in the paper. These are, first, the development of the dynamic core of a hydrostatic ocean model; second, the study of some important physical phenomena in the Arctic and approaches to their simulation; third, the results of application of the dynamic-stochastic data assimilation method in the model of the tropical region of the Atlantic Ocean.

The relations between the Euclidean algorithm, continuous fractions, and the method of leaps for Hankel matrices are studied. It is established that the denominator degrees of the convergents of continued fractions in the expansion of a half-infinite series into a continued fraction coincide with the orders of nonsingular leading submatrices in the half-infinite Hankel matrix associated with this series.