Volume 33, Issue 3

The mathematical model of the Baltic Sea dynamics developed at the Institute of Numerical Mathematics of RAS is considered. The problem of variational assimilation of average daily data for the sea surface temperature (SST) is formulated and studied with the use of covariance matrices of observation data errors. Based on variational assimilation of satellite observation data, we propose an algorithm for solving the inverse problem of the heat flux reconstruction on the sea surface. The results of numerical experiments on reconstruction of the heat flux function are presented for the problem of variational assimilation of observation SST data.

The radiation transfer through random media of three different types was simulated numerically and statistically with the same one-dimensional distributions and correlation radii. The averaged probabilities of passages of quanta and their angular distributions practically coincide, although the calculations of correlation radii and visualizations of the corresponding brightness fields give slightly distinct results. In the calculations we used the methods of ‘double randomization’ and ‘delta-scattering’ and also statistical nuclear estimates.

A new model for calculation of circulation in shallow water basins is created based on the shallow water equations taking into account the Coriolis force and quadratic friction on the bottom. Wind effects are taken into account as forcing. The main feature of the model is a new numerical method based on regularized shallow water equations allowing one to construct the simple and sufficiently accurate numerical algorithms possessing a number of advantages over existing methods. The paper provides a detailed description of all construction steps of the model. The developed model was implemented for the water area of the Azov Sea. The paper presents the modelling of extreme surges in March 2013 and September 2014, the results of calculations are compared with observation data of hydrometeorological stations in Taganrog and Yeysk.

The paper presents a Monte Carlo algorithm for the study of bidirectional angular characteristics of a scattered polarized radiation based on projection expansion of the density of the corresponding angular distribution over hemispherical harmonics. The results of numerical estimation of two-dimensional angular distributions of the intensity and the polarization degree of the radiation passed through and reflected from optically thick layers of scattering and absorbing substance are presented.

The barotropic quasi-gasdynamic system of equations in polar coordinates is treated. It can be considered a kinetically motivated parabolic regularization of the compressible Navier–Stokes system involving additional 2nd order terms with a regularizing parameter τ > 0. A potential body force is taken into account. The energy equality is proved ensuring that the total energy is non-increasing in time. This is the crucial physical property. The main result is the construction of symmetric spatial discretization on a non-uniform mesh in a ring such that the property is preserved. The unknown density and velocity are defined on the same mesh whereas the mass flux and the viscous stress tensor are defined on the staggered meshes. Additional difficulties in comparison with the Cartesian coordinates are overcome, and a number of novel elements are implemented to this end, in particular, a self-adjoint and positive definite discretization for the Navier–Stokes viscous stress, special discretizations of the pressure gradient and regularizing terms using enthalpy, non-standard mesh averages for various products of functions, etc. The discretization is also well-balanced. The main results are valid for τ = 0 as well, i.e., for the barotropic compressible Navier–Stokes system.