On some mathematical and computational problems of atmosphere modeling

The study of atmosphere phenomena is important and challenging field of research in different areas of sciences such as meteorology, geophysics, physics and mathematics. This presentation addresses some recent results obtained in mathematical and computational simulation of the atmosphere. Four problems are considered. The first one is the study of mathematical properties of atmospheric balance equations. The study of the mathematical structure of these equations revealed that certain balance relations lead to ill-posed mathematical problems. Comparison of the well-posedness conditions for balance theories of different level of complexity, made possible to conclude that the ill-posed problems arise due to applied physical simplifications, which can be inappropriate for certain atmospheric motions. The second problem is related to generation of structured grids for numerical models in spherical geometry. If the accuracy of numerical solution is required over some "limited" area of a sphere, then the most uniform flattening of a sphere can provide better properties of numerical scheme. Comparison of conformal mappings from a sphere onto a plane showed that for different "limited" area models the best stereographic projection is the stereographic one centered at the centerpoint of the chosen spherical domain. Evaluations of the advantage of the "best" stereographic grid with respect to other conformal mappings are given. The third problem is correctness of the vertical discretization in the hydrostatic atmospheric models. The positivity of spetra of different matrices arising in vertical discretization of atmospheric models was proved analytically, which assures well-posedness of initial-value problem for vertically discretized atmospheric equations. The last problem is related to application of splitting techniques in numerical atmospheric models in order to design a more efficient algorithm. Different types of physical and geometrical splitting were employed in the context of the semi-implicit scheme for shallow water and hydrostatic atmospheric models. Some techniques of reducing the additional splitting errors were also applied. Developed numerical schemes were tested with actual atmospheric data and obtained results of experiments showed high order of accuracy of forecasting fields and computational efficiency of numerical algorithms.