Equations of Hydrodynamics, Heat, and Mass Transfer in Models of Regional and Mesoscale Atmospheric Processes

The formation of meteorological processes and phenomena in the atmosphere in each specific area is influenced by the processes of different scales. However, the significance of each process in different cases is different. Therefore, there are certain priorities for the inclusion of certain processes in the hydrodynamic model. Depending on the size of the territory taken for modeling of processes or individual phenomena in the atmosphere, hydrodynamic models are divided into local and regional. In this chapter, we consider these features in terms of mathematical models.

General Characteristics and Classification of Processes in the Earth-Atmosphere System

Atmospheric processes affect the heat flows coming from above, from space, and from below, from the earth's surface. The solar radiation that comes to Earth from outer space is the main source of energy of atmospheric processes. It is the radiant energy of the Sun that is converted into heat in the atmosphere and at the Earth's surface, kinetic energy, and other forms of energy. But the Sun's rays heat the earth's surface larger than the air directly, so between the earth's surface and the atmosphere, there is lively exchange of heat as well as moisture. The structure of the earth's surface and its relief are very important for these processes. The chapter presents the picture of heat and the moisture circulation in the atmosphere and gives physical basics of atmospheric general circulation including fundamentals of air mass circulation, local physiographic impact on the atmospheric air movement, in-mass atmospheric processes, and basic laws of pollution spreading in the atmosphere.

Methods of Calculating Turbulent Processes in the Atmosphere

The hydrodynamics laws describe well the laminar motion of a continuous medium where the particle trajectories and streamlines have fully defined regular character. However, random deviations, or “disturbances”, significantly change the nature of the initial motion, and cause a transition to a new or constant movement, or to some chaotic. This chapter studies forms of motion of a viscous fluid that is very common in the atmosphere and is called the turbulent motion.

Methods of Mathematical Modeling Atmospheric Circulation

In this chapter, atmospheric processes are considered on a planetary scale that is impossible to imagine without a detailed three-dimensional modeling based on geophysical fluid dynamics equations of thermodynamics, radiation transfer, the kinetic equations describing the actual movement of air masses, conversion and transfer of energy and matter, the formation of clouds, aerosols, and rainfall and other atmospheric processes.

Numerical Methods for Solving Regional and Mesoscale Problems

Regular grids with even steps of the spatial coordinates in the whole computational domain are the most convenient for implementing numerical methods for integration of equations of weather forecasts. However, computing a local numerical weather forecast based on the global general circulation models of the atmosphere will need enormous increase in computation time exceeding reasonable limits. Moreover, as some regional weather details are well localized it is reasonable to apply high-resolution grids locally. In this chapter, we study how to use the high-resolution grids in the numerical methods for solving regional and mesoscale weather forecast problems.

Parameterization of Subgrid-Scale Processes

In Chapters 4 and 5, we considered a system hydrodynamics equations and boundary conditions that constitute the mathematical basis of the circulation models of the atmosphere of a scale. They contain terms describing sources (sinks) of mass and energy involved in the phase transformation of atmospheric moisture and radiative processes in the system atmosphere – the Earth. Direct inclusion of these microscale and mesoscale processes in the atmospheric circulation model is inappropriate as: 1) this leads to an increase in the total number of grid points in the decision area, and 2) not all these processes can be described by precise differential equations.

Equations Satisfied Physical Processes in the Atmosphere

The chapter considers theoretical foundation of the modern hydro-mechanics in their application to the atmosphere dynamics problems. General concepts, definitions, and notation needed for mathematical modeling atmospheric processes, particularly, the elements of kinematics and deformation of continuous media, and curved orthogonal coordinates are introduced. The basic equations of mechanics of viscous stratified fluid are considered including the equations expressing the law of conservation of mass, the equations expressing the law of momentum, the equations expressing the law of conservation of energy, the concentration equations for mechanics of heterogeneous medium. At the end of the chapter, the complete list of the basic equations of the model of atmosphere is given and the initial and boundary conditions in the hydrodynamic problems are described.

Brief Description of the Main Characteristics of the Earth-Atmosphere System

The chapter gives introductory concepts of the science of meteorology including astronomical data, basic information about the composition and structure of the atmosphere, determination of the main meteorological variables, solar radiation, longwave radiation of the Earth and atmosphere. There are given fundamentals of atmospheric thermodynamics and polytrophic changes in atmospheric air that include thermodynamic properties of dry and moist air and conditions for vertical air stability.

Numerical Methods of Modeling Atmospheric Circulation and Its Physical Processes

The last decades are characterized by significant progress in the development and operational use of modern numerical hydrodynamic methods of the Earth's weather and climate. This was made possible primarily due to modern understanding of the laws governing the basic physical and thermodynamic processes in the atmosphere and the emergence of more advanced mathematical models and effective methods of their implementation. In this chapter, we develop new numerical techniques used to solve the non-stationary problem of general circulation of the atmosphere with a prehistory and the problem of planetary weather forecast.