Primitive equations

Dynamic meteorological models, much like air pollution models, strive to describe the physics and thermodynamics of atmospheric motions as accurately as is feasible. Besides being used in conjunction with air quaHty models, they ate also used for weather forecasting. Like air quaHty models, dynamic meteorological models solve a set of partial differential equations (also called primitive equations). This set of equations, which ate fundamental to the fluid mechanics of the atmosphere, ate referred to as the Navier-Stokes equations, and describe the conservation of mass and momentum. They ate combined with equations describing energy conservation and thermodynamics in a moving fluid (72) ... 

Smagorinsky, J. (1963). General circulation experiments with the primitive equations I. The basic equations. Monthly Weather Review 91, 99-164. 

In this study the numerical simulations were performed with a 3-D mechanistic global Cologne Model of the Middle Atmosphere (COMMA) based on the primitive equations expressed in spherical coordinates for the horizontal and log-pressure coordinates in the vertical direction. The model equations are solved on the basis of an explicit numerical scheme (leapfrog) with a fixed time step of 450 sec. To avoid separate evolution at even and odd time steps, a Robert time filter is used. 

To conclude, it should be noted that the diagnostic (in a broad sense of this term) modeling represents the most fruitful and actively developing fine in the BSGC studies. In the future, its most developed branch, i.e. assimilation of the real-time data of satellite and field observations in full models with primitive equations (see Sect. 3.2), may become an important tool for the BSGC monitoring, at least in the upper layers of the Black Sea. 

Bryan F. (1987) Parameter sensitivity of primitive equation ocean general circulation models. J. Phys. Oceanogr. 17(7), 970-985. 

This primitive equation is often used in chemical engineering analysis. If we would like to decrease the reaction period x until we obtain the required number of... 

Stevens, D. P., 1990. On open boundary conditions for three dimensional primitive equation ocean circulation models. Geophysical and Astrophysical Fluid, Dynamics, 51, 103-133. 

Smagorinsky J (1963) General Circulation Experiments with the Primitive Equations. Monthly Weather Review 91(3) 99-164... 

Bates JR, McDonald A (1982) Multiply-Upstream, Semi-Lagrangian Advective Schemes Analysis and Application to a Multi-Level Primitive Equation Model. Mon Wea Rev 110 1831-1842... 

These equations define the basis for a theoretical treatment of the atmospheric circulation. Together with suitable boundary and initial conditions they are used in primitive equation models of the atmosphere (see Section 3.9). 

For a large class of scale of motions, the primitive equations (which are rather expensive to solve numerically) can be replaced by approximate equations for the vorticity or the potential vorticity. Relative vorticity can be thought of as a measure of the rotation of the fluid with respect to a coordinate sytem fixed to the Earth. Its vertical component is defined in spherical coordinates as... 

By cross-differentiating and combining the primitive equations (3.23a and 3.236) and using the continuity equation (3.25), one obtains the vorticity equation... 

Three-dimensional models which provide solutions to some form of the primitive equations outlined in Section 3.3 are called general circulation models (GCMs). These models can provide insight on the coupling between dynamical and radiative processes in the atmosphere. They resolve large-scale waves and synoptic eddies, and include state-of-the-... 

Ideally, the solution of the primitive equations for specified external constraints (e.g., the solar irradiance at the top of the atmosphere) and appropriate boundary conditions (e.g., observed sea-surface temperature) should provide a comprehensive representation in space and time of the atmospheric dynamical system. In practice, however, limitations in computer capabilities impose limits on the spatial resolution of these models, so that small-scale processes, rather than being explicitly reproduced, must be parameterized. The uncertainties associated with these physical parameterizations (e.g., boundary layer exchanges, convection, clouds, gravity wave breaking, etc.) often limit the overall accuracy in the model results. 

Bourke, W., An efficient, one-level primitive equation spectral model. Mon Weath Rev 100, 683, 1972. 

In the study of ordinary differential equations, we have always assumed that the given equation has been obtained by the elimination of constants from the original equation. In solving, we have sought to find this primitive equation. Partial differential equations, however, may be -obtained by the elimination of arbitrary ... 

This occurs in spite of the fact that neither of the primitive equations for the individual enzymes has this autocatalytic form. Although much over-simplified, this calculation strongly suggests that the linking of the various ceU processes may be such as to render the functioning of the enzymes effectively autosynthetic. 

The numerical model used in this study is a nonhydrostatic, primitive equation with terrain-following sigma(o) vertical coordinate MM5 version 3. The simulation performed for this study was run on two-way interactive nesting model. They consist of a 6-km grid (Doj, mesh size of 181 X 181), a 18-km grid (Doj, mesh size of 121 x 121) and a 54-km grid (Doj,... 

Mathur, M. B. (1974). A multiple grid primitive equation model to simulate the development of an asymmetric hurricane (Isabel, 1964), J. Atmos. Sci. 31, 371-393. 

---