The mathematical descriptions of the climate system widely known as GCMs are computer programs which solve, approximately, the set of coupled differential equations representing conservation of atmospheric momentum, energy, and mass. A basic form of the momentum-conservation equation is: (This and the following balance equations are expressed in condensed, vector form for brevity. Expanded forms and further description can be found in textbooks on this subject (e.g., Washington and Parkinson 1986).
where 
is the local air density, vair is its velocity vector, and g represents the acceleration of gravity.
A corresponding form for energy conservation is:
where Cv represents the specific heat of air at constant volume. Principal outputs of solutions to the momentum and energy equations, are time varying, three-dimensional fields of the wind velocity vector (v) and atmospheric temperature (T). The energy balance contains terms for the transfer of sensible and latent energy and electromagnetic radiation. Figure 1 shows some of the important energy flows that GCMs must describe.
The radiation and energy balance of the Earth's atmosphere involves a complicated set of interrelated processes. The experimental portions of the ARM Program will attempt to measure many of the energy fluxes shown in the figure. The figure demonstrates the importance of integrating the ARM program with satellite observations. The combination of ARM, with its ability to examine lower tropospheric processes in detail, with the satellite's global coverage of the outgoing radiation will make a powerful measurement system.
Usually, GCMs contain more than one conservation equation for mass. The mass-continuity equation for air, which is an essential requirement for any GCM, is:
This is accompanied by the continuity equation for water:
Equation 4 may be subdivided into separate equations for different phases of water (vapor, clouds, ice, . . .). However, computational constraints usually limit the extent of this subdivision. Solutions to material balance equations 3 and 4 result in time varying three-dimensional fields of concentrations (or densities) of the material conserved by the equations.
GCMs approximate solutions to the above differential equations using a variety of numerical- integration techniques. ( Most of today's GCMs apply standard finite differencing for integrating in the vertical dimension, and use so-called "spectral" or "pseudospectral" techniques for integration in the horizontal. The latter technique involves Fourier-transformation of the dependent variables into a frequency domain, truncation of all contributions above a specified wave number, algebraic manipulation, and then inversion back to the original solution domain. The reader should not confuse the spectral terms (e.g., wave length, wave number, frequency, . . . ) associated with this numerical technique with similar terms pertaining to electromagnetic phenomena or atmospheric motions.) Common to all, however, is the practice of dividing the computational domain (the global atmosphere, in the case of a global GCM) into a three-dimensional grid-mesh similar to that shown in Figure 2. The solution space is then tied to points on this grid. Most of today's GCMs operate with about 10 layers in the vertical, and have a horizontal grid spacing of roughly 500 km. Both the potential accuracy of the code's numerical approximations and the required computer time increase with grid resolution. Halving the grid spacing in the x,y, and z directions will generally result in an 8-fold increase in required computer storage and a 16-fold increase in computer time. (By a simple examination of numerical-analysis techniques one can demonstrate that halving the grid spacing in any direction will double the number of grid points and essentially double the amount of memory and the number of computations required for a solution. Thus, halving for three dimensions results in a 23 = 8-fold increase. For reasons of computational stability, however, one normally is required to reduce the computational time-step proportional to any reduction in the spatial grid; thus, a total computational time increase of 24 = 16-fold results.) Thus, an important trade-off exists between accuracy and computational economy. While these computational issues are a significant concern, a second manifestation of coarse grid spacing is far more important. Many meteorological phenomena that are primary determinants and manifestations of global weather occur on small scales. These scales are small enough to escape resolution by the model's grid-mesh.
Figure 3 demonstrates this effect. The figure shows a typical horizontal GCM grid superimposed on a storm system crossing the North American continent. The important processes of cloud formation, precipitation, albedo change, radiant energy transfer, and vertical and horizontal transport are taking place at small scales within the storm system. At best, these phenomena will be highly blurred by the coarse-grid model. At worst, these processes may totally escape detection or processing by the code. The effects of these unresolved phenomena are usually included in GCMs throught the process of pa-rameterization. An important issue is assuring that these parameterizations adequately represent current climate and can evolve with the climate in an appropriate physical manner.
