# Fundamentals Of The

In the GCM, all dynamic and thermodynamic processes and the radiative and mass exchanges that have been treated in Chapters 2 to 7 are modelled using five basic sets of equations. The basic equations describing the atmosphere are:

1 The three dimensional equations of motion (i.e. conservation of momentum; see Chapter 6A,B).

2 The equation of continuity (i.e. conservation of mass or the hydrodynamic equation, p. 118).

3 The equation of continuity for atmospheric water vapour (i.e. conservation of water vapour; see Chapter 4).

4 The equation of energy conservation (i.e. the ther-modynamic equation derived from the first law of thermodynamics, see Chapter 7C).

5 The equation of state for the atmosphere (p. 22).

6 In addition, conservation equations for other atmospheric constituents such as sulphur aerosols may be applied in more complex models.

Model simulations of present-day and future climate conditions involve iterating the model equations for perhaps tens to hundreds of years of simulated time depending on the question at hand. In order to solve these coupled equations, additional processes such as radiative transfer through the atmosphere with diurnal and seasonal cycles, surface friction and energy transfers and cloud formation and precipitation processes must be accounted for. These are coupled in the manner shown schematically in Figure 8.1. Beginning with a set of initial atmospheric conditions usually derived from observations, the equations are integrated forward in time repeatedly using time steps of several minutes to tens of minutes at a large number of grid points over the earth and at many levels vertically in the atmosphere; typically ten to twenty levels in the vertical is common. The horizontal grid is usually of the order of several degrees' latitude by several degrees' longitude near the equator. Another, computationally faster, approach is to represent the horizontal fields by a series of two-dimensional sine and cosine functions (a spectral model). A truncation level describes the number of two-dimensional waves that are included. The truncation procedure may be rhomboidal (R) or triangular (7); R15 (or 721) corresponds approximately to a 5° grid spacing, R30 (742) to a 2.5° grid, and 7102 to a 1° grid.

Realistic coastlines and mountains as well as essential elements of the surface vegetation (albedo, roughness) and soil (moisture content) are typically incorporated into the GCM. These are smoothed to be representative of the average state of an entire grid cell and therefore much regional detail is lost. Sea-ice extent and sea-surface temperatures have often been specified by a climatological average for each month in the past. However, in recognition that the climate system is quite interactive, the newest generation of models includes some representation of an ocean which can react to changes in the atmosphere above. Ocean models (Figure 8.2) include a so-called swamp ocean where sea-surface temperatures are calculated through an energy budget and no annual cycle is possible; a slab or mixed-layer ocean, where storage and release of energy can take place seasonally, and the most complex dynamic ocean models, which solve appropriate equations for the ocean circulation and thermodynamic state similar to 1-5 above and which are coupled to atmospheric models. Such coupled models are referred to as atmosphere-ocean general circulation models (AOGCMs). When the global ocean is considered, seasonal freezing/ melting and the effects of sea ice on energy exchanges and salinity must also be modelled. Therefore, dynamic sea-ice models, which actively calculate the thickness and extent of ice, are now replacing the specification of climatological sea ice. Because of the century-long timescale of deep ocean circulations, the use of a dynamic ocean model requires large amounts of simulation time for the different model components to equilibrate which greatly increases the cost of running these models.

Because coupled AOGCMs are used in long-term (century or millennium scale) simulations, an important

Figure 8.1 Schematic diagram of the interactions among physical processes in a general circulation model.

Source: From Druyan et al. (1975), by permission of the American Meteorological Society.

Figure 8.1 Schematic diagram of the interactions among physical processes in a general circulation model.

Source: From Druyan et al. (1975), by permission of the American Meteorological Society.

Swamp

Coupled model hierarchy

Coupled model hierarchy

SST from surface energy balance B Atmospheric GCM

SST from surface energy balance B Atmospheric GCM

Mixed layer (slab)

SST from surface energy balance, heat storage

Atmospheric GCM

Ocean GCM

SST from surface energy balance, heat storage, advection, diffusion

Figure 8.2 Schematic illustration of the three types of coupling of an atmospheric GCM to the ocean: (A) swamp ocean (B) mixed layer, slab ocean (C) ocean GCM.

concern is 'model drift' (a definite tendency for the model climate to warm or cool with time) due to accumulating errors from the various component models. These tendencies are often constrained by using observed climatology at certain high-latitude or deep ocean boundaries, or by adjusting the net fluxes of heat and fresh water at each grid point on an annual basis in order to maintain a stable climate, but such arbitrary procedures are the subject of controversy, especially for climate change studies.

Many important weather and climate processes occur on a scale which is too small for the typical GCM to simulate with a grid of several degrees on a side. Examples of this would be the radiative effects or latent heating due to cloud formation or the transfer of water vapour to the atmosphere by a single tree. Both processes greatly affect our climate and must be represented for a realistic climate simulation. Parameterizations are methods designed to take into account the average effect of cloud or vegetation process on an entire grid cell. Parameterizations generally make use of a statistical relationship between the large-scale values calculated for the grid cell in order to determine the effect of the parameterized process.

In order to gain confidence in the performance of models in predicting future atmospheric states, it is important to evaluate how well such models perform in representing present-day climate statistics. The Atmospheric Model Intercomparison Programme (AMIP) is designed to do this by comparing models from various centres around the world using common procedures and standardized data (on sea-surface temperatures, for example), as well as by providing extensive documentation on the model design and the details of model parameterizations. In this way common deficiencies can be detected and perhaps attributed to a single process and then addressed in future model versions. Figure 8.3 compares simulated zonally averaged surface temperature for January and July for all AMIP participants with the observed climatological mean. The general features are well represented qualitatively, although there can be large deviation between individual models. The evaluation of models requires analysis of their ability to reproduce interannual variability and synoptic-scale variability as well as mean conditions. A comparison project for AOGCMs similar to AMIP is now underway called the Coupled Model Intercomparison Project (CMIP). Plate G illustrates the 500-mb heights for northern winter and summer, as observed (top) and as simulated by the National Center for Atmospheric Research (NCAR) Community Climate Model (CCM) 3, and the considerable differences between them in high latitudes.

Recent models incorporate improved spatial resolution and fuller treatment of some previously neglected physical processes. However, both changes may create additional problems as a result of the need to treat accurately complex interactions such as those between the land surface (soil moisture, canopy structure, etc.) and the atmospheric boundary layer, or interactions between clouds, radiative exchanges and precipitation mechanisms. For example, fine-scale spatial resolution is necessary in the explicit treatment of cloud and rain bands associated with frontal zones in mid-latitude cyclones. Such processes require detailed and accurate representation of moisture exchanges (evaporation, condensation), cloud microphysics and radiation (and the interactions between these processes) which are all represented as averaged processes when simulated at larger spatial scales.

90 80 60 N

Eq Latitude

60 80 90 S

90 80 60 N

Eq Latitude

60 80 90 S

Figure 8.3 Comparison of zonally averaged surface temperatures for December to February (above) and June to August (below) as simulated by the AMIP models compared with observations (bold line). The shaded band shows the range of results for 17 AMIP models.

Source: AMIP website.

90 80 N

Eq Latitude

80 90 S

90 80 N

Eq Latitude

80 90 S

Figure 8.3 Comparison of zonally averaged surface temperatures for December to February (above) and June to August (below) as simulated by the AMIP models compared with observations (bold line). The shaded band shows the range of results for 17 AMIP models.

Source: AMIP website.