## Coordinate systems of numerical weather prediction models

 B. Geerts 4/'98

Grid models vs. spectral models

The three dimensions of space can be accounted for in various ways in numerical weather or climate prediction models. Most models are grid models, in which variables are computed at discrete grid points in the horizontal and vertical directions. The model resolution refers to the (horizontal) spacing between gridpoints. The grid spacing is not necessarily equidistant. For instance, some models use a longitude difference as zonal grid spacing, so near the poles the zonal grid spacing becomes zero. In the vertical direction the spacing is usually variable, the model resolution typically is highest just above sea level. Other models, in particular those whose domain is global, are spectral models (Note 15.H): these transform the variation of some variable (e.g. temperature) with latitude and longitude into a series of waves; the highest wave number retained in the Fourier transform is a measure of the model resolution. Numerical prediction models are based on the equations of motion (Note 15.G), and these involve many partial derivatives in space. Partial derivatives of wave fields (as used in spectral models) can be calculated exactly, rather than by means of a finite difference approach (used in grid models). This is the main advantage of spectral models. Of course the wave form is converted back into a spatial form after the calculations, in order to analyse the forecasts.

Vertical coordinate

The physical height is rarely used as the vertical coordinate in atmospheric simulation models. Some models use pressure as vertical coordinate, because it simplifies the equations, at least if the atmosphere is in hydrostatic balance, which is generally true for synoptic and mesoscale motion. In such models, the 500 hPa isobaric surface (which undulates in space and time) for instance is a fixed reference level. In complex terrain it is better to use ‘sigma’ than pressure, because a sigma (or ‘terrain-following’) coordinate system allows a high resolution just above ground level, whatever altitude the ground may be. Sigma (s) is defined as:

s = (ps-p)/ps

where ps is the ground-level pressure, and p the variable pressure. Sigma ranges from 0 at the ground to 1 at the top of the atmosphere. An extra equation is required in the sigma system, i.e. a prognostic equation for surface pressure.

Finally, some models employ isentropic coordinates. An ‘isentropic’ surface is one of equal potential temperature. Potential temperature generally increases with height, and potential vorticity is generally conserved on isentropic surfaces (Note 12.K). Also, as long as the air is not saturated or close to the ground, it will follow isentropic surfaces, so large-scale vertical motion can easily be diagnosed on isentropic surfaces, e.g. the 320K map. Again hydrostatic balance needs to be assumed to simplify the equations of motion in isentropic coordinates. This system provides better resolution near frontal boundaries, the tropopause, and near the ground at night, because in all these situations the potential temperature increases rapidly with height. However, boundary layer processes are poorly represented during the daytime.

Summary

• there are grid models and spectral models
• the vertical coordinate can be any of the following:
- height
• pressure
• terrain-following pressure (sigma)
• potential temperature