In the previous chapter we introduced the vertical coordinate η to handle orographic effects in mesoscale models. In the synoptic-scale models we are going to replace the height coordinate z which extends to infinity by a generalized vertical coordinate ξ. The introduction of ξ is motivated by the fact that we cannot integrate the predictive equations using z as a vertical coordinate to infinitely large heights. Replacing z by the atmospheric pressure p, for example, results in a finite range of the vertical coordinate. We will see that another advantage of the (x, y, p)-coordinate system is that the continuity equation is time-independent. There are other specific coordinate systems that we are going to discuss. Therefore, it seems of advantage to first set up the atmospheric equations in terms of the unspecified generalized vertical coordinate ξ. Later we will specify ξ as desired. We wish to point out that the introduction of the generalized coordinate is of advantage only if the hydrostatic equation is a part of the atmospheric system.

We will briefly state the consequences of the transformation from the stereographic (x, y, z)-coordinate system to the stereographic (x, y, ξ)-coordinate system, which henceforth will be called the ξ system.