Introduction

In this chapter we are going to discuss the two-level quasi-geostrophic prediction model. This model divides the atmosphere into four layers as shown in Figure 24.1. The vorticity equation is applied to levels l = 1 and l = 3 while the heat equation is applied to level l = 2. By eliminating the vertical velocity ω it becomes possible to determine the tendency of the geopotential ∂φ/∂t. Initially only the geopotential φ(x, y, p, t0 = 0) for the entire vertical pressure range 0 ≤ p ≤ p0 must be available. The discussion will be facilitated by resolving the dependent variables in the vertical direction only. The remaining differentials will be left in their original forms, which may be approximated by finite differences whenever desired.

In the second part of this chapter we are going to discuss the concept of baroclinic instability. In a rotating atmosphere this type of instability, which was first investigated by Charney (1947) and Eady (1949), arises from the vertical wind shear if the static stability is not too large. The stability properties of the Charney model are difficult to analyze. The two-level model, however, makes it possible to obtain the stability criteria in a rather simple way, with results consistent with Charney's model. Details, for example, are given by Haltiner and Williams (1980).

The mathematical development of the two-level model

The basic system consists of the vorticity equation (23.44) and the first law of thermodynamics (23.19).