Introduction

The shallow water model made it possible to examine the way to use various algorithms and to specify their respective properties (horizontal discretization, time integration). Making a more realistic model to describe the evolution of a baroclinic atmosphere requires allowance for the vertical dimension of the atmosphere and the use of suitable discretization. We commonly speak then of baroclinic models. The earliest baroclinic models used for operational forecasting were built on systems of filtered equations characterized by introducing a balance equation between the mass field and the wind field, thereby excluding the possibility of inertia-gravity wave propagation, but allowing the use of a comparatively large time step. However, the increased power of computers in the 1970s was soon to allow the primitive equations to be used to construct models for operational forecasting.

The primitive equations were used with various types of vertical coordinate. Once the drawbacks of the pressure coordinate had been identified, the normalized pressure coordinate (commonly called sigma and noted σ) was proposed for formulating the condition at the lower boundary of the atmosphere in a simple way. Introducing this coordinate, however, was not without its drawbacks for the highest layers, which were more apt to be processed by the pressure coordinate. A new hybrid coordinate (a combination of σ and p) has provided a fairly satisfactory answer to the problem and has now found its place in many models using either the primitive (hydrostatic) equations or the (nonhydrostatic) Euler equations. These different coordinates are presented in this chapter and the integral invariants of the system are studied in the context of a general formulation encompassing both the primitive equations and the Euler equations.