Weather forecasting in meteorology is based on two complementary approaches:

1. Accumulating a very large amount of data and interpreting these statistically (wind velocity, humidity, and temperature measured over very large intervals of time and over large regions of the earth). Here, the mathematical techniques that are necessary for the assimilation and the exploitation of these data are those of statistics and of stochastic processes.

2. Modeling of atmospheric phenomena by ordinary and partial differential equations and the numerical simulation of these equations. From the computational point of view, one obtains, by discretization of such partial differential equations, systems of equations with millions, even billions, of unknowns whose numerical resolution could saturate the most powerful computers currently available. The memory size and computational speed capacities needed for such calculations are very high even in the context of the teraflop (1012 operations per second), which is the next step in the foreseeable future.

In this chapter, we are interested in the second approach, and we intend to give a very modest description of the most fundamental equations widely accepted in the field. The atmosphere is a fluid whose state is described by the velocity vector, the temperature, the density, and the pressure at every point in the Eulerian description. The equations are essentially variants of the Navier–Stokes and of the temperature equations that take into account the particular aspects of the problem.

Section 12.1 is devoted to various preliminaries.