Abstract
The primitive equations are the fundamental equations of atmospheric dynamics. With the purpose of understanding the mechanism of long-term weather prediction and climate changes, the authors study as a first step towards this long-range project what is widely considered as the basic equations of atmospheric dynamics in meteorology, namely the primitive equations of the atmosphere. The mathematical formulation and attractors of the primitive equations, with or without vertical viscosity, are studied. First of all, by integrating the diagnostic equations they present a mathematical setting, and obtain the existence and time analyticity of solutions to the equations. They then establish some physically relevant estimates for the Hausdorff and fractal dimensions of the attractors of the problems.