High-Order Approximation Equations for the Primitive Equations of the Atmosphere

In this article, a family of models approximating the primitive equations of the atmosphere, which are known to be the fundamental equations of the atmosphere, is presented. The primitive equations of the atmosphere are used as a starting point and asymptotic expansions with respect to the Rossby number are considered to derive the nth-order approximate equations of the primitive equations of the atmosphere. Simple global models of the atmosphere are obtained. These higher-order models are linear and of the same form (with different right-hand sides, depending on the lower-order approximations) as the (first-order) global quasi-geostrophic equations derived in an earlier article. From a computational point of view, there are two advantages. Firstly, all the models are linear, so that they are easy to implement. Secondly, all order models are of the same form, so that, with slight modifications, the numerical code for the (first-order) global quasi-geostrophic model can be employed for all higher-order models. From a physical point of view, higher-order models capture more physical phenomena, such as the meridional flows, even though they are small in magnitude. Of course, there are still many subtle issues involved in this project, such as the convergence of the asymptotics; they will be addressed elsewhere. The article is concluded by a presentation of numerical simulations based on these models.