A note on the relation between the “traditional approximation” and the metric of the primitive equations

The so-called “traditional approximation” consists in the neglect of certain terms in the equations of motion, in particular the horizontal Coriolis components. The validity of this approximation has long been discussed in the literature. In this study, it is shown that it is possible to exploit the shallowness of the atmosphere for a simplification of the spherical metric of the considered system (“Metrische Vereinfachung”). A Lagrange equation with this simplified metric may then be used to derive the equations of motion. These equations will not contain the problematic terms neglected in the “traditional approximation”. Using this procedure, it is possible to include a simple friction model through a dissipation function.