Scaling and computation of smooth atmospheric motions

We introduce a general scaling of the inviscid Eulerian equations which is satisfied by all members of the set of adiabatic smooth stratified atmospheric motions. Then we categorize the members into mutually exclusive subsets. By applying the bounded derivative principle to each of the subsets, we determine the specific scaling satisfied by that subset. One subset is midlatitude motion which is hydrostatic and has equal horizontal length scales. Traditionally, the primitive equations have been used to describe these motions. However it is well known that the use of the primitive equations for a limited area forecast of these motions leads to an ill-posed initial-boundary value problem. We introduce an alternate system which accurately describes this type of motion and can be used to form a well-posed initial-boundary value problem. We prove that the new system can also be used for any adiabatic or diabatic smooth stratified flow. Finally, we present supporting numerical results.