The present study is designed to provide both a method whereby the details of the nonlinear exchange process which occurs in the atmosphere may be considered in a simplified form, and also to indicate from some calculations the complexity of those exchanges.

A generalization of the two-dimensional spectral forecast equations is suggested, whereby the atmospheric flow field in horizontal and vertical directions is represented in terms of orthogonal functions, which are eigenfunctions of certain differential operators in the three-dimensional equations. The technique is applied to the quasi-geostrophic potential vorticity equation.