Maximum Simplification of the Dynamic Equations

When the dynamic equations are to be used to further our understanding of atmospheric phenomena, it is permissible to simplify them beyond the point where they can yield acceptable weather predictions. Through the use of double Fourier series, and with the omission of all but the largest scales of motion, the barotropic vorticity equation may be reduced to a system of three ordinary nonlinear differential equations. The analytic solutions of these equations are elliptic functions of time. The equations may also be solved rapidly by numerical integration. Particular solutions of the equations picture the motion of finite disturbances on a zonal flow, with exchanges of kinetic energy between the zonal flow and the disturbances accompanying the meridional transport of zonal momentum by the disturbances. Other solutions picture the initial growth and eventual cessation of growth of small disturbances on an unstable zonal current. Still further solutions picture the destruction of a stable zonal flow by large disturbances, and lead to a plausible hypothesis concerning index cycles in the atmosphere. Less extreme simplifications of the dynamic equations may be used when more complicated atmospheric phenomena are to be studied. DOI: 10.1111/j.2153-3490.1960.tb01307.x