A method to obtain a series solution for the barotropic non-divergent, vorticity equation is given and is applied to the case of a vortex.A formula that gives the development of a symmetrical vortex as a function of radius, maximum velocity and latitude is obtained. A numerical example is worked out.The convergence of the series solution for the general case is discussed and shown to depend on the scale of motion to be forecast and conclusions are drawn regarding time-steps used in a forecast and the smoothing of the initial map.The possibility of improving a forecast and reducing the number of time-steps used in it by the application of this series solution, follows as a general conclusion.Finally, a theorem that gives the scaling of the variables in an experimental model of a barotropic flow is obtained. p I , = =aay/2xat, etc. 'Tellus VIII (1956), 3

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The author's thermal model for monthly and seasonal numerical prediction of temperatures is generalized, so that besides radiation other forms of heating (or the anomalies of 'heating) are generated within the model. This is done by expressing such heating a s a linear function of variables predicted in the model. The anomalies directly incorporated are those in the storages of thermal energy which are introduced by prescribing in the previous interval the temperature of the surface water in the oceans and the temperature of the midtroposphere, a s well as the anomalies in the shortwave radiation absorbed by the surface; which in turn are introduced by prescribing the albedo (snow cover) a t the end of the previous interval. The numerical experiments show that important anomalies of the evaporation a t the surface, of the vertical turbulent transport of sensible heat from the surface, of the condensation of water vapor in the clouds, and of the cloudiness are introduced by the anomalies of the computed temperature fields. Furthermore, these induced anomalies of the heating functions and of the cloudiness in turn introduce changes in the anomalies of the temperature fields. 1 This work was carried out ill. the Extended Forecast Division, National Meteorological Center, U.S. Weather Bureau, where the author is a consultant.

Applying the conservation of energy principle to the incoming solar radiation and to the long-wave radiation, a heating mechanism of the atmosphere is introduced. A simplified emission spectrum of the atmosphere with one transparent region is used. The surface of the earth and a cloud layer are considered aa black bodies.A simplified model of the atmosphere containing the coupling between dynamical and thermodynamical proceases is presented. The equations that give the temperature, the excess of radiation and the mean zonal wind for each day of the year are derived. The temperature is given by a second order differential equation that can readily be solved. The zonal wind and the excess of radiation are explicit functions of the temperature.