Williams, G. P., and D. R. Davies, 1965: A mean motion model of the general circulation. Quarterly Journal of The Royal Meteorological Society, 91 (390), 471-489.

Summary: Equations are constructed to represent quasi-stationary mean flow of momentum and heat on a spherical earth, averaged over a long period of time such as a year and over latitude circles. The crucial shearing Reynolds stress associated with meridional transfer of zonal velocity is assumed to depend linearly on a product of the earth's angular velocity, Omega, and the meridional gradient of mean temperature; the shearing stresses associated with vertical transfer of zonal velocity and of meridional velocity are assumed to depend linearly on the vertical gradients of zonal and of meridional mean velocities respectively, and the mean eddy transfer of heat along a meridian is assumed to depend linearly on the mean meridional temperature gradient. All proportionality coefficients are taken to be independent of latitude. Two forms are assumed for the non-adiabatic atmospheric heat source function, Q, used in the thermodynamic equation. In the first case, Q is assumed known (from analyses of observations) as a function of height and latitude. In the second case, Q incorporates a heating term which is partly controlled by the model itself and represents some of the characteristics of sensible and latent heat transfer. A solution of the basic equations is obtained in both cases in the form of double expansions in powers of two parameters, one depending on omega and the other on delta tau, the mean annual temperature difference between equator and pole. The solution is evaluated using Fourier techniques.
The series expansions are found to be reasonably convergent for realistic values of the various parameters involved, three terms only being required in the delta tau expansion and five terms at most in the omega expansion, but extensive numerical evaluation by digital computer is involved: the region considered is bounded by the tropopause and lies between the equator and 70 degrees latitude. The computed zonal velocity has the characteristic east-west variation with latitude and a broad band maximum of expansion, but extensive numerical evaluation by digital computer is involved: the region considered is bounded by the tropopause and lies between the equator and 70 degrees latitude. The computed zonal velocity has the characteristic east-west variation with latitude and a broad band maximum of 19 m sec^-1 and the meridional velocity the characteristic tricellular structure. A poleward eddy angular momentum flux and polar inversion are predicted.
The results, through verification of the postulates, add support to the Rossby view of the general circulation in which the cyclonic-scale eddies act to release potential energy of the atmosphere to supply their own kinetic energy and form the mean zonal kinetic energy. They further indicate the value of the reconstructed 'austausch' approach for this problem.