Schemm, C. E., and F. B. Lipps, 1976: Some results from a simplified three-dimensional numerical model of atmospheric turbulence. Journal of the Atmospheric Sciences, 33 (6), 1021-1041.

Abstract: A simplified set of subgrid-scale transport equations is used to compute the stresses in a three-dimensional model of thermal convection in the atmosphere. Terms appearing in the full transport equations thought not to be essential to the large-scale dynamics are discarded, leaving prognostic equations to be solved for the subgrid-scale energy and the virtual potential temperature variance only. Equations for the Reynolds stresses and the subgrid-scale temperature-velocity correlations are considerably simplified and can be solved algebraically. A scale analysis of the full transport equations is offered as partial justification for the present approach in the case of nearly isotropic turbulence.
The problem studied is that of a well-mixed layer bounded above by a region of strong stable stratification. The present model gives a significant improvement in the representation of the large-scale variables as compared with the more conventional eddy viscosity approach. In three experiments testing different variations of the modified transport equations, the horizontally averaged subgrid-scale energy components are found to be interrelated in much the same sense as their corresponding resolvable turbulence energy components. Above the inversion the turbulence intensity is observed to decline sharply. The temperature inversion is maintained as the thermal boundary layer rises, and in each case a countergradient upward heat transport by the subgrid-scale eddies is detected in the upper half of the well-mixed layer. In contrast, the temperature gradient at the base of the stable layer is smoothed out considerably in the eddy viscosity run.