Further studies of the properties of cellular convection in a conditionally unstable atmosphere

The problem of convection in an atmosphere which is stable for dry adiabatic descending motion but unstable for saturated ascending motion is analyzed, with a view to the determinations of the preferred areas of the ascending and descending regions and the distributions of the convection currents. Two somewhat different approaches have been formulated, one is based on an integral relation obtained from the governing linearized differential equation while the other is based on the energy integral. It is found that the area of the ascending region tends to decrease while that of the descending region tends to increase with increasing static stability for unsaturated motion, but both of them are finite under all realistic conditions. On the other hand, when the mean lapse-rate is far above the moist-adiabatic lapse-rate, the consideration of the maximum efficiency in the production of available potential energy gives a much larger preferred area ratio, which tends to approach a constant value for every given static stability factor in the descending region.

The theory predicts that for large stability factor R′₂ (> 100) in the descending region the area ratio A_a/A_d increases linearly with the ratio R₁/R′₂ of the stability factors whereas for smaller R′₂ (< 100) A_a/A_d increases more slowly.

The predicted area ratio of the ascending and descending regions may vary from 1/₂ to 1/₃₀ from nearly neutral to normal conditions in the atmosphere, which agree quite closely with that obtained from satellite observations when reasonable values of eddy viscosity and eddy conductivity are assumed. Alternatively, we may also make use of this theory and the observed area ratio to determine the appropriate eddy viscosity coefficient, if the static stability factor in either the ascending or the descending region is known. The nonlinear heat transfer has also been analyzed.