Analytical investigation of the role of lateral mixing in the evolution of non-precipitating Cumulus

A minimalistic analytical model allowing analysis of the developing and dissolving stages of non-precipitating convective clouds is proposed. We present an analytical solution of 1D equation describing effects of lateral turbulent mixing with dry surrounding air and diffusional growth/evaporation on a microphysical structure of ascending/descending cloud volume.

Spatial and temporal changes of liquid water content (LWC), the adiabatic fraction, droplet concentration, and the mean volume droplet radius are calculated at the developing stage. It is shown that the existence of a non-diluted core in a growing cumulus cloud mainly depends on the cloud width and vertical velocity. Since lateral mixing synchronously decreases both the LWC and the droplet concentration, the variation of the mean volume droplet radius is low inside the cloud. The approximate quantitative condition for cloud formation in updraft is derived. It is shown that a cloud can arise when its vertical velocity exceeds a critical value.

The temporal changes in the spatial structure of a cloud and in its immediate environment at the dissolving stage are analyzed. An equation for cloud dissolving time is derived. The comparison of the effects of a temperature increase in the course of cloud descent and of the mixing with the dry surrounding air shows that the descent is a dominating factor determining the decrease in the LWC, while mixing has a stronger effect on the cloud shape. The main parameter determining the dissolving time is the downdraft velocity. It should exceed 30 cm/s in order to provide reasonable dissolving time. After a cloud totally dissolves it leaves behind an area with humidity exceeding that of the environment.

The comparison of the model results with previously published observational data indicates a reasonable agreement. These results can be useful for parameterization purposes. Values of the environment humidity and temperature, LWC at cloud top, cloud width, vertical velocity of downdraft, and the turbulent coefficient should be used as parameters of such parameterization.