Similarity theories have played a central role in understanding turbulent transports in the atmospheric boundary layer and in the development of parametrizations of turbulent mixing. However, the similarity approach has not been used much in studies of shallow cumulus convection, in part due to the lack of suitable observations on which to base such a theory. Here a similarity hypothesis is proposed, and is tested using data from several large-eddy simulations.
In a similarity theory, a small number of parameters are assumed to determine the characteristics of the mixing process. In the present study of the cumulus layer these are assumed to be the cloud base mass flux, the depth of the cloud layer, the buoyancy excess of a parcel undergoing reversible ascent through the cloud layer and the changes in the environmental liquid water potential temperature and total water content across the layer. Using arguments based on the turbulence kinetic energy budget, a velocity scale is derived which is analogous to the convective velocity scale used in boundary layer studies, but incorporates the effects of latent heating. Scales for the liquid water temperature and total water content are also derived from the relevant variance budgets.
Four large-eddy simulation datasets have been used, one based on a case from the BOMEX trade-cumulus experiment, two modified BOMEX cases, and one based on an observed case of shallow cumulus off the coast of the United Kingdom. Profiles of the mean cloud to environment difference in vertical velocity and scalar concentration are found to collapse well when normalized with the appropriate scales. Further arguments also lead to succesful scalings for the mean buoyancy excess of the cloud ensemble over the environment, and for the fractional entrainment rate into the ensemble (which is an important quantity which has to be specified in mass flux convection parametrization schemes which are commonly used in weather and climate prediction models). Finally, the importance in describing cumulus convection of the single non-dimensional parameter available in this similarity theory (the ratio of the cloud base mass flux to the derived velocity scale) is discussed