A similarity hypothesis for shallow-cumulus transports

A similarity theory for cumulus convection is proposed, and applied to the problem of cumulus transports, using data from several large-eddy simulations to test the theory. The parameters in the similarity hypothesis include the cloud-base mass flux, the buoyancy of a parcel undergoing reversible ascent through the cloud layer and the depth of the cloud layer. Using arguments based on the turbulence kinetic-energy budget a velocity-scale is derived, in addition to the conventional Convective Available Potential Energy scale. This new scale is analogous to the convective velocity-scale used in boundary-layer theories, but incorporates the effects of latent-heat release. The ratio between the cloud-base mass flux and the velocity-scale is found to be a key parameter in describing cumulus convection. It is shown that the similarity hypothesis can be applied to mass-flux schemes to determine the fractional entrainment rate. For this the entrainment rate is assumed to be related to the rate of generation of turbulent kinetic energy in the cloud ensemble. The resulting scaling is tested against fractional entrainment rates diagnosed from the large-eddy simulations.