Abstract
Scaling of thermodynamic variance in shallow‐cumulus‐topped mixed layers is studied using large‐eddy simulation (LES). First, the performance of the top‐down scaling (the turbulent flux at mixed‐layer top divided by w~*~) is evaluated for transient shallow‐cumulus convection over land. The results indicate that this scaling fails to capture all the variance in the top half of the mixed layer when shallow cumulus clouds are present. A variance‐budget analysis is then performed, to derive a new scaling for the variance at mixed‐layer top, which differs from the standard top‐down scaling by a factor of one Richardson number. The essential new features of the proposed scaling are that the local vertical gradient is retained and that a balance is assumed between gradient production of variance and removal by transport and dissipation, using an adjustment time‐scale given by w~*~/h. Evaluation against LES for a range of different cases, including a dry convective boundary layer as well as steady‐state marine and transient continental shallow cumulus, reveals a data‐collapse of the newly‐scaled variance, for all hours and all cases in the top half of the mixed layer. The corresponding vertical structure is shown to resemble a power‐law function. The results suggest that the structure of variance in the dry convective boundary layer is similar to that in the sub‐cloud mixed layer. In transient situations, the scaling reproduces the time‐development of variance at sub‐cloud mixed‐layer top. The new cloud‐base variance scale is then further interpreted in the context of statistical cloud schemes, which depend on the variance as the second moment of the associated probability density function. The results suggest that the area fraction of the moist convective thermals uniquely depends on the ratio of cloud‐base transition‐layer depth to sub‐cloud mixed‐layer depth. This puts ‘valve’‐ or ventilation‐type closures for the cloud‐base mass flux in the context of the variance budget for the sub‐cloud layer. Copyright © 2007 Royal Meteorological Society