Entrainment into Mixed Layers. Part I: Sharp-Edged and Smoothed Tops

Abstract

A pervasive issue in the analysis of clear and cloud-containing convectively mixed layers is how to predict their growth through entrainment. This paper deals largely with the definition of the interface with the upper environment, and how that definition leads to complexities in entrainment analysis. The interface is usually quite abrupt locally, but horizontally or temporally averaged properties exhibit a more gradual transition, due to fluctuation of the interface height. The data from these transition layers produce unreliable results when applied to a mixed layer model with a flat top and zeroth-order discontinuities. Several authors have developed models with first- or higher-order discontinuities, but these vary in their structure. Presented here is a return to the zeroth-order, or ?sharp-edged,? model, but with allowance for a locally variable upper boundary by use of a transformation of the mixed layer equations into a coordinate scheme based on the local mixed layer height. The model is applied to analysis of surface heated boundary layers and stratocumulus-like smoke clouds, using data generated by large eddy simulation models. The entrainment rates predicted from the model equations and their analytic solutions are found in good agreement with most of the simulations. The effect of horizontal averaging of the interface height is approximated by use of a Gaussian transformation, based on the observed standard deviation of the mixed layer depth. With this approximation the horizontally averaged statistics are predicted and found to agree fairly well with numerical results. Some of the smoke cloud simulations show that the entrainment is much greater for cases with thin radiation cooling depths. This is probably due, at least in part, to numerical errors associated with radiative cooling in partially cloud-filled mesh boxes.