This paper by James W. Deardorff analyzes the results of a three-dimensional numerical model to study turbulence and entrainment within mixed layers containing stratocumulus clouds, with or without parameterized cloud-top radiative cooling. The model aims to eliminate many assumptions from theories of cloud-capped mixed layers but faces challenges such as poor resolution and large truncation errors in and above the capping inversion. For thick mixed layers with thick capping inversions, the cloud-top radiative cooling is found to be primarily confined within the capping inversion when it is localized to the upper 50 meters or less of the cloud. This cooling does not significantly increase buoyancy flux and turbulence in the well-mixed layer below. The optimal method for correlating the entrainment rate or mixed-layer growth rate for mixed layers with varying amounts of stratocumulus is through a functional dependence on an overall jump Richardson number, using the standard deviation of vertical velocity at the top of the mixed layer (near the center of the capping inversion) as the scaling velocity. This velocity is a fraction of the generalized convective velocity for the mixed layer as a whole, which is greater for cloud-capped mixed layers compared to clear mixed layers. The introduction highlights the common occurrence of stratocumulus-capped mixed layers over cool oceans and land behind cold fronts, noting the lack of reliable quantitative data due to measurement difficulties. The paper references pioneering work by Lilly (1968) and Schubert (1976), and Deardorff's (1976a) revisions to Lilly's theory, which allow for a more realistic representation of cloud-top cooling and entrainment.This paper by James W. Deardorff analyzes the results of a three-dimensional numerical model to study turbulence and entrainment within mixed layers containing stratocumulus clouds, with or without parameterized cloud-top radiative cooling. The model aims to eliminate many assumptions from theories of cloud-capped mixed layers but faces challenges such as poor resolution and large truncation errors in and above the capping inversion. For thick mixed layers with thick capping inversions, the cloud-top radiative cooling is found to be primarily confined within the capping inversion when it is localized to the upper 50 meters or less of the cloud. This cooling does not significantly increase buoyancy flux and turbulence in the well-mixed layer below. The optimal method for correlating the entrainment rate or mixed-layer growth rate for mixed layers with varying amounts of stratocumulus is through a functional dependence on an overall jump Richardson number, using the standard deviation of vertical velocity at the top of the mixed layer (near the center of the capping inversion) as the scaling velocity. This velocity is a fraction of the generalized convective velocity for the mixed layer as a whole, which is greater for cloud-capped mixed layers compared to clear mixed layers. The introduction highlights the common occurrence of stratocumulus-capped mixed layers over cool oceans and land behind cold fronts, noting the lack of reliable quantitative data due to measurement difficulties. The paper references pioneering work by Lilly (1968) and Schubert (1976), and Deardorff's (1976a) revisions to Lilly's theory, which allow for a more realistic representation of cloud-top cooling and entrainment.