Including radiative effects in an entrainment rate formula for buoyancy-driven PBLs

The effect of longwave radiative cooling at the planetary boundary layer (PBL) top in determining the entrainment rate was examined in this study. The entrainment rate equation that accounts for longwave radiative cooling can be formally derived from the following steps: 1) Derive the mean buoyancy budget in the thin layer between the averaged entrainment flux (i.e., the minimum buoyancy flux) level and the top of the entrainment zone where turbulent fluxes vanish. 2) Use the Deardorff entrainment closure assumption that the entrainment buoyancy flux is proportional to the vertically averaged buoyancy flux over the whole PBL, which is a generalized form of a widely accepted entrainment closure for the surface-heated convective PBL. This leads to an entrainment velocity that depends linearly on both the inverse of the interfacial Richardson number and the radiative flux divergence above the entrainment buoyancy flux level. The relative importance of these two terms was examined through large eddy simulations (LESs) of several smoke-cloud-topped PBLs with various radiative forcings, radiative properties, temperature inversion strengths, and numerical advective schemes. These PBLs are driven only by cloud-top radiative cooling. The LESs were performed with a fine-grid nesting layer in the entrainment zone where the grid size is about 16 m and 8 m in the horizontal and vertical, respectively, which should be sufficient to resolve entrainment processes, at least for cases with weak capping inversions. The LESs showed that the contribution to entrainment rate from the radiative flux divergence term was either larger than or about equal to that of the interfacial Richardson number term. The LESs also showed that this radiative flux divergence occurs within cloudy regions below the local cloud tops. The analysis was extended to the stratocumulus-topped PBL by using a stratocumulus-like smoke layer, and the result showed that the radiative flux contribution to the entrainment rate was still significant. Based on the physical understanding that this portion of the radiative flux divergence occurs within the upper half of cloud (or smoke) hummocks, the authors were able to analytically derive a relationship that links the radiative flux divergence to the cloud-top fluctuations, which were then empirically related to the interfacial Richardson number.