Parameterization of the vertical velocity equation for shallow cumulus clouds

The application of a steady-state vertical velocity equation for parameterized moist convective updrafts in climate and weather prediction models is nowadays common practice. This equation usually contains an advection, a buoyancy and a lateral entrainment term, whereas the effects of pressure gradient and subplume contributions are typically incorporated as proportionality constants a and b for the buoyancy and the entrainment terms respectively. We will present a summary of proposed values of these proportionality constants a and b, demonstrating that there is a large uncertainty in their most appropriate values. In order to shed new light on this situation we will present an analysis of the full vertical budget equation for shallow cumulus clouds obtained from large eddy simulations of three different GCSS intercomparison cases. It is found that the pressure gradient term is the dominant sink term in the vertical budget study, whereas the entrainment term only gives a small contribution. This result is at odds with the parameterized vertical velocity equation in the literature as it employs the entrainment term as the major sink term. As a practical solution the damping effect of the pressure term may be parameterized in terms of the lateral entrainment rates as used for thermodynamic quantities like the total specific humidity. By using a least square method we obtain case dependent optimal values for the proportionality constants a and b which are linearly related with each other. This relation can be explained from a linear relationship between the lateral entrainment rate and the buoyancy.