Autoconversion rate bias in stratiform boundary layer cloud parameterizations

Because of their large grid-box size, global climate models do not explicitly represent small-scale processes occurring in cloud systems in the marine boundary layer. One such process, which is thought to have an important climatological effect, is the production of warm rain. Parameterizations of this process typically partition the liquid water into a cloud and a rain component. The rate of conversion (autoconversion) of cloud to rainwater is expressed as a convex function of the local cloud liquid water content. It is well known that the distribution of cloud liquid water content within boundary layer cloud systems is spatially nonuniform. This would result in biased mean autoconversion rates if no attempt is made to model subgrid variability. Three formulations are examined, with increasing complexity, that can be used to model the distribution of liquid water content within a model grid box and assess how well each predicts the mean autoconversion rate. Assuming complete homogeneity of cloud liquid water within a model grid box results in large biases. The use of cloud fraction to partition the grid box into cloudy and clear regions substantially reduces the biases. The most significant reduction of the biases is achieved with a Gaussian distribution of saturation excess within the grid box. With this formulation, which could be facilitated using look-up tables, biases can be removed in a way consistent with the underlying distribution of saturation excess. A simple parameterization is presented that corrects much of the bias using simple algebraic expressions. It is demonstrated that to accurately calculate the mean autoconversion rate, an accurate parameterization of the width of the saturation excess distribution is required.