Evaluating Stochastic Collection Equation Solution Schemes with a Focus on Cloud Radar Doppler Spectra in Drizzling Stratocumulus

This study evaluates some proposed schemes that are designed to solve the stochastic collection equation (SCE) describing collision-coalescence of cloud particles. By comparing three schemes [Berry and Reinhardt (1974, BR74), Jacobson et al. (1994, J94), and Bott (2000, B00)] using a box model simulating warm rain formation, we found that all converge to an identical solution at sufficient grid refinement that can be regarded as the correct solution, in contrast to findings from an earlier study that B00 does not converge. In spite of the convergence itself, however, the rate of convergence is different for each scheme; the J94 scheme converges far slower than the other schemes and shows pronounced numerical diffusion at the large-drop tail of size distribution. Among the three schemes, the B00 scheme is selected for implementation in a 3D model on the basis that it is well-converged on a relatively coarse bin grid, is stable for large time steps, and is the most computationally efficient.

Using the 3D model, the J94 and B00 schemes are compared in large-eddy simulations of a drizzling stratocumulus field. A forward simulator that produces Doppler spectra from the bin microphysics model output is used to compare the model output directly with Doppler cloud radar observations. Where the J94 scheme predicts excessively large mean Doppler velocities and negatively skewed Doppler spectra compared to observations (consistent with numerical diffusion demonstrated in the box model), statistics obtained using the B00 scheme are closer to the observations, although notable differences remain. To evaluate other potential sources of numerical error, we also present some preliminary tests of sedimentation, activation, and diffusional growth schemes.