A Robust Numerical Solution of the Stochastic Collection–Breakup Equation for Warm Rain

• Published in: Journal of applied meteorology and climatology Vol. 46; no. 9; pp. 1480 - 1497
• Main Authors: Prat, Olivier P., Barros, Ana P.
• Format: Journal Article
• Language: English
• Published: Boston, MA American Meteorological Society 01.09.2007
• Subjects: Chemical engineering; Clouds; Differential equations; Drop size distribution; Earth, ocean, space; Equilibrium; Evaporation; Exact sciences and technology; External geophysics; Geophysics. Techniques, methods, instrumentation and models; Mathematical moments; Meteorology; Methods; Numerical analysis; Rain; Studies; Water in the atmosphere (humidity, clouds, evaporation, precipitation); rainfall; Water drop; atmospheric precipitation; Heavy rain; Collision; Forecast model; Dimension spectrum; Discretization method; Cloud physics; Numerical solution; Coalescence; Stochastic equation; Hydrometeor
• ISSN: 1558-8424; 1558-8432
• DOI: 10.1175/jam2544.1

The focus of this paper is on the numerical solution of the stochastic collection equation–stochastic breakup equation (SCE–SBE) describing the evolution of raindrop spectra in warm rain. The drop size distribution (DSD) is discretized using the fixed-pivot scheme proposed by Kumar and Ramkrishna, and new discrete equations for solving collision breakup are presented. The model is evaluated using established coalescence and breakup parameterizations (kernels) available in the literature, and in that regard this paper provides a substantial review of the relevant science. The challenges posed by the need to achieve stable and accurate numerical solutions of the SCE–SBE are examined in detail. In particular, this paper focuses on the impact of varying the shape of the initial DSD on the equilibrium solution of the SCE–SBE for a wide range of rain rates and breakup kernels. The results show that, although there is no dependence of the equilibrium DSD on initial conditions for the same rain rate and breakup kernel, there is large variation in the time that it takes to reach steady state. This result suggests that, in coupled simulations of in-cloud motions and microphysics and for short time scales (<30 min) for which transient conditions prevail, the equilibrium DSD may not be attainable except for very heavy rainfall. Furthermore, simulations for the same initial conditions show a strong dependence of the dynamic evolution of the DSD on the breakup parameterization. The implication of this result is that, before the debate on the uniqueness of the shape of the equilibrium DSD can be settled, there is critical need for fundamental research including laboratory experiments to improve understanding of collisional mechanisms in DSD evolution.