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Atmospheric Chemistry and Physics An interactive open-access journal of the European Geosciences Union
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Volume 15, issue 21
Atmos. Chem. Phys., 15, 12315–12326, 2015
https://doi.org/10.5194/acp-15-12315-2015
© Author(s) 2015. This work is distributed under
the Creative Commons Attribution 3.0 License.
Atmos. Chem. Phys., 15, 12315–12326, 2015
https://doi.org/10.5194/acp-15-12315-2015
© Author(s) 2015. This work is distributed under
the Creative Commons Attribution 3.0 License.

Research article 06 Nov 2015

Research article | 06 Nov 2015

An algorithm for the numerical solution of the multivariate master equation for stochastic coalescence

L. Alfonso L. Alfonso
  • Universidad Autónoma de la Ciudad de México, Mexico City 09790, Mexico

Abstract. In cloud modeling studies, the time evolution of droplet size distributions due to collision–coalescence events is usually modeled with the Smoluchowski coagulation equation, also known as the kinetic collection equation (KCE). However, the KCE is a deterministic equation with no stochastic fluctuations or correlations. Therefore, the full stochastic description of cloud droplet growth in a coalescing system must be obtained from the solution of the multivariate master equation, which models the evolution of the state vector for the number of droplets of a given mass. Unfortunately, due to its complexity, only limited results were obtained for certain types of kernels and monodisperse initial conditions. In this work, a novel numerical algorithm for the solution of the multivariate master equation for stochastic coalescence that works for any type of kernels, multivariate initial conditions and small system sizes is introduced. The performance of the method was seen by comparing the numerically calculated particle mass spectrum with analytical solutions of the master equation obtained for the constant and sum kernels. Correlation coefficients were calculated for the turbulent hydrodynamic kernel, and true stochastic averages were compared with numerical solutions of the kinetic collection equation for that case. The results for collection kernels depending on droplet mass demonstrates that the magnitudes of correlations are significant and must be taken into account when modeling the evolution of a finite volume coalescing system.

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The mathematical description of finite volume coalescing systems relies on the multivariate master equation. However, due to its complexity, it has analytical solutions only for a limited number of kernels and initial conditions. In this paper, in an effort to solve this problem, we have introduced a novel numerical approach to calculate the solution of the multivariate coalescence master equation that works for any type of kernel and initial conditions.
The mathematical description of finite volume coalescing systems relies on the multivariate...
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