Abstract
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.
Highlights
The evolution of the size distribution of coalescing particles has often been described by the kinetic collection or Smoluchowski coagulation equation, known under a number of names (“stochastic collection”, “coalescence”)
We present an algorithm that can be applied to obtain the solution of Eq (2) for any type of kernel and initial conditions
At the large end of the spectrum, results differ substantially. This is in agreement with the analytical study of Tanaka and Nakazawa (1994), who demonstrated that the true stochastic averages coincide well with those obtained from the kinetic collection Eq (1) if the bin mass k satisfies the inequality k2 M0, where M0 is the total mass of the system
Summary
The evolution of the size distribution of coalescing particles has often been described by the kinetic collection (hereafter KCE) or Smoluchowski coagulation equation, known under a number of names (“stochastic collection”, “coalescence”). Exact solutions of Eq (2) are only known for a limited number of cases (constant, sum and product kernels) and for monodisperse initial conditions For these special cases the master equation has been solved by Lushnikov (1978, 2004) and Tanaka and Nakazawa (1993) in terms of the generating function of P (n ; t). We present an algorithm that can be applied to obtain the solution of Eq (2) for any type of kernel and initial conditions By applying this method, numerical solutions of the master equation were obtained for realistic kernels relevant to cloud physics, along with calculation of the correlations for the number of droplets for different sizes.
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