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

Alfonso, L.

doi:10.5194/acp-15-12315-2015

Cited by 9 publications

(28 citation statements)

References 18 publications

(30 reference statements)

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“…These differences may be caused by the way the coalescence efficiency tables are interpolated. Another possible source of discrepancies is the numerical diffusion present in the finite-differences method of Alfonso (2015). To test whether the "one-to-one" method also gives correct fluctuations in the number of collisions, the relative standard deviation of mass of the largest droplet σ (m max )/ m max is plotted in Fig.…”

Section: The Super-droplet Methodsmentioning

confidence: 99%

“…Solving the master equation numerically is extremely difficult due to a huge phase space to be considered. Recently, Alfonso (2015) developed a method to solve the master equation numerically, but was only able to apply the method to a system of up to 40 droplets (Alfonso and Raga, 2017). Alternatively, the stochastic simulation algorithm (SSA) (Gillespie, 1975;Seesselberg et al, 1996) can be used to model a single trajectory obeying the master equation, but obtaining large enough statistics would require very long computations.…”

Section: Introductionmentioning

confidence: 99%

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Stochastic coalescence in Lagrangian cloud microphysics

Dziekan

Pawłowska

2017

Atmos. Chem. Phys.

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Abstract. Stochasticity of the collisional growth of cloud droplets is studied using the super-droplet method (SDM) of Shima et al. (2009). Statistics are calculated from ensembles of simulations of collision-coalescence in a single well-mixed cell. The SDM is compared with direct numerical simulations and the master equation. It is argued that SDM simulations in which one computational droplet represents one real droplet are at the same level of precision as the master equation. Such simulations are used to study fluctuations in the autoconversion time, the sol-gel transition and the growth rate of lucky droplets, which is compared with a theoretical prediction. The size of the coalescence cell is found to strongly affect system behavior. In small cells, correlations in droplet sizes and droplet depletion slow down rain formation. In large cells, collisions between raindrops are more frequent and this can also slow down rain formation. The increase in the rate of collision between raindrops may be an artifact caused by assuming an overly large wellmixed volume. The highest ratio of rain water to cloud water is found in cells of intermediate sizes. Next, we use these precise simulations to determine the validity of more approximate methods: the Smoluchowski equation and the SDM with multiplicities greater than 1. In the latter, we determine how many computational droplets are necessary to correctly model the expected number and the standard deviation of the autoconversion time. The maximal size of a volume that is turbulently well mixed with respect to coalescence is estimated at V mix = 1.5 × 10 −2 cm 3 . The Smoluchowski equation is not valid in such small volumes. It is argued that larger volumes can be considered approximately well mixed, but such approximation needs to be supported by a comparison with fine-grid simulations that resolve droplet motion.

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Section: The Super-droplet Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stochastic coalescence in Lagrangian cloud microphysics

Dziekan

Pawłowska

2017

Atmos. Chem. Phys.

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“…However, the SSA has difficulties in accurately reproducing the large end of the droplet size distribution. This is due to the huge number of realizations required to obtain smooth behavior at the large end of the droplet size distribution (Alfonso, 2015). The alternative approach (within the stochastic framework) is to use the master equation:…”

Section: Introductionmentioning

confidence: 99%

“…2), and the deterministic approach for an infinite system by using the KCE (Eq. 1), using the numerical algorithm reported in Alfonso (2015). By the time the gel forms, certain differences are to be expected between the two approaches at the large end of the droplet size distribution.…”

Section: Introductionmentioning

confidence: 99%

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The impact of fluctuations and correlations in droplet growth by collision–coalescence revisited – Part 1: Numerical calculation of post-gel droplet size distribution

Alfonso

Raga

2017

Atmos. Chem. Phys.

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Abstract. The impact of stochastic fluctuations in cloud droplet growth is a matter of broad interest, since stochastic effects are one of the possible explanations of how cloud droplets cross the size gap and form the raindrop embryos that trigger warm rain development in cumulus clouds. Most theoretical studies on this topic rely on the use of the kinetic collection equation, or the Gillespie stochastic simulation algorithm. However, the kinetic collection equation is a deterministic equation with no stochastic fluctuations. Moreover, the traditional calculations using the kinetic collection equation are not valid when the system undergoes a transition from a continuous distribution to a distribution plus a runaway raindrop embryo (known as the sol-gel transition). On the other hand, the stochastic simulation algorithm, although intrinsically stochastic, fails to adequately reproduce the large end of the droplet size distribution due to the huge number of realizations required. Therefore, the full stochastic description of cloud droplet growth must be obtained from the solution of the master equation for stochastic coalescence.In this study the master equation is used to calculate the evolution of the droplet size distribution after the sol-gel transition. These calculations show that after the formation of the raindrop embryo, the expected droplet mass distribution strongly differs from the results obtained with the kinetic collection equation. Furthermore, the low-mass bins and bins from the gel fraction are strongly anticorrelated in the vicinity of the critical time, this being one of the possible explanations for the differences between the kinetic and stochastic approaches after the sol-gel transition. Calculations performed within the stochastic framework provide insight into the inability of explicit microphysics cloud models to explain the droplet spectral broadening observed in small, warm clouds.

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Examination of an improved quasi‐stochastic model for the collisional growth of drops

et al. 2017

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The evolution of cloud drop size distribution due to the collision‐coalescence process is generally described by a quasi‐stochastic model that solves the stochastic collection equation in a deterministic way. In this study, an improved quasi‐stochastic (IQS) model, which is derived by rigorously considering a finite model time step, is examined in the context of comparison with the normal quasi‐stochastic (NQS) model. The IQS model allows a large collector drop to collide with a small collected drop more than one time in a model time step even if the collision probability is small. The number distribution of collector drops then follows the Poisson distribution with respect to the number of collisions. Using a box model that takes turbulence‐induced collision enhancement into account, it is found that large drops in the IQS model tend to have larger sizes than those in the NQS model and that the IQS model accelerates large‐drop formation by a few minutes compared to the NQS model. The effects of the IQS model depend on the model time step and the shape of initial drop size distribution. The IQS model is incorporated into a detailed bin microphysics scheme that is coupled with the Weather Research and Forecasting model, and a single warm cloud is simulated under idealized environmental conditions. It is found that the onset of surface precipitation is accelerated in the IQS model.

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

Cited by 9 publications

References 18 publications

Stochastic coalescence in Lagrangian cloud microphysics

Stochastic coalescence in Lagrangian cloud microphysics

The impact of fluctuations and correlations in droplet growth by collision–coalescence revisited – Part 1: Numerical calculation of post-gel droplet size distribution

Examination of an improved quasi‐stochastic model for the collisional growth of drops

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