The validity of the kinetic collection equation revisited – Part 2: Simulations for the hydrodynamic kernel

Alfonso, L.; Raga, Graciela B.; Baumgardner, Darrel

doi:10.5194/acp-10-7189-2010

Cited by 8 publications

(13 citation statements)

References 17 publications

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“…Since analytical expressions for the gelation time only exist for very simple kernels, Inaba et al (1999) proposed that it could be estimated numerically by Monte Carlo simulations (an approach followed in Alfonso et al, 2008Alfonso et al, , 2010.…”

Section: Alfonso Et Al: Part 3: Sol-gel Transition Under Turbulenmentioning

confidence: 99%

The validity of the kinetic collection equation revisited – Part 3: Sol–gel transition under turbulent conditions

2013

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Abstract. Warm rain in real clouds is produced by the collision and coalescence of an initial population of small droplets. The production of rain in warm cumulus clouds is still one of the open problems in cloud physics, and although several mechanisms have been proposed in the past, at present there is no complete explanation for the rapid growth of cloud droplets within the size range of diameters from 10 to 50 µm. By using a collection kernel enhanced by turbulence and a fully stochastic simulation method, the formation of a runaway droplet is modeled through the turbulent collection process. When the runaway droplet forms, the traditional calculation using the kinetic collection equation is no longer valid, since the assumption of a continuous distribution breaks down. There is in essence a phase transition in the system from a continuous distribution to a continuous distribution plus a runaway droplet. This transition can be associated to gelation (also called sol-gel transition) and is proposed here as a mechanism for the formation of large droplets required to trigger warm rain development in cumulus clouds. The fully stochastic turbulent model reveals gelation and the formation of a droplet with mass comparable to the mass of the initial system. The time when the sol-gel transition occurs is estimated with a Monte Carlo method when the parameter ρ (the ratio of the standard deviation for the largest droplet mass over all the realizations to the averaged value) reaches its maximum value. Moreover, we show that the non-turbulent case does not exhibit the sol-gel transition that can account for the impossibility of producing raindrop embryos in such a system. In the context of cloud physics theory, gelation can be interpreted as the formation of the "lucky droplet" that grows at a much faster rate than the rest of the population and becomes the embryo for runaway raindrops.

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Section: Alfonso Et Al: Part 3: Sol-gel Transition Under Turbulenmentioning

confidence: 99%

The validity of the kinetic collection equation revisited – Part 3: Sol–gel transition under turbulent conditions

2013

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“…However, for a finite system (with no critical behavior), the relative standard deviation (standard deviation of mass divided by mean mass) of the mass of the largest droplet σ (S max ) is expected to reach maximum for a time close to T gel = [CM 2 (t 0 )] −1 . This was explored in previous studies (Inaba, 1999;Alfonso et al, 2008Alfonso et al, , 2010Alfonso et al, , 2013, where σ was calculated for a finite system from Monte Carlo simulations in order to estimate the sol-gel transition times for the corresponding deterministic model of an infinite system. We can perform an example calculation of σ by using the species formulation of the SSA (Laurenzi and Diamond, 2002), in this case:…”

Section: Estimating the Time Of Gel Formationmentioning

confidence: 99%

“…9 for the product kernel) in this case, Monte Carlo simulations for the finite system could provide insightful information. The sol-gel transition time can be estimated approximately by calculating the time at which the time series of σ (S max ) in the SSA exhibits a maximum (Alfonso et al, 2010). As in the case of multiplicative kernel, the time evolution of σ (S max ) is calculated for a cloud volume of 1 cm 3 with an initial bidisperse distribution (20 droplets of 17 µm in radius and 10 droplets of 21.4 µm), and the time evolution of σ (S max ) is calculated from 1000 realizations (N r = 1000) of the SSA.…”

Section: Estimating the Time Of Gel Formation And The Gel Massmentioning

confidence: 99%

“…For kernels relevant to cloud physics, we have a similar situation (a decrease in the time of occurrence of the phase tran- sition as the number of particles in the initial distribution increases). A more detailed discussion of this problem for realistic collection kernels can be found in Alfonso et al (2010Alfonso et al ( , 2013.…”

Section: Estimating the Time Of Gel Formation And The Gel Massmentioning

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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“…The known limitations of the KCE are analyzed carefully in two papers (Alfonso et al, 2008 andAlfonso et al, 2010) by a direct comparison of numerical and analytical solutions of the KCE with true averages obtained with the stochastic method of Gillespie (1976). In these papers, a numerical criterion is proposed in order to calculate the validity time or breakdown time of the KCE.…”

Section: Introductionmentioning

confidence: 99%

A Monte Carlo Framework to Simulate Multicomponent Droplet Growth by Stochastic Coalescence

Alfonso¹,

Raga²,

Baumgardner³

2011

Applications of Monte Carlo Method in Science and Engineering

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The validity of the kinetic collection equation revisited – Part 2: Simulations for the hydrodynamic kernel

Cited by 8 publications

References 17 publications

The validity of the kinetic collection equation revisited – Part 3: Sol–gel transition under turbulent conditions

The validity of the kinetic collection equation revisited – Part 3: Sol–gel transition under turbulent conditions

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

A Monte Carlo Framework to Simulate Multicomponent Droplet Growth by Stochastic Coalescence

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