Stochastic simulations as a benchmark for mathematical methods solving the coalescence equation

Seeβelberg, M.; Trautmann, Thomas; Thorn, Matthias

doi:10.1016/0169-8095(95)00024-0

Cited by 20 publications

(18 citation statements)

References 17 publications

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“…In cells smaller than N 0 = 10 4 , rain develops slower than predicted by the Smoluchowski equation. Agreement of stochastic coalescence in large cells with the Smoluchowski equation for a similar initial distribution was shown using the SSA by Seesselberg et al (1996). Onishi et al (2015) present figures similar to Fig.…”

Section: Validity Of the Smoluchowski Equationsupporting

confidence: 71%

“…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: Validity Of the Smoluchowski Equationsupporting

confidence: 71%

Section: Introductionmentioning

confidence: 99%

Stochastic coalescence in Lagrangian cloud microphysics

Dziekan

Pawłowska

2017

Atmos. Chem. Phys.

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“…(The explicit form of the SCE is displayed in Appendix B (26).) This assumption seems to be correct (Seeßelberg et al, 1996), thus the result of the spectral (bin) method could be very accurate. However, the computational cost of the spectral (bin) method could be very high if we consider various microphysical processes and hence the number of attributes d becomes large.…”

Section: Spectral (Bin) Methodsmentioning

confidence: 99%

“…The motion and condensation/evaporation of droplets can be simulated by solving the ordinary differential equations (1) and (2) for all the droplets. A direct simulation of the stochastic coalescence process (3) can also be performed if we use the exact Monte Carlo method, which was developed by Gillespie (1975) and improved by Seeßelberg et al (1996). Their procedure repeatedly draws a random waiting time for which the next pair of droplets will coalesce.…”

Section: Direct Simulation Using the Exact Monte Carlo Methodsmentioning

confidence: 99%

“…The initial size distribution of the droplets follows an exponential distribution of the droplet volume X i , which is determined by the probability density p( (c) shows the results for the case of the so-called hydrodynamic kernel, which is a much more realistic coalescence kernel, defined by equations (3) and (4). The same coalescence efficiency E(R j , R k ) is adopted as described in Seeßelberg et al (1996) and Bott (1998): for small droplets the dataset of Davis (1972) and Jonas (1972) is used, and for large droplets the dataset of Hall (1980) is used. Because the analytic solution is not available for the hydrodynamic kernel, we generated the reference solution numerically using the exponential flux method (EFM) developed by Bott (1998Bott ( , 2000.…”

Section: Validation Of Our Monte Carlo Schemementioning

confidence: 99%

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The super‐droplet method for the numerical simulation of clouds and precipitation: a particle‐based and probabilistic microphysics model coupled with a non‐hydrostatic model

Shima

Kusano

Kawano

et al. 2009

Quart J Royal Meteoro Soc

217

379

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A novel, particle-based, probabilistic approach for the simulation of cloud microphysics is proposed, which is named the super-droplet method (SDM). This method enables the accurate simulation of cloud microphysics with a less demanding cost in computation. SDM is applied to a warm-cloud system, which incorporates sedimentation, condensation/evaporation and stochastic coalescence. The methodology to couple super-droplets and a non-hydrostatic model is also developed. It is confirmed that the result of our Monte Carlo scheme for the stochastic coalescence of super-droplets agrees fairly well with the solutions of the stochastic coalescence equation. The behaviour of the model is evaluated using a simple test problem, that of a shallow maritime cumulus formation initiated by a warm bubble. Possible extensions of SDM are briefly discussed. A theoretical analysis suggests that the computational cost of SDM becomes lower than the spectral (bin) method when the number of attributes -the variables that identify the state of each superdroplet -becomes larger than some critical value, which we estimate to be in the range 2 ∼ 4.

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Cloud droplets to drizzle: Contribution of transition drops to microphysical and optical properties of marine stratocumulus clouds

Glienke

Kostinski

Fugal

et al. 2017

Geophysical Research Letters

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Aircraft measurements of the ubiquitous marine stratocumulus cloud type, with over 3000 km of in situ data from the Pacific during the Cloud System Evolution in the Trades experiment, show the ability of the Holographic Detector for Clouds (HOLODEC) instrument to smoothly interpolate the small and large droplet data collected with Cloud Droplet Probe and 2DC instruments. The combined, comprehensive instrument suite reveals a surprisingly large contribution in the predrizzle size range of 40–80 μm (transition droplets, or drizzlets), a range typically not measured and assumed to reside in a condensation‐to‐collision minimum between cloud droplet and drizzle modes. Besides shedding light on the onset of collision coalescence, drizzlets are essential contributors to optical and chemical properties because of a substantial contribution to the total surface area. When adjusted to match spatial resolution of spaceborne remote sensing, the missing drizzlets bring in situ measurements to closer agreement with satellite observations.

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Stochastic simulations as a benchmark for mathematical methods solving the coalescence equation

Cited by 20 publications

References 17 publications

Stochastic coalescence in Lagrangian cloud microphysics

Stochastic coalescence in Lagrangian cloud microphysics

The super‐droplet method for the numerical simulation of clouds and precipitation: a particle‐based and probabilistic microphysics model coupled with a non‐hydrostatic model

Cloud droplets to drizzle: Contribution of transition drops to microphysical and optical properties of marine stratocumulus clouds

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