The Scalar Transport Equation of Coalescence Theory: Moments and Kernels

Drake, Ronald L.

doi:10.1175/1520-0469(1972)029<0537:tsteoc>2.0.co;2

Cited by 43 publications

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“…In Alfonso et al (2008), the numerical criteria suggested by Inaba et al (1999), to calculate the validity time for the KCE was compared with the analytical results obtained by Drake (1972) and Tanaka and Nakazawa (1994), with good agreement. Inaba et al (1999) proposed that the stochastic property of the system becomes distinct around the beginning of runaway growth.…”

Section: Introductionmentioning

confidence: 90%

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

Alfonso¹,

Raga²,

Baumgardner³

2010

Atmos. Chem. Phys.

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Abstract. The kinetic collection equation (KCE) has been widely used to describe the evolution of the average droplet spectrum due to the collection process that leads to the development of precipitation in warm clouds. This deterministic, integro-differential equation only has analytic solution for very simple kernels. For more realistic kernels, the KCE needs to be integrated numerically. In this study, the validity time of the KCE for the hydrodynamic kernel is estimated by a direct comparison of Monte Carlo simulations with numerical solutions of the KCE. The simulation results show that when the largest droplet becomes separated from the smooth spectrum, the total mass calculated from the numerical solution of the KCE is not conserved and, thus, the KCE is no longer valid. This result confirms the fact that for kernels appropriate for precipitation development within warm clouds, the KCE can only be applied to the continuous portion of the mass distribution.

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Section: Introductionmentioning

confidence: 90%

“…5 we discuss the results and possible implications for cloud physics modeling. Drake (1972) calculated the analytical solutions of the KCE for polynomials of the formK(x i ,x j ) = C(x i ×x j ). The time evolution of the second moment (with respect to the droplet distribution), M 2 (t), is given by:…”

Section: Introductionmentioning

confidence: 99%

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

Alfonso¹,

Raga²,

Baumgardner³

2010

Atmos. Chem. Phys.

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“…(12) is N (i, t)=N 0 δ i,1 . Analytical solutions of the continuous KCE have been obtained by Golovin (1963), Scott (1968), Drake (1972) and Drake and Wright (1972) for approximations of the hydrodynamic kernel given by the polynomials K(i, j )=A, B(x i +x j ) and C(x i x j ) where x i and x j are the masses of the droplets from bins i and j . For the constant kernel K(x i , x j )=A and a monodisperse initial distribution with concentration c 0 , the analytical size distribution of the discrete KCE has the form:…”

Section: Comparison Of the Monte Carlo Algorithm With Analytical Solumentioning

confidence: 99%

Monte Carlo simulations of two-component drop growth by stochastic coalescence

Alfonso¹,

Raga²,

Baumgardner³

2009

Atmos. Chem. Phys.

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Abstract. The evolution of two-dimensional drop distributions is simulated in this study using a Monte Carlo method. The stochastic algorithm of Gillespie (1976) for chemical reactions in the formulation proposed by Laurenzi et al. (2002) was used to simulate the kinetic behavior of the drop population. Within this framework, species are defined as droplets of specific size and aerosol composition. The performance of the algorithm was checked by a comparison with the analytical solutions found by Lushnikov (1975) and Golovin (1963) and with finite difference solutions of the two-component kinetic collection equation obtained for the Golovin (sum) and hydrodynamic kernels. Very good agreement was observed between the Monte Carlo simulations and the analytical and numerical solutions. A simulation for realistic initial conditions is presented for the hydrodynamic kernel. As expected, the aerosol mass is shifted from small to large particles due to collection process. This algorithm could be extended to incorporate various properties of clouds such several crystals habits, different types of soluble CCN, particle charging and drop breakup.

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“…Numerical diffusion is a fundamentally challenging problem for the sectional methods when solving the mass transfer among bins. The problem is more serious for the collision-coagulation processes, which need to be handled with advanced numerical techniques (e.g., Drake, 1972;Tzivion et al, 1987;Landgrebe and Pratsinis, 1990;Chen and Lamb, 1994;Kumar and Ramkrishna, 1996b). Also, the growth kernel in each bin is often assumed to be constant; in reality, however, the growth kernel usually is very sensitive to aerosol size and thus may vary significantly between bin limits.…”

Section: Introductionmentioning

confidence: 99%

A statistical–numerical aerosol parameterization scheme

2013

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Abstract.A new modal aerosol parameterization scheme, the statistical-numerical aerosol parameterization (SNAP), was developed for studying aerosol processes and aerosolcloud interactions in regional or global models. SNAP applies statistical fitting on numerical results to generate accurate parameterization formulas without sacrificing details of the growth kernel. Processes considered in SNAP include fundamental aerosol processes as well as processes related to aerosol-cloud interactions. Comparison of SNAP with numerical solutions, analytical solutions, and binned aerosol model simulations showed that the new method performs well, with accuracy higher than that of the high-order numerical quadrature technique, and with much less computation time. The SNAP scheme has been implemented in regional air quality models, producing results very close to those using binned-size schemes or numerical quadrature schemes.

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The Scalar Transport Equation of Coalescence Theory: Moments and Kernels

Cited by 43 publications

References 0 publications

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

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

Monte Carlo simulations of two-component drop growth by stochastic coalescence

A statistical–numerical aerosol parameterization scheme

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