Atmospheric Chemistry and Physics Discussions
www.atmos-chem-phys-discuss.net
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2007
10.5194/acpd-7-13733-2007
http://www.atmos-chem-phys-discuss.net/7/13733/2007/
http://www.atmos-chem-phys-discuss.net/7/13733/2007/acpd-7-13733-2007.html
http://www.atmos-chem-phys-discuss.net/7/13733/2007/acpd-7-13733-2007.pdf
13733
13771
2007-09-20
The validity of the kinetic collection equation revisited
L. Alfonso
lesterson@yahoo.com
G. B. Raga
D. Baumgardner
Universidad Autónoma de la Ciudad de México, México City, 09790 México
Centro de Ciencias de la Atmósfera, UNAM, México City, 04510 México
The kinetic collection equation (KCE) describes the evolution of
the average droplet spectrum due to successive events of collision and
coalescence. Fluctuations and non-zero correlations present in the
stochastic coalescence process would imply that the size distributions may
not be correctly modelled by the KCE.
<br><br>
In this study we expand the known analytical studies of the coalescence
equation with some numerical tools such as Monte Carlo simulations of the
coalescence process. The validity time of the KCE was estimated by
calculating the maximum of the ratio of the standard deviation for the
largest droplet mass over all the realizations to the averaged value. A good
correspondence between the analytical and the numerical approaches was found
for all the kernels studied. The expected values from analytical solutions
of the KCE, were compared with true expected values of the stochastic
collection equation (SCE) estimated with Gillespie's Monte Carlo algorithm
and analytical solutions of the SCE, after and before the breakdown time.
<br><br>
The possible implications for cloud physics are discussed, in particular the
possibility of application of these results to kernels modified by
turbulence and electrical processes.
Aldous, D. J.: Deterministic and stochastic models for coalescence (aggregation, coagulation): A review of the mean-field theory for probabilists, http://www.stat.berkeley.edu/users/aldous, 1997.
Bayewitz, M. H., Yerushalmi, J., Katz, S., and Shinnar, R.: The extent of correlations in a stochastic coalescence process, J. Atmos. Sci., 31, 1604–1614, 1974.
Drake, R. L.: The scalar transport equation of coalescence theory: Moments and kernels, J. Atmos. Sci., 29, 537–547, 1972.
Drake, R. L. and Wright, T. J.: The scalar transport equation of coalescence theory: New families of exact solutions, J. Atmos. Sci., 29, 548–556, 1972.
Gillespie, D. T.: The stochastic coalescence model for cloud droplet growth, J. Atmos. Sci., 29, 1496–1510, 1972.
Gillespie, D. T.: An exact method for numerically simulating the stochastic coalescence process in a cloud, J . Atmos. Sci., 32, 1977–1989, 1975.
Gillespie, D. T.: A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Comput. Phys., 22, 403–434, 1976.
Golovin, A. M.: The solution of the coagulating equation for cloud droplets is a rising air current, B. Acad. Sci. USSR, Geophys. Ser., No. 5, 482–487, 1963.
Inaba, S., Tanaka, H., Ohtsuki, K., and Nakazawa, K.: High-accuracy statistical simulation of planetary accretion: I. Test of the accuracy by comparison with the solution to the stochastic coagulation equation, Earth Planets Space, 51, 205–217, 1999.
Lee, M. H.: On the validity of the coagulation equation and the nature of runaway growth, Icarus, 142, 74–86, 2000.
Laurenzi, I. J. and Diamond, S. L.: Monte Carlo simulation of the heterotypic aggregation kinetics of platelets and neutrophils, Biophys. J., 77, 1733–1746, 1999.
Laurenzi, I. J. and Diamond, S. L.: Kinetics of random aggregation-fragmentation processes with multiple components, Phys. Rev. E, 67, 051103, doi:10.1103/PhysRevE.67.051103, 2003.
Long, A. B.: Solutions to the droplet collection equation for polynomial kernels, J. Atmos. Sci., 31, 1040–1051, 1974.
Lushnikov, A. A.: Certain new aspects of the coagulation theory, Izvestiya, Atmos. Ocean Phys., 14, 738–743, 1978, 1978.
Pruppacher, H. R. and Klett, J. D.: Microphysics of clouds and precipitation, Kluwer Academic Publishers, 1997.
Scott, W. T.: Analytic studies of cloud droplet coalescence, J. Atmos. Sci., 25, 54–65, 1968.
Tanaka, H. and Nakazawa, K.: Stochastic coagulation equation and the validity of the statistical coagulation equation, J. Geomagn. Geoelectr., 45, 361–381, 1993.
Tanaka, H. and Nakazawa, K.: Validity of the statistical coagulation equation and runaway growth of protoplanets, Icarus, 107, 404–412, 1994.
Valioulis, I. A. and List, E. J.: A numerical evaluation of the stochastic completeness of the kinetic coagulation equation, J. Atmos. Sci., 41, 2516–2529, 1984.
Wang, L. P., Xue, Y., Ayala, O., and Grabowski, W. W.: Effect of stochastic coalescence and air turbulence on the size distribution of cloud droplets, Atmos. Res., 82, 416–432, 2006.