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Atmospheric Research 52 1999 115–141 www.elsevier.comrlocateratmos
The effect of diffusion kinetics on the supersaturation in clouds
You-Suo Zou, Norihiko Fukuta
Department of Meteorology, UniÕersity of Utah, Salt Lake City, UT 84112, USA Received 23 November 1998; received in revised form 30 March 1999; accepted 7 April 1999
Abstract
To describe the nucleation–growth interaction at and above the cloud base, a cloud model has been formulated including the haze process below the cloud base and before nucleation of cloud
droplets, and a proper diffusion-kinetic droplet growth equation. Analytical equations for the w
maximum supersaturation by Twomey Twomey, S., 1959. The nuclei of natural cloud formation: Part II. The supersaturation in natural clouds and the variation of cloud droplet concentration.
Geofis. Pura Appl. 43, 243–249; Twomey, S., 1977. Atmospheric Aerosols. Elsevier, Amsterdam, x
w 302 pp. and by Fukuta and Xu Fukuta, N., Xu, N., 1996. Nucleation–droplet growth interactions
x and microphysical property development in convective clouds. Atmos. Res. 41, 1–22 , both
employing the Maxwellian growth theory, are compared with the model outputs, and the model outputs are found to be the smallest. To preserve the clarity and usefulness of the analytical
equations, a correction factor to the latter is obtained in comparison with the model results for
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different condensation coefficient b and cloud condensation nucleus activity spectra of typical continental and maritime air masses. Effect of b is found to be larger for small b and in
continental clouds. q 1999 Elsevier Science B.V. All rights reserved.
Keywords: Diffusion kinetics; Supersaturation; Nucleation–growth interaction; Clouds
1. Introduction
The microphysical properties of clouds, the droplet size distribution in particular, are known to significantly affect their radiative properties and processes leading to precipita-
Corresponding author. Fax: q1-801-585-3681; E-mail: nfukutaatmos.met.utah.edu 0169-8095r99r - see front matter q 1999 Elsevier Science B.V. All rights reserved.
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PII: S 0 1 6 9 - 8 0 9 5 9 9 0 0 0 2 5 - 3
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tion development IPCC, 1995 . Cloud droplets form on cloud condensation nuclei Ž
. CCN ; the number concentration and the size spectrum of droplets are determined by
the interaction between the supersaturation-generating updraft and the supersaturation- reducing growth process of droplets that formed on CCN according to the supersatura-
tion the interaction achieves. The interaction yields a supersaturation maximum which primarily decides the number concentration of the formed droplets, and the number
concentration remains approximately constant thereafter until precipitation processes begin. Throughout the process, entrainment and turbulence widen the size distribution.
Determination of the maximum cloud supersaturation specifically involves chemistry of nucleus particles, haze stage of their growth, nucleation, and the following droplet
growth in the dynamically changing cloud environment. The early works on the cloud Ž
. Ž
. supersaturation were carried out mostly by Howell 1949 , Squires 1952 , Mordy
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Ž 1959 and Twomey 1959 . More studies followed in the subsequent years e.g., Mason
and Chien, 1962; Mason and Jonas, 1974; Warner, 1969; Arnason and Brown, 1971; Soong, 1974; Ochs, 1978; Clark and Hall, 1979; Saxena and Fukuta, 1982; Robinson,
. 1984; Politovich and Cooper, 1988; Grabowski, 1989 . Numerical works recently carried
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out are typically Young 1993 , Chen 1994a,b , Xu 1994 , Korolov 1995 , Fukuta and Ž
. Xu
1996 . These cloud models or numerical simulations, though useful, require considerable time and effort in construction and hence are not convenient for practical
application as compared to analytical formulas. Nevertheless, cloud processes are so complicated that it is not possible to obtain an analytical solution to predict the
supersaturation maximum, including all the details of interactions involved.
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Twomey 1959 successfully worked out an analytical formula for the maximum supersaturation. It is based on the Maxwellian droplet growth theory and does not deal
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with the more realistic diffusion-kinetic DK theory Fukuta and Walter, 1970 . The Ž
. DK theory describes the effects of the condensation coefficient
b and thermal
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. accommodation coefficients a . Leaitch et al. 1986 pointed out the importance of the
DK effect stating that a change in condensation coefficient represents a change in the early droplet growth rate, which would in fact modify the supersaturation in a cloud.
Most importantly, this would alter the maximum supersaturation in the cloud, which would then modulate the droplet concentration. The DK effect was discussed by Hudson
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. 1993 for CCN spectrum measurement in his device. Under this circumstance, Fukuta
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1993 and Fukuta and Xu 1996 developed an analytical method which includes both Maxwellian and DK droplet growth theories for the maximum supersaturation predic-
tion. In derivation of both Twomey’s and Fukuta–Xu’s analytical formulas, some restricting assumptions were made to simplify the processes in clouds, omission of haze
process in particular. The problem now is, therefore, to determine how realistic these formulas are by constructing and applying a numerical model which includes all the
dominant processes. If the analytical formulas are not accurate enough to describe the cloud process, we need to obtain the correction with the help of the model to make them
practical. The purpose of this paper is then to construct a new cloud model based on the DK growth theory and to compare the results with the existing analytical solutions
involving the Maxwellian theory to obtain such correction factors.
2. Description of the new cloud model
