Editorial Type:
Article
Article Type:
Research Article

Toward the Theory of Stochastic Condensation in Clouds. Part II: Analytical Solutions of the Gamma-Distribution Type

Vitaly I. Khvorostyanov Department of Meteorology, University of Utah, Salt Lake City, Utah

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Judith A. Curry Department of Aerospace Engineering Sciences, Program in Atmospheric and Oceanic Sciences, University of Colorado, Boulder, Colorado

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Print Publication:
01 Dec 1999
DOI:
https://doi.org/10.1175/1520-0469(1999)056<3997:TTTOSC>2.0.CO;2
Page(s):
3997–4013
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Abstract

The kinetic equation of stochastic condensation derived in Part I is solved analytically under some simplifications. Analytical solutions of the gamma-distribution type are found using an analogy and methodology from quantum mechanics. In particular, formulas are derived for the index of the gamma distribution p and the relative dispersion of the droplet size spectra, which determines the rate of precipitation formation and cloud optical properties. An important feature of these solutions is that, although the equation for p includes many parameters that vary by several orders of magnitude, the expression for p leads to a dimensionless quantity of the order 1–10 for a wide variety of cloud types, and the relative dispersion σr is related directly to the meteorological factors (vertical velocity, turbulence coefficient, dry and moist adiabatic temperature lapse rates) and the properties of the cloud (droplet concentration and mean radius).

The following observed behavior of the cloud size spectra is explained quantitatively by the analytical solutions:narrowing of drop size spectra with increased cooling rate, and broadening of drop size spectra with increasing turbulence. The application of these solutions is illustrated using an example of a typical stratus cloud and possible applications for the convective clouds are discussed. The predictions of this solution are compared with some other models and with observations in stratus and convective clouds. These analytical solutions can serve as a basis for the parameterization of the cloud microphysical and optical properties for use in cloud models and general circulation models.

Corresponding author address: Vitaly I. Khvorostyanov, Department of Meteorology, University of Utah, Salt Lake City, UT 84111.

Abstract

The kinetic equation of stochastic condensation derived in Part I is solved analytically under some simplifications. Analytical solutions of the gamma-distribution type are found using an analogy and methodology from quantum mechanics. In particular, formulas are derived for the index of the gamma distribution p and the relative dispersion of the droplet size spectra, which determines the rate of precipitation formation and cloud optical properties. An important feature of these solutions is that, although the equation for p includes many parameters that vary by several orders of magnitude, the expression for p leads to a dimensionless quantity of the order 1–10 for a wide variety of cloud types, and the relative dispersion σr is related directly to the meteorological factors (vertical velocity, turbulence coefficient, dry and moist adiabatic temperature lapse rates) and the properties of the cloud (droplet concentration and mean radius).

The following observed behavior of the cloud size spectra is explained quantitatively by the analytical solutions:narrowing of drop size spectra with increased cooling rate, and broadening of drop size spectra with increasing turbulence. The application of these solutions is illustrated using an example of a typical stratus cloud and possible applications for the convective clouds are discussed. The predictions of this solution are compared with some other models and with observations in stratus and convective clouds. These analytical solutions can serve as a basis for the parameterization of the cloud microphysical and optical properties for use in cloud models and general circulation models.

Corresponding author address: Vitaly I. Khvorostyanov, Department of Meteorology, University of Utah, Salt Lake City, UT 84111.

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