Evolution of Snow-Size Spectra in Cyclonic Storms. Part II: Deviations from the Exponential Form

Abstract

Using a form of the stochastic collection equation, conservation equations for the first and second moments of the mass were parameterized to yield a height dependent one-dimensional snow growth model for unrimed stratiform snowfall. Snow-size distributions were represented by the form N(D) = N0D? exp(??D), and solutions for ? and N0 were obtained. The spectral parameter ? allows the concentration of the smaller ice particles to deviate from the exponential form and controls the degree of subexponential or superexponential behavior. The sub- and superexponential spectra analyzed in this study had ? values of 1 and ?1, respectively. A number of simple analytical relationships was developed that describes various properties of size distributions, regardless of the particle type involved. A method was developed for obtaining the three parameters of the size distribution used in the model from measured size distributions. In addition, an expression was derived to relate the two ? of an exponentially parameterized and a nonexponentially parameterized size distribution. The effect of sub- and superexponential spectra on the evolution of snow-size spectra by vapor diffusion and aggregation was examined using a steady state, fixed snowfall rate profile. Diffusional growth rates of individual ice crystals (no aggregates) were relatively low when the size distribution was constrained to be superexponential in form. This resulted in steeper spectra (smaller crystal sizes) and higher ice-crystal number concentrations. The diffusional growth rate of individual ice crystals for subexponential spectra was relatively high. Subexponential spectra were characterized by broader distributions and lower ice crystal number concentrations. Aggregation was the only growth process that substantially increased ice particle sizes for superexponential spectra, while both vapor diffusion (in the upper cloud) and aggregation (in the mid-to-lower cloud) contributed substantially to size increases for subexponential spectra. An expression for the aggregation efficiency was formulated. The primary factors governing aggregation appear to be the aggregation efficiency, the ice particle number concentration and the mean diameter. The expression may be useful in larger numerical cloud models. Mean aggregation rate constants were determined for sub- and superexponential spectra, and for exponential spectra. The mean aggregation rate constant for superexponential spectra was approximately 50% greater than for subexponential spectra. Finally, it was found that the degree of subexponential behavior predicted when ? = 1 was consistent with that observed at various levels in stratiform clouds. However, better measurements are needed to substantiate this finding.