Khvorostyanov, V. I. and J. A. Curry, 2008. Analytical Solutions to the Stochastic Kinetic Equation for Liquid and Ice Particle Size Spectra. Part II: Large-Size Fraction in Precipitating Clouds. J. Atmos. Sci., 65, 2044–2063 (original) (raw)
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The stochastic kinetic equation is solved analytically for precipitating particles that can be identified as rain, snow, and graupel. The general solution for the size spectra of the large-size particles is represented by the product of an exponential term and a term that is an algebraic function of radius. The slope of the exponent consists of the Marshall–Palmer slope and an additional integral that is a function of the radius. Both the integral and algebraic terms depend on the condensation and accretion rates, vertical velocity, turbulence coefficient, terminal velocity of the particles, and the vertical gradient of the liquid (ice) water content. At sufficiently large radii, the radius dependence of the algebraic term is a power law, and the spectra have the form of gamma distributions. Simple analytical expressions are derived for the slopes and indices of the size distributions. These solutions provide explanations of the observed dependencies of the cloud particle spectra in...
Cloud and Precipitation Microphysics(2009)
Numerous studies have demonstrated that cloud and precipitation parameterizations are essential components for accurate numerical weather prediction and research models on all scales, including the cloud scale, mesoscale, synoptic scale, and global climate scale.
Evolution of Snow-size Spectra by the Growth Processes of vapor Deposition, Aggregation and Riming
AMS, 2016
A steady-state snow growth model (SGM) is formulated by microphysical growth processes of vapor deposition, aggregation and riming. SGM is capable of predicting the temporal and vertical evolution of ice particle size distribution (PSD) by using radar reflectivity (Zw), supersaturation, temperature, liquid water content (LWC) and ice particle shape dependent mass-dimension power laws, and by solving the zeroth-and second-moment conservation equations with respect to mass. It appears that the riming process is essential in characterizing the snowfall rates and leads to the snowfall rates significantly greater than those produced by the vapor deposition and aggregation alone. Moreover, alteration in cloud condensation nuclei (CCN), due to aerosols, can modify cloud droplet SD (size distribution) and therefore change the snowfall rate. So, snowfall rate is sensitive to the shape of cloud droplet SD. Ice particle growth rates are uniquely formulated in the SGM in terms of ice particle mass-dimension (m-D) power laws (m = αD β), and in this way the impact of ice particle shape on particle growth rates and fall speeds is accounted for. These growth rates appear qualitatively consistent with empirical growth rates, with slower (faster) growth rates predicted for higher (lower) β values. It is well known that for a given ice particle habit, the m-D power law for the smallest ice particles differs considerably from the power law for the largest particles, with β being much larger for the smallest crystals. Our recent work quantitatively predicts β and α for frontal clouds as a function of maximum dimension D where the m-D expression is a second-order polynomial in log-log space. By tailoring the m-D power law to the relevant PSD moments, the SGM ice particle growth rates and fall speeds are represented more accurate and realistic. It is speculated that by implementing this new m-D treatment in any cloud resolving model or climate model, the ice particle growth rates will become more accurate. The predicted size spectra by SGM are in good agreement with observed spectra from aircraft measurement during Lagrangian spiral descents through frontal clouds.
Journal of The Atmospheric Sciences, 1999
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 r 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).
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Quarterly Journal of the Royal Meteorological Society, 2005
Particle size distributions measured by the UK C-130 aircraft in ice stratiform cloud around the British Isles are analysed. Probability distribution functions over large scales show that the zeroth, second and fourth moments (equivalent to concentration, ice water content and radar reflectivity) as well as mean particle size have monomodal distributions. Rescaling of the size distributions requiring knowledge of two moments reveals a 'universal' distribution that has been fitted with analytically integrable functions. The existence of the 'universal' distribution implies that two-moment microphysics schemes are adequate to represent particle size distributions (PSDs). In large-scale models it may be difficult to predict two moments, and so power laws between moments have been found as functions of in-cloud temperature. This means that a model capable of predicting ice water content and temperature can predict ice PSDs to use for calculations requiring knowledge of the size distribution (e.g. precipitation rate, radar reflectivity) or to make direct use of the power laws relating moments of the size distribution.
Monthly Weather Review, 2004
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Parameterizing Size Distribution in Ice Clouds
2009
Since cirrus clouds have a substantial influence on the global energy balance that depends on their microphysical properties, climate models should strive to realistically characterize the cirrus ice particle size distribution (PSD), at least in a climatological sense. To date, the airborne in situ measurements of the cirrus PSD have contained large uncertainties due to errors in measuring small ice crystals (D < ~ 60 μm). In this paper we present a method to remotely estimate the concentration of the small ice crystals relative to the larger ones using the 11 and 12 μm channels aboard several satellites. By understanding the underlying physics producing the emissivity difference between these channels, this emissivity difference can be used to infer the relative concentration of small ice crystals. This is facilitated by enlisting temperaturedependent characterizations of the PSD (i.e. PSD schemes) based on in situ measurements. An average cirrus emissivity relationship between 12 and 11 μm is developed here using the MODIS satellite instrument and is used to "retrieve" the PSD based on six different PSD schemes. The PSDs from the measurement-based PSD schemes are compared with corresponding retrieved PSDs to evaluate differences in small ice crystal concentrations. The retrieved PSDs generally had lower concentrations of small ice particles, with total number concentration independent of temperature. In addition, the temperature dependence of the PSD effective diameter D e and fallspeed V f for these retrieved PSD schemes exhibited less variability relative to the unmodified PSD schemes. The reduced variability in the retrieved D e and V f was attributed to the lower concentrations of small ice crystals in their retrieved PSD.