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Pao K. Wang

Abstract

The technique based on the concept of successive modification of simple shapes using elementary mathematical functions to represent the shape and size of ice crystals in clouds is discussed. Two hypothetical samples of ice crystals, a single-habit sample of hexagonal plates and a multihabit crystal sample, are generated using a formula developed previously to illustrate the use of this technique in generating ice crystal ensembles in cloud models. Next, a new expression representing columnar ice crystals is described. Finally, two new expressions that can be used to generate the three-dimensional combination of bullets and spatial dendrites are described. The parameters involved in these expressions are expected to be useful in characterizing the shape and size spectra of ice crystals found in cirrus clouds.

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Pao K. Wang

Abstract

Mathematical expressions based on earlier ideas of the author are given to describe the three-dimensional surfaces of hexagonal ice crystals and conical graupel and hail particles. In the former class, the expressions closely parallel the expression for prolate spheroid but the cross section is transformed into a hexagon by an expression designed previously. A special transformation containing a preset constant, ε, is implemented so that the three-dimensional, instead of two-dimensional, surface of the crystal is described. In the case of conical particles with elliptical cross sections, the equation that describes a conical body of revolution is modified to describe conical bodies with elliptical cross sections.

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Wusheng Ji and Pao K. Wang

Abstract

The ventilation coefficients for columnar, hexagonal plate, and broad branch ice crystals falling in air are computed by first solving numerically the convective diffusion equation for water vapor density to obtain its profile around these ice crystals and then determining the total vapor flux on the surface of the crystal. The ratio of this flux to the flux on a stationary crystal gives the ventilation coefficient. The local flow velocity profiles around the falling crystals necessary for specifying the convective term in the convective diffusion equation were obtained previously by numerically solving the unsteady Navier–Stokes equations subject to appropriate crystal-shaped boundary conditions. Ventilation coefficients obtained in this way are illustrated as a function of the Schmidt and Reynolds numbers and are also fitted by empirical expressions. Applications of these ventilation coefficients are discussed.

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Pao K. Wang and Wusheng Ji

Abstract

The unsteady flow fields around falling columnar ice crystals, hexagonal ice plates, and broad-branch crystals are simulated by numerically solving the time-dependent Navier–Stokes equations appropriate for these geometries in the primitive equation form. A predictor–corrector method and a quadratic interpolation for convective kinematics (QUICK) scheme are applied on nonuniform grids to determine the velocity fields. The ice crystals are held in fixed orientation but time-dependent behaviors such as eddy shedding are allowed to occur by imposing an initial perturbation with a magnitude 30% of the free-stream velocity. The computed flow fields cover a Reynolds number range from 0.1 to about 200, being slightly different for different crystal habits. Examples of velocity fields are illustrated. The computed drag coefficients for cylinders agree with experimental values to within a few percent, while those for hexagonal plates agree with experimental values and previous calculations by Pitter et al. to less than 15% even though the aspect ratios are different. The drag coefficients for broad-branch crystals are higher than those for hexagonal plates at the same Reynolds numbers. Special features of flow passing through the branch gaps of broad-branch crystals suggest that it may be possible to use a creeping flow assumption to treat flow passing through spaces in complicated dendritic crystals.

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Mihai Chiruta and Pao K. Wang

Abstract

The capacitances of seven bullet rosette ice crystals are computed based on the classical electrostatic analogy theory of diffusional growth. The rosettes simulated have 2, 3, 4, 6, 8, 12, and 16 lobes using mathematical formulas published previously. The Laplace equation for the water vapor density distribution around a stationary rosette is solved explicitly by the finite element method. The total flux of vapor toward the rosette surface and the vapor density on the surface determine the capacitance. The capacitances of these rosettes are smaller than that of spheres of equal radii but greater than columnar ice crystals of the same maximum dimensions. They can be greater or smaller than that of circular plates, depending on the number of lobes. Since many previous estimates of rosette growth rates were based on the assumption that their capacitances are the same as spheres of equal radii, the present finding implies that some of the previous rosette growth rates may be overestimated. The overestimation becomes less important if the rosettes have more lobes. Empirical power equations are given to fit the relations between the capacitance and the number of lobes, surface area, and volume of rosettes. Possible implications of rosette capacitance on the atmospheric heating by cirrus clouds are also discussed.

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Pao K. Wang and Wusheng Ji

Abstract

The efficiencies with which ice crystals at low–intermediate Reynolds numbers collide with supercooled cloud droplets are determined numerically. Three ice crystal habits are considered here: hexagonal ice plates, broad-branch crystals, and columnar ice crystals. Their Reynolds numbers range from 0.1 to slightly beyond 100. The size of cloud droplets range from a few to about 100 μm in radius. The collision efficiencies are determined by solving the equation of motion for a cloud droplet under the influence of the flow field of the falling ice crystal. The flow fields of the falling ice crystals were determined previously by numerically solving the unsteady Navier–Stokes equations. Features of these efficiencies are discussed. The computed efficiencies are compared with those obtained by previous investigators and improvements are indicated. New results fit better with the observed riming droplet sizes and cutoff riming ice crystal sizes.

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Pao K. Wang and De'er Zhang

Some historical weather records of China in the governmental archives are discussed. The records in the pre-Qing period (before 1636) are briefly summarized and their use for the reconstruction of past climate is assessed. The more-elaborate weather records of the Qing dynasty, the Clear and Rain Records and the Inches of Rain and Snow, are examined in more detail.

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David P. Sauter and Pao K. Wang

Abstract

An experimental study of the scavenging of aerosol particles of mean radius 0.75 μm by natural snow crystals of a few milimeters is carried out. Aerosol particles are spherical indium acetylacetonate particles generated by a modified La Mer generator. Snow crystals are obtained during natural snowfalls. Shapes of snow crystsals include needless column broad-branched crystals stellar crystals, and hexagonal plates. Aerosol particles are dispersed into an aerosol chamber and snow crystals fall through the chamber to scavenge aerosol particles. The collection efficiency of aerosol particles by snow crystals is found to decrease with increasing crystal size for all shapes. This can be explained by the relative strength of the inertial force of particles and the hydrodynamic drag force created by the fall of the snow crystal. Large crystals would create greater drags during the Call and force the aerosol particle to follow more closely to streamlines and hence reduce the collection efficiency.

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Joseph J. Nettesheim and Pao K. Wang

Abstract

Fluid flow fields and fall patterns of falling planar ice crystals are studied by numerically solving the unsteady, incompressible Navier–Stokes equations using a commercially available computational fluid dynamics package. The ice crystal movement and orientation are explicitly simulated based on hydrodynamic forces and torques representing the 6 degrees of freedom. This study extends the current framework by investigating four planar-type ice crystals: crystals with sector-like branches, crystals with broad branches, stellar crystals, and ordinary dendritic crystals. The crystals range from 0.2 to 5 mm in maximum dimension, corresponding to Reynolds number ranges from 0.2 to 384. The results indicate that steady flow fields are generated for flows with Reynolds numbers less than 100; larger plates generate unsteady flow fields and exhibit horizontal translation, rotation, and oscillation. Empirical formulas for the drag coefficient, 900-hPa terminal velocity, and ventilation effect are given. Fall trajectory, pressure distribution, wake structure, vapor field, and vorticity field are examined.

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Pao K. Wang, Thomas J. Greenwald, and Jianlu Wang

Abstract

The characteristics of shapes and sizes of a sample of 679 hailstones, collected on 22 June 1976 during a hailstorm at Grover, Colorado, were analyzed using a three-parameter formula developed by us previously. These parameters are a, the horizontal dimension, c, the vertical dimension, and λ, the shape parameter. Once these three parameters are specified, both the shape and size of a hailstone are fixed. It is believed that this analysis produces the most complete and quantitative information about the hailstone shape and size distributions reported so far. The dataset of the three parameter also allows the relatively good reconstruction of the sizes and shapes of the original hailstones if desired. The results for this collection of hail show that the distributions of both horizontal and vertical dimensions can be described by gamma distributions, while the shape parameter can be described by an exponential distribution. Since the shape parameter basically describes the vertical asymmetry, it may provide additional information about the physics of particles in clouds and precipitation. The distributions of axial cross-sectional areas, surface areas, and volumes are also presented. They too can be described by gamma distributions. Finally, it was found that the geometrical quantities of the hailstones are best represented by a characteristic dimension rc, defined as the average of the horizontal and vertical dimension.

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