Collision Efficiencies of Ice Crystals at LowâIntermediate Reynolds ...

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1004 JOURNAL OF THE ATMOSPHERIC SCIENCESVOLUME 57TABLE 1. Reynolds numbers, dimensions, and capacitance of columnarice crystals in the present study. The quantities are dimensionless.N Re Radius (m) Length (m) Capacitance0.20.50.71.02.05.010.020.023.532.736.641.553.477.2106.7146.467.193.3112.6138.3237.4514.9106724401.36281.36281.40541.45351.65112.01512.50673.3959TABLE 3. Reynolds numbers and dimensions of broad-branch icecrystals in the present study.N Re Radius (m) Thickness (m)1.02.010.020.035.060.090.0120.0100.0125.0350.0500.0750.01000.01250.01550.015.018.032.040.050.060.065.073.0circular symmetry but rather is a function of the azimuthalangle. The asymmetry is most easily shown bythe shape of the collision cross section A formed byconnecting y c ’s of all angles. Figures 3, 4, and 5 showexamples of these collision cross sections for dropletsof various sizes colliding with the three types of icecrystals. It is immediately clear that the shapes of theA’s are more or less similar to the ice crystal crosssections themselves. When droplets are small, their collisioncross sections (and hence the collision efficiencies)are usually (but not always) smaller. As the dropletsbecome larger, the collision cross sections becomelarger and the shapes are closer to the cross sections ofthe ice crystals. This behavior is obviously because ofthe inertia of the droplet relative the strength of thehydrodynamic drag force, a reasoning that has been discussedin great detail by Pruppacher and Klett (1997).When droplets are small, their inertias are small comparedto the drag and their trajectories are close to thestreamlines of the flow fields, which are generallycurved around the crystal. Thus the shape of the collisioncross section would be different from the crystal.When droplets are larger, their inertia becomes greaterand their trajectories are straighter; hence, the collisioncross sections have shapes closer to that of the crystal.efficiency is very small when the droplet is small dueto its small inertia, as explained previously. In the casesof Re 1.0 and 2.0, the efficiency drops to very smallvalue (10 4 ) for droplets with radii less than 9 m.For higher-Re (larger ice crystals) cases, this efficiencydrop is more gradual and there is no sharp cutoff. Thisis in contrast with earlier studies where a cutoff at a 2 5 m occurs (e.g., Pitter and Pruppacher 1974; Pitter1977). Instead, the efficiency remains finite even fordroplets as small as 2.5 m, which is in good agreementwith Kajikawa’s (1974) experimental results. Recent observationalstudies also confirm that many frozen dropletson the rimed ice crystals are smaller than 5 m.The reason that some previous field observations indicatedthe scarcity of frozen droplets with a radius lessthan 5 m on planar ice crystals (e.g., Harimaya 1975;Wilkins and Auer 1970; Kikuchi and Ueda 1979; andD’Enrico and Auer 1978) is probably due to the locala. Hexagonal platesFigures 6, 7, and 8 show the computed collision efficienciesfor the three crystal habits. The case of hexagonalice plates is shown in Fig. 6. The general featurehere is that, at a fixed crystal Reynolds number, theTABLE 2. Reynolds numbers, dimensions, and capacitance of hexagonalice plates in the present study. The quantities are dimensionless.N Re Radius (m) Thickness (m) Capacitance1.02.010.020.035.060.090.0120.080.0113.3253.3358.2473.8620.0750.0850.018.020.032.037.041.045.048.049.00.72980.69770.66390.64850.63710.62780.62210.6179FIG. 2. Trajectories of a droplet of 2 m in radius moving in thevicinity of a falling broad-branch ice crystal at Re 10. Trajectories1, 2, 6, 7, and 8 are misses, and trajectories 3, 4, 5 are hits.
15 APRIL 2000 WANG AND JI1005FIG. 3. Shape of collision cross sections for a hexagonal ice plateat Re 20, colliding with supercooled droplets of radius r. The fixedhexagon is the cross section of the ice plate: (a) r 3 m, (b) r 5 m, (c) r 11 m, and (d) r 27 m.microstructure of clouds instead of intrinsic collisionmechanism (Pruppacher and Klett 1997).As the drop size increases, the efficiency increasesrapidly. The efficiency reaches a peak or a plateau, dependingon the Reynolds number of the ice crystal, andthen drops off sharply for further increasing drop size.The drop off of efficiency is apparently due to the increasingterminal velocity of the droplet. When the collectorice crystal and the droplet have about the samevelocity, collision is nearly impossible and the efficiencybecomes very small (Pitter and Pruppacher 1974; Pitter1977; Martin et al. 1981; Pruppacher and Klett 1997).The efficiency maxima take the shape of peaks in smaller-Recases but become broader plateaus as the ice crystalRe increases, apparently because the larger crystalscan collide with droplets of broader size range and maintainfairly high efficiencies. Due to their sizes, smallercrystals are quickly ‘‘outrun’’ by droplets as dropletsbecome larger and hence are unable to perform the collision.FIG. 4. Same as Fig. 3 except for a broad-branch crystal at Re 35: (a) r 5 m, (b) r 9 m, (c) r 15 m, and (d) r 36m.The collision efficiencies of broad-branch crystals arein general smaller than those of hexagonal plates at thesame Reynolds number. The maximum efficiencies inthe plateau region are about 0.9, unlike the case of hexagonalplates, whose maximum efficiencies are near 1.0.This is probably due to the more open structure of abroad-branch crystal that would allow the droplet to‘‘slip through’’ the gap between branches. The width ofthe ‘‘plateau’’ is also much narrower than the corre-b. Broad-branch crystalsFigure 7 shows the collision efficiency for broadbranchcrystals colliding with supercooled droplets. Themain features are similar to those for hexagonal plates.The collision efficiency for Re 1.0 is practically zero,representing an inability to rime. This cutoff of rimingability will be discussed further below.FIG. 5. Same as Fig. 3 except for a columnar ice crystal at Re 2.0: (a) r 4 m, (b) r 6 m, (c) r 35 m, and (d) r 43m.
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