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<br />JUNE 1983
<br />
<br />LIN, FARLEY AND ORVILLE
<br />
<br />1075
<br />
<br />2) ACCRETION
<br />
<br />Hail grows by accretion i of other water forms in
<br />either the dry or wet growtp mode, with the applied
<br />rate being the smaller ofthe~two. The dry growth rate
<br />PGDRy is the sum of the individual hail accretion
<br />terms and can be expressed, as
<br />
<br />PGDRy = PGACW + PGACI:+ PGACR + PGACS, (39)
<br />
<br />where PGACS is given by (~9). The rates of hail ac-
<br />creting cloud water (PGACJ), cloud ice (PGACI) and
<br />rain (PGACR) are described rrtore completely in Orville
<br />and Kopp (1977) and Wisn:er et al. (1972), and may
<br />be written as
<br />
<br />. _ 7rEGwnoGlc~r(3.5) (4gPG)I/2 (40)
<br />PGACW - 4AM 3CDP'
<br />I
<br />_ 7r EGlfloGlcI~(3.5) (4gpG )1/2
<br />PGAC1 - 4Ab5 : 3CDP'
<br />
<br />PGACR = 7r2EGRnOGnokIUG - UR{p;)
<br />
<br />( 5: 2 0.5 ) 2)
<br />X ~1+52+43' (4
<br />ARAG' ARAG ARAG
<br />I
<br />In the above, EGR and EGW ~re collection efficiencies
<br />of ice for water and are assumed to be 1. EG1 is as-
<br />sumed to be 0.1 and 1 for : dry and wet growth, re-
<br />spectively. PGACS is also a S0urce term for hail when
<br />the temperature is warmer tpan OOC, allowing for the
<br />fact that melting snow could:be collected and retained
<br />by melting hail. :
<br />The equation for wet ~owth of hail, PGWET, is
<br />based on Musil (1970) modified to include the ac-
<br />cretion of snow and subsequently integrated over all
<br />hail sizes. The rate may be written as
<br />
<br />27rnodpLv1/;lirs - KaTe)
<br />PGWET = p(Lj + CwTJ :
<br />
<br />X [0.78A02 + 0.3IS~/3r(2.~5)( ~~:r4J1-1/2A02.75 J
<br />
<br />
<br />+ (PGAC1 + PGAcs)(l - L ~i~ T)' (43)
<br />I j w e
<br />
<br />where Lv is the latent heat of vaporization, t:J.rs = rsO
<br />- r, the water vapor mixingl ratio difference between
<br />the hailstone (rso) at tempe~ature OOC and the envi-
<br />ronment (r), 1/; is the molecular diffusion coefficient
<br />of water, Ka the thermal conductivity of air, Te the
<br />Celsius temperature, and Cwl, C;, the specific heats of
<br />water and ice, respectively. :As was noted earlier in
<br />the discussion of (32), it is the thermal effect of the
<br />vapor transfer that is of primary significance and we
<br />ignore the mass transfer via: the vapor phase during
<br />wet growth. :
<br />I
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<br />.~~;f,i,~~",;.~e~'iJ"'k~,l:C, ._,;..;...::, I.:.. : '_.-
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<br />
<br />We shall now discuss implications of the wet
<br />growth process in more detail. If all of the liquid that
<br />is collected cannot be frozen, wet growth results and
<br />shedding of water drops can occur. These shed drops
<br />are assumed to be larger than cloud droplets so shed-
<br />ding adds to the rain content. If PGWET is selected as
<br />the proper mode for hail growth, the amount of rain
<br />actually frozen or shed is given by
<br />
<br />PGACR = P GWET - P GACW -- PGAC1 - PGACS, (44)
<br />
<br />(41)
<br />
<br />where PGAC1 is given by (41) with EGI set equal to 1
<br />instead of 0.1, and similarly PGACS is given by (29)
<br />with EGS set equal to 1 instead of as calculated via
<br />(30). The quantity (PGWET - PGAC1 - PGACS) is the
<br />wet growth due to the liquid water collected. If PGACW
<br />is less than (PGWET - PGAC1 - PGACS), then PGACR is
<br />positive and some of the rain is frozen to hail. If, on
<br />the other hand, PGACR is negative, some of the cloud
<br />water collected by the hail is unable to freeze and is
<br />shed as rain. By way of example, if wet growth can
<br />. freeze 10 units of liquid water, of which 5 units are
<br />accreted from rain and 7 units accreted from cloud
<br />water, giving us 12 units of liquid, then 2 units of
<br />liquid are shed as rain so that only a net of 3 units
<br />of rain are frozen; we get 2 of the 5 rain units back.
<br />[In (44), PGACR is positive and some of the accreted
<br />rain is frozen.] On the other hand, ifagain wet growth
<br />can freeze 10 units and rain accreted by hail is 2 units
<br />and cloud water accretion is 13 units, then the rain
<br />content actually increases by 3 units. The rain content
<br />not only retains the 2 units apparently lost to accre-
<br />tion, but in addition gains 3 units from the shed cloud
<br />water (PGACR is negative). This shedding mechanism
<br />may cause rapid transformation of cloud to rain and
<br />occurs primarily in the 0 to -lOoC region of the
<br />cloud. In addition, any collision of hail with cloud
<br />water in cloud regions warmer than OOC results in
<br />transformation of the cloud water to rain through
<br />shedding (melting is also occurring, transforming hail
<br />to rain).
<br />
<br />3) RAINDROP FREEZING
<br />
<br />The equation for raindrop freezing is based on the
<br />work ofBigg (1953) and represents the formation of
<br />hail from raindrops due to immersion freezing, as
<br />explained in Wisner et al. (1972). It may be written
<br />as
<br />
<br />PGFR = 207r2B'floR(P;)
<br />X {exp[A'(To - T)] - l}AR7, (45)
<br />
<br />where B' and A' are parameters in the Bigg freezing
<br />process as determined from laboratory experiments.
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