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<br />JUNE 1983
<br />
<br />1069
<br />
<br />UN, FARLEY AND ORVILLE
<br />
<br />2
<br />
<br />( )1/2
<br />UDR = am; ,
<br />
<br />
<br />( )1/2
<br />UDS = em; ,
<br />
<br />, 1/2
<br />U = (4gPG) DI/2
<br />DG 3CDP G .
<br />
<br />The terminal velocity U DR bf rain is suggested by Liu
<br />and Orville (1969) who performed a least squares
<br />analysis of Gunn and Kinz~r's data (1949). The con-
<br />stants a and bare 2115 cml-b S-1 and 0.8, respec-
<br />tively. The terminal velocity U DS of snow is based on
<br />the relations suggested by Lpcatelli and Hobbs ( 1974).
<br />Specifically, UDS is that appropriate for graupel-like
<br />snow of hexagonal type, with the constants c and d
<br />being 152.93 cml-d S-I arid 0.25, respectively. The
<br />square root factor involving air density allows for in-.
<br />creasing fallspeeds with increasing altitude, similar to
<br />Foote and du Toit (1969). The terminal velocity UDG
<br />of hail is proposed by Wisner et al. (1972), with the
<br />drag coefficient CD assume:d to be 0.6.
<br />Following Srivastava (1967), we define mass-
<br />weighted mean terminal v~locities as
<br />
<br />U = f UD,l(D)dD/I, (10)
<br />
<br />where U D is the terminal yelocity of a precipitating
<br />particle of diameter D, I(~) is the mixing ratio of a
<br />precipitating particle of diapleter D, and I is the mix-
<br />ing ratio of a precipitating field. Applying (10) to each
<br />precipitating field, we obtain the mass-weighted mean
<br />terminal velocities of rain, : snow and hail:
<br />,
<br />
<br />_ af(4 +' b) (pO)1/2
<br />UR - 6)..b .
<br />R, P
<br />
<br />. _ cf(4 +:d) (pO)1/2
<br />Us - 6)..d '
<br />S' P
<br />,
<br />
<br />UG = r(4, 5):( 4gPG)1/2
<br />6)..~5 ; 3CnP
<br />
<br />,
<br />The mass-weighted mean t~rminal velocities of rain,
<br />snow and hail are shown i~ Fig. 2.
<br />
<br />2) WATER CONSERVATI~N EQUATIONS
<br />
<br />Four conservation equations are considered here:
<br />,
<br />
<br />aq V :
<br />at = -y. q + V. KhVrI - PR - Ps - PG,
<br />a~ :
<br />- = -Y.VIR + V.K VIR
<br />at m
<br />
<br />I
<br />I
<br />
<br />I
<br />
<br />
<br />l'"
<br />:.. .J...'l,,-,L., <
<br />
<br />: 1 a
<br />+'PR + - -a (URIRP),
<br />I P Z
<br />
<br />(7)
<br />
<br />
<br />(8)
<br />
<br />-- - p=o.Sxl03 9 cm-3
<br />.-.-. p=O.7 x 103 9 cm3
<br />18 ....... p=0.9xI03 9 cm-3
<br />- p=163 9 cm-3 .....................
<br />...-
<br />.--
<br />./
<br />./
<br />,/
<br />,/
<br />/
<br />/
<br />/
<br />14 I
<br />I ./
<br />I /
<br />I /
<br />/
<br />E .I
<br />- /
<br />>- 10 .
<br />l-
<br />t)
<br />S
<br />~8
<br />
<br />---
<br />
<br />--
<br />
<br />16
<br />
<br />H:.~._._._._-'-'_.
<br />.-'
<br />...-'
<br />/'
<br />
<br />(9)
<br />
<br />---
<br />---
<br />...-_---RAIN _._.-.-.
<br />....../ -.-
<br />/' ---.----.-
<br />
<br />
<br />6 ;I;;~
<br />
<br />4 V C;Nnw
<br />: SNOW _ _ _ _ _ _ _ _ _
<br />-----
<br />
<br />-~.,. .
<br />
<br />
<br />--
<br />2 ..;,~-;:-::-::-
<br />
<br />-'-'-'- -'-'-'-'
<br />..........................
<br />
<br />.-'-'-'
<br />.............
<br />
<br />00
<br />
<br />
<br />4
<br />
<br />FIG. 2. Mass-weighted mean terminal velocities for rain, snow
<br />and hail. The four curves from 9 to 19 m S-I are for hail. The four
<br />curves from 3 to 10 m S-I are for rain. The remaining four curves
<br />are for snow.
<br />
<br />(11)
<br />
<br />als
<br />-= -Y.V/s+V.K VIs
<br />at m
<br />
<br />(12)
<br />
<br />1 a
<br />+ Ps + - -a (UsIsp), (16)
<br />P z
<br />alG
<br />-= -Y.VIG+V.K V/G
<br />at m
<br />
<br />(13)
<br />
<br />1 a
<br />+ PG + - a- (UGIGP), (17)
<br />P z
<br />
<br />where q = lew + IC/ + r; lew, IC/, IR, Is, IG and rare
<br />the mixing ratios for cloud water, cloud ice, rain,
<br />snow, hail and water vapor, respectively; and PR, Ps
<br />and P G are the production terms for rain, snow and
<br />hail. These terms will be considered in more detail
<br />in the next several subsections. Only the final form
<br />of the microphysical equations will be presented here.
<br />For a more detailed explanation of the derivations,
<br />the reader is referred to Wisner et al. (1972) or Chang
<br />( 1977).
<br />The last terms in (15), (16) and (17) are the fallout
<br />terms. All of the first terms on the right-hand side are
<br />advection terms; the second terms are diffusion terms.
<br />
<br />(14)
<br />
<br />(15)
<br />
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