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CHAPTER 10 ROSENFELD AND ULBRICH
Chapter 10
Cloud Microphysical Properties, Processes, and Rainfall Estimation Opportunities
DANIEL ROSENFELD
Institute of Earth Sciences, Hebrew University of Jerusalum, Jerusalem, Israel
CARLTON W. ULBRICH
Department of Physics and Astronomy, Clemson University, Clemson, South Carolina
Rosenfeld
1. Introduction
In this work the longstanding question of the con- nections between raindrop-size distributions (RDSDs) and radar reflectivity-rainfall rate (Z-R) relationships is revisited, this time from the combined approach of rain- forming physical processes that shape the RDSD, and a formulation of the RDSD into the simplest free pa- rameters of the rain intensity R, rainwater content W, and median volume drop diameter Do. This is accom- plished through a theoretical analysis, using a gamma RDSD, of Do-R and W-R relations implied by the co- efficients and exponents in empirical Z-R relations. The results provide a means by which these Z-R relations can be classified. The most dramatic of these classifi- cations involves the relation between Do and W, which shows a remarkable ordering with the rain types. This work also summarizes the effects of various physical processes in modifying the RDSD in clouds. These individual processes are combined into concep- tual models of the way different microphysical and dy- namical rain-forming processes can build different kinds of RDSDs. Much of the physical insights that are at the heart of this study came from examining the evolution of the RDSD with respect to its ultimate mature state
of the equilibrium raindrop-size distribution described
by Hu and Srivastava (1995).
Finally, the different components of the previous sec- tions are combined in an examination of integral pa- rameters deduced from the raindrop-size distributions associated with the host of RDSD-based Z-R relations found in the literature. Only those relations are used that could be associated to the cloud microstructure and dy- namic context of the conceptual model. It is found that there exists a well-defined sequence in the transition from extreme continental to equatorial maritime for con- vective rainfall. In addition, similar behavior is found in tropical convective versus stratiform rainfall and in orographic rainfall as a function of altitude. These re- sults offer promise for the development of algorithms for classification of the rainfall with respect to type in the remote measurement of rainfall either from satellite platforms or from ground-based radars. Early attempts to explain the variability in Z-R re- lations are reviewed in section 2. Section 3 reviews the formulation of the RDSD and provides the tools to re- store RDSD parameters from published Z-R power-law relations. Section 4 describes the way the different in- dividual processes that modify the RDSD can be com-- bined into conceptual models of the rain-forming pro-
237
238 METEOROLOGICAL MONOGRAPHS VOL. 30, No. 52
TABLE 10.1. Microphysical and kinematic influences on Z-R relationships and the effect on radar rainfall estimates when no adjustment is applied (after Wilson and Brandes 1979). Change in Z = ARb Probable effect on radar rainfall if Z-R Possible region of Process A b not adjusted max influence
Microphysical Evaporation (Atlas and Chmela 1957) Increase Decrease Overestimate Inflow regions, fringe areas Accretion of cloud particles (Atlas and Decrease Increase Underestimate Downdraft Chmela 1957; Rigby et al. 1954) Collision, coalescence (Srivastava 1971) Increase Decrease Overestimate Reflectivity core Breakup (Srivastava 1971) Decrease Decrase (^) Underestimate Reflectivity core
Kinematic Size sorting (Gunn and Marshall 1955; Increase Decrease^ Tendency to Regions of strong inflow Atlas and Chmela 1957) (^) overestimate and outflow
Vertical motion Updraft Increase Decrease Overestimate Downdraft Decrease Increase Underestimate
eters were displayed for an exponential distribution of the form
where N(D) (m- 3 em:") is the number of drops per unit volume per unit size interval and No and A are the RDSD parameters. In addition, as shown by Atlas (1955), A = 3.67/Do, where Do is the median volume diameter. The Atlas-Chmela diagram is reproduced in Atlas (1964), but a more recent version is shown in Fig. 10. for an exponential distribution with isopleths of W, Do, and No, where W(g m- 3 ) is the liquid rainwater con- centration. At the time the Atlas-Chmela diagram was published, the use of Z (the single radar measurable then available) to measure R through the use of a Z-R relation was the focus of research in radar meteorology. As the field expanded the number of measurements of drop- size spectra (and Z-R relations derived from them) grew rapidly and it was discovered quickly that there was no unique relationship between Z and R; that is, there were no unique values for A and b. The advantage of the Atlas-Chmela Z-R rain parameter diagram was that for a given Z-R relation (and an exponential RDSD), it permitted the relationships between all of the drop-size distribution (DSD) integral parameters to be determined. That is, for a given Z-R relation the diagram implied corresponding relations between Do-R, W-R, Z-W, D (^) o- lY,etc.. The disadvantage was that a different diagram had tobe produced for distributions different from ex- ponential. There have been many attempts to relate the large observed variations in the coefficient A and exponent b in theZ-R law to the meteorological conditions asso- ciated with the rainfall and to the parameters of the drop- size distribution. It is well known in radar meteorology that there is a great lack of consistency in the drop-size distribution for various meteorological conditions. Even when the conditions appear to be similar the size dis- tributions can be widely different. This is apparent in
cesses. Section 5 applies the different components in the previous sections to deduce the Z-R classification scheme. Section 6 summarizes the results and offers suggestions for implementation of a dynamic Z-R clas- sification method.
2. Early attempts to classify Z-R relations
It has long been recognized that wide range of values found for the coefficient A and exponent b in Z-R re- lations of the form Z = ARb is due to variations in the form of the RDSD. Chandrasekar et al. recognize this connection between RDSD variability and the values of A and b (chapter 9 in this monograph). They also point to the importance of separating Z-R relations according to the type of rainfall. One of the earliest studies to recognize these effects was that due to Atlas and Chmela (1957) who showed that RDSD sorting at the scale of the individual rain shaft could occur due to drop sorting by wind shear and updrafts. Beyond the scale of the individual rain shafts, the causes of variability in Z-R relations were sought in differences in rainfall types, atmospheric conditions, and geographical locations (Fu- jiwara 1965; Stout and Mueller 1968; Cataneo and Stout 1968). The rationale was that different conditions would prefer different rain processes, and the effects of these processes were summarized in the form of a table in Wilson and Brandes (1979), which is reproduced here as Table 10.1. Wilson and Brandes (1979) provided this table with little discussion. Such a discussion is provided later in this work, with some explanations on the causes for the trends of the coefficient and exponent. That is done after the various analytical forms of the RDSD that have been employed in the past are introduced. To depict the relationships between the various pa- rameters of the RDSD, Atlas and Chmela (1957) pro- duced a rain parameter diagram (RAPAD) with Z plotted versus R and on which isopleths of distribution param-
N(D) = Noexp(-AD), (10.1)
I
l'
240 METEOROLOGICAL MONOGRAPHS VOL.^ 30, No. 52
TABLE 10.2. Values of A and b in Z = AR" for tropical stratiform and convective rainfall in TOGA COARE. Stratiform Convective Source A b A b
Tokay et al. (1995) 335 1.37 175 1. Tokay and Short (1996) (^367) 1.30 139 1. Atlas et at. (2000) 224 1.28 129 1. Ulbrich and Atlas (2002) 203 1.46 120 1.
thermodynamic instability were found not to be useful, but those found from the classification by rainfall type and synoptic situation displayed large systematic vari- ations in A and b, indicating the importance of using a stratification technique for measurement of rainfall us- ing Z-R relations. In spite of this finding, neither of the first two techniques was found to be superior in mea- suring rainfall amounts to the method that uses the Z- R relation given by Marshall et al. (1947). In fact, in several cases the latter was found to produce more ac- curate results than either stratification method. Some limited progress in this area has been made recently for tropical rainfall. Well-defined differences in stratiform and convective rainfall in the Tropics have been found by several investigators during Tropical Oceans Global Atmosphere Coupled Ocean-Atmo- sphere Response Experiment (TOGA COARE). Some of the results found for A^ and^ b^ by various investigators are listed in Table 10:2. It must be recognized that these relations are based on long-term temporal and spatial averages of experimental RDSDs. For individual storms and for stages of such storms the Z-R relations can vary appreciably. For example, Atlas et al. (1999) show data for tropical squall lines that are segmented into con- vective (C), transition (T), and stratiform (S) stages. The variations in A and b between different storms for each of these stages are very large and can also vary appre- ciably between storms. In any event, it is clear from Table 10.2 that there is not much difference between these relations for convective rain when plotted on the rain parameter diagram of Atlas and Chmela (1957; Fig. 10.1). The differences between the stratiform relations lie mostly in the coefficients, which Atlas et al. (2000) attribute to the inclusion by Tokay and Short (1996) of transition rain in the convective category. Nevertheless, it may be concluded from the results shown above that the principal differences between stratiform and con- vective rain in .the Tropics is that the coefficient A for stratiform rain is somewhat larger (at least 70%) than the coefficient for convective rain. Examination of these relations, when plotted on the RAPAD of Fig. 10.1, indicates that the larger coefficients A for stratiform rain are associated with larger Z values (for the same R) than convective rain and therefore also with larger values of o;
3. Formulations of the raindrop-size distribution
Raindrop-size distributions have been a subject of extensive investigation for nearly 100 years. The earliest carefully performed measurements of raindrop sizes were reported by Laws and Parsons (1943), Marshall and Palmer (1948), and Best (1950) and indicated that the distribution could be approximated well by an ex- ponential function of the form of Eq. (10.1). (In the following the term "raindrop size" is used to mean raindrop diameter). This mathematical approximation to the raindrop-size distribution has been in widespread use for decades and is especially convenient because of its simplicity. However, even in the early experimental work just cited distinct deviations from exponentiality were noted. Since these deviations are reflective of the physics of rain formation in clouds it has been consid- ered imperative that an accurate mathematical repre- sentation of the distribution be found. To account for distribution shape effects Atlas (1955) introduced a "moment" G of the distribution; which related the reflectivity factor Z to the median volume diameter Do and the liquid water concentration M. They also showed Z to be related to the rainfall rate Rand Do through the moment G. Joss and Gori (1978) also defined measures of distribution shape S(PQ), where P
and Q are any two integral parameters of the distribu-
tion. For distributions that have breadth narrower than, equal to, or broader than an exponential distribution, S is less than, equal to, or greater than 1, respectively. For the experimental distributions they investigate, Joss and Gori find that S is always less than 1, the more so the shorter the time interval used to average the data. Joss and Gori also found that considerable long-term aver- aging of disdrometer data is required for the distribu- tions to approach exponentiality; the longer the aver- aging period the closer the approach to exponentiality. Periods as long as 256 min were required to find average distributions close to exponential, regardless of the type of rainfall. Their work further demonstrates the need for RDSDs of greater generality than the exponential distribution. Other attempts to account for distribution shape have involved the use of specific mathematical forms differ- ent from exponential. One of the earliest of these was a lognormal function suggested by Levin (1954) of the form
(10.2)
with No, c, and D (^) g as parameters. This form has been applied to the analysis of cloud droplet and raindrop distributions by many investigators including Mueller and Sims (1966), Bradley and Stow (1974), and Mar- kowitz (1976). Although this function approximates drop-size distributions well, it does not allow for as broad a spectrum of RDSD shapes as other represen- tations and does not reduce to the exponential function as a special case. An alternative function that has come
/
CHAPTER 10 ROSENFELD^ AND^ ULBRICH^241
into widespread use is the gamma function having the form
N
N(D) = T DI-' exp(- AD), (10.4)
f(fL + l)AI-'+!
where NT is the total concentration of raindrops, and recommend using NT' p; and A as the distribution pa- rameters. Note that NT can.be written as
NT = Nof(fL + l)/AI-'+' (10.5)
so that this form requires that fL > -1. Values of fL ::
- 1 will produce results for NT that are undefined. Willis (1984) normalized the distribution so that it assumed the form
with No, fL, and A as parameters (Deirmendjian 1969; Willis 1984; Ulbrich 1983). The advantages of this dis- tribution are that it reduces to the exponential distri- bution when fL = 0 and it allows for distributions with a wide variety of shapes including those that are either concave upward or downward on a plot of 10g[N(D)] versus D. RDSD shapes of this type are very apparent in experimental spectra collected at the earth's surface using various sampling devices, such as drop cameras, disdrometers, 2D optical probes, and video recorders. An early example of an investigation that displays these effects is that of Dingle and Hardy (1962). More recent examples are very prevalent; one that includes extensive analysis of tropical raindrop spectra is that of Tokay and Short (1996). Such data usually consist of samples of short duration (e.g., 1 min). However, Levin et al. (1991) find such effects in disdrorneter data even when aver- aged for periods as long as 2 h. It might also be stated that these effects may not be representative of RDSD shapes observed aloft with radar. However, shape effects similar to that found with surface instruments also exist in RDSDs aloft as is apparent from the early work of Rogers and Pilie (1962) and Caton (1966), who acquired Doppler radar spectra of rain at vertical incidence. They are also apparent in the analysis by Atlas et al. (2000) of 2D optical probe data acquired aloft during TOGA COARE by an National Center for Atmospheric Re- search (NCAR) Electra aircraft. The gamma distribution has properties that provide an accurate representation to be made of these shape effects. In addition, integral rainfall parameters generally are simple to calculate with the gamma distribution. In spite of its advantages there are features that make this function troublesome. First, the coefficient No no longer has the simple units as the equivalent coefficient in the exponential distribution and, in fact, includes the parameter u: As a result No and fL are strongly correlated as shown by Ulbrich (1983), but this correlation is dem- onstrated by Chandrasekar and Bringi (1987) not to im- ply any physical basis. To avoid this problem they re- write the distribution in the form
N*(D) = Ct::,.-3 exp(-(t::,. - t::,.o)2), (10.11)
where C = (6h 3W)/1T^ 3/^2 and t::,. = VhD; For VhD (^) o > v'6 the normalized curves show one maximum and one minimum and for VhD (^) o < v'6 the curves show no extrema but one inflection point. The form of the DSD with VhD (^) o < v'6 is similar to that displayed by the experimental data of Marshall and Palmer (1948), whereas the form for VhD ° > v'6 is similar to that of the equilibrium distributions of Hu and Srivastava (1995). The disadvantage of this distribution is that it implies that the mass is distributed normally with re- spect to diameter. None of these "normalization" methods removes the strong correlation among the DSD parameters that is commonly observed in experimental data. These param- eters cannot, therefore, be considered strictly indepen- dent, a property of distribution parameters that is highly desirable in remote sensing algorithms. In an investi- gation of methods to avoid this problem, Haddad et al. (1996) found parameters of the gamma distribution that are negligibly correlated and may therefore be consid- ered independent. One of these is chosen to be the rain- fall rate R and the other two, D' and Sf, are defined in terms of R, D (^) m (the mean volume diameter), and s; (the relative standard deviation of the mass spectrum) as D' = D (^) mR-o.!55 and (10.12)
s' = smD-;;.O.2Ro.031 exp(0.017Ro.74). (10.13)
N(D) = 6 ~WD-3 exp[ -h(D - D (^) O)2], (10.10) y;
where W is the liquid water concentration, Do is the median volume diameter, and h is a parameter. The dis- tribution may be easily normalized such that it has the form
where the coefficient No is expressed in terms of the liquid water concentration W by
No = (6WA4+1-')/(1T f(4 + fL). (10.7)
In similar fashion, Testud et al. (2001) normalized the gamma distribution using Wand the mean volume di- ameter D (^) m by writing it as
N(D) = N~FI-'(X), (10.8)
where X = DID"" Nt; = 44W/(1TD~,),
F/X) = CI-'XI-' exp(-(4 + fL)X), (10.9) and C = (f(4)(4 + fL)4+1-')/(f(4 + fL)44).^ As long as value: of NT are not required, the latter two normali-
zations will yield useful results for fL > -3.
.Another analytical expression for the drop-size dis- tribution that represents some types of experimental drop-size data fairly well is that of Imai (1964). It may be written
N(D) = NoDI-' exp(- AD), (10.3)
CHAPTER 10 ROSENFELD AND ULBRICH 243
TABLE 10.3. Definitions of various raindrop integral Symbol Parameter p a (^) p Z Reflectivity factor 6 106 mms cm- 6 W Liquid water 3 0.524 g crrr? concentration R Rainfall rate^ 3.67^ 33.31 mm^ h-^ I^ m^3 cm-^3.^67 NT Total number^0 1. concentration
Exponent
b = 7 + J1.
4.67 + J1.
5 = 1 4.67 + J1. 1 + J1. 7J = 4.67 + J1.
/(= 4+J1. 4.67 + J1.
1 and
+,r(~)r 'm-ffl
a r[[3(P - q)] p [3 - 1
P = aQf3,
[3 = p + JL + 1 and q+JL+I
aprep + JL + l)NA-f
a=. [aQr(q + JL + 1)]f
p - [3q JL= [3-
A = 10 6f(7^ + J1.)Nr;2.331(4.67+p. [33.31f(4.67 + 1)](7+p.V(4.67+p. S = ___--::3,.,...6_7.,.-+--'.J1. __ [33.31Nof( 4.67 + 1)] 11(4.67+1' _ f(l + J1.)!VJ.67/(4.67+p. g - [33.31f(4.67 + J1.)] (I +1'1/(4.67+1'
_ 7Tf(4 + J1.)Ng·67/(4.67+p.
t - 6[33.31f(4.67 + J1.)](4+p.1/(4.67+p.
Coefficient
Z= ARb
Do = sR'
TABLE 10.4. Relations between integral rainfall parameters found from Eqs. (10.18)-(10.20) assuming a gamma RDSD.
This approach assumes that the parameter No is constant or at least slowly varying with R. In this work we con- sider the examples of P-Q relations of the form Z = ARb, Do = ERa, NT = gR'IJ and W = (RK. The coefficient A and exponent b in an empirical Z-R relation are used to find JL and No from which the coefficients and ex- ponents for the remaining three relations are calculated. The coefficients and exponents for all these relations are listed in the Table 10.4. The corresponding expressions for No and JL in terms of A and bare
These equations can be inverted to obtain expressions for JL and No, that is,
where
Q = a (^) Q i'" DqN(D) dD
= a f(q + JL + 1) N. Dq+p.+l Q (3.67 + JL)q+p.+l ° °.
Elimination of Do between P and Q results in the form
above for P with a (^) Q and q substituted for a (^) p and p, respectively:
P-Q relation
v(D) = 17.67Do.67 , (10.16)
Cz are constants. Note that the latter relation implies that in equilibrium rainfall a linear relation exists be- tween Z and R. Atlas and Ulbrich (2000) show that a direct proportionality between Z and R also implies that the median volume diameter Do must be constant in time during the rainfall event. Although this behavior has been observed infrequently in nature with surface disdrometer data, Atlas and Ulbrich (2000) show spectra acquired aloft for storms in TOGA COARE for which the constancy of Do and proportionality of Z and Rare evident. These spectra closely resemble the equilibrium spectra of List et al. (1987) and Hu and Srivastava (1995) and display a tendency for a peak occurring near a diameter of 1.5 mm, in agreement with the theoretical predictions. In this work the gamma RDSD will be employed to deduce the behavior of integral parameters implied by the values of the coefficient A and exponent b in the Z- R law. This is done in the manner described by Ulbrich (1983), which uses A and b to compute the gamma distribution parameters and proceeds in the following way. Any integral parameter P may be expressed in terms of the gamma distribution as
P = a (^) p i~ DpN(D) dD
= a rep + JL + 1) N. DP+P.+1 (10.15) p (3.67 + JL)P+p.+ 1 ° °.
Note that the method assumes D (^) min = 0, D (^) mn• -7 co. The effects on integral parameters and empirical relations of assuming values for D (^) min and D (^) m ax different from these values have been investigated by Ulbrich (1985, 1992, 1993). For parameters Z, W, R, and NT the values of p and a (^) p are listed in Table 10.3. In the above it has been assumed that the fall speeds of the drops in still air are given by a power law in terms of the diameter as given by Atlas and Ulbrich (1977), that is,
where v(D) is in meters per second and D is in centi- meters. This is a good approximation to the raindrop fall speeds at sea level and is sufficiently accurate for the present purposes. Consider a pair of integral parameters P and Q. The theoretical expression for Q is the same as that shown
244 METEOROLOGICAL MONOGRAPHS VOL. 30, No. 52
D
o
D
D
N(D)
(f)
N(D)
N(D)
D
D
D
o
(a)
N(D)
N(D)
FIG. 10.3. Schematic depictions of the effects of various processes on the shape of the RDSD. The processes illustrated are (a) raindrop coalescence, (b) raindrop breakup, (c) coalescence and breakup acting simultaneously, (d) accretion of cloud droplets, (e) evaporation, (f) an updraft, (g) an accelerated downdraft, and (h) size sorting.
(fO.25)
and (10.23)
[ (
2 )]b I/(I-b) A 33.31f ~ b - 1
106f(~·33b) b - 1
7 - 4.67b /L= b-l
No = )---------(
a. Coalescence (Fig. lO.3a)
Modification of the RDSD by coalescence alone de- creases the numbers of small diameter drops and in- creases those of the larger drops. Consequently, Do must increase and the total number concentration of drops NT must decrease. The process also increases /L; the amount by which it would change depends on the efficiency of the coalescence process. The result is a decrease in No and a consequent increase in A and small decrease in b. There would be an approximate parallel shift in the Z-R relation on the RAPAD upward and to the left, perpendicular to the Do isopleths. It would be necessary
Wilson and Brandes (1979) provided qualitative micro- physical and kinematic influences on Z-R relationships, presented in Table 10.1. Here, with the added benefit of the additional RDSD formulations, this can be expanded and tie the different physical processes to the parameters presented in Table 10.3 and the Z-R relation parameters. The discussion is presented for each factor acting alone, assuming everything else is held constant. Admittedly, it is rarely the case in reality; however, it serves the purpose of understanding the various processes that combine to form the actual RDSDs.
It should be noted that there are a couple of instances in which this approach will not yield useful results. First of all, if /L -s - 1, then NT is undefined. Second, if b is very close to 1 (as in the case of the equilibrium drop- size distribution), then /L becomes very large. In neither of these cases are the results found for the coefficients and exponents in the above table considered to be phys- N(O) ically meaningful. These results may now be used to illustrate the effects on the coefficient A and exponent b of each of the var- ious physical process listed in Table 10.1. For the pur- poses of illustration the RDSD before modification is shown as an exponential distribution on most of the diagrams that follow and is represented by a straight line. However, it is clear that the RDSD could have any (e) shape before modification. In describing the changes that take place, the equation for NT in terms of No, /L, and Do is used. From Eq. (10.15) the expression for NT may be shown to have the form N(D)
N = N (^) oDb+!Lf(1 + /L) T (3.67 + /L)I +1'0 •
246 METEOROLOGICAL MONOGRAPHS VOL. 30, No. 52
N
A== 169, b== 1. _ .. _ .. _ .. _ •• A==235, b==1. .... A==480, b==1. .,.,.,-,- A== 31, b==1.
0.02I 0.Q1 1 : I
Atlas-Chmela, stratiform Atlas-Chmela, stratiform Jones, heavy showers Blanchard, orographic M-P theory
0.05 i
I I !I
. .
FIG. 10.4. Rain parameter diagram for a gamma distribution with J.L = 9. Isopleths are the same as in Fig. 10.1. Also shown are the four Z-R relations displayed by Atlas and Chrnela (1957). The large filled circle corresponds to the theoretical equilibrium RDSD of Hu and
Srivastava (1995) for which J.L = 9 and Do = 1.76 rnrn. It is notable that the four Z-R relations
are close to converging to the point for the equilibrium distribution.
large rainfall rates, these four relations tend to converge to the DSDe.
a. Equilibrium DSD and Z-R relations
The DSDe is the drop-size distribution that would be developed in a rain shaft that falls a sufficiently long time for the rates of drop merging and breakup processes to reach equilibrium, assuming no gain or loss of rain- drops to other processes. The time required for reaching DSDe is shorter for greater R, because of the greater W and respectively shorter time between drop interactions. Hu and Srivastava (1995) have calculated that reaching DSDe from initial Marshall-Palmer exponential RDSD would take about 10 min for R = 90 mm h- I , but the main features of DSDe would appear already after 2. min. The time for reaching DSDe becomes longer lin- early with R:', Hu and Srivastava (1995, p. 1768) stated that "for heavy rainfall rates, say 50 mm h -I, approx- imate equilibrium between collisional processes may perhaps be expected within the usual lifetime of a con-
vecti ve shower." Supporting observations in heavy tropical rain showers were reported by Zawadzki and de Agostinho Antonio (1988). Observations at extra- tropical rain showers showed considerable deviations from DSDe (Carbone and Nelson 1978; Sauvageot and Lacaux 1995). Because DSDe is independent of R, Do is constant at 1.76 mm, and R is linear with NT' This means that the exponent in the Z-R power-law relation is unity (List 1988), and the Z-R relation is simply Z = 600 R (after Hu and Srivastava 1995). An exponent greater than 1 means that Do is increasing with R. During the growth phase of the precipitation before reaching DSDe, Do does increase with R. The conversion of cloud droplets to precipitation occurs by forming small precipitation particles that increase in size with the progress of their collection of the cloud water and so increasing R, until the drops become sufficiently large for breakup to com- pensate their additional growth. Because breakup does not occur in convective ice precipitation, that is, graupel and hail, no equilibrium is expected there and the equiv-
CHAPTER 10 ROSENFELD AND ULBRICH 247
alent melted hydrometeors would have ever-increasing Do with R. This is why rain formed from the melting of hail can reach extreme reflectivities, which translate to impossibly high R when applying to the hail Z-R relations for rainfall.
b. Evolution of warm rain
In a hypothetical rising cloud column with active co- alescence, the initial dominant process would be wid- ening of the cloud drop-size distribution into large con- centrations of drizzle drops; the drizzle continues to coalesce with other drizzle and cloud drops into rain- drops, which will continue to grow asymptotically to Doe. Therefore, during the growth phase of precipitation R increases with Do, and this would increase the ex- ponent b. Ideally, for rainfall with drops that fall from cloud top while growing, R would increase with the fall distance from the cloud top, mainly by growth of the falling drops due to accretion and coalescence, and to a lesser extent by addition of new small rain drops, until R becomes sufficiently large for breakup to become sig- nificant. Shallow orographic clouds can present condi- tions such as some distance below the tops of convective clouds. Therefore, similar evolution of R can be ob- served on a mountain slope, such as documented by Fujiwara (1965). Different values of R near cloud top or in shallow orographic clouds can come mainly from changing NT' because the drop size is bounded by the limited vertical fall distance along which they can grow. This would cause orographic precipitation to have small coefficient A, and more so with shallower clouds and stronger orographic ascent, because the stronger rising component supplies more water for the production of many small raindrops not too far below cloud top, which are manifested as a larger R.
c. Evolution of cold rain
Microphysically "continental" clouds are character- ized by narrow cloud drop-size distributions and, there- fore, by having little drop coalescence and warm rain. Most raindrops originate from melting of ice hydro- meteors that are typically graupel or hail in the con- vective elements, and snowflakes in the mature or strat- iform clouds. Graupel and hail particles grow without breakup while falling through the supercooled portion of the cloud, and continue to grow by accretion in the warm part of the cloud, where they melt. Large melting hailstones shed the excess meltwater in the form of an RDSD about which little is known. The shedding stops when the melting particles approach the size of the larg- est stable raindrops, which are later subject to further breakup due to collisions with other raindrops. In fact, new raindrop formation is limited only to the breakup of pre-existing larger precipitation particles. Therefore, we should expect that in such clouds there would be, for a given R, a relative dearth of small drops and excess
of large drops compared to microphysically "maritime" clouds with active cloud drop coalescence. Deep con- tinental convective clouds would therefore initiate the precipitation by forming large drops that, with maturing, approach DSDe from above. This is in contrast with the approach from below for maturing maritime RDSD. Recent satellite studies (Rosenfeld and Lensky 1998) have shown that microphysically maritime clouds are associated typically with a "rain-out" zone; that is, the fast conversion of cloud water to precipitation causes the convective elements to lose water to precipitation while growing. This leaves less water carried upward to the supercooled zone, so that weaker ice precipitation can develop aloft. Williams et a1. (2002) have recog- nized this as a potential cause to the much greater oc- currence of lightning in continental compared to mar- itime clouds. Williams et a1. (2002) noted that frequent lightning occurred also in very clean air during high atmospheric instability, probably because the strong up- draft leaves little time to the formation of warm rain and carries the large raindrops that manage to form up to the supercooled levels of the clouds, where they freeze and participate in the cloud electrification pro- cesses. This difference between continental and maritime clouds means that mostly warm rain would fall even from the very deep maritime convection, which reaches well above the freezing level, whereas precipitation from continental clouds would originate mainly in ice processes. Therefore, -the expected difference in RDSD between microphysically maritime and continental clouds is expected to exist also for the deepest convec- tive clouds that extend well into the subfreezing tem- peratures.
5. Proposed method of classifying Z-R relations
Equipped with this conceptual model, now we can turn our attention to actual measurements of RDSDs and their Z-R relations, which can be related to the precipitation-forming processes as discussed above. These Z-R relations are provided in Table 10.5 and are classified according to combinations of the categories of microphysically maritime and continental, convec- tive, stratiform, and orographic. The values of A and b are those listed in the source of the Z-R relation, whereas the values of the RDSD parameters No and /L were cal- culated from A and busing Eqs. (10.23) and (10.24), respectively. The coefficients and exponents (s, 8) and (" K) in the Do-R and W-R relations, respectively, were calculated using the expressions in Table 10.3. Also shown for reference in Table 10.5 are values of D (^) o(10), Do(30), W(10), and W(30), the values of Do and Wat R = 10 and 30 mm h -I. It should be recognized that the Do-R and -W-R relations derived in this way are strictly theoretical and are only approximations to those that might be found from.empirical analyses of the data from which the Z-R relations were found. However, in
CHAPTER 10
(b) Source and notes. Continental
- Joss and Waldvogel (1970)
- Foote (1966)
- Rinehart (2002)
- Sims (1964)
Moderate continental
- Petrocchi and Banis (1980)
Tropical continental
- Sauvageot (1994)
- Ulbrich et al. (1999)
- Maki et al. (2001)
Tropical maritime
- Tokay et al. (1995)
- Tokay and Short (1996)
- Stout and Mueller (1968)
- Stout and Mueller (1968)
- Tokay et al. (1995)
- Tokay and Short (1996)
- Stout and Mueller (1968) Tropical maritime aloft
- Ulbrich and Atlas (2001)
- Ulbrich and Atlas (2001) Hurricane
- Jorgensen and Willis (1982)
- Jorgensen and Willis (1982) Orographic
- Fujiwara and Yanese (1968)
- Fujiwara and Yanese (1968)
- Fujiwara and Yanese (1968)
- Blanchard (1953)
ROSENFELD AND ULBRICH
TABLE 10.5.
Thunderstorms. 25 days total, Locarno, Switzerland. Disdrometer data. Mountain thunderstorm in AZ. Filter paper measurements. 62 spectra for 37 storms. Grand Forks, ND during several autumn seasons. Filter paper measurements. Thundershowers, 1963. ISWS drop camera data.
Thunderstorm, Norman, OK. Disdrometer data.
Tropical squall line. Congo. Disdrometer data. Average of seven afternoon thunderstorms in Arecibo, PRo Disdrometer data. Darwin, Australia, 1997-98. 15 squall lines, all stages. Disdrometer data.
Tropical maritime, coastal, convective, Darwin, Australia. Disdrometer data. Tropical convective, equatorial, maritime, TOGA COARE, Disdrometer data. Marshall Islands, trade wind cumulus, warm rain, maritime. ISWS drop camera data. Marshall Islands, showers, equatorial maritime. ISWS drop camera data. Darwin, Australia. Stratiform, coastal, tropical maritime. Disdrometer data. TOGA COARE, stratiform, coastal, equatorial maritime. Disdrometer data. Marshall Islands, continuous rain. Equatorial maritime. ISWS drp camera data.
TOGA COARE Convective aloft, by updraft. PMM analysis. 2DP probe data. TOGA COARE Stratiform aloft, by updraft. PMM analysis. 2DP probe data.
Hurricane eye wall. Aircraft data aloft. 2DP probe data. Hurricane rain bands. Aircraft data aloft; 2DP probe data.
Orographic rain, Mount Fuji at altitude of 1300 m. Filter paper. Orographic rain, Mount Fuji at altitude of 2100 m. Filter-paper, Orographic rain, Mount Fuji at altitude of 3400 m. Filter paper. Orographic rain, Hawaii, Mauna Loa, altitudes between 670-920 m. Filter paper.
the vast majority of the empirical Z-R relations inves- tigated in this work there is no information available concerning the corresponding Do-R and W-R relations. The method employed in this work finds theoretical Da- R and W-R relations that are used for classifying the type of rainfall even though they may not be accurate representations of the actual empirical relations. Nev- ertheless, Atlas (1964) has shown that the Do-R rela- tions found from the Z-R relations he investigated are in good agreement with those found directly from em- pirical analysis of experimental drop-size spectra. In Figs. 10.6, 10.9, 10.11, and 10.12 that follow, theoretical Da-W relations are plotted for values of fL = -2 and 12 and for R = 10 and 30 mm h -1. These relations are found from Eq. (10.15) using the definitions in Table 10.3 and have the form
W = 0.0157(3.67 + fL)a.67 [(4 + fL) D-a.67. (10.26) R [(4.67 + fL) a The two curves in each figure represent the range of uncertainty (for given R) in nature associated with the theoretical relations. In all cases it is seen that the the uncertainty due to such variations is small, thus adding further credibility to the method of classifying the Z-R relations.
a. Maritime-continental classification
The most fundamental classification of rain clouds can be done into maritime and continental. Cloud phys- icists have traditionally designated clouds as maritime and continental based on their microstructure, where maritime clouds contain small concentrations (about 50-100 em -3) of large droplets, and continental clouds contain tenfold-larger concentrations of respectively smaller droplets. Maritime clouds precipitate easily by warm processes, whereas coalescence is often sup- pressed in continental clouds, which often have to grow to supercooled levels to precipitate by "cold" processes, that is, involving the ice phase. Some notable differ- ences have been documented between maritime and con- tinental convective clouds.
- There is distinctly less supercooled water and it is limited to warmer temperatures in the tropical mari- time compared to the continental clouds (Zipser and LeMone 1980; Black and Hallett 1986).
- The updraft velocities in- maritime clouds are char- acteristically limited to below the terminal fall veloc- ity of the raindrops, whereas no such maximum for the updraft was noted in continental convection (Zip-
250 METEOROLOGICAL MONOGRAPHS
R (Z) Convective: Continental - Maritime
VOL. 30, No. 52
'-'E a::
!!! (^)!!!
- R(40^ d^ BZ)^ ~ IllII R( 50 d BZ ) .m ";,,^ ";,, ,', :^ ,', (^) ,,~, .. ~. .~. ,', ',' '.' .....^ .,'
.'
',' .r ,', ,^ <- (^) ..
'.'
..
" ,'.^ ,',^ ,^ ,', r, ,', ,^ ,', ,^ ,', (^) " ,', ,^ ,', ,^ ,',
~
.... _.'.. ,', ,^ ,', ,^ " ,'. :,^ .. '.. :,^ .'.
.IlI, (^) .. II
'Iljl'
.. : : • : : (^) -!- :!:
,1 (^) ~ :..
...' ..
,', ,^ ,'. v,^ ,', ,^ .'. ,^ .'. (^3 4 5 6 7 8 9 1 0) 1 1 1 2 E
"
i I (^) I I I I
.... 0 0 UI^^0
~ t/)^ CII^
.. Co t/)^ = t/)^ t/)^0 'tl CIl^ t/) Type:
t/) (^) t/) (^) CII (^) t/) I- (^111) I- 111 0 l- (^) I- t/) (^) I- 111 :::Itr^0 :::I > > E (^) == 0 0 (^0 111 111) E .s tr^ C^ c^ l-^ .e^ E c .. (^) c ... UI^ t/)^ t/)^0 0 t/)^ :::I (^111 0) N^ .::.:^0 '0^^0 0 0::^0 0 111 .;:
.e (^) Cl 0. .5 w^ .e^ 'i^ U .Q .;: (^111 111) C 1:: ..J^0 «^0 :;;;:^0 :::I^ ~ 111 ~ 0::^ e .e
Z 0 0 0.^ «^111 l!!^ :::I
0 0111 0 :a^111 tr
0 :a^ w
FIG. 10.5. The Z-R relations for rainfall from maritime and continental convective clouds. The rain intensities for 40 and 50 dBZ are plotted in the figure. Note the systematic increase of R for a given Z for the transition from continental to maritime clouds. The.Z-R relations used in this figure are as follows: A b
- Swiss Locarno thunderstorms, continental (Joss and Waldvogel 1970) 830 1.
- Arizona mountain thunderstorms (Foote 1966) 646 1.
- Grand Forks, North Dakota, in autumn (R. E. Rinehart 2002, personal (^429) 1. communication)
- Illinois thunderstorms, continental (Sims 1964) (^446) 1.
- Oklahoma thunderstorms, moderate continental (Petrocchi and Banis (^316) 1.
- Congo squall line, tropical continental (Sauvageot 1994) (^425) 1.
- Puerto Rico thunderstorms, coastal, moderate maritime (Ulbrich et al. (^261) 1. . 1999)
- Darwin Squalls, coastal, tropical maritime (Maki et al. 2001) (^232) 1.
- (^) Darwin convective DSD, coastal, tropical maritime (Tokay et al. 1995) 175 1.
- COARE convective DSD, equatorial maritime (Tokay and Short 1996) (^139) 1.
- Marshall trade wind cumulus, warm rain maritime (Stout and Mueller 126 1.
- Marshall Showers, equatorial maritime (Stout and Mueller 1968) 146 1. E. Equilibrium DSD 600 1.
ser and LeMone 1980; Jorgensen and LeMone 1989; Zipser and Lutz 1994).
- The vertical profiles ofradar reflectivity in the mixed- phase region are substantially stronger in continental than in maritime clouds. This was ascribed mainly to the greater updraft velocities in the more continental conditions (Williams et al. 1992; Rutledge et al. 1992; Zipser 1994; Zipser and Lutz 1994). - All of these differences can potentially explain the dramatic contrast between the lightning over land and ocean that was revealed when observations of light- ning from space became available (Orville and Hen- derson 1986).
Given the fundamental importance of the classification of clouds into' maritime and continental, and in view of
s;^ I
252 METEOROLOGICAL MONOGRAPHS VOL. 30, No. 52
content and updraft velocities determine cloud micro- structure. Larger aerosol concentrations make the coa- lescence slower, and greater updrafts leave less time for the progress of the coalescence, so that the product of these two factors ultimately determines the "continen- tality" of the clouds, as manifested by the evolution of cloud drop-size distribution with height in the growing
Ul ::: 0 ::: 0 C c Ul
VI VI e^ :I^0 cVI^ cVI^ m^ m^ m^ ""c c
Type: C > C^ '"~ (^) :S:I^ C^ C^ :;::;:^ :;::;:^ ~ (^) .Q'" VIc
0 ~^ .cVI^ c 0 c 0 e^ 0.^ 0.^ »^ 'iii^ E
0 til 0 0 til^ >c^ - ~ wu It: 'f! c c m (^) w w (^0) - .;:^ u^ :§
.~ .~ .ce^ m.c It:« It:«^0 w^ VIw '" .;:
"" '" ':;
" '" '" e^0 0 It:^
C C :;: It:^ e^ ""^ t:r '" 0 0 «^
:;: «^ IJ..^ e^ w
00 00 IJ..
FIG. 10.8. The Z-R relations for rainfall from convective and strat- iforrn tropical rainfall and from a hurricane. The rain intensities for 30 and 40 dBZ are plotted in the figure. Note the systematic increase of R for a given Z for the transition from stratiform to convective clouds. Apparently hurricane rainfall is more similar to stratiform, even in tile eyewall. The Z-R relations used in this figure, numbered in digits for convective and characters for stratiform, are as follows:
A b
- Darwin convective DSD, coastal, tropical mari- 175 1. time (Tokay et al. 1995) a. Darwin stratiform DSD, coastal, tropical mari- 335 1. time (Tokay et al. 1995)
- Marshall showers, equatorial maritime (Stout and (^146) 1. Mueller 1968) b. Marshall continuous, equatorial maritime (Stout (^226) 1. and Mueller 1968)
- (^) COARE convective DSD, equatorial maritime 139 1. (Tokay and Short 1996) c. (^) COARE stratiform DSD, equatorial maritime 367 1. (Tokay and Short 1996)
- COARE convective aloft, by updraft (Ulbrich and 120 1. Atlas 2002) d. COARE stratiform aloft, by updraft (Ulbrich and 203 1. Atlas 2002)
- Hurricane eyewall (Jorgensen and Willis 1982) 287 1. e. Hurricane rainbands (Jorgensen and Willis 1982) 301 1. E. Equilibrium DSD 600 1.
10
E .§. 0:::
I
A R(30 dBZ) IR (Z) Convective^ -^ Stratiform e R(40 (^) dBZ)., , ,
................. ............. .............. .....• ........ .............. ..........
CD CIt^ •^ • CIt ..^ lID .................. ...•......• - -, .............. ..... ......... - •...•....... .......... .................. .............. .............. ...... ....... .. ............ .......... .................. ............. ............... .............. .............. .......... .....•..•......... .............. (^) ..... .. ....... .............. .............. ... (^) .. .. - ...... ........ .............. ............... .. ............. ............. .......... .... , ............ (^) ..A .... ··· .. · ..·A···· ..... ..............A .............. .......... A .................. (^) ......... J,;. .. ....... ..... (^) ......... :11>: ... ............. .......... A .......... ' ....A^ .............. .........A .. (^) .... - ........ - ..............A .......... A (^1) , a 2 b^3 c^4 d^5 e E , (^) I I I I I I
convective elements. The ultimate test for the role of cloud microstructure is comparing the RDSD of clouds at the same location, but at different times, when they possess maritime or continental microstructure. That is exactly what is done in Fig. 10.7. The visible and infrared scanner onboard (VIRs) the Tropical Rain- fall Measuring Mission (TRMM) satellite was used to retrieve the microstructure of rain clouds over disdrome- ter sites. The clouds were classified into continental, intermediate, and maritime, using the methodology of Rosenfeld and Lensky (1998). The DSDs from the con- tinental and maritime classes during the overpass time ± 18 h were lumped together and plotted in Fig. 10.7. Indeed, the continental and maritime DSDs are well separated in Fig. 10.7, with the continental clouds pro- ducing greater concentrations of large drops and smaller concentrations of small drops. A comparison between the directly measured disdrometer rainfall and the cal- culated accumulation by applying the TRMM Z-R re- lations (Iguchi et al. 2000) to the disdrometer measured Z resulted in a relative overestimate by more than a factor of 2 of the rainfall from the microphysically con- tinental clouds compared to the maritime clouds. The evidence shows that it is mainly the cloud mi- crostructure that is responsible to the large systematic difference in the RDSD and Z-R relations between mar- itime and continental clouds. There are several possible causes for these differences, as described in the follow- ing sections, all working at the same direction.
EXTENT OF COALESCENCE The cloud drop coalescence in highly maritime clouds is so fast that rainfall is developed low in the growing convective elements and precipitates while the clouds are still growing. The large concentrations of raindrops that form low in the cloud typically fall before they have the time to grow and reach equilibrium RDSD, thereby creating the rain-out zone (Rosenfeld and Len- sky 1998) less than 2 km above cloud-base height. Therefore, Do remains much smaller than DOe' as can be seen in Fig. 10.6b. In microphysically continental clouds with sup- pressed coalescence the cloud has to grow into large depth before it will start precipitating, by either warm or cold processes. The raindrops that fall through the lower part of the cloud grow by accretion of small cloud drops, so that they tend to break up much less than drops that grow mainly by collisions with other raindrops, as is the case for maritime clouds. This process allows Do to exceed Doe in the growing stages of the precipitation and later to approach it from above when the raindrop collisions become more frequent with the intensification of the rainfall.
WARM VERSUS COLD PRECIPITATION PROCESSES The rain-out of the maritime clouds (Rosenfeld and Lensky 1988) depletes the cloud water before reaching
0.1 0.15 0.2 0.25 0.3 0.
0.5 '--'-..l-.L...1....l-'-L.l.-J-.L...1....l-'-'-'--'-'--L.-l....L.Jl...L....L...l--L..J....l...J'-'-J
(b)
'?^ ......
E 2
.....C
0.2 0.
ROSENFELD AND ULBRICH
E
Tropical Stratiform
Hurricane
CHAPTER 10
(a)
'? E (^) 0. .....C
~ 0.
FIG. 10.9. (a) The liquid water content W vs median volume diameter Do for R = 10 mm h- I^ from convective and stratiform tropical
rainfall, and from a hurricane. The labels of the points are according to Fig. 10.8. Other details are the same as in Fig. 1O.6a. (b) The liquid water content W vs median volume diameter Do for R = 30 mm h- I^ from convective and stratiform tropical rainfall and from a hurricane. The labels of the points are according to Fig. 10.8. Note the distinct separation into convective and stratiform groups, where the convective rainfall has much smaller Do. The hurricane rainfall fits into the stratiform group. Other details are the same as in Fig. 10.6a.
the supercooled levels (Zipser and LeMone 1980; Black and Hallett 1986), so that mixed-phase precipitation would be much less developed in the maritime clouds compared to the continental. This is manifested in the smaller reflectivity aloft in the maritime clouds (Zipser and Lutz 1994), which is a manifestation of the smaller hydrometeors that form there (Zipser 1994). In contrast, the suppressed coalescence in continental clouds leaves most of the cloud water available for growth of ice hydrometeors aloft, typically in the form of graupel and hail. These ice hydrometeors can grow indefinitely with- out breakup, until they fall into the warm part of the cloud and melt. The melted hydrometeors continue to grow by accretion of cloud droplets, until they exceed the size of spontaneous breakup or collide with other raindrops. Therefore, convective rainfall that originates
as ice hydrometeors would have Do > Doe and would
approach Doe from above with maturing of the RDSD.
- STRENGTH OF THE UPDRAFTS
Updrafts are typically stronger in more continental clouds and therefore contribute to more microphysically continental clouds and less warm rain processes, as dis- cussed already above. In addition, stronger updrafts al- low drops with greater minimal size to fall through them. In addition, stronger updrafts leave less time for forming of warm rain and rain-out, and advect more cloud water to the supercooled zone. Therefore, due to the reasons already discussed in sections 5a(l) and
5a(2), the stronger updrafts are likely to lead to precip- itation with greater Do and smaller R for the same Z.
- EVAPORATION
More continental environments have typically higher cloud base and lower relative humidity at the subcloud layer. Evaporation depletes preferentially the smaller raindrops and works to increase Do.
b. Convective-stratiform classification
The mature elements of organized deep convective cloud systems often merge into widespread light to mod- erate rainfall area, which is called "stratiform," al- though it is eventually generated by convection. The classification is obvious in typical squall lines, which have a simple structure with three characteristic regions: convective, stratiform, and transition (Houze 1989). The rainfall in the convective and transition regions is formed as warm rain and graupel melt. Typically, there is more warm rain falling through the updraft, and more graupel melt falling with the downdraft toward the tran- sition zone. The stratiform precipitation is composed typically of ice particles that were advected from the convective portion of the storm, and from aggregation of newly formed ice crystals in the moderate mesoscale updraft that develops in the merged anvils over the strat- iform rainfall area. Waldvogel (1974) has shown that the onset of the stratiform precipitation is associated
CHAPTER 10
(a)
'1..... E (^) 0.
......^ Cl
.-- 0 Warm .... (^) 0.6 (^) Orographic
ROSENFELD AND ULBRICH
(b)
3
'1..... E 2 . .-- 0 Warm C') 1.5 ~hica
0.05 0.1 0.15 0.2 0.
0.05 0.1 0.15 0.2 0.25 0.3 0.
n (10) [em]
o n^ o(30)^ [em]
FIG. 10.11. (a) The liquid water content W vs median volume diameter Do for R = 10 mID h- I^ from warm rain over orographic barriers.
The labels of the points are according to Fig. 10.10 Other details are the same as Fig. 1O.6a. (b) The liquid water content W vs median volume diameter Do for R = 30 mm h- I^ from warm rain over orographic barriers. The labels of the points are according to Fig. 10.10. Note the systematic decrease of Do for increasing height. Other details are the same as in Fig. 10.6a.
of cloud droplets, both on water and ice hydrome- tears; and (ii) "stratiform," where precipitation
forms in updrafts < 1 m s -1 mainly as ice crystals
that aggregate into snowflakes and melt into rainfall in a radar "bright band;" and c) orography, where low-level orographic lifting in
clouds with active cloud drop coalescence can pro- duce extremely small Do compared to all other types of rainfall and, hence, a gross radar underestimate of R.
Figures 1O.12a, b illustrates that this classification orders
0.2 0.25 0.3 0.
Tropical Stratiform
Hurricane
0.1 0.
2
Convec ive
Maritime
- (^) Intermediate Continental
(b)
o. 5 L.L.-'-'--'-'---'---''-LJ-L-'-'--'-'-'-'-'-.l-..J..-'-'--'-J-L-.i-L-'-'--'-'
c Warm ~ 1.5 ~hic
'1..... E
0.2 0.
v e
Convec
Maritime Intermediate
(a)
0.9 (^) Hurricane
0.8 Tropical '1.....^ Stratiform E (^) 0. . (^0) .... Warm 0.6 (^) Orographic
o (10.) [em]
o n^ o(30)^ [em]
FIG. 10.12. (a) The liquid water content W for R = 10 mID h -I for all the sampled rain types. The horizontal bars show the range of Do for the various rain types: Edenotes the value of equilibrium RDSD. Other details are the same as in Fig. 10.6a. (b) The same as in (a) but for R = 30 mID h- I •
FIG. 10.13. The differential reflectivity factor ZOR as a function of the median volume' diameter Do. The' calculations assume that the axial ratio-diameter relation is that defined in Keenan et al. (2001) and that the raindrops are distributed according to a gamma distri- bution having shape parameter JL. Results are shown for values of JL
= -2,0, 2, 4, and 6. The radar wave length is assumed to be 10.
ern and the maximum raindrop diameter D max = 8 mm.
in a physically meaningful way the large range of var- iability of the Z-R and Do. This classification scheme can explain a variability of R for a given Z by a factor of 1.5-2 for convective-stratiform classification, a fac- tor of more than 3 for continental-maritime classifica- tion, and up to a factor of lOin orographic precipitation. The classification scheme reveals the potential for significant improvements in radar rainfall estimates by application of a dynamic Z-R relation, based on the microphysical, topographical, and dynamical context of the rain clouds while following known practices, such as
- determination of cloud microstructure by analyzing the dependence of the cloud drop effective radius on cloud-top temperature of growing convective ele- ments, as done by Rosenfeld and Lensky (1998);
- determination of convective-stratiform separation from existence of bright band or from the horizontal structure of the reflectivity field (e.g., Churchill and Houze 1984);
- synoptic analysis of.the low-level moisture orographic uplifting in clouds.
There are probably many more ways that can lead to this classification. This can serve as the foundation for
METEOROLOGICAL MONOGRAPHS VOL. 30, No. 52
ZoR = 10 log IO(~:),
which is a function of only Do, assuming that the RDSD is a function of only two parameters, as in the case of the exponential distribution, or, for the gamma distri- bution, that the shape of the RDSD is known. To illus- trate the latter point, the differential reflectivity ZOR is plotted in Fig. 10.13 versus Do for several values of the parameter p: in the gamma distribution. These results were calculated for a radar wavelength in the S band and assume that the maximum diameter of the RDSD is D (^) max = 8 mm. It is evident that for Do values as small as about 0.5 mm ZOR has values at least as large as 0. dB. The latter value is considered to be well within the accuracy with which ZOR can be measured so it may be concluded that it can, be determined with high accuracy for all the situations discussed in this work. A method has been developed by Zhang et al. (2001) that enables polarimetric radar measurements to determine all three of the parameters of a gamma RDSD using only the reflectivity factor at horizontal polarization ZHH and ZOR' The method involves an empirical relation between p: and .Ii and estimates raindrop median size with good accuracy, at least for storms in central Florida. It may therefore be concluded that ZOR can be used as a means of classifying rainfall by type even for median volume diameters as small as 0.5 mm as in the case of orographic rainfall and even in the case where the RDSD shape is unknown. Another recent method for determination of the RDSD parameters No, D (^) m , and f.L is described by Bringi et al. (2003). They show that these RDSD pa- rameters change systematically between microphysi- cally continental and maritime clouds along the same lines that were documented and physically explained in the current study. Spaceborne radars, however, cannot employ dual-po- larization measurements, because a falling drop remains perfectly round at the horizontal cross section regardless
a new generation of combined cloud physics-radar al- gorithms that will produce variable Z-R, which will hopefully lead to improvements in the rainfall mea- surements, not only when using reflectivity-only data, but also with polarimetric radars. The application of polarimetric radar data for the measurement of rainfall parameters and characterization of rainfall by type is covered in detail by Bringi and Chandrasekar (2001). In this work a discussion is presented only of the dif- ferential reflectivity technique. These radars possess the capability of acquiring simultaneous estimates of the two parameters of an exponential raindrop-size distri- bution through measurement of the backscattered re- flectivity factors at horizontal and vertical polarization, ZH and Zv, respectively (Seliga and Bringi 1976). One of these parameters, the median volume diameter Do, can be determined directly from the differential reflec- tivity ZOR' defined as
0.2 0.3 0.
Do (em)
a .0.0 0.
;---.. OJ
<::» 2 0:: Cl N
4
+ iL=- OiL= 0 XiL= 2 OiL= 4 (^3) OiL= 6
