Directory UMM :Data Elmu:jurnal:A:Atmospheric Research:Vol52.Issue1-2.Aug1999:
Atmospheric Research 52 Ž1999. 1–16
www.elsevier.comrlocateratmos
A double moment warm rain scheme: description
and test within a kinematic framework
Gustavo G. Carrio´ ) , Matilde Nicolini
Departamento de Ciencias de la Atmosfera,
Centro de InÕestigaciones del Mar y la Atmosfera,
Consejo
´
´
Nacional de InÕestigaciones Cientıficas
y Tecnologicas,
UniÕersidad de Buenos Aires, Buenos Aires,
´
´
Argentina
Received 14 September 1998; received in revised form 4 February 1999; accepted 10 June 1999
Abstract
A two-moment warm rain parameterization that includes prognostic equations for mixing ratios
and number concentrations is described. The number of activated condensation nuclei is also
prognosed assuming a log–linear relationship between the total number of activated condensation
nuclei and the supersaturation. Collective processes are evaluated using pre-calculated matrices of
interaction based on numerical integrations of the corresponding stochastic collection equations.
Mass transfer between cloud and raindrops due to autoconversion is evaluated using Berry’s
parameterization, its attendant raindrop number concentration change is calculated considering the
corresponding rates of change for the first and second moments of the raindrop volume
distribution. A Doppler kinematic reconstruction of a summertime Hawaiian rainband occurred on
August 10, 1990 is used to evaluate the credibility of this scheme. q 1999 Elsevier Science B.V.
All rights reserved.
Keywords: Warm rain; Autoconversion; Microphysical schemes
1. Introduction
Many cloud modelers have used the so-called Kessler Ž1969. parameterization to
evaluate the transformation between categories in which water substance is classified. Its
benefits mainly lie in its inexpensiveness in term of computer resources as it takes as
prognostic variables only the mixing ratios of the different species. On the other hand,
)
Corresponding author
0169-8095r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 9 - 8 0 9 5 Ž 9 9 . 0 0 0 3 1 - 9
2
G.G. Carrio,
´ M. Nicolinir Atmospheric Research 52 (1999) 1–16
the state of art of modeling of explicit spectral microphysics was developed during the
eighties considering detailed spectral treatments of processes such as collision–coalescence, collisional breakup and nucleation ŽKogan, 1991; Feingold et al., 1991, among
others.. Explicit treatments ensure an adequate description of the various processes,
however they require an enormous computer capacity when they are considered within a
multidimensional dynamic framework. There is a trend in recent years to include
prognostic equations for an additional moment of the species distributions, namely
number concentrations ŽFerrier, 1994; Walko et al., 1995; Meyers et al., 1997..
Multimoment schemes appear to be a compromise solution between the detail of the
description and computational expedience. A mixed phase microphysical package that
includes prognostic equations for mixing ratios and number concentrations of various
water species is being developed by the authors, its warm rain scheme is here described
and tested with observational data.
During the Hawaiian Rainband Project ŽHaRP. several aspects of a series of shallow
tropical rainbands offshore of the island of Hawaii have been studied. Dual-Doppler
radar measurements from this project have been used to document the formation,
evolution and kinematic structure of high reflectivity cores ŽSzumowski et al., 1997.. A
two-dimensional dual-Doppler kinematic reconstruction of a rainband, designed to test
warm rain microphysical packages ŽSzumowski et al., 1998a., is used here to evaluate
the credibility of the above mentioned two moment warm rain scheme. This reconstruction represents one of the best cases documenting kinematics and the time evolution of
high reflectivity cores in warm rain clouds. Within a kinematic framework, the absence
of microphysical–dynamical interactions, such as loading effects, certainly introduces
limitations, although, it allows isolating the response of different microphysical parameterizations and schemes. Some aspects of the simulated microphysical structure are
compared with those of a rainband occurred on July 23, 1985, documented and analyzed
by Szumowski et al. Ž1998b. using aircraft data. A general description of the scheme is
given in Section 2. The comparison between simulated and observed characteristics and
final comments are presented in Sections 3 and 4, respectively.
2. Scheme description
Cloud droplet and raindrop volume spectra are assumed to follow gamma distributions of the type used by Berry and Reinhardt Ž1974.. The volume corresponding to a
drop of 40 mm in radius is taken as the boundary between both spectra as it corresponds
to the nearly stationary position of a minimum when solving the stochastic collection
equation.
N Ž Õ . s Nt Ž n q 1 .
n q1
n
Ž ÕrÕf . r Õf G Ž n q 1 . exp y Ž n q 1 . ÕrÕf
Ž 1.
where N Ž Õ . is the number density of drops with volume Õ, Nt is the total number
concentration, Õf is the mean volume, n is a width parameter Ž n ) y1. and G Ž x . is the
gamma function.
G.G. Carrio,
´ M. Nicolinir Atmospheric Research 52 (1999) 1–16
3
The relation between the volume about which most of the mass is concentrated
Žpredominant volume. and the mean volume and the width parameter is given in Eq. Ž2..
Any moment of the gamma volume distribution can be evaluated in terms of the
spectrum parameters with Eq. Ž3..
Ž n q 1.
y1
s w Õf x
y1
Õg y 1 s M 1rM 0
y1
M 2rM 1 y 1
Ž 2.
where Õg is the predominant volume and M b denotes the b th moment of the volume
distribution.
Mbs
`
H0 N Ž Õ . Õ
b
b
dÕ s N G Ž n q b q 1 . r Ž n q 1 . G Ž n q 1 .
Ž 3.
The mixing ratios of cloud droplets and raindrops Ž Qc and Qr . and the corresponding
number concentrations Ž Nc and Nr . along with the concentration of activated nuclei, are
taken as prognostic variables, however the width parameters Ž nc and nr . are assumed to
remain constant during the simulation. The chosen values are consistent with observations in Hawaiian clouds and they are 0 for nc and y0.8 for nr .
The following Eqs. Ž4. – Ž8. represent the tendency equations for Qc , Qr , Nc , Nr and
the number concentration of activated nuclei Ž Nn ..
d Ncrd t s d Ncrd t < NU C q d Ncrdt < AUT q d Ncrd t < ACR
q d Ncrdt < AD V q d Ncrdt < EVA
Ž 4.
dQ crd t s dQ crdt < CO N q dQ crdt < AUT q dQ crdt < ACR
q dQ crdt < AD V q dQcrd t < EVA
d Nrrdt s d Nrrd t < AU T q d Nrrd t < SFC q d Nrrd t < SED q d Nrrdt < ADV
Ž 5.
Ž 6.
dQ rrdt s dQ rrdt < AU T q dQ rrd t < ACR q dQrrdt < SED
q dQ rrdt < AD V q dQrrdt < EVA
d Nnrdt s d Nnrdt < NU C q d Nnrd t < ADV q d Nnrdt < EVA
Ž 7.
Ž 8.
where the subscript, NUC, CON, AUT, ACR, SFC, ADV, SED and EVA stand for
nucleation, condensation, autoconversion, accretion, raindrop self-collection, advection,
sedimentation and evaporation, respectively.
2.1. Collisional processes
2.1.1. AutoconÕersion
Following Berry and Reinhardt Ž1974., an average autoconversion rate can be
evaluated relating a characteristic liquid water content change Ž L 2 . and a corresponding
time scale Žt 2 .. The resulting tendency equation for Qc due to autoconversion is given
in Eq. Ž9. as a function of the cloud drop spectrum parameters Žcgs units are used
hereafter..
dQ crd t < AU T s ydQrrdt < AUT s yr L2rt 2
Ž 9.
G.G. Carrio,
´ M. Nicolinir Atmospheric Research 52 (1999) 1–16
4
where
L 2 s 0.027 Ž rfc4 10 12 Ž nc q 1 .
r r Qc Ž rfc Ž nc q 1 .
y1 r2
y1 r6
y 0.4 . r Qc , t 2s3.7 = 10y7
y 0.00075 . ,
rfc is the mean volume droplet radius and r is the air density.
Under the constraint of a constant value for the raindrop width parameter Ž nr ., the
corresponding rates of change for the first and second moments and the number
concentration Ž M 0 . are not independent. Therefore, the time rate of change for Nr due to
autoconversion is here evaluated in terms of the corresponding rates of change of M 1
and M 2 and assuming that the newly created raindrops have a mean volume coincident
with the chosen boundary between both spectra.
The raindrop mean volume Ž Õfr . and the more physically meaningful predominant
volume Ž Õgr . are linearly related for a given nr Žsee Eq. Ž2... If rate of change for Nr due
to autoconversion is evaluated relating Eq. Ž9. to the volume of a drop of 40 mm, Õgr
and the radar reflectivity are then artificially decreased by the inclusion of these small
drops.
The raindrop predominant volume after a time interval D t can be related to
dQ rrdt < AU T using the following equation:
U U
2
Qr
M 2U s Mr2 q D MAUT
s r Ž a r Õfr . Q r q r Ž aAUT Õ40 . dQrrdt < AU T D t s r Õgr
Ž 10 .
where a r and aAUT denote the ratios between predominant and mean volumes for the
raindrop spectrum and the distribution of the new drops generated by autoconversion,
respectively. The superscript U indicates values after a time interval D t and Õ40 is the
volume of a drop of 40 mm in radius.
The rate of change for Nr due to autoconversion is obtained from Eq. Ž11., assuming
that the spectrum of the created raindrops has a unity mass relative variance Ž aAU T s 2,
nAU T s 0.. According to numerical experiments of Berry and Reinhardt Ž1974., the
second mode of the spectrum, associated to raindrops, has this value of mass relative
variance by the time its mean volume radius reaches 40 mm.
U
Õgr
s ra r Qr q dQrrdt < AU T D t r Nr q d Nrrdt < AUT D t
Ž 11 .
The formula used to evaluate mass conversion between cloud and rain spectra due to
autoconversion ŽEq. Ž8.. as well as newer parameterizations Že.g., Beheng, 1994.
perform selectively depending on the range of droplet sizes. According to the nature of
the simulation, a different formula for mass conversion could be used without affecting
this procedure to evaluate the attending change in the raindrop number concentration.
The rate of change for Nc corresponding to the combined effects of autoconversion
and cloud droplets self-collection is evaluated using the Long Ž1974. polynomial
approximation for the collection kernel for collector drop radius lower than 50 mm.
After integration, it can be expressed as
d Ncrd t < AU T s yk r 2 Q c2 Ž nc q 1 . r Ž nc q 2 .
9
y3
where k s 9.44 = 10 cm
s
y1
.
Ž 12 .
G.G. Carrio,
´ M. Nicolinir Atmospheric Research 52 (1999) 1–16
5
2.1.2. Accretion
The collision–coalescence of cloud and rain drops modifies the concentration of
cloud drops and the mixing ratios of both categories. The tendency equations for Nc and
Qc are
`
`
H0 H0 n Ž Õ . p E Ž Õ ,Õ .
d Ncrd t < AC R s y
c
c
c
r
r Ž Õc . q r Ž Õr .
2
Vt Ž Õr .
yVt Ž Õc . n r Ž Õr . dÕr dÕc
`
Ž 13 .
`
H0 H0 n Ž Õ . m Ž Õ . p E Ž Õ ,Õ .
dQcrd t < AC R s y
c
c
c
c
r
r Ž Õc . q r Ž Õr .
2
Vt Ž Õr .
yVt Ž Õc . n r Ž Õr . dÕr dÕc
Ž 14 .
where n c Ž Õc ., n r Ž Õr . are the number concentrations of droplets and drops; r, m, Vt and
E Ž Õc ,Õr . represent the radii, masses, terminal velocities and the collision efficiency
function, respectively.
Numerical solutions of Eqs. Ž13. and Ž14. were pre-computed and tabulated in
look-up tables for later interpolation for many gamma spectra characterized by different
cloud and rain drop mean volumes Ža similar procedure has been applied by Meyers et
al., 1997, although not predicting Nc .. The collision efficiency dependence on the
colliding drops sizes has been retained for evaluating the interaction matrices. They were
computed for 33 and 40 size categories for cloud and raindrop mean volumes, respectively. Expressions for terminal velocities, integration method and the set of gravitational collision efficiencies were those used by Carrio´ and Levi Ž1995..
In order to improve interpolation, the rates of change of Nc were divided by the
square of the mean volume radii and their corresponding terminal velocities. For the
mixing ratio case they were additionally divided by the droplet mean mass. The
tendency equation for Nc and Qc can then be expressed in terms of the resulting
matrices Ž AŽ Õc , Õr . and B Ž Õc , Õr .. as
d Ncrd t < AC R s Nc Nr rfr2 Vt Ž rfr . A Ž Õfc ,Õfr . Ž r 0rr .
dQ crd t < AC R s ydQrrdt < ACR s Nc Nr rfr2
0.5
Ž 15 .
Vt Ž rfr . m Ž rfc . B Ž Õfc ,Õfr . Ž r 0rr .
0.5
Ž 16 .
where r 0 is the reference air density.
The values of A and B, that are proportional to the ratios between these rates and the
corresponding continuous approximations, are obtained by a bi-linear interpolation on
the logarithms of Õc and Õr . The last factors in Eqs. Ž15. and Ž16. are introduced to take
into account the terminal velocity dependence on air density.
2.1.3. Rain self-collection
This mechanism, although it does not modify the rain mixing ratio, it is responsible
of the fast mean volume increase during the last stages of warm coalescence; this
reduction rate for Nr is
`
`
H H n Ž Õ . p E Ž Õ ,Õ .
d Nrrdt < SFC s y1r2
r
x
x
y
r Ž Õx . q r Ž Õ y .
2
6
G.G. Carrio,
´ M. Nicolinir Atmospheric Research 52 (1999) 1–16
G.G. Carrio,
´ M. Nicolinir Atmospheric Research 52 (1999) 1–16
7
Fig. 1. Idealized flow field at three key times in the simulation. When radar reflectivity reaches y20 dBZ Ža.,
and when the maximum updraft speed and the maximum radar reflectivity are attained Žb and c, respectively..
Cloud boundaries have been superimposed Ž Qc s10y6 ggy1 ..
where < DVt Ž Õ x ,Õ y .< denotes the absolute value of terminal velocity difference between
raindrops with volumes Õ x and Õ y .
Self-collection is treated in a way similar to that applied for accretion. The reduction
rate for the raindrop number concentration is evaluated as a function of the mean
volume drop radius, pre-computing and tabulating numerical solutions of Eq. Ž17.. The
tendency equation for Nr is
0.5
d Nrrdt < SFC s yNr2 C Ž Õfr . Ž r 0rr .
Ž 18 .
where the value of C is determined by linear interpolation on the logarithms of Õr .
The collisional break-up is taken into account by introducing a factor in Eq. Ž18.
related to the coalescence efficiency of the most frequently colliding drop pairs. This
factor is a function of the mean volume raindrop radius and its expression is identical to
that used by Ziegler Ž1985..
2.2. Vapor diffusion
2.2.1. Condensation of water droplet
The tendency equation for Q c due to condensation is obtained integrating the mass
rate of change of a single drop over the cloud spectrum. The latter diffusional growth
rate equation can be expressed as
d mrdt s y4p rSFT FV
Ž 19 .
G.G. Carrio,
´ M. Nicolinir Atmospheric Research 52 (1999) 1–16
8
where
FT s L vrKT Ž L v TrR v y 1 . q 1r Ž Dv rs Ž T . 4
y1
,
FV is the ventilation factor, S is the supersaturation, r the drop radius, K the coefficient
of thermal conductivity of air, L v the latent heat of evaporation, R v the gas constant for
water vapor, Dv the diffusivity of water vapor in air, rs the density of water vapor at
saturation, and T the temperature.
For cloud droplets the ventilation effect can be neglected Ž FV s 1., therefore the
integral value of Eq. Ž19. over the cloud spectrum can be easily obtained using Eq. Ž3..
Although, as an explicit condensation rate equation requires very small time steps, the
semi-implicit procedure of Soong and Ogura Ž1973. is used to approximate the
supersaturation in the resulting expression, taking into account the decrease in the water
vapor mixing ratio Žand the attendant increase in saturation mixing ratio..
2.2.2. Nucleation
The cloud droplet number concentration is modified when condensation rate is
insufficient to maintain exact saturation or when new cloud is forming. A log–linear
relation between the number of activated cloud condensation nuclei Ž NnU . and the locally
steady supersaturation is assumed. The change in Nc due to nucleation is determined
evaluating NnU and comparing it with the cumulative number of activated cloud
condensation nuclei previously processed by the parcel Ž Nn ., that is explicitly accounted.
d Ncrd t < NU C s d Nnrd t < NUC s max 0, Ž NnU y Nn . rDt 4
Ž 20 .
The relation between NnU and S, suggested by the observers for this case, is given in
Eq. Ž21..
4.78 = 10 5 S 4
NnU s 120 S 0.4
100
S F 0.001%
0.001% - S F 0.063%
S ) 0.063%
Ž 21 .
2.2.3. EÕaporation
In a subsaturated ambient, the cloud droplets are assumed to evaporate instantly. The
decrease in the cloud number concentration due to evaporation, corresponding to the
mean droplet volume, is considered as a decrease in the number of activated nuclei Žsee
Eq. Ž8...
In the case of raindrops, the integral expression used to evaluate evaporation is
obtained using Eqs. Ž3. and Ž19., although the enhancement of vapor diffusion associated to ventilation cannot be neglected Žexpressions for FV were those used by Hall,
1980..
The traditional explicit equation for drop growth rate approximates the saturation
vapor density Ž rs . with a linear function of the temperature difference between the drop
and ambient air, when solving simultaneously the diffusion equations for water vapor
and heat. Srivastava and Coen Ž1992. showed that in some cases large errors result of
G.G. Carrio,
´ M. Nicolinir Atmospheric Research 52 (1999) 1–16
9
Fig. 2. Time vertical section of simulated radar reflectivities. Abscissa indicates minutes after time 17:00 UTC.
this approximation. They proposed a more accurate explicit equation based on a second
order approach that can be expressed as
w d mrdt x 2 s w d mrdt x 1 w 1 y a S x
Ž 22 .
where a s 1r2w grŽ1 q g .x rsYrrsX rsrrsX , the primes denote differentiation with respect
to T, and g s L v DrK rsX .
This expression that relates the first and second order approaches is then used as a
correction factor in the rain evaporation integral expression. However, it introduces a
negligible change for this case in which little evaporation occurs below cloud base.
2.3. Sedimentation
Rain number concentration and mixing ratio are independently sedimentated evaluating the number and mass fluxes with the mean and mean mass terminal velocities,
respectively. These mean terminal speeds are numerically evaluated using the Uplinger
Ž1981. expression that gives Vt as a function of the drop diameter Ž D . is
Vt Ž D . s 4854D exp w y1.95D x Ž r 0rr .
0.5
.
Ž 23 .
10
G.G. Carrio,
´ M. Nicolinir Atmospheric Research 52 (1999) 1–16
3. Results
A comparison between observed and simulated characteristics of a convective cell is
made using a kinematic reconstruction based on dual-Doppler data that corresponds to a
summertime Hawaiian rainband occurred on August 10, 1990. This idealized flow field
simulates the evolution of the updraft in magnitude and tilt. In Fig. 1a, b and c, the
velocity field is shown for three key times, when the radar echo first appears, when the
maximum updraft speed is attained and when the maximum radar reflectivity occurs.
Extremely high radar reflectivities Ž59.7 dBZ. have been observed within this cloud with
a depth lower than 3000 m. The times required for the high reflectivity core to develop
from an echo free region Žy20 dBZ. up to 50 dBZ and its maximum value were 15 and
20 min, respectively. This echo free region was located in the middle of the cloud at an
altitude of approximately 1700 m and the velocity field corresponds to that of Fig. 1a.
A time-vertical section of the simulated radar reflectivity is shown in Fig. 2. The
times taken for Z to reach 50 dBZ and its maximum from the value of y20 dBZ, are
very similar to those observed. The maximum of Z remains located at about 1700 m of
altitude as the largest drops are suspended, until a strong vertical gradient associated
Fig. 3. Comparison
between 17:21 and
squares and closed
preserving M 0 and
between observed and simulated intensities of the maximum Z for the August 10 cell
17:46 UTC. Closed squares correspond to the observed evolution. Open circles, open
circles correspond to simulated evolutions using a one-moment, two-moment schemes
M 2 , respectively Žsee text..
G.G. Carrio,
´ M. Nicolinir Atmospheric Research 52 (1999) 1–16
11
with the regions of most intense drop growth develops near the top of the cloud. In Fig.
3, the observed time evolution of the maximum Z during the stage of intense
Fig. 4. Simulated Z ŽdBZ. fields, Ža. for the time at which maximum updraft is attained and Žb. 5 min after.
12
G.G. Carrio,
´ M. Nicolinir Atmospheric Research 52 (1999) 1–16
development is compared to those simulated using different schemes. Simulated curves
are superimposed making approximate updraft maximum times coincident. It can be
seen that the two-moment scheme performs better in simulating the evolution when
d Nrrdt < AU T is evaluated preserving the second moment of the raindrop volume distribution rather than strictly preserving Nr . If a one-moment parameterization is applied Žthat
used by Lin et al., 1986 and by Nicolini and Paegle, 1989., the radar reflectivity
evolution is much faster and a significantly higher value Ž63.4 dBZ. is obtained for the
maximum of Z, probably related to the overestimation of accretion rates and the
concentration of large drops in Marshall–Palmer distribution.
From the beginning to the end of the simulation, the resulting Z fields exhibit a
nearly symmetrical pattern and the maximum values of Z are located within the updraft
core. The simulated Z fields begin to show a sloped pattern associated with the strong
divergence near the inversion level by the time vertical velocity is maximum Žsee Fig.
1b.. This pattern forms because the smaller the raindrop is, the farther it is carried
outward from the updraft core and the slower it falls. The sloped base of this overhang
first deepens downward with time and finally vanishes as the raindrops within it fell
toward the surface, leading to an overall increase in the width of the cell. The simulated
Z fields corresponding approximately to the time maximum updraft is attained and 5
min after are shown in Fig. 4a and b, respectively. Fig. 5 is analogous to Fig. 4 but
corresponds to observed reflectivity fields. The overhang surrounding the cell is visible
in Fig. 4a and Fig. 5a during this stage, the largest drops, associated to the overhang, are
located on the sides of the updraft core.
The general behavior of this rainband was similar to those of several cases documented during HaRP, however the observers suggested a parallel between this cell and a
Fig. 5. Idem Fig. 3, although for observed fields. Adapted from Szumowski et al. Ž1997..
G.G. Carrio,
´ M. Nicolinir Atmospheric Research 52 (1999) 1–16
13
rainband occurred on July 23, 1985. Its time evolution, spatial dimensions and calculated
reflectivities within the rainshaft corresponded closely to the features observed with
radar in the August 10 high reflectivity core ŽSzumowski et al., 1998b., and their
soundings and heights of inversion base were very similar. They performed several
penetrations of this latter cell, each pass was targeted through the region of highest
reflectivity using an on-board radar documenting the evolution of their high reflectivity
core, drop spectra, as well as liquid water content and vertical velocity. The time
evolutions of the calculated maximum Z and its observed altitude for this cell are shown
in Fig. 6 and compared to simulated values. The July 23 sounding was used in this case
and the simulated curves are superimposed making updraft maximum times coincident.
There is a fairly good correspondence between the evolutions and positions of the
observed and simulated cores. As a result, a comparison between microphysical information documented in penetrations and simulated magnitudes for the corresponding times
and altitudes could be made. During the penetrations at which the reflectivity core
remained at an altitude of 1700 m, most of the raindrops were smaller than 1 mm in
diameter, the lowest concentrations were coincident with the updraft core and the largest
Fig. 6. Comparison between the simulated and observed intensities and the altitude of the maximum Z for the
July 23 cell between 16:22 and 16:37 UTC. Intensities are denoted by circles and altitudes by squares. Closed
and open symbols for observed and simulated values, respectively. Abscissa indicates minutes after time 17:00
UTC.
14
G.G. Carrio,
´ M. Nicolinir Atmospheric Research 52 (1999) 1–16
drops were located on the sides. Those two maxima, associated to the overhang, were
observed until 16:28. After this time, the largest raindrops were confined to narrower
regions located at the center of the updraft, as the core descended Ž0.3 km for the time at
which maximum Z occurred..
In Fig. 7, cross-sections of simulated predominant raindrop diameters are shown for
altitudes of 1700, 1300, 900 and 400 m, and the corresponding times 16:23, 16:28,
16:30 and 17:33 Žpasses 2, 5, 6 and 7, see Szumowski et al., 1998b.. The first curve, that
is considered representative of the early stage in which drops are suspended at an
altitude of 1700 m, begins to show two maxima located on the sides of the updraft core
and those drops grow as the base of the overhang descends. The largest drops tend to be
located at the updraft center as those drops, previously suspended, fall through the
nearly vertical decaying updraft associated with the highest liquid water content. Largest
simulated predominant diameters are concentrated in a very narrow region Žless than 500
m. and their maximum value Žf 3.5 mm. is attained at the time and altitude that closely
correspond to the observations. If the autoconversion scheme that strictly preserves
raindrop concentration is used, the largest simulated diameters are smaller and tend to be
more centered about the updraft since the early stages Žnot shown.. Giant drops were
documented at an altitude of 400 m when the maximum value of Z occurred Žone of 8.2
Fig. 7. Cross-sections of simulated predominant raindrop diameters for the altitudes of 1700, 1300, 900 and
400 m, and the corresponding times 16:23, 16:28, 16:30 and 16:33 are denoted by open squares, open circles,
closed squares and closed circles, respectively.
G.G. Carrio,
´ M. Nicolinir Atmospheric Research 52 (1999) 1–16
15
mm in diameter., although drops of approximately 4 mm prevailed. Szumowski et al.
Ž1997, 1998b. related the presence of these large drops, that grow by accretion as they
fall through the decaying updraft with occurrence of extremely high radar reflectivities
in summertime Hawaiian rainbands.
4. Conclusions
Even when the accuracy of a parameterized scheme could not be compared with
those that explicitly solve the stochastic collection equation, for this case, it describes in
a satisfactory manner the time required for the high reflectivity core to develop and
some general characteristics of the cell evolution.
Results suggest that the typical overestimation of autoconversion and accretion in
one-moment parameterized schemes is reduced by the presented two-moment one,
showing a much better behavior. The evaluation of time rate of change of Nr due to
autoconversion in terms of the corresponding rates for the first and second moments of
the raindrop volume distribution, enhances the performance in simulating radar reflectivities.
As simplicity is highly desirable when computer resources are rather limited, this
warm rain scheme will be included, as well as the multimoment ice phase package under
development, in the 2D non-hydrostatic convective cloud model of Nicolini and Paegle
Ž1989.. The above-mentioned model operates at the present with a one moment
complete microphysics scheme.
Acknowledgements
The authors wish to thank the Consejo Nacional de Investigaciones Cientıficas
y
´
ŽPIP 4526796., the Agencia de Promocion
ŽPICT
Tecnologicas
y Tecnologica
´
´ Cientıfica
´
´
97 N 01757. and the University of Buenos Aires ŽTX30., for financial support.
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´ G.G., Levi, L., 1995. On the parameterization of autoconversion. Effect of small-scale turbulent
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Feingold, G., Levin, Z., Tzivion, S., 1991. The evolution of raindrop spectra: Part III. Downdraft generation in
an axisymmetrical rainshaft model. J. Atmos. Sci. 48, 315–330.
Ferrier, B.S., 1994. A double moment multiple-phase four-class bulk ice scheme: Part 1. Description. J.
Atmos. Sci. 51, 249–280.
Hall, W.D., 1980. A detailed microphysical model within a two-dimensional dynamic framework: model
description and preliminary results. J. Atmos. Sci. 37, 2486–2507.
Kessler, E., 1969. On the distribution and continuity of the water substance in atmospheric circulations.
Meteor. Monogr., No. 32, Am. Meteor. Soc., 84 pp.
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Model description and sensitivity experiments. J. Atmos. Sci. 48, 1160–1189.
Lin, Y., Farley, R., Orville, H., 1986. Bulk parameterization of the snow field in a cloud model. J. Atmos. Sci.
22, 1065–1092.
Long, A.B., 1974. Solutions to the droplet collection equation for polynomial kernels. J. Atmos. Sci. 31,
1040–1052.
Meyers, M.P., Walko, R.L., Harrington, J.Y., Cotton, W.R., 1997. New RAMS cloud microphysics parameterization: Part II. The two-moment scheme. Atmos. Res. 45, 3–39.
Nicolini, M., Paegle, J., 1989. Real data deterministic forecasts of ambient motion fields upon convective
precipitation. Second Int. Cloud Modeling WorkshoprConf. Toulouse, France, WMOrTD-No. 268, pp.
207–220.
Soong, S., Ogura, Y., 1973. A comparison between axisymmetric and slab-symmetric cumulus cloud models.
J. Atmos. Sci. 30, 879–893.
Srivastava, R.C., Coen, J.L., 1992. New explicit equations for accurate calculation of the growth and
evaporation of hydrometeors diffusion of water vapor. J. Atmos. Sci. 49, 1643–1651.
Szumowski, M.J., Rauber, R.M., Ochs, H.T., Beard, K.V., 1997. The microphysical structure and evolution of
Hawaiian Rainband Clouds: Part I. Radar observations of rainbands containing high reflectivity cores. J.
Atmos. Sci. 54, 369–385.
Szumowski, M., Grabowki, W.W., Ochs, H.T., 1998a. A simple two-dimensional kinematic framework
designed to test warm rain microphysical models. Atmos. Res. 45, 299–326.
Szumowski, M.J., Rauber, R.M., Ochs, H.T., Beard, K.V., 1998b. The microphysical structure and evolution
of Hawaiian Rainband Clouds: Part II. Microphysical measurements within rainbands containing high
reflectivity cores. J. Atmos. Sci. 55, 208–226.
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Am. Meteor. Soc., pp. 389–391.
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Model development and preliminary testing. J. Atmos. Sci. 42, 1487–1509.
www.elsevier.comrlocateratmos
A double moment warm rain scheme: description
and test within a kinematic framework
Gustavo G. Carrio´ ) , Matilde Nicolini
Departamento de Ciencias de la Atmosfera,
Centro de InÕestigaciones del Mar y la Atmosfera,
Consejo
´
´
Nacional de InÕestigaciones Cientıficas
y Tecnologicas,
UniÕersidad de Buenos Aires, Buenos Aires,
´
´
Argentina
Received 14 September 1998; received in revised form 4 February 1999; accepted 10 June 1999
Abstract
A two-moment warm rain parameterization that includes prognostic equations for mixing ratios
and number concentrations is described. The number of activated condensation nuclei is also
prognosed assuming a log–linear relationship between the total number of activated condensation
nuclei and the supersaturation. Collective processes are evaluated using pre-calculated matrices of
interaction based on numerical integrations of the corresponding stochastic collection equations.
Mass transfer between cloud and raindrops due to autoconversion is evaluated using Berry’s
parameterization, its attendant raindrop number concentration change is calculated considering the
corresponding rates of change for the first and second moments of the raindrop volume
distribution. A Doppler kinematic reconstruction of a summertime Hawaiian rainband occurred on
August 10, 1990 is used to evaluate the credibility of this scheme. q 1999 Elsevier Science B.V.
All rights reserved.
Keywords: Warm rain; Autoconversion; Microphysical schemes
1. Introduction
Many cloud modelers have used the so-called Kessler Ž1969. parameterization to
evaluate the transformation between categories in which water substance is classified. Its
benefits mainly lie in its inexpensiveness in term of computer resources as it takes as
prognostic variables only the mixing ratios of the different species. On the other hand,
)
Corresponding author
0169-8095r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 9 - 8 0 9 5 Ž 9 9 . 0 0 0 3 1 - 9
2
G.G. Carrio,
´ M. Nicolinir Atmospheric Research 52 (1999) 1–16
the state of art of modeling of explicit spectral microphysics was developed during the
eighties considering detailed spectral treatments of processes such as collision–coalescence, collisional breakup and nucleation ŽKogan, 1991; Feingold et al., 1991, among
others.. Explicit treatments ensure an adequate description of the various processes,
however they require an enormous computer capacity when they are considered within a
multidimensional dynamic framework. There is a trend in recent years to include
prognostic equations for an additional moment of the species distributions, namely
number concentrations ŽFerrier, 1994; Walko et al., 1995; Meyers et al., 1997..
Multimoment schemes appear to be a compromise solution between the detail of the
description and computational expedience. A mixed phase microphysical package that
includes prognostic equations for mixing ratios and number concentrations of various
water species is being developed by the authors, its warm rain scheme is here described
and tested with observational data.
During the Hawaiian Rainband Project ŽHaRP. several aspects of a series of shallow
tropical rainbands offshore of the island of Hawaii have been studied. Dual-Doppler
radar measurements from this project have been used to document the formation,
evolution and kinematic structure of high reflectivity cores ŽSzumowski et al., 1997.. A
two-dimensional dual-Doppler kinematic reconstruction of a rainband, designed to test
warm rain microphysical packages ŽSzumowski et al., 1998a., is used here to evaluate
the credibility of the above mentioned two moment warm rain scheme. This reconstruction represents one of the best cases documenting kinematics and the time evolution of
high reflectivity cores in warm rain clouds. Within a kinematic framework, the absence
of microphysical–dynamical interactions, such as loading effects, certainly introduces
limitations, although, it allows isolating the response of different microphysical parameterizations and schemes. Some aspects of the simulated microphysical structure are
compared with those of a rainband occurred on July 23, 1985, documented and analyzed
by Szumowski et al. Ž1998b. using aircraft data. A general description of the scheme is
given in Section 2. The comparison between simulated and observed characteristics and
final comments are presented in Sections 3 and 4, respectively.
2. Scheme description
Cloud droplet and raindrop volume spectra are assumed to follow gamma distributions of the type used by Berry and Reinhardt Ž1974.. The volume corresponding to a
drop of 40 mm in radius is taken as the boundary between both spectra as it corresponds
to the nearly stationary position of a minimum when solving the stochastic collection
equation.
N Ž Õ . s Nt Ž n q 1 .
n q1
n
Ž ÕrÕf . r Õf G Ž n q 1 . exp y Ž n q 1 . ÕrÕf
Ž 1.
where N Ž Õ . is the number density of drops with volume Õ, Nt is the total number
concentration, Õf is the mean volume, n is a width parameter Ž n ) y1. and G Ž x . is the
gamma function.
G.G. Carrio,
´ M. Nicolinir Atmospheric Research 52 (1999) 1–16
3
The relation between the volume about which most of the mass is concentrated
Žpredominant volume. and the mean volume and the width parameter is given in Eq. Ž2..
Any moment of the gamma volume distribution can be evaluated in terms of the
spectrum parameters with Eq. Ž3..
Ž n q 1.
y1
s w Õf x
y1
Õg y 1 s M 1rM 0
y1
M 2rM 1 y 1
Ž 2.
where Õg is the predominant volume and M b denotes the b th moment of the volume
distribution.
Mbs
`
H0 N Ž Õ . Õ
b
b
dÕ s N G Ž n q b q 1 . r Ž n q 1 . G Ž n q 1 .
Ž 3.
The mixing ratios of cloud droplets and raindrops Ž Qc and Qr . and the corresponding
number concentrations Ž Nc and Nr . along with the concentration of activated nuclei, are
taken as prognostic variables, however the width parameters Ž nc and nr . are assumed to
remain constant during the simulation. The chosen values are consistent with observations in Hawaiian clouds and they are 0 for nc and y0.8 for nr .
The following Eqs. Ž4. – Ž8. represent the tendency equations for Qc , Qr , Nc , Nr and
the number concentration of activated nuclei Ž Nn ..
d Ncrd t s d Ncrd t < NU C q d Ncrdt < AUT q d Ncrd t < ACR
q d Ncrdt < AD V q d Ncrdt < EVA
Ž 4.
dQ crd t s dQ crdt < CO N q dQ crdt < AUT q dQ crdt < ACR
q dQ crdt < AD V q dQcrd t < EVA
d Nrrdt s d Nrrd t < AU T q d Nrrd t < SFC q d Nrrd t < SED q d Nrrdt < ADV
Ž 5.
Ž 6.
dQ rrdt s dQ rrdt < AU T q dQ rrd t < ACR q dQrrdt < SED
q dQ rrdt < AD V q dQrrdt < EVA
d Nnrdt s d Nnrdt < NU C q d Nnrd t < ADV q d Nnrdt < EVA
Ž 7.
Ž 8.
where the subscript, NUC, CON, AUT, ACR, SFC, ADV, SED and EVA stand for
nucleation, condensation, autoconversion, accretion, raindrop self-collection, advection,
sedimentation and evaporation, respectively.
2.1. Collisional processes
2.1.1. AutoconÕersion
Following Berry and Reinhardt Ž1974., an average autoconversion rate can be
evaluated relating a characteristic liquid water content change Ž L 2 . and a corresponding
time scale Žt 2 .. The resulting tendency equation for Qc due to autoconversion is given
in Eq. Ž9. as a function of the cloud drop spectrum parameters Žcgs units are used
hereafter..
dQ crd t < AU T s ydQrrdt < AUT s yr L2rt 2
Ž 9.
G.G. Carrio,
´ M. Nicolinir Atmospheric Research 52 (1999) 1–16
4
where
L 2 s 0.027 Ž rfc4 10 12 Ž nc q 1 .
r r Qc Ž rfc Ž nc q 1 .
y1 r2
y1 r6
y 0.4 . r Qc , t 2s3.7 = 10y7
y 0.00075 . ,
rfc is the mean volume droplet radius and r is the air density.
Under the constraint of a constant value for the raindrop width parameter Ž nr ., the
corresponding rates of change for the first and second moments and the number
concentration Ž M 0 . are not independent. Therefore, the time rate of change for Nr due to
autoconversion is here evaluated in terms of the corresponding rates of change of M 1
and M 2 and assuming that the newly created raindrops have a mean volume coincident
with the chosen boundary between both spectra.
The raindrop mean volume Ž Õfr . and the more physically meaningful predominant
volume Ž Õgr . are linearly related for a given nr Žsee Eq. Ž2... If rate of change for Nr due
to autoconversion is evaluated relating Eq. Ž9. to the volume of a drop of 40 mm, Õgr
and the radar reflectivity are then artificially decreased by the inclusion of these small
drops.
The raindrop predominant volume after a time interval D t can be related to
dQ rrdt < AU T using the following equation:
U U
2
Qr
M 2U s Mr2 q D MAUT
s r Ž a r Õfr . Q r q r Ž aAUT Õ40 . dQrrdt < AU T D t s r Õgr
Ž 10 .
where a r and aAUT denote the ratios between predominant and mean volumes for the
raindrop spectrum and the distribution of the new drops generated by autoconversion,
respectively. The superscript U indicates values after a time interval D t and Õ40 is the
volume of a drop of 40 mm in radius.
The rate of change for Nr due to autoconversion is obtained from Eq. Ž11., assuming
that the spectrum of the created raindrops has a unity mass relative variance Ž aAU T s 2,
nAU T s 0.. According to numerical experiments of Berry and Reinhardt Ž1974., the
second mode of the spectrum, associated to raindrops, has this value of mass relative
variance by the time its mean volume radius reaches 40 mm.
U
Õgr
s ra r Qr q dQrrdt < AU T D t r Nr q d Nrrdt < AUT D t
Ž 11 .
The formula used to evaluate mass conversion between cloud and rain spectra due to
autoconversion ŽEq. Ž8.. as well as newer parameterizations Že.g., Beheng, 1994.
perform selectively depending on the range of droplet sizes. According to the nature of
the simulation, a different formula for mass conversion could be used without affecting
this procedure to evaluate the attending change in the raindrop number concentration.
The rate of change for Nc corresponding to the combined effects of autoconversion
and cloud droplets self-collection is evaluated using the Long Ž1974. polynomial
approximation for the collection kernel for collector drop radius lower than 50 mm.
After integration, it can be expressed as
d Ncrd t < AU T s yk r 2 Q c2 Ž nc q 1 . r Ž nc q 2 .
9
y3
where k s 9.44 = 10 cm
s
y1
.
Ž 12 .
G.G. Carrio,
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5
2.1.2. Accretion
The collision–coalescence of cloud and rain drops modifies the concentration of
cloud drops and the mixing ratios of both categories. The tendency equations for Nc and
Qc are
`
`
H0 H0 n Ž Õ . p E Ž Õ ,Õ .
d Ncrd t < AC R s y
c
c
c
r
r Ž Õc . q r Ž Õr .
2
Vt Ž Õr .
yVt Ž Õc . n r Ž Õr . dÕr dÕc
`
Ž 13 .
`
H0 H0 n Ž Õ . m Ž Õ . p E Ž Õ ,Õ .
dQcrd t < AC R s y
c
c
c
c
r
r Ž Õc . q r Ž Õr .
2
Vt Ž Õr .
yVt Ž Õc . n r Ž Õr . dÕr dÕc
Ž 14 .
where n c Ž Õc ., n r Ž Õr . are the number concentrations of droplets and drops; r, m, Vt and
E Ž Õc ,Õr . represent the radii, masses, terminal velocities and the collision efficiency
function, respectively.
Numerical solutions of Eqs. Ž13. and Ž14. were pre-computed and tabulated in
look-up tables for later interpolation for many gamma spectra characterized by different
cloud and rain drop mean volumes Ža similar procedure has been applied by Meyers et
al., 1997, although not predicting Nc .. The collision efficiency dependence on the
colliding drops sizes has been retained for evaluating the interaction matrices. They were
computed for 33 and 40 size categories for cloud and raindrop mean volumes, respectively. Expressions for terminal velocities, integration method and the set of gravitational collision efficiencies were those used by Carrio´ and Levi Ž1995..
In order to improve interpolation, the rates of change of Nc were divided by the
square of the mean volume radii and their corresponding terminal velocities. For the
mixing ratio case they were additionally divided by the droplet mean mass. The
tendency equation for Nc and Qc can then be expressed in terms of the resulting
matrices Ž AŽ Õc , Õr . and B Ž Õc , Õr .. as
d Ncrd t < AC R s Nc Nr rfr2 Vt Ž rfr . A Ž Õfc ,Õfr . Ž r 0rr .
dQ crd t < AC R s ydQrrdt < ACR s Nc Nr rfr2
0.5
Ž 15 .
Vt Ž rfr . m Ž rfc . B Ž Õfc ,Õfr . Ž r 0rr .
0.5
Ž 16 .
where r 0 is the reference air density.
The values of A and B, that are proportional to the ratios between these rates and the
corresponding continuous approximations, are obtained by a bi-linear interpolation on
the logarithms of Õc and Õr . The last factors in Eqs. Ž15. and Ž16. are introduced to take
into account the terminal velocity dependence on air density.
2.1.3. Rain self-collection
This mechanism, although it does not modify the rain mixing ratio, it is responsible
of the fast mean volume increase during the last stages of warm coalescence; this
reduction rate for Nr is
`
`
H H n Ž Õ . p E Ž Õ ,Õ .
d Nrrdt < SFC s y1r2
r
x
x
y
r Ž Õx . q r Ž Õ y .
2
6
G.G. Carrio,
´ M. Nicolinir Atmospheric Research 52 (1999) 1–16
G.G. Carrio,
´ M. Nicolinir Atmospheric Research 52 (1999) 1–16
7
Fig. 1. Idealized flow field at three key times in the simulation. When radar reflectivity reaches y20 dBZ Ža.,
and when the maximum updraft speed and the maximum radar reflectivity are attained Žb and c, respectively..
Cloud boundaries have been superimposed Ž Qc s10y6 ggy1 ..
where < DVt Ž Õ x ,Õ y .< denotes the absolute value of terminal velocity difference between
raindrops with volumes Õ x and Õ y .
Self-collection is treated in a way similar to that applied for accretion. The reduction
rate for the raindrop number concentration is evaluated as a function of the mean
volume drop radius, pre-computing and tabulating numerical solutions of Eq. Ž17.. The
tendency equation for Nr is
0.5
d Nrrdt < SFC s yNr2 C Ž Õfr . Ž r 0rr .
Ž 18 .
where the value of C is determined by linear interpolation on the logarithms of Õr .
The collisional break-up is taken into account by introducing a factor in Eq. Ž18.
related to the coalescence efficiency of the most frequently colliding drop pairs. This
factor is a function of the mean volume raindrop radius and its expression is identical to
that used by Ziegler Ž1985..
2.2. Vapor diffusion
2.2.1. Condensation of water droplet
The tendency equation for Q c due to condensation is obtained integrating the mass
rate of change of a single drop over the cloud spectrum. The latter diffusional growth
rate equation can be expressed as
d mrdt s y4p rSFT FV
Ž 19 .
G.G. Carrio,
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8
where
FT s L vrKT Ž L v TrR v y 1 . q 1r Ž Dv rs Ž T . 4
y1
,
FV is the ventilation factor, S is the supersaturation, r the drop radius, K the coefficient
of thermal conductivity of air, L v the latent heat of evaporation, R v the gas constant for
water vapor, Dv the diffusivity of water vapor in air, rs the density of water vapor at
saturation, and T the temperature.
For cloud droplets the ventilation effect can be neglected Ž FV s 1., therefore the
integral value of Eq. Ž19. over the cloud spectrum can be easily obtained using Eq. Ž3..
Although, as an explicit condensation rate equation requires very small time steps, the
semi-implicit procedure of Soong and Ogura Ž1973. is used to approximate the
supersaturation in the resulting expression, taking into account the decrease in the water
vapor mixing ratio Žand the attendant increase in saturation mixing ratio..
2.2.2. Nucleation
The cloud droplet number concentration is modified when condensation rate is
insufficient to maintain exact saturation or when new cloud is forming. A log–linear
relation between the number of activated cloud condensation nuclei Ž NnU . and the locally
steady supersaturation is assumed. The change in Nc due to nucleation is determined
evaluating NnU and comparing it with the cumulative number of activated cloud
condensation nuclei previously processed by the parcel Ž Nn ., that is explicitly accounted.
d Ncrd t < NU C s d Nnrd t < NUC s max 0, Ž NnU y Nn . rDt 4
Ž 20 .
The relation between NnU and S, suggested by the observers for this case, is given in
Eq. Ž21..
4.78 = 10 5 S 4
NnU s 120 S 0.4
100
S F 0.001%
0.001% - S F 0.063%
S ) 0.063%
Ž 21 .
2.2.3. EÕaporation
In a subsaturated ambient, the cloud droplets are assumed to evaporate instantly. The
decrease in the cloud number concentration due to evaporation, corresponding to the
mean droplet volume, is considered as a decrease in the number of activated nuclei Žsee
Eq. Ž8...
In the case of raindrops, the integral expression used to evaluate evaporation is
obtained using Eqs. Ž3. and Ž19., although the enhancement of vapor diffusion associated to ventilation cannot be neglected Žexpressions for FV were those used by Hall,
1980..
The traditional explicit equation for drop growth rate approximates the saturation
vapor density Ž rs . with a linear function of the temperature difference between the drop
and ambient air, when solving simultaneously the diffusion equations for water vapor
and heat. Srivastava and Coen Ž1992. showed that in some cases large errors result of
G.G. Carrio,
´ M. Nicolinir Atmospheric Research 52 (1999) 1–16
9
Fig. 2. Time vertical section of simulated radar reflectivities. Abscissa indicates minutes after time 17:00 UTC.
this approximation. They proposed a more accurate explicit equation based on a second
order approach that can be expressed as
w d mrdt x 2 s w d mrdt x 1 w 1 y a S x
Ž 22 .
where a s 1r2w grŽ1 q g .x rsYrrsX rsrrsX , the primes denote differentiation with respect
to T, and g s L v DrK rsX .
This expression that relates the first and second order approaches is then used as a
correction factor in the rain evaporation integral expression. However, it introduces a
negligible change for this case in which little evaporation occurs below cloud base.
2.3. Sedimentation
Rain number concentration and mixing ratio are independently sedimentated evaluating the number and mass fluxes with the mean and mean mass terminal velocities,
respectively. These mean terminal speeds are numerically evaluated using the Uplinger
Ž1981. expression that gives Vt as a function of the drop diameter Ž D . is
Vt Ž D . s 4854D exp w y1.95D x Ž r 0rr .
0.5
.
Ž 23 .
10
G.G. Carrio,
´ M. Nicolinir Atmospheric Research 52 (1999) 1–16
3. Results
A comparison between observed and simulated characteristics of a convective cell is
made using a kinematic reconstruction based on dual-Doppler data that corresponds to a
summertime Hawaiian rainband occurred on August 10, 1990. This idealized flow field
simulates the evolution of the updraft in magnitude and tilt. In Fig. 1a, b and c, the
velocity field is shown for three key times, when the radar echo first appears, when the
maximum updraft speed is attained and when the maximum radar reflectivity occurs.
Extremely high radar reflectivities Ž59.7 dBZ. have been observed within this cloud with
a depth lower than 3000 m. The times required for the high reflectivity core to develop
from an echo free region Žy20 dBZ. up to 50 dBZ and its maximum value were 15 and
20 min, respectively. This echo free region was located in the middle of the cloud at an
altitude of approximately 1700 m and the velocity field corresponds to that of Fig. 1a.
A time-vertical section of the simulated radar reflectivity is shown in Fig. 2. The
times taken for Z to reach 50 dBZ and its maximum from the value of y20 dBZ, are
very similar to those observed. The maximum of Z remains located at about 1700 m of
altitude as the largest drops are suspended, until a strong vertical gradient associated
Fig. 3. Comparison
between 17:21 and
squares and closed
preserving M 0 and
between observed and simulated intensities of the maximum Z for the August 10 cell
17:46 UTC. Closed squares correspond to the observed evolution. Open circles, open
circles correspond to simulated evolutions using a one-moment, two-moment schemes
M 2 , respectively Žsee text..
G.G. Carrio,
´ M. Nicolinir Atmospheric Research 52 (1999) 1–16
11
with the regions of most intense drop growth develops near the top of the cloud. In Fig.
3, the observed time evolution of the maximum Z during the stage of intense
Fig. 4. Simulated Z ŽdBZ. fields, Ža. for the time at which maximum updraft is attained and Žb. 5 min after.
12
G.G. Carrio,
´ M. Nicolinir Atmospheric Research 52 (1999) 1–16
development is compared to those simulated using different schemes. Simulated curves
are superimposed making approximate updraft maximum times coincident. It can be
seen that the two-moment scheme performs better in simulating the evolution when
d Nrrdt < AU T is evaluated preserving the second moment of the raindrop volume distribution rather than strictly preserving Nr . If a one-moment parameterization is applied Žthat
used by Lin et al., 1986 and by Nicolini and Paegle, 1989., the radar reflectivity
evolution is much faster and a significantly higher value Ž63.4 dBZ. is obtained for the
maximum of Z, probably related to the overestimation of accretion rates and the
concentration of large drops in Marshall–Palmer distribution.
From the beginning to the end of the simulation, the resulting Z fields exhibit a
nearly symmetrical pattern and the maximum values of Z are located within the updraft
core. The simulated Z fields begin to show a sloped pattern associated with the strong
divergence near the inversion level by the time vertical velocity is maximum Žsee Fig.
1b.. This pattern forms because the smaller the raindrop is, the farther it is carried
outward from the updraft core and the slower it falls. The sloped base of this overhang
first deepens downward with time and finally vanishes as the raindrops within it fell
toward the surface, leading to an overall increase in the width of the cell. The simulated
Z fields corresponding approximately to the time maximum updraft is attained and 5
min after are shown in Fig. 4a and b, respectively. Fig. 5 is analogous to Fig. 4 but
corresponds to observed reflectivity fields. The overhang surrounding the cell is visible
in Fig. 4a and Fig. 5a during this stage, the largest drops, associated to the overhang, are
located on the sides of the updraft core.
The general behavior of this rainband was similar to those of several cases documented during HaRP, however the observers suggested a parallel between this cell and a
Fig. 5. Idem Fig. 3, although for observed fields. Adapted from Szumowski et al. Ž1997..
G.G. Carrio,
´ M. Nicolinir Atmospheric Research 52 (1999) 1–16
13
rainband occurred on July 23, 1985. Its time evolution, spatial dimensions and calculated
reflectivities within the rainshaft corresponded closely to the features observed with
radar in the August 10 high reflectivity core ŽSzumowski et al., 1998b., and their
soundings and heights of inversion base were very similar. They performed several
penetrations of this latter cell, each pass was targeted through the region of highest
reflectivity using an on-board radar documenting the evolution of their high reflectivity
core, drop spectra, as well as liquid water content and vertical velocity. The time
evolutions of the calculated maximum Z and its observed altitude for this cell are shown
in Fig. 6 and compared to simulated values. The July 23 sounding was used in this case
and the simulated curves are superimposed making updraft maximum times coincident.
There is a fairly good correspondence between the evolutions and positions of the
observed and simulated cores. As a result, a comparison between microphysical information documented in penetrations and simulated magnitudes for the corresponding times
and altitudes could be made. During the penetrations at which the reflectivity core
remained at an altitude of 1700 m, most of the raindrops were smaller than 1 mm in
diameter, the lowest concentrations were coincident with the updraft core and the largest
Fig. 6. Comparison between the simulated and observed intensities and the altitude of the maximum Z for the
July 23 cell between 16:22 and 16:37 UTC. Intensities are denoted by circles and altitudes by squares. Closed
and open symbols for observed and simulated values, respectively. Abscissa indicates minutes after time 17:00
UTC.
14
G.G. Carrio,
´ M. Nicolinir Atmospheric Research 52 (1999) 1–16
drops were located on the sides. Those two maxima, associated to the overhang, were
observed until 16:28. After this time, the largest raindrops were confined to narrower
regions located at the center of the updraft, as the core descended Ž0.3 km for the time at
which maximum Z occurred..
In Fig. 7, cross-sections of simulated predominant raindrop diameters are shown for
altitudes of 1700, 1300, 900 and 400 m, and the corresponding times 16:23, 16:28,
16:30 and 17:33 Žpasses 2, 5, 6 and 7, see Szumowski et al., 1998b.. The first curve, that
is considered representative of the early stage in which drops are suspended at an
altitude of 1700 m, begins to show two maxima located on the sides of the updraft core
and those drops grow as the base of the overhang descends. The largest drops tend to be
located at the updraft center as those drops, previously suspended, fall through the
nearly vertical decaying updraft associated with the highest liquid water content. Largest
simulated predominant diameters are concentrated in a very narrow region Žless than 500
m. and their maximum value Žf 3.5 mm. is attained at the time and altitude that closely
correspond to the observations. If the autoconversion scheme that strictly preserves
raindrop concentration is used, the largest simulated diameters are smaller and tend to be
more centered about the updraft since the early stages Žnot shown.. Giant drops were
documented at an altitude of 400 m when the maximum value of Z occurred Žone of 8.2
Fig. 7. Cross-sections of simulated predominant raindrop diameters for the altitudes of 1700, 1300, 900 and
400 m, and the corresponding times 16:23, 16:28, 16:30 and 16:33 are denoted by open squares, open circles,
closed squares and closed circles, respectively.
G.G. Carrio,
´ M. Nicolinir Atmospheric Research 52 (1999) 1–16
15
mm in diameter., although drops of approximately 4 mm prevailed. Szumowski et al.
Ž1997, 1998b. related the presence of these large drops, that grow by accretion as they
fall through the decaying updraft with occurrence of extremely high radar reflectivities
in summertime Hawaiian rainbands.
4. Conclusions
Even when the accuracy of a parameterized scheme could not be compared with
those that explicitly solve the stochastic collection equation, for this case, it describes in
a satisfactory manner the time required for the high reflectivity core to develop and
some general characteristics of the cell evolution.
Results suggest that the typical overestimation of autoconversion and accretion in
one-moment parameterized schemes is reduced by the presented two-moment one,
showing a much better behavior. The evaluation of time rate of change of Nr due to
autoconversion in terms of the corresponding rates for the first and second moments of
the raindrop volume distribution, enhances the performance in simulating radar reflectivities.
As simplicity is highly desirable when computer resources are rather limited, this
warm rain scheme will be included, as well as the multimoment ice phase package under
development, in the 2D non-hydrostatic convective cloud model of Nicolini and Paegle
Ž1989.. The above-mentioned model operates at the present with a one moment
complete microphysics scheme.
Acknowledgements
The authors wish to thank the Consejo Nacional de Investigaciones Cientıficas
y
´
ŽPIP 4526796., the Agencia de Promocion
ŽPICT
Tecnologicas
y Tecnologica
´
´ Cientıfica
´
´
97 N 01757. and the University of Buenos Aires ŽTX30., for financial support.
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