procedure. Note that particle density was taken to be independent of diameter as prescribed by the cloud model.
Ž . Let ‘i’ denote the layer index for which T z 273.2 K and let ‘i q 1’ be the layer
i
Ž .
index for which T z - 273.2. If rain production in the stratiform portion of the
iq 1
cloud is considered to be from melting only, then the produced meltwater is: M
s M z
y M z
13
Ž . Ž
. Ž .
m Žr .
r i
r iq1
The melted water from the frozen hydrometeors amounts to: M
s M
z y M z
f , j s s,g,h
14
Ž .
Ž . Ž .
Ý
m Žsqgqh .
j iq1
j i
j
j
w Ž
. Ž .x
If M s M
then Ý M z
y M z can be ingested as is by the melting
m Žr .
m Žsqgqh.
j iq1
j i
model. Otherwise, it is assumed that no more water can be produced by the melting of frozen particles than is available from already melted water, i.e. M
. It follows that:
m Žr .
M s min M , M
15
Ž .
m m
Žr . m
Žsqgqh .
Ž . correcting M z
afterwards to:
r i
M z
s M z
y M 16
Ž . Ž .
Ž .
r i
r i
m
This calculation ensures a conservative estimate of available meltwater, the conservation of water contents above and below FL as well as constant mass fluxes across the melting
layer.
3. Mie-tables
Usually, radiative transfer routines use tables of the hydrometeor optical properties relevant to the passive microwave radiative transfer to enhance computational efficiency.
These tables are computed after defining particle spectra and particle properties. Particle permittivity is a major uncertainty for melting particles. For problems of passive
Ž .
microwave radiometry, this has been recently investigated by Schols et al. 1999 , Fabry Ž
. Ž
. and Szyrmer 1999 or Meneghini and Liao 2000 . The main conclusion is that snow
particles have to be treated differently than more dense particles such as graupel and Ž
hail. According to results from various sensitivity studies Bauer et al., 1999, 2000; .
Ž .
Olson et al., 1999 , we employ a two-phase model for snow Fabry and Szyrmer, 1999 and the classical Maxwell–Garnett model with airrice inclusions in a host material
composed of water for graupel and hail. Details can be found in the referenced literature. Approximations to microwave radiative transfer usually employ bulk hydrometeor
Ž optical properties and do not require the calculation of an explicit phase function e.g.
. Kummerow, 1993 . The input parameters, i.e., extinction coefficient b , single scatter-
e
ing albedo v , asymmetry parameter g, and backscattering coefficient b integrated
o bs
over size spectra for each type, are given by:
`
p Q
cosu
s 2
b ,v , g ,b s Q ,
, ,Q
D N D D d D 17
Ž . Ž . Ž .
H
e o
bs e
s
4 Q
Q
e s
with backscattering, scattering and extinction cross-sections Q , Q and Q , average
bs s
e
Ž scattering angle cosu at frequency n and temperature T
which determine particle .
permittivity , and liquidrice water content of rain, snow, graupel, hail, cloud water and Ž
.
cloud ice, q s q which determine size spectra .
r,s,g,h,w,i
These parameters are integrated over hydrometeor types to obtain bulk cloud layer properties:
b q s b q ,
j s r,s,g,h,w,i
Ž . Ž .
Ý
e e
j
j
v b g
Ž .
Ý
o e
j
j
v q s
Ž .
o
b q
Ž .
Ý
e j
j
gv b q
Ž .
Ý
o e
j
j
g q s
Ž .
v b q
Ž .
Ý
o e
j
j
b b q
Ž .
Ý
bs e
j
j
b q s
18
Ž . Ž .
bs
b q
Ž .
Ý
e j
j
Our tables use indices for the independent variables which are: frequency, tempera- ture, hydrometeor type, and icerliquid water content, respectively. Thus, each parameter
Ž .
is a four-dimensional double-precision array of dimension 7, 6, 70, 500 as defined in Table 1.
The optical properties of melting particles are calculated only for snow and graupel. They are stored where T s 273 K, thus i s 70. For this, the melting layer is treated—as
described in the previous sections—using the frozen water content of each species as input. The coarse vertical resolution of the cloud models requires an integration through
the melting layer over small sublayers; thus, all melting stages:
z
FL
b ,v , g ,b z , q d z
Ž .
H
e o
bs j
z
FLy h
b ,v , g ,b q s
19
Ž . Ž .
e o
bs j
z
FL
d z
H
z
FLy h
where z denotes the altitude of the freezing level and h the depth of the melting layer.
FL
The increment is taken to be 20 m. At each level, z, the melting model calculates
Table 1 Specifications of optical parameters
Index Dimension
Definition Frequency
7 is1: 10.7, 2: 19.35, 3: 21.3, 4: 22.235, 5: 37.0, 6: 85.5, 7: 13.8 GHz
Type 6
is1: rain, 2: snow, 3: graupel, 4: hail, 5: cloud water, 6: cloud ice
y1 y1
Ž .
Ž .
Temperature 70
rain, cloud water: is T y233 K , others: is T y203 K
Water content 500
is q P100
r,s,g,h,w,i
meltwater fraction and corresponding mixed particle permittivity as well as all optical Ž
. properties according to Eq. 19 , which are then stored in the Mie-tables. The tables
contain the effective radar reflectivity, Z , instead of the backscattering coefficient
eff
calculated from: 10
12
l
4
n y 1 Z s
b , K s
20
Ž .
eff bs
r 5
n q 2 K p
r
where n denotes the complex refractive index and l the wavelength in centimeter. b
bs
is in units of cm
y1
. Ž
. The scaling in Eq. 19 allows application to any layer depth, given that melting is
completed within this layer. If the layer depth of the cloud model exceeds h, the optical properties can be calculated by adding the respective amount of liquid water left after
melting over the remaining layer depth. As an output of these integrations, the total melting layer depth is also stored in the tables as the depth below FL at which all
particles are completely melted.
A slight inaccuracy is introduced by the assumption that temperature is not constant throughout the melting process, as required by the melting model. The melting process
produces latent cooling and may lead to almost isothermic conditions within the melting layer; however, this was assumed to be of minor importance.
Fig. 2 shows an example of a cross-section through the stratiform part of a model cloud. The simulation was initialized over the Western Tropical Pacific on February 22,
Ž .
1993 and performed by the GCE model Tao and Simpson, 1993 on a 1-km resolution in the horizontal plane. The system represents a fast moving, bow shaped squall line
almost parallel to the y-axis, which has developed a large stratiform tail at the time of Ž
. Ž
y3
. Ž .
this example t s 210 min . The panels show the water contents in g m of rain a ,
Ž . Ž .
Ž . snow b , graupel c , and cloud liquid water d , indicating a well extended show and
graupel layer on top of a rain layer. An area with low water contents was chosen to illustrate the melting model sensitivity. Fig. 3 shows the results for b and Z
at 13.8
e eff
and 37.0 GHz for this cross-section after application of the described technique. A ‘bright band’ is visible for both parameters at z s 5.5 km, indicating increased
scattering and attenuation of radiation. In the melting layer, differences of 5–10 dBZ and doubling of b
with respect to the liquid precipitation below are found, which
e
correspond to observations that will be shown in the following section. Since bright bands occur locally and only rarely cover areas of the size of the TMI
footprints completely, the same GCE model simulation, but for a horizontal resolution of 3 km, was included for an assessment of primary resolution on net melting layer
signature. From the same initialization, another model experiment was used to estimate the influence of different model parameterizations on the melting layer effect. For this
purpose, four timesteps of the TOGA-COARE squall line simulation by the Meso-NH
Ž .
model Lafore et al., 1998 were treated in the same fashion as the GCE simulations and used for the intercomparison with TRMM measurements presented in the next section.
Major differences in the model physics of the two models are the boundary condition treatment and that Meso-NH can be run in an either two-dimensional or three-dimen-
sional model. Both used open lateral boundary conditions in the direction of propagation
Ž . Ž .
Fig. 2. GCE model cloud cross-section at time step t s 210 min. Water contents of rain a , snow b , graupel Ž .
Ž .
y3
c , and cloud water d in g m .
Ž .
Ž .
Ž Fig. 3. Extinction coefficient a, c and effective reflectivity b, d for cross-section from Fig. 2 at 37.0 GHz a,
. Ž
. b and 13.8 GHz c, d .
Table 2 Cloud model simulations
Ž .
Model Ex. reference
Cloud system Resolution
Time steps min Ž
. km
Goddard Cumulus Ensemble Tao and Simpson
squall line 1.0
120, 150, 180, 210 Ž
. Model, GCE-1
1993 TOGA-COARE
Goddard Cumulus Ensemble Tao and Simpson
squall line 3.0
180, 240, 300, 360 Ž
. Model, GCE-3
1993 TOGA-COARE
Ž .
Large Eddy Model CETPr Lafore et al. 1998
squall line 1.25
300, 360, 420, 480 Meteo-France, Meso-NH
TOGA-COARE
and include cloud microphysics for five different particle types. For further information Ž
. on mesoscale model intercomparison of this case, refer to Redelsperger et al. 2000 .
Table 2 gives a summary of the employed model simulations.
4. TRMM observations
