2. Hydrometeors
2.1. Model particle spectra The treatment of hydrometeors and their size distributions depends on the available
output from cloud models. Most of the models calculate bulk microphysical properties such as liquid water contents in the prognostic equations. In that case, hydrometeors are
usually treated as homogeneous spheres with constant densities. For the GCE, that is 0.9
y3 y3
y3
Ž g cm
for icerhail, 0.4 g cm for graupel, and 0.1 g cm
for snow Tao and .
Simpson, 1993 . Size spectra of all precipitating particles are usually calculated from the cloud model
contents by exponential formulae: N
D s N
exp yL D 1
Ž .
Ž .
Ž .
x x
o Ž x .
x x
with diameters D and where intercepts N are set to 0.08 cm
y4
for rain and to 0.04
x o
Žx. y4
Ž . cm
for both snow and graupel in our examples. The index ‘ x’ refers to rain r , Ž .
Ž .
melting m , or snowrgraupelrhail s,g,h particles. The slope, L , is a function of
x
Ž
y3
. waterrice content, q , and particle density, r , both in g m
:
x x
1r4
pr N
Ž . x
o x
L s 2
Ž .
x
ž
6 q
x
Ž . Eq. 2 stems from the solution of the integral over size distribution for the calculation
of liquid water contents. The solution is found assuming diameter limits of zero and infinity. Since we aim at the investigation of the effects of inhomogeneous particles
only, the model inherent size distributions were not modified and used as defined above.
Ž . Ž .
Modifications to Eq. 1 may have different reasons. 1 The cloud models provide Ž
. more information on size distributions than the single-moment techniques as the above .
Ž .
For example, a double-moment representation Ferrier, 1994; Swann, 1998 calculates water contents and total number concentrations over time, or a triple-moment representa-
Ž . tion which may calculate the particle volume of frozen species. 2 The ‘a posteriori’
adjustment of particle spectra and size-dependent densities as a result of radiative Ž
transfer simulations compared to observations Panegrossi et al., 1998; Schols et al., .
1999 . The latter, however, may be in conflict with the cloud model parameterizations but seems justified by the lack of better cloud model simulations.
2.2. Profiles in the melting layer Ž
. In this study, the melting process is simulated using the model of Mitra et al. 1990 ,
which derives mass and volume fractions of meltwater in particles depending on their initial density, size, ambient temperature and moisture. Having available only liquidrice
Ž .
water contents in predefined layers above and below the freezing level FL , the melting process has to be inserted with certain limitations.
v
Particle aggregation and break-up are neglected.
v
The vertical resolution of mesoscale cloud model profiles requires a recalculation of the melting process on a finer layer structure followed by an integration of the radiative
transfer parameters back to the coarse layers. Here, the finer structure was taken to consist of layers with 20-m thickness, respectively.
v
In stratiform regions where our model is applied, updrafts are considered negligible and a constant mass flux of precipitation through the melting layer is assumed.
v
Constant mass flux profiles results in varying liquidrice water content and number concentration profiles due to the difference in terminal fall velocities among the species.
v
Fall velocity parameterizations of frozen particles above and melted particles below melting layer have to be checked for consistency so that the conversion from one state to
the other does not produce discontinuities in the transfer between size spectra.
v
Liquid water and ice from the layers above and below the FL determine the available meltwater.
The last three items are discussed in more detail because they determine the microphysical framework of our technique.
2.2.1. Mass flux Ž
y2 y1
. The assumption of constant mass flux in units g m
s requires:
r N D V D D
3
d D s const. 3
Ž . Ž
. Ž .
x x
x x
x
where r denotes particle density and V denotes fall speed. Following an individual
x
particle, the terminal fall velocity during melting is either calculated with respect to the rain spectrum below or the snowrgraupelrhail spectrum above the melting layer when
the drag force exerted on the particle is taken to be constant:
D
r
V D
s V D 4
Ž .
Ž .
Ž .
m m
r r
D
m
D
s ,g ,h
V D
s V D
5
Ž .
Ž .
Ž .
m m
s ,g ,h s ,g ,h
D
m
Melting particle density results from: r r
y r
r s 6
Ž .
m
f r q 1 y f r
Ž .
m y
m r
where r denotes frozen particle density and f
is the melted mass fraction of the
y m
particle. Obviously, the results depend on the employed fall speed parameterization. A review
Ž . of fall speed parameterizations for rain and snow particles suggests that Eq. 4 gives the
Ž .
best results together with the parameterization of Sekhon and Srivastava 1971 for rain: V D s 16.9D
0 .6
7
Ž .
Ž .
r r
r
Ž .
and the parameterization of Ferrier 1994 for snow, graupel, and hail:
0 .5
r
0 .42
V D s 1.296 D
Ž .
s s
s
ž
r
0 .5
r
0 .37
V D s 3.512 D
Ž .
g g
g
ž
r
0 .5
r
0 .6384
V D
s 10.943 D 8
Ž .
Ž .
h h
h
ž
r where r and r denote actual and surface air density, respectively.
Ž . Ž .
Fig. 1 shows an intercomparison of the formulae 4 and 5 for the above fallspeed Ž .
Ž . Ž .
parameterizations. Only Eq. 4
together with Eqs. 7
and 8
gives an almost Ž .
continuous transfer between spectra. The use of Eq. 5 always causes a large discrep- ancy between almost melted and liquid spectra. In the case of other fall speed
Ž parameterizations, discontinuities for both microphysical and optical properties e.g.
. radar reflectivities are even stronger. Profiles from Doppler radar retrievals of fall
speeds also indicate a steep increase of fallspeeds at FL right after the onset of melting with a rather smooth transition into the liquid rain layer below. In conclusion, the
Ž insertion of melting layers
under the above assumptions of flux constancy and .
negligence of particle aggregation and break-up into prescribed hydrometeor profiles requires a continuous evolution of particle spectra during melting to avoid discontinuous
profiles of microwave optical properties, e.g., radar reflectivity.
2.2.2. Water contents Since the standard cloud model output provides only liquid water and ice contents
above and below the FL, the water content available for melting has to be determined. Ž .
Using the constant mass flux assumption in Eq. 3 and the definition of total flux integrated over the spectrum:
`
pr
x 3
R s N D
V D D d D
9
Ž . Ž
. Ž .
H
x x
x x
x
6 the amount of rain water resulting from melting a certain amount of ice can be
Ž .
calculated. With the definition of water mass above the FL e.g. for snow :
`
pr
s 3
M s N D D d D
10
Ž .
Ž .
H
s s
s s
6 Ž
. Ž
. Ž .
and by inserting N D into Eq. 10 using Eq. 3 , assuming constant particle mass:
m
D r
1r3
s D r
1r3
s D r
1r3
11
Ž .
r r
m m
s s
it follows that:
1r3 1r3
r r
s s
M s M ,
M s M
12
Ž .
m s
r s
ž ž
r r
m r
or M s 0.464 M , M s 0.737M , M s 0.965M using the above particle densities.
r s
r g
r h
w x
These conversion ratios, f s 0.464, 0.737, 0.965 , have to be used in the following
s,g,h
Fig. 1. Terminal fallspeeds of precipitation during melting using combinations of different fallspeed parameter- izations.
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
Kategori