3960
MONTHLY WEATHER REVIEW
VOLUME 137
Cloud and Precipitation Parameterization in a Meso-Gamma-Scale Operational
Weather Prediction Model
LUC GERARD
Royal Meteorological Institute of Belgium, Brussels, Belgium
JEAN-MARCEL PIRIOU
CNRM-GAME, Météo-France, Toulouse, France
RADMILA BROŽKOVÁ AND JEAN-FRANÇOIS GELEYN*
Czech Hydro-Meteorological Institute, Praha, Czech Republic
DOINA BANCIU
National Meteorological Administration, Bucharest, Romania
(Manuscript received 8 August 2008, in final form 28 April 2009)
ABSTRACT
The paper assesses the difficulties of running an operational NWP model in the resolution range of 3–8 km.
In this case, deep convection cells are neither much smaller than the grid box as assumed by most parameterization schemes, nor completely resolved as would be required for them to be treated explicitly. A specific
approach is proposed, with an integrated sequential treatment of resolved condensation, deep convection,
and microphysics together with the use of prognostic variables. It currently allows for the production of
consistent and realistic results at resolutions ranging from a few tens of kilometers down to less than 4 km.
Model skill scores and an example of an operational forecast at different resolutions are presented.
1. Introduction
The geometrical characteristics of deep convective
systems make them difficult to represent in numerical
operational weather prediction or climate models. Convective cells stay subgrid with gridbox lengths larger
than 1 or 2 km in the horizontal, while they reorganize
significantly the atmospheric layers along the vertical. Deep precipitating convection is not compatible
with isotropy assumptions or down-gradient diffusion,
and cannot just rely on the ‘‘conservative’’ variables frequently used in nonprecipitating convection or
turbulence.
* Additional affiliation: Météo-France, Toulouse, France.
Corresponding author address: Luc Gerard, Royal Meteorological Institute of Belgium, Dept R&D, 3 Av. Circulaire, B 1180
Brussels, Belgium.
E-mail: [email protected]
DOI: 10.1175/2009MWR2750.1
Ó 2009 American Meteorological Society
As long as the motions in the convective cells are not
resolved by the model grid, a parameterization can be
used, relying on a simple model of the processes occurring at the subgrid scale to provide source terms in
the mean-flow equations. Most deep convection parameterizations have been developed under the hypotheses of large grid boxes and long time steps, and
while they provide source terms for the vertical transport of some of the model variables, they often treat
precipitation and cloud more directly, in a simple diagnostic way.
These simple schemes, when used at finer resolution
and shorter time steps, tend to produce an intermittent
on–off behavior of deep convection, and a too early onset
of rainfall in the convective diurnal cycle (Guichard et al.
2004); on the other hand, assuming explicit convection at
resolutions coarser than 1 or 2 km produces excessive
vertical transports, the up- and downdrafts being extended to the size of entire grid boxes (e.g., Deng and
Stauffer 2006). Running a model at resolutions between
NOVEMBER 2009
GERARD ET AL.
7 and 2 km (qualified as the ‘‘gray zone’’) remains then
particularly delicate.
The sources of most of these problems can be divided
in two categories: first, the missing or unsatisfying representation of some physical phenomena; second, using
the parameterization in conditions where its assumptions become invalid.
In the first category, the scheme needs to provide source
terms for all relevant model variables. In early models, the
source terms were limited to a mean gridbox heat source
and a moisture sink (Yanai et al. 1973). Later, momentum has also been treated (Moncrieff and Miller 1976;
Kershaw and Gregory 1997). If the model includes an
evolution equation for cloud condensates, corresponding
source terms should also be provided by the deep convection scheme. Molinari and Dudek (1992) qualify this
approach as ‘‘hybrid parameterization’’ (as opposed to
‘‘traditional’’ where no such terms are provided and to
‘‘fully explicit’’ with no parameterization) and point out
its impact on the acute representation of mesoscale convective systems. Frank and Cohen (1987) proposed such a
parameterization for mesoscale models at resolutions
coarser than 20 km and implying steady-state updrafts and
downdrafts. These source terms can be handled consistently in schemes where the convective updraft condensates are not precipitated directly, but detrained to be
handled by a microphysical package (Boville et al. 2006;
Wilson et al. 2008; Gerard 2007, hereafter G07).
The formulation of the source terms in the resolved flow
equations can affect the accuracy but also the structure of
the parameterization. The commonly used expression of a
quite arbitrarily parameterized detrainment and compensating subsidence can be advantageously replaced by
the approach of the Microphysics and Transport Convective Scheme (MTCS; Piriou et al. 2007) using the convective condensation and transport terms directly.
Another set of generally incomplete representations
concerns the interactions between updrafts and downdrafts and with other parameterizations, like stratiform
condensation, shallow convection, or the boundary
layer. The coexistence in a model of a subgrid deepconvection scheme producing its own precipitation, with
another scheme handling resolved/stratiform condensation and microphysics, can lead to a double representation of a same physical process, inducing an excessive
consumption of the available resource, hence a double
counting of the moisture that can condense and precipitate. G07 addressed this point by using a cascaded organization of the related parameterizations with a neat
separation of the different contributing processes. Interactions with other parameterizations are an ambitious
challenge toward the unification of physical parameterizations (Arakawa 2004).
3961
Conditions where parameterization assumptions can
become invalid concern time- and space-scale inconsistencies. A closure assumption is required to scale the
subgrid system in function of the resolved model variables.
The closure has been the object of extended discussions
about causality and the delineation of the ‘‘convective’’
character of processes. Arakawa and Schubert (1974), in
the frame of coarse-resolution models, separated a slow
‘‘larger-scale or dynamic forcing’’ from a faster ‘‘convective response.’’ The difference of time scales allowed us to
consider that convective processes were, at all time steps,
in quasi-equilibrium (QE) with the larger-scale resolved
flow. Very often, the closure of a mass-flux scheme uses
moisture convergence toward the grid column, or buoyancy considerations. In coarse-resolution models, a closure by resolved convergence of moisture did not seem to
be able to alone produce enough precipitation. Kuo (1974)
combined to it the surface evaporation; Krishnamurti et al.
(1983) considered an additional unmeasurable mesoscale
or subgrid-scale source of moisture supply. Bougeault
(1985) introduced into the moisture convergence term a
local contribution from the turbulent diffusion scheme. In
such cases, it becomes less sustainable to assert that this
forcing contribution would be much slower than the
convective process (Gerard and Geleyn 2005). Pan and
Randall (1998) also objected that cumulonimbus clouds
produce anvils interacting with radiation. The radiative
feedback can affect the temperature profiles and the
vertical exchanges over an area significantly wider than
the updraft, modifying the input conditions of the deep
convection scheme and the local CAPE, but also gradually the larger-scale flow contributing to the closure. The
time scale of these effects covers a continuous range
starting from the ones of the convective process to longer
values following the extension of the anvil clouds. The
separation of time scales becomes then quite artificial.
The problem of QE is to indicate a cause and an effect
(control and feedback) between the convective updraft
and the larger-scale circulation; Mapes (1997) suggested
to rather separate between dry dynamics versus moist
convective and precipitating processes. A possible cure
for the unsatisfying hypothesis of quasi-equilibrium is to
use a prognostic equation for closure, instead of a budget
equation. Pan and Randall (1998) introduced a prognostic
convective kinetic energy equation in the Arakawa–
Schubert scheme, resulting in a significantly lighter calculation than QE. Gerard and Geleyn (2005) completed
Bougeault (1985)’s mass-flux scheme with a prognostic
updraft vertical velocity and a prognostic updraft mesh
fraction, allowing to depart from the QE hypothesis.
Using prognostic variables (and subsequently providing
a memory of earlier activity) addresses both spaceand time-scale inconsistencies, allowing precipitation to
3962
MONTHLY WEATHER REVIEW
fall in another grid box and at another time step
than where and when the convective conditions initially
appeared.
Finally, assuming that the updraft environment is
identical to the mean grid box cannot hold below around
10-km resolution, when the updrafts can occupy a significant part of the grid box. Expressing explicitly an
updraft mesh fraction like in Gerard and Geleyn (2005)
allows us to compute average properties of the updraft
environment.
The current study tries and addresses a number of the
above-described problems, while presenting below a
scheme avoiding double counting, increasing the memory
of parameterized convective cells, and proposing a common microphysical treatment of subgrid- and resolvedcondensation condensates. Section 2 describes the features
of the model we have used, with a particular focus of the
ones dedicated to the gray zone problems. Section 3
demonstrates the behavior of the scheme at meso-gamma
scale, compared with a classical convective scheme and the
case with no deep-convection parameterization. Operational model scores are presented in section 4, while section 5 shows an example of operational forecast at three
resolutions. Conclusions are drawn in section 6.
2. Description of the model
Our framework ‘‘Alaro-0’’ is a version of the Action
de Recherche Petite Echelle Grande Echelle–Aire
Limitée Adaptation Dynamique Développement International (ARPEGE–ALADIN) operational limitedarea model (e.g., Horányi et al. 2006) with a revised and
modular structure of the physical parameterizations.
The latter can include memory from one time step to the
next, either through prognostic variables (using an evolution equation and advected by the resolved flow, e.g.,
the up/downdraft vertical velocities and mesh fractions,
turbulent kinetic energy) or ‘‘pseudohistoric’’ (not advected, e.g., the convective cloud fraction).
There are five prognostic model variables for water
species, represented by specific contents: water vapor (qy),
cloud ice (qi), cloud droplets (ql), rain (qr), and snow (qs).
The vertical diffusion of cloud condensate variables is
derived from the diffusion of ‘‘conservative variables’’
(total suspended water and liquid–ice static energy).
a. Main building blocks
1) RESOLVED CLOUD AND CONDENSATION
Called at the beginning of the time step, this scheme
estimates stratiform-like condensation/evaporation and
cloud fraction (i.e., supposing a nonskewed distribution
of total moisture over the grid box). Hanging cloud
condensates of convective origin (produced at earlier
VOLUME 137
time steps) are protected against reevaporation at this
stage [see section 2b(3)]. The condensates generated by
this scheme are kept hanging until they are combined
with those produced by the subsequent updraft; autoconversion and other microphysical processes are applied in one shot for the combined condensates.
Alternatively to a scheme similar to Smith (1990)’s
scheme (described in G07), a new scheme for stratiform
cloud fraction f st and resolved condensation–evaporation
has also been implemented. It uses an approximation of
the formula proposed by Xu and Randall (1996) for the
cloud fraction (noted here with superscript asterisk values
for the nonconvective part of the grid box):
rh
i
d
q*
y
f st 5 y 1 eaq*c /(qwq*)
qw
r
q*
aq*c
’ y
,
(1)
qw aq* 1 (q q*)d
c
w
y
with a 5 100, r 5 0.25, and d 5 0.5. Following a proposal
of E. Bazile (1997, personal communication) the condensates are estimated by assuming that in all clouds of
the grid box, the intensive condensate qc/f is the same
and the vapor content qy 5 qw (wet-bulb specific humidity), while in the clear part qy/qw 5 RHc, a prescribed critical relative humidity profile (depending on
the gridbox size). A balanced state in terms of f st, q*y, and
q*c is obtained by solving a Newton iterative loop.
The ice fraction in the cloud is taken as
(
)
[T t min(T t , T)]2
,
(2)
ai (T) 5 1 exp 2(T t T x )2
where Tt 5 08C and Tx is the temperature of the maximum difference between the saturation vapor pressures
with respect to ice and to liquid.
2) DEEP CONVECTIVE UPDRAFT
The mass-flux scheme replaces the horizontal gridbox
variety by a single equivalent updraft. Starting from wetbulb conditions, an entraining ascent is computed by
alternating isobaric mixing and moist adiabatic ascent
segments. The mixing uses a prescribed entrainment
profile depending on the vertically integrated buoyancy
(Gerard and Geleyn 2005). At all stages the conservation of moist static energy is ensured.
A layer is considered ‘‘active’’ if the elevated air is
saturated, stays warmer than its environment, is buoyant
and if there is positive resolved moisture convergence
(integrated from the surface); otherwise, a new ascent is
started after remixing the previous outcome with the
environment.
NOVEMBER 2009
3963
GERARD ET AL.
The total updraft condensation is later combined to
the resolved one before feeding the microphysical
scheme. The condensate hanging in the updraft qc_u is
estimated (similarly to G07) as
qc u 5 f0
›(qyu 1 qc u )
,
›f
(3)
where f0 is a critical (geopotential) thickness beyond
which the condensates are assumed to be detrained and/or
precipitate.
The use of prognostic variables allows the updraft to
have its own response time, and prevents an intermittent
on–off behavior. The updraft mass flux is given by
Mu 5 su
v*u
, with
g
v*u 5 vu ve ,
(4)
where both su, the updraft mesh fraction, and v*u, the
updraft vertical velocity relative to its environment, are
given by prognostic equations. Here v 5 dp/dt, and the
updraft-environment velocity ve 5 v su vu . Here, v*u
is obtained from a vertical motion equation in the form
(similar to the one of Gerard and Geleyn 2005):
›v*u
›v*2
5 B 1 Dv*u 2 A u ,
›t
›p
(5)
where B is the buoyancy force, D includes entrainment
and the drag coefficient, and the term with A accounts
for autoadvection. Consistency of the partial tendency
expression with the resolved dynamical calculation requires that B and A depend on su; here we use the value
of su advected from the previous time step.
A two-dimensional closure is used, currently based on
vertically integrated moisture convergence. It is expressed
by a vertical budget over active updraft layers:
ð
ð
›su
dq
dp
(hu he ) 5 L su (vu ve ) ca
g
›t
g
ð
dp
1 L CVGQ ,
g
flux profile is homothetic to vu. In this context, su is first
a scaling parameter for the updraft mass flux. Considering it as a horizontal extension of the updraft is a
convenient approximation.
The prognostic expressions for vu and su allow a
gradual setup of convective activity, from one time step
to the next. On the other hand, the updraft properties
(i.e., moist static energy, momentum, and water phases)
are rediagnosed at each time step, considering a parcel
starting from cloud base with current environment wetbulb properties (requiring that the environment does
not change very rapidly). The mesh fraction evolution
only concerns its horizontal extension and this is assumed to take the same value up to the detrainment
level. While the behavior of the scheme appears satisfying down to the resolution of 4 km, this last simplification was found unsatisfactory at 2 km, where it leads to
an excessive stabilizing convective transport contribution in layers the updraft would actually not be able to
reach, preventing the resolved scheme to take over. A
solution to this is under development.
Stability closures, implying usually a relaxation time
for CAPE, were earlier reported to behave better than
moisture convergence closures in a diagnostic scheme
(e.g., Grell 1993). Here the temporal aspect is introduced by the prognostic approaches and the combination of resolved-convergence closure with a prognostic
updraft vertical velocity profile including the effects of
turbulence on local buoyancy appears to perform very
well in most situations.
Following the MTCS approach [section 2b(1)], the
updraft scheme outputs condensation fluxes and separate transport fluxes for qy, qi, ql, s 5 cpT 1 gz, and the
horizontal velocities u and y. In addition it outputs a
detrainment area increment dsD, based on a local mass
budget and used to compute the convective cloud fraction [Eq. (12)].
3) MICROPHYSICAL PACKAGE AND CLOUD
GEOMETRY
(6)
where g is acceleration of gravity, h 5 cpT 1 f 1 Lqy
is the moist static energy, L is the local latent heat of
vaporization, and CVGQ is the resolved moisture convergence. The first term of the rhs (moisture consumption by the updraft) uses the condensate increment dqca
obtained in the ascent calculation. Equation (6) states
that the water vapor converging to the grid column
during the time step is either condensed into the active
updraft layers (the saturated buoyant layers of the updraft) or stored in an increase of the updraft mesh
fraction. It yields a scalar su, hence, the updraft mass-
The gross condensates from both sources—resolved
and updraft—are combined before being submitted to
further microphysical processes. The intensive concentration of condensates may be higher in clouds of convective origin. This is roughly accounted for by estimating
the intensive cloud condensate as
"
#
(1 aco )2
a2co f st 1 f cu
1 cu
,
q^c 5 qc
f
f
f st
(7)
where aco is the ratio of convective to total condensation and f cu, f st, and f are the convective, stratiform, and
3964
MONTHLY WEATHER REVIEW
total cloud fractions, respectively. The arbitrariness of
this formulation has actually a small impact, the actual
intensive value only playing some role when the condensates are not much bigger than the scaling thresholdtype parameters for autoconversion.
Computing the microphysics between updraft and
downdraft allows a space–time separation of cause and
effect (e.g., precipitation falling from stratiform anvils
can maintain a downdraft after the updraft has decayed).
The microphysical package (see the appendix) handles the five prognostic water phases plus a diagnostic
pseudograupel. It uses a statistical sedimentation with
probability distribution functions (Geleyn et al. 2008),
and handles autoconversion (including the Bergeron–
Findeisen effect), collection, and evaporation processes,
calculated one level at a time.
In Alaro-0’s target range of model resolutions, realism
commands to use partial cloudiness, especially in the
presence of deep convection. For this, some overlapping
strategy has to be chosen, and the microphysics of falling
precipitation needs to be computed by layer, from top to
bottom of the atmosphere. In each layer, one considers
(for internal purpose only) four areas: cloudy, clear,
seeded (i.e., receiving precipitation from above generated at current time step), or not seeded. Precipitation
can occur in all of them, because the clear nonseeded
area can still contain precipitation species from earlier
time steps. The fractions are estimated by assuming
maximum cloud overlap between connected layers, and
random overlap of not connected layers. Noting Pc and
Pe, respectively, the fraction of the cloud and of clear
environment seeded from above is
Pcl
5
Pel 5
min( f l , f l1 ) 1 [ f l min( f l , f l1 )]Pel1
fl
and
[max( f l , f l1 ) f l ] 1 [1 max( f l , f l1 )]Pel1
,
(1 f l )
(8)
where level l is below level l 2 1. In Eq. (8) it is assumed
that each layer rehomogenizes the precipitation at the
bottom of its cloudy area.
While this classification may appear quite approximate,
this has no heavy consequences because all areas are
treated with equal care in the microphysics computations.
At each level, a separate calculation of autoconversion–
collection–evaporation is done in the four fractions when
relevant. The goal here is to get the most correct possible
rates of evaporation of falling precipitation of convective origin, which is essential in the feedback with other
physical effects (e.g., within the PBL). Ignoring subgrid
geometry was found detrimental to the model behavior.
VOLUME 137
4) MOIST DOWNDRAFTS
The downdraft calculation (G07) follows the same
principles as the updraft, but in a slightly simplified
manner. The prescribed entrainment rate is taken constant over the vertical. The prognostic vertical equation
yields the absolute downdraft velocity vd; the prognostic
closure yielding the downdraft mesh fraction sd, is
written as
ð
›sd
›t
[(hd he ) 1 (kd ke )]
ð
5
Fb
dp
g
vd ve dp
1 MHS,
g
rg
(9)
where k is the kinetic energy, MHS is the microphysical
heat sink, is a tunable parameter, partitioning between
the fraction of MHS used in the closure and the remainder contributing to the input profile (the neat separation
prevents double counting of this sink). The first term of
the rhs represents the work of the buoyancy force Fb.
b. The Modular Multiscale Microphysics and
Transport (3MT) features
We gathered the features more specifically dedicated
to address the gray zone or multiscale challenges under
the name 3MT. The package was made modular to be
able to use alternative individual components, 3MT
providing the host structure. It can also be switched off
for comparisons of model behavior.
1) THE MTCS CONCEPT
Expressing the effect of convective updrafts on the
mean gridbox variables is commonly done, since Yanai
et al. (1973), through pseudosubsidence and detrainment, for instance, for the model variable c representing
either a water phase or the dry static energy:
›c
›t
cu
5 su (vu ve )
›c
1 Du (cu ce ) 1 evaporation,
›p
(10)
where subscript cu index stands for convective updraft and
Du is the updraft detrainment rate. The term ‘‘evaporation’’ is used because of the evaporation of cloud condensates or precipitation.
Following Piriou et al. (2007), the MTCS uses the expression directly (for updraft, and similarly for downdraft):
›c
›t
5
cu
›[su (vu ve )(cu c)]
›p
1 condensation 1 evaporation
(11)
NOVEMBER 2009
GERARD ET AL.
3965
FIG. 1. Package organization chart. The square brackets mark the successive updates of the
internal state: specific contents of water vapor (qy), cloud ice (qi), cloud droplets (ql), falling
snow (qs), falling rain (qr), and temperature (T).
(the terms condensation and evaporation having opposite signs) in which the separation of condensation and
transport directly appears. This separation is used both
as a way to introduce into the parameterization a more
explicit causal link between all involved processes and as
a vehicle for an easier representation of the memory of
convective cells: the stationary cloud mass budget assumption is not required in a MTCS approach, the latter
being therefore more consistent with the introduction of
memory through prognostic closures for cloud properties.
2) THE CASCADING APPROACH
Double counting of condensable water vapor is avoided by cascading the parameterizations feeding on it,
updating an internal state (temperature and the five
prognostic water species) after each of them (G07). On
Fig. 1, the updates of the internal state ()* are marked
between square brackets.
The mean gridbox input profile passed to the updraft
is a mean-balanced state obtained after vertical diffusion
and stratiform condensation. The local vertical turbulent
diffusion contributes to the profiles of vapor, condensates,
and temperature, affecting the buoyancy profile and subsequently, the updraft vertical velocity evolution. In case
of closure by moisture convergence, the inclusion of the
effects of local evaporation into buoyancy can be considered as an advantageous alternative to the more arbitrary
addition of a ‘‘subgrid’’ contribution (Krishnamurti et al.
1983), or of the vertical turbulent diffusion moisture flux
(Bougeault 1985) into the closure.
The explicit expression of updraft condensation in the
closure [Eq. (6)] allows its accurate accounting in the
evolution of the resolved flow, to prevent double
counting; this contributes to the consistent behavior of
the scheme at different resolutions.
The convective cloud fraction is computed as f cu 5
su 1 sD. While, as stated above, the value of su results
from various feedback processes and it cannot pretend
to reflect reality, its weight in f cu is generally small; in
nature, the true extent of a cumulonimbus cloud does
3966
MONTHLY WEATHER REVIEW
not say either a lot about the fraction of it actually active
in precipitation. The accumulated detrainment area sD
is given by a budget between the increments dsD of each
time step (directly related to a mass budget, hence to the
good evaluation of the updraft mass flux) and a dilution
effect (when the condensates have mixed far enough to
be considered as stratiform):
1
(dt/t D )
5 min(1 su1, s
1 dsD )
sD
De
(12)
where superscripts 2 and 1 stand, respectively, for t and
t 1 dt. The e-folding time of the detrained area is the
only additional tuning parameter of 3MT compared to
more classical schemes. Its value was tuned at 900 s by
comparing satellite images and direct model outputs for
the spring of 2008 over central Europe in an application
at 9-km mesh size. In the interaction between time steps
[section 2b(3)], the ratio of convective to stratiform
areas plays an important role.
3) HANDLING INTERACTIONS WITHIN AND
BETWEEN TIME STEPS
The large-scale cloud scheme applied at the beginning
of the time step assumes a nonskewed moisture distribution within a grid box; convective clouds produced at
earlier time steps cannot be handled correctly this way
(assuming horizontal homogeneity implies diluting
these, which can lead to significant reevaporation).
Wilson et al. (2008) addressed this by skewing the assumed subgrid variability distribution. In 3MT, we
protect a fraction f cu/f of the condensates (where the
total cloud fraction f is given by a random overlap relation f 5 f cu 1 f st 2 f cuf st). Here the gradual erosion of
the detrainment area [Eq. (12)] is a way to represent the
gradual transformation of convective condensate into
stratiform.
Stratiform condensation as well as turbulent vertical
diffusion modify the state input to the updraft scheme;
the microphysics and the downdraft at current time step
do not (we hope to later improve the life cycle on the
basis of an version in early test that uses density currents
associated with downdrafts to control the subsequent
intensity of updraft mixing).
To allow microphysical processes to feed back on the
updraft, a call to a simplified microphysics is included in
the updraft scheme. This adjusts the condensation amount
in response to the heat sink/source associated with convective precipitation melting/freezing. If not treated directly, the heat sink induces a cold pool (around 08C) in
the mean gridbox temperature profile, entraining additional resolved condensation at the next time step, but
also a significant reduction of the updraft buoyancy at
this level, with artificial detrainment of the convective
VOLUME 137
updraft and its restart just above. The problem is that the
heat sink and the subsequent condensation are then
distributed over the whole grid box while in reality the
sink localized near the updraft is rather likely to induce
increased updraft condensation, preventing a resolved
cooling. The simplified microphysics assumes a complete autoconversion of the convective increments of
cloud ice and droplets. Sedimentation is accompanied
by melting/freezing. The melting heat sink is then compensated by increased updraft condensation at the corresponding levels.
The downdraft, powered by the microphysical heat
sink, induces an additional evaporation, which has to be
reflected on precipitation contents. To avoid iterating
the full microphysics, the impact is applied locally (using
local mean fall velocity wP computed in the microphysics), with no modification of collection/evaporation
in downstream layers.
In summary, compared to other schemes, the important features of 3MT are the sequential organization, the
MTCS separation, the disappearance of the need to
parameterize convective detrainment rates, the use of
prognostic variables in convective up- and downdrafts,
the estimation, accumulation and decay of the detrainment area, the calculation of a microphysical feedback
in the updraft, the protection of convective fraction
against reevaporation, and the internal use of cloudgeometry considerations in microphysics.
3. Compared study of model behavior
A study using a single-column model was presented in
G07, together with some three-dimensional tests. Here
we want to illustrate the distinctive features of 3MT
especially its behavior in the ‘‘gray zone’’ compared to
other solutions. Figures 2 and 3 show an instantaneous
radar image and model forecasts for an episode of severe
thunderstorm over Belgium. On the radar image, intense
precipitation cores cover at most 1 km 3 4 km (smaller
than the model resolution), with an inner variability
suggesting they already include several convective cells.
Figure 3 compares the mean gridbox vertical velocity
field at 4-km resolution, with either the 3MT scheme, no
deep convection parameterization, or a classical diagnostic parameterization [the diagnostic scheme based on
Bougeault (1985), improved in various ancillary aspects
as described in Gerard and Geleyn (2005)]. In the two
latter cases, we could qualify as ‘‘gray-zone signature’’ the
wide resolved updraft areas accompanied by wide subsidence areas sometimes taking the shape of a crescent
at the rear of the updraft, and inducing significant perturbations in the wind field. The same fields with the
3MT package are much smoother, the uprising and
NOVEMBER 2009
GERARD ET AL.
3967
FIG. 2. Instantaneous radar image: case of violent thunderstorms over Belgium at 1700 UTC
10 Sep 2005.
subsidence areas keeping more realistic horizontal extents and a location close to the precipitation kernel on
the radar image.
With no parameterization the spinup from zero condensates is around 3 h instead of 30 min (as illustrated
in G07).
Figure 4 shows vertical cross sections across a convective region as indicated in Fig. 3. We adapted the
location of the section to sample the same convective
system as forecast by the different model configurations.
The resolved vertical velocity fields appear quite smooth
with 3MT (Fig. 4a). These are complemented by the upand downdrafts mass fluxes represented as equivalent
vertical velocities (Fig. 4d): the updraft flux reaches
11 Pa s21, and takes significant values over 4 or 5 gridbox lengths (i.e., 15–20 km). It represents the action of
several updrafts over this area. In addition, with the
parameterization, downdraft and updraft can coexist in
the same grid box. The vertical velocities in the 3MTparameterized up- and downdrafts (not illustrated) can
reach vu ; 245 Pa s21 and vd ; 140 Pa s21, but only
over a small fraction of the grid box.
With no parameterization scheme (Fig. 4c) the resolved updraft reaches 15 Pa s21 and values above
10 Pa s21 extent over more than 4 grid boxes (i.e., 16
km), which unlike the parameterized case, represents an
ensemble movement of the atmosphere in this region.
As illustrated with another section more to the south
(Fig. 4f), the resolved downdraft can also be strong over
a large area.
The section with the diagnostic scheme also shows
very large resolved up- and downdrafts. In this configuration, the condensation in the updraft does not impact
directly on the resolved condensation, but is supposed to
precipitate in one time step. Interaction of the scheme
with the mean gridbox values passes through detrainment and pseudosubsidence, and it appears that the
subgrid condensation and associated latent heat release
do not produce a sufficient feedback to prevent the ‘‘grayzone syndromes.’’ This seems to confirm the benefit of
the above-mentioned ‘‘hybrid parameterization’’ (like
MTCS) for correctly handling the gray zone. The subgrid
up- and downdraft regions are narrower than with
3MT, and the superimposition of a strong downdraft with
3968
MONTHLY WEATHER REVIEW
VOLUME 137
FIG. 3. Case from Fig. 2, as forecast by 5-h model integration. Horizontal wind and vertical velocity fields at
4-km resolution, model levels (left) 22 (;500 hPa) and (right) 32 (;850 hPa). Updraft (solid line), downdraft
(dashed line) using 1, 5, 10, 15, and 20 Pa s21 isolines. (a),(b) 3MT scheme; (c),(d) no deep convection scheme;
and (e),(f) classical diagnostic deep convection scheme. Horizontal segments mark the vertical cross sections of
Fig. 4.
NOVEMBER 2009
GERARD ET AL.
3969
FIG. 4. Vertical cross section in convective events at 4-km resolution (along solid line in Fig. 3) with isobars (hPa; dotted lines). (a) 3MT:
mean gridbox vertical velocity (Pa s21). (b) As in (a), but with diagnostic scheme. (c) As in (a), but with no deep convection scheme.
(d) 3MT: subgrid updraft and downdraft mass flux divided by g (Pa s21). (e) As in (d), but with diagnostic scheme. (f) Section along dashed
line in Figs. 3c,d with no deep convection scheme.
the updraft seems to poorly contribute to the restabilization.
The ice cloud (Fig. 5) is comparable in the three
configurations, while slightly shallower and wider with
no parameterization; the liquid cloud is also wider
and less concentrated in this case. On 1-h accumulated
precipitation charts (not shown) the case with classical
parameterization presents very large areas of weak precipitation, with only a few kernels of very intense showers.
The maximum values for 3MT and no convection are
similar, but the structure with the 3MT is closer to the
accumulated radar image. The wet-bulb potential temperature u9w (Fig. 6) confirms that 3MT, while producing
intense precipitation, is efficient at restabilizing the
atmosphere. With no scheme or with a diagnostic parameterization (Figs. 6b,c) the atmosphere appears more
perturbed over several gridbox lengths, wider than the
observed structures. The good skill of 3MT down to
4 km suggest this stabilization is correct; at higher resolutions it can become too strong, preventing the resolved scheme to take over, which will be addressed in
another version.
3970
MONTHLY WEATHER REVIEW
VOLUME 137
FIG. 5. Vertical cross section in convective events at 4-km resolution (along solid line in Fig. 3) with isotherms (8C; dotted lines). Cloud ice
and droplets (g kg21): (a) 3MT scheme, (b) no convection, and (c) with classical diagnostic mass-flux scheme.
4. Operational model scores
A systematic verification of 3MT with respect to
observations at 9-km resolution has been done since
April 2008 at the Czech Hydro-Meteorological Institute.
Figures 7–9 present comparative skill scores over Europe
with the full 3MT or with the diagnostic convection
scheme over the 2 first (preoperational) months. The diagnostic scheme is the one based on Bougeault (1985),
further described by Ducrocq and Bougeault (1995), and
improved as described in Gerard and Geleyn (2005). This
scheme can be considered as up to date in the NWP
community, as shown by the fair benchmark results it
produced in the European Cloud Systems Study/Global
Energy and Water Cycle Experiment (GEWEX) Cloud
System Studies (EUROCS/GCSS) intercomparison study
(Derbyshire et al. 2004). Nearly all the difference between
the two configurations resides in the convection scheme
FIG. 6. Vertical cross section in convective events at 4-km resolution (along solid line in Fig. 3). Wet-bulb potential temperature (K):
(a) 3MT scheme, (b) no convection, and (c) with classical diagnostic mass-flux scheme.
NOVEMBER 2009
GERARD ET AL.
3971
FIG. 7. Verification scores (rmse and bias) with respect to observations for relative humidity (%) over a domain
around 2700 3 2500 km over Europe at 9-km resolution from analysis at 0000 UTC between 2 Apr and 5 Jun 2008;
forecast range from 0 to 54 h: with 3MT (solid line) and with diagnostic convection scheme (dashed line). Standard
pressure levels: (a) 200, (b) 500, (c) 700, and (d) 850 hPa.
and its interactions with the moist physics (i.e., the 3MT
developments). At the chosen resolution of 9 km we can
consider to be out of the gray zone.
For the upper-air fields, we observe that while the
relative humidity error (Fig. 7) is slightly increased for
3MT, the temperature rms error and bias (Fig. 8) are
both reduced at all ranges, except the temperature bias
at 200 hPa. For geopotential (not illustrated), both
rmse and bias are improved by 3MT at those standard
levels. Wind speed and direction (not shown either)
both yield a slightly better rmse, and a similar bias.
Near the surface (Fig. 9), the 2-m relative humidity
scores are improved by 3MT, while the 2-m temperature
scores are similar. The total cloud improvement also
appears in rms error and bias (Fig. 10).
The ensemble of the scores of 3MT could be considered at least as good as the reference scheme at 9-km
resolution (and better for clouds), while, as illustrated in
the next section, the evolution of precipitation is frequently better represented.
The vertical profiles of mean horizontal tendencies
of 3MT are very similar for gray-zone and coarser
resolutions, as illustrated in previous section for 7 and
4 km. In addition, surface charts obtained at 4 and
7 km for a few months of preoperational use at the
Royal Meteorological Institute of Belgium (RMIB),
Brussels, have always been consistent. The scores in
the gray zone compared to the same network of observations are thus expected to be close to the ones at
9 km.
3972
MONTHLY WEATHER REVIEW
VOLUME 137
FIG. 8. As in Fig. 7, but for temperature (K).
5. Example operational forecast at varying
resolution
Figure 11 shows a situation of small-scale convective
cells of medium intensity over central Europe developing after about 0900 UTC and rapidly advected within a
slightly anticyclonic flow. The shape of precipitation
corresponds here to the trace of the cells’ displacement,
not to a line organization. The 9-km domain was around
2000 km 3 2500 km coupled to the ARPEGE GCM,
the 4.5- and 2.3-km domains were coupled in a cascade
over smaller domains each time, the smallest (i.e., 2.3
km) is the highlighted area. Given the model time lag
before the start of active convection no risk of spinup
impact exists. With all three physical setups (e.g., 3MT,
classical scheme, and no convective scheme) the same
ALARO-0 schemes were used for the microphysical
calculations, either only for the ‘‘resolved’’ part or for
the 3MT accumulated-condensation-input specific occurrence.
One first notices the good multiscale signature of 3MT
(Figs. 11a–c). The tracking of the displacement is always
present while the scale of the events gets closer and
closer to the truth as resolution increases. The individual
events are roughly positioned along the correct largescale tracks (a more exact matching of individual events
would require, e.g., high-resolution data assimilation,
beyond the scope of this study).
In the case of the classical diagnostic scheme (Figs.
11d–f), the precipitation is smeared over the whole area
of interest (at all scales) and there is not even a clear
signature of the displacement of the maxima due to
advection. The gray-zone syndrome shows some manifestations (noisy patterns over the Austrian Alps and
within some more intense rainfall areas over the Czech
Republic) at 4.5 km but also at 2.3 km, suggesting that
NOVEMBER 2009
GERARD ET AL.
3973
FIG. 9. As in Fig. 7, but for 2-m temperature (K) and relative humidity (%).
the convective systems are not resolved at 2.3 km. This
is confirmed by the test with ‘‘no convection’’ at 2.3 km
(Fig. 11 h): an ‘‘explicit’’ treatment of the precipitation
without any parameterization of subgrid organized moist
convection fails to represent what happens in reality,
given the relatively small size of the nevertheless quite
intense cells. As such this case can be considered as a
favorable one for 3MT (cells characteristics plus the
advantage of the memory of prognostic convection handling for tracking their displacements). It also demonstrates that 3MT behaves like anticipated in its design,
for a situation where neither the classical convection at
9-km mesh size nor the explicit solution at 2.3-km mesh
size offer correct alternatives.
The above-described consistent multiscale behavior
of 3MT repeats itself (not shown) in numerous situations
that we tested (good or bad overall forecast alike),
also independently of its quality with respect to other
solutions.
MTCS approach is then a natural way to cleanly obtain
the convective condensation source. Once in the MTCS
framework, having a predictive equation for the area
fraction in which the ‘‘convective’’ condensation occurs
suppresses the need to arbitrarily prescribe any rate of
detrainment, the various budget for moist species being
handled by the (single) microphysical computations.
Putting the closure under the shape of area fraction
evolution fits nicely with the idea developed in G07 (on a
similar basis to Pan and Randall) where separate prognostic equations for the ascent area fraction and for the
ascent updraft velocity recombine their result to produce a prognostic mass flux, this ensuring a ‘‘convective
memory’’ during model integrations. This also leads to
drive downdrafts on the basis of an evaporation forcing term coming from both resolved and convective
6. Final remarks and conclusions
We believe that the consistent treatment of deep convection at various resolutions from fully subgrid to fully
explicit would help to improve the work of both climate
and operational NWP models. Common current options
of either keeping a standard parameterization or assuming explicit convection do not appear to give universal
solutions working for various atmospheric situations.
Our approach explores different ways to overcome
the ‘‘gray-zone syndromes.’’ Avoiding double counting
of precipitable water quite naturally leads to the idea of
summing various condensation sources before their
handling by a single microphysical computation. The
FIG. 10. As in Fig. 7, but for total clouds (octas).
MONTHLY WEATHER REVIEW
FIG. 11. Accumulated precipitation over central Europe between 0600 and 1200 UTC 2 May 2008. Forecasts from initial conditions of 0000 UTC at (a),(d) 9- and (b),(e) 4.5-km resolution
(hydrostatic) and at (c),(f),(h) 2.3-km resolution (nonhydrostatic). (g) Scaled radar composite image. (a)–(c) 3MT, (d)–(f) diagnostic, and (h) no convection scheme.
3974
VOLUME 137
NOVEMBER 2009
3975
GERARD ET AL.
precipitation sources; evaporation below anvils detrained
by convective clouds may indeed lead to downdraft activity in nature.
The prognostic treatment for the up/downdraft mass
fluxes ensures that the associated convective activity (up
and/or down) is not strictly bound to instantaneous favorable conditions (but such conditions of course determine its perpetuation). It also addresses the problem
of intermittence of diagnostic schemes in the gray zone.
Putting in direct connection the ‘‘memory’’ part concerning the macrophysical aspects of convective activity
(updraft and downdraft separately) and the microphysical memory of a fully prognostic scheme for clouds and
hydrometeors makes the main cement of the 3MT proposal. This allows handling feedback mechanisms present in nature and up to now only treated in models
where the mesh size permits a fully explicit simulation of
convective clouds.
Finally, our practice in implementing 3MT showed
that one can insert in the resulting algorithmic framework important parts of various existing ‘‘classical’’ independent schemes, which then bring in their intrinsic
know-how for the simulation of the relevant individual
physical processes.
The validation tests at 9-km resolution have shown
good performance compared to the earlier diagnostic
schemes, with a better representation and evolution of
the precipitation. Results at higher resolutions are consistent with the coarser resolution. Operational use of
3MT at 9-km resolution has started in Prague, Bratislava,
Ljubljana, and Vienna, and at 7 and 4 km in Brussels.
Some weaknesses of the current package now have to
be addressed. To realize a satisfactory convergence with
explicit convection at resolutions finer than 2 km, we are
developing a version where the updraft vertical growth
in one time step is limited by its vertical velocity. It will
also include a vertical variation of the updraft mesh
fraction that better represents the presence of clouds of
different heights in a grid box, or a single cloud with
varying sections.
The expression of the detrainment area contributing to
the convective cloud fraction has a significant impact in the
scheme; we plan to refine it using a cloud-system resolving
model (CSRM) or large-eddy simulation (LES) data. The
moisture convergence closure appears quite successful in
most situations; a prognostic mixed closure (CAPE and
moisture convergence) would allow for handling situations with no moisture convergence. Concerning life cycle,
the prognostic approach allows a gradual transition to
deep convection. Relating the updraft entrainment/mixing
to the accumulated impact of the density currents associated with earlier downdrafts in the way proposed by
(Piriou et al. 2007) is presently further investigated.
Acknowledgments. Alaro-0 and 3MT benefited from
the work of several persons. We want to thank all of
them here and in particular I. Stiperski, B. Catry,
C. Wittmann, M. Tudor, J. Cedilnik, and N. Pristov. We
also thank the two anonymous reviewers for their constructive comments.
APPENDIX
General Features of the Microphysical Scheme
The current package is single moment, based on specific contents. Beside the five prognostic water species
(i.e., water vapor, cloud ice, cloud droplets, snow, and
rain) advected with the resolved flow, a diagnostic
pseudograupel specific content is estimated each time
step by computing a proportion of the precipitating ice
phase that sediments with the same speed and collects
with the same efficiency as rain. Source terms for the
pseudograupel quantities are the Bergeron–Findeisen
process, a proportional amount of the collection output
rescaled by the differential efficiency, and the result of
rain freezing. Sink terms are the proportional amounts
of evaporation and melting processes.
The internal state of gross mean gridbox cloud condensates resulting from stratiform and convective condensation enters the autoconversion routine, together
with an equivalent cloud fraction, combining the convective and stratiform mesh fractions (G07).
a. Autoconversion, mixed layer, and
Wegener–Bergeron–Findeisen treatment
The autoconversion in the homogeneous phase follows Sundqvist (1978):
o
p
2
dqi
q n
5 i
1 e 4 [qi /qicr (T )]
dt au
t i (T)
(A1)
for cloud ice, and a similar relation for cloud droplets, but
with a constant threshold qlcr and a constant time scale t l.
The Bergeron–Findeisen process in the mixed phase is
parameterized as an autoconversion from cloud droplets
to snow, following the same form:
dql
dt
wbf
5 F a
D
ql ql qi
t l (q 1 q )2
3 1
l
i
E
p
e 4 fqi ql /[F b qlcr F b qicr (T )]g .
(A2)
Equation (A2) was derived from van der Hage (1995),
using part of the analysis G07 made of it. Here we consider
3976
MONTHLY WEATHER REVIEW
dql
dt
dql
N
q r 3
5G i ’G i d ,
dt au
Nd
ql ri
wbf
where Ni and Nd are the number concentrations of ice
crystals and droplets, rd and ri their respective radius,
assuming no difference in spectral selectivity of the
conversion process and a constant density ratio of ice to
liquid water. Following (van der Hage 1995), the gain
factor G can be expressed as
G}
ri ci si
ql qi ri
}
,
r3d
(ql 1 qi )2 r3d
dql
dt
"
#
ql
ql qi
qicr (T)
ql qi
}
,
2 (T)q
2
t
q
q
(T)
r
(q
wbf
i
l lcr icr
lcr
l 1 qi )
where the bracketed term is taken as constant. Leaving
the asymptotic proportionality and introducing parameters this equation leads to Eq. A2.
b. Prognostic sedimentation
We use the prognostic sedimentation scheme described in Geleyn et al. (2008). The scheme expresses
the precipitation crossing the bottom of any model layer
by means of three probabilities of transfer: P1 for precipitation present in the layer at the beginning of the
time step, P2 for precipitation coming from the layer
above, and P3 for precipitation generated (or destroyed)
in the layer during current time step. All three probabilities are functions of Z 5 dz/wdt, where dz is the layer
depth and the mean fall velocity w at this level depends
on the phase and the intensity of the precipitation flux.
Computations done in a similar way to those leading to
Eqs. (B3) and (B4) of Lopez (2002) yield in our case
(with somewhat different simplifying assumptions and
scaling choices), respectively, for rain and snow:
vr 5 13.4
(vr qr )1/6
2/3
(r)
and vs 5 3.4e[0.0231(TT t )]
(vs qs )1/6
.
(r)2/3
(A3)
The rain flux at the bottom of the layer is expressed by
P bot
l 5 P1 qr
col
eva
are the increments of rain
where dqau
r , dqr , and dqr
specific contents during the time step dt associated, respectively, to autoconversion, collection, and evaporation processes. A similar equations is used for snow.
These various microphysical processes are parameterized again in a manner similar to Lopez (2002) for the
various collection occurrences and, as in the Aladin diagnostic precipitation package, for evaporation and
melting (Geleyn et al. 1994).
REFERENCES
where si is the supersaturation of vapor with respect to
ice and ci is nearly constant. The second proportionality
translates the observed symmetry of the curve cisi, Fig. 2
of van der Hage (1995). Taking the asymptotic behavior
of Eq. (13) for droplets, we get
VOLUME 137
Dp
top
col
eva Dp
1 P2 P l 1 P3 (dqau
,
r 1 dqr dqr )
gdt
gdt
(A4)
Arakawa, A., 2004: The cumulus parameterization problem: Past,
present, and future. J. Climate, 17, 2493–2525.
——, and W. Schubert, 1974: Interaction of a cumulus cloud ensemble with the large-scale environment, Part I. J. Atmos. Sci.,
31, 674–701.
Bougeault, P., 1985: A simple parameterization of the large-scale
effects of cumulus convection. Mon. Wea. Rev., 113, 2108–2121.
Boville, B. A., P. J. Rasch, J. J. Hack, and J. R. McCaa, 2006:
Representation of clouds and precipitation processes in the
Community Atmosphere Model version 3 (CAM3). J. Climate, 19, 2184–2198.
Deng, A., and D. R. Stauffer, 2006: On improving 4-km mesoscale
model simulations. J. Appl. Meteor. Climatol., 45, 361–381.
Derbyshire, S. H., I. Beau, P. Bechtold, J.-Y. Grandpeix,
J.-M. Piriou, J.-L. Redelsperger, and P. M. M. Soares, 2004:
Sensitivity of moist convection to environmental humidity.
Quart. J. Roy. Meteor. Soc., 130, 3055–3079.
Ducrocq, V., and P. Bougeault, 1995: Simulations of an observed
squall line with a meso-beta scale hydrostatic model. Wea.
Forecasting, 10, 380–399.
Frank, W. M., and C. Cohen, 1987: Simulation of tropical convective systems. Part I: A cumulus parameterization. J. Atmos.
Sci., 44, 3787–3799.
Geleyn, J.-F., and Coauthors, 1994: Atmospheric parametrisation
schemes in Météo-France’s ARPEGE NWP model. ECMWF
Seminar 94: Physical Parametrisations in Numerical Models,
Reading, United Kingdom, ECMWF, 385–402.
——, B. Catry, Y. Bouteloup, and R. Brožková, 2008: A statistical
approach for sedimentation inside a micro-physical precipitation scheme. Tellus, 60A, 649–662.
Gerard, L., 2007: An integrated package for subgrid convection,
clouds and precipitation compatible with the meso-gamma
scales. Quart. J. Roy. Meteor. Soc., 133, 711–730.
——, and J.-F. Geleyn, 2005: Evolution of a subgrid deep convection parametrization in a limited area model with increasing
resolution. Quart. J. Roy. Meteor. Soc., 131, 2293–2312.
Grell, G. A., 1993: Prognostic evaluation of assumptions used by
cumulus parameterizations. Mon. Wea. Rev., 121, 764–787.
Guichard, F., and Coauthors, 2004: Modelling the diurnal cycle of
deep convection over land with cloud-resolving models and
single-column models. Quart. J. Roy. Meteor. Soc., 130, 3139–
3172.
Horányi, A., S. Kertesz, L. Kullmann, and G. Radnóti, 2006: The
ARPEGE/ALADIN mesoscale numerical modelling system
and its application at the Hungarian meteorological service.
Idöjárás, 110, 203–227.
Kershaw, R., and D. Gregory, 1997: Parameterization of momentum transport by convection. I: Theory and cloud modelling
results. Quart. J. Roy. Meteor. Soc., 123, 1133–1151.
NOVEMBER 2009
GERARD ET AL.
Krishnamurti, T. N., S. Low-Nam, and R. Pasch, 1983: Cumulus
parameterization and rainfall rates II. Mon. Wea. Rev., 111,
815–828.
Kuo, H., 1974: Further studies of the parameterization of the influence of cumulus convection on the large scale flow. J. Atmos. Sci., 31, 1232–1240.
Lopez, P., 2002: Implementation and validation of a new prognostic large-scale cloud and precipitation scheme for climate
and data assimilation purposes. Quart. J. Roy. Meteor. Soc.,
128, 229–258.
Mapes, B. E., 1997: Equilibrium vs. activation control of large-scale
variations of tropical deep convection. The Physics and Parameterization of Moist Atmospheric Convection, R. K. Smith,
Ed., Kluwer Academic Publishers, 321–358.
Molinari, J., and M. Dudek, 1992: Parameterization of convective
precipitation in mesoscale numerical models: A critical review. Mon. Wea. Rev., 120, 326–344.
Moncrieff, M. W., and M. J. Miller, 1976: The dynamics and simulation of tropical cumulonimbus and squall lines. Quart. J.
Roy. Meteor. Soc., 102, 373–394.
Pan, D., and D. Randall, 1998: A cumulus parametrization with a
prognostic closure. Quart. J. Roy. Meteor. Soc., 124, 949–981.
3977
Piriou, J.-M., J.-L. Redelsperger, J.-F. Geleyn, J.-P. Lafore, and
F. Guichard, 2007: An approach for convective parameterization with memory, in separating microphysics and transport
in grid-scale equations. J. Atmos. Sci., 64, 4127–4139.
Smith, R. N. B., 1990: A scheme for predicting layer clouds and
their water content in a general circulation model. Quart. J.
Roy. Meteor. Soc., 116, 435–460.
Sundqvist, H., 1978: A parameterisation scheme for non convective
condensation including prediction of cloud water content.
Quart. J. Roy. Meteor. Soc., 104, 677–690.
van der Hage, J., 1995: A parametrization of the Wegener-BergeronFindeisen effect. Atmos. Res., 39, 201–214.
Wilson, D. R., A. C. Bushell, A. M. Kerr-Munslow, J. D. Price, and
C. J. Morcrette, 2008: PC2: A prognostic cloud fraction and
condensation scheme. I: Scheme description. Quart. J. Roy.
Meteor. Soc., 134, 2093–2107.
Xu, K. M., and D. A. Randall, 1996: A semi-empirical cloudiness
parameterization for use in climate models. J. Atmos. Sci., 53,
3084–3102.
Yanai, M., S. Esbensen, and J.-H. Chu, 1973: Determination of
bulk properties of tropical cloud clusters from large-scale heat
and moisture budgets. J. Atmos. Sci., 30, 611–627.