QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY
Q. J. R. Meteorol. Soc. 00: 1–19 (2007)
Published online in Wiley InterScience
(www.interscience.wiley.com) DOI: 10.1002/qj.06.42
An integrated package for subgrid convection, clouds and
precipitation compatible with meso-gamma scales.
L. Gerard∗
Royal Meteorological Institute of Belgium, 3 Av. Circulaire, B 1180 Brussels, Belgium
Abstract: The integration into a coherent package of the main ”moist” parametrizations – deep convection, resolved condensation,
microphysics – of a limited area model is presented. The development of the package is aimed at solving efficiently the problem
of combining ’resolved’ and ’subgrid’ condensation at all resolutions, in particular the range between 10 km and 2 km where
deep convection is partly resolved, partly subgrid. The different schemes of the package are called in cascade, with intermediate
updating of internal variables, so that for instance the initial profiles passed to the deep-convection scheme are already balanced
with respect to resolved condensation effects. Further on, the clean separation of the contributions to the closure of the updraught
and downdraught from the initial vertical profile from which they evolve prevents double counting. The convective parametrization
works with a prognostic mass-flux scheme, and acts on the resolved variables through condensation and convective transport. It
detrains condensates that are added to the prognostic resolved condensates. A sensitivity study in a single-column model, and
c 2007 Royal Meteorological
further validation in three-dimensional experiments at different resolutions, are presented. Copyright Society
KEY WORDS
Local area modelling; grey-zone resolutions; prognostic convection; prognostic microphysics.
Received 29 March 2006; Revised 1 December 2006; Accepted 6 February 2007
1
Introduction
In a previous paper (Gerard and Geleyn, 2005), we
assessed and tried to address different limitations of a
classical mass-flux parametrization of the deep convection, initially developed for a global circulation model
with grid boxes bigger than 20 km, when used in a highresolution limited-area model and reducing the grid-box
lengths down to around 7 km. Beside various enhancements of the diagnostic approach, we were led to relax
the hypothesis of quasi-equilibrium between the convective activity and the large-scale processes feeding it, by
using prognostic equations for the vertical velocity in the
updraught and for the fraction of the grid-box area covered
by it. This updraught mesh fraction also influences the
mean grid-box variables when the convective cells cover
a significant part of the grid-box area.
The prognostic approach has significant advantages.
Tiedtke (1993) advised that the explicit representation
of anvil and cirrus clouds associated with cumulus convection was a strong argument in favour of prognostic
schemes. Knowledge of the updraught’s vertical velocity and mesh fraction is important for understanding the
microphysics, as well as for parametrizing mesoscale
updraughts and downdraughts (Leary and Houze, 1980;
Donner et al. 2001); this means that the prognostic scheme
is also useful at coarser resolutions. Arakawa (2004) sees
the closure of the convective parametrization as evolving
from diagnostic to prognostic to stochastic.
While we obtained some improvement, we were confronted with two major difficulties. The first was the challenge of combining different precipitating schemes working in parallel in such a way as to avoid double counting,
while remaining independent of the model resolution in
space and time. The second difficulty was the need to handle cloud condensates as model variables and to replace
the rough diagnostic parametrizations of clouds and precipitation by a more elaborate microphysical package
treating all condensates, whatever their origin – resolved
or subgrid (convective).
In this paper, we describe an integrated package that
addresses these difficulties. Such an approach is in accord
with the recent trends in parametrization. The need to
treat all cloud processes in a unified consistent way has
already been emphasized by Tiedtke (1993). Arakawa
(2004) insists that the artificial separation of the physical
processes causes most of the direct small-scale interactions between those processes to be lost. He recommends
working on the unification of deep convection with other
parametrizations: the use of mass transport by cumulus
convection in the microphysics; the generation of liquid and ice phases of water leading to a unified cloud
parametrization; the interactions with the boundary layer
(diurnal cycle, shallow clouds); the coupling of radiation
and cloud processes on the cloud scale; consideration of
the effect of momentum transport on the mean flow and
the problem of clouds organization; and the inclusion of
non-deterministic effects. The eventual challenge would
∗ Correspondence to: Research Department, Royal Meteorological Institute of Belgium, 3 Av. Circulaire, B 1180 Brussels, Belgium. E-mail: be to develop a ’physics coupler’ in which all these processes are fully coupled.
[email protected].
c 2007 Royal Meteorological Society
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2
L. GERARD
The scheme described in (Gerard and Geleyn, 2005)
already produced convective transport fluxes, including
the transport of momentum. However, the effect of the
updraught on the resolved variables was estimated through
a computation of detrainment and pseudo-subsidence. In
our new scheme, following the proposal of Piriou (2005),
we express these effects directly through transport and
condensation. One of the basic features of our package
is the cascading approach, which not only allows a clean
separation of the closure contributions, but also avoids
competition between resolved and subgrid parametrizations. The convective transport impacts on the microphysics, which is computed subsequently. In our new convective scheme, the updraught does not produce precipitation directly, but it detrains cloud condensates, which are
combined with the resolved condensates to feed the microphysics. This approach allows one to take into account
the anvil and cirrus clouds generated by deep convection.
The updraught entrains condensates from the environment
as well. The package we propose here corresponds, in
Arakawa’s terms, to the step of the ’unified cloud scheme’.
In the context of climate and general circulation models, several authors (Del Genio et al., 1996, Tiedtke, 1993,
Ose, 1993, Fowler et al., 1996) add the detrained condensed water from the convective updraughts to the prognostic stratiform cloud water. In Tiedke’s scheme, convective precipitation is generated through a drain term
similar to the one used for the autoconversion of stratiform condensates to precipitation. But it is then difficult
to keep the same coherence when replacing or refining
the microphysical package. Fowler and Randall (2002)
proposed to go one step further in the coupling of convection and microphysics, by entraining cloud water into the
convective updraughts. Still, their convective parametrization converts diagnostically most of its condensates to
precipitation, which is sent promptly to the surface, thus
following a path different from that of the large-scale
microphysics. They assess alternate ways to handle the
convective snow – either detraining it at the top and passing it through the microphysics, or precipitating it outside
or inside the updraught – and show the sensitivity of the
results to this choice. Boville et al. (2006), for the CAM3
climate model, use two separate schemes for shallow and
deep convection, both of which detrain condensates into
the stratiform clouds; but the majority of the condensate
formed by the deep convection is directly precipitated
rather than detrained, and they explain that for this reason,
the detrained water that may feed anvil clouds is significantly underestimated.
The greatest advantage of our integrated package is
that it can work at all resolutions, including the ’grey
zone’ where deep convection is partly resolved, partly
subgrid. A few authors have approached the problem of
convection in high resolution models, particularly when
the mesh size becomes comparable to the size of the convective systems. Weisman et al. (1997) present a systematic academic study of the explicit representation of convection with grid-box lengths ranging from 2 km to 12
km. They use a non-hydrostatic model, and show that for
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resolutions coarser than 8-km, it behaves the same way as
a hydrostatic model. Their study uses a semi-infinite 2kmdeep surface cold pool, which suppresses the problem of
triggering and for which even a 50 km grid would be
sufficient to resolve a portion of the mature system-scale
structure. In this case, they show that the 4 km resolution
still gives quite satisfying results compared with the 2 km
grid, while there is a progressive degradation of the timing and the extension of the development for coarser grids.
Deng and Stauffer (2006) present sensitivity experiments
with the non-hydrostatic model MM5 at 4 km resolution. The use of a convective parametrization improves the
results, despite the fact that the 4-km resolution violates
the underlying assumption of the two tested parametrization schemes that the size of subgrid deep convection is
well below the grid-box length. They show that a convective scheme is required because the explicit microphysics alone cannot represent deep convection on a 4 km
grid: convective updraughts are forced on a coarser-thanrealistic scale; the rainfall and the atmospheric response
are too strong; and the evaporative cooling and the
downdraughts are too vigorous, causing widespread disruption of the low-level winds and spurious advection
of the simulated tracer. Grabowski (2001) proposes a
completely different approach, the ’cloud-resolving convection parametrization’, consisting of applying a twodimensional (zonal–vertical) cloud-resolving model with
a resolution around 1km, inside each mesh of a largerscale model, with horizontal grid-length around 100 km.
This approach is mainly relevant to climate models; in
operational forecast, the three-dimensional description of
smaller features remains essential.
The present work uses the hydrostatic version of the
Aladin model, and this may explain that a convective
parametrization is required down to 2 km mesh sizes. The
convective scheme we presented in (Gerard and Geleyn,
2005) removed assumptions that would be violated at
high resolutions, so that it would be a good candidate
to complement Deng and Stauffer’s study. But this is
not sufficient, because there would still be competition
between two schemes generating precipitation in different
ways.
The solution presented in this paper works at all
resolutions (for instance, as coarse as 100 km or as fine as
2 km, including all intermediate resolutions) , thanks to a
coherent coupling of convection with the resolved cloud
and precipitation scheme – including a microphysical
package with prognostic liquid and ice cloud condensates
– and avoidance of the problems of double counting.
In section 2, we describe the general organization
of our package. Section 3 gives more details on its main
components. In Section 4 we comment on the sensitivity
to the different parameters. In Section 5 we present two
case studies at different resolutions, and in Section 6 we
present our conclusions.
Q. J. R. Meteorol. Soc. 00: 1–19 (2007)
DOI: 10.1002/qj
A PACKAGE FOR CONVECTION, CLOUDS AND PRECIPITATION
2
2.1
Components and hypotheses
General layout
A numerical weather prediction model computes the evolution of ’model variables’ corresponding to the mean values of the meteorological fields over the area or volume
of its grid boxes. Physical parametrizations are needed to
evaluate the effects of the different subgrid processes on
these model variables. The corresponding ’physical tendencies’, act as source terms in the mean-flow dynamical
equations.
Since all processes are acting simultaneously, a first
possibility it to call the different parametrizations in parallel, based on the same initial state (’parallel split’). On the
other hand, the physical processes are interacting with one
another, so that a sequential call of the parametrizations,
each of them working on an updated state can be justified. An ordering of the processes has then to be chosen.
To take into account the two-way interactions between
the processes, a symmetrized sequential-split method can
be used, where a fractional updates are performed in
sequence until the last process is reached, and residual
updates are then computed in reverse order. This approach
is quite expensive.
Dubal et al. (2004) show that the parallel-split
method can introduce significant errors on the steady
state when an implicit discretization is used. The sequential split can, under certain conditions, yield an accurate
steady state. However, the maintenance of the organization of the different parametrizations is significantly less
flexible than with the parallel split. In Arpège-Aladin,
some of the parametrizations are accurate to the first order
anyway and the modularity is found to be more important than the steady-state accuracy. What is essential is
the coherence between location and time: the physics is
computed at the origin point of the semi-Lagrangian trajectories and at the time t − △t. So the turbulent diffusion
scheme, the radiation scheme, the cloud and precipitation or the surface scheme are all working in parallel. A
detailed discussion of the Arpège-Aladin time-stepping
has been done by Termonia and Hamdi (personal communication, submitted paper).
Now, if the different schemes implied in the moist
physics are called in parallel, referring to the same (not yet
balanced) mean grid-box state (water contents and phases,
temperature, pressure, wind), each of them will produce
a response that ignores the work of the others towards
the final state; when combining these contributions, the
response is excessive, implying a multiple counting of
some phenomena. This is the main source of the difficulty
we had in (Gerard and Geleyn, 2005) when trying to combine precipitation from separate large-scale and subgrid
schemes.
On the other hand, the physical processes occurring
’inside’ a package are often cascaded in time. Within the
microphysical package for instance, transient values of
cloud condensates (and other variables) after condensation are passed to an autoconversion calculation, which
c 2007 Royal Meteorological Society
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3
modifies them; the result is then passed to the calculation
of collection and evaporation–melting processes.
If we want to obtain a wholly coherent treatment of
cloud and precipitation processes, we must consider these
as forming part of a single package: that is, we must apply
them in cascade rather than in parallel. In a way, this is a
step towards the ”sequential-split”, with an ordering of the
processes based on physical considerations.
To this end, we introduce internal variables for the
water phases and the temperature that copy the initial
state of the time step but are subsequently updated by
each scheme of the package to yield the initial state for
the next part. The cascaded parts (see Figure 2) are:
the turbulent diffusion; the resolved condensation; the
convective updraught; the autoconversion of condensates
to precipitation; the evaporation, melting and collection
processes associated with precipitation; and finally a moist
downdraught (associated with cooling by precipitation
evaporation and melting).
Clean handling of the closure contributions is also
important to prevent double counting with the cascading
part of the scheme. The source of condensation is not
the same for the resolved and the subgrid schemes. The
resolved scheme condenses the excess water vapour that
is present at the beginning of the time step. The convective
updraught scheme essentially condenses the excess water
vapour brought to the grid box during the time step by
the resolved convergence of moisture. The local vertical
turbulent diffusion of moisture may be either added to this
moisture convergence or put in the initial state before both
condensation schemes (see section 3.3).
2.2
Geometrical subdivision
The package distinguishes different fractions of the gridbox area, associated with different properties.
The resolved condensation produces a ”resolved” or
”stratiform” cloud fraction f st . The updraught covers a
fraction σu , and detrains condensates into a fraction σD , of
the grid-box area. The resulting ”subgrid” or ”convective”
cloud fraction is f cu = σu + σD .
The total cloud fraction at a given level is taken as
f = f cu + f st − f cu f st .
(1)
Below we also define an equivalent cloud fraction f eq
linking the mean in-cloud condensate densities with the
mean grid-box condensate densities. We assume that precipitation falls over a fraction σP of the grid-box area,
equal to the maximum of f eq over the layers above.
2.3
Water variables and fluxes
Several parts of our microphysical scheme have been
based on that of Lopez (2002). Unlike him, we do not use
separate variables for the precipitation contents, preferring
a much simpler (and lighter) approach based on the
precipitation fluxes (section 3.4.2).
The scheme uses 3 prognostic water species (which
are advected from one time step to the next by the resolved
Q. J. R. Meteorol. Soc. 00: 1–19 (2007)
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4
L. GERARD
qv
F vi
F sv
F vl
qi
Frv
Ps
F is
Fli
ql
Pr
F lr
Fsr
Figure 1. Water fluxes in the integrated scheme. Fluxes in dashed
lines are derived from the others.
flow), in the form of specific contents: vapour (qv ), cloud
ice (qi ) and cloud droplets or ”liquid” (qℓ ). Below, we also
use the total condensate qc = qi + qℓ . The precipitation
contents are directly related to the precipitation fluxes
(snow or solid Ps and rain Pr ), which are diagnosed at
each time step. All parts of the parametrization produce
contributions to the transfers between the different phases
as presented in Fig. 1. Fluxes qualified below as ”net”
may occur in either direction, and are taken positive from
the first to the second index. The fluxes shown on Fig. 1
are: Fvi , net condensation to ice; Fvℓ , net condensation
to liquid; Fℓi , net condensate freezing; Fis , generation of
solid precipitation from cloud ice and Fℓr , generation of
rain from cloud droplets; Fsr , net precipitation melting;
Fsv , snow evaporation; and Frv , rain evaporation.
To convert cloud droplets into solid precipitation, we
first convert them to ice with Fℓi and then the ice to solid
precipitation with Fis .
The heat fluxes associated with phase changes can be
derived from the water fluxes by using appropriate latent
heats.
The evaporation of solid and liquid precipitation, Fsv
and Frv are obtained from the budgets:
and
Ps = Fis − Fsv − Fsr
Pr = Fℓr − Frv + Fsr .
(2)
The precipitation melting flux is zero at the top of the
atmosphere, and receives an increment at the levels where
precipitation melting takes place:
△Fsr = (△Pr − △Ps ) − (△Fℓr − △Fis )
+ (△Frv − △Fsv ) /2.
(3)
Precipitation melts rapidly after crossing the triple-pointtemperature level. Assuming that across the transition the
evaporation of snow △Frv and rain △Fsv are equal, the
melting flux Fsr may be deduced from Eq. 3.
2.4 Tendencies
In Arpège-Aladin, the physical routines output vertical
”diffusive” fluxes, whose vertical divergence contributes
to the mean grid-box tendency. For a model variable ψ
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these fluxes include turbulent diffusion fluxes Jψtd , convective transport fluxes Jψcu , fluxes associated with phase
changes (for water variables or heat), radiation (for heat),
drags (for heat or momentum), and so on. In the scheme
described in (Gerard and Geleyn, 2005), precipitation was
directly generated from water vapour condensation and
there was no suspended cloud phase. Therefore we could
directly bind the precipitation flux and the associated heat
flux through a bulk latent heat (depending on the option
taken for ”mass conservation”). This is no longer possible
here, because precipitation is generated from the (usually
adiabatic) conversion of cloud condensates to precipitation, while the reduction of the precipitation is associated
with diabatic processes of evaporation. Instead, we consider the local latent heat (assumed to be a function of
the temperature). Conservation is guaranteed by the clean
formulation of all water fluxes and the associated heat
exchanges.
In the current version we assume that falling precipitation is replaced by an equivalent quantity of dry air from
the surface (the ’mass conservation hypothesis’). A more
precise barycentric formulation is now being developed
(Catry et al., 2007).
2.5 General organization of the package
Figure 2 shows the sequence of the calculation. The
resolved advection calculation can produce at some places
non-physical values of the water contents: negative contents, or non-zero ice above the triple-point temperature.
This must be fixed before performing any further calculation with them, and the corrections must also be transfered
to the physical tendencies of the model variables. Since
the bad values result from numerical rather than physical processes, we consider that the subsequent fix must
remain adiabatic. To fix the internal state, the missing condensates are taken from the water vapour, and the negative
water vapour values are replaced by zero. To reflect this on
the model variables, a corrective water vapour diffusion
flux Jvcor is added (at the end of the parametrizations) to
the turbulent diffusion, fetching the missing vapour from
below and even from the surface. Condensate correction
fluxes Jℓcor , Jicor are similarly added to the condensate
turbulent diffusion fluxes – but in this case they are simply corrections, the missing condensate having been taken
from the vapour.
The resolved cloud fraction f st is presently computed using the scheme of (Smith, 1990) (see section
3.2). The model may either use a simple radiation scheme
(inspired by (Ritter and Geleyn, 1992)), called at each
time step, or a more elaborate one (Morcrette, 1991),
called less frequently. In the first case, the stratiform cloud
fraction f st is directly combined with the convective cloud
fraction f cu− kept from the previous time step, yielding a total cloudiness value f ∗ to pass to the radiation
scheme. Radiation affects the surface temperature which
is passed to the turbulent diffusion scheme. This scheme
computes turbulent fluxes of conservative variables: total
water qt = qv + qc , ”liquid static energy” sℓ = s − Lqc ,
Q. J. R. Meteorol. Soc. 00: 1–19 (2007)
DOI: 10.1002/qj
A PACKAGE FOR CONVECTION, CLOUDS AND PRECIPITATION

(Fix negative contents 
from advection)
−→ Jℓcor , Jicor , Jvcor

(Adiab. phase adjustment)
ւ
[qv∗ , qi∗ , qℓ∗ ]
Resolved cloud fraction
f st , f cu− −→ f ∗
ց
−→ (Radiation)
→ (Tsurf , Turbulent diffusion) −→
internal state, and, if necessary, prevents the specific contents of condensates from being negative by an additional
condensation of water vapour. After the updraught, an upto-date value f of the total cloud fraction is obtained,
combining f st and f cu = σD + σu .
We define the convective fraction as the ratio of the
convective to the total condensation flux:
cu
cu
cu
cu
st
st
αcu = (Fvl
+ Fvi
)/(Fvl
+ Fvi
+ Fvl
+ Fvi
),
Jℓtd ,
Jitd ,
Jvtd ,
ւ
st
st
Resolved condensation −→ Fvi
, Fvℓ
ւ
[qv∗ , qi∗ , qℓ∗ , T ∗ ]
cu
cu
Fvi
, Fvℓ
,
moisture
cu
cu
conver- → Deep convection −→ Jv , Ji , Jℓcu ,
Jscu , JVcu
gence
f cu = σD + σu
→ αcu → f eq
ւ
[qv∗ , qi∗ , qℓ∗ , T ∗ ]
Autoconversion −→ Fis , Fℓr , Fℓi
and we use it to decide what fraction of the condensates,
as well as of the total precipitation flux may be declared
’convective’.
Autoconversion of condensates to precipitation and
condensate collection by precipitation, depend on the
local densities of the condensates, not on the mean gridbox densities.
The density of condensate in the convective clouds
may be significantly higher than in the resolved clouds,
because the ratio f cu /f is often much smaller than αcu .
The local specific condensate content is estimated by
ւ
[qi∗ , qℓ∗ , T ∗ ]
FhP ւ
Fis , Fℓr , Fℓi ,
Pr , Ps
ւ
[qv∗ , qi∗ , qℓ∗ ]
ց
Downdraught −→
Pr , Ps , Jvcu , Jicu ,
Jscu , JVcu
ւ
[qv∗ , qi∗ , qℓ∗ ]
(Final diabatic
−→ Fℓi
phase adjustment)
αcu −→
P cu , Pscu ,
(Convective/
−→ rst
Pr , Psst
Resolved)
Figure 2. Package organization chart. The square brackets mark the
successive updates of the internal state
momentum. The turbulent fluxes of water vapour, cloud
condensates and dry static energy are finally derived from
these. The internal state is updated following the turbulent
diffusion. An alternative would be to combine the water
vapour turbulent diffusion flux with the resolved moisture
convergence flux in the closure of the updraught. In this
case, only the internal variables of temperature and condensates should be updated with the turbulent diffusion
before entering the updraught routine.
The resolved condensation is described in section
3.2. Again, it updates the internal variables. The resulting
state is input to the updraught (section 3.3), which procu
cu
duces ”convective” condensation fluxes (Fvi
, Fvℓ
) and
convective transport fluxes of water (Jvcu , Jℓcu , Jicu ), heat
(Jscu ) and horizontal momentum (JVcu ). It also updates the
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Copyright Prepared using qjrms3.cls
(4)
Jstd
[(qv∗ , )qi∗ , qℓ∗ , T ∗ ]
Precipitation −→
5
qˆc = qc
2
(1 − αcu )2
αcu
+
f st
f cu
qc
f st + f cu
= eq , (5)
f
f
where f eq is an ”equivalent” cloud fraction that would
occur
Jℓcu
, if the condensate density was the same in all clouds
in the grid box. The factor in square brackets prevents f eq
from being greater than f .
The autoconversion algorithm (section 3.4.1)
receives the values of moisture, condensates and temperature as output by the updraught, together with the
equivalent cloud fraction. It computes the contribution of
the nucleation, coalescence and the Bergeron-Findeisen
processes to the precipitation generation fluxes, the third
one also implying a a condensate freezing flux.
The box labelled ”precipitation” computes the collection
and evaporation effects. It outputs the precipitation fluxes,
and contributes to the precipitation generation fluxes and
also to the condensate freezing flux where the riming
process is active.
The downdraught scheme (section 3.5) is driven by the
heat sink associated to falling precipitation evaporation,
melting or heating. The corresponding heat flux FhP is
passed to the downdraught, which uses it in its closure.
To avoid double counting, the internal temperature input
to the downdraught is not yet affected by this flux. The
downdraught further modifies the precipitation fluxes, the
water vapour, and the convective transport fluxes.
A final diabatic phase adjustment of the condensate is performed before the advection to the next time step, while
the precipitation flux is partitioned between ”convective”
and ”resolved” precipitation:
P cu = αcu P,
P st = (1 − αcu )P.
(6)
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L. GERARD
3 Components of the integrated scheme
3.1 Mixed phase partition
We assume that the fraction of ice αi in the mixed phase
is a function of the temperature only:
"
#
2
(Tt − min(Tt , T ))
αi (T ) = 1 − exp −
(7)
2(Tt − Tx )2
by simply (in addition to the limitation explained above)
smoothing the relative-humidity profile around the triple
point. Starting from the top and moving downwards, at
each level where the Celsius temperature is positive we
replace the relative humidity by that obtained by mixing
the air with the adjacent levels, when this mixing is not
unlikely: we mix two or three levels below if they would
be buoyant when raised along a dry adiabat, and two or
three levels above if they would sink when moved down
along a dry adiabat.
At the lowest model level, a cooling can occur from
the precipitation evaporation and the downdraught. Subsequent exchanges with the surface should temper this
cooling, but again this would require an iterative calculation. To prevent unrealistic condensation, we compute
the saturation moisture at the lowest model level using the
arithmetic mean between the surface and air temperatures.
To prevent excessive variations of the intensive condensate values (Eq. 5) when the resolved scheme is active
alone, we impose that the cloud fraction be non-zero wherever qc is not negligible; in addition, we limit its increment
or decrement △f st between adjacent model levels to a
tunable maximum value.
where Tt = 273.15K is the triple-point temperature and
Tx is the temperature of the maximum difference between
the saturation vapour pressures with respect to ice and to
liquid.
The water vapour condenses into ice and liquid
phases according to this ratio, computed using for T the
mean grid-box temperature T for the resolved part, and
the updraught temperature Tu for the subgrid part. But
the autoconversion and collection processes, as well as the
advection by the large-scale flow, modify the ice fraction.
As shown on Fig. 2, we readjust the phases to the
ratio αi after both of these processes, the correction contributing to the freezing-melting flux Fℓi . This systematic
readjustment is convenient for controlling the biases due
to advection, but since it reduces the independence of the
two cloud condensate variables, we intend to relax it in a 3.3 The updraught mass flux scheme
future version.
The prognostic updraught scheme described in (Gerard
and Geleyn, 2005) has been revised extensively. In the
3.2 Resolved cloud fraction and condensation
following, the term ’environment’ designates the part of
the grid box which is outside the updraughts, and the
The scheme for the resolved cloud fraction and conden’updraught’ is a composite representation by a single
sation is based on (Smith, 1990). It removes the resolved
mass flux of all the updraughts present in a grid box.
saturation by condensing the excess moisture. We split the
Eventhough the new scheme does not explicitly check
scheme in two parts: the first part outputs a ”stratiform”
the conservation of the vertically integrated moist static
cloud fraction, f st ; the second part, which is called after
energy, this is achieved through the clean construction of
the turbulent diffusion scheme, updates the internal state
the scheme, as we will show in section 4.
st
.
and outputs a ”stratiform” condensation flux Fvc
The scheme assumes a triangular probability distribution function for the total water specific content inside the 3.3.1 Precipitation
grid box. The saturation moisture is calculated, based on In the current scheme the updraught does not produce
the mean grid-box temperature T and pressure p, accord- precipitation by itself, but it contributes to the gross cloud
ing to the phase partition αi (T ). Thus the variation of condensates passed to the autoconversion routine.
the saturation moisture following the local increase of
the temperature induced by condensation is neglected. If
3.3.2 Mass-flux transport scheme
heating by condensation becomes significant, the calculation becomes invalid. Currently we limit the resolved In (Gerard and Geleyn, 2005), we expressed the concondensation in one time step at any level to the amount tribution of the deep convection scheme to the physical
that implies a mean grid-box heating of 1K. More heat- tendency by means of a pseudo-subsidence (associated
ing would require the use of an iterative formulation; with with the channelling effect of the updraught, the actual
our limitation, we avoid unrealistic condensation and let vertical velocity in the environment being much smaller
the adjustment take place in more than one time step. than in the updraught) and a detrainment of the cloud
A linearization of the saturation moisture could also be material into the environment. Piriou (2005) and Piriou
envisaged.
et al. (2007) proposed a much more direct formulation
At the level of the triple-point temperature, the melt- through the convective transport fluxes Jψcu and the net
ing of the precipitation induces a local cooling; the mean convective condensation-evaporation fluxes, whose vertigrid-box temperature is lowered, as is the saturation mois- cal divergences directly contribute to the tendencies. This
ture. As a result, condensation increases, and there is sub- formulation avoids the calculation of pseudo-subsidence,
sequent reheating. To represent this accurately we would as well as the difficult problem of estimating detrainment
need an iterative calculation. We have left in our scheme while using a prescribed entrainment profile. We use it
the possibility to prevent an unrealistic peak of condensate here, as it brings a substantial gain of accuracy, and is
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Q. J. R. Meteorol. Soc. 00: 1–19 (2007)
DOI: 10.1002/qj
A PACKAGE FOR CONVECTION, CLOUDS AND PRECIPITATION
7
a more logical method now that the scheme includes an where the condensate detrainment rate Dcu is obtained by
evaluation of the condensation.
a local mass budget in the updraught. As a first guess,
we take qcD = qcu ; if it yields too large a detrainment
area
(σD > (1 − σu )), then we increase qcD . This one is a
3.3.3 Updraught profile
gross quantity whose major part will be precipitated when
The updraught profile is obtained by alternating satu- passing through the microphysical package.
rated pseudo-adiabatic ascent segments with isobaric mixing. The decrement of water vapour resulting from the
ascent equals the total condensate increment, whose accu- 3.3.6 Outputs of the updraught routine
mulation yields the ”convective” condensation flux as
The convective condensation fluxes are obtained by accuexplained below. In (Gerard and Geleyn, 2005), the conmulating the water vapour decrements △qva along the
densate staying in the updraught was estimated through
successive ascent segments. Noting Mu = −σu (ωu − ωe )
the relation
the convective mass flux, the increments to the condensa∂(qvu + qcu )
qcu
=−
,
(8) tion fluxes over a model layer are given by
∂φ
φ0
where φ0 was a cloud critical thickness beyond which
the condensate was assumed to precipitate. We apply here
the same kind of limitation to the updraught condensates,
but their excess is supposed to be detrained instead of
precipitated. We also distinguish ice clouds from liquid
clouds, with a bigger critical thickness in ice phase.
The mixing of air from the environment with the
cloud air works the same way as in (Gerard and Geleyn,
2005), with a diagnostic entrainment depending on height
and the local vertical integral of the buoyancy. The condensate contents in the updraught environment are taken
to be equal to the mean grid-box values, assuming (for
lack of a better solution) no a priori spatial correlation of
the clouds at the subgrid scale with the updraught.
The virtual temperature in the updraught Tvu and the environment Tve take into account their respective condensate contents; if these are different, the buoyancy force is
affected. A cloud layer is declared ”active” when there is
both upward buoyancy and moisture convergence.
cu
△Fvi
= −αi △qva Mu /g,
cu
△Fvℓ
= −(1 − αi )△qva Mu /g.
The updraught transport is given by
∂Jψcu
∂ψ
∂
= − Mu (ψ − ψu ) = −g
,
∂t
∂p
∂p
1
Jψcu = Mu (ψ − ψu )
g
Prognostic variables and closure
As in (Gerard and Geleyn, 2005), we use prognostic
variables for the updraught vertical velocity ωu and its
mesh fraction σu . The presence of the condensates does
not affect the shape of the equations. The prognostic
closure is based on the convergence of water vapour
towards the grid box. In (Gerard and Geleyn, 2005) we
added the contribution of the local turbulent diffusion.
Here, if we use this contribution to update the initial
profile passed to the updraught, we no longer have to
include it in the closure.
3.3.5
Detrainment area
The contents of the detrainment area must be combined
with the output of the resolved condensation scheme,
and they form the main part of the convective cloud
fraction. Hence we need an estimation of the fraction
σD of the grid-box area covered by them and of their
condensate concentration qcD . If δσD is the detrainment
area extension over the time step △t, we have
Dcu △t · qcu = δσD · qcD ,
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(11)
with Mu > 0. An implicit discretization is used to ensure
stability.
Finally, the internal variables are updated, so that
they include at this stage the effects of both resolved
and subgrid condensation. The ”convective” cloud fraction f cu = σD + σu is combined with the resolved cloud
fraction f st to yield the total cloud fraction f .
3.4
3.3.4
(10)
3.4.1
Precipitation generation
Autoconversion
This part represents the effect of the initial growth and
collision processes allowing the conversion of the cloud
particles to falling precipitation. In this section we work
with local, in-cloud specific contents, and we drop the hats
in the notation:
qi = qi /f eq ,
qℓ = qℓ /f eq
(12)
The scheme proposed by Lopez (2002) used a formula of
the type described by Kessler (1969), which is written, for
autoconversion of liquid cloud water to rain:
dqℓ
= −Eℓ (qℓ − qℓ∗ )
dt
(13)
where Eℓ is the autoconversion efficiency, and qℓ∗ is a
threshold below which no conversion occurs. A similar
formula was used for the conversion of ice to snow, but
with an efficiency Ei which was a function of the temperature. In the mixed phase, the threshold of autoconversion
(9) was simply reset to zero.
Q. J. R. Meteorol. Soc. 00: 1–19 (2007)
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8
L. GERARD
Van der Hage (1995) proposed a more general for- Hence,
mula based on the volumetric concentrations Nc = Nℓ +
Ni :
∂Nc
= −φ1 Nℓ2 − φ2 Nℓ Ni − φ3 Ni2
(14)
∂t
= −φ1 Nc2 (1 − ni )2 + Gni (1 − ni ) − φ3 n2i Nc2
where ni = Ni /Nc is the ice fraction, supposed to remain
unaffected by the processes, and G is an autoconversion
gain associated with the Wegener-Bergeron-Findeisen
(WBF) effect occurring in the mixed phase. (Because of a
lower saturation pressure, vapour condenses on ice nuclei
while evaporating from droplets: some droplets vanish
while the bigger ice particles are removed by precipitation).
Outside the mixed phase, Kessler’s formulation may
be seen as a local linearization of Eq.14. Considering
droplets of mass md , ice crystals of mass mi , and noting
ρ the local air density, we have the relations
qℓ =
N ℓ md
,
ρ
qi =
N i mi
,
ρ
αi =
qi
qi + qℓ
1 − ni
ni
∂qc
= −Gφ1 qℓ qi ρ(
+
)
∂t
mi
md
−Gφ1 ρ
=
qℓ qi
mi + αi (md − mi )
(18)
The underlined term is a function of αi . If we assume that
the ice particles are bigger at the bottom than at the top
of the mixed layer, we may replace this whole term by a
linear function of αi . This yields:
∂qc = −G0 (1 + G1 αi )qi qℓ
(19)
∂t WBF
We use this formula for the mixed phase (with the tunable parameters G0 ∼ 0.5, G1 ∼ 0.5). The autoconversion
tendencies are then
∂qℓ
∂qc ∗
= −Eℓ (qℓ − qℓ ) + (1 − αi )
∂t
∂t WBF
(20)
∂qi
∂qc ∗
= −Ei (T )(qi − qi ) + αi
(15)
∂t
∂t WBF
In the liquid phase, assuming that the average mass of 3.4.2 Collection and evaporation
individual droplets does not vary during the time step, the
linearization of the parabola from a concentration Nℓ⋄ is We assume that the generation of precipitation at any level
immediately affects the precipitation flux at all the levels
written
below. This would be the case with an infinite fall veloc∂qℓ
md ∂Nℓ
md 2
ity – but also if the autoconversion process only varies
≈
= −φ1
N
slowly in time, so that what is presently generated does
∂t
ρ ∂t
ρ ℓ
(16)
⋄
not differ too much from what was generated some time
md
N
≈ −2φ1 Nℓ⋄
(Nℓ − ℓ ) ≡ −Eℓ (qℓ − qℓ∗ )
ago and has now reached the lowest levels. The gross
ρ
2
precipitation-generation flux resulting from autoconverWe see that the autoconversion efficiency Eℓ and the sion is partitioned into a gross solid precipitation flux Ps0
threshold qℓ∗ are both proportional to Nℓ⋄ : hence it is and a gross liquid precipitation flux Pr0 , assuming that
inconsistent to set the threshold to zero, because the the melting occurs over a few levels when reaching the
linearization of the parabola at its top yields zero. For this triple-point temperature.
Precipitation falls over a fraction σP of the grid-box
reason, we propose to keep the Kessler formulation with
its threshold in pure ice and in pure liquid phases, while area, taken equal to the maximum of the equivalent cloud
finding another expression for the mixed phase, where the fraction f eq over the layers above. The instantaneous
densities ρr and ρs of rain and snow in this area and at a
WBF effect is predominant.
For the total condensate, the ice and the droplets, the model level l are then (if l represents the lower boundary
of level l):
WBF term alone is written:
∂Nc
= −Gφ1 Ni Nℓ ,
∂t
∂Nc
∂Nℓ
= (1 − ni )
∂t
∂t
∂Ni
∂Nc
= ni
,
∂t
∂t
(17)
To translate these relations to specific contents, we assume
that the mass of individual hydrometeors does not vary
during the time step: the droplets evaporate at once; the
ice particles grow at once to a precipitable size; and only
the number concentrations vary. Then
md ∂Nℓ
1 − ni
∂qℓ
≈
= −Gφ1 qℓ qi ρ
,
∂t
ρ ∂t
mi
mi ∂Ni
ni
∂qi
≈
= −Gφ1 qℓ qi ρ
.
∂t
ρ ∂t
md
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ρlr = σP
l−1
l−1
l
+ Pr0
Pr0
P l + Ps0
, ρls = σP s0
2wr
2ws
(21)
where wr and ws are the fall speeds of rain and snow
respectively. Here, contrary to the estimation of the gross
precipitation flux, we consider finite velocities (around 0.9
m s−1 for snow, and 5 m s−1 for rain).
Evaporation is likely to occur in the clear part of the
precipitating area: this is given at any level by (σP − f ).
The evaporation and collection processes are computed as in the scheme of Lopez(2002), using the densities defined by Eq. (21). The major differences are
that we impose a fixed relation between the precipitation
fluxes and the precipitation contents, and that we avoid his
expensive vertical-advection calculation.
Q. J. R. Meteorol. Soc. 00: 1–19 (2007)
DOI: 10.1002/qj
A PACKAGE FOR CONVECTION, CLOUDS AND PRECIPITATION
Another difference is in the riming process: this
converts liquid condensate to solid precipitation, which
we translate into a condensate-freezing flux followed by
a solid-precipitation generation flux. The heat associated
with this freezing may then be properly accounted for in
the scheme.
3.5
Moist downdraught
The moist downdraught is calculated after the microphysics, and results from the heat sinks accompanying the
precipitation: evaporation, melting, and vertical advection.
The downdraught occurs in the precipitation area. The vertical velocity in its environment is taken equal to the mean
vertical velocity outside the updraught, ωe .
9
at the full model levels where ωd is calculated, and ωd is
prevented from taking negative values.
The downdraught mesh fraction σd is assumed to be constant over the whole height of the downdraught, and is
obtained with the prognostic equation
Zpb
dp
∂σd
· (hd − he ) + (kd − ke )
∂t
g
pt
{z
}
|
storage
Zpb
=
Fb
pt
|
(ωd − ωe ) dp
+ ε| · MHS
{z },
ρg
g
input
{z
}
(23)
−consumption
where h is the moist static energy and k the kinetic energy.
This states that a fraction ε of the microphysical heat sink
The computation for the downdraught is similar to that
MHS either contributes to the downdraught activity (work
for the updraught, being composed of saturated pseudoof the buoyancy force Fb ) or is stored in an increase of σd
adiabatic downward segments alternating with isobaric
mixing. The entrainment rate is assumed to be constant. The local latent heat of evaporation follows the 3.5.3 output fluxes and properties
phase of the precipitation, which presents a quick tran- The transport fluxes are similar to those of the updraught.
sition at the triple point. The profile construction yields The downdraught evaporation flux induces a reduction of
a precipitation-evaporation flux. In principle we cannot the precipitation fluxes (Eq. 2)
have saturation in the downdraught, but condensate may
exist temporarily; for the virtual temperature we assume
the same condensate content as in the environment. Down- 4 Single column model tests
draught activity is decreed where there is negative buoyancy and the downdraught is colder than the wet-bulb For a first test and sensitivity study, we have performed a
set of experiments with the single-column (SCM) version
temperature of the environment.
of Aladin, using the TOGA-COARE dataset (e.g. Bechtold et al., 2000) to provide a forcing every hour. This
3.5.2 Prognostic schemes
forcing consists of mean profiles over a radius of 111km,
The downward acceleration results from the balance produced by a cloud-resolving model. This experiment is
between negative buoyancy and drag. The initial velocity suitable for observing the onset of deep convection and
of the entrained parcels is difficult to guess: the precipi- the associated precipitation. The cloud condensates are
tation itself has a downward velocity, but a return upward not coupled and are initialized to zero. The time step is
current is intertwined with it. Currently, we suppose the 10 minutes, in an Eulerian leap-frog scheme.
Fig. 3 shows the vertical profiles of cloud condenoriginal vertical velocity of the entrained material to be
ωe (the updraught environment), but it would be possible sates and of the updraught and downdraught mass-fluxes
o
to modulate this in function of the precipitation fall speed. at 3 different forecast ranges. The 0 C isotherm is at a
As the downdraught approaches the surface, its flow has height of around 4,5km, which is the lower boundary of
to bend, under the influence of the local high created by cloud ice (Fig. 3(a)). Above this level, qi takes values up
−1
the accumulation of air near the surface, and eventually to 24 mg kg . The cloud droplets are observed up to a
take the horizontal direction. To avoid the complication height of 11km, in agreement with the mixed phase delimo
of calculating three-dimensional effects, we introduce an itation between 0 and -40 C. The maximum value of qℓ
−1
additional braking term in the one-dimensional vertical can reach 85 mg kg , i.e. concentrations 3 to 4 times
equation, representing the effect of the local high. The higher than qi . Towards the end of the run (400 min), we
observe a peak of condensate below the 0o C isotherm,
equation then becomes
because the local cooling from rapid melting of the precipitation flux induces a local cold anomaly (which at
∂ωd
aωd2
= negative buoyancy − drag −
, (22) this range has extended to the mean grid-box value, in
∂t
(ps − p)b
the absence of horizontal transport in the SCM) reducwhere ps is the surface hydrostatic pressure, ωd the (abso- ing the saturation moisture. A similar peak appears in the
lute) downdraught vertical velocity (positive downwards, updraught mass flux (Fig. 3(b)), the locally cooler enviin Pa s−1 ). We presently use b=2 so that a represents a ronment increasing the buoyancy. The updraught mass
reference pressure thickness above the surface for decel- flux reaches values around 0.1 kg m−2 s−1 . The downerating ωd . Note that ps is always higher than the pressure draught mass flux is around one-tenth of the updraught
3.5.1
Downdraught profile
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Q. J. R. Meteorol. Soc. 00: 1–19 (2007)
DOI: 10.1002/qj
10
L. GERARD
Figure 3. Evolution of SCM profiles in time. (a) Cloud ice (upper part) and droplets (lower part) at forecast ranges 100 min (dashdot), 250min (dash) and 400min (solid) (mg kg−1 ). (b) Updraught mass flux (kg m−2 s−1 ), same ranges. (c) Downdraught mass flux (kg
m−2 s−1 ). The horizontal dotted lines represent the limits of the mixed phase (0 and -40o C).
Figure 4. (a) Contributions to cloud fraction: σD (dot), f cu (alternated dashes), f st (dash-dot), f (dash), f eq (solid). (b) Evolution of
surface precipitation: total and subgrid parts, with time steps 600s (solid/dash) and 180s (alternate dash/dash-dot). (c) Updraught mass flux
after 6:40h forecast, with time steps 600s (solid), 300s (dash) and 180s (dash-dot).
mass flux, and its top is much lower. For the 400 min
range we observe that a first downdraught, starting at
around 5.5 km, is stopped when passing the 0o C isotherm,
because the local cold anomaly cancels the negative buoyancy. A new downdraught flux restarts immediately below,
extending down to the surface. The reduction of the mass
flux towards the surface is associated to the deceleration
in the downdraught prognostic equation (22).
The components of the cloud fraction at a range
of 400 min are shown in Fig. 4(a). The subgrid cloud
fraction f cu = σD + σu ∼ 0.4 is close to the detrained
fraction σD . The resolved fraction f st ∼ 0.6 to 0.8 is
here slightly larger. The total f is computed with Eq. (1).
The solid line shows the equivalent cloud fraction used
to estimate intensive condensate concentrations (Eq. (5)):
this is always smaller than f and it allows us to account for
more important concentrations in the convective clouds
than in the resolved ones.
Fig. 4(b) and (c) show the effects of varying the integration time step. With a shorter time step the subgrid
precipitation (Fig. 4(b)) is weaker, associated to a smaller
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updraught mass flux (Fig. 4(c)). However, the total precipitation is nearly unchanged, because there is a compensation between the subgrid and the resolved condensation
schemes. At the beginning of the run, there is only subgrid precipitation; the resolved precipitation starts after
around 3 hours’ integration. In three-dimensional tests, we
observed a similar delay of the cloud condensates and precipitation when the convective parametrization is switched
off (see Fig. 10). This long spin-up of the resolved scheme
also explains the small deficit of precipitation in the first
three hours here, when using the shorter time step.
We mentioned above that the clean construction of
the scheme ensures the conservation of the verticallyintegrated moist static energy in the updraught (and it
is also the case for the downdraught). Fig. 5 shows the
contribution of the updraught to the apparent heat source
Q1 and moisture sink Q2 , as introduced by Yanai et al.
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DOI: 10.1002/qj
A PACKAGE FOR CONVECTION, CLOUDS AND PRECIPITATION
11
Figure 5. SCM profiles after 6:40 h run. Contribution of the updraught to Q1 and Q2 (K h−1 ). (a) Q1 (dash), −Q2 (dot), Q1 − Q2 (solid).
(b) Decomposition between the condensation-evaporation term (dash), and the transport terms for dry static heat (solid) and moisture (dash
dot).
(1973). Here, they are computed as:
∂(s′ ω ′ )
∂s
cp Q1u =
= L(c − e) −
,
∂t u
∂p
∂((Lq)′ ω ′ )
∂(Lqv )
= L(c − e) +
.
cp Q2u = −
∂t
∂p
u
(24)
where (c − e) is the net condensation-evaporation and
index u stands for updraught. The inclusion of the latent
heat in the vertical divergence is necessary to take into
account its variation with the local temperature and phase.
We use here the pressure on the vertical axis, to make it
more apparent that the vertical integral of Q1 − Q2 (Fig.
5a) is zero:
Zt
b
dp
1
(Q1u − Q2u )
=
g
cp
Zt
∂h dp
= 0.
∂t g
b
(25)
(Gerard and Geleyn, 2005). The tuning is quite indirect,
as appears on Fig. 6(a): dividing the minimum entrainment by two increases the maximum entrainment in the
lower layers, and multiplying it by two has the opposite
effect. The updraught mass flux (Fig. 6(b)) shows a small
increase where the entrainment is bigger, a small decrease
where it is smaller. The updraught vertical velocity wu
(Fig. 6(c)) is decreased by a bigger entrainment (because
the entrained air has to be accelerated), and increased by a
smaller one. All these effects are small and have very little
impact on the surface precipitation.
The downdraught closure (Eq. 23) assumes that the
downdraught uses a fraction ε of the cooling associated to
the precipitation evaporation, melting and transport. This
parameter has a direct impact on the downdraught mass
flux, as illustrated in Fig. 7(a). However, in SCM, the
impact on the precipitation evolution is unclear, and very
small.(Moreover, the downdraught remained very weak in
these experiments.) We observe a greater impact in threedimensional runs (see section 5.2).
To see an example of the microphysical tunings,
we take a closer look at the new parametrization of the
Bergeron-Findeisen effect we have developed (Eq. (19)).
Operationally we use G0 =0.5 and G1 =0.5. Setting G0 =0
(Fig.7(b)) deactivates the parametrization. There is then a
significant accumulation of cloud ice between 6 and 9 km.
On the other hand, with G0 =1, the cloud ice is reduced to
nothing at those levels.
Fig. 7(c) keeps G0 =0.5. With G1 =0, the downward
decrease of qi is slow, while with G1 = 1 it is a little too
quick. Again, the value 0.5 seems a good compromise.
The other parameters in the microphysics have been
tuned in a similar manner, to obtain realistic profiles
of condensates. None of them significantly affects the
surface precipitation.
This equation states that the updraught induces a vertical
reorganization of moisture and heat, while conserving the
total moist static energy. (The limits of the integral are
the bottom and the top of the updraught.) Fig. 5(b) shows
that the large variations along the vertical are associated
with transport. Vapour and heat are transported from levels 900-1000 hPa to the levels 750-900hPa, and from levels 650-750 hPa to the levels 550-650 hPa, concerned by
the cold (and dry) pool induced by precipitation melting;
conversely, the pseudo-subsidence in the updraught environment cools and dries levels at 650-750 hPa.
Further diagrams, including ones showing the evolution of the internal moisture variables and the temperature
tendencies are given in section 5, for a three-dimensional
experiment.
The sensitivity to the tuning of the parameters
remains quite limited. To illustrate this, we present effects
of the updraught entrainment, the downdraught closure 5 Three-dimensional validation
and the Bergeron-Findeisen parameters.
In Fig. 6 we have tried to modify the settings of Deep convection is composed of cells with a diameter
the updraught entrainment. The equations were given in of a few kilometers, which may interact with each other
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Q. J. R. Meteorol. Soc. 00: 1–19 (2007)
DOI: 10.1002/qj
12
L. GERARD
Figure 6. SCM profiles after 6:40 h run. Variation of entrainment parameter: reference (solid), En × 2, (dash), En /2 (dash-dot). (a)
relative entrainment profiles; (b) updraught mass flux profile (kg m−2 s−1 ); (c) Updraught vertical velocity (m s−1 ).
Figure 7. SCM profiles after 6:40 h run. (a) Effect of downdraught closure parameter (Eq. 23) on downdraught mass flux (kg m−2 s−1 ).
Reference profile : ε=0.25, (solid), ε=0.5 (dash). Effects of the Bergeron-Findeisen parameters (Eq. 19) on the cloud condensate profiles
(mg kg−1 ): (b) G0 =0: qi (solid), qℓ (dots), G0 =1: qi (dashes), qℓ (alternate dashes). (c) G1 =0: qi (solid), qℓ (dots), G1 =1: qi (dashes), qℓ
(alternate dashes).
in a wider convective system. In models with grid boxes
longer than 7–10km, it can be assumed that the convective
cells are subgrid, and their effects on the resolved model
variables must be computed by a parametrization. However, the effects of the convective system extend over a
much wider area, as far as the Rossby radius of deformation (Mapes, 1998). Unlike turbulence, convection is
determined by both large-scale and local phenomena and
the coupling of the parametrization with the resolved processes (which also provides the closure of the scheme)
is essential. For this reason, the functioning of a package
including a convective parametrization cannot be assessed
thoroughly with single-column tests.
With grid-box lengths below 2 km, one usually
assumes that the convective cells are resolved by the
model grid. In this case, deep-convection parametrization
may be suppressed; even so, we will show below that it
can be worth keeping it.
At intermediate resolutions (between 7 and 2km) , a
parametrization is required, but it is complicated by the
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fact that a significant part of the process already contributes to the resolved cloud and precipitation processes.
Our integrated package attempts to solve these difficulties,
apparently with some success.
Our goal was to access high-resolution operational
numerical weather prediction while maintaining consistency across the whole range of resolutions, from the
coarser (30–100km) to the finer (kilometre-scale) ones.
5.1 Convective case over Belgium: horizontal fields
Intense convective showers and lightning were observed
in Belgium on 10 September 2005 between 1600 and 2230
UTC. The satellite picture (Fig. 8(a)) shows the complex
low-pressure zone, with a main depression (1005 hPa) on
the Golfe de Gascogne, prolonged by a trough (1007–
1010 hPa) over western and central Europe. During the
afternoon, a more stable flow from northeast remained
over Denmark and the Netherlands, as far south as northern Belgium. The radar pictures (Fig. 8(b)) shows a mesoscale convective system moving slowly to the northwards.
Q. J. R. Meteorol. Soc. 00: 1–19 (2007)
DOI: 10.1002/qj
A PACKAGE FOR CONVECTION, CLOUDS AND PRECIPITATION
13
(b)
(a)
Figure 8. Convective case of 10/9/2005, (a) NOAA infrared satellite image at 16:25 utc (b) 1h accumulated radar picture at 18:00 utc
(b)
(a)
5 m/s
max=41.7, mean= 0.46
max=12.1, mean= 0.37
0.2
3
6
9
12
15
18
21
24
27
30
5 m/s
max=23.3, mean= 0.51
5 m/s
(e)
(d)
max=16.2, mean= 0.55
(f)
5 m/s
max=36.1, mean= 0.71
5 m/s
5 m/s
(c)
max=38.4, mean= 0.53
0.2
3
6
9
12
15
18
21
24
27
30
Figure 9. Convective case of 10/9/2005, 6-hour forecast (for 18:00 utc). 1-hour cumulated precipitation (mm), mean sea-level pressure
(hPa), 10-m wind. (a, b, c) Full package; (d, e, f) no convection (see main text). Resolution: (a, d) 6.97 km; (b, e) 4.01 km; (c, f) 2.18 km.
With the operational configuration (using the diagnostic convection and condensation schemes with no microphysics), the model missed the event completely.
We ran a set of experiments over a domain of around
700 × 700 km with 41 vertical hybrid levels, starting from
the analysis of 1200 UTC. The model was coupled to the
results of the Aladin-France limited-area model, running
at a resolution of 9.5km, itself coupled on the Arpège
global circulation model. The coupler models where not
rerun with our package. The horizontal resolutions were
6.97 km (time step 300s), 4.01 km (time step 180s) and
2.18 km (time step 100s). All other tunings were kept
identical; in particular, the model dynamics was kept
hydrostatic.
We compare the results obtained when switching off
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the updraught and the downdraught parametrizations, so
that the resolved condensation alone feeds the microphysics (’no convection’ – lower three panels of Fig. 9),
with those with the complete integrated package (’full
package’ – upper three panels of Fig. 9). The cloud condensate variables are initialized to zero (the coupling models, using diagnostic cloud schemes, had no such variables); there is actually no need to couple them because
their advection has a minor impact.
Comparison to the observed radar images The 1hour accumulated precipitation on the radar image (Fig.
8 (b)) shows some very narrow and intense maxima. Such
maxima are also observed on the 2 km forecasts (Fig. 9
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(a)
(b)
(c)
(d)
(e)
(f)
Figure 10. Spin-up of the total surface precipitation, resolution 2.18 km. (a,b,c) no convection, 2h, 3h and 4h forecast range. (d,e,f) full
package, 2h, 3h and 4h forecast range. Same palette as Fig. 9.
(c)), though their location is a little different. Note that
with no convection (Fig. 9 (d) and (e)), the precipitation at
7 km and 4 km stays essentially at the borders of Belgium,
while the full package (Fig. 9 (a) and (b)), lets it enter
more inside the country (also verified at other forecast
ranges), in agreement with the 2 km forecast (Fig. 9 (c)
and (f)).
At 2.18 km resolution, we still observe differences
between the full-package run and the run with no convection. The location and geometry of the events is very
similar, but the widening of the precipitation area is still
evident. The source of the difference here is the convective transport, which is not well represented by the hydrostatic model dynamics. The prognostic convection scheme
computes vertical acceleration and can handle some nonhydrostatic effects whereas the resolved scheme alone,
with the hydrostatic model dynamics, cannot.
At 2.18 km resolution with the full package, the convective condensation is of the same order of magnitude as
the resolved part, while their sum (Fig. 9(c)) is is comparable to what is obtained with no convection scheme (Fig.
9(f)). This demonstrates ’good collaboration’ between the
two schemes: the convective scheme supplies what the
resolved scheme does not, and conversely.
Effect of resolution on precipitation amounts The
full package produces a gradual increase in precipitation
with increasing resolution: this is consistent with the
fact that the model outputs mean grid-box precipitation
amounts, and that the area of the active systems becomes
an increasing proportion of the grid-box area as the
resolution decreases. In addition to this averaging effect,
the correlation between high moisture and high vertical
velocity is more finely represented at higher resolution:
this may imply more condensation in some places and less
at others. Finally, the model orography is less smoothed at Consistence between resolutions At some forecast
high resolution; this may also affect the results.
time ranges (not illustrated), the ’no convection’ runs produce some displacement of the maximum precipitation
between the 4 km and the 2 km resolutions, while the ’full
Structure and width of the precipitation areas With
package’ runs simply produce no such variation, but only
no convection scheme (Fig. 9 (b)), the model produces
an intensification.
unrealistically wide areas of very intense precipitation:
this is especially clear at the 4 km resolution, i.e. in the
middle of the ’grey-zone’. This corroborates the observa- Spin-up The effects of the convective transport are of
tion of Deng and Stauffer (2006), that when not handled particular importance for the model spin-up, when starting
by a specific parametrization, the convective updraughts with cloud condensates initialized to zero. Fig. 10 shows
are forced on a coarser-than-realistic scale, producing a that in the case with no convection, the precipitation is
too strong atmospheric response. The full package (Fig. 9 underestimated during the first 3 hour of forecast, whereas
(a)) does not produce this excessive behaviour.
there is an overestimation at 4 hours, when the clouds
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A PACKAGE FOR CONVECTION, CLOUDS AND PRECIPITATION
15
Figure 11. Vertical cross section (resolution 7km): (a) qi (dash) and qℓ (solid) (g kg−1 ), temperature (dot) (o C). (b) Updraught (solid) and
downdraught (dash) mass flux (kg m−2 s−1 ). (c) Profile of the updraught (solid) and downdraught (dash) vertical velocity (m s−1 ) along
the vertical of the maximum mass flux.
Figure 12. Evolution of the mean profile of the internal water variables (g kg−1 ) along the package calculations. Total cloud condensate
qc : (a) initial value (solid line), after turbulent diffusion (dot), after resolved condensation (dash), after updraught condensation (dash-dot);
(b) after autoconversion (solid), after collection processes (dash). Difference between the final and the initial condensate (dash-dot). (c)
Water vapour differences: total difference between initial and final contents (solid), part due to resolved condensation (dash), part due to
updraught condensation, transport and precipitation processes (dash-dot), part due to downdraught (dot).
have finally built up and precipitated the excess moisture
accumulated before. When the subgrid scheme is active,
we have a much quicker spin-up (around half an hour).
In this case, both schemes (subgrid and resolved) produce
precipitation from the beginning. Thus the presence of the
subgrid scheme at 2.18 km has two benefits: it allows a
quick spin-up without coupling the condensates; and it
allows hydrostatic dynamics (which is cheaper because
the calculation is simpler and the time step may be longer)
to be run.
Pressure and wind fields The wind fields remain consistent between the different resolutions. The mean sealevel pressure field is less smooth at higher resolutions;
this is a result of the detail in the surface pressure field
(with a rougher mean orography) and the surface temperature field.
5.2
Vertical profiles and cross sections
To better demonstrate the functioning of the package, we
present here the vertical structure of different features.
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We have created an east–west vertical section across
a precipitation area at 1800 UTC in the full-package
experiment at 7 km resolution. The section location is
marked with a thick black horizontal line on Fig. 9(a).
The cloud ice (Fig. 11(a)) rises up to 0.07 g kg−1 , the
cloud droplets up to 0.5 g kg−1 . We observe a mixed phase
above the 0o C isotherm. The updraught mass flux (Fig.
11(b)) is up to 6 times bigger than the downdraught mass
flux. The prognostic updraught and downdraught vertical
velocities (Fig. 11(c)) take reasonable values (maxima at
10 m s−1 and -6 m s−1 ).
Fig. 12, 13 and 14 present mean profiles over the 13
horizontal points of the vertical cross section.
For cloud condensate qc = qi + qℓ (Fig. 12(a)), the
initial profile (advected from the previous time step) is
first moved slightly upwards by the turbulent diffusive
transport. This is consistent with the upward turbulent
diffusion flux Jℓtd , shown on Fig. 13(b).
The resolved condensation scheme increases significantly the condensate contents (Fig. 12(a)),and an additional increment is provided by the convective updraught,
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Figure 13. Mean vertical profiles of the transport flux (kg m−2 h−1 ). Fluxes are counted positive downwards. Water vapour: (a) Jvtd (dashdot), Jvcu after updraught (dash) and final (solid). Cloud condensates: (b) Jℓtd (dot) (Jitd ∼ 0 not drawn), Jℓcu after updraught (dash) and
final (solid), Jicu (dash-dot). (c) Jitd+cor (solid), Jℓtd+cor (dash).
Figure 14. Mean vertical profiles: components of the temperature tendency. (a) convective transport (solid), turbulent diffusion (dash).
(b) convective condensation (dash), resolved condensation (dot), liquid to ice conversion (dash-dot), total (solid). (c) total phase changes
(dash), total transport (dot), brought by precipitation flux (dash-dot), final tendency (solid) associated to the moist processes.
also in the upper part. Fig. 13(b) shows that the updraughts
transport condensates upwards (Jℓcu ).
Fig. 12(b) shows that the autoconversion process
has reduced qc , and the collection processes reduce it
further. The relative differences between the final and
initial values of qc remain less than 10%.
The relative variations of the water-vapour profile are
small, so we prefer to plot the differences in qv between
different stages of its internal evolution (Fig. 12(c)).
The turbulent diffusion and the resolved condensation induce a decrease of qv in the cloud, but also a a
slight increase of qv at the lower levels, corresponding to
the upward (and upwards-converging) turbulent diffusion
flux Jvtd , shown on Fig. 13(a).
The updraught and collection-evaporation processes
in the microphysics result in a further decrease higher up
(clearly associated to the fact that the updraught condensates vapour at higher levels). We see on Fig. 13(a) that the
updraught induces a negative (upward) transport of moisture at medium levels, but this moistening is completely
cancelled by the larger drying by convective condensation
flux, which is around 1 kg m−2 h−1 at level 21 and reaches
3.5 kg m−2 h−1 at level 33.
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After the subsequent downdraught, there is a further
decrease in the lower levels, and an increase above (Fig.
12(c)): this is associated with the downdraught-induced
circulation, which brings drier air from above to the
lower layers. This is also seen on Fig. 13(a), where the
effect of the downdraught is the difference between Jvcu
after the updraught and at the end: there is a significant
negative moisture flux in the lower layers associated to the
downdraught. The difference between the final and initial
qv is also plotted on Fig. 12(c).
The vertical transport fluxes of cloud droplets by turbulent diffusion and by the updraught and downdraught
circulations are shown on Fig. 13(b). The upward transport of ice by the updraught, Jicu , remains less than the
transport of droplets, Jℓcu . The downdraught reinforces
the upward transport of droplets at levels 26 to 36; since
we assume that there is no condensate within the downdraught itself, the effect on condensate transport is entirely
associated to its upward return current.
Fig. 13(c), shows the final profiles of Jitd+cor =
Jitd + Jicor , Jℓtd+cor = Jℓtd + Jℓcor : we observe that there
has been a phase correction between levels 17 and 26,
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A PACKAGE FOR CONVECTION, CLOUDS AND PRECIPITATION
(b)
min= 0.0, max= 1.0
mean= 0.27
(a)
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
17
Figure 15. (a) NOAA infrared satellite image at 17:40 utc (b) 9 km forecast for 18:00 utc: total cloud fraction.
where some liquid water had to be frozen, to maintain our
statistic profile of the ice mixing ratio αi (T ).
The downdraught has a significant impact on reducing the precipitation flux – a decrease of 0.4 kg
m−2 h−1 bringing the surface precipitation to around 5.5
kg m−2 h−1 , while the evaporation in the microphysics
remains around 10−5 kg m−2 h−1 .
Finally, Fig. 14 shows different components of the
temperature tendency. The effect of the net ’convective’
transport (updraught plus downdraught) of sensible heat
(Fig. 14(a)) may be compared to Fig. 5(b): here, the
cooling of the layers 33 to 41 is due to the downdraught,
which brings colder air from above. The other peak of
cooling, at levels 27 to 33, below the 0o C isotherm (level
27), is induced by the updraught circulation, which starts
(following the locations in the section) between level 36
and 33: so the updraught entrains warm air from these
levels upwards. The levels above 27 are warmed by the
warmer air which is detrained from the updraught.
On Fig. 14(b), the heating associated with the convective condensation is situated higher than the part due
to the resolved condensation. The effect of conversion of
liquid to ice (Bergeron effect and riming) is small (0.1 K
h−1 ).
The totals of Fig. 14 (a) and (b) are shown on Fig.
14(c), together with the cooling by the precipitation flux
(strongest at level 28, from precipitation melting). The
total tendency due to the moist processes is here much
greater than that due to radiation (not illustrated).
5.3
A frontal case
As a further test, we choose a very active cold front at the
border of Bohemia on 8 July 2004. Fig. 15 compares the
satellite picture and a cloudiness forecast at a resolution
of 9.0 km, with 43 vertical levels.
The 6-hours accumulated rain by rain-gauges (Fig.
16(a)) amounted up to 72 mm in one station, and 15–49
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mm in several others, with quite a big variability between
the stations. The forecast at 9 km resolution ( Fig. 16(b))
yields high amounts, up to 47.7 mm along the border
between Czech Republic and Germany.
When the mesh size is reduced to 4.5 km (Fig. 17(a)),
the maximum value reaches 55 mm at the same location;
and with a mesh size of 2.28 km, it reaches 75 mm (Fig.
17(b)) – agreeing well with the observations. There is a
gradual increase of the grid-box average precipitation with
resolution, for the same reasons as explained in section
5.1: reduction of the averaging area, better representation
of the correlation between high moisture and vertical
velocity,and more detailed model orography. Again, the
4.5 km forecast, which is in the middle of the ’grey zone’,
stays completely consistent with the forecasts at coarser
and finer resolutions.
6 Conclusions
We propose a practical solution to the delicate problem
of the coherent treatment of the condensation, clouds and
precipitation at all resolutions in an operational numerical weather prediction model. We have shown how this
development fits into a logical evolution towards unified
physics. We have presented the essential features that
allow us to combine the subgrid and resolved contribution
at all scales: the cascading approach and clean separation
of closure contributions; the prognostic approach; the production of condensates by the updraught, which is closed
by moisture convergence; and the interface through massflux transport and condensation fluxes. Besides these features, we have also included a treatment of the BergeronFindeisen effect, and a prognostic approach involving a
moist downdraught that is completely independent of the
updraught.
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(b)
min=−0.0,
max=47.7
mean= 6.49
0.1
0.3
1
4
10
20
30
40
60
80
Figure 16. (a) rain gauge observations; (b) 9 km forecast for 24:00 utc: 6h-accumulated precipitation.
0.1
0.3
1
4
10
20
30
40
60
80
(b)
min=−0.0,
max=75.6
mean= 6.53
(a)
min=−0.0,
max=55.2
mean= 7.59
0.1
0.3
1
4
10
20
30
40
60
80
Figure 17. Forecast for 24:00 utc: 6h-accumulated precipitation. (a) resolution 4.54 km (b) resolution 2.28 km
We have assessed the general behaviour in single
column model and shown the effects (with all restrictions applying to SCM) of the most sensitive tunings:
the entrainment in the updraught; the downdraught closure; and as an example for the microphysics, the new
Bergeron-Findeisen parametrization that we have introduced. All these tunings have very little impact on the surface precipitation. Important time-step variations changed
it only slightly, in the transition part associated with the
long spin-up of the resolved condensation. We observe
good ’teamwork’ between the resolved and the subgrid
parts, in two ways: first, the smaller subgrid precipitation
at short time steps is compensated by a larger resolved
precipitation; secondly, the long spin-up of the resolved
precipitation (around 3 hours) is compensated for by an
increased subgrid precipitation at the beginning of the run.
Three-dimensional tests confirm the behaviour of our
scheme, in that forecasts at ’grey zone’ resolutions are
consistent with forecasts at coarser and finer resolution.
Additional tests (not shown) performed at the coarser
resolution of 40 km also behaved consistently.
We have used the hydrostatic version of the Aladin
model for all of our runs. Recent research on the dynamics in the Aladin community has shown that the nonhydrostatic effects do not fundamentally change the
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results (at least in dry situations) for resolutions coarser
than 1 or 2 km. Here, we observed that the use of our prognostic scheme, which is internally non-hydrostatic, while
keeping hydrostatic model dynamics, might be a cheap
alternative to using the non-hydrostatic dynamics at 2-km
resolution at least outside mountainous regions.
Further work is being initiated to use a prognostic
mixing following developments of Piriou (2005), a more
complete microphysics (with 3 prognostic falling species),
and refining the triggering of convection. An adapted
version of our package is being installed in the frame of
the international project ALARO-0 aiming at producing
operational forecasts at resolutions between around 4 km
and 10 km within the Aladin community.
Acknowledgements
Many thanks to Jean-François Geleyn and to all those who
gave me advices or test materials. I also want to thank the
two anonymous reviewers for their constructive criticisms.
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DOI: 10.1002/qj