10
Large-scale precipitation
1 Description of the scheme
This scheme results from the statistical precipitation scheme of Smith (1990).
It must be associated with the statistical cloud scheme (ACNEBR, see Chapter 9). This scheme has been used in the former versions of Arpege-climat.
Ricard (1992) explained why the first tests of the Kessler-type precipitation scheme (1969), did not give satisfaction when the old cloud scheme
ACNEBT were replaced by ACNEBR. The cloud amount was too weak,
reaching 30% in global average. The Kessler-type scheme type eliminated
any supersaturation at the mesh scale. Coarse tests to allow supersaturation
up to 110% showed positive impacts and it was decided to connect the rate
of precipitation to the liquid water amount which is calculated by statistical
scheme ACNEBR, thanks to Smith (1990) scheme, which itself results from
a simplification of the Sundqvist (1978) scheme.
Thus, in ACPLUIS, one takes into account the quantity of condensate (PQLIS=liquid+solid) and of stratiform cloudiness (PNEBS) in each mesh, both
already calculated by statistical scheme ACNEBR. ACPLUIS thus fits in
same approach as the statistical cloud generation scheme, in the sense that
it includes the concept of sub-grid variance.
The atmosphere is scanned from top to bottom, in the direction of precipitation. These calculations led in a layer cannot be independent of calculations
of the directly upper layer, because there is a mass transport. Therefore,
there is a necessarily transmission of the information from layer to layer
(precipitation flux and proportion of snow).
The random overlap assumption is made here, which supposes that the precipitation flux coming from the upper layer is the same in term of unit of
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10. Large-scale precipitation
mesh area as in term of unit saturated (or unsaturated) mesh area. This assumption makes it possible to treat independently the saturated part and the
unsaturated part of the mesh, since each one receives the same precipitation
flux at the top of the mesh.
It should be noted that in Arpege-climat, the maximum-random overlap assumption is used in ACNEBR to calculate “radiative cloudiness”, (for
transmission to the radiation scheme).
The parametrization used is mainly that of Smith (1990), however the Kessler
(1969) formulation concerning the evaporation of precipitations has been
used.
Three processes are involved: precipitation, evaporation and melting or freezing. They are dealt in the next three sections.
2 Precipitation in a cloud layer (Smith)
The Smith (1990) model requires a parametrization of the reduction in cloud
water due to precipitations. The conversion rate of cloud water into precipitation depends on the phase of water. One supposes here an abrupt transition between the liquid state and the solid state at the temperature of triple
point Tt .
One uses a partition of condensate qc (PQLIS) in liquid water ql and solid
water qi
f (T ) = δTt
where Tt is the temperature of triple point and δTt Heavyside function ,
yielding 1 if T < Tt , 0 if not.
It should be noted that in the other parts of the code of Arpege-climat
one uses a Gaussian and continuous function in the transition from the liquid
form to the solid form (with a characteristic width ∆T = 11.82 K (RDT)
and with a coexistence of the two phases between Tt − ∆T and Tt ).
The value of f (T ) being known, one calculates the partition liquid/ice qc by:
• ql = [1 − f (T )] qc
• qi = f (T ) qc
10. Large-scale precipitation
157
2.1 Liquid phase
The parametrization of the tendency of liquid cloud water due to precipitation is defined as (Smith, 1990):
"
(
"
∂ql
ql /ns
= − Ct 1 − exp −
∂t
Cw
2 #)
#
+ Ca FPh ql = − [ A ] ql
(1)
where
• Ca , Ct and Cw are constants, respectively TCA, TCT and TCW in
namelist
• ns is stratiform cloud fraction (PQLIS) and ql stratiform liquid water
• FPh is precipitation flux per unit area coming from the upper layer
The exponential factor inhibits the liquid water conversion per unit of cloud
mesh ql /ns into precipitations, if this one is weak compared to Cw .
2.2 Solid phase
The tendency of cloud solid water (per unit of mesh) due to precipitation is
expressed in the following way (Smith, 1990):
∂qi
=
∂t
FPh
ρ ∆z
−
vf
∆z
qi = B − D q i
(2)
where
• ∆z is layer depth
• vf = TVF is falling speed for ice/snow. One supposes here that TVF =
1 m/s
.
2.3 Precipitation flux at the base of the saturated part of the
mesh
In the cloud part, the principle consists in calculating the tendencies of
moisture due to the process of condensation in the sub-grid cloud, then to
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10. Large-scale precipitation
convert these tendencies into precipitation flux. One can write the formulas
(1) and (2) in the form:

∂ql



 ∂t
= −A ql



 ∂qi
= B − D qi solid case
∂t
liquid case
(3)
where A, B and D are defined in Equations (1) and (2).
In the current version of ACPLUIS, the total value qc = ql + qi intervenes
everywhere in Equation (3). One obtains tendencies (∂qc /∂t)(l) = −A qc and
(∂qc /∂t)(i) = B − D qc which undergo the later recombination:
∂qc /∂t = [1 − f (T )] (∂qc /∂t)(l) + [f (T )] (∂qc /∂t)(i)
(see the following sub-section). One should use separately ql and then qi in
System (3), to obtain directly
∂ql /∂t = −A ql and ∂qi /∂t = B − D qi
as envisaged in Smith (1990). There would be more simply
∂qc /∂t = ∂ql /∂t + ∂qi /∂t.
The scheme remains consistent as long as it does not make coexist the solid
and liquid phases, as it is currently the case in this part of the code which
uses Heavyside function f (T ) = δTt .
For numerical stability reasons, one uses an implicit scheme where the second
member of each one of these two equations is taken at time t + ∆t. For
example, the first equation of System (3) is written as:
∂ql
∂t
(t)
"
= −A (ql )t+1
∂ql
= −A (ql )t + ∆t
∂t
(t) #
(4)
One then obtains classically the Smith (1990) equations of moisture tendency
in their implicit form:

∂ql





 ∂t
−A qlt
≡
1 + A ∆t
=



∂qi



B − D qit
=
≡
1 + D ∆t
∂t
∂qc
∂t
∂qc
∂t
liquid case
(l)
(5)
solid case
(i)
10. Large-scale precipitation
159
2.4 Recombination of the total condensation flux
With the above values for function f (T ), the total tendencies (solid and
liquid) due to the condensation of the Smith (1990) scheme are obtained
starting from the combination of the partial tendencies given by System (5):
∂qc
∂qc
= [1 − f (T )]
∂t
∂t
+ [f (T )]
(l)
∂qc
∂t
(i)
The conversion of the moisture tendencies into precipitation flux is given by:
∂qn
g
=
(FPs − FPh )
∂t precip ∆p
(6)
where ∆p is the layer thickness in pressure (positive).
In the code, the moisture tendencies are positively counted if they contribute
to precipitations.
One keeps the total precipitations flux of Smith for the values of qv which
are greater than QSMIN = 10−4 . For values less than QSMIN, a Kessler
(1969) scheme is coded and replaces that of Smith (1990). Consistently, the
same threshold QSMIN is used in ACNEBR to put cloudiness at a residual
value (10−12 ) if qv < QSMIN. The motivation of this threshold is to avoid
having too strong cloudiness and precipitation in higher troposphere and polar stratosphere. Elsewhere, for high altitudes, safety relies on a limitation
along the vertical which is ensured through maximum levels given in NAMTOPH (where ETNEBU=100 hP a is the highest level for both ACNEBR
and ACPLUIS).
3 Evaporation in the unsaturated parts (Kessler)
A precipitation flux falling into an unsaturated zone in the lower layer is
reduced at the base of this sub-mesh by evaporation phenomena for which
Kessler (1969) proposes the following parametrization:
√
∂ FP
= Evap (qw − q)
∂ (1/p)
where
(7)
Evap = CEvap · [1 − rme (1 − RV )]
Here, CEvap = EVAP = 0.48 107 is an empirical coefficient, RV = REVGSL =
80 is the ratio of evaporation speed between snow and liquid water, and rme
are the fictitious proportion of snow in precipitations.
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10. Large-scale precipitation
By integrating Equation (7) between the top and the base of each layer, one
obtains FPc , precipitation flux, at the base of the layer per unit of area of
unsaturated mesh:
FPc =
q
FPh − Evap (qw − q)
∆p
p2
2
(8)
Applying these two processes, namely condensation and evaporation, leads
to the knowledge of the precipitation flux at the base of the layer per unit
of area of unsaturated mesh (FPc ) and of saturated mesh (FPs ). Obtaining
the precipitation flux at the base of the layer per unit of area of mesh (FPb )
is done by:
FPb = ns FPs + (1 − ns )FPc
(9)
4 Melting or freezing (Kessler)
In the configuration which consists in taking into account the cryoscopic
cycle, the equation governing the phenomenon of snow melt or freezing of
liquid water is:
∂rf
T − Tt
= Fmelt √
,
∂ (1/p)
FP
(10)
where Fmelt represents the melting of precipitation:
Fmelt = CF onte · [1 − rme (1 − RV )] .
(11)
Here, CF onte = FONT = 0.24 105 , with the same definitions as previously
for RV = REVGSL = 80 and for rme .
The component of the snow proportion (rf ) due to melting or freezing of
new precipitation results from the integration of Equation (10) between the
top and the base of the layer, whose result is:
rf = Fmelt
Tt − T
∆p
p
0.5( FPh + FPb ) p2
p
(12)
To obtain the actual snow proportion (rn ), one adds to rf the snow proportion in the precipitation flux (rg ).
The term rg is calculated from the snow proportion rn of the upper layer
and precipitation flux. It depends on the part of the mesh in which it is
calculated: saturated part or unsaturated part.
10. Large-scale precipitation
161
4.1 Saturated part of the mesh
Two cases arise according to whether one is above or below the triple point:
• If T > Tt
There cannot be condensation in solid phase. If vapor condensates, liquid
water is formed. Noting rn the snow fraction in the upper layer and FPbc
the precipitation flux at layer base, due only to condensation in the cloudy
sub-mesh, the snow fraction produced by precipitation (rg ) is:
rg = rn
FPh
FPbc
(13)
• If T < Tt
If there is condensation, snow is formed; rg is expressed using the
equation:
rg = 1 − (1 − rn )
FPh
FPbc
(14)
4.2 Unsaturated part of the mesh
The snow proportion due to precipitation does not change in the non-cloudy
zone: the evaporation and the sublimation of precipitation are done in the
same proportions.
The process of melting or freezing intervenes in the determination of the
nature of the precipitation. One obtains stratiform precipitation fluxes in
liquid form and solid form by multiplying the precipitation flux at the base
of each layer by (1-rn ) and rn respectively.
One uses here a partition of condensate in rain and/or snow according to a
Gaussian and continuous formulation:
n
h
io
f (T ) = δTt 1 − exp −(T − Tt )2 /(2 ∆T 2 )
One uses for ∆T the difference between the temperature of triple point and
the abscissa of the maximum of ew (T ) − ei (T ), i.e. ∆T = 11.82 K, which
provides a good approximation of the integral. The variable used in the code
for ∆T is noted RDT in YOMCST.
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10. Large-scale precipitation