Table of contents
Chapter 1. Overview
Chapter 2. Radiation
Chapter 3. Turbulent diffusion and interactions
with the surface
Chapter 4. Subgrid-scale orographic drag
Chapter 5. Convection
Chapter 6. Clouds and large-scale precipitation
Chapter 7. Land suface parametrization
Chapter 8. Methane oxidation
Chapter 9. Climatological data
REFERENCES
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As cloud processes are rapidly varying in time, care must be taken when
Eq. (6.9) and Eq. (6.10) are integrated over the relatively
large model time steps. Therefore terms that depend linearly on  and  are integrated analytically. Eq. (6.9) and Eq. (6.10) can be written as
where C is defined by Eq. (6.13),
Eq. (6.17), Eq. (6.21), Eq. (6.25), Eq. (6.27) and Eq. (6.29) and D is defined by Eq. (6.48) or Eq. (6.49) respectively, and
with  ,  , and  defined by Eq. (6.14), Eq. (6.18), Eq. (6.24), Eq. (6.28) and Eq. (6.31). Analytical integration of Eq.
(6.57) and Eq. (6.58) yields
and
Since there are terms in Eq. (6.57)
that depend on  and terms in Eq. (6.58) that depend on  the following
method to solve the two equations is adopted. First Eq. (6.59) is solved using the values for
 at time  . Then a time-centred value for the cloud fraction,
 is calculated as
Then Eq. (6.60) is divided by  and solved yielding a new value for the in-cloud value  which is converted into the grid-mean value by re-multiplying with
 .
Special care has to be taken in the numerical calculation of  from Eq.
(6.19). Since the saturation water vapour pressure depends exponentially
on temperature, straightforward numerical integration of Eq. (6.19) would produce large truncation errors.
Therefore the average of  over the time
step is determined by the means of moist adjustment (e.g. Haltiner
and Williams 1980). This is achieved by first extrapolating the cloud
temperature to time-level  and then adjust
temperature and moisture toward saturation conditions.
The vertical discretization of equations (6.13) and (6.14) is achieved with
a simple upstream scheme, i.e.,
and
Although two of the terms in equation (6.62) depend linearly on  it was decided to treat the convective source (like any other source
of condensate) fully explicitely, i.e., (6.62) is added into (6.57) as a
contribution to  only. For cloud
fraction it is obvious that the first term on the right hand side of (6.63)
can be added to  in equation (6.58)
wheras the second term can be split into a contribution to  and  .
It is evident from (6.24) that the stratiform source of cloud cover is quadratically
dependent on  and can therefore not easily be integrated
analytically following (6.58). To overcome this problem one factor of  is integrated into  , i.e., treated
explicitely, before carrying out the analytic integration of (6.58), i.e.,
The method to determine  and  is as follows. If precipitation is generated
in a level through the processes of autoconversion or ice sedimentation,
it is assumed to be generated at all portions of the cloud uniformly and
thus at the base of level k,  . The precipitation generated in this cloudy region is
given by:
and the cloudy precipitation flux at the base of level k is given
by  , where the twiddle symbol indicates the value of  at the top of level k. Because the cloud is assumed to be
internally homogenous, (6.65) simplifies to  , where  is the generation rate of precipitation inside the cloud. If only
accretion occurs in the clouds of level k,  equals  , the fractional area that contains cloudy precipitation
flux at the top of level k.
Because the clear precipitation flux is assumed to be horizontally uniform,
evaporation does not alter the area containing clear precipitation flux
such that  . Only in the case that all of the clear precipitation
flux evaporates in level k does  . The clear-sky
precipitation flux at the base of level k is given by  , where  is the clear-sky precipitation flux at the
top of level k, and
where  represents precipitation evaporation. Note that precipitation evaporation
is a function of  guaranteeing that
precipitation generated in a level cannot evaporate in the same level. This
will guarantee consistency with the assumption that clouds where present
fill the vertical extent of the grid cell and that horizontal transfer of
precipitation mass from cloudy to clear regions of the grid cell is not
possible.
At the interfaces between levels, precipitation mass that is in cloud of
the upper level may fall into clear air of the lower level, or precipitation
mass that is in clear air of the upper level may fall into cloud of the
lower level. Thus at level interfaces an algorithm is needed to transfer
precipitation and its area between the cloudy and clear portions of the
grid box. The algorithm is constructed by determining the amount of area
associated with each transfer and then transferring precipitation fluxes
between clear and cloudy components according to the assumption that the
precipitation flux is horizontally uniform but with different values in
the clear and cloudy regions containing precipitation.
There are four possible areas to be defined (see schematic in Figure ??):
the area in which cloudy precipitation flux falls into cloud of the lower
level, the area in which cloudy precipitation flux falls into clear air
of the lower level, the area in which clear precipitation flux falls into
clear air of the lower level, and the area in which clear precipitation
flux falls into cloud of the lower level. To determine these areas, the
cloud overlap assumption is applied to determine the relative horizontal
placements of clouds in the upper and lower levels. For the ECMWF model,
the cloud overlap assumption is expressed in terms of an equation which
relates the total horizontal area C covered by clouds in levels 1
to k (where k = 1 is the top level of the model), to the total
horizontal area cover by clouds in levels 1 to k-1:
where  is a tiny number set to 10-6. Equation (6.67) gives maximum overlap for clouds in
adjacent levels and random overlap for clouds separated by clear levels.
From this equation, one can determine the portion of clouds of the lower
level which is unoverlapped by clouds at all higher levels; this area,  , cannot have
any precipitation falling into it. Using this assumption, the area for which
cloudy precipitation flux falls into clear air of the level below is given
by:
Equation (6.68) makes the further
assumption that there is maximum overlap between the area covered by cloudy
precipitation at the base of the upper level and the portion of the lower
level cloud which lies beneath clouds in higher levels,  . With the assumption that the precipitation flux is horizontally
uniform, the amount of cloudy precipitation flux of the upper level that
falls into clear air of the level below is:
The area in which clear precipitation flux of the upper level falls into
cloud of the level below is:
which assumes maximum overlap between the portion of the cloud in the lower
level k which has cloud at some higher level other than k-1,
and the area covered by the clear precipitation flux. Again, with the assumption
that the precipitation flux is horizontally uniform, the amount of clear
precipitation flux of the upper level that falls into cloud of the level
below is:
Finally, the areas and fluxes at the top of level k can be related
to those at the base of level k-1 by:
From these equations it is obvious that total precipitation area,  , and precipitation flux,  , are conserved
at level interfaces.
After the integration of Eq. (6.60)
the fallout of condensate (represented by the term  in Eq.
(6.57)) out of model level  is determined as
The condensate falling out of model level  is then distributed
into rain, snow or cloud ice in the level below using the following assumptions:
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(i) Pure water clouds. In the case of
pure water clouds ( ) all condensate falling out of a model level is
converted into rain, i.e., |
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(ii) Mixed phase clouds. In the case of
mixed phase clouds ( ) all condensate falling out of a model level is
converted into rain or snow whereby the partitioning between the two
phases is determined using Eq. (6.6), i.e., |
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(iii) Pure ice clouds. In the case of
pure ice clouds ( ) the condensate falling out of a model level is partitioned
into a source of cloud ice in the level below and snow. There are
two sources of snow from falling cloud ice; i) all ice content in
particles larger than 100 µm is converted into snow, and ii)
of the falling cloud ice in particles smaller than 100 µm, ice
falling into clear sky is converted into snow, while ice in falling
into cloud remains cloud ice. This is implemented in the code as follows.
First (6.44) is solved to determine the ice water content in particles
smaller than 100 µm. Then (6.60) is solved for layer  using |
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 represents all sources and sinks of cloud ice not
related to precipitation processes, whereas  is the source of
cloud ice through settling of particles from the layer above (for
definition see below). Then (6.76) is solved to determine the fallout
of ice out of layer  ,  . Of the ice water content falling into layer  in particles
smaller than 100 µm, i.e.,  , the part falling
into overlapping cloud area is treated as source of cloud ice,  . The area of cloud overlap is determined as |
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where  is the change of total cloud cover from layer
 to layer  as described above. Hence, |
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With these defintions the generation of snow
in level  becomes |
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After the definitions above the precipitation
at the surface can be written as |
Since the evaporation of precipitation has a threshold value of relative
humidity at which the process should cease to exist (see quation (6.55))
an implicit treatment is applied when solving (6.54). If (6.54) is written
as
the implicit solution becomes
where  refers to the time level at the beginning of timestep  . (6.88) ensures that evaporation of precipitation never leads to
 . To ensure the maximum relative humidity after evaporation
does not exceed the threshold value defined in (6.55) the maximum change
in specific humidity is calculated as
The smaller of the values given by (6.88) and (6.89) is then chosen as the
true value of evaporation of precipitation.
After parametrizing the entrainment flux as in Eq. (6.32) and the entrainment velocity as in
Eq. (6.33) the tendency equations
for the two levels involved in the entrainment process are solved simultaneously
using an implicit formulation.
The tendency equation for the cloudy model level,  , can be written
as
where  is the flux of  taken at half-level  . A similar equation
can be written for the level immediately above the cloud,  . Since only the transport between levels  and  are considered only the flux at half level  is non-zero. The
solution for  at both model levels for time  given the values at time  can then be found
by solving the system of two linear equations
and
After the calculation of the liquid water/ice tendency and the corresponding
tendencies of temperature and moisture a final test for supersaturation
is performed. If any supersaturation is found the grid box is re-adjusted
to saturation (using the moist adjustment formulation) and the moisture
excess is converted into precipitation.
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