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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Cloud and large-scale precipitation processes are described by prognostic
equations for cloud liquid water/ice and cloud fraction and diagnostic relations
for precipitation. The scheme is described in detail in
Tiedtke (1993).
The grid-mean specific cloud water/ice content is defined as
where  is the density of cloud water,  is the density
of moist air and  is the volume of the grid box. The fraction
of the grid box covered by clouds is defined as
Furthermore, the definition of the specific cloud water content per cloud
area (in-cloud water/ice content) is
The saturation specific humidity is expressed as a function of saturation
water vapour pressure as
where the saturation water vapour pressure is expressed with the Tetens
formula
where  and  are different depending on the
sign of  (i.e. water or ice phase with  )
In the scheme only one variable for condensed water species is used. The
distinction between the water and ice phase is made as a function of temperature.
The fraction of water in the total condensate is described as
 and  represent the threshold temperatures
between which a mixed phase is allowed to exist and are chosen as  and  . The saturation thermodynamics are calculated
according to the mixture of water and ice obtained with Eq.
(6.6) so that the saturation specific humidity becomes
where  and  are the saturation specific humidities with respect to
water and ice, respectively. The latent heat of phase changes is described
as
With these definitions and the usual assumption that clouds encountered
extend vertically over the whole model layer depth the equations for the
time change of the grid-box averaged cloud water/ice content and the cloud
fraction are obtained as
and
The terms on the right-hand side of Eq. (6.9) and Eq. (6.10) represent the following processes:
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•  ,  - transport of
cloud water/ice and cloud area through the boundaries of the grid
volume |
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•  ,  - formation
of cloud water/ice and cloud area by convective processes |
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•  ,  - formation
of cloud water/ice and cloud area by boundary-layer turbulence |
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•  ,  - formation
of cloud water/ice and cloud area by stratiform condensation processes |
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•  - rate of evaporation of cloud water/ice |
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•  - generation of precipitation from cloud
water/ice |
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•  - dissipation of cloud water/ice by
cloud top entrainment |
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•  - rate of decrease of cloud area due
to evaporation. |
The large-scale budget equations for specific humidity  , and dry static
energy  after introduction of the scheme are modified to
and
where  and  represent all processes except those related to clouds
and radiation.  is the latent heat of freezing,  is the rate of snowmelt,  and  are the radiative heating rates in cloud-free and cloudy areas. The
flux-divergence terms represent the effects of cloud top entrainment.
Clouds formed by convective processes are parametrized by considering them
to be condensates produced in cumulus updraughts and detrained into the
environmental air. This approach, besides being part of the cloud parametrization,
represents also an important extension of the model's cumulus parametrization.
It is applied for all types of convection, namely deep, shallow and mid-level.
The source of cloud water/ice content is
and the source of cloud area is described as
where  is the detrainment of mass from cumulus updraughts,  is the specific cloud water/ice content in cumulus updraughts and
 is the updraught mass flux (see chapter 5). The factor  in Eq. (6.14) appears because
updraught air detrains simultaneously into cloud-free air as well as into
already existing clouds.
This part of the scheme considers stratocumulus clouds at the top of convective
boundary layers. They are distinguished from shallow cumuli by making the
assumption, that the cloud depth must not exceed one model-layer depth.
All clouds deeper than one layer are represented as convective clouds by
the cumulus convection scheme. The scheme follows the mass-flux approach,
so that the cloud transport for moisture is written as
where  and  are updraught and downdraught specific humidity, respectively,
and  ( ) is the cloud mass flux,  being the updraught velocity and  the fractional
area of updraughts. Note that in contrast to convection, stratocumulus cloud
circulations contain roughly equal ascending and descending branches. The
cloud-base mass flux is determined by reformulating the moisture transport
at cloud base produced by the boundary layer parametrization  (see Sections 3.3 and 3.4 of Chapter 3 `Turbulent diffusion and interactions
with the surface' ) into the mass-flux concept so that
The subscripts `0' and `top' refer to model levels near the surface and
close to the cloud top (i.e. next level above cloud base), respectively,
indicating that the updraughts start close to the surface and the downdraughts
close to the cloud top. Above cloud base the assumption is made that  decreases linearly to zero at cloud
top. The net generation of cloud water/ice due to condensation in updraughts
and evaporation in downdraughts then becomes
and the source of cloud air in terms of cloud cover is
Here the formation of clouds by non-convective processes (e.g. large-scale
lifting of moist air, radiative cooling etc.) is considered. The parametrization
is based on the principle that condensation processes are determined by
the rate at which the saturation specific humidity decreases. This rate
is linked to vertical motions and diabatic cooling through
where  is the change of  along a
moist adiabat through point  ,  is the area-mean generalized vertical velocity,  is the cumulus-induced subsidence between the updraughts, and
 is the net temperature tendency due to radiative
and turbulent processes. Two cases of condensation are distinguished
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(a) in already existing clouds and |
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(b) the formation of new clouds |
Condensation in already existing clouds is described as
New clouds are assumed to form, when the grid-averaged relative humidity
exceeds a threshold value which is defined as a function of height as
where  ,  with  being the pressure
and  the pressure at the surface,  ,  is the height of the tropopause in  -coordinates and  . The increase in cloud cover is determined
by how much of the cloud-free area exceeds saturation in one time step which
in turn depends on the moisture distribution in the cloud-free area and
how fast saturation is approached. The moisture is assumed to be evenly
distributed within the range  around the mean environmental value  , while
the approach to saturation is determined by  . The increase
in cloud cover then becomes
which can be expressed in terms of grid averages (using the definition  ) as
For the application of Eq. (6.24)
at values of  close to saturation, the constraint  is imposed to ensure realistic values of  .
The generation of cloud water/ice in newly formed clouds is then
where  is the fractional cloud cover produced in the time step
by Eq. (6.24).
The scheme describes evaporation of clouds by two processes in connection
with large-scale and cumulus-induced descent and diabatic heating and by
turbulent mixing of cloud air with unsaturated environmental air.
The first process is accounted for in the same way as stratiform cloud formation
except that  . Hence
Assuming a homogeneous horizontal distribution of liquid water in the cloud,
the cloud fraction remains unaltered by this process except at the final
stage of dissipation where it reduces to zero.
The parametrization of cloud dissipation as cloud air mixes with environmental
air is described as a diffusion process proportional to the saturation deficit
of the environmental air:
where the diffusion coefficient is
The decrease in cloud cover is parametrized as
where  is the specific cloud water/ice content per cloud area
as defined in Eq. (6.3). Note that
because of Eq. (6.3) the parametrizations Eq. (6.29) and Eq. (6.31) imply a reduction in cloud area while
 remains unchanged.
Fluxes of heat, moisture, cloud water/ice, and momentum through cloud top
due to the cloud top entrainment process are described as
where  stands for any of the transported variables and  is the entrainment
velocity.  stands for the change of  between two
model levels. The parametrization of cloud top entrainment is currently
only used if the level above a cloudy model level is entirely cloud free
and if  is positive (stable
layer), where  represents the virtual dry static energy
between the two layers. There are two parametrized contributions to the
entrainment velocity
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(i) Clouds at the top of convective boundary
layers. In the case of clouds at the top of convective boundary
layers the parametrization of the entrainment velocity follows
Deardorff (1976). The entrainment velocity is represented
as |
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is the average buoyancy flux in the mixed layer
of height  and  . |
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(ii) All cloud tops. The second contribution
to the entrainment velocity is parametrized as |
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where  is the longwave radiative flux divergence and
 . |
Cloud water/ice transported into the cloud free layer above by entrainment
is assumed to evaporate immediately.
Similar to radiation, precipitation processes are treated seperately in
clear and cloudy skies. This owes to the fact that the microphysical processes
in these two regions are very distinct from each other, with conversion,
collection and accretion processes being relevant in clouds whereas evporation
of precipitation is the relevant process outside clouds. Therefore the precipitation
flux is written as
with
and
where the step function,  , marks the portion of the grid-cell containing
cloud with a condensate specific humidity  and A is the area
of the grid-cell.
The precipitation fraction in the gridbox is then described as
with
and
Precipitation sources are represented differently for pure ice clouds and
for mixed phase and pure water clouds.
The distinction is made as a function of temperature according to Eq.
(6.6). The rain and snow formed is removed from the column immediately
but can evaporate, melt and interact with the cloud water in the layers
it passes through.
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(i) Pure ice clouds. The precipitation
process in ice clouds is treated separately for two classes of particles.
The separation is made by size at a threshold of 100 µm. First
the ice water content in particles smaller than 100 µm is determined
following a parametrization proposed by McFarquhar and Heymsfield
(1997) as |
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is the total ice water content in g m-3,
 is set to 1 g m-3, b1=0.252 g m-3
and b2= 0.837. The fallout of the so diagnosed  (now in kg m-3) is treated as a
sedimentation of the ice particles with a terminal fall speed of |
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based on Heymsfield and Donner (1990);
the constants currently chosen are c1=3.29 and c2=0.16. |
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The ice content in particles larger than 100
µm |
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is converted into snow within one timestep. |
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Given the fallspeed and the separation by particle
size the contribution to  from pure ice clouds can be written
as |
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where  is the time-step of the model. For the samll
particle ice settling into cloudy area is treated as source of cloud
ice in the layer below whereas ice settling into clear sky is converted
into snow (see Subsection 6.2.3). Note that the minus sign in
the first term of the right hand side of (6.48) appears since the
fall velocity of ice is assumed to be positive downwards. |
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(ii) Mixed phase and pure water clouds.
For mixed phase and pure water clouds a parametrization following
Sundqvist (1978)
is used. The generation of precipitation is written as |
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where  represents a characteristic time
scale for conversion of cloud droplets into drops and  is a typical cloud water content at which the release of precipitation
begins to be efficient. These disposable parameters are adjusted as
follows |
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to take into account the effect of collection
of cloud droplets by raindrops falling through the cloud ( ) and the Bergeron-Findeisen mechanism ( ). Here  and  are defined as |
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where  is the local cloudy precipitation rate ( ) and  is the
temperature at which the Bergeron-Findeisen mechanism starts to enhance
the precipitation. The values for the constants are those used by
Sundqvist (1978),
namely  ,  ,  ,  , and  . |
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(iii) Evaporation of precipitation. The
parametrization of rain and snow evaporation is uncertain. A scheme
following Kessler
(1969) is used. It describes the evaporation rate as |
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where  is the clear-sky precipitation fraction. Evaporation
of rain/snow only takes place when the grid mean relative humidity
is below a threshold value. The choice of the threshold value is not
straightforward for numerical reasons. Here, the assumption is made
that the clear-sky relative humidity (= grid mean relative humidity
in the absence of clouds) that can be reached by evaporation of precipitation
is a function of the fractional coverage with precipitation of the
clear sky part of the grid-box. Hence, the threshold value is parametrized
as |
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(iv) Melting of snow. The melting of snow
is parametrized by allowing the part of the grid box that contains
precipitation to cool to  over a time scale  , i.e., |
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where  and |
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