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Chapter 5. Convection
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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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5.3 Cloud model equations
5.3.1 Updraughts
The updraught of the cloud ensemble is assumed to be in a steady state. Then the bulk equations for mass, heat, moisture, cloud water content and momentum are
|
(5.3) |
where
and 
are the rates of mass entrainment and detrainment
per unit length, 
is the updraught
cloud water/ice content, and 
is precipitating
rain and snow in the updraughtss. The treatment of the cloud microphysical
processes is described in Section 5.6. The vertical integration of (5.3) requires knowledge of the cloud-base mass flux and of the mass entrainment and detrainment rates. Cloud-base mass flux is determined for the various types of convection from the closure assumptions discussed in Section 5.4.
Entrainment of mass into convective plumes is assumed to occur (1) through turbulence exchange of mass through the cloud edges and (2) through organized inflow and detrainment is assumed to occur (1) through turbulent exchange and (2) through organized outflow at cloud top. The superscripts (1) and (2) are used to denote the components of the entrainment and detrainment due to turbulent and organized exchanges, respectively;
|
(5.4) |
5.3.1 (a) Entrainment and detrainment rates
Turbulent entrainment and detrainment are parametrized as
|
(5.5) |
where the fractional entrainment/detrainment rates depend inversely on cloud radii in the updraughts (
) (Simpson
and Wiggert, 1969; Simpson, 1971):
|
(5.6) |
By assuming typical cloud sizes for the various types of convection, average values of entrainment/detrainment rates are defined; deep convection is assumed to have a larger radius and so a smaller entrainment rates than shallow convection. In order to keep the scheme simple we use fixed values of turbulent entrainment/detrainment rates for each of the various types of convection:
  
|
(5.7) |
For penetrative convection and mid-level convection we deliberately impose a very small value typical for tropical thunder clouds (Simpson, 1971) so as not to inhibit the penetration of clouds to large heights. For shallow convection we use a value typical for the larger trade wind cumuli (Nitta, 1975), noting that small clouds with much larger entrainment/detrainment rates which detrain immediately above cloud base are not represented in our parametrization. In order to take into account enhanced turbulence in the lower part of the clouds,
and 
are increased
in the lowest 150hPa of the cloud in the case of deep and shallow convection.
The enhancement factor varies linearly from 4 at cloud base to 1 at 150hPa
above cloud base. Turbulent entrainment is only applied over the lowest
half of the cloud layer.5.3.1 (b) Organized entrainment and detrainment
Organized entrainment is applied to deep and mid-level convection. The formulation used is discussed in Subsection 5.4.1 below.
Organized detrainment is estimated from the vertical variation of the updraught vertical velocity
, which is estimated from the budget
equation for the updraught kinetic energy
|
(5.8) |
with
|
(5.9) |
where
is the updraught kinetic energy, 
is the virtual
temperature of the updraught and 
the virtual
temperature of the environment. 
is a mixing
coefficient which is equal to the entrainment maas flux (
), or the detrainment mass flux (
) if this is
larger. As entrainment is set to zero in th upper part of the cloud layer,
use of detrainment mass flux in this region better represents the effect
of mixing and vertical pressure gradients in the upper part of deep convective
clouds, reducing vetical velocity and reducing overshoot of convective towers
into the lower stratosphere. 
(
) is the virtual mass coefficient (Simpson and Wiggert 1969),
the factor 
(
) is introduced
because the flow is highly turbulent (Cheng et al. 1980) and for the drag
coefficient a value of 
is used (Simpson and Wiggert 1969). The
value for 
is 1.875. The cloud base value of the
updraught velocity is chosen as 1 
.
enters the scheme in several ways: (i) for the generation and fallout
of rain (Section 5.6), (ii) to determine penetration
above the zero-buoyancy level and the top of cumulus updraughts and (iii)
to specify detrainment below the top of the updraught.
determines the level to which convection penetrates (where it reduced
to zero). This allows convection to penetrate above its level of neutral
buoyancy. Organized detrainment is estimated by equating the decrease in
updraught vertical velocity due to negative buoyancy at the top of the cloud
to the decrease in mass flux with height:
|
(5.10) |
This assumes that the cloud area remains constant in the detraining layer and neglects the vertical variation of buoyancy. Eq. (5.10) defines the reduction of mass flux with height, which combined with the updraught continuity equation (Eq. (5.3)) gives the organised detrainment rate.
5.3.2 Downdraughts
Downdraughts are considered to be associated with convective precipitation from the updraughts and originate from cloud air influenced by the injection of environmental air. Following Fritsch and Chappell (1980) and Foster (1958), the Level of Free Sinking (LFS) is assumed to be the highest model level (below the level of minimum moist static energy) where a mixture of equal parts of cloud and saturated environmental air at the wet-bulb temperature becomes negative buoyant with respect to the environmental air. The downdraught mass flux is assumed to be directly proportional to the upward mass flux. Following Johnson (1976, 1980) the mass flux at the LFS is specified from the updraught mass flux at cloud base as
|
(5.11) |
The vertical distribution of the downdraught mass flux, dry static energy, moisture and horizontal momentum below the LFS are determined by entraining/detraining plume equations similar to those for the updraught;
|
(5.12) |
is the evaporation of convective rain to maintain a saturated descent;
the moistening and cooling of the environmental air injected at LFS is also
due to evaporating rain. Entrainment and detrainment in downdraughts are highly uncertain as relevant data are not available. As for the updraught, both turbulent and organized entrainment/detrainment are considered.
5.3.2 (a) Turbulent entrainment and detrainment
For turbulent mixing
|
(5.13) |
5.3.2 (b) Organized entrainment and detrainment
Organized entrainment for the downdraught is based upon a formulation suggested by Nordeng (1994);
|
(5.14) |
where
is the vertical velocity in the downdraught at the LFS (set
to 1 m s-1).The scheme has no explicit rain water equation for the downdraught and so
is estimated by
|
(5.15) |
Organized detrainment from the downdraught occurs when either the downdraught becomes positively buoyant or approaches the surface. If the downdraught remains negatively buoyant until it reaches the surface then the mass flux is decreased linearly over the bottom three model levels for L31 and L50 versions of the IFS. In L60 versions the downdraught is detrained over the lowest severn model levels, maintaining an outflow depth of about 50hPa as in lower resolution versions of the model. However if a downdraught becomes positively buoyant during its descent, it is detrained over one level, except where this occurs at cloud base, when the downdraught fluxes are decreased linearly (deep convection) or quadratically (mid-level convection) to zero at the surface.
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05.04.2002 |
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