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Chapter 3. Turbulent diffusion and
interactions with the surface
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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
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3.3 The exchange coefficients above the surface layer
3.3.1 General
A first order closure specifies the turbulent flux of a given quantity
at a given model level proportional to the vertical
gradient of that quantity:
|
(3.26) |
The exchange coefficients
are estimated at half model levels. The computation of the exchange
coefficients depends on the stability regimes (locally and at the surface)
and on the vertical location above the surface. Fig.
3.1 summarizes the various areas where each scheme (non-local diffusion
from Troen and Mahrt, local diffusion dependent on the Richardson number,
local diffusion with Monin-Obukhov functions) is applied. First, the local
Richardson number is computed in each vertical layer:
|
(3.27) |
Given the value of
, in stable local conditions the stability
parameter 
is deduced from precomputed tables giving 
. A cubic spline interpolation is performed (Press et al. 1992, pp107-111).
In unstable local conditions, we simply set 
. 3.3.2 The exchange coefficients
3.3.2 (a) Turbulence length scale
The mixing lengths
used in the surface layer are bounded
in the outer layer by introducing asymptotic length scales 
and 
(Blackadar, 1962)
|
(3.28) |
The underlying idea is that vertical extent of the boundary layer limits the turbulence length scale. Since the results in the boundary layer are not very sensitive to the exact value of the asymptotic length scales, these parameters are chosen to be constants:
 .
|
(3.29) |
Parameter
is 1 in the boundary layer but reduces the length
scales above the boundary layer in order to prevent excessive mixing to
occur in and around the jet stream. The following expression is used
|
(3.30) |
where
and 
. 3.3.2 (b) M-O similarity with Ri< 0 (Area 1 in Fig. 3.1 )
In this regime, the exchange coefficients
are based on local similarity (Nieuwstadt, 1984) stating that the expressions
of the surface layer similarity can be used in the outer layer (strictly
speaking only valid for stable condtions):
|
(3.31) |
Here it is used for the unstable regime above the boundary layer, basically to provide strong vertical mixing in statically unstable situations.
3.3.2 (c) Revised Louis scheme for Ri > 0 (Area 1 in Fig. 3.1 )
The use of Eq. (3.31) to define the exchange coefficients in the stable regime was found to be detrimental to the scores of the model (Beljaars, 1995) because of insufficient turbulent exchange in the lower troposphere. Therefore a revised version of the Louis scheme is used (Beljaars and Viterbo, 1999; Viterbo et al., 1999):
|
(3.32) |
The functional dependencies of
and 
with 
are:
|
(3.33) |
with
and 
(these functions are revised versions of the Louis et
al., 1982 functions and were introduced in September 1995 in order to
enhance turbulent transport in stable layers, see
Viterbo et al., 1999). 3.3.2 (d) Unstable at the surface (Area 2 in Fig. 3.1 )
In unstable surface conditions (
), the exchange coefficients are expressed as integral profiles for
the entire convective mixed layer. This K-profile closure is based on a
proposal from Troen and Mahrt (1986). This
approach is more suitable than the local diffusion one when the length scale
of the largest transporting turbulent eddies have a similar size as the
boundary layer height itself (unstable and convective conditions). It also
allows for an explicit entrainment parametrization in the capping inversion
(Beljaars and Viterbo, 1999).
First a characteristic turbulent velocity scale
is computed:
|
(3.34) |
The velocities
and 
are defined by equations (3.25) and (3.16) respectively.Since the most energetic transporting scales of turbulent motion in the convective boundary layer are thermals, their strength is defined by a temperature excess with respect to their surrounding. The dry static energy excess is written as
|
(3.35) |
The mixed-layer depth,
, is then defined in terms of the first
level k above the surface where 
, i.e. 
.• Area 2.1 in Fig. 3.1 . In the surface layer
above the first atmospheric level,  , the exchange coefficients are prescribed as follows |
|
(3.36) |
• Area 2.2 in Fig. 3.1 . In the unstable outer
layer ( ), similar expressions are used: |
|
(3.37) |
The Prandtl number  is evaluated at  . |
• Entrainment zone. Entrainment
at the top of the convective boundary layer is taken into account
explicitly. The buoyant flux at  is assumed to be proportional to the surface heat flux: |
|
(3.38) |
where the entrainment constant  is determined from experimental data. The numerical value of 0.2
is chosen from Driedonks and Tennekes (1984). |
Knowing the flux at the top of the mixed layer, the exchange coefficient can be expressed as:
|
(3.39) |
Then at
:
|
(3.40) |
Instead of the exchange coefficients
themselves, the scaled quantities 
are computed
|
(3.41) |
where
is the implicitness factor of the finite difference
scheme (see equation (3.44)).Next Section
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05.04.2002 |
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