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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The surface layer approximation is applied between the lowest model level
(about 10 m above the surface in the 60-level model) and the surface and
for each tile separately. It is assume that the turbulent fluxes are constant
with height and equal to the surface values. They can be expressed, using
Monin-Obukhov similarity theory, in terms of the gradients of wind, dry
static energy and specific humidity, which are assumed to be proportional
to universal gradient functions of a stability parameter:
The scaling parameters  ,  and  are expressed in terms of surface fluxes:
The stability parameter  is the Obukhov length defined as
 is the virtual temperature flux in the surface layer,  is the Von Kármán constant ( ),  is a reference temperature taken as
a near-surface temperature (the temperature of the lowest atmospheric level)
and  , where  and  are the gas constants for water vapour and dry air, respectively.
In the surface layer, the gradient functions (3.4) can be integrated to profiles
 ,  and  are the roughness lengths for momentum, heat and moisture. The stability
profile functions  are derived from the gradient functions (3.4) with the help of the relationship
 . These profiles are used for the surface
atmosphere interaction as explained in the following sections and also for
the interpolation between the lowest model level and the surface (postprocessing
of 10 m wind and 2m temperature and moisture).
In extremely stable situations, i.e. for very small positive  , the ratio  is large, resulting in unrealistic profile shapes with standard stability
functions. Therefore the ratio  is limited to 5 by defining a height  such that  . If  , then the profile functions described above, are used up to  and the profiles are assumed to be uniform above that.
This modification of the profiles for exceptionally stable situations (no
wind) is applied to the surface transfer formulation as well as to the interpolation
for postprocessing.
Surface fluxes for heat and moisture are computed separately for the different
tiles, so most of the surface layer computations loop over the tile index.
Here a general description is given of the aerodynamic aspects of the transfer
between the surface and the lowest model level. The description of the individual
tiles can be found in Chapter 7.
Assuming that the first model level above the surface is located in the
surface boundary layer at a specified height  , the gradient functions (3.4) can be integrated to profiles
for wind, dry static energy and specific humidity. The surface fluxes are
expressed in terms of differences between parameters at level  and surface quantities (identified by the subscript `surf'; the tile
index has been omitted in this general description).
where  ,  and  are provided by the land scheme,  , and  is the apparent surface humidity also provided by the land surface
scheme (the humidity equation simplifies over water where  ,  and  .
The transfer coefficients can be expressed as follows
The wind speed  is expressed as
with  the free convection velocity scale defined by
The parameter  is a scale height of the boundary layer depth
and is set to constant value of 1000 m, since only the order of magnitude
matters. The additional term in equation
(3.15) represents the near surface wind induced by large eddies in the
free-convection regime. When the surface is heated, this term guarantees
a finite surface wind-forcing in the transfer law even for vanishing  and  , and prevents  and  from becoming zero. Beljaars (1994) showed that this
empirical term, when added into the standard Monin-Obukhov scaling, is in
agreement with scaling laws for free convection. When used with the roughness
lengths defined below, it provides a good fit to observational data, both
over land and over sea.
The empirical forms of the dimensionless gradient functions  (equations (3.4)) have been deduced from field
experiments over homogeneous terrain.
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(a) In unstable conditions,  , the gradient functions proposed by Dyer and Hicks are used (Dyer, 1974; Hogström, 1988): |
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These functions can be integrated to the universal
profile stability functions,  , (Paulson, 1970): |
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with  . The  -functions are used in the surface layer
and the  -functions for unstable stratification are used above the surface
layer for local closure. |
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(b) For stable conditions,  , the code contains gradient function  as documented by Hogström (1988), and  as derived from the Ellison and Turner relation for the ratio
 : |
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These functions were meant to be used for local
closure above the surface layer, but are not used at all in the current
model version, because Richardson number dependent functions are used
instead (see section on exchange coefficients above the surface layer). |
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The stable profile functions as used in the surface
layer, are assumed to have the empirical forms proposed by Holtslag
and De Bruin (1988), with a modification to allow for the effects
of a critical flux Richardson number for large  : |
The transfer coefficients needed for the surface fluxes require the estimation
of stability parameter  , itself a function of the surface fluxes. Therefore, an implicit
equation, relating  to bulk Richardson number  , is solved:
with
where  and  are the virtual potential temperatures at level  and at the surface, and  is a virtual potential temperature within the surface layer. Equation
(3.22) can be expressed
in terms of dry static energy:
Knowing  at time  , a first guess of the Obukhov length is made from fluxes computed
at the previous time step. Equation (3.21) is solved numerically using
the Newton iteration method to retrieve  .
In contrast to the previous formulation used in the model (Louis et al., 1982), the
present scheme allows a consistent treatment of different roughness lengths
for momentum, heat and moisture. The revised stability functions also reduce
diffusion in stable situations resulting in more shallow stable boundary
layers.
The integration constants  ,  and  , in the equations for the transfer coefficients  ,  and  , (equations (3.12)-(3.14)) are called roughness lengths
because they are related to the small scale inhomogeneities of the surface
that determine the air-surface transfer.
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• Over land, roughness lengths are assumed
to be fixed climatological fields as described in Chapter 9. They
are derived from land-use maps, with an extra contribution dependent
on the variance of subgrid orography. |
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• Over sea, the specification of surface
roughness lengths is particularly important. Because of the fixed
boundary conditions for temperature and moisture the sea is, in principle,
an infinite source of energy to the model. The surface roughness lengths
are expressed by (Beljaars, 1994): |
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These expressions account for both low and high
wind regimes: |
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• At low wind speed the sea surface becomes
aerodynamically smooth and the sea surface roughness length scales
with the kinematic viscosity  . |
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• At high wind speed the Charnock relation
is used. The chosen constants are  ,  , and  (Brutsaert, 1982). The Charnock coefficient,
 , is set equal to 0.018 for the uncoupled
model, and is provided by the wave model in coupled mode. |
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The smooth-surface parametrization is retained
in high wind speed regimes for heat and moisture because observations
indicate that the transfer coefficients for heat and moisture have
very little wind-speed dependence above 4  (Miller et al., 1992;
Godfrey and Beljaars, 1991). In Eqs. (3.24), friction velocity  , is calculated from |
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with  from equation (3.16) using fluxes from the previous
time step. |
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