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Chapter 7. Land surface parametrization
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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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7.4 Snow
The snow scheme represents an additional "layer" on top of the upper soil layer, with an independent, prognostic, thermal and mass contents. The snow pack is represented by a single snow temperature, Tsn and the snow mass per unit area (snow mass for short) S. The net energy flux at the top of the snow pack,
, is the residual of the skin energy
balance from the snow covered tiles and the snow evaporation from the tile
with high vegetation over snow (Eq. (7.15)). The basal heat flux,
, is given by equation a resistance
formulation modified in case of melting. The absorbed energy is used to
change the snow temperature or melt the snow, when Tsn
exceeds the melting point.The heat capacity of the snow deck is a function of its depth and the snow density, which is a prognostic quantity depending on snow age following (Douville et al. 1995). The snow thermal conductivity changes with changing snow density. The snow albedo changes exponentially with snow age. For snow on low vegetation it ranges between 0.50 for old snow and 0.85 for fresh snow (to which it is reset whenever the snow fall exceeds 1 mm hr-1). The albedo for high vegetation with snow underneath is fixed at 0.15.
7.4.1 Snow mass and energy budget
The snow mass budget reads as:
|
(7.20) |
where F is snowfall (units kg m-2s-1), S is snow mass (sometimes referred as snow water equivalent) grid-averaged (units 103 kg m-2),
is the water density (units kg m-3), Esn
and Msn are snow evaporation and melting, respectively
(units kg m-2s-1), and csn is the
snow fraction (see Eq. (7.2)), i.e. the sum of tiles
5 and 7 (see Eq. (7.4)). In Eq. (7.20) and in the remaining
of this section, all surface fluxes are per unit area and apply only to
the snow area (i.e. tile 5 and 7). The snow equivalent water S applies
to the entire grid square and therefore occurs in the equation divided by
the total snow fraction. The snow flux from the atmospheric model, F,
is again for the entire grid square. As a general rule, all quantities with
subscript sn will refer to the snow area. In Eq. (7.20), the snow evaporation
is defined as
|
(7.21) |
Snow mass and snow depth are related by
|
(7.22) |
where Dsn is snow depth for the snow-covered area (units m; Dsn is NOT a grid-averaged quantity) and
is the snow density (units kg m-3).The snow energy budget reads as
|
(7.23) |
where
and 
are the ice and snow volumetric heat capacities, respectively
(units 
), 
is the ice density (units kg m-3, 
is the net radiation absorbed by the snow pack (units
W m-2), Ls is the latent heat of sublimation
(units J kg-1), Hsn, 
, and Qsn represent,
respectively, the snow sensible heat flux, basal heat flux (at the bottom
of the snow pack), and energy exchanges due to melting (units W m-2).
Eq. (7.23) neglects the thermal
energy brought by precipitation. The snow is composed of an ice fraction,
a liquid water fraction and an air fraction, 
, 
and 
, respectively, where typically 
and the liquid water fraction is significantly different from zero
in melting conditions. The following approximations are made in Eq. (7.23)
|
(7.24) |
The melting term couples the mass and energy equation
|
(7.25) |
where Lf is the latent heat of fusion (units J kg-1) and the subscrit m represents melting.
7.4.2 Prognostic snow density and albedo
Following Douville et al. (1995) snow density is assumed to be constant with depth and to evolve exponentially towards a maximum density (Verseghy, 1991). First a weighted average is taken between the current density and the minimum density for fresh snow
|
(7.26) |
The exponential relaxation reads
|
(7.27) |
where timescales
, and 
corresponding to an e-folding time of about 4 days, with minimum
density 
kg m-3 and maximum density
kg m-3 (see Table 7.4).
Snow albedo in exposed areas evolves according to the formulation of Baker et al. (1990), Verseghy (1991) and Douville et al. (1995). For non melting-conditions:
|
(7.28) |
where
, which will decrease the albedo by 0.1 in 12.5
days. For melting conditions 
:
|
(7.29) |
where
and 
. If snowfall 
kg m-2hr-1, the snow albedo is reset
to the maximum value, 
.The above formulae are inadequate to describe the evolution of the surface albedo of snow cover with high vegetation. Observations suggest a dependence on forest type but, by and large, the albedo changes from a value around 0.3 just after a heavy snowfall to a value around 0.2 after a few days (see Betts and Ball (1997) and the discussion in Viterbo and Betts (1999)). This change reflects the disappearance of intercepted snow, due to melt (for sufficiently warm temperatures) or wind drift (for cold temperatures). Ways of describing those two mechanisms would involve either a separate albedo variable for the snow in the presence of high vegetation, or the introduction of an interception reservoir for snow. In the absence of any of the two, we define
for the snow in the presence of high vegetation.
This value was chosen to match the overall forest albedo in the presence
of snow from the results of Viterbo and Betts (1999).7.4.3 Additional details
7.4.3 (a) Limiting of snow depth in the snow energy equation
Initial experimentation with the snow model revealed that the time evolution of snow temperature was very slow over Antartica. The reason is rather obvious; the snow depth over Antartica is set to a climatological value of 10 m which can respond only very slowly to the atmospheric forcing due to its large thermal inertia. In previous model versions, the properties of layer 1 were replaced by snow properties when snow was present, which kept the timescale short. A physical solution would have been to introduce a multilayer snow model, with e.g. four layers to represent timescales from one day to a full annual cycle. As a shortcut, a limit is put on the depth of the snow layer in the thermal budget,
. The energy equation reads:
|
(7.30) |
7.4.3 (b) Basal heat flux and thermal coefficients
The heat flux at the bottom of the snow pack is written as a finite difference in the following way:
|
(7.31) |
where rsn is the resistance between the middle of the snow pack and the middle of soil layer 1, with two components: the resistance of the lower part of the snow pack and the resistance of the top half of soil layer 1:
|
(7.32) |
where the second term is the skin layer conductivity for bare soil (tile 8), which can be seen as an approximation of
. The snow thermal conductivity, is related to the ice thermal
conductivity according to Douville et al. (1995):
|
(7.33) |
Table 7.4 contains the numerical values of the ice density and ice heat conductivity.
7.4.3 (c) Numerical solution for non-melting situations
The net heat flux that goes into the top of the snow deck is an output of the vertical diffusion scheme
|
(7.34) |
In the absence of melting, the solution of Eq. (7.30) is done implicitly. The preliminary snow temperature, prior to the checking for melting conditions,
, is given by
|
(7.35) |
|
(7.36) |
where superscript t refers to the current time step and superscript * to the preliminary value at the next time step. The solution for
is obtained from
|
(7.37) |
The basal snow heat flux to be used as input for the thermal budget of the soil (in the snow covered fraction only) is
|
(7.38) |
Finally, a preliminary new value for the snow mass,
, is computed from snow fall and snow evaporation
|
(7.39) |
7.4.4 Treatment of melting
7.4.4 (a) No melting occurs
If
no melting occurs and the preliminary values 
and 
become the t+1 values, while the basal heat flux is
given by Eq. (7.38).7.4.4 (b) Melting conditions
If
, snow melting occurs and the time step is divided
in two fractions, 
, where the first fraction, 
brings the temperature to T0 with no melting:
|
(7.40) |
while, during the second fraction,
, melting occurs with no resultant warming of the snow:
|
(7.41) |
If not all the snow melts, i.e., if St+1>0, the following heat flux is passed to the soil
|
(7.42) |
When all the snow melts, i.e., if St+1<0, the melting time step is redefined as:
|
(7.43) |
and the basal heat flux is redefined as
|
(7.44) |
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05.04.2002 |
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