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Chapter 7. Land surface parametrization
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Chapter 7. Land suface parametrization
Chapter 8. Methane oxidation
Chapter 9. Climatological data
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7.5 Soil heat transfer
In the absence of internal phase changes, the soil heat transfer is assumed to obey the following Fourier law of diffusion
|
(7.45) |
is the volumetric soil heat capacity 
, 
is the soil temperature (units K), 
is the vertical coordinate-the distance from the surface, positive
downwards-(units m), and 
is the thermal conductivity 
. The above equation assumes that heat fluxes are predominantly
in the vertical direction, that the effects of phase changes in the soil
and the heat transfer associated with the vertical movement of water in
the soil can be neglected (de Vries 1975), and that the effects
of hysteresis can be neglected (Milly 1982).The boundary condition at the bottom, no heat flux of energy, is an acceptable approximation provided that the total soil depth is large enough for the time-scales represented by the model or, in other words, the bottom of the soil is specified at a depth where the amplitude of the soil heat wave is a negligible fraction of its surface amplitude (see de Vries (1975) and next section).
7.5.1 Discretization and choice of parameters
For the solution of Eq. (7.45) the soil is discretized in four layers, of depths
, 
, the temperatures are defined at full layers (
), and the heat fluxes, at half layers (
is the heat flux, positive downwards, units 
, at the interface between layer 
and 
). An energy-conserving implicit algorithm is used, leading
to a tridiagonal system of equations with solution detailed in
Section 7.8.The boundary condition at the bottom is:
|
(7.46) |
At the top, the boundary condition is the soil heat flux at the surface, computed as a weighted average over the tiles. For the snow free tiles, the flux into the soil consists of two parts. Apart from the diffusion of heat governed by
(see Eq. (7.19)), the net shortwave
radiation not absorbed by the skin layer (fRs,i)
provides energy to the soil. Table 7.2 lists the values
of 
and fRs,i
for each of the tiles. For the snow tiles, the heat flux into the soil is
the snow basal flux, calculated using a resistance formulation and modified
in the case of partial melting (see Eqs. (7.31), (7.38), (7.42), and
(7.44)).The net heat flux into the soil is given by:
 ,
|
(7.47) |
where the summation scans all snow free tiles.
The volumetric soil heat capacity is assumed constant, with value
(see Table 7.5 for a list of constants
used by the model). The heat conductivity, 
, depends on the soil-water content following Peters-Lidard
et al. (1998) (see also Farouki 1986; Johansen 1975) and
is given by a combination of dry 
and saturated 
values, weighted by a factor known as the Kersten number, Ke:
 ,
|
(7.48) |
and
 ,
|
(7.49) |
and the thermal conductivity of water is 
. Eq. (7.49) represents a simplification
of Peters-Lidard formulation, neglecting the changes in conductivity due
to ice water and assuming the quartz content typical of a loamy soil. Finally,
the Kersten number for fine soils was selected in Peters-Lidard et al.
(1998):
|
(7.50) |
The depths of the soil layers are chosen in an approximate geometric relation (see Table 7.5 ), as suggested in Deardorff (1978). Warrilow et al. (1986) have shown that four layers are enough for representing correctly all timescales from one day to one year. Using the numerical values of the heat capacity and soil depths defined in Table 7.5, the amplitude and phase response of the numerical solution of Eq. (7.45) were analysed by Viterbo and Beljaars (1995) for typical values of soil moisture in Eq. (7.48), and for harmonic forcings at the surface with periods ranging from half a day to two years. The analysis points to an error in the numerical solution of less than 20% in amplitude and 5% in phase for forcing periods between one day and one year.
7.5.2 Soil-water phase changes
At high and mid latitudes the phase changes of water in the soil have an important effect on the water and energy transfer in the soil. A proper consideration of the solid phase of soil water requires modifications including, in order of importance:
| (a) The thermal effects related to the latent heat of fusion/freezing (e.g. Rouse 1984); |
| (b) Changes in the soil thermal conductivity due to the presence of ice (e.g. Penner 1970, not included in TESSEL as mentioned in the previous section); |
| (c) Suppression of transpiration in the presence of frozen ground (e.g. Betts et al. 1998) and already described in Eq. (7.11); and |
| (d) Soil water transfer dependent on a soil water potential including the effects of frozen water (e.g. Lundin 1989), represented in a proxy way by Eq. (7.66). |
The latent-heat effects are described in the following. The main impact will be to delay the soil cooling in the beginning of the cold period, and to delay the soil warming in spring, although the latter effect is less important because it occurs when the solar forcing is significant. Both effects make the soil temperatures less responsive to the atmospheric forcing and damp the amplitude of the annual soil temperature cycle. More details on the soil-freezing scheme and its impact on forecasts and the model climate are described in Viterbo et al. 1999.
The soil energy equation, Eq. (7.45), is modified in the presence of soil water phase changes as
|
(7.51) |
where
is the volumetric ice-water content. Without loss
of generality, for the grid squares characteristic of NWP models it can
be assumed that
|
(7.52) |
where
is the total soil-water content (liquid + ice),
and
|
(7.53) |
where Tf1 and Tf2 are characteristic temperatures limiting the phase change regime. In reality, the values of Tf1 and Tf2 and the function ffr(T) have complicated dependencies on soil texture and composition (see e.g. Williams and Smith 1989), but here they are approximated in a simple way. For an idealized homogeneous, one-component soil, ffr(T) would be a step-function. The physical reasons for having an interval over which melting/freezing is active, rather than a threshold temperature, include (Williams and Smith 1989):
| (a) Adsorption, resulting from forces between the mineral parts of the soil and the water; |
| (b) Capillarity, related to the fact that the water-free surface is not plane; |
| (c) Depression of the freezing point due to the effect of dissolved salts; and |
| (d) Soil heterogeneity. |
To avoid an undesirable coupling between the temperature and water equations in the soil, Eq. (7.52) is simplified to
|
(7.54) |
where
is a constant, representing the amount of soil water
that can be frozen (thawed). For simplicity, 
. The scaling with the vegetated fractions is the simplest
way of distinguishing between dry (vegetation-sparse areas, e.g. deserts)
and wet (vegetated) areas. Combining Eq.
(7.54) with Eq. (7.51) results in
|
(7.55) |
showing that the effect of freezing can be interpreted as an additional soil heat capacity, sometimes referred in the literature as the `heat-capacity barrier' around freezing; not considering the process of soil water freezing/melting can lead to very large artificial temperature changes that do not occur in nature when sufficient soil water is available.
Finally, function ffr(T), is given by
|
(7.56) |
with Tf1 = T0 + 1, Tf1 = T0 - 3.
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05.04.2002 |
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