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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
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7.8 Numerical solution of the surface equations
7.8.1 Recap of the analytical equations
The water budget (Eqs. (7.57)-(7.58), with boundary conditions given by Eq. (7.67)), the soil energy budget (Eq. (7.45), with boundary conditions given by Eqs. (7.46)-(7.47)) and the ice energy budget (Eq. (7.70)) can be rewritten in a generalised form as:
|
(7.72) |
The meaning of the different variables in each individual equations is summarized Table 7.6, together with the respective upper and lower boundary conditions,
.| Equation |
 |
 |
 |
 |
 |
UBC |
LBC |
|---|---|---|---|---|---|---|---|
| Soil moisture |
 |
1 |
 |
 |
 |
 |
 |
| Soil temperature |
 |
 |
 |
0 |
0 |
 |
 |
| Ice temperature |
 |
 |
 |
0 |
0 |
 |
 |
UBC and LBC stand for upper and lower boundary condition, respectively |
|||||||
7.8.2 Implicit numerical solution
Eq. (7.72) is time discretized in the following way:
|
(7.73) |
where
|
(7.74) |
and the semi-implicit coefficient,
. If the prognostic variable 
is defined at full levels and the fluxes 
are defined at half-levels (the interface between layers),
Eq. (7.73) can be discretized
in space to give:
|
(7.75) |
where the horizontal brace means that the term exists only for the ice temperature equation (because of the bottom temperature boundary condition for ice) and
, 
, 
, and 
represent the thickness of layer k, and the depths of
its centre, the top and the bottom interface, respectively:
|
(7.76) |
Eq. (7.75) leads to a triadiagonal system of equations:
|
(7.77) |
with the generalized modified diffusivities,
, defined as:
|
(7.78) |
where
is the depth of the deepest soil layer. The discretization
above conserves water (energy) and is linearly stable. The coefficients
and 
are a function of variable at the current
time step, 
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05.04.2002 |
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