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Chapter 3. Turbulent diffusion and
interactions with the surface
IFS documentation Front Page
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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3.4 Solution of the vertical diffusion equation
The equations for turbulent transfer are solved with the tendencies from the adiabatic (subscript `dyn') and radiative processes (subscript `rad') as source terms in the right hand side:
|
(3.42) |
Since the thickness of the model layers
is small near the ground, the time-stepping procedure must be implicit
in order to avoid numerical instability when 
. However, the interaction with the adiabatic and radiative
processes is treated implicitly, and Janssen et al. (1992) have shown
that if the tendencies are not added to the right hand side of equation
(3.42) a time step dependent equilibrium,
and a too low numerical drag coefficient for high wind speeds, arise. By
applying a `fractional-steps' method (Beljaars,
1991), the discretization of equation (3.42) becomes, for 
,
|
(3.43) |
where
|
(3.44) |
The parameter
determines the implicitness of the scheme.
For 
the scheme is explicit, for 
we have a Crank-Nicholson and for 
we have an implicit backward scheme. In the model, 
, to avoid non-linear instability from the K-coefficients.
The exchange coefficients are computed from the mean variables at 
.The previous equation can be written as
|
(3.45) |
leading to the inversion of a tridiagonal matrix to solve for
. The coefficients 
are defined from (3.41).At the lowest level (
) the equation includes the surface
fluxes which are obtained by averaging over 
tiles:
|
(3.46) |
with
and
|
(3.47) |
Eq. (3.46) can be re-written
|
(3.48) |
Term at
on the right hand side is obtained from coupling
this last equation with the calculation of the surface energy budget through
the computation of the skin surface temperature (see equation (3.62) in Section 3.5). At the top of the atmosphere (
) turbulent fluxes at set to zero and we have
|
(3.49) |
which can be re-written
|
(3.50) |
The tridiagonal matrix equation is solved by a downward elimination scan followed by an upward back substitution (Press et al., 1992, pp 42-43).
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05.04.2002 |
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