All surface data are processed in the routine SURFACEO. Preparations for the vertical interpolation is done as for all other data in PREINT (see Subsection 5.3.2), and for surface data there are a few additional tasks which are performed in a separate routine, PREINTS. In PREINTS surface roughness over sea, dry static energy (SURBOUND), Richardson number, drag coefficients and stability functions (EXCHCO), are computed, as detailed in the following.

5.7.1 Mathematical formulation

An analytical technique (Geleyn, 1988) is used to interpolate values between the lowest model level and the surface. When Monin-Obukhov theory is applied:

(5.19)

(5.20)

(5.21)

where

are wind and energy variables,

are friction values and

is von Kármán's constant.

The temperature is linked to the dry static energy

by:

(5.22)

(5.23)

Defining the neutral surface exchange coefficient at the height

as:

(5.24)

The drag and heat coefficients as:

(5.25)

(5.26)

we can set the following quantities:

(5.27)

and considering the stability function in stable conditions as:

(5.28)

we obtain integrating Eqs. (5.19) and (5.20) from 0 to

(the lowest model level):

(5.29)

(5.30)

In unstable conditions the stability function can be expressed as:

(5.31)

and the vertical profiles for wind and dry static energy are:

(5.32)

(5.33)

The temperature can then be obtained from

as:

(5.34)

When

is set to the observation height, Eqs. (5.29) and (5.30) and Eqs. (5.32)-(5.34) give the postprocessed wind and temperature. To solve the problem, we have to compute the dry static energy at the surface

(Subsection 5.7.2), with

and

values depending on the drag and heat exchange coefficients Eq. as detailed in Subsection 5.7.3.

5.7.2 Surface values of dry static energy

To determine the dry static energy at the surface we use Eqs. (5.22) and (5.23) where the humidity at the surface is defined by:

(5.35)

is given by (Blondin, 1991):

(5.36)

with

(5.37)

where

is the soil moisture content and

is the soil moisture at field capacitiy (2/7 in volumetric units). Eq. (5.36) assigns a value of 1 to the surface relative humidity over the snow covered and wet fraction of the grid box. The snow-cover fraction

depends on the snow amount

where

m is a critical value. The wet skin fraction

is derived from the skin-reservoir water content

,

where

with

m being the maximum amount of water that can be held on one layer of leaves, or as a film on bare soil,

is the leaf-area index, and

is the vegetation fraction.

5.7.3 Transfer coefficients

Comparing the Eqs. (5.19) - (5.20) integrated from

with Eqs. (5.24) to (5.26),

and

can be analytically defined:

(5.38)

(5.39)

Because of the complicated form of the stability functions, the former integrals have been approximated by analytical expressions, formally given by:

(5.40)

where

is given by Eq. (5.24). The bulk Richardson number

is defined as:

(5.41)

where

is the virtual potential temperature. The functions

and

correspond to the model instability functions and have the correct behaviour near neutrality and in the cases of high stability (Louis, 1979; Louis et al. 1982)

(a) unstable case