#### Aerodynamic resistance with stability correction below vegetation
canopy

The aerodynamic resistance above the soil surface, *r*_{as}, is calculated as a sum of two
components – a function of wind speed and temperature gradients,
*r*_{aa}, which is corrected for atmospheric stability, and an
additional resistance representing the influence of the crop cover,
*r*_{ab} (see viewing function Aerodynamic Resistance,
r_{as}):

(4.12)

The influence of the crop canopy on the aerodynamic
resistance above the soil surface is made proportional to the leaf area index,
*A*_{l}:

(4.13)

where *r*_{alai} is
an empirical parameter (see viewing function Aerodynamic
Resistance below canopy, r_{ab}).

The influence of atmospheric stability on the aerodynamic
resistance, *r*_{aa}, can be calculated either as (*I*)
an analytical function of the Richardson number or (*II*) as a
function of the Monin-Obukhov stability parameter (see switch Stability
Correction). Method (*I*) is preferred from a
computational point of view, since (*II*) involves an iterative
solution of the relation between the Richardson number and the Monin-Obukhov
stability parameter (Eq.(4.19)). However, only method (*II*)
allows for a consistent treatment of variations in the roughness lengths for
momentum and heat.

*(I)* The aerodynamic resistance at neutral
conditions is multiplied by an analytical stability function:

(4.14)

where *u* is the wind
speed at the reference height, *z*_{ref}, *d* is the zero level displacement height (c.f.
Potential Transpiration in Plant Water Processes), *R*_{ib} is the
bulk Richardson number (eq.(4.17)), *k* is the von Karmans constant and
*z*_{0M} and *z*_{0H} are the surface roughness
lengths for momentum and heat respectively. If *z*_{0M} is
exchanged to *z*_{0M,snow} the equation can be used for snow
surfaces. *f(R*_{ib}) is a function that governs the influence of
atmospheric stability:

(4.15)

where *a*_{ri,1}, *b*_{ri,1}, *a*_{ri,2} and *b*_{ri,2} are empirical parameters.

The surface roughness length of momentum,
*z*_{0M}, can either be given as a specific parameter for
different sub-surfaces (i.e. bare soil, snow and canopies) or as a function of
canopy height (c.f. “Potential transpiration” in Plant Water Processes). The
surface roughness length of heat, *z*_{0H}, is then derived
from:

(4.16)

where *kB*^{-1} is a
parameter with a default value 0 (implies
*z*_{0H}=*z*_{0M}). The parameter is the product of a
von Karmans constant, *k*, and a
parameter, *B*, but since it is often found in the literature as *kB*^{-1} we have kept it as such in the
model.

The bulk Richardson's number is calculated as:

(4.17)

*(II)* The aerodynamic resistance as a function
of the Monin-Obukhov stability parameter, (adopted from Beljaars and
Holtslag,1991):

(4.18)

where *L*_{O} is the
Obukhov length and *Ψ*_{Μ} and *Ψ*_{Η} are empirical
stability functions for momentum and heat respectively (unfortunately the
nomenclature coincides with that for latent heat of vaporisation and water
tension). The relation between the Obukhov length and the Richardson number is
specified by the following equation:

(4.19)

which is solved by an iterative procedure following Beljars
and Holtslag (1991). The empirical stability functions is calculated for
*unstable conditions* ((*z*_{ref}-d)/L_{O}<0) by:

(4.20)

and

(4.21)

where

(4.22)

where the non-optional
parameter value *a*_{z/L}=19 was taken from Högström (1996).

For *stable conditions*
((*z*_{ref}-d)/L_{O}>0) the empirical
stability function is instead calculated as:

(4.23)

(4.24)

following Bejaars and Holtslag (1991), with the
non-optional parameter values *α*=1, *β*=0.667, *γ*=5 and
*δ*=0.35.

Furthermore, an upper limit of the aerodynamic resistance
in extreme stable conditions is set by the “windless exchange” coefficient, *r*_{a,soil,max}^{-1}, adopted from
Jordan (1991). It is applied in both (**I**) and (**II**):

(4.25)