It follows that C aM increases with z 0M , that is, the drag per unit area is larger for
rougher canopies.
By repeating the same approach, for calculating r M , based on Eqs. (2.25) and
(2.27), it is possible to determine the aerodynamic resistance of the canopy z aM , to
the momentum transfer between a height z, in the boundary layer of a uniform
canopy, where the concentration is qu(z), and the height of the elements at the
surface where the concentration is zero, is given by
r M ¼
quðzÞ
s
ð2:32Þ
where s is the downward momentum flux. The Eq. (2.32) describing the resistance
of momentum transfer to a canopy, is equivalent to Eq. (2.29) for the resistance of
momentum transfer towards a single element.
By combining Eqs. (2.30), (2.31), and (2.32), the following equation can be
derived:
r M ¼
1
u z
ð ÞC aM
¼
u z
ð Þ
u 2
Ã
¼
ln z À d
ð
Þ=z oM
f
g
2
K 2 u z
ð Þ
ð2:33Þ
In general, according to Eq. (2.33), the aerodynamic resistance, r M , decreases
with increased wind speed and surface roughness. Oke (1992) provided values for
aerodynamic resistance for typical terrain types. Some of these in sm
−1 units were:
200 for aqueous surfaces; 70 for turf; 20–50 for agricultural crops; and 5–10 for
forests. Measurements for cork oak forests in Portugal gave values of about 15.2
sm
−1 (Rodrigues 2002).
Equation (2.18) shows that the units for turbulent diffusivity are L
2 T
−1 . The units
for 1/K are either TL
−2 or TL
−1 /L, equivalent to the aerodynamic resistance per unit
length, or aerodynamic resistivity. For the momentum transfer, the integral of 1/K M
between heights z 1 and z 2 , where the airspeed u has values u 1 and u 2, equals the
aerodynamic resistance to the momentum transfer between these two heights, given
by r M (z 1 , z 2 ). Indeed, by combining Eqs. (2.10), (2.19), and (2.33), the following is
obtained (Thom 1975):
Z z 2
z 1
dz
K M
¼
1
u 2
Ã
Z u 2
u 1
du ¼
u 2 À u 1
u 2
Ã
¼ r M z 1 À z 2
ð
Þ
ð2:34Þ
so that the analogy between 1/K M and r M is confirmed.
2.2 Turbulent Diffusivity Coefficients
23
rougher canopies.
By repeating the same approach, for calculating r M , based on Eqs. (2.25) and
(2.27), it is possible to determine the aerodynamic resistance of the canopy z aM , to
the momentum transfer between a height z, in the boundary layer of a uniform
canopy, where the concentration is qu(z), and the height of the elements at the
surface where the concentration is zero, is given by
r M ¼
quðzÞ
s
ð2:32Þ
where s is the downward momentum flux. The Eq. (2.32) describing the resistance
of momentum transfer to a canopy, is equivalent to Eq. (2.29) for the resistance of
momentum transfer towards a single element.
By combining Eqs. (2.30), (2.31), and (2.32), the following equation can be
derived:
r M ¼
1
u z
ð ÞC aM
¼
u z
ð Þ
u 2
Ã
¼
ln z À d
ð
Þ=z oM
f
g
2
K 2 u z
ð Þ
ð2:33Þ
In general, according to Eq. (2.33), the aerodynamic resistance, r M , decreases
with increased wind speed and surface roughness. Oke (1992) provided values for
aerodynamic resistance for typical terrain types. Some of these in sm
−1 units were:
200 for aqueous surfaces; 70 for turf; 20–50 for agricultural crops; and 5–10 for
forests. Measurements for cork oak forests in Portugal gave values of about 15.2
sm
−1 (Rodrigues 2002).
Equation (2.18) shows that the units for turbulent diffusivity are L
2 T
−1 . The units
for 1/K are either TL
−2 or TL
−1 /L, equivalent to the aerodynamic resistance per unit
length, or aerodynamic resistivity. For the momentum transfer, the integral of 1/K M
between heights z 1 and z 2 , where the airspeed u has values u 1 and u 2, equals the
aerodynamic resistance to the momentum transfer between these two heights, given
by r M (z 1 , z 2 ). Indeed, by combining Eqs. (2.10), (2.19), and (2.33), the following is
obtained (Thom 1975):
Z z 2
z 1
dz
K M
¼
1
u 2
Ã
Z u 2
u 1
du ¼
u 2 À u 1
u 2
Ã
¼ r M z 1 À z 2
ð
Þ
ð2:34Þ
so that the analogy between 1/K M and r M is confirmed.
2.2 Turbulent Diffusivity Coefficients
23