Geoscience Reference
In-Depth Information
In the surface layer, it is assumed that kinetic momentum flux is constant with
height and, recalling the momentum flux
t
k
is con
ventio
nally selected to be in the
opposite direction to the kinematic turbulent flux
uw
¢¢
, it is usual to set:
(19.19)
τ=
u
2
*
=−
uw
¢¢
k
where
u
*
is the '
friction velocity
. Combining Equations (19.18) and (19.19) and
taking the square root of the resulting equation gives:
∂
u
uz
z
(19.20)
=
*
∂
Integrating Equation (19.20) with the boundary condition that the mean wind
speed is zero at the aerodynamic roughness length,
z
0
, gives:
u
⎡⎤
τ
⎡⎤
z
z
*
ln
k
(19.21)
u
=
=
ln
⎢⎥
⎢⎥
k
z
k
z
⎣⎦
⎣⎦
0
0
By simultaneously measuring the shape of this logarithmic wind profile above an
aerodynamically rough surface and the momentum transfer to that surface (to
give the value of
u
*
from Equation (19.9)), it is possible to obtain an estimate of
k
. The constant
k
is called the von Kármán constant and is believed to be
approximately 0.4.
Equation (19.21) provides the description of the logarithmic profile of
average wind speed observed in neutral conditions over aerodynamically
rough surfaces when covered with comparatively short roughness elements. It
is relevant over rough soil and short turf, for example. However, over taller
vegetation such as agricultural crops or forests, the apparent height at which
the average wind speed appears to go to zero when deduced by extrapolating
the observed wind speed profile measured above the canopy is greater, see
Fig. 19.6. It disappears at the level (
z
0
+
d
), where
d
is the
zero plane displacement
.
Consequently, the more general form of Equation (19.21) that is applicable
over all natural surfaces is:
*
l
u
zd
⎡
⎤
(
−
)
(19.22)
u
=
⎢
⎥
k
z
⎣
⎦
0
Equation (19.20) must also be adjusted to be consistent with this shift in the origin
of the
z
axis, and the resulting more general expression for the friction velocity in
neutral conditions is:
u
ukzd
z
∂
=−
∂
(19.23)
(
)
*
