Geoscience Reference
In-Depth Information
respectively, and are obtained by integrating the reciprocal of the appropriate
eddy diffusivity between the two levels.
Quite commonly, however, it is the total
aerodynamic resistances
to flow that are
of interest. These resistances act between the reference level above the canopy,
where atmospheric variables are measured, and the effective source of fluxes
within the canopy of vegetation. In neutral conditions this aerodynamic resistance
is comparatively simple to calculate for momentum transfer if mixing length
theory is assumed, and if (on the basis of extrapolating above-canopy profiles) it is
assumed that the wind speed profile goes to zero at a height
z
0
above the zero
plane displacement, see Fig. 19.6. In this case
K
M
1, see
Equation (19.3). Consequently
r
a
M
, the aerodynamic resistance between the sink
of momentum in the canopy and the height
z
m
, is given by:
=
ku
*
(
z
-
d
) and
f
M
=
z
⎛
⎞
⎡
⎤
m
⎛
⎞
1
ln(
zd
−
)
1
zd
−
z
M
∫
m
m
r
=
.
dz
=
⎢
⎥
=
ln
(20.32)
⎜
⎟
⎜
⎟
a
ku z d
(
−
)
ku
ku
z
⎝
⎠
dz
+
⎢
⎥
⎝
⎠
o
*
⎣
*
⎦
*
0
dz
+
o
In neutral conditions the wind speed profile is given by Equation (19.21), which
equation can be rearranged to give the value of
u
*
in terms of the wind speed
u
m
measured at height
z
m,
thus:
−
⎡
zd
⎤
(
−
)
(20.33)
uu
1
m
=
ln
⎢
⎥
*
m
z
⎣
⎦
0
Consequently, Equation (20.32) becomes:
⎛
⎞
1
zd
−
(20.34)
M
2
m
r
==
ln
⎜
⎟
a
2
z
ku
⎝
⎠
m
0
Similarly, if
r
a
H
and
r
a
V
are respectively the aerodynamic resistance to sensible and
latent heat transfer between
z
m
and the source/sink of these two heat fluxes in the
canopy, these resistances can be simply derived in neutral conditions. If the source/
sink heights for sensible and latent heat are assumed to be at heights
z
0
H
and
z
0
V
above the zero plane displacement, respectively, the two resistances are given by:
⎛
⎞
⎛
⎞
zd zd zd
−
⎛
−
⎞
−
1
1
H
r
=
ln
m
=
ln
m
ln
m
(20.35)
⎜
⎟
⎜
⎟
⎜
⎟
a
H
2
H
ku
z
k u
⎝
z
⎠
z
⎝
⎠
⎝
⎠
*
0
0
m
0
and
⎛
⎞
⎛
⎞
1
zd zd zd
−
1
⎛
−
⎞
−
V
r
=
ln
m
=
ln
m
ln
m
(20.36)
⎜
⎟
⎜
⎟
⎜
⎟
a
V
2
V
ku
z
k u
⎝
z
⎠
z
⎝
⎠
⎝
⎠
*
m
0
0
0
Deriving these aerodynamic resistances in other stability conditions is more
difficult and somewhat circuitous. It involves integrating eddy diffusivities that
