Agriculture Reference
In-Depth Information
Derivation of the equation for wind profile inside a canopy in the “sandwich” approach
Let us consider an element of the canopy volume having an area
S
and height
H
. The loss of
air particles' momentum due to close contact with the plant leaves comes from the drag force
arising on the leaf surface. This drag force,
F
d
, produces a shearing such that
dτ/dz
, the
vertical gradient of shear stress,
τ
, is equal to the drag force per volume
V
, i.e.,
d
τ
.
(9)
=
F
/
V
d
dz
The drag force per leaf unit area,
S
l
, is parameterized to be proportional to the wind
speed,
u
, i.e., the volumetric kinetic energy
1/2ρu
2
with the coefficient of proportionality
C
d
-
the leaf drag coefficient. So,
F
1
.
(10)
d
=
C
ρ
u
2
S
2
d
l
Note that
S
l
is the area of all leaves in the considered volume. Following the definition of
leaf area index (
LAI
), we can write
LAI=S
l
/2S
, since
LAI
is defined in terms of only one side
of the leaf. Using
τ = ρK
s
du/dz
(where
K
s
is the turbulent transfer coefficient and
z
is the
vertical coordinate), Eqs. (9)-(10), and keeping in mind that the volume occupied by plants is
S·H
, after some manipulation we arrive at
(
)
2
u
d
du
C
L
H
−
h
⎛
⎞
.
(11)
⎜
⎝
K
⎟
⎠
=
d
d
s
dz
dz
H
where
d
L
is the area-averaged canopy density and
h
is the canopy bottom height (the height
of the base of the canopy).
2.2. Deriving Aerodynamic Resistances for Calculating air Temperature
Inside the Canopy
The canopy air space temperature,
T
a
, in land surface schemes can be determined
diagnostically from the energy balance equation. This procedure comes from the equality of
the sensible heat flux from the canopy to some reference level in the atmosphere, and the sum
of the sensible heat fluxes from the ground and from the leaves to the canopy air volume
[21,24], i.e.,
2
T
T
T
f
+
g
+
r
r
r
r
.
(12)
T
=
b
d
a
2
1
1
a
+
+
r
r
r
b
d
a
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