Geoscience Reference
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soil surface area (
˃
0
=
ʲ
k
m
k
), m
k
is the mass of roots beneath unit soil area,
ʲ
k
is an
empirical constant,
ˈ
0
is the water potential within the root system, s(z) is the
function of the vertical distribution of the root system (s(0) = 0, s(z
0
) = 1).
To complete the synthesis of the tree
'
s water regime, let us parameterize the
processes of water
fl
flow in the plant and transpiration. It is assumed that from the root
system the water
flows by the trunk to canopy (branches, leaves) and then evaporated
into the atmosphere (Lai et al. 2000; Lal et al. 1998) and the above ground part of the
tree of the height H is divided into n equal layers: H = n
fl
z. Across the boundaries of
the ith layer the water moves due to the difference between water potentialsψi
ʔ
ˈ
i
and
ˈ
i
−
1
overcoming the resistance of the xylem vessels:
1
r
ks
ð
z
Þ
¼ S
ks
ð
z
Þn
ks
;
c conductivity of the xylem, S
ks
is the cross-section of the tree
at a height z. As a result, with the area of the cross-section supposed to be vertically
constant, the following formula describing the rate of water
where
ʾ
ks
is the speci
fl
flow in the trunk across
the ith layer is obtained:
v
i
1
;
i
¼
n
ks
S
ks
D
z
1
ð
w
i
w
i
1
þ q
g
D
z
Þ
According to Kirilenko (1990), the intensity of transpiration from the ith layer
can be described by the formula:
d
i
r
i
L
i
¼
S
L
c
where d
i
is the de
cit of saturation of an absolute air humidity in the atmosphere, S
L
is the leaf area of the ith layer, r
i
L
¼ r
st
þ
r
a
;
r
i
dt
is the stomatal resistance, r
a
is the
air boundary layer resistance.
To complete the description of the model after Kirilenko (1990), a standing
wood of the density
ρ
F
is considered. The model equations are written as
d
2
u=
d
g
¼ Uv
ðgÞuðgÞ; g 2
½0
; a;
ð
8
:
4
Þ
and the boundary conditions are
0
0
¼
G
ðu
s
u
0
Þþ
F
u
½
;
d
u=
d
g
¼
F for
g 2
½
a;
1
;
ð
8
:
5
Þ
u
0
a
¼
F
For
ˆ
0
≤
0, a system of equations (
8.4
) and (
8.5
) becomes:
8
<
d
d
¼
F
u
0
¼
g
:
u
s
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