Geoscience Reference
In-Depth Information
Equation (
6.6
) can be combined with Eqs. (
6.1
) to (
6.3
) to yield the following second-
order differential equation:
T
D
2
−=−−
∂
q
r
∂
=
∂
q
r
M
rr
∂
+
∂
M
r
p
(6.7)
∂
2
r
Equation (
6.7
) can be solved for the governing boundary conditions (Appendix D):
T
D
2
2
rr
−
r
r
+
( )
p
MM
−=
0
rr
2
2
ln
(6.8)
0
m
0
2
2
r
0
where
r
0
is the root radius, and
r
m
is equal to the half mean distance between roots
(
Figure 6.4
). The latter is related to the root length density
R
(m m
-3
) by:
1
1
R
=
or
r
=
(6.9)
m
2
π
r
π
R
m
Metselaar and de Jong van Lier (
2007
) showed by numerical analysis that
M
(
r
) under
limiting soil hydraulic conditions (or falling rate phase) has the same shape as under
nonlimiting conditions and may be described with an expression equivalent to Eq.
(
6.8
), with
T
p
replaced by the actual transpiration rate
T
a
and
M
0
equal to the mat-
ric lux potential at permanent wilting point, which by deinition is equal to zero
(Eq. (
6.3
)). Therefore, in the falling rate phase,
T
D
rr
2
−
2
r
r
+
( )
2
2
M
=
a
0
rr
ln
(6.10)
m
0
2
2
r
0
To account for water uptake per soil layer, we apply an equation similar to Eq. (
6.8
),
substituting the
T
p
/
D
r
term by the root water uptake per unit of soil volume at depth
z, S
z
(m
3
m
-3
d
-1
):
−=
−
r
2
r
2
S
r
+
( )
0
,
z
MM
z
r
2
r
2
ln
(6.11)
z
0
,
z
m
,
z
0
,
z
2
2
r
0
,
z
in which the index
z
refers to layer-dependent parameters.
At a radial distance from the root surface
r
a
(m), the water content will be equal
to the mean (bulk) soil water content in the rhizosphere, and
M
a
is the corresponding
matric lux potential. Therefore a coeficient
a
z
can be deined as:
r
r
=
a,
m,
z
(6.12)
a
z
z
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