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radial water flux towards roots
h
r
m
r
0
root surface
midway between roots
Figure 6.4
Strong gradients ∂
h
/∂
r
near root surface due to radial low and low
hydraulic conductivity.
accurately when we use the matric lux potential, instead of the soil water pressure
head, as driving variable. The matric lux potential
M
(m
2
d
-1
) is deined as:
h
=
(
∫
d
w
Mk hh
h
(6.3)
where
h
w
is the pressure head corresponding to plant wilting point. Inserting the
matrix lux potential in the Darcy equation gives:
qk
h
r
=−
∂
∂
=−
∂
M
r
∂
(6.4)
When we numerically solve Eq. (
6.1
), resulting
M
(
r
) proiles are more linear than
h
(
r
) proiles, which is obvious from the linear character of Eq. (
6.4
). Use of the mat-
ric lux potential allows us to derive an analytical solution for microscopic root water
extraction, and we might even upscale this approach to entire root zones (De Jong
van Lier et al.,
2008
). The general theoretical background is explained below.
When the soil is relatively wet, root water extraction is not limited by soil hydrau-
lic resistances and is equal to the potential transpiration rate
T
p
(
Section 8.1
). In this
so-called
constant rate phase
, mass conservation yields:
T
D
∂
∂
=−
θ
a
p
(6.5)
t
r
where
θ
a
is the average soil water content in the rhizosphere (m
3
m
-3
) and
D
r
is the
thickness of the root zone (m). Numerical solution of radial soil water low to roots
showed that the change of water content with time d
θ
/d
t
is more or less independent
of
r
. Therefore Eq. (
6.5
) can be generalized for any
r
:
T
D
∂
∂
=−
θ
t
p
(6.6)
r
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