Published online 1 August 2007
Published in Vadose Zone J 6:524-526 (2007)
DOI: 10.2136/vzj2007.0036
© 2007 Soil Science Society of America
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COMMENTS
Comment on "Root Water Extraction and Limiting Soil Hydraulic Conditions Estimated by Numerical Simulation"
Tom Schrödera,
M. Javauxb,*,
J. Vanderborghtc and
H. Vereeckenc
a Central Institute for Applied Mathematics, Forschungszentrum Juelich GmbH 52425 Juelich, Germany
b Dep. of Environ. Sci. and Land Use Planning, Univ. Catholique de Louvain, B-1348, Louvain-La-Neuve, Belgium
c Agrosphere, ICG-IV, Forschungszentrum Juelich GmbH, 52425 Juelich, Germany
* Corresponding author (javaux{at}geru.ucl.ac.be).
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INTRODUCTION
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In a recent paper, de Jong van Lier et al. (2006) developed a methodology to determine the limiting averaged soil pressure head and water content under which plant transpiration is reduced. They presented a numerical finite difference method describing the flow from soil to root and validated it by a derived analytical solution. However, we noticed an error in the derivation and final definition of their analytical solution (Eq. [45] in de Jong van Lier et al., 2006).
To derive the analytical solution, the continuity equation describing water flow in the soil (Richards' equation) is approximated by steady-rate behavior, which means the time derivative term is assumed constant. The differential equation in radial form, under the condition that no gravity and no sink term are taken into account, is
 | [1] |
where q (m d–1) is the Darcy's flow denoted by q=–K(h)
, h (m) is the pressure head, r (m) is the radius, and t (d) is the time. The volumetric water content is denoted by 
(m3 m–3), and the hydraulic conductivity is denoted by K (m d–1).
De Jong van Lier et al. (2006) stated in their Eq. [37] that
 | [2] |
where Tp is the potential transpiration rate and z is the rooting depth. Correct substitution of the constant time derivative term and the usage of the Kirchhoff potential M=
K(h)dh leads to the subsequent common solution of the radial Richards' equation
 | [3] |
which was wrongly deducted by the authors (see their Eq. [38]). Solving the common solution for the subsequent set of boundary conditions as given by their Eq. [41] and [42]
 | [4] |
yields
 | [5] |
and
 | [6] |
Equation [6] must be compared with Eq. [45] in de Jong van Lier et al. (2006); both equations are plotted in Fig. 1
. It can be seen that their behavior is equal near the root surface but differs farther away from the root surface. The same numerical results are reproduced as in their Fig. 14a.
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FIG. 1. The pressure head is plotted vs. the radius (clay soil, medium root density, high transpiration rate) for the numerical solution (solid line) at different times after the start of water extraction (Fig. 14a in de Jong van Lier et al. [2006]). The results at the moment of first occurrence of limiting hydraulic conditions at the root surface are indicated by the dashed lines (numerical = black, incorrect analytical = red); in addition, the correct analytical solution (dots) is given.
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Despite the analytical error, the authors found a surprisingly good agreement between their analytical and numerical solutions (their Fig. 15). It should be noted that we assume that in their Fig. 15, the range of numerically simulated pressure heads at first occurrence of limiting hydraulic conditions at the root surface is plotted versus the range of pressure heads obtained with the analytical solution and not the mean limiting pressure head, which should be a single value.
Figure 2
shows the comparison between their analytical solution with a numerical solution of Richards' equation for different soil types (Table 1 in de Jong van Lier et al. [2006]). The same figure shows a comparison between the correct analytical solution and the numerical solution. For all soil types, the analytical solution of de Jong van Lier et al. (2006) does not match the numerical solution, especially for the smaller absolute valued pressure heads. On the other hand, the correct analytical solution now corresponds quite well to the numerical solution. Furthermore, a similar, though an opposite, discrepancy for the sandy soil as in de Jong van Lier et al. (2006) has been found between the analytical and numerical solutions. These results are obtained by imposing similar transpiration rates for both the numerical and analytical models. It seems that the potential transpiration rate chosen by de Jong van Lier et al. (2006) for the analytical model in the case of a sandy soil was not chosen identically as its corresponding numerical value.
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FIG. 2. Analytical vs. numerical solution at limiting soil pressure head (||-150 m ||) for three different soil types using a low root density profile and a high transpiration rate. Left panel shows the total pressure head range; the right panel shows a magnification. The red lines indicate the analytical solution as derived in Eq. [45] of de Jong van Lier et al. (2006); the blue lines indicate the correct analytical solution (Eq. [6]).
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For other cases as given in their Table 2 with different root density and transpiration rate, the discrepancies between the erroneous and correct analytical solutions may be larger than suggested in their Fig. 15. For instance, for a medium root density and low transpiration rate, the discrepancies are shown in Fig. 3
. Again for all soil types, the analytical solution of de Jong van Lier et al. (2006) does not match the numerical solution, especially for the smaller absolute valued pressure heads. The correct analytical solution, however, corresponds fairly well. Even for the sandy soil, both solutions correspond relatively well, in contrast to the discrepancy found for a low root density system with high transpiration rate.
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FIG. 3. Analytical vs. numerical solution at limiting soil pressure head (||-150 m ||) for three different soil types using a medium root density profile and a low transpiration rate. Left panel shows the total pressure head range; the right panel shows a magnification. The red lines indicate the analytical solution as derived in Eq. [45] of de Jong van Lier et al. (2006), and the blue lines indicate the correct analytical solution (Eq. [6]).
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Due to the change in their Eq. [45], the subsequent Eq. [46] is incorrect as well. Substitution of r by arm in Eq. [6], multiplication with a factor of 2 due to double transpiration rates (see de Jong van Lier et al., 2006), and the assumption that rm >> r0 leads to
 | [7] |
Eventually, the simplification from their Eq. [46] to [47] yields the same result as the same simplification applied to Eq. [7].
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Conclusions
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A mistake in derivation of the analytical solution led to an incorrect analytical solution in de Jong van Lier et al. (2006). This error should cause a deviation between the analytical and numerical results, which, however, was not observed by de Jong van Lier et al. (2006). The right derivation is given by Eq. [6] and was shown to match the numerical model well.
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REFERENCES
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This article has been cited by other articles:
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Macroscopic Root Water Uptake Distribution Using a Matric Flux Potential Approach
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T. Schroder, M. Javaux, J. Vanderborght, B. Korfgen, and H. Vereecken
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Q. de Jong van Lier, K. Metselaar, and J. C. van Dam
Response to "Comment on 'Root Water Extraction and Limiting Soil Hydraulic Conditions Estimated by Numerical Simulation'"
Vadose Zone J.,
August 1, 2007;
6(3):
527 - 528.
[Full Text]
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TOP
INTRODUCTION
Conclusions
