Published online 24 January 2007
Published in Vadose Zone J 6:124-139 (2007)
DOI: 10.2136/vzj2006.0086
© 2007 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
ORIGINAL RESEARCH
The Shape of the Transpiration Reduction Function under Plant Water Stress
Klaas Metselaara and
Quirijn de Jong van Lierb,*
a Dep. of Environmental Sciences, Wageningen Univ., Droevendaalsesteeg 4, 6708 PB Wageningen, the Netherlands
b Exact Sciences Dep., Esalq-Univ. of São Paulo, 13418-900 Piracicaba (SP), Brazil, currently at Dep. of Environmental Sciences, Wageningen Univ., the Netherlands
* Corresponding author (qdjvlier{at}esalq.usp.br)
Funding provided by the framework of the Dutch National Research Programme Climate Changes Spatial Planning and by CAPES-MEC, Brazil.
Received 30 June 2006.
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ABSTRACT
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Assuming transpiration to be reduced after a critical pressure head (usually chosen as –1.5 MPa or –150 m) at the root surface has been reached, transpiration rates in this so-called falling-rate phase were analyzed numerically for soils described by the van Genuchten–Mualem equations (numerical soils). The analysis was based on the differential equation describing radial flow to a single root. Numerically, the system was simulated by an implicit scheme. It is shown that, at limiting hydraulic conditions, relative transpiration (ratio between actual and potential transpiration) is equal to relative matric flux potential (ratio between actual matric flux potential and matric flux potential at the onset of limiting hydraulic conditions). Given this equality, transpiration reduction functions as a function of soil water content and as a function of time are presented for five types of analytical soils: a constant diffusivity, Green and Ampt, Brooks and Corey, versatile nonlinear, and exponential soil. While in the case of constant diffusivity, relative transpiration decreases as a linear function of water content, for the remaining four cases the decrease is a concave function of soil water content. Numerical simulations also result in a concave shape, unless the difference between water content at the onset of limiting hydraulic conditions and at permanent wilting is very small, for example, at high root densities. These discrepancies may be explained by the relative importance of a transition period between the constant- and falling-rate phases.
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INTRODUCTION
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ALTHOUGH detailed root architecture modeling (Wu et al., 2005) and small-scale measurement of root water uptake (Aylmore, 1993; Doussan et al., 2006) is progressing by leaps and bounds, the development of root water uptake functions to modify transpiration fluxes in global meteorological models (Sellers et al., 1997), and to modify the crop growth rate in land use models (e.g., van den Berg and Driessen, 2002) at present seems to lag behind. This gap can be bridged either by summarizing complex root architecture models to such an extent that their input requirements match the input available for the model of interest, or by further developing the theoretical basis for the root uptake models presently used, while retaining their modest input requirements. Actual research topics (climate change, water use efficiency in agriculture) provide a strong motivation to follow one of these research tracks. This study focused on the theoretical basis of the root water uptake functions presently used.
In hydrological models, the approach to root water uptake is that it reaches a maximum level, potential transpiration, under nonlimiting hydraulic conditions (van den Berg and Driessen, 2002). Under these conditions, transpiration occurs at a rate that does not depend on soil water content, and it is also referred to as the constant-rate phase. This maximum uptake is partitioned throughout the rooting depth, where the partitioning is assumed to be proportional to root density (Feddes and Raats, 2004). Unless water is added by rainfall or irrigation, transpiration during the constant-rate phase will cause soil water depletion and hydraulic conditions will turn from nonlimiting into limiting. From this point, under limiting hydraulic conditions, transpiration will decrease with soil water content and therefore this phase is often referred to as the falling-rate phase (e.g., Palmer et al., 1964; Feddes and Raats, 2004; Kozak et al., 2005).
The behavior of transpiration under both nonlimiting and limiting hydraulic conditions expressed as a function of a soil state variable (water content or pressure head) and normalized by potential transpiration is known as the transpiration reduction function, sometimes as the transpiration extraction function (Molz, 1981), or the moisture availability function (Sellers et al., 1997). It is assumed not to vary with depth.
The use of spatial averages is referred to as the macroscopic approach; a local analysis of the hydraulic conditions near a single root is referred to as the microscopic approach, or as the mesoscopic approach (Raats, 2007).
Mathematical analyses for the microscopic approach have been presented by Philip (1957) and Gardner (1960) for the steady-state case and by Cowan (1965) for the steady-rate case. The use of these results has been extended to the falling-rate phase by a sequence of steady-rate (or steady-state) solutions with iteratively adapted values of the soil physical characteristics (Passioura and Cowan, 1968). The mathematical analyses have been reviewed by Tinker (1976) and more recently by Raats (2007).
After the first theoretical analyses, leading to root water uptake functions reviewed by Molz (1981), subsequent approaches to estimate transpiration rates under limiting hydraulic conditions adopted the macroscopic approach. Practical applications and problems of the macroscopic reduction functions actually used have been reviewed by van den Berg and Driessen (2002).
An important issue is to establish when and which hydraulic conditions first become limiting. Passioura (1980) discussed early studies, and the discussion, notably about the location of the most important hydraulic resistance within the soil–plant–atmosphere continuum, is ongoing (Sperry et al., 2002). Brisson (1998) presented an analysis using an exponential relation between hydraulic conductivity and pressure head; Slabbers (1980) presented an analysis using a critical leaf water potential. Sadras and Milroy (1996) reviewed the threshold value of plant-available water in the soil at which tissue expansion or gas exchange (photosynthesis, evapotranspiration and transpiration, and stomatal conductance) starts to decrease. Jong van Lier et al. (2006) based their analysis of the threshold value at which transpiration starts to decrease on matric flux potential (Raats, 1970) in the soil. They showed by numerical simulations that the matric flux potential corresponding to the volume-averaged water content at first occurrence of limiting hydraulic conditions is independent of soil type and depends only on the potential transpiration rate and root density. For a known matric flux potential function, the threshold pressure head or the threshold water content can be calculated.
Another issue is the experimental and theoretical definition of a lower uptake threshold below which transpiration is zero. A value for the lower limit of soil water availability that is often used, but is criticized as well (Savage et al., 1996), is a pressure head of –1.5 MPa.
Once the lower and upper threshold have been defined, one remaining question is that regarding the shape of the reduction function between these two threshold values. Macroscopic approaches to estimate transpiration rates under limiting hydraulic conditions (reviewed by Molz, 1981) adopted an empirical piecewise linear relation with a spatial average of pressure head h (e.g., Feddes et al., 1988) or with a spatial average of soil water content 
(e.g., Doorenbos and Kassam, 1986). Nonlinear empirical relations have also been proposed (e.g., Minhas et al., 1974; Skaggs et al., 2006).
Whereas Jong van Lier et al. (2006) analyzed the upper threshold value, the objective of this study was to provide an estimation of transpiration rates in the falling-rate phase, i.e., to provide insight in the shape of the reduction function between the wilting point and the point where transpiration starts to decrease. To do so, we fixed the value of the wilting point at a pressure head of –1.5 MPa, and assumed root water uptake to be uniformly distributed throughout a layer of interest, i.e., the rooting depth. In terms of soil physical characteristics, our approach is based on Raats (2001), who distinguished two groups of soils that are often encountered in hydrology: one group referred to as the "analytical soils" and another group referred to as "numerical soils." The first group is characterized by soil physical properties that yield flow equations that can be solved analytically, in most cases as a result of linearization following one or more transformations. The second group is used in numerical studies and is based on soil nonlinear hydrodynamic characteristics not tractable analytically (pore size distribution combined with Poiseuille flow). Numerical simulations were be performed to analyze soils described by the van Genuchten–Mualem equations (Wösten and van Genuchten, 1988). Root water extraction was also investigated for the analytical soils, based on findings from the numerical simulations.
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MATERIALS AND METHODS
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Numerical Simulation
Simulations were executed using the model described by Jong van Lier et al. (2006), with the van Genuchten equation set (Wösten and van Genuchten, 1988) to define hydraulic characteristics. Hydraulic data for three typical topsoils from the Dutch Staring series (Wösten et al., 2001) were used, as listed in Table 1. The main simulation scenario parameters are listed in Table 2. We assumed a constant daily transpiration.
A summary of the main model characteristics is presented here; more details are given by Jong van Lier et al. (2006).
The root system throughout the rooting depth z was supposed to be composed of vertically oriented cylindrical roots, all with the same diameter and equally distributed (Fig. 1a
). All roots had the same water potential at their surface, and soil-to-root and within-plant resistances to water flow were not computed. There were no vertical differences in root density. Figure 1b shows a schematic representation of the root and axial segments considered for the simulations, including the distance from the root center, r, the root radius, r0, and the radius of the root extraction zone, rm.
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Fig. 1. Schematic representation of (a) spatial distribution of roots in root zone and (b) root and axial segments, as used for the simulations (rm is the radius of the root extraction zone, r0 is the root radius, dr is the root segment size, and z is the rooting depth).
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Segment size (dr) was chosen smaller near the root and larger at greater distance, according to
 | [1] |
with dr,min = 10–8 m, dr,max = 5.10–4 m, and S = 0.5. Distances r, r0, and rm, in meters, are the distance from the root center, the root radius, and the radius of the root extraction zone (equal to the half-distance between roots). The chosen segment size distribution resulted in 24, 73, 228, and 717 segments for the high, medium, low, and very low root density simulations, respectively.
Radial root water extraction was simulated by an implicit numerical scheme, adapted from van Dam and Feddes (2000) to simulate water movement in the rhizosphere. Main differences compared with the van Dam and Feddes (2000) algorithm are that the gravitational component was disregarded, that root water extraction was analyzed in an axisymmetric radial geometry, and that the pressure head at the root surface had a minimum value, hl, assumed to be –150 m. The outer boundary condition is a no-flow condition.
The implicit numerical scheme to calculate the water flow between segments and into the root was solved iteratively for each time step. The time step was optimized for convergence purposes but limited to a minimum of 1 s and a maximum of 1 h. For details, refer to Jong van Lier et al. (2006).
Mean water content 
(m3 m–3) was calculated as the weighted average of all segment water contents:
 | [2] |
where (r) is the water content at r and rm was calculated from rooting depth z (m) and total root length per unit area L (m–1) as
 | [3] |
Pressure head at mean water content (h) and matric flux potential at mean water content (M) were calculated from the mean water content using the van Genuchten hydraulic functions (Wösten and van Genuchten, 1988).
Basic Assumptions for Analytical Solutions
Considering a drying soil with no water recharge by any means, and averaging water content throughout rooting depth z, actual flux density during limiting hydraulic conditions toward the idealized root (qa, m d–1) can be calculated from actual transpiration rate (Ta, m d–1), area per plant Ap (m2), and total root area Ar (m2). On the other hand, it can also be calculated from the Darcy equation, therefore:
 | [4] |
where K (m d–1) is the hydraulic conductivity and h (m) is the pressure head.
Matric flux potential (M, m2 d–1) was defined as the integral of unsaturated hydraulic conductivity [K(h), m d–1] over pressure head, equivalent to the integral of diffusivity [D(), m2 d–1] over water content (, m3 m–3). It is convenient to choose the permanent wilting point in terms of pressure head (hw, m) or water content (w, m3 m–3) as the lower bound of the integral:
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Rewriting Eq. [4] in terms of the matric flux potential defined by Eq. [5] yields
 | [6] |
with M0 being the matric flux potential at the root surface and 
r a small increment of r. During the falling-rate phase, we assumed the soil at the root surface to be at the permanent wilting point, and therefore M0 = 0. At the time of first occurrence of limiting hydraulic conditions (tl), when transpiration is only just potential transpiration, and matric flux potential at the root surface has just become zero, a similar equation holds:
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where Tp (m d–1) is the potential transpiration rate and qp (m d–1) is the flux density toward the root under nonlimiting hydraulic conditions. Dividing Eq. [6] by [7] yields
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We now made a very important assumption to be tested by numerical means: we assumed that Eq. [8] holds not only very close to the root surface, but across the entire range from r0 to rm:
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where Ml (m2 d–1) is the mean matric flux potential at the onset of the falling-rate phase.
Some support for Eq. [9] is provided by a theoretical result regarding the maximum steady-state flow rate to roots (Lang and Gardner, 1970; Whisler et al., 1970; Nye and Tinker, 1977; Personne et al., 2003). The differential equation for the annular flow Iw from a distance rb (Eq. [2].12 from Nye and Tinker, 1977)
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can be solved for steady-state conditions in terms of the matric flux potential M:
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In the quoted analysis, a sink of infinite strength is assumed: the root surface is at a constant potential that does not decrease further. Assuming this potential to be equivalent to the permanent wilting point, the matric flux potential at the root surface is zero (M0 = 0), as defined above. Defining an additional scaling value—the value Mbp of M at the distance rb at which the annular flux is only just potential (Iwp), it can be shown that
 | [12] |
which should hold at all distances rb. Equation [12] should also hold for the matric flux potential associated with the volume-averaged water content, which is the variable of interest in macroscopic models. Using results presented by Cowan (1965), but also by Passioura (1980), and Moldrup et al. (1992), Eq. [9] can also be derived for a steady-rate condition and a no-flux boundary condition at the outer radius of the soil cylinder enveloping the root (results not presented).
While the results for both steady-state and steady-rate conditions are supportive of our hypothesis, to argue why a steady-state (or -rate) assumption should hold in the falling-rate stage is not straightforward, and the validity of Eq. [9] still needs to be tested.
If tested and shown to be valid, Eq. [9] can be used to obtain an analytical solution for the mean water content in the cylinder surrounding the root in the falling-rate phase, as from mass conservation it follows that the amount of water lost by transpiration in this phase is
 | [13] |
where z (m) is the rooting depth. Substitution of Eq. [9] into this expression yields
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This differential equation describes the decrease of mean water content as a function of M/Ml at any r and, thus, Ta/Tp as a function of time after the time of first occurrence of limiting hydraulic conditions (tl, d). Its specific solution is determined by the choice of the M– relation. We will discuss analytical solutions of this equation, for five soil types.
Types of Analytical Soils
Analytical solutions for as a function of time and for Ta/Tp as a function of are derived for the following special soils:
- Constant diffusivity soil, for which
 | [15] |
where D* (m2 d–1) is the constant diffusivity and w (m3 m–3) is the soil water content at the permanent wilting point, here considered to be equivalent to the water content at a pressure head of –150 m.
- Green and Ampt soil, for which the diffusivity is defined by a scaled Dirac delta function:
 | [16] |
where S (m d–1/2) is the desorptivity and l (m3 m–3) is the soil water content in the root water extraction sphere, the drying front water content. Matric flux potential, as the integral of diffusivity over , is then defined by the Heaviside function (Abramowitz and Stegun, 1972, p. 1020):
 | [17] |
- Brooks and Corey power function soil, for which
 | [18] |
where Ds (m2 d–1) is the hydraulic diffusivity at saturation, r (m3 m–3) and s (m3 m–3) are the residual and saturated water contents, respectively, a is an empirical parameter, 
is the relative water content defined by
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and w is at = w. For this soil, diffusivity is defined in the appendix (Eq. [A17]).
- Versatile nonlinear soil, for which diffusivity is defined by
 | [20] |
and for which
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where b is a constant.
- Exponential diffusivity and matric flux potential soil, for which
 | [22] |
where c is a constant. For this soil, diffusivity is defined in the appendix (Eq. [A42]).
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RESULTS AND DISCUSSION
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Numerical Evidence of Validity of Equation [9]
In Eq. [9], M/Ml was assumed to be equal to Ta/Tp at any r within the rhizosphere. To verify this assumption, we simulated different combinations of transpiration rate, root density, and soil (scenarios from Table 2). It was confirmed that in the falling-rate phase, M profiles are independent of soil type; similar results were obtained by Jong van Lier et al. (2006) for the constant-rate phase. In Fig. 2
, M/Ml was plotted as a function of distance at selected times for the clay soil at Tp = 6 mm d–1 and low root density. Visual inspection of this figure suggests that, for any t > tl, M/Ml has a constant value independent of r.
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Fig. 2. Relative matric flux potential M/Ml as a function of distance from the root center at selected times, obtained from numerical simulations with the van Genuchten hydraulic relations for the clay soil at a potential transpiration rate of 6 mm d–1 and low root density (tl = time of first occurrence of limiting hydraulic conditions).
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To take a more detailed look at the deviations of M/Ml with distance from the root center, a relative deviation Mr (%) was calculated as
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where Ml is the matric flux potential at t = tl. In Fig. 3
, Mr obtained from the same simulations used for Fig. 2 is plotted for several times within a period of 2 wk immediately after the first occurrence of limiting hydraulic conditions. Relative deviations increase as a function of time and are largest close to the root surface; however, deviations do not exceed 1.3%. Some small numerical irregularities can be detected in this figure as a result of rounding errors in Eq. [23].
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Fig. 3. Relative deviation of matric flux potential ratio M/Ml from its value at the outside of the rhizosphere at selected times from time of first occurrence of limiting hydraulic conditions (tl) between tl + 0.16 d and tl + 14 d, as a function of distance from the root center at selected times, obtained from numerical simulations with the van Genuchten hydraulic relations for the clay soil at a potential transpiration rate of 6 mm d–1 and low root density.
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Figure 4
shows transpiration as a function of mean matric flux potential for very low, low, and medium root density and for some combinations of soils and transpiration rates. As can be seen, at M 
Ml, data points can be described by a straight line with a slope equal to Tp/Ml, in agreement with Eq. [9]. In some cases (at high root density, as well as at medium root density in combination with the loam soil), the value of the mean water content at first occurrence of limiting hydraulic conditions (l, m3 m–3) becomes almost equal to w. As a result, the decrease of water content from l to the permanent wilting point w at potential transpiration (tl–w) requires <1 d (Table 3). Because we assumed a constant daily transpiration level, the results for these very small differences were regarded as unrealistic. These cases were excluded from Fig. 4, and not considered for further analysis.
In view of these results, we consider Eq. [9] to be a reasonable assumption. This is very useful for current ecohydrological models using macroscopic root water uptake functions. In these models, actual transpiration is calculated from average water content or hydraulic head at a particular depth. Given the matric flux potential as a function of or h, the corresponding relative transpiration Ta/Tp can be estimated with Eq. [9] at any or h. If the integral of conductivity or diffusivity defining the matric flux potential does not easily yield a fully evaluated symbolic expression, numerical integration offers an alternative.
Numerical Simulation with van Genuchten Hydraulic Functions
Numerical simulations with the van Genuchten hydraulic relations for the three soils (Table 1) were done at two potential transpiration rates and four root densities (see Table 2). Resulting Ta/Tp vs. graphs are shown in Fig. 5
for Tp = 3 mm d–1. At very low root density, the functions are concave, whereas at higher root densities their shape becomes less concave or even convex. Similar functions, albeit with a stronger concavity, are obtained when plotting Ta/Tp vs. h (results not presented). During water depletion, Ta/Tp values come very close to zero as water content approaches w. It can also be seen that in one of the simulated cases (clay soil, very low root density), hydraulic conditions are limiting even at field capacity, defined as –1 m pressure head.
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Fig. 5. Relative transpiration Ta/Tp as a function of mean soil water content , obtained from numerical simulations with the van Genuchten hydraulic relations for the three soils (Table 1) at four root densities (RD, Table 2) and low transpiration rate (3 mm d–1).
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In Fig. 5, we observe that there seems to be a relation between the shape of the curve and the difference l – w: the smaller this difference, the less concave the curve. For very small values of l – w (as occurred for some soils at high and medium root density, Table 3), the curves become convex. To analyze this relationship at the same scale, we plotted Ta/Tp as a function of 
, which is scaled as
 | [24] |
Rescaling Fig. 5, excluding the cases for which tl–w was <1 d (all soils at high root density and loam soil at medium and high root density) results in Fig. 6
. In this figure, the change of curvature from concave to convex with a decreasing difference l – w (corresponding to increasing root density) can be observed in more detail. When interpreting this and subsequent figures, one should keep in mind that = 0 corresponds to w, which in this study is assumed to depend on soil properties only. However, = 1 corresponds to l, which, apart from soil properties, also depends on transpiration rate and root density. Low values of Tp and high root densities result in low values of l (Jong van Lier et al., 2006).
At present, the following hypothesis is suggested to understand the change in curvature: transpiration rate as a function of soil water content can be divided into three phases—the constant-rate phase, the falling-rate phase, and a relatively short transition phase in between. We hypothesize that, in the transition phase, the ratio Ta/Tp is a convex function of water content and, in the falling-rate phase, it is a concave function of water content. If the difference l – w is small, and therefore the total period for the transpiration reduction is short, transition time dominates flow processes and the reduction function is predominantly convex.
Analytical Solutions
Given that the numerical analysis supports Eq. [9], analytical solutions for as a function of time (based on Eq. [9]) and for Ta/Tp as a function of (based on Eq. [14]) for the special soils defined by Eq. [15], [17], [18], [21], and [22] are presented in Table 4. The derivation of the equations in this table can be found in the appendix.
For a constant-diffusivity soil, the result is a linear reduction of Ta/Tp with (Eq. [A4]). Therefore, a possible assumption implicitly underlying piecewise linear transpiration reduction functions as proposed by, e.g., FAO (Doorenbos and Kassam, 1986) is a constant diffusivity. For the other types of analytical soils, the expressions result in a concave dependence of Ta/Tp on , as shown in Fig. 7
. In this figure, boundary conditions from Table 2 with low root density and s, r, and w from the sand soil (Table 1) were used.
Green and Ampt Soil
For the Green and Ampt soil (Eq. [A16]):
 | [25] |
This equation has a concave shape with an exponent that should be >1, as Ml, z, Tp, and (rm – r0) have positive values. According to Jong van Lier et al. (2006), analyzing results from numerical simulations for the soils, transpiration rates and root densities used in this study, mean hydraulic conditions occurred at around 0.53rm from the root surface and a regression analysis showed that (n = 7, r2 = 0.9997):
 | [26] |
with p = 23.5 and q = 2.367. Using these values in combination with z = 0.5 m and r0 = 0.5 mm (Table 2), the exponent in Eq. [25] equals 5.17, 3.84, and 3.12 for low (rm = 56.4 mm), medium (rm = 17.8 mm), and high (rm = 5.64 mm) root density, respectively. The resulting relations between and Ta/Tp are shown in Fig. 7a. As shown in this figure, a higher coefficient, corresponding to a lower root density, leads to a sharper decrease of Ta/Tp with decreasing . To show the implications of this in a specific case, Fig. 8a
is the transformation of Fig. 7a to a scale for the sand soil at low transpiration rate. The value of l was obtained from Jong van Lier et al. (2006). As l approaches w with increasing root density, the scaled curves from Fig. 7a represent an ever smaller range of . Figure 8b shows the transformation of the concave –Ta/Tp curves from Fig. 8a to result in even more concave h–Ta/Tp curves.
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Fig. 8. Relative transpiration Ta/Tp (a) as a function of volumetric water content and (b) as a function of pressure head h, for three root densities (RD) according to the analytical solution for the Green and Ampt soil, applied to the sand soil at low transpiration rate (3 mm d–1).
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Brooks and Corey Soil
For the Brooks and Corey soil, both the general solution (Eq. [A24]) and the case with w = 0 (Eq. [A25]) show Ta/Tp to depend on l, w, and a. If a > 0, both solutions result in concave Ta/Tp– relations. From the theory presented by Brooks and Corey (1964), the parameter a equals 3 + 2
, where is the pore-size index characterizing the retention curve. Rawls et al. (1992) present a pedotransfer function to estimate and found values between 0.1 and 0.7 for selected texture classes, corresponding to values of a in the order of 3 to 5. In Fig. 7b, the relation between and Ta/Tp is shown using a = 4, for the general solution, and with values of l according to
 | [27] |
for A equal to 0.1, 0.5, and 0.9, and for the case with w = 0. Lower values of A, corresponding to lower values of l (closer to w) caused by either higher root densities or lower transpiration, are shown to have a more linear, less concave decrease of Ta/Tp with , similar to the Green and Ampt solution. At very low A, when l 
w, corresponding to a very high root density, Eq. [A24] can be shown to become equivalent to Eq. [A4], i.e., linear in . For the case of w = 0, the decrease of Ta/Tp with is even sharper.
Versatile Nonlinear Soil
The ratio Ta/Tp for versatile nonlinear soils depends on l, w, and b. Ten Berge (1990) reported an extensive list of soils with their corresponding hydraulic properties. Values of b varied between 1.0046s and 1.1670s with a median value of 1.0402s, which is the value used for Fig. 7c. The same values for A to calculate l (Eq. [27]) as in the Brooks and Corey soil were evaluated. Results are in Fig. 7c, with a concave shape similar to that of Green and Ampt and Brooks and Corey soils. Compared with the Brooks and Corey soils and at the same value of A, however, curves are closer to the linear (constant-diffusivity) soil, especially for the low values of A, which correspond to high root densities.
Exponential Soil
For these soils, Ta/Tp is dependent on l, w, and c. Values for c (fitted across a specific range of ) can be found in Suleiman and Ritchie (2003), varying within the range of 10 to 100, with a median value of 26.03, which was used to make Fig. 7d. A similar response of curvature to the value of A as in the previous three soils can be observed. The sensitivity to variations in A (and hence in root density), however, is much higher for this type of soil.
Discussion of Model Assumptions
The parameters used in the analysis were calculated on the basis of root system data and transpiration levels to more easily define representative values. Restrictions imposed by the assumption of a constant root density throughout a rooting depth constant in time should be taken into consideration when interpreting model output. A model test should consist of predicting measured actual transpiration; using data on average root density, rooting depth, potential evapotranspiration, and soil physical characteristics; and establishing predictive quality.
As the input data required are rarely available simultaneously, and the way in which to generate, e.g., average root density values from a root density profile are as yet unknown, a quantitative comparison between our simulation results and reported experimental data is not straightforward and outside the scope of this study. The ideal study for this type of verification would be one in which Ta/Tp has been established as a function of pressure head or water content, and where in addition rooting density and soil physical characteristics are known, e.g., Lawlor (1972). The root density in his experiment is very high, however, and transpiration reduction becomes a stepwise transition.
Personne et al. (2003) tested a model along similar lines, but with a root density varying with depth, against field experiment data. In addition, real root system functioning is much more complicated: roots are neither equidistant nor do they have an equal diameter and sink strength (Doussan et al., 2006); in fact they tend to clump or cluster (Tardieu et al., 1992; Logsdon and Allmaras, 1991) and the contact with the soil is often restricted by shrinking (Van Noordwijk et al., 1993; North and Nobel, 1997). Also, resistance to water transport in the soil–plant–atmosphere system may not reside entirely in the soil (Sperry et al., 2002). Concerning root length density and root activity, these are shown to be variable with depth (Schenk and Jackson, 2002; Pregitzer et al., 2000), and root water uptake is a function of temperature (Cox and Boersma, 1967; Mellander et al., 2006). The root system grows, and grows differently according to water availability (Zerihun et al., 2006) and soil structure (Droogers et al., 1997) or soil density (Lipiec and Hatano, 2003). Furthermore, transpiration levels were constant, and no diurnal course was assumed.
Notwithstanding the simplifying assumptions, the analysis offers theoretical support to the empirical approaches actually used in hydrological and crop production models (van den Berg and Driessen, 2002). At the level of the concepts used in these models, qualitative comparisons can certainly be made, as experimentally determined reduction functions have been presented in the literature. According to our theoretical results, these relationships should be predominantly concave between the critical water content and wilting point. The shape should be related to the soil physical characteristics or to root density.
Sadras and Milroy (1996) reviewed studies in which the (evapo)transpiration reduction function was analyzed. Their overview showed that, in roughly half of the sources analyzed (10 out of 18), a piecewise linear function was used to describe the reduction function. As the measurement of transpiration is difficult and not without error, and as in field experiments day-to-day variability adds complexity, available results show a large amount of scatter. The problem posed by experimental scatter can be recognized in results presented, e.g., by Tomar and Ghildyal (1973) for rice (Oryza sativa L.), by Meyer and Green (1980) for wheat (Triticum aestivum L.), by Rosenthal et al. (1987) for sorghum [Sorghum bicolor (L.) Moench] and cotton (Gossypium hirsutum L.), and by Masinde et al. (2006) for three nightshade (Solanum ssp.) genotypes. These researchers described their results using piecewise linear functions, neither contradicting nor supporting the theory presented here. In 7 out of 18 sources reviewed by Sadras and Milroy (1996), a nonlinear function was used to describe the reduction function. A logistic function—typically a convex–concave function—was fitted in four sources (Al-Khafaf et al., 1978; Mason et al., 1980; Hammer and Muchow, 1990; Muchow and Sinclair, 1991; Milroy and Goyne, 1995).
As the analysis presented here focuses on the shape of the curve between the critical point and wilting point, this aspect may be lost in curve fitting across the full water content range, notably when using lower order polynomials or convex functions. Extending the above review, relationships manually drawn by Gardner and Ehlig (1963) suggest concave h–Ta/Tp reduction functions for birdsfoot trefoil (Lotus corniculatus L.); convex–concave –Ta/Tp reduction functions were fitted for different crops by Morison and Gifford (1984) and presented for siratro [Macroptilium atropurpureum (DC.) Urb.] and wheat. Miller and Gardner (1972) presented convex–concave visual fits for snap bean [Phaseolus vulgaris L. var. vulgaris]. More recently, another four studies (Lecoeur and Sinclair, 1996; Wopereis et al., 1996; Ray and Sinclair, 1998; Sarr et al., 2004) presented an empirically determined reduction function and fit a logistic function.
Given these eight instances, we conclude that there is experimental support for the combined convex–concave reduction functions as predicted by the numerical model for Ta/Tp as a function of both (Fig. 5) and h. Considering also the numerical evidence shown in this study, a combined convex–concave reduction function is a more likely shape for the reduction function than a linear function. In addition, a linear reduction function is derived for a soil with a constant diffusivity, which is unlikely to occur under real conditions. It is interesting to note that Sinclair (2005), applying Cowan (1965) using a Clapp and Hornberger (1978) soil, found that root length density, transpiration rate, and rooted depth had little effect on the shape of the reduction function. For the Brooks and Corey soil, a power function identical to Clapp and Hornberger's function, we also found that the shape of the reduction function only depends on the soil physical characteristics. Root density and potential transpiration rate, however, do have an effect on the critical pressure head at which reduction starts (Jong van Lier et al., 2006). The scaling used by Sinclair (2005) removed this dependence. Sinclair's Eq. [10] can be rewritten to show that the reduction is linear in pressure head between wilting point and critical matric head. This hypothesis was also used by Feddes et al. (1988). In contrast to our results, Sinclair's (2005) analysis leads to a convex shape of the reduction function as a function of water content.
Whereas there seems to be an experimental and a theoretical basis for a nonlinear relation between soil water content and transpiration reduction, the predictive quality of this approach, as well as the sensitivity of plant growth and hydrological models to different curvatures of reduction functions should be evaluated in future research. Perhaps as Sperry et al. (2002) hypothesized, this approach only allows quantifying the system behavior as bounded by the soil physical properties, the soil hydraulic envelope.
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CONCLUSIONS
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- Numerical simulation shows that, under limiting hydraulic conditions, relative transpiration equals relative matric flux potential.
- On the basis of the relation M/Ml = Ta/Tp, analytical solutions for Ta/Tp as a function of and for as a function of time could be derived for five types of analytical soils.
- The analytical results show that, between the critical condition and wilting point, relative transpiration Ta/Tp decreases as a concave function of soil water content. Combining these with the usual –h relations, the decrease is concave also with respect to pressure head.
- The analytical results show that the reduction function approaches a linear form for increasing root densities or decreasing potential transpiration levels.
- The transpiration reduction function is a (piecewise) linear function of water content only if diffusivity is constant.
- Numerical simulations show that a convex reduction function occurs shortly after the onset of limiting conditions, which may be explained by a transition period between the constant- and falling-rate phases. If the difference between water content at the onset of limiting hydraulic conditions and at permanent wilting is small, e.g., due to high root densities, this transition period gains relative importance, resulting in a convex reduction function.
- Experimental support for the combined convex–concave reduction functions found in literature, combined with the numerical evidence shown in this study, makes this a more likely shape than the linear function sometimes assumed.
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APPENDIX
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Derivations of Analytical Solutions for Mean Water Content as a Function of Time and for Relative Transpiration as a Function of Mean Water Content
Soil 1: Constant Diffusivity
In the case of a constant diffusivity D* and the associated linear matric flux potential function M():
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| [A1] |
 | [A2] |
where w is the water content at permanent wilting. Defining
 | [A3] |
and combining Eq. [9] with [A1]
to [A3] yields
 | [A4] |
The differential equation for the decrease of water content with time becomes
 | [A5] |
with
 | [A6] |
Equation [A5] can be rewritten and solved to yield a linear reservoir-type decrease for (t) at any r:
 | [A7] |
Soil 2: Green and Ampt Soil
Equation [17] defines M for these soils; M can also be described by a logistic approximation:
 | [A8] |
where the logistic defines the Heaviside step function in the limit of S 0. Equating the change in water content in the soil to the change in l, solving both the change of with time and the relative transpiration rate becomes a case of describing sorptivity, as
 | [A9] |
and
 | [A10] |
Sorptivity is defined as the integral of the Boltzmann variable ():
 | [A11] |
and () describes the wetting–drying front and is the intermediate variable used to determine diffusivity D from horizontal infiltration or evaporation experiments. For the Green and Ampt soil in this root uptake geometry, the sorptivity is
 | [A12] |
which yields
 | [A13] |
This integral can be solved to yield
 | [A14] |
where the dimensionless time constant 
g is defined as the derivative in t = tl:
 | [A15] |
Using the definition of the sorptivity and the derived time course of shows that the relative transpiration is equal to
 | [A16] |
Soil 3: Brooks and Corey Power Function
Diffusivity D (m2 d–1) as a power of relative water content is
 | [A17] |
with
 | [A18] |
where r and s are the residual and saturated water contents, respectively. Integration of Eq. [A17] yields M:
 | [A19] |
The integral to be solved becomes
 | [A20] |
This integration is straightforward if w is equal to r, in which case w = 0 and integration yields
 | [A21] |
which can be shown to be equivalent to
 | [A22] |
where 
b is the time constant, calculated as the value of the derivative at t = tl:
 | [A23] |
Substituting Eq. [A19] in Eq. [9] yields the ratio Ta/Tp:
 | [A24] |
or, for the case with w = 0:
 | [A25] |
Soil 4: Versatile Nonlinear Soils
Integration of Eq. [20] yields
 | [A26] |
from which the ratio Ta/Tp can be derived by applying Eq. [9]:
 | [A27] |
with
 | [A28] |
To derive the –t relationship, we first apply Eq. [14], substituting Eq. [A26] and integrating, to yield
 | [A29] |
Its solution is
 | [A30] |
which cannot be rewritten explicitly in terms of .
Alternatively, we may use a series approximation of the log-term:
 | [A31] |
which yields the following second-order approximation:
 | [A32] |
Equation [A32] can be rewritten as
 | [A33] |
or
 | [A34] |
Equation [A34] has two solutions, of which the negative root yields = l for t = tl:
 | [A35] |
which can also be written as
 | [A36] |
with
 | [A37] |
This is the time constant for versatile nonlinear soils, defined as the derivative with respect to t at t = tl.
Soil 5: Exponential Diffusivity and Matric Flux Potential
From the definitions of diffusivity D and matric flux potential M, it follows that
 | [A38] |
which, combined with Eq. [14]
TOP
ABSTRACT
INTRODUCTION
