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ORIGINAL RESEARCH

Root Water Extraction and Limiting Soil Hydraulic Conditions Estimated by Numerical Simulation

^a Exact Sciences Dep., Esalq-Univ. of São Paulo, 13418-900 Piracicaba (SP), Brazil, currently at Dep. of Environmental Sciences, Wageningen Univ., Wageningen, the Netherlands
^b Dep. of Environmental Sciences, Wageningen Univ., Nieuwe Kanaal 11, 6709 PA Wageningen, the Netherlands
^* Corresponding author (qdjvlier{at}esalq.usp.br)

Received 13 April 2006.

	ABSTRACT

TOP EXECUTIVE SUMMARY ABSTRACT INTRODUCTION MATERIAL AND METHODS Results and Discussion REFERENCES

 
Root density, soil hydraulic functions, and hydraulic head gradients play an important role in the determination of transpiration-rate-limiting soil water contents. We developed an implicit numerical root water extraction model to solve the Richards equation for the modeling of radial root water extraction. The average soil water content at the moment root water potential dropped below a defined critical value was then estimated. The dependence of average water content at the onset of plant water stress on potential transpiration and root density was compared with an analytical solution for hydraulic conditions in the root sphere. The critical value was a function of potential transpiration rate, soil hydraulic properties, and root density. Matric flux potential appears to be a convenient hydraulic property to determine the onset of limiting hydraulic conditions, as numerical simulations showed that, at onset, matric flux potential vs. distance from the root surface is independent of soil type. This was also determined analytically under the constant-rate assumption. Mean water content occurs at about 0.53 times the half-distance between roots. This allows calculation of the mean limiting soil water content and pressure head from the matric flux potential at this distance, which is a function of transpiration rate and root density only. A nomogram was developed that—given the transpiration rate, the root density, and the soil hydraulic functions—allows determination of the values of mean water content and mean pressure head that occur at the onset of transpiration reduction.

	INTRODUCTION

TOP EXECUTIVE SUMMARY ABSTRACT INTRODUCTION MATERIAL AND METHODS Results and Discussion REFERENCES

 
THE INTERACTION between crop transpiration rates and soil water availability is a complex issue. In an engineering approach used in irrigation scheduling and hydrologic modeling, a threshold value for water content () or pressure head (h) is often assumed below which transpiration rates decrease below maximum values. Theoretical considerations (Gardner, 1960; Cowan, 1965) and experiments (Carbon, 1973; Zur et al., 1982) have established that root density, soil hydraulic functions, and hydraulic head gradients play an important role in the determination of this threshold value, and that the permanent wilting point (based on soil bulk values) may also vary as a function of these variables. At present, these factors are not considered in management or hydrological models. To illustrate this, Fig. 1 shows the widely used FAO irrigation guidelines for estimating limiting water content as a function of potential transpiration rate (Doorenbos and Kassam, 1986). Although four crop groups are defined and limiting water content is a function of water content at field capacity and at the permanent wilting point, root characteristics and soil hydraulic properties are not considered in a direct way.

View larger version (15K): [in this window] [in a new window]   Fig. 1. Mean water content () relative to field capacity (fc) and the permanent wilting point (pwp), * = ( – pwp)/(fc – pwp), at first occurrence of limiting hydraulic conditions, as a function of potential transpiration rate for four crop groups, as defined by Doorenbos and Kassam (1986).

Microscopic models (e.g., Moldrup et al., 1992; Personne et al., 2003; Roose and Fowler, 2003; Raats, 2006) describe water extraction at an individual root level. The most common approach in these models is to consider the root as a straight cylindrical water sink, provoking a radial water flow toward it. This flow can be modeled with the Richards equation, which has been applied to solve numerous problems of unsaturated flow. A major constraint with respect to this equation is the highly nonlinear behavior of involved parameters, which limits analytical solutions to strict assumptions and boundary conditions. Analytical solutions for the microscopic approach have been presented by Gardner (1960) and Cowan (1965). A family of analytical solutions involves the quasilinear approximation (Pullan, 1990), but the advent of powerful computers now allows iterative numerical calculation techniques with less restrictive boundary conditions.

A mixed approach in which an analytical solution of the microscopic uptake profile has been combined with the overall behavior of the root system in a numerical three-dimensional model has been presented by Heinen (2001). At present, combined modeling of root architecture and flow patterns is envisaged (Darrah et al., 2006).

Explicit numerical models, in which flow processes are simulated in discrete steps in space and time, allow simulation of any kind of scenario but the resulting calculation procedures are usually very time consuming. Implicit schemes, in which a flow problem is described by a system of equations to be solved iteratively through time, may be adapted to many situations and require much less computer time to be processed. Van Dam and Feddes (2000) presented an implicit finite difference solution using the Richards equation for the simulation of infiltration, evaporation, and shallow groundwater levels. Their solution allows a flux-controlled or head-controlled surface boundary condition.

In this study, we developed an implicit numerical scheme that solves the Richards equation for radial root water extraction. The scheme uses an algorithm adapted from Van Dam and Feddes (2000). By defining a critical root water potential, the model was used to estimate the mean soil water content at the moment root water potential drops below the critical value. The dependence of this value on potential transpiration and root density was evaluated for three soils of the Dutch Staring series (Wösten et al., 2001) with different textures. An analytical solution for hydraulic conditions in the root sphere at first occurrence of water stress was used to support the conclusions. The overall objective was to derive critical parameter values for macroscopic root water extraction models from the microscopic (single-root) models.

	MATERIAL AND METHODS

TOP EXECUTIVE SUMMARY ABSTRACT INTRODUCTION MATERIAL AND METHODS Results and Discussion REFERENCES

 
Soils and Hydraulic Properties
Root water extraction and limiting hydraulic conditions were analyzed using hydraulic data for three topsoils from the Dutch Staring series (Wösten et al., 2001) as listed in Table 1. For these soils, K––h relations can be described by the van Genuchten (1980) equation system:

[1]

[2]

in which = ( – _r)/(_s – _r); , _r, and _s are water content, residual water content and saturated water content (m³ m^–3), respectively; h is pressure head (m); K and K_sat are hydraulic conductivity and saturated hydraulic conductivity, respectively (m d^–1); and (m^–1), n, and are empirical parameters.

View this table: [in this window] [in a new window]   Table 1. Van Genuchten parameters (Eq. [1] and [2]) for three soils from the Dutch Staring series used in simulations.

[3]

Soil water content, hydraulic conductivity, and matric flux potential (integration started at h₀ = –150 m) as a function of hydraulic head are shown for the three soils in Fig. 2 .

View larger version (20K): [in this window] [in a new window]   Fig. 2. (a) Soil water content, (b) hydraulic conductivity, and (c) matric flux potential as a function of pressure head for sand, clay, and loam soils.

[4]

where t is time (d), C is the differential water capacity (d/dh, m^–1), q is the water flux density (m d^–1), r is the distance from the axial center (m), S is a sink or source term (d^–1), and H is the hydraulic head (m). To solve the Richards equation for the case of radial root water extraction, a numerical scheme based on the scheme presented by Van Dam and Feddes (2000) was used with the following modifications:

(i) the gravitational component is not considered in the root extraction model

(ii) fluxes are adapted to the one-dimensional axisymmetric nature of root extraction, i.e., fluxes converge as they approach the root, proportional to the distance from the axial center

(iii) there is no sink (the only water exit is the root surface located at the inner side of the first compartment)

(iv) flux density at the outermost compartment is set to zero

(v) flux density at the innermost compartment (the one immediately neighboring the root) is set equal to flux density entering the root, which is determined by transpiration rate and total root area

(vi) if the pressure head in the innermost compartment drops below a critical value h_lim (in this study h_lim = –150 m), pressure head is fixed to this critical value and the flux density adjusted to that

Segment size (d_r) was chosen smaller near the root and larger at greater distance, according to

[5]

with d_r,min = 10^–8 m, d_r,max = 5.10^–4 m, S = 0.5, r₀ (m) is the root radius and r_m (m) is the radius of the root extraction zone, equal to the half-distance between roots (r_m), which relates to the root density R (m m^–3) as

[6]

It can also be shown that

[7]

and

[8]

in which L (m) is the root length, z (m) is the total rooted soil depth, A_p (m²) is the surface area and A_r (m²) is the root surface area. Combining these equations, it follows that

[9]

Figure 3 shows a schematic representation of the root and axial segments considered for the simulations, including r, r₀ and r_m. The chosen segment size distribution resulted in 24, 73, and 228 segments for the high, medium, and low root density simulations, respectively. Further reduction of the segment size did not lead to significant differences in simulation results. In addition, intermediate K values were calculated using arithmetic or harmonic means, but no significant difference was found, confirming the chosen space step to be sufficiently small. Time steps were automatically adapted by the computer program to reach convergence of the iterative matrix solution in three to five iterations. In this way, observed time steps ranged from 1 s to 1 h.

View larger version (22K): [in this window] [in a new window]   Fig. 3. Schematic representation of root and axial segments as used for the simulations; rm is the half-distance between roots, r0 is the root radius, dr,i is the segment size for segment i, and ri is the distance from the axial center for segment i.

[10]

Equation [10] can be discretized by explicit linearization of K and C:

[11]

where i is the segment number (Segment 1 closest to the root) and j is the time step. In this equation, indexes refer to axial segments as illustrated in Fig. 3.

Analogous to Van Dam and Feddes (2000), Eq. [10] can be discretized using the finite difference scheme of Eq. [11], yielding the following equation to be solved iteratively:

[12]

where p is the iteration level.

A solution to the problem can be found by applying Eq. [12] to each segment and solving the resulting equation system described by a tridiagonal matrix as follows:

[48]

in which _i, ß_i, _i, and f_i are defined as follows:

1. Intermediate segments (i = 2 to i = n – 1):

[13]

[14]

[15]

[16]

2A. Root segment (i = 1) with flux boundary condition (applied while h₁ > h_lim, i.e., when transpiration meets potential transpiration demand):

[17]

[18]

[19]

in which q_root = T_pA_p/A_r, A_p being the surface area (m²) and A_r the root area (m²).

2B. Root segment (i = 1) with head boundary condition (applied while h₁ = h_lim, i.e., when transpiration is less than potential transpiration demand):

[20]

[21]

[22]

3. Outer border segment (i = n):

[23]

[24]

[25]

Intermediate K values (at positions i – 0.5 or i + 0.5) were calculated from K values at positions i – 1, i, and i + 1 obtained by Eq. [2] using the geometric mean.

Relations between Bulk Properties and Microscopic Properties
Mean water content (_mean) was calculated as the weighted average of all segment water contents:

[26]

Mean pressure head (h_mean), mean hydraulic conductivity (K_mean) and mean matric flux potential (M_mean) were calculated from the mean water content by Eq. [1] to [3]:

[27]

Transpiration demand, a function of the atmospheric conditions, together with soil hydraulic state variables, determines pressure head at the root surface. Its value, however, is limited to a minimum, h_lim, here assumed to be equal to the commonly used pressure head at the permanent wilting point, –150 m. The instant of first occurrence of h_lim at the root surface is very important, as from that moment on, water root uptake no longer meets transpiration demand. As depletion continues, relative transpiration will decrease and stop completely when the pressure head has dropped to the permanent wilting point everywhere in the soil cylinder surrounding the root. When the root potential reaches h_lim for the first time, h_mean as calculated by Eq. [26] and [27] is not equal to h_lim but, instead, has a higher value that depends on the entire root extraction history. We will refer to the hydraulic conditions at this instant as limiting hydraulic conditions, which will be defined by the subscript mean,lim, as in h_mean,lim. Thus, h_mean,lim is h_mean at the first occurrence of h_lim at the root surface. The determination of h_mean,lim and _mean,lim is a major challenge in soil physics. Simulation of flux phenomena in the root zone can help in doing so, as we will explain.

Simulation Scenarios
The main simulation scenario parameters are listed in Table 2. The variation in root densities (0.01–1 m m^–3) and transpiration rates (3–6 mm d^–1) was chosen in such a way that main crop types and climatic conditions are included (see, e.g., Stalham and Allen [2001], Pietola and Alakukku [2005], and Graveel et al. [2002] on root density, and Allen et al. [1998] on transpiration rates). Potential transpiration (T_p, mm d^–1) was supposed to vary daily according to a sine-wave function with its amplitude equal to its mean value T_p,mean. In this way, during a 24-h period, the transpiration rate varied between a minimum of zero and a maximum of 2T_p,mean. In the simulations, we assumed pressure head at the root surface to be variable, but not able to fall below –150 m. The value of –150 m is assumed to be independent of all other parameters.

	Results and Discussion

TOP EXECUTIVE SUMMARY ABSTRACT INTRODUCTION MATERIAL AND METHODS Results and Discussion REFERENCES

 
Root Pressure Head, Mean Pressure Head, and Transpiration as a Function of Time
Simulation results are shown in Fig. 4 for the two potential transpiration rates and three root densities for the clay soil. The daily fluctuation of the pressure head in the root (h_root) to match varying transpiration can be observed in this figure, especially at low and medium R. For the low R, within a few days h_root reaches much more negative values than h_mean, with a minimum every 24 h simultaneous with the maximum in the daily transpiration curve. In contrast, at high R, h_root remains almost equal to h_mean from beginning to end of the simulations.

View larger version (22K): [in this window] [in a new window]   Fig. 4. Simulated pressure head at the root surface (hroot), mean pressure head (hmean) and relative transpiration as a function of time for low and high potential transpiration rates and low, medium, and high root density in a clay soil.

View larger version (20K): [in this window] [in a new window]   Fig. 5. Pressure head as a function of distance from the axial center (hr) and mean pressure head (hmean) for low and high potential transpiration rates (Tp) for simulations with low, medium, and high root density, at the moment of first occurrence of limiting root water potential; rm is the radius of the root extraction zone or the half-distance between roots.

After the first occurrence of h_lim, root water uptake no longer meets the potential transpiration demand, as shown by the simultaneous reduction in relative transpiration in all simulations (cf. Fig. 4). Comparing the three root densities at one transpiration level, it is clear that a higher R increases the time to first occurrence of limiting root water potential. For the case of low T_p (Fig. 4a–4c) it takes 14, 31, and 34 d at low, medium, and high R, respectively; at high T_p (Fig. 4d–4f) this changes to 3, 14, and 17 d, respectively. Therefore, in this soil and as far as water uptake is concerned, there is a great benefit for a crop in increasing R from low to medium, whereas the gain of an increase from medium to high R is small.

These results depend on the actual soil hydraulic functions, although the overall tendency is not expected to change: higher transpiration demand and lower root density cause greater discrepancy between h_root and h_mean and, therefore, limiting hydraulic conditions are reached at higher values of h_mean and _mean. Low hydraulic conductivity in the root zone water content range will cause greater discrepancy between h_root and h_mean. Under these conditions, root density becomes more important for root water extraction than in soils with a higher conductivity.

Once the limiting conditions are reached, relative transpiration decreases. In the low-R scenarios, at this moment much water is still available farther away from the roots, as can also be seen in Fig. 5. During the night, transpiration decreases but established pressure head gradients maintain water movement toward the root, increasing the water content near the root. This results in a temporary relief of water stress: root pressure head becomes less negative and, during the first part of the next day, extraction can cope with potential transpiration. Due to this effect, relative transpiration decreases more slowly in the low-R scenarios. In contrast, in high-R scenarios, the exploited soil volume is almost completely depleted at the first occurrence of limiting pressure head at the root surface. As a consequence, the transpiration rate decreases much more abruptly from its maximum value to zero.

Dependence on Transpiration Rate: Diurnal Fluctation vs. Constant Rate
To analyze the dependence of h_mean,lim and the corresponding _mean,lim on potential transpiration rates, a series of simulations were performed with the three soils and different root densities.

Results for the clay soil at transpiration rates from 2 to 8 mm d^–1 are shown by the sawtooth curves identified with A in Fig. 6 . The variable shape of these lines is caused by the fact that daily variations in transpiration are simulated: when root water potential approaches its limiting value at the beginning of a night period with low potential transpiration, the occurrence of the limiting value is postponed to the next day, consequently allowing an additional small reduction in water content and pressure head. On the other hand, if the limiting root potential is reached simultaneous with a high daytime transpiration, the first occurrence of water stress will tend to be sooner and at less negative h_mean.

View larger version (19K): [in this window] [in a new window]   Fig. 6. Mean pressure head at first occurrence of limiting hydraulic conditions (hmean,lim) as a function of potential transpiration rate (Tp) for low, medium, and high root density in a clay soil simulated with and without daily transpiration amplitude (for lines identified with "A", mean transpiration equals Tp and amplitude equals Tp; for lines identified with "B", mean transpiration equals Tp and amplitude is zero; for lines identified with "C", mean transpiration equals 2Tp and amplitude is zero).

Selection of a Key Soil Physical Variable Controlling Transpiration Rate
Macroscopic limiting soil water conditions are often defined with respect to some spatial average of h (e.g., Feddes et al., 1988) or some spatial average of (e.g., Doorenbos and Kassam, 1986). Equating these spatial averages with the mean values calculated using the numerical results, Fig. 7 and 8 show that these mean values are a function of transpiration rate, soil type, and root density. A function relating relative transpiration to matric head or soil water content should then include transpiration rate, soil type (soil hydraulic parameters), and root density as parameters. Comparing Fig. 8 with the FAO guidelines (Doorenbos and Kassam, 1986) for estimating limiting water content depicted in Fig. 1 shows similarity in order of magnitude between the FAO guidelines and the simulated low root density. The FAO approach, however, does not include soil physical factors.

View larger version (14K): [in this window] [in a new window]   Fig. 7. Mean pressure head at first occurrence of limiting hydraulic conditions (hmean,lim) as a function of potential transpiration rate (Tp) for low, medium, and high root density in the sand, loam and clay soils (dotted lines refer to minimum root pressure head h = –150 m).

View larger version (15K): [in this window] [in a new window]   Fig. 8. Mean water content () relative to field capacity (fc) and the permanent wilting point (pwp), * = ( – pwp)/(fc – pwp), at first occurrence of limiting hydraulic conditions (*mean,lim) as a function of potential transpiration rate (Tp) for low, medium, and high root density in the sand, loam, and clay soils.

View larger version (17K): [in this window] [in a new window]   Fig. 9. Hydraulic conductivity at first occurrence of limiting hydraulic conditions (Kmean,lim) as a function of potential transpiration rate (Tp) for low, medium, and high root density in the sand, loam and clay soils. (Kpwp is the hydraulic conductivity at the permanent wilting point for the respective soil).

View larger version (15K): [in this window] [in a new window]   Fig. 10. Matric flux potential at first occurrence of limiting hydraulic conditions (Mmean,lim) as a function of potential transpiration rate (Tp) for low, medium, and high root density. Lines for all three soils are coincident.

[28]

with p = 23.5 and q = 2.367 (n = 7, r² = 0.9997).

View larger version (8K): [in this window] [in a new window]   Fig. 11. Ratio between mean matric flux potential at first occurrence of limiting hydraulic conditions and potential transpiration rate (Mmean,lim/Tp) as a function of half-distance between roots (rm).

View larger version (20K): [in this window] [in a new window]   Fig. 12. Nomogram for the determination of pressure head at first occurrence of limiting hydraulic conditions (hmean,lim) and relative water content at first occurrence of limiting hydraulic conditions (mean,lim) for three soils from transpiration and root density data. Dotted arrow lines illustrate nomogram use for a high transpiration rate (6 mm d–1) and a medium root density (0.1 m m–3) for the sand soil.

Comparison of Simulation Results to Analytical Solutions
The simulation results show that, at the first occurrence of limiting head at the root surface when transpiration rate is still equal to potential rate but about to drop below that, matric flux potential as a function of the space coordinate r is in good approximation independent of soil characteristics (Fig. 13 ). The difference between the three soils can be expressed as the relative r-weighted root mean square difference. Comparing M for clay and loam to sand, calculated values are in the range of 0.8 to 3.0%. The weighted mean water content corresponds to the water content in the root zone at some location between 0.53r_m and 0.57r_m for most cases, slightly depending on soil type, transpiration rate, and root density (Table 3). Further analysis of these values shows a mean of 0.56, a standard deviation of 0.06, and a median of 0.53.

View larger version (21K): [in this window] [in a new window]   Fig. 13. Matric flux potential (M) at first occurrence of limiting hydraulic conditions as a function of distance from axial center for low, medium, and high root density at low and high potential transpiration (lines for three soils coincide at this resolution); rm is the radius of the root extraction zone or the half-distance between roots.

View this table: [in this window] [in a new window]   Table 3. Fraction of the half-distance between roots (rm) at which volumetric water content () matches water content at first occurrence of limiting hydraulic conditions (mean,lim) in numerical simulations.

[29]

assuming that the average water content _mean is found at a distance r_mean. In this equation, _i is the initial water content and is the Euler–Mascheroni constant ( 0.5772).

Defining t = t_lim as the moment of first occurrence of h_lim at the root surface, this equation can be rewritten to express r_mean as a function of _mean at t_lim:

[30]

The average water content can be determined from the r-weighted integral of the (r) profile over r according to Eq. [26] as follows:

[31]

Substitution of Eq. [31] into [30] yields

[32]

which, for r_m >> r₀ reduces to

[33]

indicating that, under the assumptions made, the mean value of corresponds to at a distance of 0.6065r_m from the axial center.

For each scenario of transpiration rate and rooting density, and while the transpiration rate matches potential transpiration rate, flux density at the root surface is independent of soil characteristics. Numerical simulation results show that this independency can be extended to the whole root sphere for the three evaluated soils (Fig. 13). These results also show that the weighted mean water content corresponds to the water content in the root zone at some location between 0.53r_m and 0.57r_m for most cases, slightly depending on soil type, transpiration rate, and root density (Table 3).

The simulation-derived values in Table 3 are in rough agreement with Eq. [33], except for some values for the high-root-density scenarios. The assumption of a constant diffusivity causes analytical (r) profiles to be less depleted close to the root surface and, consequently, _mean,lim to occur at a greater distance from that surface than in the numerical simulations. For the high-root-density scenarios, it is probably not realistic to expect Eq. [33] to be valid, as one of the assumptions is an infinite extension of the root zone.

If an analytical expression for M(r) could be found, it would allow the calculation of M at the distance of mean , providing a good estimation of _mean,lim or h_mean,lim. The continuity equation for one-dimensional, axisymmetric flow (see also Eq. [4]) is

[34]

From mass conservation, it follows that

[35]

Ignoring the root volume (assuming r_m >> r₀), Eq. [35] reduces to

[36]

To solve Eq. [34], an assumption has to be made about the behavior of d/dt as a function of r. Figure 14 shows h–r and –r profiles from numerical simulations with clay soil, medium root density, and a high transpiration rate. In this figure, h–r profiles show a rather sharp increase of dh/dr near the root surface up to the moment of first occurrence of h_lim, whereas –r profiles are much more parallel with time, suggesting that d/dt is independent of r and constant during the period in which transpiration is equal to its potential rate. We assume that as long as and up to the moment the transpiration is (still) potential, the profiles of the soil hydraulic properties are in equilibrium with a constant uptake rate at the root surface. De Willigen and Van Noordwijk (1987) argued that, in the case of a roughly constant D, this implies that water content decreases at equal rates independent of the distance; corresponding matric flux potential profiles show a parallel downward shift with time (for studies using the same assumption, see, e.g., De Willigen and Van Noordwijk, 1987; Heinen, 2001; Raats, 2006). Therefore Eq. [36] can be generalized for any r:

[37]

View larger version (29K): [in this window] [in a new window]   Fig. 14. (a) Pressure head and (b) water content as a function of distance from axial center at several times after start of water extraction (clay soil, medium root density, high transpiration rate). Dotted lines indicate pressure head or water content profile at the moment of first occurrence of limiting hydraulic conditions at the root surface.

[38]

for which the following general solution is found:

[39]

where C₁ and C₂ are integration constants. This solution is known as the steady-rate solution. It was presented by Van Noordwijk and De Willigen (1987) and is also used in the model described by Heinen (2001).

Values for C₁ and C₂ can be found assuming the following boundary conditions to be valid at the moment of first occurrence of limiting root water potential:

[40]

[41]

Equation [40] combined with Eq. [9] yields

[42]

Combining these boundary conditions with Eq. [39] yields

[43]

[44]

and

[45]

Equation [45] shows that, under the assumption that d/dt has the same value for all r, M(r) is indeed independent of soil characteristics, as shown in the numerical simulations (Fig. 14). This, combined with the fact that _mean,lim occurs at a fraction of the root mean distance that is not very sensitive to soil properties, allows estimation of M_mean,lim with Eq. [45] and determination of the corresponding h_mean,lim and _mean,lim.

Figure 15 shows the values of h_mean,lim calculated from values of M at first depletion estimated by the analytical model, compared with estimates by the numerical model for the three soils at low root density and high transpiration rate. Among the simulated combinations of root density and transpiration rate, and then especially for the sand soil, this was the only case in which a small discrepancy between numerical and analytical estimates could be observed. For all other cases, discrepancies were much smaller and lines approximately matched the 1:1 line. This confirms that the assumption resulting in Eq. [37] is reasonably well supported by the numerical simulation results.

View larger version (10K): [in this window] [in a new window]   Fig. 15. Pressure head at the first occurrence of limiting root water potential (hmean,lim) calculated from analytical and numerical matric flux potential (Manalytical and Mnumerical, respectively) for the three soils at low root density and high transpiration rate.

[46]

As a is between 0 and 1, and as generally ar_m >> r₀, it is reasonable to neglect the a/2 term, further simplifying Eq. [46] to

[47]

Evaluated for a = 0.53 and simulation parameters r₀ = 0.5 mm and z = 0.5 m, values calculated by Eq. [47] for r_m ranging from 15 to 75 mm fit to Eq. [28] with p = 18.1 and q = 2.277 (n = 61, r² = 0.999) are similar to the values obtained with numerical simulation.

These results indicate that analytical and numerical methods are in good agreement. As a first approximation, the analytical solution may therefore be used. Results from the numerical method do not depend on assumptions about d/dt, however, and are therefore preferable. The difference between the numerically estimated distance of mean water content, 0.53r_m, and its analytical equivalent, 0.6065r_m, is an indicator of the degree of dependency between d/dt and r. Further investigation of this value for soils with different hydraulic properties, as well as detailed evaluation of the depletion phase after the first occurrence of limiting hydraulic conditions, will be important to provide a physically supported way to estimate soil water availability and actual transpiration rates.

CONCLUSIONS

1. The limiting values (water content or pressure head) of macroscopic root water extraction models are a function of potential transpiration rate, soil hydraulic properties, and root density.
   
2. In contrast, numerical simulations showed the matric flux potential at first occurrence of limiting hydraulic conditions (M_mean,lim) to be independent of soil type and to depend only on potential transpiration rate and root density. Analytically, it was shown that this can be explained if water depletion d/dt has the same value within the entire root sphere. In this case, M(r) is a soil-independent function of transpiration rate and root density.
   
3. Analytically, mean soil water content is shown to be expected to occur at a distance from the axial center 0.6065 times the half-distance between roots (r_m). Numerical simulations showed that, in fact, this distance varies according to soil type, root density, and transpiration rate. For most simulations, however, the value was around 0.53.
   
4. Combining the above two conclusions, we suggest that mean soil water content and pressure head at first occurrence of limiting hydraulic conditions (h_mean,lim and _mean,lim) can be calculated from the matric flux potential at 0.53 r_m, which is a function of transpiration rate and root density and which depends on a soil-specific K(h) function.
   
5. Thus, either the described numerical model, nomograms as depicted in Fig. 12, or the analytical solutions can be used for a comprehensive determination of h_mean,lim and _mean,lim.
   

	ACKNOWLEDGMENTS

 
Quirijn de Jong van Lier is funded by CAPES-MEC, Brazil; Klaas Metselaar is funded by the framework of the Dutch National Research Programme Climate changes Spatial Planning. We thank Pieter A.C. Raats and Sjoerd E.A.T.M. van der Zee for a valuable discussion of the analytical solutions.

	REFERENCES

TOP EXECUTIVE SUMMARY ABSTRACT INTRODUCTION MATERIAL AND METHODS Results and Discussion REFERENCES

 

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