A one-dimensional model of water flow in soil-plant systems based on plant architecture
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Abstract
The estimation of root water uptake and water flow in plants is crucial to quantify transpiration and hence the water exchange between land surface and atmosphere. In particular the soil water extraction by plant roots which provides the water supply of plants is a highly dynamic and non-linear process interacting with soil transport processes that are mainly determined by the natural soil variability at different scales. To better consider this root-soil interaction we extended and further developed a finite element tree hydro-dynamics model based on the one-dimensional (1D) porous media equation. This is achieved by including in addition to the explicit three-dimensional (3D) architectural representation of the tree crown a corresponding 3D characterisation of the root system. This 1D xylem water flow model was then coupled to a soil water flow model derived also from the 1D porous media equation. We apply the new model to conduct sensitivity analysis of root water uptake and transpiration dynamics and compare the results to simulation results obtained by using a 3D model of soil water flow and root water uptake. Based on data from lysimeter experiments with young European beech trees (Fagus silvatica L.) is shown, that the model is able to correctly describe transpiration and soil water flow. In conclusion, compared to a fully 3D model the 1D porous media approach provides a computationally efficient alternative, able to reproduce the main mechanisms of plant hydro-dynamics including root water uptake from soil.
Keywords
Transpiration Plant hydro-dynamics model Root water uptake European beech Porous media equationIntroduction
An approach to model plant water relations has developed during the last two decades based on two main concepts: the cohesion-tension theory (Tyree and Zimmermann 2002) and the electrical circuit analogy applied for simulating water transport in plants using resistances, capacitances, water potentials and flow (Cruiziat et al. 2002). By applying these two concepts, the description of tree hydraulic architecture has made a strong improvement towards a more realistic and comprehensive vision of tree water relationships (Cruiziat et al. 2002). Only recently, this approach was further improved by substituting the electrical circuit analogy by a porous media description for sap flow in wood that is based on Darcy’s law and includes a water capacity term to better account for the dynamic behaviour of the hydraulic storage in trees in a way that mass conservation is considered, e.g. by Bohrer et al. (2005), see also Arbogast et al. (1993), Früh and Kurth (1999), Kumagai (2001) and Aumann and Ford (2002). This avoids calculations of negative water contents and unlimited water withdrawal from the tree as occurring with applications of the electrical circuit model (Chuang et al. 2006). By using a three-dimensional (3D) representation of the tree structure and the physically based representation of tree hydro-dynamics, questions can now be adressed of how the variability in crown architecture (i.e. inter-specific, age or ecosystem dependent structure) would lead to different transpiration responses (Bohrer et al. 2005). In particular by considering the stomatal response to the corresponding branch water potential, we can then simulate the stomatal closure, the corresponding change in leaf water conductance and the resulting actual transpiration rate. Moreover, considering a loss of conductivity with decreasing water potential allows the representation of vulnerability to cavitation at critical highly negative water potential values (Hölttä et al. 2005; Cochard et al. 1999).
Since water transport from the soil through the plant into the atmosphere takes place in a soil-plant-air continuum that is interconnected by a continuous film of water, modelling of plant water transport has to consider both the water exchange at the leaf-air interface and the water flux at the soil-root interface. Therefore, also root water uptake including root hydraulic architecture has to be described to integrate the tree hydro-dynamic model into ecosystem models (Doussan et al. 2003), a step still to be taken to replace current representations of plant water transport, e.g. big leaf or resistor-capacitor approaches (Bohrer et al. 2005) that are often coupled to one-dimensional (1D) effective root water uptake models (Cowan 1965; Gardner 1960; Nimah and Hanks 1973; Feddes et al. 1978; Campbell 1985). New root water uptake models are available that explicitly describe root architecture and related soil-plant processes in three dimensions by explicitly considering the 3D distribution of the uptake (Clausnitzer and Hopmans 1994; Somma et al. 1998; Vrugt et al. 2001) and also considering water flow in the root system (Doussan et al. 2006; Javaux et al. 2008). They can be based on root growth models that allow the integration of a great diversity of environmental conditions and their impact on root system development (Dunbabin et al. 2002).
Despite the more realistic assumptions for predicting soil-root interactions, a disadvantage of the 3D models is their high level of complexity and consequently their high computational demand. Therefore, to describe and simulate transpiration at the stand level a less complex but still realistic approach is needed. By extending the hydro-dynamics model for aboveground parts of trees of Bohrer et al. (2005), we developed a simple root water uptake and transpiration model that couples a 1D soil water flow model with a 1D plant xylem water flow model. Notwithstanding its 1D character the xylem water flow model takes 3D plant architecture into account and can consider different root properties for each root node and different root types. Moreover, by specifying different properties of the soil directly surrounding each root, to a certain extent also horizontal variability might be represented.
Model development
In our model both the water flow within the plant and the water flow in the soil are described by applying the 1D porous medium or Richards equation (Richards 1931). This is based on the assumption that similar to the soil matrix, also the plant xylem can be conceived as a porous medium. The xylem is seen as a bundle of small parallel capillary pores, which are filled with water and air. A capillary can conduct water only if it is completely saturated. With increasing suction head more and more pores cavitate and the ability of the xylem to conduct water decreases. To simulate the xylem water flow and the water uptake from the soil by the roots we need a representation of the plant architecture as model input to define the flow domain and the interfaces between the plant and its environment consisting of the soil and the atmosphere.
Description of plant architecture
Water flow within the plant
Hydraulic characteristics of the xylem
The elastic modulus E (mm), the porosity ϵ_{x} (m^{3} m^{ − 3}), the air entry value a (mm) and the exponent λ (–) that determine the xylem water retention curve, are model input parameters and assumed to be fixed values for the whole tree.
Input parameters
Soil or plant | Reference parameters | Perturbed parameters |
---|---|---|
Loam | θ_{s,res} = 0.078 | \(k_{\rm sat}=2.496\cdot 10^3\) mm d^{ − 1} |
θ_{s,sat} = 0.43 | ||
α = 0.036 cm^{ − 1} | ||
n = 1.56 | ||
k_{sat} = 249.6 mm d^{ − 1} | ||
Clay loam | θ_{s,res} = 0.095 | k_{sat} = 624.0 mm d^{ − 1} |
θ_{s,sat} = 0.41 | ||
α = 0.019 cm^{ − 1} | ||
n = 1.31 | ||
k_{sat} = 62.4 mm d^{ − 1} | ||
Clay | θ_{s,res} = 0.068 | k_{sat} = 480.0 mm d^{ − 1} |
θ_{s,sat} = 0.38 | ||
α = 0.008 cm^{ − 1} | ||
n = 1.09 | ||
k_{sat} = 48.0 mm d^{ − 1} | ||
Root | k_{ max } = 432.0 mm d^{ − 1} | \(k_{\max}=4.32\cdot 10^3\) mm d^{ − 1} |
k_{ max } = 43.2 mm d^{ − 1} | ||
\(k_{r}=3.0 \cdot 10^{-6}\) d^{ − 1} | \(k_{r}=3.0\cdot 10^{-5}\) d^{ − 1} | |
Shoot | \(k_{\max}=1.3\cdot 10^3\) mm d^{ − 1} | \(k_{\max}=13.0\cdot 10^3\) mm d^{ − 1} |
k_{ max } = 130.0 mm d^{ − 1} | ||
Leaf | b = 1.0 · 10^{5} mm | b = 3.0 · 10^{5} mm |
c = 3.5 | ||
Stem | d_{stem} = 10.0 mm |
Water flow in the soil
Soil water uptake and transpiration
Similar to water uptake, also transpiration, the loss of water to the atmosphere, is described by a sink term. At the outer branches, where the leaves are located, the corresponding water flux is prescribed by the atmospheric evaporative demand and by the stomatal hydraulic conductivity, which similar to the xylem hydraulic conductivity is assumed to depend on the xylem matric potential of the branch element (Bohrer et al. 2005). For simplicity the atmospheric water demand is prescribed by a constant flux condition or calculated from the potential evapotranspiration rate (mm s^{ − 1}) estimated by the FAO Penman-Monteith method (Allen 2000) and is distributed according to the leaf area. Therefore, in contrast to the water exchange with the soil, the exchange with the atmosphere is prescribed and no direct feedback of transpiration to the micro-climate at the leaf level is considered.
Numerical implementation
The soil and tree water flow equations are coupled via the below ground water uptake sink terms and each solved by iterations using a standard Newton–Picard fixed point iteration scheme (Celia et al. 1990; Priesack 2006) until tree and soil water contents and potentials converge below a threshold value (Huang et al. 1996). Although both flow equations may be coupled by a further fix-point iteration via the water uptake term to account for the non-linearity of the systems, similar to Tseng et al. (1995) we use a single-pass solution scheme without iteration between the two subsystems, since for the considered simulation scenarios the additional iteration does not lead to significantly different results. The solution procedures of both flow equations 1 and 12 are based on a finite element discretisation following the approach of the HYDRUS-1D model (Priesack 2006; Simunek et al. 2008). Since the solution method of the Richards equation on a graph as represented in Fig. 1b has not been documented in more detail, we describe the applied finite element method in Appendix A.
Statistical criteria
Simulation scenarios and parameterisation
The new model of water flow in soil-plant systems is applied to three different scenarios mainly to illustrate that important features of water flow in plants and soils can be simulated. In a first scenario, the drying soil scenario, we simulate soil water uptake and transpiration under conditions of a prescribed constant transpiration rate without refill of soil water. By this scenario the effects of stomatal conductivity, xylem conductivity, radial root conductivity and soil conductivity on water uptake and transpiration are considered. In particular the onset of cavitation is examined. In a second scenario, the hydraulic lift scenario, we demonstrate the basic ability of the model to simulate transport of water through the roots from wetter to dryer soil regions. In a third scenario, the lysimeter scenario, we test if the model can simulate daily and seasonal water balances based on measured input parameters. In this case for the xylem water flow model only parameters of the tree architecture are changed: the stem diameter and the distributions of leaf and root area index based on measurements from the lysimeter experiment.
In the drying soil scenario, we simulate soil water uptake and transpiration also for comparison with a 3D root water uptake modelling approach. In the scenario, the water is taken up by the roots from a homogeneous soil column of 0.4 m depth and of 0.1 m by 0.1 m soil surface, which is assumed to be initially in hydrostatic equilibrium with an aquifer located 3.0 m below the soil surface (Javaux et al. 2008). The soil boundary conditions are a no flux condition at the bottom of the soil profile and no rainfall or evaporation at the soil surface. For reasons of comparison with the numerical experiment of Javaux et al. (2008) we use the same parameters to study the effect of different soil types and to analyse the sensitivity of different soil and xylem hydraulic conductivity values (Table 1). For the reference case input parameter values are taken from the second column of Table 1, and for the sensitivity analysis only the conductivity parameters, k_{sat} for the soil, k_{ max } for the xylem and k_{r} for root water uptake, were changed as indicated in the third column. This perturbation of parameter values corresponds to a realistic variation in observed conductivity values for soil or xylem (Javaux et al. 2008; Doussan et al. 1998). In both cases for the above-ground tree a constant potential transpiration rate of 1.0 mm d^{ − 1} was assumed and prescribed along the tree at the ends of stem or branch elements according to the assumed leaf area distribution. Related to the 0.01 m^{2} soil column surface this corresponds to a volumetric water flux rate of 1.0 ·10^{4} mm^{3} d^{ − 1} from the soil to the atmosphere.
In the hydraulic lift scenario, instead of the no flux of the first scenario we consider a Dirichlet boundary condition which prescribes saturated conditions at the lower boundary of the soil column now assumed to be 1.5 m deep. Furthermore, we now prescribe a daily sinusoidal cycle of the potential transpiration according to (Childs and Hanks 1975; Priesack 2006), but keep the daily rate at 1.0 mm d^{ − 1}. All other conditions and parameters are kept the same as in the loam reference case of the first scenario.
Soil input parameters of the third scenario
Soil horizon texture | Depth (cm) | θ_{s,sat} (–) | α (cm^{ − 1}) | n (–) | k_{sat} (mm d^{ − 1}) |
---|---|---|---|---|---|
Sandy loam | 0–5 | 0.52 | 0.11 | 1.13 | 1,381 |
Sandy loam | 5–30 | 0.504 | 0.12 | 1.14 | 1,285 |
Silt loam | 30–45 | 0.475 | 0.1 | 1.05 | 1,276 |
Clay loam | 45–90 | 0.395 | 0.05 | 1.13 | 632 |
Sand | 90–200 | 0.417 | 0.04 | 1.25 | 5,000 |
Results
Numerical accuracy and convergence
Summary of numerical results for the xylem water flow model in case of the soil drying scenario with reference parameters
δ_{tol} (mm) | No. elem. (–) | CPU (s) | MB (%) | No. iter. (–) | No. steps (–) |
---|---|---|---|---|---|
10.0 | 243 | 3,046 | 0.05 | 60,945 | 21,219 |
1.0 | 243 | 3,237 | 5.0 · 10^{ − 3} | 94,732 | 22,004 |
0.1 | 243 | 3,615 | 5.0 · 10^{ − 4} | 145,330 | 22,222 |
0.01 | 243 | 3,961 | 5.0 · 10^{ − 5} | 199,583 | 22,414 |
0.001 | 243 | 4,458 | 5.0 · 10^{ − 6} | 266,459 | 22,443 |
0.001 | 482 | 13,167 | 4.3 · 10^{ − 6} | 303,137 | 24,132 |
0.001 | 903 | 26,918 | 3.6 · 10^{ − 6} | 301,279 | 23,173 |
Drying soil scenario
Due to the evaporative water flux from the leaves to the atmosphere, the simulated xylem matric potentials show a strong vertical gradient along the tree height (Fig. 2) having lower potentials at the tree crown, where the leaf area is highest and higher potentials towards the roots and within the roots which take up the water from the soil.
The perturbation of conductivity values gives only very small differences between the simulations of the reference case and the cases with one-tenth- or ten-fold maximal xylem conductivity (Fig. 5). This small deviation is similar to results obtained by Javaux et al. (2008) when using the same reference conductivity values for root water uptake simulations (Table 1). Additionally, no difference can be detected if we neglect the second term of the relative conductivity curve in the case θ_{x} > θ_{x,a}, i.e. if we neglect the impact of an increased mean radius of the water-filled xylem vessel elements. Moreover, for the small trees the exact form of the conductivity curve, which is difficult to evaluate considering the high variability of the measured conductivity data (Fig. 3), may play a less important role, if only the decrease of the curve with decreasing xylem matric potential is appropriately described. Because of the low sensitivity of maximal xylem conductivity variation on the simulated actual transpiration in case of the considered small trees, the relative xylem conductivity curve given by Eq. 9 could be further simplified in the saturated range above the air entry value by omitting the second term of the right hand side without significantly changing our simulation results of xylem water flow dynamics (results not shown). This low sensitivity also means, that the impact of the elastic change on the diameter and hence the conductivity of water-filled tubes of the capillary bundle representing the xylem can be neglected in our model description of xylem water flow in small European beech trees. In this case the low sensitivity of the axial xylem conductivities seems to be caused by their relatively high values compared to the low stomatal and radial root conductivity values.
When compared to the reference case in the cases of increased soil or radial root conductivity the begin of limited water supply for root water uptake is 2–5 days delayed (Fig. 5a). Applying the radial rhizosphere conductivity the delay of the onset of limitation of transpiration is 9–10 days (Fig. 5b). In the sequel, the higher conductivities lead to earlier soil water exhaustion and reduced transpiration rates below 0.2 mm d^{ − 1}.
To represent effects of horizontal soil variability we introduce a possibility to account for different soil properties in the direct surroundings of different roots or root elements. This representation is achieved by considering the radial rhizosphere conductivity defined by Eq. 14. The root water uptake rates for the three soil types applying either the constant radial root conductivity k_{r} or the rhizosphere conductivity k_{rs} are shown in Fig. 5c and d. Generally, we observe, that if we apply the radial rhizosphere conductivity k_{rs} instead of the radial root conductivity k_{r} in Eq. 13, the simulated root water extraction decreases earlier and remains slightly higher when the soil gets drier (Fig. 5b and d). For k_{r} the root water uptake limitation appears first for the clay, then for the loam and eventually for the clay loam (Fig. 5c). For k_{rs} the picture changes in case of the loam soil, since the time-span until transpiration gets limited is now shortest for this soil type (Fig. 5d). In both cases the clay loam soil supplies the transpiration water demand by far the best, up to 15 d longer than the clay and loam soils.
In the cases with increased soil and radial root conductivity, similar to the 3D root water uptake simulations of Javaux et al. (2008), the limitation of root water uptake is significantly delayed by 2–5 days (Fig. 5a). Furthermore, the finding that the clay loam best sustains the evaporative demand of the plant, about 10–15 d longer than the other soil types could be confirmed by our simulations (Fig. 5c and d). But, in contrast to the 3D water uptake simulations under the second collar boundary condition (CBC 2) by Javaux et al. (2008), in our 1D simulations water stress occurred earlier for clay than for loam soil, if the radial root conductivity was used (Fig. 5c). Only if the radial rhizosphere conductivity is applied, the same order of soil types concerning the appearance of water stress is found, i.e. transpiration is reduced first for loam, then for clay, and eventually for clay loam soil (Fig. 5d).
Hydraulic lift scenario
Lysimeter scenario
Discussion
A new model approach is introduced to describe water flow in the soil-plant system. The model extends previous approaches that were developed to describe hydro-dynamics in above-ground plants (Früh and Kurth 1999; Aumann and Ford 2002; Bohrer et al. 2005) by additionally including the hydraulic root system architecture in combination with a soil water flow model. In this way a complete water flow model for the whole soil-plant system is obtained, which then vice versa extends existing root water uptake modelling approaches (Doussan et al. 1998; Javaux et al. 2008) by including the above-ground water dynamics of the plant. Similar to the soil water flow model, which is based on the continuum approach as described by the Richards’ equation, also the plant water flow is described by this porous media equation following Bohrer et al. (2005). This description is in contrast to more discrete approaches that consider the plant as built up of storage compartments and describe the water flow by water exchange processes between these compartments (Perämäki et al. 2001; Steppe and Lemeur 2007). But, besides the advantages of a continuous mathematical formulation representing mass conservation (Früh and Kurth 1999), due to the similarity between the porous media approaches to describe water flow in the soil and the xylem, it seems also to be conceptually easier to derive a consistent model of hydro-dynamics of soil-plant systems based on the continuum approach.
In a first analysis, the new model was applied to investigate model behaviour and model sensitivity by scenarios with young European beech trees that have already been analysed in two previous studies (Javaux et al. 2008; Gayler et al. 2009). Model tests and numerical results show, that the model is successfully implemented and can efficiently simulate water flow in the soil-plant system based on the a simple representation of plant architecture. Mass balance errors are low and in an acceptable range. But since the global mass balance is a necessary but not a sufficient criterion for having a correct solution (Celia et al. 1990), the model needs further analysis, which may be based on comparison with exact solutions (Lehmann and Ackerer 1998) or as in our case by evaluating simulations of practical examples using experimental data.
In the case of the 3D simulation by Javaux et al. (2008) the effect of the steeper slope of the loam hydraulic properties generates a heterogeneous distribution of soil water potentials in the horizontal direction with drier soil near the roots. This distribution leads to a larger conductivity drop and produces an earlier water stress (Javaux et al. 2008). In the case of our 1D simulation this effect is mimicked by amplifying the direct impact of the lower xylem potential on the soil matric potentials directly near the roots resulting in a lower radial hydraulic conductivity between root axis and soil, which is modelled by Eq. 14 based on the definition of the radial rhizosphere conductivity k_{rs}. If the distribution of radial conductivities of the roots is adapted in this way, the main conclusions about effects of soil type and conductivity values remain, although the root architecture was different in the drying scenario simulations we compared. But amplitudes or local values of considered variables are different due to the 1D simulation of soil water flow. Therefore, a full data comparison cannot be performed, even if locally the water uptake by the root system may be of similar magnitude.
Besides the model sensitivity of the radial root or rhizosphere conductivity, the most significant model sensitivity on simulated soil water uptake and transpiration is due to the stomatal hydraulic conductivity. This model behaviour is mainly caused by eventually rather low conductivity values, reflecting the significant role of both conductivities in controlling tree water flow which has been shown in several experimental studies (Steudle 2000; Lemoine et al. 2002; Sperry et al. 2003). Moreover, the water has to flow through nonxylary pathways in the root across the root cortex and endodermis and in the leaf through the mesophyll. These flow paths have low hydraulic conductivities and can account to a high proportion of the overall root respectively shoot hydraulic resistance (Sperry et al. 2003). In particular, the radial root conductivity may be strongly changed by aquaporin activity (Ehlert et al. 2009), which itself may be controlled by metabolism or be triggered by environmental factors (Steudle 2000). Also changes of leaf hydraulic conductivity may be mediated by plasma membrane aquaporins (Cochard et al. 2007).
Hydraulic lift has been shown to occur in more than 60 species (Espeleta et al. 2004), “but there is no fundamental reason why it should not be more common as long as active root systems are spanning a gradient in soil water potential and that the resistance to water loss from roots is low” (Caldwell et al. 1998). However, for European beech trees it seems to be an open question whether the radial hydraulic root conductivity from inside the roots towards the soil is large enough to permit considerable water flow to dry soil (Rewald 2008). Therefore our simulations of hydraulic lift only show that in principle the model is able to describe this kind of water redistribution from wet to dry soil under almost optimal conditions for hydraulic lift, i.e. assuming water saturation at the bottom of the soil profile with a resulting simulated leaf xylem matric potential ranging from −40,000 to −70,000 mm. Nevertheless, the hydraulic lift could be significant in mitigating plant water stress and need to be accounted for in models describing root water uptake and the onset of plant water stress in a water limited ecosystem (Siqueira et al. 2008).
The lysimeter scenario simulations confirm the applicability of the new model to describe daily water balance dynamics in soil-plant systems. Also the diurnal course of transpiration seems to be realistically described. In particular, the simulated diurnal changes of xylem diameter up to 0.194 mm (Fig. 9) are within the range of xylem or stem diameter changes that were observed for young European beech and Norway spruce (Pinus abies L.) trees (Steppe and Lemeur 2007; Perämäki et al. 2001). This agreement results from the fact that based on the parameterisation of the xylem water retention curve obtained from the experimental curve of Oertli (1993), we get a value for the elastic modulus E which is similar to values found by Steppe and Lemeur (2007) and Perämäki et al. (2001). However, this value might have been underestimated, since we assume that changes of xylem water potential correspond only to changes in the xylem water content neglecting related changes in water contents of phloem, cambium and heartwood.
Since the new plant water flow model focuses on the water flow in the sapwood tissue in the longitudinal direction of the stem and branch axes, the model still needs to be extended to describe water flow in the phloem and to account for the radial exchanges of water between xylem, heartwood, phloem and cambium. In particular the active uploading and unloading of sugars according to Münch’s hypothesis (Münch 1928) needs to be considered, because near the sources of sugars considerable water flow from the xylem to the phloem can occur and vice versa close to sugar sinks significant amounts of water can flow from the phloem to the xylem. Locally this water exchange can strongly influence the xylem water flow dynamics which also can lead to xylem diameter changes (Sevanto et al. 2002).
At the present state of development the model is supposed to be already useful to simulate the impact of differences in plant architecture and soil properties on transpiration and water uptake under various climatic conditions, in particular considering differences of leaf area and root area distribution or of root and shoot branching. The model may be also helpful to describe the impact of different distributions of conductivity values along the tree architecture in particular of larger trees to study their role in protecting the xylem against cavitation, while maintaining transpirational water flow. This description may lead to a better understanding of observed differences in xylem structures and functions between deep and shallow roots, roots, stem and branches (Nadezhdina 2010; North 2004) and of the relevance of nonxylary pathways for water uptake and transpiration (Sperry et al. 2003). The model may also be applied to better describe differences of stand water budgets between mixed forest stands of different species composition.
Conclusion
The newly proposed model of water flow in soil-plant systems presented in this study combines approaches to simulate water flow in the above-ground plant based on the porous media equation with root water uptake and soil water flow models also based on the porous media equation. In this way the model concept exploits the similarity between water flow in the soil and the xylem of plants. It could be shown that the model can simulate water balance dynamics in soil-plant systems with young European beech trees including some main known features of water flow in plants and soil such as the diurnal cycle of plant water content and xylem diameter change, stomatal control of transpiration, cavitation due to water stress, sensitivity of root water uptake to radial root conductivity, hydraulic lift, interaction of hydraulic soil properties, root distribution and plant water availability. Due to the simplified representations of plant architecture and distribution of soil properties that lead to 1D representations of water flow domains, the model is numerically very efficient and may therefore be also applicable to simulate water flow in canopies accounting for spatial and temporal variability of canopy structures based on rather simple representations of individual plant architecture.
Acknowledgements
We are grateful to the Deutsche Forschungsgemeinschaft which funded this study within the frame of Forschergruppe 788 ‘Competitive mechanisms of water and nitrogen partitioning in beech-dominated deciduous forests’. We also want to thank an anonymous reviewer whose comments helped to considerably improve the manuscript and we thank Sebastian Bittner for his help during the revision of the manuscript.
Open Access
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