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Institute of Forest Biometry and Informatics

Faculty of Forest Sciences and Forest Ecology
at the
University of Göttingen

PLANT MODELLING GROUP

PROJECT:

Numerical simulation of the hydraulic system of trees: The case of Durmast Oak (Quercus petraea [Matt.] Liebl.)

Overview:

State of the art
Aims of the project
Working programme: Experimental part
Working programme: Theoretical and modelling part
References
Cooperation partners

State of the art

Recent research has given evidence for a close interrelation between function and structure of branching systems of trees. Stem and branches form an open network of flow paths for water. This network bears spatial patterns of hydraulic conductivity which stand in correlation with parameters of the branching structure (Zimmermann 1978, Tyree et al. 1983, Ewers & Zimmermann 1984).

If the water potential in the conducting tissue decreases steadily, for example due to drought stress, a progressive loss of conductivity will be the result (Edwards & Jarvis 1982, Tyree & Dixon 1986, Sperry & Tyree 1988, 1990, Sellin 1991, Cochard 1992), and this effect is reversible only to a limited degree (cf. Waring et al. 1979, Sperry et al. 1987, Borghetti et al. 1991). The dependence upon water potential is species-specific and correlates with the requirements of a tree species concerning the water conditions at its site (Tyree & Ewers 1991, Cochard 1992). Given an unrestricted transpiration, the losses of conductivity are reinforced by a positive feedback (Tyree & Ewers 1991). The studies give support to the conjecture that drought stress triggers a nonlinear, spatio-temporal dynamics in the hydraulic network of the tree, which involves the danger of a breakdown of the hydraulic system (Tyree & Sperry 1988). The spatial distribution of hydraulic conductivities in the tree crown appears as the result of a strategy to keep this dynamics under control (hypothesis of hydraulic segmentation, Zimmermann 1978, 1983). Only gradually it is recognized that stomatal regulation plays eventually an important role in this context (Cochard 1992, Jones & Sutherland 1991, Tyree & Cochard 1996).

A theoretical synthesis of the empirical observations could now clarify how an integration at the level of the whole tree is obtained. What is the appropriate formalism for such a synthesis?

Essential aspects of the hydraulic system of trees can be summarized in an initial boundary value problem (IBVP), i.e. a certain type of system of differential equations with prescribed initial values and boundary values. This IBVP reflects the branched architecture of the tree and is based on a nonlinear diffusion equation (Früh 1995). An equation of the same type is also used in numerical simulation models of water flow in the soil (Hornung & Messing 1984). There, a methodical standard is obeyed which ensures a theoretically concise translation of the basic assumptions into the mathematical and numerical procedures used to solve the problem on a computer (cf. Hornung & Messing 1984, Hörmann & Schmidt 1995). However, in tree physiology the degree of mathematization is much lower. Moreover, water flow in a porous medium which is spatially organized in a network poses special mathematical and physical problems. Hence it is not astonishing at all that the first branching-oriented tree water flow model by Tyree (1988) still had inconsistencies between the basic assumptions and the numerical realisation (see Früh 1995).

Based on elements of Tyree's approach (1988), in the Plant Modelling Group (University of Güttingen) a numerical simulation model (HYDRA) of tree water flow has been developed which improved the translation of the basic assumptions into a numerical algorithm in terms of exactness, reproducibility and efficiency (Früh 1995, 1997, Früh & Kurth 1999). Via a data interface, HYDRA can take structural information from the software GROGRA (Kurth 1994a, 1998) and from the AMAP system (de Reffye & Blaise 1993; Kurth 1999, Früh & Kurth 1999, Lanwert et al. 1997). Hence it can be used as a tool to compare the hydraulic performance of different tree architectures (see examples of simulated water potential profiles). Propositions concerning the dependence of water flow dynamics in the tree upon architectural parameters can be deduced on a sound theoretical basis.

However, there is still a gap concerning the mutual dependence between the hydraulic systems of tree and soil. Soil water potential sets the boundary value for the tree-internal gradient of water potential (Weatherley 1970); a biochemical signal (ABA) seems to generate a dependence between stomatal conductivity and rhizospherical water status in many species (Gollan et al. 1985, Zhang & Davies 1989, Dreyer et al. 1995). In the other direction, the water uptake by the tree triggers a strongly nonlinear dynamics of flow in the soil regions close to the tree roots (Weatherley 1975, Hainsworth & Aylmore 1986). Unfortunately, most contemporary models treat either the water flow in the tree (Tyree 1988, Früh & Kurth 1999) or in the soil (Hauhs 1985, Lafolie et al. 1991) in a spatio-temporally high resolution, but the strong interdependence between both systems is normally not sufficiently taken into account. The model designed by Barataud et al. (1995) considers this dependency, but tree architecture and water capacities in the above-ground part of the tree are neglected. Only the models of Clausnitzer & Hopmans (1994) and of Doussan et al. (1998) link soil water flow, root structure and water uptake at a spatial high-resolution level. However, both models have only been applied to herbaceous plants so far, not to trees, and they do not consider the above-ground architecture.

Pedunculate oak (Quercus petraea Matt. Liebl.) is a tree species which is described in the literature as tolerant against drought stress, with secure xylem sap flow and maintenance of a significant stomatal conductivity and transpiration even under considerable drought stress (Bréda et al. 1993, Leuschner et al. 1997). This property is seen in connection with the deep rooting behaviour as well as with the above-ground hydraulic architecture and an efficient stomatal regulation (Bréda et al. 1993, 1995a, 1995b; Granier et al. 1994, Cochard et al. 1996). The involvement of a root-born ABA signal in stomatal control is discussed somewhat controversely for this tree species (Dreyer et al. 1995, Fort et al. 1997); anyway, stomatal aperture seems not to be determined by ABA concentration in xylem sap alone (Triboulot et al. 1996). Concerning hydraulic architecture, the cited studies are restricted on calculations of hydraulic conductivity and transpiration at the whole-tree level. For the conductivities of single woody axes and at branch junctions in the crown, only very partial results have been obtained (e.g. by Tyree & Alexander 1993 at Quercus velutina Lam.). But knowledge of conductivity patterns in the branching system would be relevant for an assessment of the significance of tree architecture for hydraulic performance, e.g. in comparison with other tree species. The mentioned simulation model HYDRA has until now been applied to coniferous species (Picea and Thuja, see Früh & Kurth 1999) and uses in these cases empirically-obtained relations between shoot diameter and conductivity. In view of the differing inner structure of the axes in Pedunculate oak (ring-porous architecture), this approach will probably not directly work for this species. It is a task in our project to obtain empirically a relation between axial hydraulic conductivity of segments from trunk and branches and biometrical parameters (age, diameter, position in the branching system). E.g., in the coffee tree (Coffea arabica L.) Rapidel (1995) has identified a significant impact of branching order upon hydraulic conductivity. Oliveira et al. (1996) investigated spatial variations of stomatal conductivity in the crowns of Quercus suber trees and came to the conclusion that leaves adapt themselves at the microclimatic conditions of their respective positions in the crown. Only a 3D-model with high spatial resolution with the possibility to link a microclimatic model can take such effects into account and would help to assess their significance in quantitative terms.

The simulation model HYDRA can read tree structures from the morphological meta-model GROGRA. These structures can be generated artificially from a growth grammar, or they can come from an empirical mapping of whole trees (see Früh 1995, 1997, Früh & Kurth 1999, Kurth 1999). These data transfer processes are currently generalized in a related project on linking structural and functional tree models. A data exchange with the AMAP model from CIRAD (Montpellier) was already prepared by Lanwert et al. (1997). Hence HYDRA can be used to investigate dependencies between hydraulic system dynamics of trees (e.g. reaction to drought) and architectural properties (e.g. shoot-root ratio, ratio between total leaf area and trunk diameter, secondary growth, crown shape, stem shape, topological indices). Most of these properties can be described in a rather compact form - concerning stem shape curves, see e.g. Gaffrey et al. 1998. First studies give evidence for clear dependencies upon structural parameters, for unstressed hydraulic regime as well as for the nonlinear system dynamics under drought stress (Früh & Kurth 1998). We hope to provide forestry with new approaches to explain the unexpected sensitivity against drought stress which some tree species show when competitors are removed.

Aims of the project


Working programme: Experimental part

1. Measurement and digital representation of crown structure

The simulation model HYDRA needs complete specifications of the branching structure of the crown of the tree for which the simulation shall be carried out. These structural data can be obtained from artificially-created trees on the basis of growth grammars, but in order to start with samples as close to reality as possible, we decided to use measured real oak trees instead.
We have chosen the neighbourhood of the established research area of Unterlüss (Lüneburger Heide, Northern Germany) as the main area where our sample trees come from. The central research plot is equipped with a tower for meteorological measurements, and the team of our cooperation partner in Kassel (Prof. Leuschner) has already conducted several long-term projects on this plot at oak and beech trees, including research in tree water flow, roots, competition between the trees, and succession processes (see Leuschner 1993, 1994, 1997a,b, Coners et al. 1998, Leuschner & Senock 1998). Our investigations concern trees around this research site which have ages of 5 years, 20 years and (approx.) 200 years. The main focus is on the two younger age classes (5 and 20 years). If possible, trees of an intermediate class (10 years) will also be investigated in the future.
During the last year, the above-ground branching system has already been measured (destructively) in 2 of the small trees (5 years; total mapping of the crown) and in 3 of the 20-years-old trees (total mapping of branch samples, mapping of the positions and directions of order 1 branches). Topological and geometrical structure has been coded in a way which enables the reconstruction of the mapped tree parts with the software GROGRA. The necessary parameters for each growth unit include length, diameter, branching angle, position and direction at the supporting shoot, and (partially extrapolated from samples) leaf areas and leaf dry masses. The method was utilized already in earlier work on conifers and beech trees (Kurth 1994a, Kurth & Lanwert 1995, Steilmann 1996, Anzola Jürgenson 1998). Interpolation of the branch samples to approximate the whole 20-year-old trees is currently done with the help of statistical and topological analysis. From three 200-years-old oak trees, which were cut and totally dissected according to crown layers and branching structure, parameters for a somewhat coarser model have been measured. Namely, the positions of the order 1 branches and their respective leaf and wood masses in different horizontal layers have been determined, in cooperation with M. Hagemeier from Kassel.
Further trees have to be analyzed during the next vegetation period, i.e. in summer 2000. It is planned to map at least 8 further 5-years-old trees, 5 of them non-destructively to have objects for sap-flow monitoring (see below). From the stand with the 20-years-old trees, at least 3 more specimen will be mapped according to the coarser method and non-destructively, with some branch samples selected for finer analysis.
These measurements and the corresponding architectural data evaluation will be conducted by W. Kurth, using the GROGRA tools, together with students from the forestry faculty.

2. Measurements of axial hydraulic conductivity of sample segments

In the 5-year-old oak trees, such measurements have already been conducted in 1999 by Stefan Frühauf (Kassel), according to the gravity flow method (Dixon 1914). Detailed architectural information about the positions of the measured samples has been taken into account. Similar measurements will be done in 2000 at samples from the 20-years-old trees and at samples from the old trees. This experimental work is done by our cooperation partners in Kassel (see below).

3. Daily courses of transpiration and conductivity of leaves

Also by specialists from Prof. Leuschner's group, measurements of quantum flux, transpiration and water vapor exchange at selected leaves have been carried out at trees of all three age classes. Daily courses have been obtained on August 4, 1999, and September 2, 1999. LI-COR steady state porometers (LI-1600) have been used for these measurements.
It has still to be decided what amount of data of this kind has to be measured in 2000. The monitoring of complementary gas exchange data will preferably been done in connection with the planned sap flow measurements (see below).
It is not planned to parameterize a detailed model of stomatal control on a physiological or biochemical basis (this would require a considerably higher amount of measurements and expensive experimental equipment which is currently not available for this project). Instead, the porometer data will help to perform a consistency check which can serve to assess if the model assumptions concerning stomatal control (see below) give results which are of a reasonably realistic order of magnitude.

4. Xylem sap flow

We are currently planning a measurement campaign together with our cooperation partner in Brno (Czech Republic), aiming at obtaining daily patterns of xylem sap flow from several trees of each age class (trunk and main branches). The basic method will be the trunk tissue heat balance method (THB; Cermák et al. 1973, 1976, 1982, Kucera et al. 1977); in our case, new sensors suitable for small diameters (EMS minisystem T693, which are still under development, cf. Lindroth et al. 1995) and special sensors for measuring radial differentiation of flow (Heat field deformation method: HFD, Nadezhdina & Cermák 1998, Nadezhdina et al. 1998) will come to application.
This campaign will be connected with manipulation experiments at the trees:
Other parameters, including soil water potentials, will also be monitored during this campaign. The sap flow experiments will be a joint action with both of our cooperation partners, from Brno and from Kassel.

Working programme: Theoretical and modelling part

1. Adaptation of the software HYDRA (and of the data interface GROGRA - HYDRA) to the demands of deciduous trees

Special procedures and case distinctions for the treatment of leaves (including their resistances and capacities) have to be designed and implemented. Furthermore, the visual representation of encoded, measured trees by GROGRA will be improved by capabilities to take leaves (and other tree organs) of various, given shapes into account. The software AMAP (from CIRAD) provides also such facilities, but the direct availability in GROGRA is practical for the utilization of the software in training courses with students or for quick evaluation of measured architectural data.
The work concerning GROGRA and the data interface will be done by W. Kurth, possibly with the support of students or programmers with short-term obligation.

2. Implementation and test of the model linkage between tree and soil

Until now, the design and conceptualization of the interaction between the models for tree water flow and soil hydraulic regime has followed the subsequent philosophy (which could, however, be modified if there is reason to do so):
The whole hydraulic system tree-soil is considered as essentially homogeneous concerning the laws governing water flow. For both parts, soil and tree, and also for the exchange between them, the same fundamental physical principles hold (particularly, Darcy's law). Consequently, water flow is everywhere described by one and the same type of differential equation (nonlinear diffusion equation). Possible special phenomena of a more biological character concerning the water uptake by the roots, like e.g. an impact of fine root metabolism on hydraulic properties of the surrounding soil (Passioura 1988) or drought-induced reductions of contact surfaces between roots and soil are neglected. In the moment, such phenomena cannot be assessed empirically in our project, and their inclusion would introduce new, unknown parameters into the model, which would diminish falsifiability and pose the danger of an uncertain "trial-and-error" fitting in a too large parameter space.
The structure of the root system is represented by a prescribed spatial distribution of fine root mass (cf. Gardner 1964, Tinker 1976, Büttner & Leuschner 1994). The branched structure of the root system and root-internal water flow are currently not represented in the model. One reason is that this would require a time-consuming 3D flow simulation (in contrast to the above-ground system which can be treated as locally 1-dimensional). Another reason is the lack of structural data from root systems. However, a 3D simulation of the roots would lead to an even more homogeneous model structure and is a possible option for the future.
Root water potential is tied to the water potential at stem basis and is assumed to be homogeneous in the whole root system. The neglect of axial gradients seems to be justified by the high axial conductivities observed in root systems; cf. Lafolie et al. (1991). The transfer resistance between roots and soil is also considered as spatially homogeneous. Spatial distribution of water demand in the soil is calculated according to the amount of root mass located in the respective soil cell.

A software which links the two models, HYDRA for the tree and SilVlow for the soil, has already been implemented under LINUX (but yet without a possibility of graphical output). It has still to be tested for internal consistency and validity. The soil water flow model SilVlow has been developed by Chr. Blendinger (Bonn), M. Hauhs (Bayreuth) and J. Schmidt (formerly at the Institute of Soil Science and Forest Nutrition, Göttingen) and has been used in several hydrological investigations (Lange et al. 1996, Schmidt 1997). SilVlow and HYDRA (and their connecting prototype software, SilHyd) are implemented in C and are based on similar physical and numerical considerations. The SilHyd concept has been designed by Th. Früh, Chr. Blendinger and J. Schmidt commonly.

3. Simulation of water flow in the measured trees and comparison with the empirically-obtained hydraulic data

By a comparison of model results with the measured xylem sap flow data, a check of the underlying theory (represented by the linked models HYDRA and SilVlow) will be done. (This test is not a validation of the model in the strict sense, since we have no control over all of the boundary flows and soil structure under field conditions.)
Systematic sensitivity analysis shall be carried out with the model. This will help to assess how great the influence of the uncertain parameters on the model output is. One of the parameters will be varied in discrete steps, the other parameters being fixed. Resulting changements in the output (flow profiles along paths in the crown, time series of flow or potential) will be quantified and analyzed. Likewise, changements due to different representation of the leaves (see above) will be checked.
The manipulation experiments in the field (see above) will be mirrored in the model and do also serve as a means to check the used assumptions. They are designed according to system-analytic methods: The response of the system to a simple (step-shaped) input is investigated. Furthermore, the use of several tree specimen of different size (age) and the manipulations in some of the crowns provide systematic variations of the architectural structure, which can be complemented by model experiments with artificially created trees (from the software GROGRA). - A comparison of simulated and measured sap flow has already been done for spruce trees from the Lange Bramke research plot (Früh 1995).

4. Model experiments with HYDRA concerning the relation between microclimatic variables and tree/soil water status

With tree architectures from the 20-years-old oak trees (and, for purposes of comparison, also with spruce data), the effect of a spatially inhomogeneous transpiration rate in the crowns shall be studied. This requires modifications in HYDRA and the creation of a connection to microclimatic models, which will be carried out together with the project "Linking structural and process-oriented tree models". In a second step, diverse proposed mechanisms of stomatal regulation shall be included in HYDRA (as alternatives which can be switched on and off). In model experiments, they will be compared with each other. We think about modelling the quick propagation of a root-born signal (representing ABA) on the one hand, a control of stomatal closure by purely physical signal transport (decrease of turgor) on the other hand. The resulting hydraulic dynamics in scenarios of drought stress shall be compared with each other and with the model version without stomatal control.

5. Simulations concerning the dependence of the dynamics of the hydraulic system upon tree architecture

This will be a main task of the project. It is planned to check hypotheses concerning patterns and laws in the spatio-temporal dynamics under undisturbed conditions and under drought stress, and concerning the functional significance of some features of tree architecture (cf. "State-of-the-art"). Using the morphological meta-model GROGRA, variants of trees can be generated systematically according to precisely specified architectural laws. These variants will be compared in terms of their functional performance using the water-flow simulator HYDRA. We will try to deduce some statements of a general character from such modelling experiments. As in the sensitivity tests described above, simple (step-shaped) signals will be used as input. When tree architecture is varied, system dynamics as emerging from such simple input signals will be characterized in condensed form and has to be analyzed statistically for dependence upon architectural parameters of various kind. The statistical part of this work will be carried out by W. Kurth, eventually in joint work with other colleagues at our Institute.
Some model experiments will also focus on damages resulting from severe drought stress. The nonlinear response of the system when leaf mass is lost due to a breakdown of (parts of) the system, resulting from cavitation events, is already included in HYDRA. The amount of damage resulting from certain scenarios will be quantified at the whole-tree level, yielding an index of drought stress induced damage suitable for comparisons of trees of different architecture, age and species.

6. Comparison with J. Dauzat's water flow simulation model

(see "Aims").


Among the further activities which took place in the framework of this project was the organisation of a workshop on individual-based and functional-structural models (webpage in German).


References


Cooperation partners:

Prof. Dr. Christoph Leuschner and Stella Landwehr, Department of ecology and ecosystem research at the Albrecht-von-Haller-Institute, University of Göttingen,

Prof. Dr. Jan Cermák, Institute of Forest Ecology at the University of Brno,

Dr. Jean Dauzat, CIRAD-amis, Plant Modelling Programme, Montpellier.

It is also planned to (re-)establish contact with research groups in Nancy and Clermont-Ferrand (INRA, France).


Start of project: April 1, 1999.
Duration: 3 years.
End of project: August 31, 2002.
(The project was interrupted for 5 months and was resumed in June, 2000.)
Funding: DFG project.

Final report (short version) (in German)
Final report (long version) (in German)

This project was carried out by Michael Schulte.

 


 

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Last modifications: August 13, 2003