

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
- Until now, the simulation model HYDRA has been applied only
to conifers. Despite its genericness as far as the fundamental
assumptions and the model structure are concerned, two problems have
to be solved before the model can be applied to deciduous trees:
- The axial hydraulic conductivity of the segments of branches and
trunk, which is used by the models as an input parameter for each
tree segment, has to be determined in dependence upon parameters which
are related to architecture (e.g. diameter of the segment, branching
order, age of the segment). Such an empirical relation is also
interesting in itself because it gives information about the inner
differentiation of the tree crown according to functional requirements.
- In the case of conifers, the leaf areas have been associated directly
to the leaf-supporting axis segments. Given the significant resistances
in the axial conducting system of the leaves of deciduous trees, it
will probably become necessary to model the leaves as elements of their
own (cf. Rapidel 1995, Chemouny 1998). Based on measured daily courses
of leaf conduction, both variants - with and without separate modelling
of the leaves - shall be compared with each other and with the empirical
data.
- Hydraulic performance of spruce trees (Picea abies) and
durmast oak shall be compared in the model. For spruce, there
exist already architectural data and HYDRA model runs (cf. Früh
& Kurth 1999).
- A spatially inhomogeneous transpiration rate shall be taken into
account in HYDRA. Microclimatic models will provide the necessary
information to parameterize such a refined model (until now, the
transpiration rate is assumed as equal for all branches in the tree,
and varies only in time according to a given pattern). This extension
shall be realized in connection with the above-mentioned project
on linking structural and process-oriented tree models.
- The simulation of water flow in the tree shall be linked with
the simulation of soil water movement. This linkage is already
prepared in a concept developed together with soil
hydrologists.
- The simulation model HYDRA shall be used as a tool to check
hypotheses concerning the patterns of nonlinear flow dynamics in
the tree and their dependence upon architectural parameters.
These hypotheses concern "hydraulic segmentation"
(Zimmermann 1978, 1983), the mechanism of "running embolism"
in case of drought-stress induced loss of conductivity (Tyree &
Sperry 1988, Tyree & Ewers 1991), the possible optimization of
the root-shoot-ratio (cf. Weatherley 1970, Tyree & Ewers 1991,
Roloff & Römer 1989), assumptions of the so-called
"pipe model" (Shinozaki et al. 1964a,b, Whitehead 1978,
Gruber 1992). So far, most of these hypotheses have been considered
only in qualitative terms. Now, with improved numerical tools, a
quantitative analysis of the function-structure relationship becomes
tractable.
- Related to this hypotheses-testing, a reduction of the complexity
of tree water flow modelling is desirable to make the obtained results
accessible for applications at coarser scales (whole tree stands,
forestry practice). Here we have two sub-aims:
- The obtained relations between architectural properties of trees
and hydraulic properties shall be condensed into "secondary
models" which could be the shape of quickly-calculable regression
equations or rule systems.
- A comparison between HYDRA and the simpler tree water flow model
HYDRO of the group of J. Dauzat (CIRAD) shall be made (cf. Rapidel 1995).
Because HYDRO uses simpler basic assumptions, its calculation times are
smaller. The technical preconditions for such a comparison (a common
data interface for both models) are currently prepared in the
above-mentioned parallel project on linking models.
- Further possible extensions concern the inclusion of nutrient
transport in models of this type, and dynamics and spatial structure
of root systems.
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:
- In (at least) two of the 20-year-old trees, approximately 50 percent
of the order 1 branches in the crown will be cut and removed. This
severe changement in architecture can easily be reconstructed in the
model, and the effect on water flow dynamics can be studied in the
model and in reality, and will be compared.
- Some of the 5-year-old trees will quickly be covered and uncovered
by an opaque cloth, triggering a sudden interruption of the radiation
input which drives transpiration. We will measure the reaction of the
hydraulic system to such a sharp input signal and compare it with
model predictions.
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.
Other projects of the research group
Back to Plant Modelling Group Homepage
Last modifications: August 13, 2003