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Quality of Life and Management of Living Resources
fSilvoarable Agroforestry For Europe (SAFE) European Research contract QLK5-CT-2001-00560
Deliverable 5.1
Water partition and uptake modules in HySAFE
Nick Jackson1,
Christian Dupraz2, Alain Fouéré3,
Martina Mayus4 and Harry Ozier-Lafontaine3
1 Centre for Ecology and Hydrology (Natural Environment Research Council), UK 2 Institut National de la Recherche Agronomique (INRA) SYSTEM, Montpellier, France 3 Institut National de la Recherche Agronomique (INRA) APC, Guadeloupe, Antilles 4 Wagenigen University, Wageningen, Netherlands
Version 1 – September 2003
This document is a working document of the SAFE project, and will be updated regularly until the SAFE project completion.
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1. Introduction and initial modelling decisions:
It should be noted that this document details the current status of the modules delivered by Workpackage 5 (Belowground Interactions) to the SAFE project. As it is to be expected that the modules will be further refined and extended, so should this document be seen as a work in progress, which will be similarly updated and revised to reflect the changes made in the course of model development. HySAFE is one of several models being developed within the SAFE project, but is the one with the greatest physiological complexity. As such, it must fulfil one of the objectives of the project – to design and validate a mechanistic model of tree-crop interactions in silvoarable plots, simulating the processes involved the growth of both tree and crop system components. The model must be able to accurately model these major processes in terms of the access and use of the available resources (water, carbon and nutrients etc.); but also how these processes are driven and limited by environmental variables (e.g. soil structure, temperature) and system management interventions (cropping season, ploughing and pruning etc.). A modelling workshop early in the project compared several existing models developed by previous European or International programmes, with a view to considering their relevance and usefulness in the context of European silvoarable agroforestry systems. Two agroforestry models in particular were examined: HyPAR, developed by the Centre for Ecology and Hydrology (Mobbs et al., 1999), and WaNuLCAS, developed by the International Centre for Research in Agroforestry (van Noordwijk, and Lusiana, 2000). The decision of the workshop was that the SAFE project would attempt to use the HyPAR model, reviewing and adjusting the various component modules as necessary. The WaNuLCAS model would be considered as a stopgap measure in the event that the model developed by the project (HySAFE) did not meet its objectives. The HySAFE model is required to accurately represent sensitivity to various environmental and biological factors, but also conserve computation time (to allow for simulation runs of several years). For this reason, the modelling workshop also prioritised the need for new modules, in cases where they either did not exist in the current agroforestry models, or in which the SAFE team considered that the implicit processes were either more complicated or too simplistic than the requirements of the SAFE project required. However, the developers of HyPAR cautioned the SAFE team concerning the crop model (PARCH) currently used within HyPAR, as it was no longer being supported and as a tropical model, was not parameterised for the temperate latitude crops that the SAFE project would consider. Replacing PARCH with the DSSAT crop modelling environment was investigated, but considered too time consuming to be completed within the timescale in which the HySAFE model was required. After much deliberation it was decided to use the STICS crop model developed by INRA.
1.1. Objectives of the workpackage: The objectives of Workpackage 5 (Belowground Interactions) were to design and validate submodules for belowground tree-crop interactions, relevant to both crop and tree growth. Trees and crops in mixed plots compete for soil resources (water, nutrients), but also explore resources that would be unavailable in monocultures. The spatial and temporal distribution of tree and crop root systems and their uptake of water and nutrient resources form the key to understanding inter-specific relationships in mixed cropping systems. This knowledge can explain why sustained yields of intercrops were observed in our experimental plots, making silvoarable systems with widely spaced trees a sustainable arable system, and not a stepping-stone to afforestation.
The description of the modelling requirement (as specified in the SAFE Project Technical Annex) states: T5.1. Design and writing of a simplified model for water extraction and sharing between a tree and a crop, taking into account water interception by the canopies, water redistribution by stem-flow and throughfall, transpiration, and water redistribution in the soil profile by water migration and water transportation by the rooting systems. This model will be able to take into account the dynamic colonisation of the soil by the crop roots, which is specific to silvoarable systems with annual crops. The model will allow assessment of the possibility of silvoarable systems in reducing nitrate leaching to water tables. Two modifications to this specification have been made:
• The adoption of STICS as the crop model within HySAFE (and the reluctance to substantially alter the code of this model) lead to a decision to use the existing water and nitrogen dynamics routines contained within STICS. Therefore, the STICS model would monitor all water dynamics except water extraction. Separate modules for water extraction and nitrogen competition between the tree and crop would be built to operate under the CAPSIS shell, but which would each interact with the STICS model.
• The crop root growth would continue to be simulated by the STICS model, with a
separate module being developed to simulate tree root growth in a dynamic fashion, in response to local soil conditions.
This report will therefore consist of descriptions of the development of the individual modules. This will include the modelling objectives (i.e. the necessary outputs from each module); the ‘state of the art’ (e.g. the approaches taken by previous models); the eventual module developed and incorporated into HySAFE; the necessary input parameters; and planned future improvements.
1.2. The question of scale Modelling in the SAFE project will take place at a variety of scales, but the belowground modules described in this report must operate at relatively small horizontal and vertical scales, over which local conditions can vary significantly. The following diagrams describe the process by which we move from the field scale through to the soil layer and compartment scale at which the three belowground modules will operate.
Figure1: Spatial resolution – from the field scale to the compartment scale in the HySAFE model:
Scene scale: Unspatialised Homogenous soil layers Average tree Variable crop Toric symmetry
Cell scale: Spatialised Homogenous soil layers Homogenous tree influenceHomogenous crop
Field scale: Heterogeneous soil Variable trees Variable crop
Layer and compartmentscale: Homogenous soil physicalproperties Homogenous tree and croprooting
Plot sampling
Crop + Tree root disaggregation
Soil disaggregation
Figure 2: Definition of terms used in modules describing soil processes in the HySAFE model:
It is at the above scale that the three modules (water repartition, nitrogen competition and tree root growth) will operate, as well as the STICS model, as it will be run on separate CAPSIS cells or on groups of cells ‘clustered’ on the basis of similar characteristic conditions.
CAPSIS cell: Considered for modelling purposes as thecell into which the plot is divided and the soilcolumn lying beneath it
Compartment: Defined as the intersection between the CAPSIScell (column) and one of the soil pedological layers(shown here in different colours
Voxel: Compartments consist of one or more voxels,each of which will have the same soil properties.Voxels have a maximum allowed depth
STICS layers (minicouches): Voxels consist of numerous1cm soil layers used by theSTICS model
2. Water repartition module
2.1. Objectives and inputs The water repartition module is required:
• To simulate the extraction of water from different soil compartments, by the (single) tree and by areas of surrounding crops exposed to different environmental conditions (with consequently different water demands).
• Following execution, to feed the correct values of soil water extraction from each compartment back to the STICS model, and how much is directed towards the crop.
As inputs, the module should use the following daily data: • Water demands from the crop and/or the tree in each soil compartment • The availability of water in each soil compartment • The tree and crop fine root density in each soil compartment
2.2. Challenge to be modelled
Currently, the HySAFE ‘scene’ is considered only in terms of one single tree and one single accompanying crop species. The tree is considered as being able to ‘access all areas’, as it can theoretically extract water from any voxel within the soil block. In contrast, and due to the nature of the STICS model, the crop plants growing within each CAPSIS ‘cell’ can only extract water from the voxels that lie beneath that cell. This applies even if the individual crop plant might lie close to a border between two CAPSIS cells. In addition, the location of each CAPSIS cell with respect to the location of the tree causes the local conditions in each of these cells to differ significantly. The shading and rainfall interception by the tree will affect the crop state, the crop behaviour and the crop water demand quite differently from one cell to another. Figure 3: Diagram illustrating the different zones from which the tree and the crop (on a single CAPSIS cell) may extract water (depending on tree fine root density).
2.3. Tree and crop water uptake in HyPAR and WaNuLCAS In HyPAR, the soil is divided into vertical layers, but will only be divided horizontally into columns (similar to that shown in Figure 2 above), if a particular light interception sub-model is selected, i.e. the water uptake depends on the choice of aboveground simulation options.
The initial demand for soil water by the tree and crop is partitioned between the soil cells on the basis of the relative root densities for the two components. The two original tree and crop models that were incorporated into HyPAR each calculate water uptake as if the roots of the other plant were not present. These computed values are considered as maximum transpiration demands, and the combined maximum demand is compared with the available water in the soil. Water demand is calculated by summing the potential extraction that may occur from each layer of the HyPAR water balance sub-model, assuming a maximum water uptake rate per unit soil depth – a value of 0.1mm (water extracted) mm-1 (of soil depth) d-1, obtained from work by Robertson et al. (1993b) and Jansen & Gosseye (1986). If there is enough water in the soil the no competition occurs and the demand from both tree and crop components are satisfied. However, if the combined maximum demand for uptake in any soil cell is more than can be satisfied by the water content of that cell (or if the combined extraction exceeds a maximum rate), then competition between the tree and the crop occurs. The process of competition begins at the first soil layer at the surface in each plot independently (analogous to the CAPSIS cell). The combined maximum demand is compared with the available water in the soil layer, and water is extracted to satisfy as much of this demand as possible given the constraint of the maximum uptake rate described above. However, this uptake rate is only applies to water extraction from a layer if it the water in that layer is at or above field capacity, and is so 'saturated' with roots that further root development roots would not increase water extraction significantly. If either the soil water content, or the root density are below these values then potential extraction is reduced. Any remaining, unsatisfied demand is then compared to the available water in the next soil layer below, and the process continues until the demands have been satisfied, or there is no more available water in any of the soil layers that can be extracted in that simulation step. A diagram showing the process of water extraction in HyPAR is shown below.
Figure 4: Water Competition between trees and cropswhere supply from each cell (plot x layer) isallocated to trees or crops depending ontheir comparative 'optimum demand' androot length ratio. (From Mobbs et al. 1999)
Potential water uptake by the trees and crops in the WaNuLCAS model is similarly driven by the transpirational demand, with the actual water extraction being determined by the root length densities and by the soil water content in each of the various cells (e.g. voxels) in which the plant has roots. The actual amount of water that it is possible to extract is calculated according to a procedure that approximates that used by De Willigen and Van Noordwijk (1987, 1991), which was an iterative procedure, solving simultaneous equations for soil + plant resistance as a function of flow rate. The approximation was used, as the actual iteration process is not possible to model using the Stella environment that WaNuLCAS is written in. The potential transpirational demand is estimated using potential biomass production, which is reduced according to shading and leaf area index values, and multiplied by the water use efficiency. Tree and crop water potentials are derived from soil water potentials and root lengths, and the uptake resistance to satisfying the maximum transpirational demand calculated as proportional to soil water potential. The water potentials are then used to calculate a transpiration reduction factor. Potential water uptake rates for each soil layer are calculated. Actual water uptake is calculated from the minimum demand and total available water in all soil layers, and is then apportioned to each of the layers on the basis of the potential uptake rates, and the water contents in each of the layers is updated ready for the next simulation cycle. The model calculates a 'water stress factor' by comparing the actual water uptake with the potential transpirational demand.
2.4. The water uptake module in HySAFE Initial decisions One of the first decisions to be taken was to build a water repartition module under the CAPSIS shell that would interact with the STICS model (where all water dynamics except water extraction would be calculated). This decision was made partly due to the complexity that would arise if water dynamics were simulated both inside and outside the STICS crop model – trying to reconcile these two approaches would inevitably lead to computational errors. In addition, the SAFE modelling team felt extremely reluctant to alter the STICS code, as this would render HySAFE unable to make use of any future refinements and improvements made to STICS by INRA modellers working elsewhere. Indeed, STICS has been improved during the course of the SAFE project and we have been able to ‘move up’ to the new version of the model.
Figure 5: Diagramshowing at which point theSTICS crop model will beinterrupted in order for thewater repartition moduleto operate.
In looking at the approaches taken by both HyPAR and WaNuLCAS, neither of the water competition routines appeared to suit the specific objective of the belowground modules of HySAFE as described in the SAFE contract, i.e. that the model should be able to describe the opportunism of plants in heterogeneous environments, and especially when heterogeneity results from plant competition. Indeed the preferential extraction of water from the surface layers in HyPAR, before extracting from deeper layers, would seem to preclude this. Therefore, the second major decision was to suggest a new approach to water uptake and repartition between the trees and crops – one that would better suit our objectives. A different concept was suggested, that based upon the ‘minimisation of energy’ approach. This simply means that the water will be extracted where it is the easiest to extract, ranking the soil compartments on the basis of similar soil water (matric) potential, and attribute the water demand progressively to this group of compartments. This allows us to extract water from the soil in a manner that is much more representative of what actually happens in the field situation. Indeed, there was a concern that adopting the minimisation of energy approach would rapidly lead to homogeneous soil water potentials within the rooting zone, which seemed both unsatisfactory and unrealistic. However, recent soil moisture measurements from one of the field sites (Vézénobres, France) during a period where no significant inputs from rainfall occurred, suggest that the volume of soil where tree roots are present, is effectively becoming very homogeneous in terms of soil water potential. One key decision was to consider the crop component in terms of a series of independent ‘crop cells’, each with a cell-specific growth stage, crop water demand etc., rather than to consider the crop as a single system, more or less evenly distributed across the plot. To achieve this, the STICS model will be run on the each CAPSIS cell where the crop is planted, or on ‘clusters’ of cells where the environmental conditions are sufficiently similar to justify the savings in computation time that will accrue. The water competition module will use as inputs the tree water demand and the crop water demand from each CAPSIS cell (or clusters of cell families). Therefore, it will only be activated once the crop water demands have been calculated by STICS, which will be run on each of these cells/clusters sequentially, each interrupted after the crop demand calculation module. After completion, the module will feed back into STICS both the mass of water extracted by the crop and the soil moisture content after extraction by both the tree and the crop. It was necessary to decide at what scale (see Figure 2) this module would operate. Should it calculate the extractions from each of the minicouches used by STICS for calculating the water balance; or from the whole soil compartments (the intersections of the horizontal soil layer with the CAPSIS cell/column)? The first option would prove almost computationally impossible given the size/depth of the minicouches (1cm). The possibility of modifying STICS to increase the depth of these layers was discussed with the model developers, but they were reluctant to do this. The SAFE team did not consider that modifying the STICS code by ourselves to achieve the change in layer depth was advisable, given the concerns about future compatibility issues mentioned previously. Applying the module at the soil compartment scale was considered too likely to produce errors, as the volumes of soil concerned would most likely be very large and it would be unrealistic to make assumptions such as homogeneity of root densities over such distances. Soil layers are defined by their structural properties (texture, organic matter, bulk density etc.). However, in many situations there may only be two distinguishable soil layers (e.g. at the INRA Restinclières site: 0-30cm; and 30-300cm). While some of the compartment properties can be assumed homogenous, it would be incorrect to assume uniform water content across such a volume. A vertical gradient of humidity would never develop in such a simple column of compartments – we needed some form of subdividing deep compartments. The decision was reached to calculate water extraction from intermediate sub-layers within the compartments, called voxels. The term is a contraction of ‘volume element’ (by analogy
with ‘pixel’), and is commonly used in three-dimensional modelling. It is defined as ‘the smallest distinguishable box-shaped part of a three-dimensional space’. The voxels will differ in terms of their water content, even if they share similar soil structural parameters. Further discussions centred on whether to consider only voxels of uniform dimensions (e.g. 1m X 1m X 1m), or whether it was necessary to be able to have non-cubic voxels. Eventually it was decided that the horizontal X-Y dimensions of voxels in HySAFE would be uniform (i.e. square), but that the depth (Z-dimension) could vary. This was necessary in order to be able to divide the compartments (of variable depth due to the heterogeneity of the soil pedological layers) into discrete voxels. To match STICS minicouche and voxel depths, a simple rounding rule should be applied: all soil layers and voxels should have a thickness that is a multiple of the 1cm minicouche. Rounding all soil layer depths to 10cm seemed an acceptable compromise. It was suggested that we should define soil layers not only based on structural characteristics, but also on a maximum thickness of voxels suitable for the cellular automata module (for tree root growth) being simultaneously developed. The final definition of this maximum depth is still under discussion, but should be in the order of 50cm. The surface soil voxel would naturally depend on the thickness of the ploughed layer. A rule for soil converting soil layers/compartments into voxels would therefore be: Any soil layer with a thickness < 50cm is considered as a single voxel Any soil layer with a thickness > 50cm is split into a number of voxels e.g. if 50 < thickness < 100cm then compartment split into 2 voxels if 100 < thickness < 150 cm then compartment split into 3 voxels etc. It was highlighted early in the discussions on the development of the module that the manner in which the extracted water is shared between the tree and the crop needed to be defined. The evaporative demand from the two coexisting species (tree and crop) must be ‘realistically’ shared, and this demand should be spread between all the voxels where tree or crop roots are present in such a way as to simulate where the plants actually extract water in the soil profile. In order to prevent the calculation procedure from becoming unstable, the module has to assume that when the demand is distributed throughout a group of voxels (with the same water potential), the actual extraction from each voxel is ‘independent’ of the extraction from the other voxels. There are several ways by which sharing the water resource in each compartment could be simulated; sharing it according to the potential water demands of the tree and the crop, respectively; according to the fine root densities of the two species. In this case, it would be necessary to make an extra calculation, of the order in which the soil compartments are accessible by the tree; or according to a function that combines both the fine root density and the potential water demands. The decision was taken to share the extracted water according to the third option, incorporating both root densities and relative water demands. How the decisions reached have been implemented in the water repartition module The module operates in the following manner. Available water is extracted from the various soil compartments by both the tree and the crops in a continuous fashion (and in parallel), throughout the day. Water extraction proceeds in a manner that is proportional to each of the tree and crop water demands. Water extraction must also take account of both the tree and crop root density in each compartment when defining the potential rate of extraction. Firstly, the module examines all soil voxels within the plot (i.e. within the horizontal limits of the plot, and the vertical limit set by maximum simulated rooting depth) to identify those containing tree and/or crop roots. Naturally, only these voxels are considered in terms of water extraction by roots. These voxels need to be ranked in order of the ‘availability’ of the water they contain for extraction. As the water content (θ) does not correctly explain the relative ease with which water can be extracted, the value of θ (m3 m-3) in each of these voxels is used to calculate the soil water potential, ψ (kPa). This is achieved by using an inverted form
of the van Genuchten pedotransfer function (van Genuchten, 1980) to obtain the soil water potential (cm H2O) from the water content (m3 m-3):
where: θs is the saturated water content, θr is the residual water content, h is the ‘pressure head’ (matric potential), α and n are parameters describing the shape of the curve.
Values for the parameters θs, θr, α and n are obtained from the database of European soils data from a previous EU project HYdraulic PRoperties of European Soils (HYPRES), which derived a series of class pedotransfer functions (PTFs) capable of predicting hydrological characteristics of some 3890 European soil horizons (Wösten et al., 1999). HYPRES uses 11 soil type classes (5 topsoils, 5 subsoils and 1 organic soil type). It defines them according to the table below. The water potential can be simply converted from cm H2O to kPa if required. Table 1: Definition of 11 soil classes in the HYPRES database (from Wösten et al., 1999). Soil type name Description Topsoils Coarse Clay% less than 18%; and Sand% more than 65%
Medium Clay% between 18% and 35%; and Sand% more than 15%; or Clay% less than 18%; and Sand% between 15% and 65%
Medium fine Clay% less than 35%; and Sand% less than 15% Fine Clay% between 35% and 60% Very fine Clay% more than 60% Subsoils Coarse Clay% less than 18%; and Sand% more than 65%
Medium Clay% between 18% and 35%; and Sand% more than 15%; or Clay% less than 18%; and Sand% between 15% and 65%
Medium fine Clay% less than 35%; and Sand% less than 15% Fine Clay% between 35% and 60% Very fine Clay% more than 60% Organic soils FAO (1990) description Voxels with water available for extraction may well come from very different regions of the soil, indeed this is what this module is intended to be able to simulate. Therefore different PTFs are used to accurately convert water contents to water potentials before ranking can take place, although the diagram above assumes the voxels are all of the same soil type for clarity of illustration. Voxels also vary in size (as shown in Figure 6 below), with different quantities of water to contribute, even if their soil water potentials are similar. This is taken into account in the water extraction process. The rooted voxels are ranked according to the soil water potential – with the voxels with the least negative values of ψ being the easiest, and therefore the first, from which extraction can take place. The maximum amount of water that it is possible to extract from this first group of voxels is defined as the amount of water which can be extracted before the soil water potentials in these voxels equilibrate with the next group (with more negative soil water potentials). The actual amount of water that can be extracted from each group of voxels will be only a fraction of this maximum amount of water, as not all the extractable water from a voxel can realistically be extracted by the roots in the given simulation timestep (i.e. 24 hours). This is a logical assumption to make and could arise for a number of reasons. Firstly, the hydraulic conductivity of the coarse root system may simply not be sufficient to allow this amount of water to be transported over the time available. This may be true in some cases but is unlikely to be the main reason. Secondly, in large unsaturated voxels the soil hydraulic
hs r
nn
n
=−
−
−11
1
1
αθ
θ θ
conductivities may be insufficient to allow all of the extractable water to move towards the roots and be taken up before the 24-hour period is over. Lastly, it is possible that the conductivity at the interface between the soil and the fine roots decreases under dry conditions, causing the system to self-limit.
In the current version of the module a ‘brake function’ is included, which was intended to simulate the fact that water is more difficult to extract when the water potential is more negative. This took the form of a simple linear function that allows the calculation of the extractable water from the voxel. However, this form of brake function based solely on water potential may be unnecessary as in situations where the soil water potential becomes increasingly negative, the quantity of ‘available’ water decreases significantly. However, as long as the soil water potential is still less negative than the wilting point, and active roots are present, that water will be extracted. It may be preferable to accept that this bias should balance out over the longer term (several days), as the wetter voxels will dry more rapidly, and further voxels will progressively contribute to the water extraction process, with soil water potentials within the rooted zone gradually equilibrating. In the current version of the model, the extracted water is determined by comparing each of the tree and crop water demands from the voxels with the quantity of water available in these voxels (Figure 7 below). Once extracted, the amount of water actually removed from each individual voxel is determined by the root density coefficients of the tree and crop in each voxels, respectively. The crop + tree water demands (from STICS and the external tree growth module) are considered in term of mass of water (i.e. grams) and therefore the
Figure 6. Simpleillustration of how therooted voxels areranked for stepwiseextraction, and how thequantity of wateravailable for extractionis currently defined.
amount of water extracted from each group of voxels should similarly be calculated in terms of mass flow (grams), rather than in terms of water contents (millimetres or %). The fact that the voxels within each group may also be of different volumes also makes this mass flow consideration essential.
Once the actual amount of water possible to extract from the first group of voxels has been extracted, the module moves on to the next voxel (or group of voxels) with the more negative soil water potentials. Extraction proceeds as before, and the process stops only once one of two conditions are met. Extraction ceases when both of the external modules determining crop and tree water demands decide that water demand is zero. (If either the water demand is still greater than zero, then extraction continues until that single water demand is satisfied, but obviously, no partitioning of this extracted water is required). Extraction will also cease if all the water that it is possible to extract from all rooted voxels has been exploited, without the tree and/or crop water demands being satisfied. In this case, one (or both) of these demands will remain unsatisfied. Further improvements to the water repartition module Despite the reservations expressed previously, some form of ‘brake function’ coefficient to limit extraction may be needed. This revised function should take account of the average distance between the fine roots, which could be calculated from the root length density, and compared to optimum values available from the literature. The function may also need to take account of the unsaturated soil water conductivity, which can be obtained from the pedotransfer functions and database parameters from HYPRES – as the project determined conductivity values (Ks) for the soils studied.
Figure 7: Water extractionby tree and crop roots:repartition between thetree and the crops.
In the current version of the module, extraction occurs simultaneously from the first group of voxels that were identified as having the most easily extracted water, calculating the volume of water available for extraction as a single amount from the group of voxels. This has been found to be incorrect as the probability exists that some of these voxels will come from soil columns beneath different CAPSIS cells. The crop water demands from these cells are expected to vary (as discussed previously) so sharing the extracted water should not be performed on a ‘combined’ volume of water, but should instead be performed on each voxel within the group separately. This will be corrected in the next version of the module. In nature, water is extracted throughout the rooting zone, where water extraction is a process that occurs simultaneously from both wetter and drier voxels (admittedly at different rates). Applying the ‘minimisation of energy approach’ does not simulate this. Instead, water is extracted from one part of the zone, before the module allows extraction from other, drier areas. In this sense, it makes only a slight improvement on the approach adopted in HyPAR, where priority is given to water extraction from the surface before deeper layers are accessed. At least with the current module, soil voxels are prioritised on the basis of water availability, rather than their spatial location. Nevertheless, it is possible that the approach would lead to unrealistic results, at least in the short term. One such possibility is that the water demand could, in theory, be completely satisfied by over-extraction from one or two ‘water-rich’ voxels that are the first to be considered during the simulation run, ignoring extraction from drier voxels because demand is satisfied before all voxels are examined. (In reality, root water extraction from a heterogeneous soil occurs concurrently from many different voxels at very different water potentials, with even dry voxels contributing to the extraction process.) The situation would not be expected to occur in the field, as when a new voxel with a favourably high water content is first encountered by the expanding growing system, only a few roots are able to exploit what is effectively a very large water resource in this humid (but not saturated) voxel. Over just 24 hours, it is very unlikely that this water could be completely extracted, due to the slow water movement in the unsaturated soil. The assumption implicit in the ‘minimisation of energy’ approach that all the available water from this voxel will be extracted until its soil water potential equilibrates with the next ranked voxel would be expected to drive the rooting zone to develop a homogeneous water potential field throughout the soil. Despite recent observations at one of the INRA field sites (Vézénobres, France), that suggest this is occurring, this does not mean that is always will be the case and the module should not assume that it will be. The suggestion for the next version of the module is to ‘freeze’ the water that the root systems were unable to extract during the calculation step, i.e. the quantity remaining after the ‘brake function’ was applied. This amount of water will no longer be considered ‘available voxel water’ for the remaining calculation steps. The revised extraction process is summarised in Figure 8 (below). If we were to adopt this ‘freezing’ procedure, we would still be keeping to the original ‘minimisation of energy’ approach that was the core concept of the water extraction and repartition module, i.e. water will still be preferentially extracted from voxels where the water is more easily extractable. The advantage comes from the greater number of voxels that will be considered in the calculation, more closely mimicking the natural situation in the field. As a result, it will take the extraction process longer to produce a homogeneous water potential field within the rooting zone, and situations of water stress would take less time to develop. This improvement will be implemented directly in the Java code by the computer programming team, and checked by the developers of the water repartition module. The results of the modules ‘with’ and ‘without’ this modification will be compared in a methodological study in 2004.
Step 0:Voxels (with roots present) ranked by decreasing water potential
Step 1:Some water is extracted from the first voxel, (depending on the fine root density, the water potential, and the species ‘ability to extract water’). The remaining water is ‘frozen’ and considered ‘unavailable’ to the roots during this extraction run
Water available for extraction during first step
Step 2:Again, some water is extracted from the first (and now second) voxels. The remaining water is ‘frozen’ and considered ‘unavailable’ to the roots during this extraction run.
Step 3:The process continues with some water extracted from each of the first three voxels, and some water still considered ‘unavailable’ to the roots.
Step 4:In this case, the difference between the value of ψ in neighbouring voxels is much smaller than in the preceding steps. It is possible for all the available water to be extracted, with none remaining ‘unavailable’.
Step 5:When there are no more voxels left to extract from the remaining water contents of the voxels are recalculated ready for the next computing run.
Water used
Water remaining
Step 0:Voxels (with roots present) ranked by decreasing water potential
Step 1:Some water is extracted from the first voxel, (depending on the fine root density, the water potential, and the species ‘ability to extract water’). The remaining water is ‘frozen’ and considered ‘unavailable’ to the roots during this extraction run
Water available for extraction during first step
Step 2:Again, some water is extracted from the first (and now second) voxels. The remaining water is ‘frozen’ and considered ‘unavailable’ to the roots during this extraction run.
Step 3:The process continues with some water extracted from each of the first three voxels, and some water still considered ‘unavailable’ to the roots.
Step 4:In this case, the difference between the value of ψ in neighbouring voxels is much smaller than in the preceding steps. It is possible for all the available water to be extracted, with none remaining ‘unavailable’.
Step 5:When there are no more voxels left to extract from the remaining water contents of the voxels are recalculated ready for the next computing run.
Water used
Water remaining
Figure 8: Revised process of voxelwater extraction by tree and crop roots,including ‘freezing’ non-extracted water
Further improvements are required to better handle the conversion between the STICS minicouches and the voxels used by the water repartition module. After the STICS model is interrupted, following the calculation of crop water demands, water contents per minicouche are transferred to the water repartition module. Converting the state variables in each minicouche (water content, nitrogen content, root pools etc.) into voxel state variables is simply to calculate the average value for all state variables for all the minicouches that comprise the voxel in question. After the water repartition module completes the water extraction process, water contents per voxel have to be disaggregated back into the appropriate minicouche in STICS. This voxel to minicouche conversion is more complicated. It is suggested that the state variable values in the surface minicouche from STICS are copied to the water repartition module, and are kept unchanged. These will act as an ‘anchor point’ from which some form of interpolation algorithm is employed to break the voxel back into the constituent voxels. Further work is involved in determining the most appropriate form of algorithm, but it may include a curvature minimisation procedure. Although it is not considered a priority for further refinement, observations from one of the field sites has raised the question of how the module would be able to correctly simulate the occurrence and persistence of water-saturated voxels resulting from water tables that are within reach of the tree root system. It is not a priority because the tree species currently considered by HySAFE are not species known to be resistant to water. It is therefore safe to assume that any fine roots in waterlogged voxels would die after a given time due to anoxia and suffocation. However, the modelling team has always hoped that the model would be capable (at some future date) of being generic, and as such might be applied to situations incorporating tree species such as Alnus whose root systems have evolved to withstand continued exposure to waterlogged conditions. In cases where the tree root system grows down to meet the water table, or where there is a sudden rise in the water table inundating previously unsaturated voxels, the tree has access to an unlimited supply of water available with no potential gradients to contend with (Figure 9). Water uptake is only limited by the hydraulic conductivity of the coarse root system. Even if this is a short-lived effect as one assumes the fine roots will die rapidly from lack of oxygen, the fine roots in the soil directly above the interface with the water table will benefit from water moving upwards through capillary rise, and should be modelled. Capillary rise is modelled by STICS and this model should therefore adjust the water contents of the minicouches (and hence the voxels) directly affected. However, care must be taken in aggregating the numerous minicouche water contents into the constituent voxel water content as the effect of a couple of water-enhanced voxels at the bottom of the voxel will be ‘diluted’ following aggregation. Following discussion it has been decided that a voxel must be considered to be entirely ‘saturated’ or ‘not saturated’. To ensure this happens, the height of the water table must be adjusted by the water repartition module to the nearest horizontal boundary between voxels. Further work is required to examine how this will work, and to consider introducing a function determining fine root survival in saturated voxels.
voxels directly affected by capillary rise
water table height
Figure 9: Saturated andunsaturated voxelsaffected by theexistence of a localwater table in anagroforestry system
In addition, soil water transport from one voxel to another via the root system (i.e. ‘hydraulic lift’) is a well-documented process that is known to occur in some agroforestry systems, and it would be good if the HySAFE model could incorporate this in cases where the soil water potential gradients make it possible. As the architecture of the coarse root system is not modelled in HySAFE, it is impossible to know which voxels are directly ‘connected’ by a coarse root channel. However, it is logical to assume that water moves from moist voxels to dry voxels, and therefore the transport distance could be ascertained from the distances between the voxels concerned, although the exact manner in which this is calculated will vary between trees with heart-shaped rooting systems (such as wild cherry) and those with tapping rooting systems (such as walnut). Hydraulic lift is not considered a priority issue for immediate incorporation. Finally, it has been suggested that the ability to extract water per unit length of fine roots may vary between species, e.g. between Quercus and Fagus sp. If this is true in the case of the four tree species considered by the SAFE project (or for other tree species to which the model may eventually be applied) then the repartition of water extraction from the voxels should not be driven simply by the ratio of fine roots. A more sophisticated approach may be warranted, e.g. by converting each fine root length into ‘effective’ fine root length before the ratio (of effective root extraction) is calculated. Until data is available to calibrate this parameter, the modelling team is cautious to introduce this modification, as it is likely that the module may be very sensitive to this parameter, given that water extraction is strongly dependent on root dynamics. A literature review is underway to see if differences in the ability to extract water per unit root length have been recorded for the four species (Quercus, Prunus, Juglans and Populus). While not strictly the responsibility of the water repartition module, some additional hydrological issues should be given consideration, and result from the nature of the STICS model, and specifically the water dynamics routines within the model. These focus on the horizontal redistribution of water between voxels/compartments/columns. STICS assumes a uniform crop growing on a given plot area, and any movement of water ‘out’ of the plot (e.g. runoff) is considered as a loss from the system. There does not appear, for example, to be any way in which there is an equivalent ‘run-on’ input to the system. Horizontal redistribution of water between voxels can take two forms; subsurface lateral flow and soil surface run-off. The first of these is not currently implemented in STICS as it is assumed that no horizontal gradients will arise under the uniform crop for which the STICS model was designed. As mentioned above, run-off is considered as a net loss of water to the system. In HySAFE, we use multiple STICS model runs on each CAPSIS cell (or groups of cells); we are not applying one STICS model run to the whole of the HySAFE plot. More work is needed to see if run-off from one CAPSIS cell can be used as input for an adjacent cell, but this would have major implications for the cell-clustering routine developed in HySAFE. It may be even more complicated when considering subsurface lateral transfers of water between cells.
2.5. Application + coding Alain Fouéré (INRA-APC) coded the current version of the water repartition module in C++, and Isabelle Lecomte (INRA-SYSTEM) has translated code into JAVA, incorporating aspects such as the pedotransfer functions along the way. Initial difficulties included finding both a fast and accurate sorting algorithm for ranking the voxels with regard to water availability, and also deciding at which point the STICS crop model could (and should, correctly) be stopped to allow interaction with the water repartition module. Figure 10 (below) shows a flowchart representation of the stepwise processing of the voxels for root water extraction, including the interaction between it and both STICS and the tree growth module.
2.6. References Food and Agriculture Organisation (FAO). 1990. Guidelines for soil description, 3rd edn.
FAO/ISRIC, Rome. Mobbs, D.C., Lawson, G.J., Friend, A.D., Crout, N.M.J., Arah, J.R.M., and Hodnett, M.G.
1999. HyPAR: Model for Agroforestry Systems. Technical Manual Model Description for Version 3.0. Centre for Ecology and Hydrology, Edinburgh, United Kingdom
van Genuchten, M.T. 1980. A closed form equation for predicting hydraulic conductivity in
unsaturated soils. Soil Science Society of America Journal, 44, 892-898. van Noordwijk, M. and Lusiana, B. 2000. WaNuLCAS version 2.0, Background on a model of
water nutrient and light capture in agroforestry systems. International Centre for Research in Agroforestry (ICRAF), Bogor, Indonesia
Wösten, J.H.M., Lilly, A., Nemes, A. and Le Bas, C. 1999. Development and use of a
database of hydraulic properties of European soils. Geoderma, 90, 169-185.
19
Figu
re 1
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3
Feedback to Tree root model: voxel water content
3. Tree root growth module
3.1. Objectives and inputs The simplistic representation of the shape and density of tree root systems in agroforestry systems was identified as a major weakness of many models, including the two examined in detail as possible to include in the HySAFE model (HyPAR and WaNuLCAS). The objectives of the HySAFE demand a subtler means of simulation that reacts to local conditions that vary over quite a small scale. The module must simulate the fact that, logically, the tree must colonise the most favourable areas of soil, taking account of the influence that neighbouring crops exert on the development of its root system.
3.2. Challenge to be modelled A universal root growth module was suggested, which would combine a degree of genetic control and the local pedo-climatic conditions affecting root growth. In effect, if this growth were purely dependent on the local soil conditions, all trees would have identical root systems (a purely ‘opportunistic’ system). This should be possible to simulate via the relative allocation of carbon to belowground growth, and it would allow the uniqueness of each tree species to be properly represented, as this allocation would depend on parameters that are specific to each species, contained within the modules of carbon acquisition and allocation. Under the same local pedo-climatic conditions, and faced with the same crop competition, a purely opportunistic form of root growth simulation would allow two trees of different species to develop slightly different root systems, even if the same rules governing root growth were applied. In this way, a cherry tree that flushes a month earlier than walnut would have the time exploit the water available in the soil, some of which would have disappeared by the time that the walnut comes into leaf, and would therefore develop roots in parts of the soil that the walnut is unable to colonise. However, if taken to the extreme, this approach might predict that ‘early’ temperate trees should develop root systems closer to the soil surface. It would also suggest that later-developing temperate trees have more deep rooted systems, with the proviso that they have evolved surrounded by vegetation that is earlier to develop each year, and in a climate where summer rainfall is ineffectual. Bearing this in mind, it was considered necessary to introduce some form of genetic control that takes account of the diverse propensity of trees to prioritise certain directions of root colonisation over others. It was proposed to classify the types of root systems of the four tree species considered by HySAFE and to describe each type by a series of parameters that is as short as possible. Such a classification should take into account the following:
• preference between deeper or shallower growth • tendency to explore the area of soil closer or further away from the collar of the tree • the influence of the distance from the trunk and the limit to root system expansion (in
the absence of limiting local pedo-climatic conditions) • the maximum density of colonisation of the rooted zones; and the aptitude to colonise
upwards (negative geotropism), etc. The various forms of soil colonisation are illustrated in the following diagram.
Figure 11: different forms of tree root system possible –‘balanced’ (left), ‘superficial’ (middle) and ‘deep’ (right)
The roots within any given volume of soil explored by the tree root system can be distributed in several different ways. This is illustrated in the following diagram.
In addition, the module should take account of the differences in (topological) coarse root architecture that exists between the four HySAFE tree species, in order to determine the impacts of interventions such as root pruning.
Figure 12: Different ways in which tree may distribute roots within a rooted volume.From left to right – uniform, proximal, distal, central and peripheral root systems
Walnut root system Cherry root system
Before root pruning
After root pruning
Figure 13: Illustration of the relativeeffects of root pruning ontwo tree species withdifferent root systemarchitectures. Pruninglocation and depth isrepresented in red.
3.3. Tree root growth in HyPAR and WaNuLCAS HyPAR models tree root growth in two stages. Firstly, the carbon allocated to the root system from aboveground is distributed among roots in each soil layer within the rooted zone. The ‘rooting front’ progresses down the soil profile from the surface, thus extending the rooted zone. The processes of colonisation (of new soil layers) and proliferation (within a soil layer) follows the hypothesis of Robertson et al. (1993), i.e. when roots first colonise a soil layer, little water or nutrients is extracted until the proliferation of further roots occurs within that layer. This approach improves on earlier attempts (Monteith, 1986), it was assumed that roots in a soil layer were ‘fully expanded’ immediately after colonisation occurred. The rate of new root length production, R (cm root cm-2 ground d-1), is calculated from the assimilate available for growth (fBGG), using:
where: η is the linear density of roots (10-4 g.cm-1) and Fη an adjustment used during the juvenile stage when roots are long and thin.
The tree model incorporated into HyPAR (Hybrid v3.0), did not require the spatial distribution of the fine and coarse roots to be specified, but rather considered them as discrete known biomass pools. It was necessary to distribute coarse and fine roots between horizontal plots and vertical soil layers within HyPAR due to the 3-dimensional soil modelling approach adopted, and to account for tree/crop interactions in agroforestry systems. The way in which this achieved in HyPAR depends on the user choice of aboveground interactions, namely the method of light interception. If the user considers the aboveground canopy to be homogeneous then the roots are assumed to be horizontally distributed in a uniform manner. If the canopy is heterogeneous, then root distribution is represented in a spatially explicit and 3-dimensional manner. However, regardless of which option is selected, the model assumes exponential decreases in root length with depth, and with distance from the tree if the canopy is ‘disaggregated’ see Figure 14). This is common to many modelling approaches to tree root distribution. The maximum soil depth, RTD (mm) that roots can exploit is dependent on the tree height (m), and is limited by the maximum depth of soil. It also depends on a species-specific parameter frd. For each tree, the fraction of the total biomass of fine roots, Fi
f , found in each vertical layer of soil is given by:
where di is the depth to the bottom of layer i, and Rhd is the depth at which the fine root length per unit volume (RLV) has declined to half that at the surface. Rhd is a function of the potential maximum depth (unconstrained by the physical limit of the soil).
Figure 14: exponential decrease in rootlength density with depth and distance
The maximum radial rooting distance (m) from the tree, RTW, is also dependent on tree height in the same way as maximum root depth. The fraction of root located below each plot square (analogous to the CAPSIS cell in HySAFE) is found by calculating a weighting factor, RWF, for each plot (‘cell’), p, normalised using the sum of the RWF over all plots within the potential rooting zone (ignoring the field boundaries). Tree root length density in the WaNuLCAS model can either be distributed according to an elliptical function: Where:
LraX0 is the total root length per unit area (cm cm-2) at distance X from the tree. Rt_TDecDepth (m-1) determines decreasing root length density with depth. Rt_TDistShape is a parameter specifying the shape of the tree root system. If <1 root systems are wide but shallow; values = 1 indicate symmetrical root systems, and values >1 indicate deep root systems which do not expand horizontally significantly.
A second approach is to allow the user to enter values for root length density for each soil compartment. A third approach in WaNuLCAS for obtaining root length density is to implement a ‘functional equilibrium’ response (Van Noordwijk and Van de Geijn, 1996), allowing the relative allocation of growth to roots to increase when water and/or nitrogen limit plant growth. This response is regulated by two parameters T_RtAllocResp and Rt_TdistResp. It uses (inverse) allometric equations relating proximal root diameters to the total root biomass, and drives the specific root length (length per unit biomass) as a function of this diameter.
3.4. The tree root growth module in HySAFE
Initial decisions
In HySAFE, it was proposed to simulate roots as two components: fine and coarse roots. Fine roots in each soil voxel will be characterised in terms of root length densities (m.m-3). Coarse roots in each soil voxel will be characterised in terms of their biomass (kg.m-3). It was not envisaged that the coarse root system should be simulated in an architectural manner, and it would therefore not be possible to say which voxel contained a coarse root of a certain diameter; it would be more realistic to adopt a probability-distribution approach to modelling the coarse roots. Any explicitly topological approach would effectively only provide a description of an individual that would not be ‘representative’. In any case, the simulation process in HySAFE focuses on an average tree, and besides, it would be extremely calculation-intensive. It was clear that this distinction must allow the taking account of various interventions such as root pruning, as this is part of the remit of the HySAFE model. Root pruning suppresses both fine and coarse roots in the distal compartments in comparison with the plane of pruning. Coarse roots have a function of regulating the flow of carbon allocated to the belowground part of the system. They must be subject to a rule of mortality that is not tied to pruning; in terms of carbon-limitation; by anoxia caused by waterlogging etc. A root-pruning event does not destroy a quantity of large roots in proportion to the quantity of fine roots that are destroyed. In effect, it only increases the distal portion of the large roots, which represent a small fraction of the coarse roots. It is the same for mortality due to a rise in the water table. The ratio between the fine and coarse roots destroyed by pruning therefore depends on the manner in which the density of fine roots is deployed in the soil around the tree (see Figure 13 above). It was therefore important to look for a probabilistic description of the density of coarse roots as a function of their position in the soil. Although not currently implemented in the module, it may be possible to use a simple fractal approach, which would
define a linear relationship between fine root density and a volume of coarse roots. From a map/distribution of fine roots, it should be possible to deduce a map/distribution (blurred, and non-topological) of coarse roots.
The HySAFE tree root voxel cellular automata
The HySAFE tree root module aims at providing a realistic tree root growth modelling in a 3-dimensional heterogeneous soil. It should be able to model the opportunistic growth of tree root systems, reacting to local soil conditions for root proliferation, colonisation and death. We decided to adopt a 3-D voxel cellular automata running at the 24-hour timestep. The rationale for this choice is presented below, followed by an elaboration of the transition rules of the voxel cellular automata. Cellular automata (CA) are dynamical systems in which space and time are discrete (Gutowitz, 1991). The states of cells (2D) or voxels (3D) are updated synchronously according to deterministic interaction rules (Figure 16). CA are extremely useful idealisations of the dynamical behaviour of many real complex systems with non linearly-interacting components. They have been used extensively in many domains in mathematics and physics (Rucker and Walker 1997, Ilachinski, 2001). CA were introduced by Von Neumann (1966) in an attempt to model self-reproducing biological systems. The most famous and simple CA is the life game (Conway, 1970). CA differ largely from all other computation models: it is difficult to separate the dynamics of the CA from the computation itself. Most CA models usually possess 5 generic characteristics (Ilachinski, 2001; Rennard, 2000):
• Discrete lattice of cells or voxels • Discrete states: each cell assumes 1 of a finite number of possible discrete states • Homogeneity principle: all cells are equivalent • Locality principle: each cell interacts only with cells that are in its local
neighbourhood
Fine root density (m.m-3)
Coarse root density (kg.m-3)
Fractal rule description?
Figure 15: Proposed processfor mapping coarse rootbiomass from fine rootdistribution.
Figure 16: cells and voxels used in 2-D and 3-D cellular automata modelling
• Parallelism principle: at each discrete unit time, each cell updates its current state according to a transition rule taking into account the states of cells at the previous timestep in its neighbourhood (discrete dynamics)
Voxel automata model processes that sense and react to a voxel (3-D) environment. Voxel cellular automata modelling was suggested by Greene (1989) to model 3-D above-ground growth of plants. No dynamic modelling of tree root growth using voxel cellular automata could be found in the literature.
The rationale for adopting a voxel automata for tree root modelling in HySAFE
Dynamic tree root models can be classified as architectural or continuum models (Pages et al., 2000) – see Figure 17. The root state variables in architectural models are the coordinates and size of each root segment. The root state variables in continuum models are root mass or root length per unit of soil volume. Architectural models were discarded for the HySAFE model: 3-D models can’t meet the requirements of HySAFE i.e. for rapid computation at the day timestep, and require field data that are not available for the four tree species targeted by the SAFE project. Continuum models usually include a root colonisation submodel, which can be solved with: differential equations describing a diffusion-like process (Page and Gerwitz, 1974), exchanges of root masses between adjacent cells of finite dimensions (Chopart and Vauclin, 1990), or as changes in root concentration as a function of depth and time. This latter approach is included in the STICS crop model that is incorporated in HySAFE. No three dimensional root system simulators have been developed using continuum models so far (Acock and Pachepski, 1996). 2-D models using differential equations require a fine description of the soil volume that is not compatible with the HySAFE scene discretisation. For example, the diffusion-convection modelling approach of Pachepsky et al. (1996) was not considered appropriate, as it requires numerical methods that are not compatible with the computation requirements of HySAFE. It was therefore decided to consider the approach of Jones et al., (1990), exploring diffusion from cell to cell. No publication using the cellular automata approach for this purpose could be found. A requisite for the tree root model in HySAFE is the ability to react locally to the variations of soil variables like soil water content or soil nitrogen content. Such variables exhibit variability in both time and space. In silvoarable systems, horizontal gradients in soil water content or soil nitrogen content are induced by competition between plants. Those gradients will differ along the tree row or across the tree row. The problem is therefore truly 3-dimensional. Root activity is considered opportunistic, i.e. it will be driven by local conditions. However, root activity is coordinated, in that the amount of C for the whole tree root system is fixed at the daily timestep. Root growth in a voxel is not independent from root growth in an other voxel. After reviewing all published papers on dynamic modelling of a whole plant root system, it was decided to adopt voxel cellular automata for the following reasons:
Figure 17: simulating tree root distributionusing architectural (left) and continuumapproaches (right)
No current available model could fit the needs of the HySAFE model CA allow local response of root activity to be described by simple transition rules CA provides a flexible and generic root model that can be easily coupled to the water budget module; both use the same lattice of voxels that is already implemented CA programming is simple CA computation times and memory costs are low CA allow simulation of complex behaviour with simple rules – this is expected from the root voxel automata (RVA). It should be able to model interdependent root growth in the rooted soil volume. It should be able to model the fine root dynamics after disturbing events like temporary waterlogging, root pruning or branch pruning (these events will affect either C allocation to the root system or local activity of the root system or both). In fundamental physics, CA are powerful tools for performing calculations of complex systems that would be impossible with the most powerful computers available to date using conventional differential equations. In HySAFE, CA will allow modelling dynamic tree root growth with ordinary PCs, with short computing time, meeting part of the HySAFE challenge. How the decisions reached have been implemented in the tree root growth module Two teams (INRA-SYSTEM and INRA-APC) worked on implementing the initial decisions and two initial versions of the RVA were developed. A combined RVA was the developed, combining the best aspects of each model. Following the cellular automata method, both modelling approaches assumed that roots colonise adjacent voxels according to set of rules that determine which voxels are the ‘best’ ones to grow into. The approaches of the two teams were similar in most respects, but differed in a number of ways, which are briefly discussed below, after which a full description of the current version of the RVA is given. Definition of ‘neighbouring’ voxels: INRA-SYSTEM and INRA-APC modelling approaches differed in the number of neighbouring voxels into which root growth could occur within a single simulation step. This is illustrated in the diagram below. The INRA-SYSTEM approach considered root colonisation could only proceed via the faces between the 6 adjacent neighbouring voxels (i.e. in the x, y and z directions). In contrast, the INRA-APC approach allowed colonisation to also occur via the edges or corners between neighbouring voxels, giving 26 voxels into which root growth could spread. This involved taking distances between the centres of the various voxels into account as they varied. The current combined version of the tree root growth module allows root colonisation only in the x, y and z directions.
If root expansion is allowed to occur via the flat faces of the central voxel, only six other voxels are truly adjacent and therefore available for colonisation
If root expansion is allowed to occur also via the linear edges and corners of each voxel, then a further 20 directions are possible, giving 26 total accessible voxels for expansion
If root expansion is allowed to occur via the flat faces of the central voxel, only six other voxels are truly adjacent and therefore available for colonisation
If root expansion is allowed to occur also via the linear edges and corners of each voxel, then a further 20 directions are possible, giving 26 total accessible voxels for expansion
Figure 18. Voxel colonisation by tree root growth in the two modelling approaches.
Carbon allocation to voxels: The INRA-SYSTEM approach involved distributing carbon allocated to fine root growth according to existing soil conditions (voxel carbon, nitrogen and water contents etc.) while that of the INRA-APC team involved distributing carbon solely according to existing voxel carbon, but used the other voxel characteristics (water, nitrogen, temperature etc.) when considering which neighbouring voxels are prioritised for fine root colonisation. The current, combined module follows the INRA-SYSTEM approach to carbon allocation. Further refinements to the model may include allocating carbon to voxels on the basis of the rates of water extraction from each voxel the previous day. Initiating colonisation to other voxels: The two approaches differed in terms of which conditions in the parent voxel were required before colonisation of neighbouring voxels could begin. Under the INRA-APC strategy root growth in neighbouring voxels did not begin until the parent voxel was ‘saturated’, i.e. no more carbon (fine roots) could be added to this voxel and was added to neighbouring voxels instead (in a manner similar to ‘tipping bucket’ hydrological models.) The approach adopted by the INRA-SYSTEM team was to set a ‘threshold’ level of fine root biomass in the parent voxel, which, once crossed, would initiate root spread into adjacent voxels, even though carbon allocation could continue in the parent voxel. This latter approach has been maintained for the current, combined root growth module. Direction of colonisation (‘tropisms’): Both the INRA-SYSTEM and INRA-APC approaches took the ‘genetic’ (or innate) strength of rooting direction (iso/anisotropism) into consideration. Whereas the INRA-SYSTEM module considered soil water gradients between neighbouring voxels (and differences in soil bulk density), the INRA-APC module also took other factors into consideration (mentioned in 2, above), including the concept of ‘available space’ in the destination voxels into which roots could grow, and the distance between voxels. As the combined module relies on a ‘threshold’, rather than ‘saturation’ for voxels spreading to occur, the concept of available space was abandoned. However it must adopt part of the INRA-APC approach in as much as it must take the concept of ‘voxel-to-voxel distance’ into account. This is because even though the module will only consider the x, y and z directions for colonisation, it will have to deal with voxels that are no longer ‘cubic’ (as in Figure 5.1B.1) but ‘rectangular cubic’, i.e. the x and y dimensions are the same, but the vertical depth of the voxel will be different. Fine root senescence and/or conversion to coarse roots: The INRA-APC approach to root senescence was to give a ‘birth date’ to a voxel as it was newly colonised, and then apply a fixed life-expectancy duration function. The module would use an array to record root ages in voxels, and then if the life-expectancy duration were exceeded, all related fine roots would die. The process is complicated and acts in concert with the processes governing voxel filling by roots and fine root conversion (maturity) to coarse roots. The combined processes are illustrated in Figure 19 below. In contrast, the INRA-SYSTEM approach was to simply apply a fixed fraction that is not explicitly related to root age (e.g. dead root mass considered to be proportional to leaf abscission). An alternative approach has recently been presented, which is the one in the WaNuLCAS agroforestry model, and is based on the idea of the root ‘half-life’ (estimated from repeated observations) and referred to as the ‘root decay’. One advantage to this approach would be that the number of living roots in a cell would never reach zero, and would therefore provide a method by which we could simulate root flushes in the next new growing season. The combined model currently uses the more simple approach adopted by the INRA-SYSTEM team, but further refinements to the model may well include elements of the INRA-APC and/or WaNuLCAS model approaches if they provide a more realistic representation of fine root duration and behaviour.
Departures from the Cellular Automata approach
When modelling tree root dynamics using the root voxel automata, three basic assumptions of elementary CA will not be ‘respected', as the RVA will have the following features: The RVA may be asynchronous. CA transition rules are typically defined so that all lattice sites are updated simultaneously at each timestep (synchronous updates). However, computers update voxels sequentially. If the transition result is independent of the order in which the voxels are processed, then the CA can be considered synchronous. If the result of the transition depends on the order of processing the voxels, the CA should be considered asynchronous; and then special care must be taken when designing transition rules (Ingerson, 1984). The RVA will use a coupled-map lattice (Kaneko, 1993). Coupled-map lattices are CA models in which continuity is restored to the stet space. In the RVA, each voxel will not be limited to discrete zero or one values, but will be allowed to take any arbitrary real value (e.g. the fine root length). Such systems are still simpler than partial differential equations (i.e. the reason for their introduction in HySAFE), but are more complex than generic CA. One of the key advantages of simple CA is no longer valid: truncations and round-off errors are not a worry in simple CA with only discrete values of the cells. They will be a limit in the RVA.
Figure 19: Combiningthe processes governingvoxel filling by roots,root senescence andfine to coarse rootconversion proposed bythe INRA-APC team.
The RVA will be a non-homogeneous CA. The state-transition rules will vary from one voxel to the other, for two reasons: Carbon allocation to each voxel will differ, as a result of a C-allocation rule that is external to the RVA (in the carbon allocation module of the aboveground model) ‘Border voxels’ will require modified transition rules. Soil surface voxels should logically not permit upward root colonization; voxels at the bottom (maximum soil depth) should not permit further downward root colonization; and lateral border voxels will require special transition rules to reflect the Toric symmetry of the system.
3.5. The transition rules of the HySAFE root voxel automata The water budget and the root automata will share the same lattice of cuboid voxels. The typical size of these voxels is about 1 x 1 x 0.2m. HySAFE Root systems will have two pools: fine roots and coarse roots. Both will be defined at the voxel scale, each with 3 root state variables: fine root density (m.m-3), fine root biomass (kg.m-3), coarse root biomass (kg.m-3). Four main processes will be modelled at the daily time step: Fine root proliferation (increase in the voxel fine root density) Fine root colonisation (exportation of fine roots to neighbour voxels) Fine root decay (death of fine roots) Coarse root biomass variation (increase and possible decay after root pruning or durable waterlogging) Root proliferation (i.e. the increase of fine root length in a voxel) and root colonisation (the fine roots growing from one voxel into its neighbour voxels) are intimately linked processes. Available carbon at the voxel scale is used for both processes. Some simple transition rules are being designed to represent this. The RVA will be informed by other HySAFE modules about the other voxel state variables and the carbon allocation: voxel state variables will be updated by both the water repartition module and the nitrogen competition module; while total carbon availability for the root system will be delivered daily by the C allocation module
Sharing the daily carbon allowance between fine and coarse roots
A total carbon pool is daily allocated to the root system by the tree growth module. It must be split between the coarse roots and the fine roots. The rule for this split is not considered as part of the root voxel automata. Instead, it will be included in the C-allocation module, and will include functional equilibrium responses. A conventional approach, i.e. a fixed ratio between the fine root and the coarse root pools, will not be used as it would not be able to model the flush of fine roots at various times of the year.
The allocation of fine root carbon to individual voxels
Carbon that has been allocated to fine root growth should be distributed throughout all the rooted voxels where conditions allow this to occur. To develop a physiologically sound rule
1. CUBOID
(Terminology used in HyPAR)
A closed box composed of three pairs ofrectangular faces placed opposite eachother and joined at right angles to eachother, also known as a rectangularparallelepiped. The cuboid is also a rightprism, a special case of the parallelepiped,and corresponds to what in everydayparlance is known as a (rectangular) ‘box’. http://mathworld.wolfram.com/Cuboid.html
for this is a challenge; and it will not be a transition rule of the RVA. Instead, this will be a rule making the RVA strictly non-homogeneous. Transition rules will still be homogeneous, but will use a specific input figure for each voxel. Three generic approaches can be considered: Carbon apportioned proportionally to the existing fine root pool (diffusive process) – the argument that with more roots meristems present in the voxel, more growth is expected to occur, with a greater carbon demand Carbon apportioned inversely to the existing fine root pool (convective process) –greater available space, would make roots more likely to proliferate. Carbon apportioned proportionally to local favourable conditions for fine root growth (irrespective of the existing fine root pool) – the argument that the most important factors would be the availability of resources, and local limitations to growth (e.g. soil bulk density). A rule combining the diffusion process weighted by local conditions was suggested for HySAFE. The values of water extraction per voxel from the preceding day will be used to apportion the carbon – an ‘index’ of water availability (local condition) and fine root densities (Figure 20). Each ‘rooted’ voxel will get the same percentage of the carbon allocated for fine roots as the percentage to which it contributed to the total water extraction the previous day. It should be noted that unrealistically high fine root densities should not be generated by this module. In a voxel with a high fine root density, extraction will be active, and local conditions (e.g. water, nitrogen) will rapidly become unfavourable, and result in lower carbon allocation to that voxel. The spring flush fine root flush may occur while roots still exist; in that case the algorithm is operational. If no roots exist (due to short life span, or winter waterlogging), C should be allocated proportionally to coarse roots carbon on the first day.
Fate of fine root C in a voxel: linking proliferation and colonisation
Available carbon in a voxel will be used for both proliferation and colonisation. As mentioned previously, two approaches were explored: Colonisation begins when proliferation has filled totally the voxel: This requires a parameter for the maximum value for fine root density. The question of whether this value was sensible was discussed, and it was decided that this parameter would probably be impossible to calibrate. It might also generate distortion in the automata; if several neighbouring cells reach saturation, the excess C should be distributed among neighbouring cells using an iterative calculation that would no longer respect the locality and simultaneity principles of cellular automata. In particular, the result would depend on the order of processing the voxels. It would prove very difficult to find a rule where the final carbon allocation was independent of the order of processing the voxels, when spreading the excess carbon. Colonisation begins when proliferation has filled partially the voxel: This second approach has been retained for HySAFE. The module uses two parameters. An ‘exploration threshold’ triggers colonisation, and when fine root density in the voxel reaches this threshold, the colonisation begins. A ‘generosity parameter’ fixes the proportion of excess carbon that is allocated to colonisation once this threshold is reached. With this option, no iteration is needed, and the result is independent of the order for processing the voxels. These two parameters define four colonisation strategies by tree roots (Figure 21 – below):
Figure 20: illustration of how waterextraction might vary between voxelswithin the rooted volume, and is usedto allocate C to voxels for root growth
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Orienting fine root colonisation in the neighbour voxels
As mentioned previously, three sorts of factors have been considered to drive the colonisation in adjacent voxels: Local gradients in soil conditions (water potential, nitrogen availability, soil bulk density (soil strength), and temperature) Genetic species-specific factors (that predispose vertical/horizontal colonisation) Sink strength of adjacent cells, which includes the emptiness of the cell (more roots to empty cells) and the distance of the cells (when voxels are not cubes). Upon examination, sink strength was excluded, as this can only be calculated if a maximum filling of the voxel is known. Therefore, a combination of local gradients and genetic factors will be used. The distance factor will also be included to account for the variability in voxel dimensions (voxels will have constant horizontal dimensions but variable vertical dimensions in HySAFE). The question of what would happen when neighbouring voxels are all rich in fine roots was raised. Neighbouring voxels will mutually give C to their neighbours, and this will result in an enhanced proliferation as reciprocal colonisation = proliferation. This is a logical assumption, and would correctly model what occurs in restricted volumes such as containers or shallow soils. However, as the local conditions for root growth will rapidly become unfavourable for growth (as water and nitrogen will rapidly be depleted), C allocation to these voxels will rapidly decrease, and therefore the algorithm will remain stable.
The phenology of fine roots
A phenology module for fine roots will be included, with two trigger dates: a date at which fine root flushing begins, and a date at which fine root growth is considered to end. These dates may or may not be simultaneous with bud burst and leaf fall. They could be temperature dependent. The cellular voxel automata will be activated by the phenology module. The spring flush of fine roots will be driven by the C allocation module. A large quantity of C will be allocated to the root system by tapping the carbon reserve pool. This aspect is dealt with in the C allocation module (aboveground).
Fine root decay
Although discussions are ongoing, there is no agreement yet on how to model this process. However, a fixed fraction applied at the daily time step is an easy way to start. Others possibilities have been suggested. Perhaps it would be possible to use a fixed life expectancy, or an exponentially decaying process. This would probably require that in each voxel an average age of the fine roots should be tracked. This age would be computed as the weighted average of fine root biomass – at day d, an additional fine root pool frd is added to the existing pool FRd-1. The age of the existing pool is: AgeFRd-1. A fraction %FRd-1 of FRd-1 is suppressed by the decaying function (using AgeFRd-1 to calculate it). FRd = FRd-1 – %FRd-1. The new age of the fine roots in the voxel at the end of the day would be: AgeFRd = (a + FRd* AgeFRd-1)/(a+ FRd). Our knowledge on fine root dynamics is very limited, e.g. it is uncertain as to whether some (or many) fine roots survive during the winter period.
Coarse root dynamics
A nominal (balanced) fine roots to coarse roots ratio will be used by the C allocation module (see Grégoire Vincent: Suggested modifications to the first draft of a C allocation module, September 2003). This ratio will be used to split the C daily allowance to the root system between the fine root and the coarse root pools. Daily additional C for coarse roots needs to be dispatched between all the rooted voxels. As the HySAFE root module is not an architectural model, it can be done only with a simplified hypothesis on the structure of the root system. We suggest distributing the additional C available for coarse roots at the daily time step in the following way: Each voxel is located by its distance from the trunk. Voxels are sorted with this distance. Clusters of voxels (all voxels at the same distance from the bottom of the tree, a cluster is approximately a spherical cup – volume between two concentric spheres) are defined. In each, the quantity of fine roots and coarse roots in more distant voxels is computed. These roots have to be fed by the coarse roots in the cluster. Finally, the daily additional C available is dispatched to all the voxels proportionally with their feeding role. With this algorithm, it is no longer necessary to introduce a conversion process from fine roots to coarse roots.
Suggestion of an algorithm for allocating coarse roots C to voxels:
The total daily C allowance for coarse roots is A. All rooted voxels are ranked according to their distance from the tree: C1, C2,… Cn. C1 is the voxel at the bottom of the tree; Cn is the most distant voxel. Voxels at the same distance are grouped in clusters. We obtain p clusters, each cluster groups a number of voxels N1, N2,… Np Each cluster feeds a biomass of fine and coarse roots in more distant clusters. DRB1, DRB2, … DRBp (Distant Root Biomass for cluster p). DRB can be expressed as a percentage of the total root biomass. The calculation of daily additional C to the coarse root pool of cluster DACCRi is computed as DACCRi = (DRBi-DRBi+1)*A The daily additional C to each voxel of the cluster is DACCRi/Ni The calculation should start from the more distant cluster. It is assumed that the more distant cluster has no coarse roots and therefore receives no C for coarse roots. With some distorted root system shapes, this algorithm may not be adapted. For example, if the root system is superficial and large, deep voxels under the tree trunk will be assumed to feed superficial voxels far away from the trunk, and get a non-sensible amount of C for their coarse root pool. This could be avoided by applying the algorithm successively to spherical sectors of the belowground hemisphere. With this algorithm, no decrease of the coarse root biomass in a given voxel will happen, which is fine and consistent. Root pruning can be easily modelled, as a root pruning event can be described by a depth and a position on the scene (usually a distance from the tree row, but circular root pruning around isolated trees can be considered). An algorithm to decide if a given voxel is affected by root pruning should be developed. In the case of root pruning along the tree row line, it will be necessary to calculate the distance of each voxel from its tree row, which is easy. A voxel will be root pruned if its distance to the tree row is within than the pruning distance, and its depth is within the pruning depth. Root pruning indices that both fine root and coarse root biomass are incorporated into the soil organic matter pools. After root pruning, colonisation would be correctly modelled, as a consequence of higher resource concentrations in the voided root voxels (attractiveness)
and a high root content in the unpruned voxels. The C allocation module will also calculate an unbalanced coarse/fine root ratio, as root pruning would affect the fine root pool more than it would the coarse root pool. It will therefore allocate more C to the fine root pool.
3.6. The geometric validity of the voxel automata Two issues have been raised: voxel size and shape; number of neighbours considered in the colonisation process
The voxel size and shape
It is important to ascertain the influence of the voxel size and shape on the performance and results of the RVA. HySAFE voxels defined for the water budget (and above ground processes, i.e. the horizontal dimensions) are large (typically 1 x 1 x 0.2 m), and using relatively few, large voxels may result in unrealistic rooting system dynamics. Ideally, the voxel automata should use small cubic voxels. This would limit geometric effects on root growth results. If strong biases are shown to result from using large metric voxels, a more detailed voxellisation of the soil for the voxel automata (typically with 0.1 x 0.1 x 0.1m minivoxels) may be necessary. However, this would require greater computation time, and additional calculations for averaging minivoxel values to update voxels. This danger must be evaluated by comparing voxel automata using different sizes of voxels with the same soil conditions and C allocation to the root system. This will be performed by RM in the next months. The default setting of HySAFE is to use the same large voxels as for the water budget.
The number of neighbour voxels considered in the colonisation process
As mentioned briefly previously, two options were suggested: Limiting colonisation to the 6 voxels sharing a common face with the parent voxel Allowing colonisation to occur to all 26 voxels that share either a common face (6), edge (12) or corner (8) with the parent voxel (see Figures 18 and 22). For the sake of simplicity, the default setting in HySAFE will be using the six-voxel approach. Simulations will be run to test the impact of this simplification on the automata behaviour. If only six voxels can be colonised, this will delay the colonisation into the other 18 voxels. These voxels will be colonised via the six voxels. If a spherical expansion of the root system from the centre of the parent voxel is assumed, root colonisation in the 26 voxels can is proportional to the intersection of the volume of an expanding sphere and the volume of the neighbouring cells, but the calculation of the volumes intersected by cubes and spheres is complex.
Figure 22: The delay that incurs dependingon whether colonisation is allowed via sixneighbour voxels (i.e. only via faces), or viaall 26 neighbour voxels (i.e. through edgesand corners). The red voxel is the parentvoxel from which colonisation initiates
3.7. The parameters of the voxel automata The current version of the voxel automata requires six structural parameters and a number of additional parameters. This part of the module is still under development.
3.8. Further improvements to the water repartition module The voxel cellular automata and the families of cells algorithm As mentioned previously (in the discussion of the water repartition module), in order to reduce computation time, various CAPSIS cells will be grouped in ‘clusters’. The STICS crop model will be applied to each of these cell clusters. All voxels located at the same depth, but in different cells of the same cluster, will be identical. Their state variables will be computed as the weighted average of the different cells at clustering. However, there is a question as to what effects this clustering process will have on the functioning and results obtained from the voxel cellular automata. Two possibilities will be explored: Option 1: All of the state variables, including fine root densities, are considered to be uniform within a cluster. In this case, the functioning of the voxel automata may be strongly perturbed. If the individual fine root density values of in voxel are replaced by a weighted average, some spatial patterns that are important for root colonisation may be modified. Option 2: All of the state variables, except for the fine root densities, are considered to be uniform within a cluster. In this case, the model will have to deal with two values of fine root density for each of the voxels. The voxel cellular automata will use the detailed (individual cell) values, while the STICS crop model will use the cluster-average values. However, it is accepted that local numerical discrepancies may occur with this approach, as local water extraction will be calculated with the average value of the cluster, and may not be exactly consistent with local fine root densities. For the time being, Option 2 will be implemented as the default setting, pending further exploration of the consequences that averaging the fine root densities may have on the functioning and results from the voxel automata. The two possibilities must be compared.
3.9. Consistency of the tree and the crop root models in HySAFE The tree root growth and distribution will be modelled by 3-D voxel automata, while that of the crop root will be simulated by the STICS root module. The STICS root module is 1-D, and assumes an exponentially decreasing root profile with depth, in much the same way as the HyPAR tree model. However, it includes a factor that reflects ‘sensitivity’ to local soil conditions that can modify the rooting profile. The STICS model is therefore also sensitive to local conditions. The tree and crop root systems will interact in HySAFE, albeit indirectly. A single unique tree root system will interact with numerous different crop root systems, on the different CAPSIS cells within the scene. In addition, if several trees are considered within the scene, their root systems may also interact. This interaction will be controlled by modified soil conditions due to root activities. Therefore, each root system should react to soil conditions as modified by the activity of all other root systems present in the scene; however, specifically ‘allelopathic’ interactions between root systems are not considered by HySAFE. If extremely favourable conditions persist within a given soil volume, then the HySAFE model would be expected to predict similarly high root densities of both trees and crops. Field experiments should be designed to check to see if this predicted result actually occurs.
3.10. Conclusion Dynamic root models that use the continuum approach usually adopt contrasting modelling hypotheses. The hypotheses adopted by the HySAFE Voxel Root Automata are summarised below, following the criteria classification description of Acock and Pachepski (1996).
Aspect Hypothesis currently
adopted in the HySAFE VRAPossible alternative hypothesis
Carbon allocation to roots
Carbon is allocated to voxels with the best conditions for root activity (based on amount of water extracted the previous day)
Carbon is allocated to voxels that are already ‘rich’ in roots (i.e. with a large number of root meristems present)
Is there an upper limit to root proliferation within a voxel?
No maximum root concentration is specified in the VRA
A maximum root concentration is specified, with implications for further allocation of carbon
Dependence of the proliferation rate on local soil conditions
Rate depends on the soil state variables (but this is determined by the carbon allocation rule)
Constant rate of proliferation
Dependence of the proliferation rate on root density
Rate depends on the fine root concentration (again determined by the carbon allocation rule)
Rate of proliferation depends on the mature root concentration
Rate of colonisation of new, neighbouring voxels
Rate depends on a threshold value for fine root density
Rate depends on a threshold value for mature root density
The number of separate root categories considered, each with distinct growth and colonisation parameters
Two root categories considered – fine and coarse (diameter >2mm)
Between 1 to 4 separate categories could be considered
Geotropism considered as a factor determining growth
Geotropism considered; as the difference between vertical and horizontal voxel colonisation rates
No preferential direction of growth is considered
Root colonisation is drive by a convective process (local source root density) or by a diffusive process (sink gradient of root density)
Root colonisation driven mostly by convective processes, but uses active root density – cells rich in roots but with unfavourable conditions will not grow.
Root colonisation is often driven either by purely convective means (e.g. Marani et al., 1992), or purely diffusive processes (e.g. Page and Gerwitz, 1974)
Root growth dependency on the distance from tree base
Root growth independent of distance
Root growth often dependent on distance (e.g. Jones et al., 1990)
3.11. Application + coding
Alain Fouéré coded the INRA-APC version of the root growth module in C++, and provided a 3-D viewer by which root growth could be visualised in a dynamic fashion. Rachmat Mulia used Microsoft Excel and Visual Basis to code the preliminary version of the INRA-SYSTEM module. Isabelle Lecomte (INRA-SYSTEM) has translated the current, combined version of the root module from the Excel/VB code into JAVA. The current, combined module is able to simulate a tree root system that is asymmetrical in all directions in response to found heterogeneous soil conditions around the tree trunk. It comprises four different processes:
• Carbon allocation to rooted voxels • Proliferation in the voxel and/or root colonisation to neighbours voxels • Root senescence • Fine root evolution to coarse roots (> 2 mm)
3.12. References HyPAR and WaNuLCAS modelling approaches Mobbs, D.C., Lawson, G.J., Friend, A.D., Crout, N.M.J., Arah, J.R.M. and Hodnett, M.G. 1999. HyPAR. Model for agroforestry systems. Technical Manual. Model Description for Version 3.0. Monteith, J.L. 1986. How do crops manipulate water supply and demand? Phil. Trans. R. Soc. London, A. 316: 245-259. Robertson, M.J., Fukai, S., Ludlow, M.M. and Hammer, G.L. 1993. Water Extraction by Grain Sorghum in a Sub-Humid Environment. I. Analysis of the water extraction pattern. Field Crops Research, 33: 81-97. Elsevier Science Publishers BV, Amsterdam. Van Noordwijk, M. and Van de Geijn, S.C. 1996. Root, shoot and soil parameters required for process-oriented models of crop growth limited by water or nutrients. Plant and Soil 183: 1-25. Cellular automata Arvo J. and Kirk D. 1988. Modelling plants with environment-sensitive automata. In: Proceedings of Ausgraph’88. pp. 27-33. Conway, J. 1970. Mathematical games. Scientific American. October, 120-127 Greene, N. 1991. Detailing tree skeletons with voxel automata. In: Siggraph Course Notes for photorealistic volume modelling and rendering techniques, pp 7.1 – 7.15. Greene, N. 1989. Voxel Space Automata: Modeling with stochastic growth processes in voxel space. Computer Graphics, 23 (3), 175 – 184. Gutowitz, H. 1991. Cellular Automata: theory and experiment. MIT press, London, 483pp. Ilachinski, A. 2001. Cellular automata: a discrete universe. World Scientific Publishing Company, Singapore, 808p. Ingerson, T.E. and Buvel, R.L. 1984. Structure of asynchronous cellular automata. Physica, 10D, 59-68; Kaneko, K. (editor). 1993. Theory and application of coupled map lattices. Wiley & Sons. Lindemayer A. and Pruzinkiewicz P. 1989. Developmental models of multicellular organisms: a computer graphics perspective. In: Artificial life, proceedings of the Los Alamos workshop. [Langton C.G. editor]. Addison-Wesley Publ. Pp 221 – 249. Prusinkiewicz P. 1993. Modelling and Visualisation of Biological Structures. Proceedings of graphics interface’93. University of Calgary 15p. Rennard, J.P. 2000. Introduction aux automates cellulaires. www.rennard.org/alife/ Rucker, R. and Walker, J. 1997. CelLab manual. http://www.fourmilab.ch/cellab/manual/ Von Neumann J., 1966. Theory of self-reproducing automata. [Ed. A.W. Burcks]. Univ. of Illinois Press, Champaign, IL.
Root modelling Acock B. and Pachepsky Y.A. 1996. Convective-diffusive model of two-dimensional root growth and proliferation. Plant and soil, 180: 231-240. Brouwer, R. 1983. Functional equilibrium : sense or nonsense? Netherlands Journal of Agricultural Science, 31: 335-348 Chikushi, J. and Hirota, O. 1998. Simulation of root development based on the dielectric breakdown model. Journal of Hydrological Science, 43(4): 549 – 560. Chopart J.L. and Vauclin M. 1990. Water balance estimation model : field test and sensitivity analysis. Soil Sci. Soc. Am. J. 54, 1377 – 1384. Coder, K.J. 1998. Soil constraints on root growth. Warnell School of Forest Resources. The University of Georgia. http://www.forestry.uga.edu/ Ericsson, T. 1990. Dry matter partitioning at steady state nutrition. In: Above and below ground interactions in forest trees and acidified soils, pp 236 – 243. Uppsala Air Pollution Report 32, Commission of the European Communities and Swedish Agricultural University. Jones C.A., Bland W.L., Ritchie J.T. and Williams J.R. 1990. Simulation of root growth. In: Modelling Plant and Soil Systems. [Eds. J.Y. Ritchie and R.J. Hanks]. Pp 91 – 123. American Society of agronomy, Madison, WI. USA Lacointe, A. 2000. Carbon allocation among tree organs: A review of basic processes and representation in functional-structural tree models. Ann. For. Sci. 57, 521 – 533. Lawson, G.J. and Mobbs, D.C. 1998. Carbon allocation in individual tree models: a literature review and description of recent modification made to the HyPAR model. Institute of terrestrial Ecology, Bush Estate, Penicuik, Midlothian EH 26 0QB Le Roux, X., Lacointe, A., Escobar-Gutiérrez, A., and Le Dizès, S. 2001. Carbon-based models of individual tree growth: a critical appraisal. Ann. For. Sci. 58: 469 – 506. Mobbs, D.C., Lawson, G.J., Friend, A.D., Crout, N.M.J., Arah, J.R.M. and Hodnett, M.G. 1999. HyPAR. Model for agroforestry systems. Technical Manual. Model Description for Version 3.0. Page E.R. and Gerwitz, A., 1974. Mathematical models based on diffusion equations to describe root systems of isolated plants, row crops and swards. Plant and Soil, 41, 243 – 254. Pagès, L., Asseng, S., Pellerin, S., and Diggle, A. 2000. Modelling Root system growth and architecture. In: Root Methods. [A.L. Smit et al. Eds.]. Springer-Verlag Berlin Heidelberg, pp. 113 – 146. Schroth, G. 1999. A review of belowground interactions in agroforestry, focussing on mechanisms and management options. Agroforestry Systems 43: 5 – 34. Thaler, P. and Pagès, L. 1998. Modelling the influence of assimilate availability on root growth and architecture. Plant and soil, 201: 307 – 320.
4. Nitrogen competition module
4.1. Objectives and inputs In a silvoarable system, the available amount of mineral nitrogen has to be distributed between trees and crops in their interface zone. The approach suggested in this report follows the method applied for water repartitioning.
4.2. Nitrogen uptake and competition in HyPAR In HyPAR the nitrogen demand for the tree and crop components are each calculated as if in isolation from the other component. If the combined demand exceeds the available soil nitrogen, then the uptake of both plants is modified due to nitrogen competition. HyPAR assumes that no leaching of nitrogen from the plant occurs; and that there are no additional inputs of nitrogen into the tree via atmospheric deposition. The estimated N demand from each tree is the nitrogen uptake required to maintain the tree at optimum C:N ratios, and is calculated by comparing the current C:N ratio in the foliage, wood (except heartwood) and fine roots with a theoretical optimum value. Initial tree nitrogen demand is distributed throughout each (rooted) soil layer, or soil cell, proportional to the root density distribution and summed over all trees to give a total per cell. Within each soil cell, the tree demand is added to the demand by the crop, where present, and the combined demand compared with the available soil mineral nitrogen in that cell. If N is limited, or the combined potential extraction of nitrogen exceeds a maximum rate, then competition must occur. Uptake of nitrogen by each tree occurs every day, and is positively related to fine root mass (taking account of the tree root distribution across many plots), soil mineral N content, and the C:N ratio of the entire plant (excluding the C and N bound in the heartwood). It is modified by competition with the crop. This uptake is accumulated in a store for allocation at the end of each day. Maximum nitrogen uptake by crop roots is calculated for each soil layer within each plot and is dependent upon the concentration of nitrate within the layer, the root density, the water content and the nutrient stress of the crop. The combined demand is first modified depending on the concentration of nitrate within the layer, the root density and the water content. For each soil cell the nitrogen demand, DN, is given by,
Where: [Ni] is the nitrate concentration in the cell i, ρT+C
i is the total root length density, ρmax is the maximum root length density, awi is the available water and awfc is the available water when the soil is at field capacity.
The tree and crop demands are reduced in proportion to the initial demand if necessary. As was the case with the HyPAR water extraction methodology, the process of nitrogen competition begins with the surface cells. The combined demand is compared with the available N, and if there is sufficient supply, the full amount is removed from the soil and added to the tree and crop pools. If the supply is limited then all of the available N is removed from the soil and apportioned to tree and crop in proportion to their relative root densities. As with water extraction, any unfulfilled demand is applied to the next soil layer down. The total tree uptake is re-allocated to individual trees according to their initial demand (Figure 23).
4.3. The nitrogen competition module in HySAFE
Initial decisions
A preliminary concept for a soil nitrogen dynamic module soil was developed, based on earlier work of both van Keulen and de Willigen, and the WaNuLCAS model (van Noordwijk and Lusiana, 2000). The approach was compared that adopted by the STICS crop model. The concept, including all major equations, was discussed with the computer scientists, with a view to including it in the HySAFE model. However, the requirements of both coding and validation, plus the possible instability that might result from modifying the STICS code lead to a decision to continue using the STICS approach to nitrogen dynamics. It would have been difficult to use the STICS model for simulating the water dynamics within the HySAFE model, but to treat the nitrogen dynamics in a different manner. However, there is a problem similar to that discussed in the water repartition module, namely the disaggregation of post-processed voxels back into the STICS minicouche layers. A major decision was to consider only nitrogen for tree and crop nutrient uptake. Agroforestry models often also consider phosphorus uptake as well, as they must be applicable to many tropical situations where P is in short supply. It is possible that HySAFE may consider P limitation in future versions, but for the time being, only N competition is simulated Two forms of mineral N occur in most soils, ammonium and nitrate, which differ in their effective adsorption to the soil and therefore both the leaching rate and the movement to roots (van Noordwijk and Lusiana, 2000). Plant species differ in their relative preference for ammonium relative to nitrate in uptake, with only specialised plants able to survive on a pure ammonium supply. The STICS crop model offers the user an option of considering both forms of soil nitrogen, or of only considering soil nitrate. Following discussions with colleagues at Wageningen University, it was decided that the HySAFE nitrogen competition
Figure 23: Nitrogen competitionin HyPAR occurs between treesand crops where supply fromeach cell (plot X layer) isallocated to individual trees orcrops depending on theircomparative 'optimum N-demand' and root length ratios
modules should only handle soil nitrate (as a first step), in common with many other models (e.g. SWHEAT, van Keulen & Seligman, 1987) It was decided that the HySAFE nitrogen competition module would assume that under conditions of ample transpiration and nitrogen supply the plant demand for nitrogen can be satisfied, and no competition will occur, as is assumed by the HyPAR model. The nitrogen uptake is very closely related to the plant water uptake, since a large part of nitrogen is transported to the plant roots by mass flow resulting from the transpiration stream. The nitrogen competition module will define the plant nitrogen demand as the amount of nitrogen needed to achieve maximum nitrogen concentration in all plant components. Nitrogen demand comprises the nitrogen needed for:
a) existing biomass (maintenance) b) current growth of biomass. (Conijn, 1995).
The module must respect the other modules that provide input data used to determine nitrogen competition. The STICS crop model calculates crop nitrogen demand according to a relationship established from the upper envelop of the soil nitrogen dilution curves (Lemaire and Gastal, 1997). In HyPAR, the tree demand is the nitrogen required to obtain an optimum C/N ratio in the tree organs (foliage, stem, fine roots), as mentioned above. It was decided that the ‘available nitrogen’ should be defined as that part of the total amount present in the root zone which can be taken up by the plant when transport through the soil is not limiting (definition after de Willigen and van Noordwijk, 1987). The supply of nitrogen should then be defined as the available amount of nitrogen that can be transported by both mass flow and diffusion to the plant. The module should assume that trees and crops should only extract nitrogen from soil compartments in which roots are present. Active absorption of nitrogen would not be considered as it is in the STICS crop model. It was decided that the module would only consider nitrogen uptake by two processes: mass flow and diffusion. If adequate levels of soil moisture allow the movement of nitrogen towards roots, then all nitrogen (assumed to be in the form of nitrate) in the rooted soil volume should be considered as ‘instantaneously available’ for uptake and would be extracted. It is possible that this will occur even if the soil compartments concerned have small root length densities (e.g. SWHEAT model; van Keulen & Seligman, 1987; and PAPRAN; Seligman and van Keulen, 1981). This assumption needs to be further investigated. However, if soil moisture contents reach very low values (i.e. the effective diffusion coefficient becomes very small) the diffusion rate may be the major limiting factor for nitrogen uptake.
How the decisions reached have been implemented in the N competition module
Nitrogen supply or uptake is divided into two phases: Nitrogen can be taken up by the tree and crop through two processes – mass flow transport (i.e. via the transpiration stream), or via diffusion slowly through the soil water matrix. The first of these processes (mass flow) has been added to the HySAFE model (‘couched’ within the water repartition module), whereas the second (diffusion-based uptake) will be added when the model is further refined. The nitrogen demands for trees and crops are outputs from the Tree Growth and STICS modules respectively. The available N supply of the defined (rooted) soil volume is computed from the residual values from the previous day. If the N demand exceeds the supply of nitrogen to trees and crops, N is limiting and competition occurs between the two species. Nitrogen is extracted via mass flow (water extracted in water repartition stage) stepwise from voxels that have been ranked on the basis of their soil water potential. Using the values for
mass flow of water per voxel as input, computing the nitrogen extraction for trees and crops simply proceeds by multiplying mass flow per species with the concentration of available N:
iS
n
i
iplantmfwplantmfN NUU ∑
=
=1
,,,,
where: UN,mf,plant, the total uptake rate by mass flow resulting from transpiration flux (kg ha-1 d-1) UW,mf,plant, the water extraction by or mass flow to plant roots (mm d-1) NS, the concentration of mineral nitrogen in the soil (kg mm-1) plant refers to trees and crops and i indicates the voxel number. The process of nitrogen extraction continues in step with, but ultimately limited by, the process of extracting enough water to satisfy actual (rather than potential) tree + crop water demand. Therefore, satisfaction of nitrogen demand via mass flow is contingent on the water demands being satisfied. Nitrogen concentration of each voxel is subsequently reduced, ready for ranking with regard to diffusion uptake (if required), or the next day’s computation step. The process is shown diagrammatically in Figure 24.
The total nitrogen uptake, (currently by mass flow alone, but eventually including diffusion), as calculated for trees and crops is transferred as input for the tree and crop module. Extraction ceases either
• when both of the external modules determining crop and tree nitrogen demands decide that demand is zero. If either of the demands is still greater than zero, then extraction continues until that single demand is satisfied.
• when all of the nitrogen that it is possible to extract from all rooted voxels via diffusion has been exploited, without the tree and/or crop nitrogen demands being satisfied. In this case, one (or both) of these demands will remain unsatisfied.
Further improvements to the nitrogen competition module
Then diffusion of nitrogen towards roots may supply an additional amount. However, it is important to note that diffusion will only occur when the soil moisture potential is less negative than the wilting point, or else the discontinuities in the water layer prevents effective diffusion towards the root surface. The manner in which the HySAFE model will be further refined, to incorporate additional nitrogen extraction via diffusion, is shown in Figure 25.
Water extracted
Stage 2
Nitrogen (x2/N) extracted
Water extracted Stage 1
Nitrogen (x1/N) extractedVoxel has certain N concentration (g.cm-3)
Water extracted
Stage 2
Nitrogen (x2/N) extracted
Water extracted
Stage 2
Nitrogen (x2/N) extracted
Water extracted Stage 1
Nitrogen (x1/N) extractedVoxel has certain N concentration (g.cm-3)
Water extracted Stage 1
Nitrogen (x1/N) extractedVoxel has certain N concentration (g.cm-3) Figure 24: Stepwise extraction of nitrogen by
mass flow from rooted voxels that occurssimultaneously with the extraction of voxel water.
All voxels containing roots are ranked, but this time with regard to decreasing nitrogen concentration per voxel (i.e. the gradient driving the diffusion process). This is an elegant way of distributing the total N diffusion in proportion to the amount of available nitrogen within the root profile. Nitrogen is extracted by diffusion, taking into account root length, voxel size, soil moisture content, and the maximum nitrogen uptake per root length. The amount of nitrogen removed is then divided to satisfy both Tree and Crop N demand according to the relative root length densities of each component. Nitrogen extraction by diffusion UN,diff per voxel i is described by: NN
ir
iplantdiffN RUpU max,, =
where: pr, the root length per voxel (m), UNmax the maximum uptake rate of nitrogen
per unit root length (kg m-1 d-1) a required plant species-specific parameter.
RN nitrogen uptake reduction factor for dry soils.
It is proposed that nitrogen should be extracted via diffusion in a similar way as for the water repartition module. Nitrogen is extracted from the 1st voxel (or ‘cluster of voxels), to a point where the nitrogen concentration in that voxel (and therefore the ‘driving gradient’) is equivalent to the next ranked voxel. From that point, nitrogen is extracted from the 1st and 2nd voxels until they reach a point where their nitrogen concentrations are equal to the next ranked voxel, and so on. As with future refinements to the water repartition module, the proposed approach for nitrogen extraction via diffusion may have to be examined in order to ensure that any remaining nitrogen demand is not simply satisfied via diffusion only from the first voxel considered, but rather than all possible voxels are able to contribute some nitrogen. Discussions are ongoing on this topic. The combination of the two processes, mass flow and diffusion, and their integration with the other HySAFE modules, is illustrated in the Figure 26 (below). ‘Luxury uptake’ of nitrogen. As nitrogen uptake via mass flow is dependent on the amount of water extracted, it is possible that when both soil nitrogen concentrations and transpiration rates are high, more nitrogen is extracted than is required (from the tree and crop demands). There appear to be two ways to deal with this, either by allowing leaching of nitrogen back into the soil, or by storing excess nitrogen temporarily in the tree/crop component, linking this to the re-computation of the C:N ratios, so that the forecasted nitrogen demand for the next day is reduced. If we were to allow nitrogen leaching back from the plant to the soil, the problem is to know into which voxels to allow leaching to take place. It is proposed to adopt the second approach of storing excess nitrogen, which would most likely lead to an oscillating system where nitrogen demand fluctuated day to day as excess nitrogen accumulated on one day was balanced by sub-optimal nitrogen the following day etc.
Stage 2
Stage 1Nitrogen (d1/N) extracted
Nitrogen (d2/N) extracted
Nitrogen (d3/N) extracted
(d4/N) extractedStage 4
Stage 3
Stage 2
Stage 1Nitrogen (d1/N) extracted
Nitrogen (d2/N) extracted
Nitrogen (d3/N) extracted
(d4/N) extractedStage 4
Stage 3
Figure 25: the stepwise extraction of soilnitrogen by diffusion from a successivelygreater number of rooted soil voxels.
Plant growth stage and diffusion-extraction. Nitrogen extraction by diffusion is dependent on plant growth stage. For crops it is assumed that diffusion ceases after flowering (Seligman and van Keulen 1981). At present we investigating ways in which we might consider this aspect for N extraction by trees. Species-specific parameters. The literature will be searched for reported examples of maximum nitrogen uptake per unit root length in order to parameterise the diffusion component for the species considered in the HySAFE model. Finally, it should be noted that ‘hand-in-hand’ with the future refinement of the water repartition module, the nitrogen competition module will need to be refined to reflect the fact that each CAPSIS crop cell will have a different crop nitrogen demand resulting from the separate STICS model runs on each cell. The proposed cell-clustering approach adopted in HySAFE will also need to be checked in order to examine the effects that it will have on the performance and results obtained from the nitrogen competition module.
Application + coding
Although only partially implemented, the nitrogen competition module is as yet incomplete. The first stage of nitrogen uptake (via mass flow) was incorporated into the water-repartitioning module (a version from February 2003). However, this version does not include the most recent corrections in terms of limiting water extraction per voxel in correspondence with the following voxel conditions: water potential, fine root length density, and species root water extraction strength. This needs to be corrected. Nitrogen extraction by diffusion is not yet implemented, but is expected to be completed in the near future. Input for the calculating nitrogen extraction via mass flow: The input for nitrogen repartition within the water repartition module is per voxel: e.g. mineral nitrogen concentration in kg m3. Input data are obtained:
1. For the first simulation run: either input values as initialised in CAPSIS, which is the ‘initial nitrogen concentration in kg m3’ or as provided within the soil data file.
2. For simulation on day 2 onwards, the input values from the STICS horizons (kg N ha -1) is used. Values are transferred to the soil data file, where they are converted into kg m3 and read into the water repartitioning module.
An example of the conversion process from the STICS mineral nitrogen contents per horizon (kg N ha -1) to the soil data file (i.e. to the water repartitioning module) mineral nitrogen concentration per voxel (kg N m-3) is given as follows:
i) conversion from the amount of nitrogen (kg in 1 ha: STICS) to amount of nitrogen (kg in 1 m2 : HySAFE) – multiply by (1/10000)
ii) conversion from soil layer to soil voxels: e.g. if there are 10 layers of 0.1m in one voxel of 1 m – multiply by 10
Therefore, the overall conversion would be achieved by multiplying the input data provided from STICS by 1/1000
45
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Table 2: Some inputs related to water and nitrogen dynamics
Description Units In CAPSIS Infiltration mm d-1 In STICS Quantity of nitrates (NO3) at initialisation kg N ha -1. Quantity of ammonium (NH4) at initialisation kg N ha -1. Quantity of nitrogen in plant at initialisation kg N ha -1. Mineral nitrogen concentration in irrigated water kg N mm -1. Mineral nitrogen concentration in rain water kg N mm -1. Quantities of mineral nitrogen input kg N ha -1. Quantities of nitrogen fertiliser kg N ha –1 Minimum soil concentration in NH4 kg N ha –1 mm -1. Minimum quantity of nitrogen in the plant to calculate kg N ha –1 OUTPUT STICS Quantity of nitrates (NO3) in each horizon kg N ha -1. Quantity of ammonium (NH4) in each horizon kg N ha -1. Quantity of mineral nitrogen in depth of measurement kg N ha -1. Quantity of nitrogen in plant (fruits) kg N ha -1. Instantaneous nitrogen demand of plant kg N ha -1. Cumulative nitrogen demand of plant kg N ha -1. Nitrate concentration in drainage water mg l-1 Nitrogen absorption flux with limiting transfer soil ->root kg N ha –1 d-1
References
Seligman, N.G. and van Keulen, H. 1981. PAPRAN: A simulation model of annual pasture
production limited by rainfall and nitrogen. pp. 192-221. In: Simulation of nitrogen Behavior of Soil-Plant Systems. M. J. Frissel and J. A. van Veen (eds.) PUDOC, Wageningen, Netherlands.
van Keulen, H. 1975. Simulation of water use and herbage growth in arid regions. Simulation
Monographs. Pudoc, Wageningen, The Netherlands. van Keulen, H and Seligman, N.G. 1987. Simulation of water use, nitrogen nutrition and
growth of a spring wheat crop. Simulation Monographs, Pudoc, Wageningen, 310 pp. van Noordwijk, M. and Lusiana, B. 2000. WaNuLCAS version 2.0, Background on a model of
water nutrient and light capture in agroforestry systems. International Centre for Research in Agroforestry (ICRAF), Bogor, Indonesia.
5. The belowground modules in the context of the HySAFE model
The following diagram has been adapted from the second Annual Report of the SAFE project and illustrates where the WP5 modules sit within the daily loop of in HySAFE processes, decided at the modelling meeting held in Plasencia, Spain earlier in 2003. It shows the water repartition and nitrogen modules as separate, although at present, the nitrogen extraction (mass flow only) is a function embedded within the water repartition module. The root growth module (RVA) is not shown as separate, as it may be considered part of the Tree Growth Module (when completed), which combines both above- and belowground elements of tree growth.
The modules that directly interact with the belowground modules include the STICS crop growth model, the tree growth module (root-shoot interactions), and the carbon allocation module. Likely additional linkages between the modules include one between tree phenology and the RVA (and nitrogen competition).
Figure 27: The daily loop of HySAFE model processes, indicating when the waterrepartition, nitrogen and tree root growth modules are processed in relation toother model processes.
6. Conclusions and further work
In the next phase of the project, important work will continue on the linkages between these modules as the modelling process continues and the model is improved. This includes attempts to improve various calculation algorithms mentioned in this report with a view to further reducing the computing time of the overall HySAFE model, and the SAFE project computing scientists plan on implementing a batch mode approach to allow interactive runs of the model with modified parameters. It is intended to set up specific field protocols and/or experiments to test the some of the assumptions implicit within the belowground modules, as mentioned throughout this report. These protocols should aim at testing two important aspects of HySAFE:
• The balance between the aboveground and belowground portions of the tree species considered by the SAFE project (Walnut, Cherry, Poplar and Oak).
• Adaptation of the tree root system to the competition from the crop.
7. Acknowledgements
Input and contributions to the workpackage are acknowledged from the following: Rachmat Mulia (INRA-SYSTEM) Isabelle Lecomte, Angélique Adivèze, Jean-Philippe Crépeau, Lydie Dufour, Jonathan Mineau and many other staff from INRA-SYSTEM Nadine Brisson and Dominique Ripoche (INRA-Avignon) Grégoire Vincent (IRD, Montpellier) Deena Mobbs (CEH Edinburgh) Gerry Lawson (NERC) John Roberts, Martin Hodnett and Eleanor Blyth (CEH Wallingford)
