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A soil-plant-atmosphere continuum (SPAC) model for simulating tree transpiration with a soil multi-compartment solution

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An Erratum to this article was published on 06 June 2017



A soil-plant-atmosphere continuum (SPAC) model for simulating tree transpiration (Ep) with variable water stress and water distribution in the soil is presented. The model couples a sun/shade approach for the canopy with a discrete representation of the soil in different layers and compartments.


To test its performance, the outputs from the simulations are compared to those from an experiment using trees of olive ‘Picual’ and almond ‘Marinada’ with the root system split into two. Trees are subjected to different irrigation phases in which one side of the root system is dried out while the other is kept wet.


The model is able to accurately predict Ep (R2 and the efficiency factor (EF) around 0.9) in the two species studied. The use of a function that modulates the uptake capacity of a root according to the soil water content was necessary to track the fluxes observed from each split part. It was also appropriate to account for root clumping to match the measured and modelled leaf water potential.


Coupling the sun/shade approach with the soil multi-compartment solution provides a useful tool to explore tree Ep for different degrees of water availability and distribution.

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This work was supported by project AGL-2010-20766 of the Spanish Ministry of Economy and Competitiveness (former Ministry of Science and Innovation) and by the European Community’s Seven Framework Programme-FP7 (KBBE.2013.1.4-09) under Grant Agreement No. 613817 (MODEXTREME, The authors wish to thank both the “FPI” programme of the aforementioned ministry and the JAE programme of the Spanish Research Council (CSIC) for providing the Ph.D. scholarships granted to the first and the second author, respectively. We thank Manolo Gonzalez, Jose Luis Vazquez, Ignacio Calatrava and Rafael del Río for the excellent technical assistance provided. The authors also thank the constructive suggestions from the anonymous reviewers which enabled us to improve the final version of the manuscript.

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Correspondence to Omar García-Tejera.

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Appendix 1. Finding equilibrium point for Ci

Before describing the iterative procedure for obtaining g co2 and Ψ l , we need to derive the equation of maximum stomatal conductance for CO2 at no limiting leaf water potential (g co2max ). To compute gross photosynthesis one can use the general diffusion function which multiplies g co2 to the CO2 concentration gradient between the substomatal cavities (C i ) and the atmosphere (C a ) plus R d :

$$ {A}^{\prime }={g}_{co2}\left({C}_a-{C}_i\right)+{R}_d $$

A’ can also be obtained from Farquhar’s approach for biochemical photosynthesis (Farquhar et al. 1980). In its general form, Farquhar’s equation for gross assimilation is:

$$ {A}^{\prime }=\frac{B\left({C}_i-\Gamma \right)}{E{C}_i+D} $$

The coefficients B, E, D are different when the rate of carboxylation is limited by the saturation of the ribulose biphosphate carboxylase/oxigenase or by the regeneration of the ribulose biphosphate according to the rate of electron transport.

Substituting Eq. 17 in Eq. 16 and rearranging for g co2 :

$$ {g}_{co2 max}=\frac{B\left({C}_i-\Gamma \right)-{R}_d\left(E{C}_i+D\right)}{\left(E{C}_i+D\right)\left({C}_i-{C}_a\right)} $$

The iteration starts with an initial value for Ci set as 0.7C a , the initial value is introduced in Eq. 18 to compute g co2max , then, A’ from Eq. 17 is computed using g co2max . In the next step Ψ l is calculated from Eq. 12. Then, the Ψ l obtained is used to compute actual g co2 from Eq. 9. Finally a new C i is calculated as:

$$ {C}_{inew}={C}_a-\frac{A^{\prime }-{R}_d}{g_{co2}} $$

If the convergence criterion is not satisfied, C inew becomes C i and the loop starts again. The process is repeated until the difference between Ci and C inew is less than 1 micromol mol−1. Figure 9 represents a schematic diagram of the iterative process.

For each leaf class, two iterative processes are performed using the coefficients B, E and D for the limitation by the saturation of the ribulose biphosphate carboxylase/oxigenase on the one hand or by the regeneration of the ribulose biphosphate on the other. Final value for g co2 and Ψ l would be chosen from the alternative giving lower A’.

Fig. 9
figure 9

Iteration procedure followed to find g co2 and Ψ l

Appendix 2

Tab. 5

Table 5 List of symbols, units and sources for parameters used in the model

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García-Tejera, O., López-Bernal, Á., Testi, L. et al. A soil-plant-atmosphere continuum (SPAC) model for simulating tree transpiration with a soil multi-compartment solution. Plant Soil 412, 215–233 (2017).

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