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Ecological Modelling 221 (2010) 1779–1797

Contents lists available at ScienceDirect

Ecological Modelling

journa l homepage: www.e lsev ier .com/ locate /eco lmodel

Linking intra-seasonal variations in climate and tree-ring �13C: A functionalmodelling approach

Thomas Eglin ∗, Christophe Francois, Alice Michelot, Nicolas Delpierre, Claire DamesinUniversité Paris XI, Laboratoire d’Ecologie, Systématique et Evolution, UPRESA n◦ 8079, Bâtiment 362, F 91405 Orsay Cedex, France

a r t i c l e i n f o

Article history:Received 26 May 2009Received in revised form 8 April 2010Accepted 17 April 2010Available online 22 May 2010

Keywords:Stable carbon isotope discriminationTree ringProcess-based modelIntra-seasonal variationQuercus petraea

a b s t r a c t

Stable carbon isotopic composition (�13C) in tree rings is a widely recognized tool for climate reconstruc-tion, and several works suggest that seasonal information can be extracted from intra-ring �13C variations.In this study, we explored the link between climate and intra-seasonal oak ring �13C using a process-based modelling approach. The ISOCASTANEA model was developed to compute the seasonal dynamicsof tree-ring �13C for deciduous species from half-hourly climatic data by accounting for photosyntheticdiscrimination and carbon translocation and allocation at the tree scale and in tree rings.

The model was applied from March 2005 to December 2007 in a 150-year-old deciduous oak forest.Canopy photosynthesis and stomatal conductance were calibrated using H2O and CO2 fluxes measuredby the eddy flux technique, and simulated �13C values were compared to seasonal patterns of totalorganic matter �13C measured in tree rings for 2006 and 2007 at the same site. With the inclusionof carbon translocation and with regard to 13C enrichment of starch compared to soluble sugars, themodel can reasonably simulate the intra-seasonal and inter-annual variability of tree-ring �13C using thesame parameter values for 2006 and 2007. The amplitude of the seasonal carbon isotope pattern in treerings was influenced by both photosynthetic and post-photosynthetic processes (starch enrichment andreserve use). The �13C variations in the early part of the ring, i.e., mainly in the earlywood, were relatedmostly to carbohydrate metabolism, although diluted information about environmental conditions dur-ing the previous year could also be found. The last part of the ring, consisting mainly of latewood, wasfound to be a good recorder of current-year environmental conditions, in particular relative humidity,at a fine temporal resolution when the growth rate was high. The sensitivity of the �13C in the earlypart of the ring to carbohydrate metabolism suggests that intra-ring �13C could be used to explore therelationship between tree decline or mortality and carbohydrate deficiency.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Stable carbon isotopic composition (�13C) in tree rings is awidely recognized tool for climate reconstruction (Raffalli-Delerceet al., 2004). Carbon isotope composition is known to capture infor-mation both on environmental conditions (McCarroll and Loader,2004) and on anthropogenic changes in the �13C content of atmo-spheric carbon dioxide (February and Stock, 1999).

The link between the climate and tree-ring �13C is usually ana-lyzed using a model of photosynthetic discrimination at the leafscale (Farquhar et al., 1982). This model mechanistically relatesthe �13C of photoassimilates to the ratio of intercellular and atmo-spheric CO2 concentrations (Ci/Ca) and, therefore, to the ratiobetween CO2 assimilation and stomatal conductance or intrinsicwater use efficiency (WUEi). In this way, the stable carbon isotopic

∗ Corresponding author. Tel.: +33 1 69 15 56 75; fax: +33 1 69 15 72 38.E-mail addresses: [email protected], [email protected] (T. Eglin).

composition of tree rings provides a time-integrated estimate ofWUEi. Numerous studies have found correlations between tree-ring �13C and environmental factors known to influence WUEi,such as soil water availability, rainfall, vapor pressure deficit,relative humidity, temperature, solar radiation, and nutrient avail-ability (Farquhar et al., 1989; Leavitt, 1992; Dupouey et al., 1993;Saurer et al., 1997; Porté and Loustau, 2001; Barbour et al.,2002).

Growth rings are also known to exhibit radial (e.g., seasonal)variation in �13C (Leavitt and Long, 1986; Leavitt, 2002; Barbouret al., 2002; Helle and Schleser, 2004). Although some trends sug-gest that these variations are related to variations in temperature,soil water availability, radiation, atmospheric CO2 concentration, oratmospheric �13C over the growing season (Leavitt, 2002; Barbouret al., 2002), other studies suggest that the seasonal variation intree-ring �13C is not fully explained by this model of photosyn-thetic discrimination (Hemming et al., 2001; Ogée et al., 2009),particularly in deciduous species (Helle and Schleser, 2004). Infact, the relationship between climate and �13C in leaf assimi-

0304-3800/$ – see front matter © 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.ecolmodel.2010.04.007


1780 T. Eglin et al. / Ecological Modelling 221 (2010) 1779–1797

lates could be altered before organic-matter deposition in treerings by numerous physiological and biochemical processes, suchas post-photosynthetic fractionation (Badeck et al., 2005) duringthe transport of assimilates (Damesin and Lelarge, 2003), duringthe biosynthesis of wood compounds (Panek and Waring, 1997) orduring respiration (Duranceau et al., 1999; Tcherkez et al., 2003;Damesin and Lelarge, 2003), or by mixing with carbon reserves(Damesin and Lelarge, 2003; Kagawa et al., 2006; Keel et al., 2007;Ogée et al., 2009).

One of the main difficulties in correlating �13C in tree ringswith environmental parameters arises from the translocation andstorage of carbohydrates (Gessler et al., 2007). The use of carbonreserves for structural growth and the mixing of all sequentially-assimilated carbon within the mobile carbon pool dampens the linkbetween climate and structural tissue �13C (Keel et al., 2007) andinduces a temporal lag between photosynthesis uptake and theuse of photosynthates for ring formation in the stem (Kagawa etal., 2006). In a modelling study, Hemming et al. (2001) stressedthat the carbon source for growth may be extremely importantin determining the �13C value of the resulting wood; in particu-lar, it may affect the relative contributions of photosynthates andstored starch that is generally 13C-enriched compared to solublesugars (Brugnoli et al., 1988). Helle and Schleser (2004) describedthe �13C pattern in rings of broadleaf species as a recurrent triphasepattern that is largely independent of seasonal variations in pho-tosynthetic fractionation and may be related to the relative useof 13C-enriched stored carbon reserves (mainly starch) and of13C-depleted new photosynthates. According to this argument,earlywood is assumed to contain photoassimilates from previousyears, while latewood is composed mainly of photoassimilatesfrom the current growing season (Kagawa et al., 2006). This isparticularly the case in oaks where trunks are known to beginradial growth before leaves are fully expanded (Breda and Granier,1996).

The aim of this paper is to develop a model that links intra-seasonal �13C variations in deciduous oak tree rings and climaticconditions by doing the following:

(1) Integrating a leaf gas exchange model of photosynthetic dis-crimination (Farquhar et al., 1989) and a quasi-topologicalmodel accounting for carbon translocation and distribution atthe tree scale and in tree rings within CASTANEA, which isa physiological multilayer forest ecosystem model simulatingenergy, water, and carbon balances at the stand scale from half-hourly climatic data (Dufrêne et al., 2005);

(2) Calibrating the model to obtain a best fit for carbon and waterfluxes in the Barbeau forest from March 2005 to December2007. The Barbeau forest site is equipped with a flux towermeasuring half-hourly climatic data and exchanges of carbonand water between the atmosphere and the ecosystem;

(3) Comparing simulated ring �13C outputs to high temporal res-olution measurements of intra-seasonal ring �13C sampled inoaks at the study site and assessing the necessity of accountingfor the main processes leading to post-photosynthetic dis-crimination, i.e., starch synthesis (Gleixner et al., 1993) andrespiration (Tcherkez et al., 2003; Kodama et al., 2008).

Tree rings corresponding to the years 2006 and 2007 were sam-pled. The model was run from March 2005 to provide consistentinitial �13C values for starch and soluble sugars at the beginning of2006.

1.1. Model description

The main difficulty in the development of the model wasthe modelling of C translocation and partitioning within the tree(Fig. 1A). For example, the phloem pathway is explicitly describedand organ compartmentalization for structural and mobile carbon

Fig. 1. Description of the C translocation and allocation model structure: (A) the C translocation module and (B) canopy layers in which photosynthesis occurs. The circlednumbers associate physiological processes to the C fluxes in the C translocation model.


T. Eglin et al. / Ecological Modelling 221 (2010) 1779–1797 1781

Fig. 2. Description of the modelled structural matter deposition in tree rings.

is taken into account (Fig. 1A). C allocation in our model differs fromC allocation in CASTANEA because we explicitly introduced thetransport of C within the phloem, and we distinguish between dif-ferent C storage compounds (starch and soluble sugars). However,we kept the same amount of total C allocated to growth, respirationand storage.

In this section, we will first describe the modelling of C translo-cation and partitioning within the tree. Then, we will presentthe modelling of phenological events, photosynthetic processes(Fig. 1B), tree ring maturation (Fig. 2) and post-photosynthetic frac-tionations. All model parameters are listed in Table 1 .

1.2. Carbon exchange between the canopy and other tree organs

In CASTANEA, the canopy is divided into a fixed number (ncanopy)of compartments, which enables distinctions to be made betweenlower and higher canopy layers in terms of carbon balance andexport (Fig. 1A). Each canopy compartment is fed each half-hourby a defined number of canopy layers (Fig. 1B). Photoassimilatesare mixed with older soluble sugars in leaves and allocated firstto leaf respiration and growth. Starch synthesis/breakdown andleaf export/import into the phloem pathway is calculated from theremaining soluble-sugar content. When carbon assimilation andlocal reserves in a canopy compartment are not sufficient to meetleaf demand for growth and respiration, carbon is taken directlyfrom the phloem pathway. Otherwise, carbon is loaded actively intothe first phloem compartment according to a Michaelis–Mentenfunction (Moing et al., 1994):

Li = Lmax(Si − Smin)

Kexportm + (Si − Smin)

, (1)

where Li and Si are the export rate and the soluble sugar concen-tration, respectively, of canopy compartment i, Lmax and Kexport

m

are the apparent Michaelis parameters, and Smin is the minimumsoluble-sugar concentration.

1.3. Sieve flow in the phloem pathway (Fig. 1A)

For trees, it is now widely accepted that carbon translocationfrom sources to sinks is caused by a mass flow resulting fromthe osmotic pressure gradient generated by solute accumulationwithin the sieve tubes at the source level and by its consumptionat the sink level (Münch, 1930; Lalonde et al., 2003). The phloempathway is explicitly described and envisioned as a simple linearstructure composed of 1-m-long conducting elements. The soluble

sugars are assumed to be only in the form of sucrose and moveby mass flow. The osmotic potential ˘ j (MPa) is described in eachelement by the Van’t Hoff law as:

˘j = −RTCp (2)

where R is the universal gas constant (MPa m3 mol−1 K−1), T thetemperature (◦C), and Cp the sucrose concentration in the phloemelement j (mol m−3).

Some simplifications are made to facilitate the numerical res-olution of the sieve translocation: (1) the waterflow between thexylem and the phloem is assumed to be sufficiently rapid (that is,that water potential in the sieve tube lumen is approximately thatof the apoplast, and that the water potential of the apoplast is thesame as in the adjacent xylem vessels), and (2) matric and gravi-tational potentials are neglected. These simplifications should notgreatly impact the results. First, the first assumption implies thatthe impact of transpiration on sugar fluxes is not taken into account.As shown by Höllta et al. (2006) in a modelling approach, phloemflow decreases when transpiration increases because transpirationreduces the axial water-pressure gradient in the phloem by induc-ing an opposite-direction pressure gradient in the xylem. However,the effect of transpiration on the daily amount of sugar transportshould only be significant at high rates of transpiration (Höllta etal., 2006). Secondly, Thompson and Holbrook (2003) consider thatthe gravitational component of transport can be ignored becausethe standing pressure gradient in the phloem will cancel the stand-ing pressure gradient in the xylem. Indeed, the two sap densitiesare approximately the same, and water transport between xylemand phloem can be assumed to be sufficiently rapid. Therefore, thewater potential is assumed to be constant among the conductingelements and summarized as:

�j = Pj + ˘j = constant value (3)

where Pj (MPa) and � j (MPa) are the turgor pressure and the waterpotential, respectively, in phloem element j.

Sap flow ˚wj(m3 h−1 m−1) between two successive elements is

expressed by Darcy’s law:

˚wj= Kaxial(Pj − Pj+1) = −Kaxial(˘j − ˘j+1), (4)

where Kaxial (m3 MPa−1 h−1 m−1) is the axial conductance to thesieve solution for a conducting element.

Any flux of water is directly associated with a corresponding fluxof sucrose, ˚p

Sj(mol h−1 m−1). These two quantities can be related

as:

˚pSj

= Cpj

˚wj, (5)

The calculation of the dynamics of solute concentrations inphloem elements is based on exact time-dependent equationsusing a backward Euler scheme (Press et al., 1992). We consider azero-flux boundary condition at both ends of the phloem pathway.The axial conductance and the volume of each conducting elementwere parameterized from a range of realistic values, as describedin Appendix A and Table 2. Values were checked for compatibilitywith the ranges reported in the literature: a sieve velocity up to1.25 m h−1 for trees (Sheehy et al., 1995; Thompson and Holbrook,2003) and a sap turgor pressure gradient in the phloem pathway intrees between 0.02 and 0.05 MPa m−1 (Hammel, 1968; Sheehy etal., 1995).

1.4. Sucrose exchange between the phloem and the sink elements(Fig. 1A)

Exchanges of sugars between the phloem and surrounding tis-sue have been shown to occur through both symplasmic andapoplasmic routes (Lalonde et al., 2003). Symplasmic transport



Table 1List of input parameters and ranges used for sensitivity analysis.

Symbol Description Units Value (range forsensitivity analysis)

Reference

Canopy photosynthesis and photosynthetic discriminationCa Atmospheric carbon dioxide

concentration�mol CO2 mol air−1 Variable Measured at the site

�13Cair Carbon isotopic composition ofatmospheric CO2

‰ Variable (±10%) Extrapolated from NOAAmeasurements at Mace Head (Ireland)

˛ Quantum yield mol electron (mol quanta)−1 0.24 (±10%) Le Maire (2005)� Curvature of the quantum response of

the electron transport rateDimensionless 0.7 (±10%) Le Maire (2005)

gbCO2Leaf boundary layer conductance mol CO2 m−2 s−1 1 (±10%) Dufrêne et al. (2005)

g0 Leaf cuticular conductance mol H2O m−2 s−1 0.001 (±10%) Dufrêne et al. (2005)g1max Slope of the Ball relationship

(maximum value)Dimensionless 11.8 (±10%) Medlyn et al. (2001)

g1min Slope of the Ball relationship(minimum value)

Dimensionless 0.001 (±10%) Dufrêne et al. (2005)

Agreg Clumping factor Dimensionless 0.8 (±10%) Le Maire (2005)agi Slope of the relationship between

Vcmax and gi

Dimensionless 4.4 (±10%) Piel (2002)

bgi Intercept of the relationship betweenVcmax and gi

mol CO2 m−2 s−1 −18.6 (±10%) Piel (2002)

ab Carbon isotope fractionation duringdiffusion across the boundary layer

‰ 2.9 (±10%) Farquhar et al. (1989)

a Carbon isotope fractionation duringdiffusion across the stomatal cavity

‰ 4.4 (±10%) Farquhar et al. (1982)

al Carbon isotope fractionation duringdiffusion in water

‰ 0.7 (±10%) Oleary (1984)

es Carbon isotope fractionation as CO2

enters solution‰ 1.1 (±10%) Oleary (1984)

b Carbon isotope fractionation byRubisco CO2 fixation, assuming 5% ofPEP fixation

‰ 28.2 (26.2–30.3) Farquhar et al. (1989), Suits et al.(2005), McNevin et al. (2007)

e Carbon isotope fractionation duringdecarboxylation

‰ 6 (±10%) Ghashghaie et al. (2003)

f Carbon isotope fractionation duringphotorespiration

‰ 8 (±10%) Gillon and Griffiths (1997)

� * Chloroplastic CO2 compensation point Mbar 37.4 Bernacchi et al. (2002)LAImax Maximum leaf area index m2 m−2 5.1 (±10%) Soudani and Dufrêne (pers. comm.)LMAmax Maximum leaf mass by area g m−2 Variable (±10%) Maunoury-Danger Florence (pers.

comm.)Nleaf Maximum leaf nitrogen content mg N gD M−1 24 (±10%) Maunoury (pers. comm.)BBDAY Starting date of budburst Julian day Variable (±5 days) Fixed

Tree architectureHtree Tree height M 27 (25–30) Measurednd Branching ratio Dimensionless 3 (±10%) Bary-Lenger and Nebout (1993)atp Parameter relating the evolution of the

branch radius with the branching levelDimensionless 1 (±10%) Fixed

btp Parameter relating the evolution ofliving wood proportion with thebranching level

Dimensionless 1 (±10%) Fixed

rpetiol Leaf petiole radius M 0.005 (±10%) Measuredlpetiol Leaf petiole length M 0.02 (±10%) Measuredrfineroots Fine roots radius M 0.005 (±10%) Fixedlfineroots Fine roots length M 0.02 (±10%) FixedBaerialwood Trunk and branches biomass g m−2 of soil 23,000 (±10%) Estimated from allometric relationshipBcoarse root Coarse roots biomass g m−2 of soil 2500 (±10%) Estimated from allometric relationship

Carbon allocationGrowthInit Carbon demand for aerial growth

before sink to source transition inleaves

gC d−1 1 (±10%) Fixed

AGaerial wood Aerial wood allocation coefficientbefore and after Julian day 215

Dimensionless 0.6–0.14 (±10%) Calibrated for nearly constant reservesand fine roots biomass at a yearly scale

AGcoarseroots Coarse roots allocation coefficientbefore and after Julian day 215

Dimensionless 0.12–0.03 (±10%) Calibrated for nearly constant reservesand fine roots biomass at a yearly scale

AGstorage Storage allocation coefficient beforeand after Julian day 215

Dimensionless 0.18–0.3 (±10%) Calibrated for nearly constant reservesand fine root biomass at a yearly scale

AGfineroots Fine roots allocation coefficient beforeand after Julian day 215

Dimensionless 0.1–0.53 (±10%) Calibrated for nearly constant reserveand fine root biomass at a yearly scale

CRleaves Leaf construction cost gC gC−1 1.2 (±10%) Niinemets (1999)CRwood Wood construction cost gC gC−1 1.38 (±10%) Damesin et al. (2002)CRfine roots Fine roots construction cost gC gC−1 1.28 (±10%) Agren and Axelsson (1980)MRN Nitrogen dependency of maintenance

respirationmol CO2 gN

−1 h−1 4e−4 (±10%) Le Maire (2005)



Table 1 (Continued )

Symbol Description Units Value (range forsensitivity analysis)

Reference

Carbon translocation in conducting elementsncanopy Number of canopy compartments Dimensionless 5 FixedLmax Maximum carbon export rate mol C m−2 leaf s−1 5e−6 (±10%) Moing et al. (1994)Km export Apparent Michaelis parameter for

carbon exportmol C m−3 4 (±10%) Moing et al. (1994)

Smin Minimum soluble sugar concentrationin a canopy compartment

mol C gD M−1 0.03 (±10%) Fixed

Kaxial Axial conductance to water for aconducting element

m3 h−1 MPa−1 m−1 3e−4 (1e−4–4e−4) Calculated from Sheehy et al. (1995)and calibrated for reliable values ofsieve flow

Vphloem Conducting element volume mL m−1 6 (3–6) Calculated from Sheehy et al. (1995)and calibrated for reliable values ofsieve flow

ksucrose Sucrose diffusivity in water m2 s−1 4e−10 (±10%) Sheehy et al. (1995)˛st Parameter related to sink activity Dimensionless 1 (±10%) FixedStmax Maximum starch content in organ g gDM−1 0.2 (±10%) Barbaroux (2002)rtrunk Trunk radius at 1.30 m M 0.25 (±10%) Fixed

Carbohydrate compartmentalization in organ elementPSymplasm Symplasm proportion in living wood

biomassmL gD M−1 1 (±10%) Fixed

VMSt Maximum rate for starch synthesis mmol mL−1 h−1 0.1 (±10%) Daudet et al. (2002)KMSt Michaelis–Menten constant of starch

synthesismmol sucrose mL−1 1 (±10%) Daudet et al. (2002)

kstmin Minimal starch lysis activity h−1 0.07 (±10%) Fixedkstmax Maximal starch lysis activity h−1 0.04 (±10%) Fixed˛SS Parameter relating starch hydrolysis

rate with day lengthDimensionless 1 (±10%) Fixed

dSS Parameter relating starch hydrolysisrate with day length

H 10 (±10%) Fixed

Organic matter deposition in tree rings�OM Duration of organic matter deposition Days 30 (0–30) Fixed˛OM Parameter for holocellulose deposition h−1 0.005 (±10%) Fixed

Post-photosynthetic processes affecting carbon isotope composition�SS Carbon isotope discrimination during

starch synthesis‰ 0 (−2 to 0) Fixed

�R Carbon isotope discrimination duringrespiration

‰ 0 (−6 to 0) Ghashghaie et al. (2003)

is likely to conform to diffusion or mass flow kinetics, whereasapoplasmic transport (facilitated membrane transport) can bemodelled in terms of a solute-saturable component described byMichaelis–Menten kinetics combined with a solute non-saturablecomponent obeying first-order kinetics (Lalonde et al., 2003).Sucrose exchanges between the phloem and surrounding organsother than mature leaves (i.e., young developing leaves, branches,trunk, coarse roots, and fine roots) were modelled as a diffusion-likeprocess according to Fick’s formulation:

˚SSj = K lateral

j (CSj − CP

j ), (6)

where ˚SSj

(mol h−1) and K lateralj

(m3 h−1) are the diffusive flowand the lateral conductivity, respectively, between the conduct-ing element and the surrounding organ j, and CS

j(mol m−3) is the

soluble-sugar concentration in the organ. Results are not impactedby the use of a diffusion-like process instead of a mass flow.

The ability of a sink element j to import soluble sugars (sinkstrength) is considered to depend not only on the concentration gra-dient but also on modifications of conductivity with sink activity.The transfer of sugars from the sieve tubes to the adjacent cells canoccur via cytoplasmic connections, the plasmodesmata or trans-membrane transport via an apoplasmic route (Lalonde et al., 2003).Changes in the structure and number of plasmodesmata (reviewedby Roberts and Oparka, 2003) or in the regulation of the mem-brane transporters (Ho, 1988) could alter the apparent conductivitybetween sieve tubes and sink cells. Moreover, enzymes related tosucrose conversion in sink cells (e.g., sucrose synthase, which catal-yses the conversion of sucrose into glucose and fructose) participate

Table 2Calculations of Kaxial and Vphloem according to Sheehy et al. (1995). Data for anatomical parameters are those for tree type A defined in Sheehy et al. (1995). Stocking densityis calculated from mean trunk radius and basal area. Range for sieve viscosity values comes from Bancal and Soltani (2002) and Sheehy et al. (1995).

Symbol Description Units Values

ast Radius of a sieve tube �m 18npore Number of pores between two sieve tubes Dimensionless 50f1 Ratio of sieve pore length to sieve tube length m m−1 0.0056f2 Ratio of sieve pore radius to sieve tube radius m m−1 0.111� Sieve viscosity Pa s 0.002–0.003ne Number of active layers in the phloem Dimensionless 6ST Basal area m2 m−2 25–30rtrunk Mean trunk radius at 1.30 m m 0.2–0.3Sd Stocking density arbres ha−1 88–239Kaxial Axial conductance to water for a conducting element m3 h−1 Mpa−1 m−1 1.8e−0.4–4.8e−04Vphloem Conducting element volume mL m−1 3.5–6.2



in the control of sucrose import, and thereby affect sink strength (LeHir et al., 2005). Thus, the lateral conductivity K lateral

jvaries along

the phloem pathway according to its average total circumferenceCircj (m), i.e., the average surface available for exchange, and to thesink activity parameter �starch

j:

K lateralj = ksucroseCircj�

starchj (7)

where ksucrose is the diffusivity between conducting elements andthe surrounding organs in the trunk, which is chosen to be closeto the sucrose diffusivity in water. The average total circumferenceCircj for each element j is expressed using a fractal model (West etal., 1999; see Appendix B).

The dynamics of sink activity are related principally to thedynamics of reserve storage and structural growth. Growth pro-vides room for reserve storage (Lacointe, 2000), and reservesappear to act as an active sink rather than a buffer for excess carbon(Barbaroux and Breda, 2002; Lacointe et al., 2004). Therefore, thesink activity parameter can be related to the starch content STj inthe sink organ j as:

if CPj > CS

j , 0 < �starchj =

(1 − STj

STmax

)˛ST

< 1, (8)

where STmax determines the starch content limit above which theorgan under consideration cannot import carbon. It is advantageousto impose a limitation on sugar import through the relationshipwith starch because it prevents the accumulation of sugar in uppersink organs.

1.5. Growth and respiration

The amount of carbon allocated to respiration and growth is cal-culated following Dufrêne et al. (2005). Maintenance respiration(MR) depends on temperature, living biomass, and the nitrogencontent of various organs, while growth respiration (GR) is asso-ciated with the synthesis of new structural matter and dependson the biochemical composition of the organs under consideration(Dufrêne et al., 2005). Maintenance respiration is given priority andtaken first from the sugar pools. The remaining carbon is partlyallocated to growth (G) and growth respiration (branch, trunk,coarse roots, and fine roots) as a proportion of the daily assimilatesminus leaf C demand and total maintenance respiration. An excep-tion is made at the beginning of the growth season because in thespring, stem growth in oaks begins before leaf expansion (Breda andGranier, 1996). Thus, the growth of aboveground woody biomassis prescribed during this period (parameter Growthinit in Table 1).In our tree model, growth is distributed among the organ elementsto keep the architecture given by the fractal model described inAppendix B.

In each organ element, respiration and growth have direct accessto the local pool of soluble sugars (Fig. 1A). The carbon that is notallocated to these processes is split into soluble sugar and starch asdescribed in the next paragraph. Therefore, the carbon balance ofan organ element can be expressed at each time step as:

dSSj

dt= ˚S

Sj− MRj − Gj − GRj − dSTj

dt, (9)

where SSj (gC) is the soluble-sugar content, Gj (gC h−1) the growthrate, MRj (gC h−1) the maintenance respiration rate, GRj (gC h−1)the growth respiration rate, and STj (gC) the starch content in theorgan under consideration.

In our case, growth is the major carbon sink from growth initi-ation to August (day of year 215), and then storage is favored untilthe end of the vegetation period (Barbaroux, 2002).

1.6. Buildup of reserves in organ and canopy compartments

Starch stored in any organ compartment is modeled as a result ofthe dynamic equilibrium between starch synthesis from availablesoluble sugars and starch hydrolysis rates (Escobar-Gutierrez et al.,1998):

dSTj

dt= STsyn

j− SThyd

j, (10)

where STsynj

(molC h−1) and SThydj

(molC h−1) are the starch synthe-sis and hydrolysis rates, respectively.

The starch synthesis rate obeys Michaelis–Menten kinetics:

STsynj

=VmStC

Sj

KmSt + CSj

, (11)

where VmSt and KmSt are the apparent Michaelis parameters. Thestarch hydrolysis rate is assumed to be proportional to the existingstarch content of each organ:

SThydj

= kStSTj, (12)

where kSt is the starch hydrolysis constant. The starch synthesisand hydrolysis rates are then converted into gC.

Carbohydrate composition in woody plant tissues is known tochange seasonally as a result of the interchange between solublesugars and starch (Ashworth et al., 1993). The concentration ofsugar increases in the autumn, reaches a maximum in the mid-winter, and declines in the spring, whereas starch content followsthe reverse trend. The increase in starch in the spring coincideswith budburst and is followed by a steep decline, presumably asa result of carbon demand for spring growth (Ashworth et al.,1993). Low temperatures promote this starch-to-sugar conversion(Sauter, 1988). For simplicity and model robustness, this seasonalinterchange was determined by varying the starch hydrolysis ratewith day length according to a sigmoid curve:

kSt = kSt max + kSt min

2+

(kSt min − kSt max + kSt min

2

)tanh(˛SS(dlength − dSS)). (13)

The maximum starch hydrolysis rate kSt max occurs when daylength is short, i.e., in winter, and the minimum hydrolysis ratekSt min occurs during the leafy period. The parameters ˛SS and dSSdetermine the curvature and symmetry point, respectively, of thesigmoid.

The main parameters for carbohydrate compartmentalizationwere chosen to simulate starch and soluble-sugar contents in amanner consistent with seasonal values for oaks from the literature(Barbaroux and Breda, 2002; Hoch et al., 2003). Using the param-eterization presented in Table 1, the simulated starch content intrunk-living wood varies seasonally between 2 and 5 g 100 g−1 ofdry matter and the soluble-sugar content between 0.5 and 3.5 g100 g−1 of dry matter (see also Fig. SP1 in the supplementarymaterials). The minimum values are reached after budburst for bothcarbohydrate pools. The starch content is maximal at the end of thegrowing season and the soluble-sugar content in the winter.

1.7. Phenological events

Budburst and leaf senescence starting dates are calibrated onmeasured ecosystem carbon and water fluxes. Other phenologicalevents (full leaf area development, leaf mass per area evolution,leaf senescence) are described as functions of degree days and dayduration (Dufrêne et al., 2005).



1.8. Canopy photosynthesis and photosynthetic discrimination(Fig. 1B)

The canopy is assumed to be homogenous horizontally and ver-tically and to be subdivided into a variable number of layers, whereeach layer contains the same amount of leaf area (0.1 m2 m−2 ofsoil). Photosynthesis is then calculated according to Dufrêne etal. (2005). A variable mesophyll conductance has been shown toinfluence photosynthetic gas exchange at the scales of the ecosys-tem (Cai et al., 2008) and tree-crown (Le Roux et al., 2001) and istherefore included in calculations of carbon isotope photosyntheticdiscrimination. The leaf mesophyll conductance gi was estimatedaccording to the relationship defined by Piel (2002) for Juglans regiaL.:

gi = agiVc max − bgi

, (14)

agiand bgi

are the parameters of the linear regression, and Vc max isthe maximal Rubisco carboxylation rate.

In each canopy layer, the carbon isotope composition of thenewly formed photosynthates can be calculated as:

�13Cphotosynthates = �13Cair − �p

1 + �p. (15)

Then, �p is the carbon isotope discrimination during photosyn-thesis at the leaf scale according to Farquhar et al. (1989):

�p = abCa − Cs

Ca+ a

Cs − Ci

Ca+ (es + al)

Ci − Cc

Ca+ b

Cc

Ca− (eRd/A) + f� ∗

Ca,

(16)

where ab expresses the fractionation during diffusion across theboundary layer; a the fractionation during diffusion across thestomatal cavity; es the fractionation occurring as CO2 enterssolution; al the fractionation due to diffusion in water; b thefractionation by Rubisco CO2 fixation; e and f denote overall dis-criminations during day respiration (Rd) and photorespiration,respectively; A is the net assimilation; and � * is the CO2 com-pensation point in the absence of day respiration (Farquhar et al.,1989). Values of atmospheric CO2 �13C (�13Cair) were estimated asdaily values interpolated from flask measurements at Mace Head inIreland (53◦33′N, 9◦9′W, 25 m elevation, NOAA/CMDL atmospheric[CO2] monitoring site) between 2005 and 2007 using a polynomialequation. Interpolation from flask measurements at a continentalstation (Heggyhatsal, Hungary) was also tested and did not signif-icantly change the results presented in the following sections (seeFig. SP3 in the supplementary materials).

1.9. Carbon isotope composition of the carbon pools

The isotopic composition of each carbon pool is calculatedaccording to a mixing model:

dı13Ci(t)dt

= ı13Ci(t)(˚out(t) − ˚in(t)) + ˚in(t)((ı13CSource(t) − �in)/(1 + in)) − ˚out(t)((ı13Ci(t) − �out)/(1 + �out))Ci(t)

, (17)

where �13Ci is the isotopic composition of pool i; Ci the carboncontent of pool i; �13CSource the isotopic compositions of the carbonsources; ˚in and ˚out the carbon fluxes entering or leaving the pool;and �in and �out the fractionation intensities associated with thesefluxes. For example, a fractionation during respiration (�out = �R)impacts directly the isotopic composition of the soluble sugar pool(�13Ci) in which carbon is taken relative to the amount of respiredC (˚out = GR + MR) and the amount of C in the soluble sugar pool(Ci).

1.10. Starch pool compartmentalization

Lacointe et al. (1993) showed that recently accumulatedreserves were used before older storage pools in walnut trees(Juglians regia L.). It is hypothesized that this is related to the struc-ture of starch grains that are made of successive layers, to thedistance from the vascular system, or to both; the most recentlyformed reserves are more readily available than older ones (lastin, first out). This pattern was modelled by considering starch ineach organ as composed of successive layers (i.e., starch structureis coded as a table including both starch amount and �13C). Whenstarch was synthesized during a time step, a new layer was added.Conversely, when starch was hydrolyzed, the most recently formedlayers were removed in priority order.

1.11. Tree-ring maturation (Fig. 2)

Tree-ring maturation is not instantaneous and occurs in threestages: division of cambial cells, cell enlargement, and cell-wallthickening (Samuels et al., 2006). Completion of these variousphases, in particular cell-wall thickening, requires some time. InAbies balsamea, Deslauriers et al. (2003) found cell enlargementdurations from less than 1 week to 10 days and cell-wall thickeningdurations of 10–20 days. Holocellulose is synthesized during pri-mary cell wall and secondary cell wall production, whereas lignindeposition begins in the later stages of polysaccharide biosyn-thesis (Donaldson, 2001). Recent work suggests high correlationsbetween �13C values in cellulose, lignin, and whole wood (Loaderet al., 2003; Eglin et al., 2008), and this suggests that lignin issynthesized from substrates isotopically similar to those used forholocellulose biosynthesis. The maturation of tree-ring cells isexpressed in the model with a decreasing exponential functionwithout differentiating the time of deposition of wood components,as described in Appendix C. Seasonal variations in the dynamics oforganic-matter deposition is known to occur. For example, in A. bal-samea, the cell-wall thickening phase lasts approximately 10 dayslonger in latewood than in earlywood, and the rate and duration ofthe cell differentiation process could be influenced by climatic con-ditions (Deslauriers et al., 2003). In this study, we chose to ignorethis process because this information was not available for oaks.

1.12. Post-photosynthetic carbon fractionation

All processes modelled here can be associated with constantor seasonally variable values of carbon fractionation. In this study,only constant discriminations during starch synthesis and respi-ration are considered. High �13C values for starch (Brugnoli et al.,1988; Damesin and Lelarge, 2003) and respired CO2 (Duranceau etal., 1999; Maunoury et al., 2007) compared with those for solublesugars are usually found in various plant tissues. If fractionationprocesses occur during starch synthesis and respiration in woodyorgans, they have therefore been considered to favor 13C in the

model (�SS ≤ 0 and �R ≤ 0). Potentially fractionating processesduring phloem transport or loading are not taken into accountbecause the literature does not contain consistent results (Damesinand Lelarge, 2003; Brandes et al., 2006; Gessler et al., 2007).

1.13. Study site, meteorological data, and modelparameterization

The experimental site is located in the Barbeau National Forest(48◦29′N, 02◦47′E, 90 m elevation), 60 km southeast of Paris, France.



The climate is temperate with long-term (1960–1990) annual rain-fall and temperature equal to 690 mm and 10.6 ◦C, respectively. Thesoil is 80–90 cm deep, hydromorphic with gley and millstone grit.The trees are rarely subjected to water stress. The site is man-aged as a mature 100- to 140-year-old oak forest with a denseunderstory of coppiced hornbeam (Carpinus betulus) and is a part ofthe ECOFOR and CARBOEUROPE-IP projects. The average height ofthe dominant trees is 29 m. Initial aboveground and belowgroundtree biomasses have been determined from a C130 (circumferenceat 1.30 m) inventory by empirical allometric models (Vallet et al.,2006, for aboveground biomass; Drexhage et al., 1999, for below-ground biomass).

The model was run and tested using half-hourly meteorolog-ical data and eddy covariance measurements (Baldocchi, 2003).Climatic variables used in the model were air temperature (Ta

in ◦C), photosynthetically active radiation (PAR in �mol m−2 h−1),air relative humidity (RH in %), wind speed (m s−1), and precipita-tion (mm). Meteorological data, atmospheric CO2 concentrations(Ca), and net carbon and water exchange measurements usingthe eddy covariance technique were routinely acquired on a half-hourly basis following the standard methodology recommended byAubinet et al. (2000). All available continuous data over the March2005–December 2007 period were quality-controlled and gap-filled according to CarboEurope database standards (Reichstein etal., 2005; Papale et al., 2006).

Realistic ranges of the empirical parameters for the photosyn-thesis and stomatal conductance equations (the quantum yield ˛,the slope of the Ball relationship g1max and the curvature of thequantum response of the electron transport rate ) were chosento provide the most accurate simulation of the carbon and waterfluxes measured at the site (see Table 1). It was also assumed thatsoil water content did not limit stomatal conductance or photosyn-thesis. This crude assumption is supported by the good agreementbetween modelled and measured fluxes (see Section 2) and couldresult from deep-water uptake by the tree. The influence of theunderstory vegetation on the CO2 �13C gradient was also ignored.First, our “high-canopy” model corresponds well to our site becauselower oak branches were at about 10 m height. Secondly, thisassumption may not be crucial because the canopy air is well mixedat our site. Indeed, the CO2 gradient between 20 cm above the soiland the top of the canopy (29 m) measured at the flux tower loca-tion did not exceed 10 ppm during the daytime since 2005 (datanot shown).

1.14. Model validation with experimental intra-seasonal ı13Cpattern in tree rings of oaks

Three rings were sampled on three different dominant andco-dominant trees during the winter of 2006/2007 (covering the2006 season) and were taken from the study by Michelot et al.(submitted for publication). Two trees were resampled in thewinter 2007/2008 to obtain rings covering both the 2006 and2007 seasons (see Figs. 7 and 8 in Section 2). High-resolutionintra-annual sampling was performed using a sledge microtome.Samples were cut in contiguous slices of 40-�m down to 20-�mthicknesses.

Carbon isotope composition was measured on whole organicmatter because previous studies showed similar patterns in wholewood and cellulose (Helle and Schleser, 2004). Each sample wasenclosed in 4 mm × 6 mm tin capsules (Säntis Analytical) and com-busted in an elemental analyzer (Model NA-1500; Carlo Erba, Milan,Italy). Carbon isotope composition was determined using a stableisotope ratio mass spectrometer (VG Optima; Fison, Villeurbanne,France, precision ±0.2‰). Carbon isotope ratios were expressed interms of the conventional ı (‰) with reference to the Vienna-PDBstandard.

Stem increment data were not available for the trees sampled inthis study. Similar to Barbour et al. (2002), the timing of the sam-ples was estimated from stem increment data measured on otheroaks at the study site. Stem increments at 1.30 m were measuredby automatic band dendrometers (stainless steel band associatedwith a movement sensor MM30; Megatron, Allinges, France; reso-lution <0.01 mm, stroke 30 mm) on 2 co-dominant oaks. The growthdynamics of these oaks corresponded to the manual dendrometerdynamics measured every week on 61 oaks distributed in the studyforest and were used to determine the starting date for growth ofabove-ground woody biomass.

1.15. Sensitivity analysis methodology

ISOCASTANEA uses many input parameters. It is important toidentify the input parameters that have the greatest effect on modelpredictions. Two output variables were chosen to study the modelbehavior:

- The mean tree-ring �13C in 2006;- The difference between earlywood and latewood �13C in 2006 as

an indicator of the seasonal variation (the boundary between ear-lywood and latewood was defined as occurring on Julian day 166,which was consistent with binocular observations). This proxyof the seasonal variation was chosen because it is comparable tonumerous existing time series of ring �13C data that include onlyearlywood and latewood �13C.

In order to substantially reduce computing time, a sensitivityanalysis was conducted on a statistical model built to have the samebehavior as the process-based model. First, using Monte Carlo sim-ulations, a set of 10,000 simulations of ISOCASTANEA was createdusing a random selection of input parameter values. Each param-eter was associated with a defined range of values and assumeda uniform distribution (Table 1). Secondly, the outputs of thesesimulations were used for building multiple regression trees or a“random forest” (Breiman, 2001; Pappenberger et al., 2006). Therandom forest method is based on combining multiple regressiontrees constructed using different bootstrap samples of both theoutput variables and the input parameters (Breiman, 2001). Thisstatistical model behaves like the process-based model in the givenrange of the parameters (r2 > 0.99). Finally, key parameters wereidentified using the sensitivity analysis on the statistical model. Theimportance of each input parameter was tested by permuting theparameter values in all data sets. An indicator of the importance ofeach parameter was defined as the root mean square error (RMSE),which was computed as:

RMSE =√∑

(YP − YNP)2

n(18)

YNP and YP are the values of the prediction variable before andafter permutation, and n is the size of the data set. The results ofsuch a study depend on the site, soil, and climate that are consid-ered and are therefore only valid for the specific conditions underconsideration.

2. Results

2.1. Inter-annual and seasonal variations in climatic variables atthe Barbeau forest site

The overall seasonal variations in daily air temperature, relativehumidity, and precipitation over 2005, 2006, and 2007 are shownin Fig. 3. The yearly maximum air temperature and the yearly mini-mum RH always occurred during the period of trunk radial growth.



Fig. 3. Moving averages over 20 days of variation in (A) minimum (grey line) and maximum (black line) air temperature, and (B) minimum (grey line) and maximum (blackline) air relative humidity (RH) and daily precipitation (black bars) at the Barbeau forest site from March 2005 to December 2007.

The maximum air temperature generally occurred in July, and thedate of minimum RH varied from year to year. Considering onlythe period of trunk radial growth, the magnitude of variation inmean daily air temperature was highest in 2006 and lowest in 2007,with ranges of 17.3 ◦C, 21.0 ◦C and 15.5 ◦C in 2005, 2006, and 2007,respectively. For RH, the magnitude of variation was lowest in 2005and highest in 2006, with ranges of 45%, 54%, and 47% in 2005,2006, and 2007, respectively. However, when only the leafy seasonwas considered, the variation in RH was highest in 2007 (Fig. 3).The lowest values of RH corresponded generally to periods of lowprecipitation, as in April 2007, or high temperatures, as in July 2006.

2.2. Inter-annual and seasonal variations in ecosystem gasexchange

The seasonal trends in evapotranspiration (ETR) and diurnalnet ecosystem carbon exchange (NEE) fluxes from March 2005to December 2007 are shown in Fig. 4A and B. Both of thesevariables show a seasonal trend typical of temperate deciduousforests: a large uptake of CO2 and a large release of water dur-ing the spring-summer period and the release of CO2 and reducedrelease of water during the late autumn–winter period. DiurnalNEE and ETR during the growing season were highest in 2007,intermediate in 2005, and smallest in 2006. The model simula-tions were in good agreement with eddy covariance data for bothETR (r2 = 0.91; RMSE = 0.7 mm d−1; Bias = −0.17 mm d−1) and diur-nal NEE (r2 = 0.9; RMSE = 1.2 g C m−2 d−1; Bias = −0.03 g C m−2 d−1).However, during the 3 weeks preceding the fitted starting datesof budburst, the model always overestimated diurnal NEE. Thediscrepancies between the model and the measurements at thebeginning of the leafy season could be explained by the presenceof coppices of hornbeams at the study site, which had an earlierdate of budburst than oaks. The understory coppice was not taken

into account because CASTANEA was designed to be a monospe-cific forest stand model (Dufrêne et al., 2005). Therefore, oak GPP isprobably overestimated. The fitted starting dates of budburst weredifferent for each year: Julian days 115, 120, and 100 in 2005, 2006,and 2007, respectively. Maximum leaf mass area was 117 g m−2 for2005 and 2006 and was increased to 130 g m−2 for 2007 to best fitthe diurnal NEE. These values are in the range of the measurementsmade on the site in 2006 and 2007.

2.3. Inter-annual and seasonal variations in modelledphotosynthetic discrimination at the canopy scale (ıcanopy)

There was high variability in daily photosynthetic discrimina-tion at the canopy scale with values ranging from 16.4‰ to 24.4‰(Fig. 4C). Minimum values occurred at the beginning and end ofthe leafy season, and maximum values occurred both at the begin-ning of the leafy season and during July and August. The seasonalpatterns of variation in ıcanopy clearly differed from year to year.This variability was highly and positively correlated with varia-tion in air relative humidity (canopy = 0.12RH + 12.5; r2 = 0.89;p < 0.05) as expected from the Farquhar model (Eq. (16)) coupledwith the model of leaf stomatal conductance from Ball et al. (1987).The correlations with daily photosynthetically active radiation andair temperature were significantly lower, with r2 values of 0.7 and0.33, respectively. Similar to RH, the magnitude of daily variationin ıcanopy was highest in 2006.

2.4. Modelling ı13C variation in soluble sugars

The model enabled simulation of variation in soluble-sugar �13Cin the canopy and at various heights in the phloem pathway andtree organs. Results of model simulations are presented in Fig. 5for the 2006 vegetation period. To characterize the impact of the



Fig. 4. Daily variation in measured and modelled ETR and diurnal NEE and modelled photosynthetic discrimination (�canopy) in the canopy from 2005 to 2007.

carbon allocation and translocation model on the �13C temporalsignal, comparisons were made (1) between �13C of soluble sugarsin the top and bottom canopy, between �13C of soluble sugars inthe canopy compartments and in the highest phloem element thatinteracts directly with these compartments, and (2) between �13Cof soluble sugars in this phloem element and in the stem elementat 1.50 m (that is, in the trunk at core sampling height).

The �13C levels of soluble sugars in the top and bottom canopycompartments were similar during the first part of the leaf growthperiod (15 days after budburst) and then differed substantially dur-ing the rest of the growing season (up to 4‰ in mid-July), with 13Cenrichment in the top canopy compartment (Fig. 5A). This lattercompartment also showed greater temporal variability in soluble-sugar �13C resulting from a steeper response of the photosyntheticdiscrimination intensity and the stomatal conductance to climaticvariations. The slope of the linear regression between daily pho-tosynthetic discrimination and air relative humidity values was

1.7 times steeper for the top canopy compartment (�canopy =0.13RH + 10.7; r2 = 0.9) than for the bottom canopy compartment(�canopy = 0.08RH + 16.4; r2 = 0.75). This is because the assimila-tion rate saturates with increasing irradiance level (Farquhar et al.,1980). In fact, the top canopy generally receives more irradiancethan the bottom canopy. Ci in the bottom canopy was thereforemore sensitive to irradiance variability than Ci in the top canopy.The relationship between RH and ıcanopy was thus weakened inthe bottom canopy. The �13C variation and the level of solublesugars in the highest phloem element were generally intermedi-ate between those for the top and bottom canopy compartments,and soluble-sugar �13C was more similar to the top canopy com-partment. During the first part of the growing season (15 daysafter budburst), the pattern of variation in soluble-sugar �13C inthe highest phloem element differed from that in the canopy com-partments because the export of photosynthates from the canopycompartments was null or low during this period.



Fig. 5. Modelled variation in soluble-sugar �13C during the 2006 leafy season: (A)in top and bottom canopy compartments and in the highest phloem-conductingelement, and (B) in the organ element at height 1.50 m. No post-photosyntheticfractionation was taken into account.

Soluble-sugar �13C in the organ element at 1.50 m differed fromsoluble-sugar �13C in the highest phloem element in terms of bothlevel and temporal variability (Fig. 5B). The soluble-sugar �13C inthe organ element was about 1‰ higher than in the phloem ele-ment for the whole growing season, and the seasonal range ofvariation was half that of the phloem element, with values of 2.9‰and 5.9‰, respectively. This can be explained by the progressivemixing of newly assimilated C with older C along the transloca-tion pathway and by the short residence time of C in the phloemdue to the small size of this compartment (6 mL, see Table 1 andFig. SP2 in the supplementary materials for diurnal variations).Moreover, the amount of newly assimilated C entering the organcompartment diminished with increasing starch content (see Fig.SP1 in the supplementary material) leading to a lower decrease in�13C at the end of the leafy season than in the phloem element.

2.5. Determination of key input parameters

The 10 most sensitive input parameters, as determined by thesensitivity analysis, are presented in Fig. 6 for each output vari-able. The whole tree-ring �13C simulated for 2006 was sensitiveprimarily to fractionation during Rubisco CO2 fixation (b), discrim-ination during respiration (�R), the carbon isotope composition ofatmospheric CO2 (�13Cair), the maximum slope of the ball relation-ship (g1max), wood construction cost (CRwood), and the slope of therelationship between Vcmax and gi (agi

).The difference between earlywood and latewood �13C as sim-

ulated for 2006 (which was considered to be representative ofseasonal variations) was sensitive primarily to the values of the

Fig. 6. Key parameters influencing: (A) the simulated whole-tree-ring �13C in 2006and the difference between simulated early- and latewood �13C in 2006. For symboldefinition, see Table 1.

post-photosynthetic discriminations during respiration (�R) andstarch synthesis (�SS) and the starting date of budburst (BBDAY).In contrast to the sensitivity analysis on whole-ring �13C, no param-eter influencing photosynthetic discrimination was found to be akey parameter.

2.6. Sensitivity of the modelled intra-seasonal ı13C values to keyparameters

The results of the sensitivity analysis on the three key param-eters with the greatest influence on the difference betweenearlywood and latewood �13C are presented in Fig. 7 for the mod-elled 2006 and 2007 rings. The parameters �R, �SS, and BBDAYrepresent different physiological mechanisms that can greatlyaffect the carbon isotope composition of earlywood: respired CO2enrichment, starch enrichment and the use of reserve or current-year assimilates for growth.

The introduction of a discrimination during respiration induced13C-depletion of whole tree-ring organic matter (Fig. 7A) becauseenriched CO2 was released by respiration. This effect was moreimportant during earlywood formation because the arrival rate ofnewly formed photosynthates was low.

The �SS directly influenced starch enrichment compared withsoluble sugars, and a negative �SS resulted in 13C enrichment inearlywood (Fig. 7B). This parameter also had an impact on late-wood �13C by depleting soluble sugars during its formation asstarch reserves were built simultaneously. Therefore, �SS acted inan opposite manner to �R on the intra-seasonal �13C pattern dur-



Fig. 7. Modelled 2006 and 2007 tree-ring �13C sensitivity to: (A) the discrimination during respiration (�R), (B) the discrimination during starch synthesis (�SS) and (C) thestarting date of budburst (BBDAY).

ing the period of earlywood formation but in a similar way duringlatewood formation.

BBDAY affected the timing of the transition from the use ofreserves to the use of current assimilates for growth. An earlyBBDAY resulted in limited use of enriched starch for tree-ring syn-thesis in the earlywood growth period and an earlier use of thecurrent-year photosynthates for wood formation. The �13C in ear-lywood and the magnitude of �13C in latewood were thereforereduced for both the 2006 and 2007 simulated tree rings.

To study the role of �R and BBDAY, �SS was defined as equalto −2‰ (Fig. 7A, C, and D). BBDAY had the greatest impact on ring�13C when starch �13C differed greatly from soluble-sugar �13C.This parameter primarily influenced the �13C pattern in earlywood,as shown in Fig. 6.

2.7. Model comparison with measurements of intra-seasonalvariation in tree-ring ı13C

The simulated intra-seasonal variation in tree-ring �13C wascompared with measured values for tree rings from 2006 at the

Barbeau forest site. Measured intra-seasonal variation in tree-ring �13C differed between trees. The results for three differentrings chosen to represent the between-tree variability in seasonal�13C pattern are presented in Fig. 8. The range of the measuredvariation was between 1‰ (ring B) and 2.5‰ (ring C). Whereasrings A and B showed concave patterns, ring C had a convexpattern.

Comparison among the simulations (Fig. 8) showed the impor-tance of a few key parameters (�SS, BBDAY and �OM the durationof organic matter deposition in tree rings). Respiratory discrimina-tion was not included in the fitting of measured ring �13C becauseit implied a large increase in �SS, a substantial depletion of whole-ring �13C, and a �13C decrease with height, which is opposite tothe positive �13C gradient usually found within trees (Weigl et al.,2008).

To fit the �13C variations measured on the three rings, it wasnecessary to introduce a significant value for discrimination duringstarch synthesis (Fig. 8). Otherwise, it was impossible to explain themagnitude of 13C enrichment observed in the early part of rings Aand B (Fig. 8Aleft and Bleft). The differences in �13C pattern between



Fig. 8. Comparison between simulated and measured intra-seasonal variations in tree-ring �13C in 2006. Three different rings (A–C) sampled in different trees are presentedwith the key parameters that enabled a good fit to each of them: the starting date of budburst (BBDAY), the value of the discrimination during starch synthesis (�SS) and theduration of organic matter deposition (�OM). In each graph, the left part represents the model outputs and the right part the comparison between the measurements and themodel integrated on the same points. The left axis refers to the simulations and the right axis to the measurements. The three phases of the seasonal pattern, as defined inthe discussion, are represented as P1, P2 and P3. Early wood (EW) and late wood (LW) length have also been replaced in the graphs (the earlywood to latewood transitionwas not determined in ring C). The circumferences at 1.30 m were 1.99 m, 2.21 m and 1.73 m on the sampling date for trees A, B, and C, respectively.

the three trees could be fit by varying the starting date of budburst(BBDAY). Thus, rings A, B, and C were successively fitted assuminga decreasing BBDAY with r-values of 0.87, 0.48, and 0.63, respec-tively. The duration of organic-matter deposition (�OM) diminishedthe magnitude of temporal variation and caused the simulated pat-tern to occur earlier in time. For ring C, growth was considered tohave stopped in early August instead of the end of August becausethis ring did not show the �13C decrease observed for ring A and B inAugust. The model underestimated the �13C levels for rings A andC by approximately 1.4‰ and 1‰, respectively, but it was accuratefor ring B.

2.8. Model comparison with inter-annual variations in tree-ringı13C

The model outputs were compared to 2006 and 2007 intra-seasonal variations in �13C, which were measured on the samecores because resampling was done in the winter 2007–2008on two different trees. The results are shown in Fig. 9. The2 years were characterized by clearly different �13C patterns,which were reproduced in both trees sampled. The overall mag-nitude of variation in �13C was approximately 1–1.5‰ for therings built in 2006 and approximately 2.5–3‰ for those built in2007.

The models simulations were done using key parameters valuesthat were chosen to provide the best fit to the 2006 �13C variationsand were not changed for 2007. In these simulations, �SS was fixedat −1.8‰ and −1.5‰ for trees D and E, respectively, and BBDAYwas 2 and 7 days earlier than in the initial simulations. The modelunderestimated �13C levels by approximately 1.5‰ for D and 1.2‰for E. However, it accurately simulated the measured decrease ofabout 2‰ in �13C between 2006 and 2007. It also reproduced wellthe difference between the two patterns for both trees. However,the model showed discrepancies with the measurements in thelate wood of 2007 for both trees, where the model indicated anincrease in �13C that was not observed in the measurements. Thelarge decrease in �13C measured in the first part of the rings builtin 2007 was overestimated by 0.3‰ in the model for both trees(Fig. 9).

It is noteworthy that trees D and E differed in ring width. The2006 and 2007 rings of tree D measured 3.5 and 5 mm, whereas the2006 and 2007 rings of tree E measured 2 and 2.2 mm, i.e., the differ-ent rings were cut into 58, 79, 20, and 31 sections, respectively. Therings sampled on tree D showed �13C variation at a higher temporalresolution than the rings sampled on tree E. For example, the 2007ring sampled on tree D had two peaks in �13C between mid-Mayand the end of June, which were not observed in the ring sampledon tree E.



Fig. 9. Comparison between simulated and measured intra-seasonal variations in tree-ring �13C in 2006 and 2007 for two trees (D and E). Earlywood (EW) and latewood(LW) lengths have also been replaced in the graphs. Dotted arrows indicate two �13C peaks occurring in the latewood of ring D2.

3. Discussion

3.1. Model consistency with literature data

The mean level and the high day-to-day variability of the simu-lated discrimination at the canopy scale (�canopy) was in agreementwith a modelling study in a boreal forest in which predicted valuesof �canopy ranging from 13‰ to 25‰ at a seasonal scale (Chen etal., 2006). The mean �13C level simulated in leaves was low withrespect to literature values, in particular for the bottom canopyand at the end of the vegetation season. As shown by the sensitiv-ity analysis on simulated-ring �13C (Fig. 6), this could be correctedin the model by changing the values of the discrimination dur-ing carboxylation (b), the maximum slope of the Ball relationship(g1max), or the relationship between Vcmax and mesophyll conduc-tance (gi). However, changing the discrimination during respiration(�R) or assuming re-fixation of 13C-depleted respired CO2 cannothelp because these processes would further deplete the �13C con-tent of the soluble-sugar pool. An underestimation of atmosphericCO2 �13C (�13CO2) would also be a possible cause.

The simulated enrichment of leaf soluble-sugar �13C with heightin the canopy agrees with existing modelling studies at the crownscale (Le Roux et al., 2001) and with previous experimental studies(Livingston et al., 1998; Damesin and Lelarge, 2003; Scartazza etal., 2004) that showed differences of up to 5‰ between the top andthe bottom canopy layers. In the course of the 2006 growing season,the model simulated both enrichment and depletion between thehighest phloem element and the trunk element at 1.50 m. This vari-ability reflected temporal variations due to leaf-to-stem transport,as previously assumed by Gessler et al. (2007).

When the canopy acted as a carbon source for the whole tree,carbon translocation and mixing with older carbohydrates induceda time lag and a decrease in the variability of soluble-sugar �13C

between the canopy and the base of the trunk (Fig. 5B). Themagnitude of these processes varied seasonally in relation to thesoluble-sugar level within the tree, which was monitored by pho-tosynthate production at the canopy level, carbon demand forgrowth, and storage of starch (sink strength). According to themodel parameterization, the time lag was approximately 1–3 days(i.e., 0.3–1.2 m h−1) when canopy was fully developed and trunkwas a carbon sink. This is in agreement with the previously cal-culated time lag value associated with phloem transport of about1 m h−1 in Eucalyptus globulus (Gessler et al., 2007). This attenua-tion of �13C variations has also been observed in E. globulus (Gessleret al., 2007) and in Pinus sylvestris (Brandes et al., 2006; Kodama etal., 2008).

3.2. Understanding the seasonal carbon isotope pattern in treerings

The model reasonably simulated the measured seasonal �13Cpattern within rings for both years studied. The use of certain keyparameters to improve the match between the measurements andthe model provides insight into the processes influencing �13C vari-ation within tree rings. Three phases may be distinguished (seeFig. 8 and Fig. SP4 in the supplementary materials).

At the beginning of the growing season, ring formation dependson reserves stored as soluble sugars and starch. Initially, solu-ble sugars are preferentially used but are rapidly consumed. Thenthe starch pool is mobilized. The progressive use of 13C-enrichedstarch explains the observed �13C increase during the early grow-ing period. The 13C enrichment in starch can be partly explained bythe 13C gradient between starch layers, but starch enrichment suffi-cient to fit the measured values can be achieved only by assuming adiscrimination during starch synthesis (from −1‰ to −1.8‰). Thishypothesis is supported by the literature, which describes �13C val-



ues up to 3‰ higher in starch compared with those in soluble sugarsin woody organs (Brugnoli et al., 1988; Damesin and Lelarge, 2003).Moreover, repeated measurements of starch �13C and phloem sol-uble sugars �13C of current-year twigs are available at our site fromApril to December 2003 (Claire Damesin, unpublished results). Thedifference between starch and soluble sugar �13C was 2.4‰ for theearlier measurement (Julian day 119), reached a minimum of 0.5‰(Julian day 225) and increased up to 2.2‰ in December (Julian day343). This variability is not easily explained and may be related toclimatic variability and its direct influence on soluble sugar �13C;however, it is noteworthy that starch is always more 13C-enrichedthan soluble sugars.

The second phase of the pattern is related to the transition ingrowth substrates from carbon reserves to current-year photoas-similates. As new assimilates are generally more 13C-depleted thanremaining reserves, the result is a �13C decrease during this period.Such a decrease was apparent in all rings sampled for both years.However, this assumption needs refinement and would requireexamining each sampled ring separately. Ring A showed a 1‰decline in �13C in the second part of June. This decrease had alreadybeen simulated without considering starch to be 13C-enriched(Fig. 8A). Therefore, the decrease was most probably caused byvariations in photosynthetic discrimination (Fig. 4). The transitionfrom reserves to current assimilates occurred earlier in the ring(beginning of June) and was partly hidden because current photo-synthates had high values of �13C that were similar to those foundin starch. Rings B and C also showed a decrease in �13C in the sec-ond part of June. However, this decrease was preceded by anotherdecrease between May and the beginning of June (Fig. 8B and C).The earlier decrease could be related to the transition from reservesto current photoassimilates because it occurred in the simulationonly if enrichment during starch synthesis was assumed. There-fore, this transition occurred earlier for these two rings (B and C)than in ring A, and it may have occurred because current photo-synthates had low �13C values during the transition period (Fig. 4).For the rings formed during 2007 (D2 and E2), the source transitionseemed to occur during May. However, during the same period, thephotosynthetic discrimination showed a large increase (Fig. 4). Forthese rings, the 2‰ �13C decline in May could be related both tothe source transition and to variation in photosynthetic discrim-ination. Hence, �13C during this phase may not always decreasebecause it depends on the timing of the transition from reserve tocurrent assimilates and the respective �13C levels of the two carbonsources.

The third phase begins when current photosynthates becomethe only carbon source for ring formation, and the seasonal �13Cvariations within the rings are related to photosynthetic discrimi-nation. During this period, variations in measured and simulatedresults were in good agreement for all sampled rings except atthe end of the 2007 growing season (from mid-July to the end ofAugust). This discrepancy cannot be explained by the presence ofhighly depleted mobile compounds at the date of sampling becauselipid content is low in oak wood, and starch or soluble sugarsare usually 13C-enriched compared with the structural compounds(Eglin et al., 2008). Rather, the results may be explained by therelease of 13C-enriched CO2 as the respiration intensity decreasedduring this period or by a peak in starch synthesis for which theseasonal change could not be accurately modelled. This could alsobe due to the seasonal variation in the delay for organic-matterdeposition (Deslauriers et al., 2003) because �OM was defined asconstant. However, the model did not show such a divergence fromthe measurements during the same period in 2006.

The �13C variations in the first two phases were more sensi-tive to the model parameters than in the third one. In fact, thethird phase differs from the other two in its high rate of exportof photosynthates by the canopy and its high carbon demand for

growth and starch reserves. During this period, starch is not usedfor growth, and the discrimination processes have less influenceon the �13C content of the soluble-sugar pools because of their fastturnover as they are constantly replenished by new photosynthates(Fig. 7A and B). The relative length of each phase in the rings mayvary with the total reserve content, the starch-to-soluble-sugarsratio, the current-assimilate production rate, the carbon demandfor growth and starch storage within the tree, and the synchronic-ity between the beginning of growth and the date of budburst. Thebetween-tree variability in the seasonal pattern may therefore bepartly linked to carbohydrate metabolism and the timing of thetransition between reserves and current assimilates as the sourceof carbon for growth. These processes may also explain the circum-ferential variability previously observed in other studies (Tans andMook, 1980; Ramesh et al., 1985).

3.3. Climatic significance of seasonal ı13C variations

The link between climate and ring �13C differs between thevarious phases of the pattern. The first part of the ring is built ofreserves of �13C stored from July to leaf fall and should be related tothe climate in the previous year during the carbon storage period(Porté and Loustau, 2001). Because soluble sugars and starch arepartially mixed during the winter interchange between these twocompounds, it may be difficult to obtain more precise climatic infor-mation. The second part of the pattern is related to the transitionbetween reserves and current assimilates. The resulting mix cannotbe easily interpreted in terms of climatic or environmental condi-tions. However, previous studies have shown strong relationshipsbetween fine-scale isotope measurements and climate even in theearlywood of oaks (Ogle and McCormac, 1994; Loader et al., 1995).Such relationships may be explained by an early use of currentassimilates, as supposed for ring C in our study.

Variation in the last phase of the pattern may be due mostlyto varying environmental influences on the leaf carbon isotope.Indeed, during this period, the modelled variations are littleinfluenced by the parameters related to carbon allocation or post-photosynthetic discrimination (Fig. 7). This phase covers a differentperiod relative to the timing of the transition from reserves tocurrent assimilates. For example, the �13C variations in currentassimilates were recorded earlier in ring C (Fig. 8C) than in ringA (Fig. 8A). However, the signal was largely dampened by mix-ing with previously assimilated carbon. Hence, the environmentalsignal exhibited by this phase of the seasonal �13C pattern was amoving average whose length depended on the turnover of solublesugars all along the translocation pathway between the canopy andthe cambium and on the duration of cell-wall synthesis. The tem-poral resolution of the recording is also dependent on ring widthbecause isotopic analysis requires a minimum amount of materialto be reliable. For example, rings D1 and D2 had a higher tempo-ral resolution than rings E1 and E2 (Fig. 9). The �13C variationsmeasured during this period were related mostly to air relativehumidity, which is the main climatic factor influencing variation inphotosynthetic discrimination because the trees were not signifi-cantly water-stressed. Indeed, water stress may largely impact ring�13C through stomatal closure and growth stop.

3.4. Neglecting respiratory discrimination

Fractionation processes during respiration may have influ-enced the �13C content of tree rings in terms of both seasonalvariations and absolute values (Fig. 7A). Respired CO2 is usu-ally 13C-enriched compared with its potential substrates in bothautotrophic (Duranceau et al., 1999; Tcherkez et al., 2003) and het-erotrophic organs (Damesin and Lelarge, 2003; Brandes et al., 2006;Gessler et al., 2007; Kodama et al., 2008). However, the difference



in �13C content between the organic source and the released CO2should vary seasonally and between tree organs according to tem-perature (Tcherkez et al., 2003; Maunoury et al., 2007), the rateof respiratory metabolism (Tcherkez et al., 2003; Maunoury et al.,2007; Kodama et al., 2008), the importance of the respired CO2re-fixation by phosphoenol-pyruvate carboxylase (Badeck et al.,2005), and the composition of the wood organic matter synthesis(Bowling et al., 2008). During the growing season, discriminationduring respiration should be greatly reduced because temperatureand respiration rate (Tcherkez et al., 2003; Kodama et al., 2008)are high and should have a relatively low impact on ring �13C.The �13C content of the reserves should be more strongly impactedduring the dormant period because discrimination increases withlow temperature and low respiratory intensity. On the other hand,re-assimilation during the leafy season of previously respired CO2may directly impact the �13C of photosynthates, particularly in thelower canopy near the soil surface. Our model could potentially beimproved in the future by considering seasonal variation in respi-ratory discrimination and the recycling of respired CO2, but furtherunderstanding of the influence of these factors is needed.

4. Conclusion

In this study, the ISOCASTANEA model was developed tosimulate the seasonal dynamics of the stable carbon isotopic com-position of carbohydrate reserves and tree-ring organic matter indeciduous species. With the inclusion of carbon translocation, ourmodel showed more consistent ranges of �13C variation within aring than the model of Hemming et al. (2001). It reasonably sim-ulated the intra-seasonal and inter-annual variability of tree-ring�13C and provided insight into the processes influencing �13C vari-ation within and between tree rings. Intra-ring �13C in oaks couldnot have been explained without the distinction between solublesugars and starch �13C and a negative discrimination during starchsynthesis. These results contrast with results from a similar mod-elling study showing that a single substrate model is able to explainmost of the �13C variations within a ring of Pinus pinaster Ait. (Ogéeet al., 2009). Therefore, in the near future, the model will be testedon a bigger dataset and a longer time series.

Helle and Schleser (2004) previously described the seasonal car-bon isotope pattern in tree rings of broadleaf species as divided intothree different phases: enrichment in 13C during the early growingperiod, �13C decline during the main vegetation period, and a newand slight �13C increase at the very end of the vegetation period.In the present study, all the sampled rings exhibited the first twophases for both years studied, but the third phase was not observed.Moreover, the �13C did not decline throughout the main vegeta-tion period but rather showed significant variations that appearto be linked to climatic variability. Contrary to observations byHelle and Schleser (2004), the seasonal carbon isotope pattern intree rings was similarly influenced by both photosynthetic andpost-photosynthetic processes with regard to the magnitude ofvariations, in particular for rings formed during 2006. The �13Cvariations in the early part of the ring, i.e., mainly in the earlywood,were related mostly to carbohydrate metabolism, although dilutedinformation about environmental conditions during the previousyear could be found. The last part of the ring, which mainly con-sisted of latewood, was found to be a better recorder of current-yearenvironmental conditions, in particular relative humidity, at a finetemporal resolution when growth intensity was high. The lengthof the recorded period depended on the timing of the transitionfrom reserves to current photoassimilates and the duration of thering growth period. These results emphasize that intra-ring �13C ofdeciduous species is not simply interpretable in terms of climateand should be examined carefully with regard to tree functioning.

Interestingly, these results also suggest that intra-ring �13C maybe used to explore the responses of tree functioning to extremeevents, such as drought. First, �13C variation in latewood providesinformation on the direct response of photosynthetic activity to dryperiods. Secondly, the length of the portion of the ring built from Creserves (as inferred from �13C values) may be helpful in determin-ing the long-term impact of C starvation on tree growth. Indeed, itis known that drought may reduce the amount of stored carbohy-drates at the end of the growing season (Breda et al., 2006) andimpact the recovery of the growth potential in the following years(Battaglia et al., 1998; Breda et al., 2006). More generally, intra-ring �13C could be used in future work to examine the relationshipbetween long-term tree decline and carbohydrate deficiency.

Appendix A.

The axial conductance of phloem elements can be calculatedaccording to Poiseuille’s law (Sheehy et al., 1995):

Kaxial = nst�a4st

8�[f1/nporef2], (A1)

where nst is the number of sieve tubes in a conducting element,ast (m) the radius of an individual sieve tube, � (MPa h−1 m−1) thesieve coefficient of viscosity, f1 the ratio of sieve pore length to sievetube member length, f2 the ratio of sieve pore radius to sieve tuberadius, and npore the number of pores in a sieve plate.

The volume of a conducting element Vphloem in m3 (needed forcalculation of sucrose concentration) is:

Vphloem = nst�(nporef1(ast f2)2 + a2st) (A2)

Only crude estimates of the axial conductance and the volumeof a conducting element can be obtained because of the variabilityin anatomic parameters (Thompson and Holbrook, 2003). Difficul-ties include knowing which phloem sieve tubes are involved in thepathway from sources to sinks, variations in resistance along thephloem pathway (in particular variations in viscosity), and the con-sideration that the phloem cells are impermeable cylindrical tubes(Bancal and Soltani, 2002).

The number of phloem sieve tubes in a conducting element persquare meter of soil can be calculated according to Sheehy et al.(1995) as:

nst = ne2�(rtrunk − bark)Sd

20000ast(A3)

where rtrunk (m) is the trunk radius at 1.30 m, bark (m) is the barkwidth, ne is the number of active layers in the phloem, and Sd(trees ha−1) is the stocking density. For calculations, the range ofvalues for all the parameters were chosen according to Sheehy et al.(1995) except for stocking density, trunk radius, and bark thickness.

Appendix B.

The model can be described as a continuously branching hierar-chical network running from the trunk level to the petioles or to thefine roots. The tree architecture is characterized by simple scalinglaws:

nbranchingk+1

nbranchingk

= nd, (B1)

rk+1

rk= n−atp/2

d , (B2)

Lnk+1branching

Lnkbranching

= n−1/3d

, (B3)



where nd is the branching ratio, i.e., the number of daughterbranches derived from a parent branch, nbranching

k, rk, and L

nbranchingk+1

are respectively the number, the radius, and the length of branchesin the kth level, and 1/3 is the value determined by West et al. (1999)for a volume-filling network. The average total circumference foreach element is calculated according to its branching level and thescaling laws described in the main text. The number of branchinglevels is therefore defined by:

nbranchingmax = 2 ln(rtrunk/rpetiol/fine root)

atp ln(nd)(B4)

where rpetiol/fine roots is the radius of the terminal unit of the branch-ing network, i.e. the petiol or the fine root.

At each branching level kth total circumference can be written:

Circnbranching

k

= 2�rtrunknnbranching

k(1−(atp/2))

d(B5)

The length of each branching level LNbranching

k

(m) is given by:

Lnbranching

k

= LTotal

⎛⎝ 1 − n1/3d

1 − nnbranching

max /3d

⎞⎠nnbranching

max −nbranchingk

/3

d(B6)

The height of each node Hnbranching

k

(m) is:

Hnbranching

k

= H0 + Lnbranching

k

for the branches’ network (B7)

Hnbranching

k

= H0 − Lnbranching

k

for the roots’ network (B8)

Several branching levels can be comprised in a conducting ele-ment. Thus, the average total circumference for each element jlength is therefore:

∀k((Hmaxj > H

nbranchingk

> Hminj )Y(Hmax

j < Hnbranching

k

< Hminj ))

Circj =

∑kCirc

nbranchingk

(min(Hnbranching

k

, Hmaxj

) − max(HnNbranching

k−1

, Hminj

))

Hmaxm − Hmin

j

(B9)

Appendix C.

The tree ring is divided into a number of growth layers whichdiffer by their initial date of formation. Organic-matter depositionbegins immediately at growth layer initiation. A new layer is initi-ated at each time step if there is sufficient C for maturation of olderlayers. In each growth layer k, rates of deposition for organic matter,VOM,k (gC h−1), are calculated according to a decreasing exponentialfunction:

VOM,k(t) = AOMk e−˛OMt, (C1)

where AOMk

(gC h−1), ˛OM are parameters characterizing respec-tively the initial value and the time dependence of the depositionrate. t is the time since growth layer initiation.

The amount (gC) of organic matter formed in each layer k duringeach model time step �t are:

OMk =∫ t+�t

t

VOM,k(t)dt. (C2)

At each time step, the carbon allocated to a new growth layer k′,IG (gC) is therefore:

IG = G −nk∑i=1

(OMi) =∫ �t

0

VOM,k′ (t)dt, (C3)

where nk is the number of growth layers still in the maturationphase and G the total C available for tree ring growth.

The total amount of carbon that is allocated to the new layer k′

during its maturation, OMTotk′ (gC), can be expressed as:

OMTotk′ =

∫ �OM

0

VOM,k′ (t)dt = IGe−˛OM�OM − 1e−˛OM�t − 1

, (C4)

where TOM is the time needed for organic-matter deposition.The parameter AOM

k′ can therefore be written for each new layerk′ as:

AOMk′ = −˛OM

e−˛OM�OM − 1OMTot

k′ = IG−˛OM

e−˛OM �t − 1(C5)

Eq. (C5) is obtained by integrating Eq. (C1) on the interval [0;TOM]and by isolating the variable AOM

k′ . When growth begins, no growthlayer is in maturation phase and IG is equal to G the total C availablefor growth.

Appendix D. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.ecolmodel.2010.04.007.

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