modelling canopy co2 fluxes: are ‘big-leaf’ simplifications justified?

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© 2001 Blackwell Science Ltd. http://www.blackwell-science.com/geb 603 ETEMA SPECIAL ISSUE Global Ecology & Biogeography (2001) 10 , 603–619 Blackwell Science, Ltd Modelling canopy CO 2 fluxes: are ‘big-leaf’ simplifications justified? A. D. FRIEND Center for Environmental Prediction, Rutgers University, 14 College Farm Road, New Brunswick, NJ 08901–8551, U.S.A. ABSTRACT 1 The ‘big-leaf’ approach to calculating the carbon balance of plant canopies is evaluated for inclu- sion in the ETEMA model framework. This approach assumes that canopy carbon fluxes have the same relative responses to the environment as any single leaf, and that the scaling from leaf to canopy is therefore linear. 2 A series of model simulations was performed with two models of leaf photosynthesis, three distributions of canopy nitrogen, and two levels of canopy radiation detail. Leaf- and canopy-level responses to light and nitrogen, both as instantane- ous rates and daily integrals, are presented. 3 Observed leaf nitrogen contents of unshaded leaves are over 40% lower than the big-leaf approach requires. Scaling from these leaves to the canopy using the big-leaf approach may underestimate canopy photosynthesis by ~20%. A leaf photo- synthesis model that treats within-leaf light extinc- tion displays characteristics that contradict the big-leaf theory. Observed distributions of canopy nitrogen are closer to those required to optimize this model than the homogeneous model used in the big-leaf approach. 4 It is theoretically consistent to use the big-leaf approach with the homogeneous photosynthesis model to estimate canopy carbon fluxes if canopy nitrogen and leaf area are known and if the dis- tribution of nitrogen is assumed optimal. How- ever, real nitrogen profiles are not optimal for this photosynthesis model, and caution is necessary in using the big-leaf approach to scale satellite estimates of leaf physiology to canopies. Accurate prediction of canopy carbon fluxes requires canopy nitrogen, leaf area, declining nitrogen with canopy depth, the heterogeneous model of leaf photo- synthesis and the separation of sunlit and shaded leaves. The exact nitrogen profile is not critical, but realistic distributions can be predicted using a simple model of canopy nitrogen allocation. Key words big-leaf, canopy nitrogen profile, canopy photosynthesis, carbon flux, optimization, scaling. INTRODUCTION Understanding and predicting the responses of carbon exchange between vegetation and the atmosphere to environmental forcings is central to many areas of ecological research, and is a funda- mental component of the ETEMA model frame- work (Sykes et al ., 2001, this issue). Unfortunately, the non-linear dependencies of photosynthesis on leaf nitrogen and absorbed light, and changes in leaf microenvironment with canopy depth, com- plicate the task of scaling leaf physiology to the canopy. Therefore a simplification in which the whole canopy is treated as if it were one big leaf has been widely adopted (e.g. Farquhar, 1989; Kull & Jarvis, 1995; Friend et al. , 1997), particularly after an elegant linearization of the scaling between leaf and canopy photosynthesis was presented by Sellers et al . (1992). This scaling method is referred to here as the ‘big-leaf approach’. Despite its wide acceptance, subsequent developments in the modelling of leaf photosynthesis and measurements Correspondence: A.D. Friend, NASA Goddard Institute for Space Studies, 2880 Broadway, New York, NY 10025–7886, USA. E-mail: [email protected]

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© 2001 Blackwell Science Ltd. http://www.blackwell-science.com/geb

603

ETEMA SPECIAL ISSUE

Global Ecology & Biogeography

(2001)

10

, 603–619

Blackwell Science, Ltd

Modelling canopy CO

2

fluxes: are ‘big-leaf’ simplifications justified?

A. D. FRIEND

Center for Environmental Prediction, Rutgers University, 14 College Farm Road,

New Brunswick, NJ 08901– 8551, U.S.A.

ABSTRACT

1

The ‘big-leaf’ approach to calculating the carbonbalance of plant canopies is evaluated for inclu-sion in the ETEMA model framework. Thisapproach assumes that canopy carbon fluxes havethe same relative responses to the environment asany single leaf, and that the scaling from leaf tocanopy is therefore linear.

2

A series of model simulations was performedwith two models of leaf photosynthesis, threedistributions of canopy nitrogen, and two levels ofcanopy radiation detail. Leaf- and canopy-levelresponses to light and nitrogen, both as instantane-ous rates and daily integrals, are presented.

3

Observed leaf nitrogen contents of unshadedleaves are over 40% lower than the big-leaf approachrequires. Scaling from these leaves to the canopyusing the big-leaf approach may underestimatecanopy photosynthesis by ~20%. A leaf photo-synthesis model that treats within-leaf light extinc-tion displays characteristics that contradict thebig-leaf theory. Observed distributions of canopy

nitrogen are closer to those required to optimizethis model than the homogeneous model used inthe big-leaf approach.

4

It is theoretically consistent to use the big-leafapproach with the homogeneous photosynthesismodel to estimate canopy carbon fluxes if canopynitrogen and leaf area are known and if the dis-tribution of nitrogen is assumed optimal. How-ever, real nitrogen profiles are not optimal for thisphotosynthesis model, and caution is necessaryin using the big-leaf approach to scale satelliteestimates of leaf physiology to canopies. Accurateprediction of canopy carbon fluxes requires canopynitrogen, leaf area, declining nitrogen with canopydepth, the heterogeneous model of leaf photo-synthesis and the separation of sunlit and shadedleaves. The exact nitrogen profile is not critical,but realistic distributions can be predicted usinga simple model of canopy nitrogen allocation.

Key words

big-leaf, canopy nitrogen profile,canopy photosynthesis, carbon flux, optimization,scaling.

INTRODUCTION

Understanding and predicting the responses ofcarbon exchange between vegetation and theatmosphere to environmental forcings is central tomany areas of ecological research, and is a funda-mental component of the ETEMA model frame-work (Sykes

et al

., 2001, this issue). Unfortunately,the non-linear dependencies of photosynthesis on

leaf nitrogen and absorbed light, and changes inleaf microenvironment with canopy depth, com-plicate the task of scaling leaf physiology to thecanopy. Therefore a simplification in which thewhole canopy is treated as if it were one big leafhas been widely adopted (e.g. Farquhar, 1989;Kull & Jarvis, 1995; Friend

et al.

, 1997), particularlyafter an elegant linearization of the scaling betweenleaf and canopy photosynthesis was presented bySellers

et al

. (1992). This scaling method is referredto here as the ‘big-leaf approach’. Despite itswide acceptance, subsequent developments in themodelling of leaf photosynthesis and measurements

Correspondence: A.D. Friend, NASA Goddard Institute forSpace Studies, 2880 Broadway, New York, NY 10025–7886,USA. E-mail: [email protected]

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of canopy physiology have brought into questionthe assumptions behind this approach. The workpresented here is therefore an evaluation of thebig-leaf approach for inclusion in the ETEMAflux module (Woodward & Lomas, 2001, thisissue). The following questions are addressed. Arethere circumstances in which the big-leaf approachis invalid? If so, how accurate is it likely to be?If it is unacceptable, is there an alternativeapproach that gives improved accuracy, yet issimple enough for ecosystem modelling?

The big-leaf approach, as defined here, assumesthat canopy net photosynthesis (

A

can

) can beapproximated by:

A

can

=

A

0

f

PAR

/

k

(1)

where

A

0

is the rate of net photosynthesis in anunshaded leaf,

f

PAR

is the fraction of incidentphotosynthetically active radiation (PAR) absorbedby the canopy, and

k

is the canopy PAR extinc-tion coefficient (Sellers

et al.

, 1992).

f

PAR

and

k

are mean values over a suitable time period. Theutility of this approach for making estimates ofproductivity from satellite-borne instruments wasemphasized by Sellers

et al

. (1992), who claimedthat spatial and temporal variation in

f

PAR

can bemeasured from space with sufficient accuracy.The other two parameters can be either assignedvalues based on observed vegetation type or canbe modelled biogeochemically.

This approach can be extended to ecosystemmodels that either prescribe or predict total canopynitrogen and leaf area. The nitrogen content ofan unshaded leaf (

N

0

) is given by (Friend

et al.

,1997):

N

0

=

N

can

k

/

f

PAR

(2)

where

N

can

is canopy nitrogen on a ground areabasis.

N

0

can then be used to calculate

A

0

usinga model of leaf net photosynthesis.

The relationships expressed by equations 1 and2 depend on three important assumptions: theresponse of net photosynthesis to PAR is quali-tatively the same for all leaves, net photosynthesisat any PAR level is a linear function of leafnitrogen, and leaf nitrogen is linearly propor-tional to the ratio of leaf-absorbed PAR to totalPAR absorbed by the canopy, averaged over asuitable time period. The photosynthesis model

used by Sellers

et al

. (1992) has the first twoproperties, and an optimization argument basedon the first two assumptions, together with someexperimental evidence, support the final assump-tion (e.g. Field, 1983; Hirose & Werger, 1987;Farquhar, 1989).

This approach is assessed herein using the leafphotosynthesis model used by Sellers

et al

. (1992),and a more recent approach that takes accountof the heterogeneity of light within leaves. Inaddition, the implications of the distribution ofcanopy nitrogen and the level of model detailwith respect to the distribution canopy radiationare analysed.

MODEL SIMULATIONS

The analysis presented here utilizes two modelsof leaf photosynthesis, two models of canopyradiation and three patterns of canopy nitrogenin order to test fully the big-leaf approach. Allequations and parameters for the net photo-synthesis and canopy radiation models have beenpublished elsewhere and are described in theAppendices.

Leaf net photosynthesis

Model I: the homogeneous model

The model of leaf net photosynthesis used bySellers

et al.

(1992) is referred to here as ModelI. This is perhaps the most commonly used physio-logical approach to modelling leaf photo-synthesis, and was first described by Farquhar

et al.

(1980) and Farquhar & von Caemmerer (1982).All chloroplasts in a leaf are assumed to experi-ence the same light level, and an empiricalsmoothing function is often used to make thetransition from light-limited to light-saturatednet photosynthesis more closely resemble obser-vations (Collatz

et al.

, 1990). Light-limited netphotosynthesis is calculated as a linear functionof the rate of whole-leaf electron transport,which is a saturating function of absorbed lightand electron transport capacity (von Caemmerer& Farquhar, 1981). Electron transport capacity isa linear function of leaf nitrogen (Friend, 1995).Light-saturated net photosynthesis is assumed tobe a linear function of the amount and turnoverof the carboxylation enzyme Rubisco (Farquhar

et al.

, 1980), and the amount of this enzyme is

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also assumed to be a linear function of leaf nitro-gen (Farquhar

et al.

, 1980). In addition, bothlight-limited and light-saturated rates are saturat-ing functions of internal leaf CO

2

concentrationand dark respiration is a linear function of thecapacity of Rubisco to fix CO

2

, an index of leafmetabolic activity (Collatz

et al.

, 1990). A fulldescription of Model I and the parameter valuesused in the simulations presented here are givenin Appendix 1. In these and all subsequentsimulations, net photosynthesis is predicted as afunction of leaf nitrogen and PAR; all otherparameters are fixed. Stomata are assumed tomaintain the internal leaf CO

2

partial pressure at24 Pa. Possible natural variation in this parameterdoes not significantly affect model comparisons.

In Model I, the fixed linear dependencies ofelectron transport capacity and Rubisco contenton leaf nitrogen, and the assumed homogeneityof light throughout the leaf, produce the sameshaped net photosynthesis-light response curves

at different nitrogen levels (Fig. 1). In the big-leafapproach, it is assumed that the distributionof canopy nitrogen in the canopy matches thegradient in mean PAR (Sellers

et al.

, 1992). Thelinearity expressed by equation 1 arises becauseeach leaf always operates at the same point ofcurvature on its net-photosynthesis /PAR rela-tionship curve; that is, as PAR incident on thecanopy varies, net photosynthesis increases ordecreases by the same relative amount in eachleaf. Therefore the behaviour of only one leaf isrequired to estimate the behaviour of the wholecanopy.

However, Kull & Kruijt (1998) concluded thatthe theoretical assumptions for the relationshipsbetween light, nitrogen, and photosynthesis behindthe scaling scheme expressed in equation 1 do nothold when examined using a more mechanisticmodel of leaf photosynthesis. In addition,observed canopy nitrogen is not distributed inproportion to mean PAR. Given the frequency

Fig. 1 Predicted responses of leaf net photosynthesis to PAR penetrating leaves with different levels of leafnitrogen using Model I (Appendix 1). The linear increase in light harvesting and carboxylation capacities withnitrogen, together with the empirical smoothing coefficient between light-limited and light-saturated rates ofphotosynthesis and assumed homogeneity of within-leaf light, cause the curves to all have the same form.

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with which the big-leaf approach is used, it istherefore important to know the extent to whichit might be in error.

Model II: the heterogeneous model

The leaf photosynthesis model of Kull & Kruijt(1998) is referred to here as Model II. This modelis also based on the equations of von Caemmerer& Farquhar (1981), but separates regions spa-tially within leaves limited by electron transportcapacity and light harvesting, depending on lightlevel, by treating the extinction of PAR throughthe leaf. This obviates the need for an empiricalsmoothing function and is thus claimed to bemore mechanistic. At low light all chloroplastsin a leaf will be light-limited, and the rate ofphotosynthesis is proportional to the amount ofPAR absorbed by the light harvesting apparatusand the intrinsic quantum efficiency for CO

2

uptake. As incident light increases, a point willbe reached where the uppermost chloroplastsreceive more energy than can be utilized to fixCO

2

because of limitations due to the capacityfor electron transport or the maximum rate ofcarboxylation by Rubisco. Model II uses thesame formulation as Model I for Rubisco-limited, light-saturated conditions, but the secondlimitation in Model I combines the light harvest-ing and electron transport limitations into oneequation, necessitating a more empiricalapproach. Kull & Kruijt (1998) found that ModelII produces a good fit to observed responses ofnet photosynthesis to light.

Although there are three potentially limitingrates, Model II reduces to one analytical equa-tion when all rates are expressed as functions ofnitrogen and it is assumed that the ratio of chloro-phyll to total nitrogen is constant throughout theleaf (Kull & Kruijt, 1998). The light harvesting-limited rate is a function of nitrogen becausePAR absorption is assumed to be proportional tochlorophyll content which, like the amount ofRubisco and the maximum rate of electron trans-port, is assumed to be linearly proportional tonitrogen. For each light level there exists a leafnitrogen content, and hence chlorophyll content,above which absorption of light within the leafcauses some chloroplasts to be light-limited. Thisnitrogen content is calculated as a function ofleaf internal CO

2

concentration, the intrinsicefficiency for CO

2

uptake and the ratio between

chlorophyll and total nitrogen, and is useddirectly in the analytical solution for leaf grossphotosynthesis (Kull & Kruijt, 1998). This modeland its parameterization are described fully inAppendix 2.

In Model II, the separation of regions in theleaf limited by light alone, or by the ability totransport electrons and fix CO

2

given abundantenergy, results in the lack of a requirement forempirical smoothing coefficients between limitingrates, with important implications for the assump-tions of the big-leaf approach. The response ofnet photosynthesis predicted by Model I topenetrating PAR shown in Fig. 1 is repeated inFig. 2 for Model II. Model II is parameterized asdescribed in Kull & Kruijt (1998) from data on

Populus tremula

L.,

Corylus avellana

L. and

Tiliacordata

Mill. canopies measured by Kull &Niinemets (1998) (see Appendix 2). All parameterscommon to both models (i.e. leaf Rubiscocontent- and electron transport capacity-nitrogenrelationships, Rubisco kinetic parameters, leafinternal CO

2

and O

2

partial pressures, and theratio of dark respiration to Rubisco) are set tothe same values, and the ratio of chlorophyll tonitrogen is constant throughout the leaf in ModelII. The most noticeable differences betweenFigs 1 and 2 are that net photosynthesis saturateswith respect to light more strongly in Model II,but peak rates are higher than in Model I,despite the same total quantity of photosyntheticmachinery. Model II saturates when all chloro-plasts are limited by electron transport capacityor Rubisco content. In Model I, the empiricalsmoothing between the light- and Rubisco-limited rates causes no such saturation, but reducesthe overall rate at high light. At maximum PARand nitrogen, net photosynthesis is 26% higher inModel II than Model I.

Model I predicts that the limitation to photo-synthesis by light is greater than the limitationdue to Rubisco throughout the leaf, and the finalrate is reduced a further 15% due to the colim-itation function (see Appendix 1). Model IIpredicts that light-saturated photosynthesis occursdown to a nitrogen depth of 90% of total leafnitrogen, with light harvesting-limited photo-synthesis below this. Light-saturated net photo-synthesis in Model II is determined by Rubiscocontent and turnover, and is the same per unitnitrogen as in Model I. However, the light-limited

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rate is 9% lower than the Rubisco-limited rate inModel I, resulting in Model II having almost thesame rate of net photosynthesis in the light-saturated portion as Model I does in total. ButModel II also predicts 10% of nitrogen in thesubsaturated portion of the leaf, which contrib-utes a further 8% to leaf net photosynthesis.Without the co-limitation function in Model I(i.e. the total rate is equal to the limiting rate),the difference between the two models is reducedfrom 26% to 7%.

The behaviour of Model II suggests that thebasic assumptions of the big-leaf approach out-lined above are incorrect. As is evident in com-paring Figs 2 and 1, if nitrogen falls at the samerate as PAR down through the canopy, Model IIwill predict that at sufficient radiation intensitiesthe upper leaves will operate on the saturatedportion of their light response curves while lowerleaves will be at least partially light-limited. Con-sequently, as the total amount of light incident

on the canopy varies, the rate of net photo-synthesis in different leaves will vary by differentrelative amounts. Measurements confirm varyingconvexity in the photosynthetic light responsewith the amount of photosynthetic machinery (e.g.Leverenz, 1988). The implications of the resultingnon-linearity for instantaneous canopy net photo-synthesis, total daily canopy net photosynthesisand the big-leaf approach are assessed below.

Instantaneous canopy photosynthesis

The implications of Model I or II for instant-aneous total canopy net photosynthesis are nowconsidered using different assumptions regardingprofiles of nitrogen and different levels of com-plexity in the treatment of canopy radiation.Integration of leaf net photosynthesis is performedwith Simpson’s Rule over 20 canopy layers,although results are accurate to within 1.5% withonly four layers. Leaf area index (

LAI

) is set to

Fig. 2 As Fig. 1, but using Model II and with chlorophyll to nitrogen ratio fixed at 2.5 µmol [chlorophyll]mmol–1 [N]. The shape of each curve results from an increasing fraction of the leaf that is light-saturated asPAR increases. When all chlorophyll is light-saturated, photosynthesis no longer varies with PAR. The pointof this discontinuity does not vary linearly with nitrogen, resulting in each curve having a different form.

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3 m

2

[sum of individual projected leaves] m

–2

[ground], which produces the same total PARextinction (i.e. 88%) observed by Kull &Niinemets (1998) using a detailed canopy radia-tion model (see below) at mid-morning. Whenrequired, leaf nitrogen is distributed with relativePAR using a fit to the data of Kull & Niinemets(1998) for the three species mentioned above. Thesedata and the fit are shown in Fig. 3. Observednitrogen fell slowly with PAR until about 60%was absorbed, below which nitrogen fell moresteeply, and so two straight lines are used toapproximate these measurements. The observedratio between leaf chlorophyll and nitrogen (

n

3

;

µ

mol [chlorophyll] mmol

–1

[N]) increases withcanopy depth, and when required is also setusing a fit to the observations of Kull &Niinemets (1998):

n

3

= 0.48 + 300/

N

(3)

where

N

is observed leaf nitrogen content (mmolm

–2

). Otherwise

n

3

is set to 2.5

µ

mol [chlorophyll]mmol

–1

[N], as measured at the top of the can-opies. Total observed canopy nitrogen integratedover an

LAI

of 3 is 337 mmol [N] m

–2

[ground].Depending on the simulation, canopy nitrogenis distributed uniformly, in parallel with meanPAR, or to optimize canopy net photosynthesis,but in each case total canopy nitrogen is keptthe same.

Simple radiation

Predicted total canopy net photosynthesisincreases more steeply with PAR flux at the topof the canopy using Model II than Model I,regardless of how canopy nitrogen is distributed

Fig. 3 Leaf content profiles in Tilia cordata Mill., Populus tremula L. and Corylus avellana L. canopiesmeasured by Kull & Niinemets (1998). Canopy position described by relative PAR irradiance. Also shownare two lines fitted to the observations (solid lines), the same total nitrogen (i.e. 337 mmol [N]) distributedin proportion to relative PAR (short dashes), predicted optimal nitrogen distribution with Model II (longdashes; see text), and predicted nitrogen distribution using a dynamic model of leaf nitrogen with Model II(dot-dashes; see text). The fitted lines are: 115 + 55 rPAR (rPAR > 0.376); 361rPAR (rPAR < = 0.376), whererPAR is relative PAR.

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(Fig. 4). These predictions are obtained using asimple Beer’s Law expression to calculate PARextinction (Appendix 3). The PAR extinctioncoefficient (k) is calculated to be 0.78 from amore complex radiation model at mid-morning(extinction coefficients vary with sun zenithangle), where the vertical profile of PAR irradi-ance is calculated using the method of Spitters(1986), in which direct and diffuse light, and sunand shade foliage are separated (Appendix 3).Model II predicts greater rates of canopy netphotosynthesis at all but the lowest light levelsdue to the leaf-level differences explained in theprevious section (i.e. more efficient use of nitro-gen within each leaf and lack of an empiricalsmoothing function). Canopy net photosynthesisis higher in both models when nitrogen falls inproportion to PAR down through the canopythan if it is distributed uniformly (e.g. by 7%at 800

µ

mol m

–2

s

–1

PAR in Model II). If theobserved relationship between nitrogen and PAR

is used (Kull & Niinemets, 1998), net photo-synthesis is slightly reduced in Model I, whereasModel II predicts higher rates. This shows thatthe optimum distribution of nitrogen for Model IImust be closer to that observed than the optimumfor Model I. Observed nitrogen at the top ofthe canopy is 42% lower than when the sametotal canopy nitrogen is distributed in proportionto mean PAR. Including the observed increasein the fraction of nitrogen in chlorophyll withcanopy depth increases Model II predictionsbelow 800

µ

mol m

–2

s

–1

slightly (e.g. by 6% at 400

µ

mol m

–2

s

–1

).The relatively small impact of the distribution

of nitrogen on canopy photosynthesis is surpris-ing, especially given the attention it has receivedin the literature. This implies that the inconsist-ency between observed nitrogen profiles and‘optimal’ profiles may not matter for applicationof the big-leaf approach. Of importance, how-ever, is the choice of leaf photosynthesis model,

Fig. 4 Response of canopy net photosynthesis to horizontal PAR irradiance at the top of the canopypredicted with both leaf net photosynthesis models, but different assumptions of leaf nitrogen profiles. PARprofile calculated using simple Beer’s Law (Appendix 3). All simulations use the same total canopy nitrogen(i.e. 337 mmol [N]), LAI of 3, and PAR extinction coefficient of 0.78.

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the total amount of canopy nitrogen, and thevalue and linearity of the scaling coefficient fromleaf to canopy.

Complex radiationIt is possible that the importance of the distri-bution of nitrogen and the effect of the twophotosynthesis models on modelled canopyphotosynthesis are functions of the level ofdetail used to describe canopy radiation. There-fore the responses presented in Fig. 4 arerepeated using a much more detailed model ofcanopy radiation, where the vertical profile ofPAR irradiance is calculated using the methodof Spitters (1986), in which direct and diffuselight, and sun and shade foliage are separated(Appendix 3). Extinction coefficients for thismethod assume spherical leaf angle distributionwith random location in space. The rates of photo-synthesis in sun and shade foliage in each canopylayer are calculated and summed before integrationacross layers, and the results shown in Fig. 5.

All canopy net photosynthesis rates arereduced by 5–10% at the highest PAR, and to

18.5% at intermediate PAR in Model II, com-pared to the simple canopy radiation scheme.These reductions are due to the non-linearresponse of photosynthesis to PAR, as shownby Spitters (1986). The positive effect of theobserved nitrogen distribution compared with theone with proportionality to PAR is reduced forModel II, and the beneficial effect of theobserved decline in the ratio of chlorophyll tonitrogen with canopy depth is increased at inter-mediate light levels. When observed quantities areused, the predicted rate of canopy net photo-synthesis at the highest PAR is 34% greater inModel II than Model I. The effect on canopyphotosynthesis of declining nitrogen with canopydepth is greater with the more complex radiationscheme, but the precise pattern of decline is stillnot important for either model. Comparison withFig. 4 demonstrates the importance of consider-ing the separation of sunlit and shaded foliage(cf. Reynolds et al., 1992). The effects of thesedifferent model approaches and nitrogen distri-butions on the validity of the big-leaf approachare considered next.

Fig. 5 As Fig. 4, but using the complex canopy radiation scheme described in Appendix 3.

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Daily carbon uptake

To test the extent to which the modelled canopyacts as a big-leaf, and how this is influenced bymodel choice and nitrogen distribution, ModelsI and II are used to predict total canopy netphotosynthesis on a theoretical July 1 at 40°Nlatitude under clear skies and with the same totalcanopy nitrogen for each representation. Tem-poral change in PAR at the top of the canopy iscalculated using the equations in Appendix 4 andcanopy radiation is treated using the complexscheme (Appendix 3).

When nitrogen is distributed in proportion tomean PAR, as assumed in the big-leaf approach,Model I predicts 7.2 g C uptake over the 24-hperiod, and Model II predicts 8.9 g C (chlorophyllto nitrogen ratio is fixed at 2.5 µmol [chlorophyll]mmol–1 [N]) (Fig. 6). This large difference is dueto the empirical smoothing coefficient in Model

I and the more efficient use of nitrogen in ModelII as discussed above. If nitrogen is prescribedin relation to the mean PAR profile as observedby Kull & Niinemets (1998), and the chlorophyllto nitrogen ratio for Model II also varies asobserved, total carbon uptake predicted by ModelI is reduced by 4% and increased for Model IIby 2% (Fig. 6). These are modest changes eventhough leaf nitrogen at the top of the canopyis increased by 72%. Therefore, the big-leafapproach is a reasonable approximation if ModelI is considered to be correct, LAI and totalcanopy nitrogen are known, and the distributionof nitrogen is assumed to match mean PAR.

Ratio of canopy to unshaded leaf net photosynthesis

The big-leaf approach implies a constant ratiobetween canopy net photosynthesis and the rate

Fig. 6 Canopy net photosynthesis at 0.5-h intervals on a July 1 under clear skies at 40°N latitude using ModelsI and II with different canopy nitrogen profiles, but the same total canopy nitrogen. Canopy radiation modelledwith the complex scheme (Appendix 3), and chlorophyll to nitrogen ratio for Model II increasing with canopydepth as observed (Kull & Niinemets, 1998). Horizontal PAR irradiance above the canopy modelled from timeof day as in Appendix 4. All simulations use the same total canopy nitrogen (i.e. 337 mmol [N]) and an LAI of 3.

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in unshaded leaves, given by fPAR/k in equation 1.This ratio is derived from the rates of photo-synthesis calculated for Fig. 6, and is shown in Fig. 7.LAI is 3, and k is 0.78, and therefore the ratioshould be about 1.2 if the big-leaf approach isvalid. When nitrogen is distributed in proportionto mean PAR, the ratio is fairly constant for bothmodels during the daylight period, suggestingthat a mean daytime value may be adequate.When the observed distribution of canopy nitro-gen (and chlorophyll in Model II) is used, thisratio is highly variable in both models, and farhigher than the theoretical value (upper lines inFig. 7). The ratio has a peak at solar noon, anddeclines significantly with irradiance in the morn-ing and afternoon. Observed leaf nitrogen at thetop of the canopy is 42% lower than obtained iftotal canopy nitrogen is distributed in proportionto mean PAR, resulting in top leaves respondingrelatively less to changes in irradiance than leaveslower down, which are operating on steeper partsof their photosynthesis to light response curves.If scaling of canopy photosynthesis is performed

from observed top leaf nitrogen, and a meanPAR extinction coefficient of 0.78 is used, pre-dicted canopy net photosynthesis is ~20% lowerfor both models than predictions using the actualdistribution. Therefore, the big-leaf approachcannot be used to scale canopy photosynthesisfrom the rate at the top of the canopy with eithermodel because canopy nitrogen profiles do notmatch those required. This conclusion was alsoreached by Raulier et al. (1999), who used leaf-level measurements in an Acer saccharum Marsh.canopy to compare the big-leaf approach with amultilayer model.

The ratio between canopy and unshaded netphotosynthesis is investigated further usingsimple Beer’s Law PAR extinction to calculateinstantaneous net photosynthesis canopy profilesat a range of PAR levels with leaf nitrogen eitherscaled with PAR or scaled with relative PAR asobserved (Fig. 8). The ratio is constant and closeto the theoretical value for Model I when nitro-gen is scaled with PAR, as expected. For ModelII, however, the ratio varies significantly up to

Fig. 7 Ratio between total canopy net photosynthesis and the rate of an unshaded leaf for the samesimulations shown in Fig. 6.

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PAR of about 600 µmol m–2 s–1. The ratio varieseven more in both models if the observed nitro-gen profile is used (Fig. 8). Thus, the canopydoes not operate as a big leaf if Model II is usedat low to moderate PAR, even if it is assumedthat nitrogen is distributed in proportion to meanPAR. This is not evident in Fig. 7 where thepredictions are for a sunny day.

Optimal nitrogen profiles

Many measurements of canopy nitrogen profilesshow that nitrogen does not decline as steeply asPAR (e.g. Dang et al., 1997; Kull & Niinemets,1998; Stenberg et al., 1998). A possible explana-tion is provided by using Model II to calculatethe optimal distribution of canopy nitrogen forthe uptake of carbon. This is shown in Fig. 3 fora PAR irradiance at the top of the canopy of1000 µmol m–2 s–1, PAR extinction coefficient of0.78, total canopy nitrogen of 337 mmol [N] m–2

[ground], observed ratio of chlorophyll to nitro-gen with canopy depth, and simple Beer’s LawPAR extinction. Also shown are the profiles ofcanopy nitrogen observed by Kull & Niinemets(1998). The optimum distribution for Model I isgiven by the short dashes, and matches the dis-tribution of mean PAR. The optimal distributionfor Model II is clearly closer to the observations.This is because Model II saturates more stronglywith leaf nitrogen than Model I due to leafinternal shading. The response of Model II tothe interaction of PAR with leaf nitrogen is con-sistent with actual canopy nitrogen distributionsif plants have evolved to use nitrogen efficiently.The comparison with the profile of mean PARprovides indirect evidence in support of Model IIover Model I and helps explain measured nitro-gen profiles. This conclusion was also reachedby Badeck (1995) using an alternative treatmentof within-leaf light heterogeneity. Badeck (1995)pointed out that the optimal canopy nitrogen

Fig. 8 Ratio between total canopy net photosynthesis and the rate of an unshaded leaf for Models I and IIas functions of horizontal PAR above the canopy for different leaf nitrogen distributions. All simulations usethe same total canopy nitrogen (i.e. 337 mmol [N]) and an LAI of 3. Chlorophyll to nitrogen ratio for ModelII increasing with canopy depth as observed. Canopy radiation using Beer’s Law with k = 0.78.

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distribution is sensitive to the photosynthesisfunction used, and that real distributions donot match those implied by Model I. He showedthat treatment of within-leaf light heterogeneitycauses greater saturation of photosynthesiswith increasing incident PAR and leaf nitrogencontent than occurs using Model I, resultingin a reduction in the optimal gradient of canopynitrogen and a better match to observations.These important findings are corroboratedhere.

Predicting nitrogen profiles

The simulations presented here show that usingModel II with observed nitrogen profiles is likelyto provide a significantly more accurate predic-tion of canopy CO2 exchange than Model I withany nitrogen profile. Model I responds unreal-istically to high leaf nitrogen, resulting in opti-mal nitrogen contents of unshaded leaves farhigher than those observed. The requirement forobserved nitrogen distributions makes dynamicprediction of realistic canopy nitrogen profiles adesirable component of a vegetation/ecosystemmodel incorporating Model II.

Pons & Bergkotte (1996) found a strong cor-relation between transpiration rate and leafnitrogen per unit area following changes in lightor CO2 concentration. Cytokinins are transportedin the xylem, retard leaf senescence and stimulatethe synthesis of the photosynthetic apparatus,and are therefore invoked as mediators in leafnitrogen responses to shading (Pons & Bergkotte,1996). Based on these observations, a simplefirst-order model of photosynthetic leaf nitrogendynamics includes nitrogen uptake by leavesas a function of transpiration rate, and break-down of this nitrogen due to ongoing proteinturnover:

(4)

where Nl is photosynthetic leaf nitrogen content,E is transpiration rate, Nc is an available plantnitrogen pool (common to all leaves), t is time,and α and β are constants. Transpiration isgenerally linearly related to photosynthesis, andtherefore this model will reach the optimaldistribution of nitrogen at equilibrium becauseNl is also linearly related to photosynthesis.

Setting dNl/dt to zero gives equilibrium leafnitrogen as a function of transpiration and thesize of the common nitrogen pool, which fallsas leaves assimilate nitrogen. The ratio betweenthe α and β was set to give the optimum topleaf nitrogen content consistent with a rate ofnet photosynthesis of 10 µmol m–2 s–1 and acommon nitrogen pool of 40 mmol [N] m–2

[ground]. Equation 4 was then run to equilibriumwith Model II to calculate net photosynthesisin each layer for different initial common poolsizes and small initial leaf nitrogen contents.An initial common pool size of 381 mmol [N]m–2 [ground] gives a total observed canopynitrogen of 337 mmol [N] m–2 [ground], and theresulting equilibrium profile is shown in (Fig. 3).This prediction is very close to both the obser-vations and that required to optimize Model II atthe canopy level, suggesting that this simplemethod could be used to predict canopy nitro-gen profiles in a dynamic vegetation/ecosystemmodel, and may in fact be close to the actualmechanism used by real plants. It is possiblethat species differences in leaf nitrogen profiles,such as those measured by Kull & Niinemets(1998), may be accounted for by variation inNc, which is controlled by rates of nitrogenuptake, allocation to different plant parts andturnover.

Kull & Kruijt (1999) presented a mechanisticmodel of canopy nitrogen profiles similar to theone suggested here. However, their model is morecomplex and relies on the local carbohydratepool and resistances between common nitrogenand carbohydrate pools and the leaf pools. Theirmodel works because incorporation of nitrogeninto the photosynthetic apparatus is a functionof photosynthesis through the size of the localcarbohydrate pool. In reality it is likely that hor-monal control dominates the passive responsesassumed in the model of Kull & Kruijt (1999),with cytokinins playing a central role (Pons &Bergkotte, 1996).

In addition to this treatment of nitrogendynamics, leaf area could be controlled mech-anistically within the same model structure. Leafsenescence is enhanced in shaded conditions and iscontrolled by nitrogen export (Woledge, 1986; Pons& Pearcy, 1994). Plants appear to shed leaveswhen their nitrogen contents reach some minimumvalue, and this could easily be incorporated into

dNl

dt-------- αENc βNl–=

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a dynamic growth model to predict plant leafarea dynamics.

CONCLUSIONS

The big-leaf approach to modelling canopycarbon fluxes is attractive because it greatly reducescomputing time and data requirements. One rateof photosynthesis is scaled over all leaves in thecanopy using a simple coefficient. However, itis the unique properties of the homogeneousmodel of leaf photosynthesis, Model I, thatresult in this simplification, and a more mech-anistic approach; Model II, which treats the het-erogeneity of light within leaves, does not displaythese properties. Moreover, real canopies do notdisplay the theoretical distribution of canopynitrogen required by the big-leaf approach. Realcanopies therefore never operate as big leaves,even if Model I is assumed to be correct. If ModelII is correct, even theoretical canopies can neveroperate as big-leaves regardless of the distributionof nitrogen because all leaves operate on differentportions of their photosynthesis/light responsecurves. The big-leaf approach can be used, however,if only the total amount of canopy nitrogen and leafarea are known, and Model I is adopted, withnitrogen assumed to be distributed in proportionto mean PAR. However, it is likely that Model IIis a better representation of the response of leafphotosynthesis to light and nitrogen. Calcula-tions that scale canopy photosynthesis fromobserved rates, or the observed nitrogen contentof unshaded leaves, using the big-leaf approachwith either leaf photosynthesis model will signi-ficantly underestimate carbon fluxes.

Accurate modelling of canopy photosynthesisrequires separate calculations for canopy layers,declining nitrogen down through the canopy, andModel II. Separation of sun and shade foliageand changing chlorophyll/nitrogen ratios withcanopy depth improve predictions, but the choiceof leaf photosynthesis model, the decline in leafnitrogen with canopy depth, and the totalamount of canopy nitrogen are more important.The actual pattern of decline in leaf nitrogenwith depth is not critical. Realistic nitrogendistributions can be predicted using a simpleapproach that links leaf nitrogen uptake to tran-spiration rate. Critical evaluation of Model I vs.Model II using high quality data is required.

ACKNOWLEDGMENTS

This work was sponsored by the EU under theEuropean Terrestrial Ecosystem Modelling Activity(ETEMA) contract (No: ENV4-CT95-0052), andby the NASA/Rutgers University cooperativeagreement Research in Regional and GlobalClimate Variability. I would also like to thankFranz Badeck for stimulating discussions andthe comments of two anonymous reviewers.

REFERENCES

Badeck, F.-W. (1995) Intra-leaf gradient of assimila-tion rate and optimal allocation of canopy nitro-gen: a model on the implications of the use ofhomogeneous assimilation functions. AustralianJournal of Plant Physiology, 22, 425–439.

von Caemmerer, S. & Farquhar, G.D. (1981) Somerelationships between the biochemistry of photo-synthesis and the gas exchange of leaves. Planta,153, 376–387.

Collatz, G.J., Ball, J.T., Grivet, C. & Berry, J.A.(1991) Physiological and environmental regulationof stomatal conductance, photosynthesis and tran-spiration: a model that includes a laminar boundarylayer. Agricultural and Forest Meteorology, 54,107–136.

Collatz, G.J., Berry, J.A., Farquhar, G.D. & Pierce, J.(1990) The relationship between the Rubisco reac-tion mechanism and models of photosynthesis.Plant, Cell and Environment, 13, 219–225.

Dang, Q.L., Margolis, H.A., Sy, M., Coyea, M.R.,Collatz, G.J. & Walthall, C.L. (1997) Profiles ofphotosynthetically active radiation, nitrogen andphotosynthetic capacity in the boreal forest:implications for scaling from leaf to canopy. Journalof Geophysical Research, 102, 28,845–28,859.

Farquhar, G.D. (1989) Models of integrated photo-synthesis of cells and leaves. Philosophical Transac-tions of the Royal Society of London B, 323, 357–367.

Farquhar, G.D. & von Caemmerer, S. (1982) Model-ling of photosynthetic response to environmentalconditions. Physiological plant ecology II: waterrelations and carbon assimilation, vol. 12b, (ed. byO.L. Lange, P.S. Nobel, C.B. Osmond and H. Ziegler),pp. 549–587. Springer Verlag, Berlin.

Farquhar, G.D., von Caemmerer, S. & Berry, J.A.(1980) A biochemical model of photosyntheticCO2 assimilation in leaves of C3 species. Planta,149, 78–90.

Field, C. (1983) Allocating leaf nitrogen for themaximisation of carbon gain: leaf age as a controlon the allocation programme. Oecologia, 56, 341–347.

Friend, A.D. (1995) PGEN: an integrated model ofleaf photosynthesis, transpiration, and conductance.Ecological Modelling, 77, 233–255.

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Friend, A.D., Stevens, A.K., Knox, R.G. &Cannell, M.G.R. (1997) A process-based, terrestrialbiosphere model of ecosystem dynamics (Hybridv3.0). Ecological Modelling, 95, 249–287.

Hirose, T. & Werger, M.J.A. (1987) Maximizingdaily canopy photosynthesis with respect to theleaf nitrogen allocation pattern in the canopy.Oecologia, 72, 520–526.

Kull, O. & Jarvis, P.G. (1995) The role of nitrogenin a simple scheme to scale up photosynthesisfrom leaf to canopy. Plant, Cell and Environment,18, 1174–1182.

Kull, O. & Kruijt, B. (1998) Leaf photosyntheticlight response: a mechanistic model for scalingphotosynthesis to leaves and canopies. FunctionalEcology, 12, 767–777.

Kull, O. & Kruijt, B. (1999) Acclimation of photo-synthesis to light: a mechanistic approach. Func-tional Ecology, 13, 24–36.

Kull, O. & Niinemets, Ü. (1998) Distribution of leafphotosynthetic properties in tree canopies: com-parison of species with different shade tolerance.Functional Ecology, 12, 472–479.

Leverenz, J.W. (1988) The effects of illuminationsequence, CO2 concentration, temperature andacclimation on the convexity of the photo-synthetic light response curve. Physiologia Plantarum,74, 332–341.

Pons, T.L. & Bergkotte, M. (1996) Nitrogen allocationin response to partial shading of a plant: possiblemechanisms. Physiologia Plantarum, 98, 571–577.

Pons, T.L. & Pearcy, R.W. (1994) Nitrogen realloca-tion and photosynthetic acclimation in responseto partial shading in soybean plants. PhysiologiaPlantarum, 92, 636–644.

Raulier, F., Bernier, P.Y. & Ung, C.-H. (1999) Canopyphotosynthesis of sugar maple (Acer saccharum):comparing big-leaf and multilayer extrapolationsof leaf-level measurements. Tree Physiology, 19,407–420.

Reynolds, J.F., Chen, J.L., Harley, P.C., Hilbert, D.W.,Dougherty, R.L. & Tenhunen, J.D. (1992)Modeling the effects of elevated CO2 on plants:extrapolating leaf response to a canopy. Agricul-tural and Forest Meteorology, 61, 69–94.

Sellers, P.J., Berry, J.A., Collatz, G.J., Field, C.B. &Hall, F.G. (1992) Canopy reflectance, photo-synthesis, and transpiration. III. A reanalysis usingimproved leaf models and a new canopy integra-tion scheme. Remote Sensing of the Environment,42, 187–216.

Spitters, C.J.T. (1986) Separating the diffuse anddirect component of global radiation and itsimplications for modeling canopy photosynthesis.Part II. Calculation of canopy photosynthesis.Agricultural and Forest Meteorology, 38, 231–242.

Spitters, C.J.T., Toussaint, H.A.J.M. & Goudriaan, J.(1986) Separating the diffuse and direct componentof global radiation and its implications for modelingcanopy photosynthesis. Part I. Components ofincoming radiation. Agricultural and ForestMeteorology, 38, 217–229.

Stenberg, P., Smolander, H., Sprugel, D. &Smolander, S. (1998) Shoot structure, light inter-ception, and distribution of nitrogen in an Abiesamabilis canopy. Tree Physiology, 18, 759–767.

Sykes, M.T., Prentice, I.C., Smith, B., Cramer, W. &Venevsky, S. (2001) An introduction to the Euro-pean Terrestrial Ecosystem Modelling Activity.Global Ecology and Biogeography, 10, 581–593.

Woledge, J. (1986) The effect of age and shade onthe photosynthesis of white clover leaves. Annalsof Botany, 57, 257–262.

Woodward, F.I. & Lomas, M. (2001) Integratingfluxes from heterogeneous vegetation. GlobalEcology and Biogeography, 10, 595–601.

BIOSKETCH

Andrew Friend’s research interests include ecosystem–atmosphere interactions (especially moisture and CO2 fluxes); plant ecophysiology (especially stomatal conductance, phenology, growth processes, and physiological differences between general plant types); soil hydrology and carbon turnover processes; the ecology of African ecosystems (especially savannas of Southern Africa); montane ecosystem ecophysiology; theory of biological evolution and adaptation; the global carbon cycle; Earth system science; climate modelling; and remote sensing and ecophysiological instrumentation. He is a member of the AIBS, AGU, and the American Museum of Natural History in New York.

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APPENDIX 1

Leaf net photosynthesis: the homogeneous approach (Model I)

Model I uses the approach of Farquhar et al.(1980) and Farquhar & von Caemmerer (1982) tocalculate leaf photosynthesis as a function ofphotosynthetically active radiation (PAR) andnitrogen as interpreted by Friend (1995). ModelI is parameterized to match the model of Kull &Kruijt (1998; see Appendix 2).

Maximum electron transport (Jmax; µmol [CO2]m–2 s–1) depends on leaf nitrogen (Kull & Kruijt,1998):

Jmax = anlN (A1.1)

where a is the number of electrons required to fixone molecule of CO2 (i.e. 4), nl is the proportion-ality coefficient between Jmax and N (assumed tobe 0.12 µmol [CO2] (mmol [N])–1 s–1), and N isleaf nitrogen (mmol m–2).

Potential electron transport (J; µmol m–2 s–1)depends on Jmax and PAR (Farquhar & von Cae-mmerer, 1981):

(A1.2)

where Ia is PAR penetrating the leaf (µmol m–2 s–1).RuBP regeneration-limited net photosynthesis

(ARuBP; µmol m–2 s–1) is then given by (Farquhar &von Caemmerer, 1982):

(A1.3)

where Ci is leaf internal partial pressure (assumedto be 24 Pa), Γ* is the photorespiratory com-pensation point (assumed to be 4 Pa), and Vc,max isthe maximum rate of carboxylation (µmol [CO2]m–2 s–1). The second term gives dark respiration(Collatz et al., 1991). Vc,max depends on leaf nitro-gen (Kull & Kruijt, 1998):

(A1.4)

where n2 is the proportionality coefficientbetween Vc,max and N (assumed to be 0.23 µmol[CO2] (mmol [N])–1 s–1).

Carboxylation-limited net photosynthesis(Acarb; µmol [CO2] m–2 s–1) is given by (Farquharet al., 1980):

(A1.5)

where Kc is the Michaelis–Menten constant forCO2 (assumed to be 30 Pa), Oi is the internal leafO2 partial pressure (assumed to be 20.9 kPa), andKo is the Michaelis–Menten constant for O2

(assumed to be 30 kPa).The final prediction of leaf net photosynthesis

(A; µmol [CO2] m–2 s–1) is calculated as thesmoothed minimum of these two rates, which istaken as the smallest root of the following quad-ratic (Collatz et al., 1990):

(A1.6)

where Θ is an empirical co-limitation factor(assumed to be 0.95).

APPENDIX 2

Leaf net photosynthesis: the heterogeneous approach (Model II)

Leaf photosynthesis in Model II uses theapproach of Kull & Kruijt (1998):

(A2.1)

where Ag is gross photosynthesis (µmol [CO2] m–2

s–1), Γ* is the CO2 compensation point in theabsence of dark respiration (assumed to be 4 Pa),Ci is leaf internal CO2 partial pressure (assumedto 24 Pa), msat is light-saturated carboxylation perunit nitrogen (µmol [CO2] (mmol [N])–1 m–2 s–1),Nsat is cumulative leaf nitrogen at which lightharvesting-limited photosynthesis starts (mmol[N] m–2), α is intrinsic quantum efficiency for CO2

uptake (assumed to be 0.08 mol [CO2] (mol[quanta])–1), m1 is equal to Ci /(Ci + 2Γ*), rr is leafreflectance (assumed to be 0), I0 is incidentphotosynthetically active radiation (PAR) (µmol[quanta] m–2 s–1), ka is PAR extinction on chloro-phyll coefficient (assumed to be 0.0055), n3 is thechlorophyll to nitrogen ratio (µmol [chlorophyll](mmol [N])–1), and Np is leaf nitrogen (mmol [N] m–2).

msat is the minimum of the electron transportcapacity-limited and Rubisco-limited rates ofcarboxylation:

JJmaxIa

Ia 2.1Jmax+---------------------------=

ARuBPJ Ci Γ*–( )

4.5Ci 10.5Γ*+-------------------------------------- 0.015Vc max,–=

Vc max, n2N=

AcarbVc max, Ci Γ*–( )

Ci Kc 1 Oi Ko⁄+( )+------------------------------------------------- 0.015Vc max,–=

θA2 A Acarb ARuBP+( )– AcarbARuBP+ 0=

Ag 1 Γ*Ci------–

[msatNsat

+ αm1 1 rr–( )I0 ekan3Nsat–

ekan3Np–

–( )]

=

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(A2.2)

where n1 is the ratio between electron transportcapacity and leaf nitrogen (assumed to be 0.12µmol [CO2] (mmol [N])–1 m–2 s–1) and n2 is theratio between maximum carboxylation rateand leaf nitrogen (assumed to be 0.23 µmol[CO2] (mmol [N])–1 m–2 s–1). m2 is equal to Ci/(Ci + Kc(1 + Oi/Ko) ), where Kc and Ko are theMichaelis–Menten constants of CO2 and O2,respectively (assumed to be 30 Pa and 30 kPa,respectively), and Oi is leaf internal oxygenpartial pressure (assumed to be 20.9 kPa).

Nsat is calculated as the point where the rate ofchange in light harvest-limited photosynthesiswith nitrogen equals the light-saturated rate ofcarboxylation per unit nitrogen (i.e. msat):

(A2.3)

Nsat is limited to the range 0 ≤ Nsat ≤ Np. Darkrespiration (µmol [CO2] m–2 s–1) is assumed toequal 0.015n2Np (after Collatz et al., 1991), and issubtracted from Ag to give leaf net photosynthesis.

APPENDIX 3

Canopy radiation

The simple Beer’s Law approach uses one extinc-tion coefficient for all PAR:

(A3.1)

where Il is PAR irradiance under leaf area index(LAI ) L, I0 is PAR irradiance above the canopy,and k is the PAR extinction coefficient.

In the more complex approach, profiles ofdirect and diffuse PAR are calculated using themethod of Spitters (1986). The irradiance profileresulting from the horizontal diffuse irradiance atthe top of the canopy is given by:

(A3.2)

where Idf is diffuse irradiance below cumulativeLAI L, ρ is the canopy reflection coefficient, I0,df

is horizontal diffuse PAR irradiance above thecanopy, and kdf is the net extinction coefficientfor diffuse irradiance. The corresponding extinc-tion of direct irradiance is given by:

(A3.3)

where Idr is irradiance under cumulative LAI Lresulting from the direct irradiance above thecanopy, I0,dr is horizontal direct PAR irradianceabove the canopy, σ is the leaf scattering coeffi-cient (assumed to be 0.2), and kbl is the extinctioncoefficient for ‘black’ leaves. For overcast condi-tions all PAR is diffuse and the irradiance profileis given by Idf and PAR flux penetrating theleaves at L on a leaf area basis (i.e. Idf,a) is:

(A3.4)

However, when there is also direct beam irradi-ance the situation is more complex because directlight becomes scattered whenever it encounterscanopy objects such as leaves, and this diffusedcomponent must be added to Idf,a. Mean penetra-tion of PAR flux into leaves at L, on a leaf areabasis, resulting from direct irradiance above thecanopy, is given by:

(A3.5)

and, of this, part is not scattered:

(A3.6)

Therefore, PAR irradiance penetrating shadedleaves (Ish,a) is:

(A3.7)

and because sunlit leaves receive both directbeam and diffuse irradiance, PAR irradiancepenetrating these leaves is:

(A3.8)

The fraction of sunlit leaves under L is simplythe fraction of direct beam:

(A3.9)

and therefore if the rate of net photosynthesis ofsunlit leaves is Asl, and shaded leaves is Ash, thenthe mean rate of net photosynthesis for leavesimmediately under L (Al) is:

(A3.10)

msat min m1n1,m2n2( )=

Nsatln msat α 1 rr–( )I0kan3m1[ ]⁄{ }

kan3-----------------------------------------------------------------------–=

Il I0ekL–=

Idf 1 ρ–( )I0 df, ekdf L–

=

Idr 1 ρ–( )I0 dr, e 1 σ–( )1 2⁄ kblL–=

Idf,a kdfIdf=

Idr a, 1 ρ–( )I0 dr, 1 σ–( )1 2⁄ kble1 σ–( )1 2⁄ kblL–

=

Idr dr a, , 1 σ–( )I0 dr, kble1 σ–( )1 2⁄ kblL–

=

Ish a, Idf a, Idr a, Idr dr a, ,–( )+=

Isl a, Ish a, 1 σ–( )kblIo dr,+=

fsl ekblL–

=

Al fslAsl 1 fsl–( )Ash+=

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APPENDIX 4

Incoming radiation

The description of incoming solar radiation istaken from Spitters et al. (1986). Extra-terrestrialsolar irradiance (Scs) is assumed to be 1370 W/m2.Extra-terrestrial solar irradiance at a plane par-allel to the earth’s surface (So) varies with solarelevation (β, degree):

(A4.1)

where td is day since January 1. The sine of solarelevation at hour th (solar time) is:

(A4.2)

where λ is latitude (degree) and δ is solar declina-tion (degree). Declination varies with day of year:

(A4.3)

Total PAR irradiance (µmol m–2 s–1) parallel tothe surface above the canopy is given (assuming80% atmospheric transmittance) by:

PAR = 2.3 × 0.8So (A4.4)

The ratio between diffuse and total solar irradi-ance (R) is assumed to be:

(A4.5)

and the equivalent ratio for PAR is taken as 1.4R.

So Scs 1 0.033 cos 360td 365⁄( )+[ ] sin β=

sin β sin λ sin δ cos λ cos δ cos 15 th 12–( )[ ]+=

sin δ sin 23.45( )cos 360 td 10+( ) 365⁄[ ]–=

R 0.847 1.61 sin β– 1.04 sin2β+=

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