Ee

Sa

b

c

d

a

ARRA

KPSTSBO

1

bsSir2e

U

0d

Agricultural and Forest Meteorology 151 (2011) 1672– 1689

Contents lists available at ScienceDirect

Agricultural and Forest Meteorology

j ourna l ho me pag e: www.elsev ier .com/ locate /agr formet

mpirical and optimal stomatal controls on leaf and ecosystem level CO2 and H2Oxchange rates

amuli Launiainena,b,∗, Gabriel G. Katulc, Pasi Kolarid, Timo Vesalab, Pertti Harid

Finnish Forest Research Institute, Joensuu Research Unit, P.O. Box 66, FI-80101, Joensuu, FinlandDepartment of Physics, P.O. Box 48, FI-00014, University of Helsinki, Helsinki, FinlandNicholas School of the Environment, Duke University, Durham, NC, USADepartment of Forest Ecology, P.O. Box 27, FI-00014, University of Helsinki, Helsinki, Finland

r t i c l e i n f o

rticle history:eceived 29 June 2010eceived in revised form 23 June 2011ccepted 7 July 2011

eywords:hotosynthesistomatal conductanceranspiration ratecots pineall–Berry modelptimality stomatal hypothesis

a b s t r a c t

Linkage between the leaf-level stomatal conductance (gs) response to environmental stimuli and canopy-level mass exchange processes remains an important research problem to be confronted. How variousformulations of gs influence canopy-scale mean scalar concentration and flux profiles of CO2 and H2Owithin the canopy and how to derive ‘effective’ properties of a ‘big-leaf’ that represents the eco-systemmass exchange rates starting from leaf-level parameters were explored. Four widely used formulationsfor leaf-level gs were combined with a leaf-level photosynthetic demand function, a layer-resolving lightattenuation model, and a turbulent closure scheme for scalar fluxes within the canopy air space. The four gs

models were the widely used semi-empirical Ball-Berry approach, and its modification, and two solutionsto the stomatal optimization theory for autonomous leaves. One of the two solutions to the optimizationtheory is based on a linearized CO2-demand function while the other does not invoke such simplification.The four stomatal control models were then parameterized against the same shoot-scale gas exchangedata collected in a Scots pine forest located at the SMEAR II-station in Hyytiälä, Southern Finland. Thepredicted CO2 (Fc) and H2O fluxes (Fe) and mean concentration profiles were compared against multi-level eddy-covariance measurements and mean scalar concentration data within and above the canopy.It was shown that Fc comparisons agreed to within 10% and Fe comparisons to within 25%. The optimalityapproach derived from a linearized photosynthetic demand function predicted the largest CO2 uptake

and transpiration rates when compared to eddy-covariance measurements and the other three models.Moreover, within each gs model, the CO2 fluxes were insensitive to gs model parameter variability whereasthe transpiration rate estimates were notably more affected. Vertical integration of the layer-averagedresults as derived from each gs model was carried out. The sensitivities of the up-scaled bulk canopyconductances were compared against the eddy-covariance derived canopy conductance counterpart. Itwas shown that canopy level gs appear more sensitive to vapor-pressure deficit than shoot-level gs.

. Introduction

The primary pathway by which gas molecules are exchangedetween a leaf and the atmosphere was proposed to be throughtomatal pores more than a century ago (e.g. Blackman, 1895).ince then, the precise regulatory mechanism controlling the open-ng and closure of a stoma continues to be the subject of intense

esearch activity (Cardon et al., 1994; Parkhurst, 1994; Fan et al.,004). Stomatal action is often studied with measurements of gasxchange conducted at a single leaf level. These measurements gen-

∗ Corresponding author at: Finnish Forest Research Institute, Joensuu Researchnit, P.O. Box 66, FI-80101, Joensuu, Finland. Tel.: +358 40 8015323.

E-mail address: samuli.launiainen@metla.fi (S. Launiainen).

168-1923/$ – see front matter © 2011 Elsevier B.V. All rights reserved.oi:10.1016/j.agrformet.2011.07.001

© 2011 Elsevier B.V. All rights reserved.

erally demonstrate that the response of stomatal conductance (gs)to environmental stimuli is well behaved when aggregated overnumerous stoma (Schulze et al., 1972; Cowan and Farquhar, 1977).This finding invites the interpretation of gs in the context of aneffective or ‘big stoma’ with an effective aperture (Mott and Peak,2007). With conifers, a common practice is to use the shoot as thebasic unit to determine the responses of stomatal conductance tomicroclimate (Gower and Norman, 1991), which involves aggrega-tion of processes occurring at the needle surface elements. Whenaggregating to ecosystem level, the response of the canopy-scaleconductance to environmental stimuli is more variable and not as

well behaved as leaf- or shoot-level gs. However, clear determinis-tic trends do emerge also at such a scale (Jarvis and McNaughton,1986) and these are commonly exploited in the so-called big-leafrepresentation. The aim of this work is to explore the connections

Forest

bmt

tcisoeraecrgCBasrremhto2wctvc

eccie

(wpt

td(pmdoi

S. Launiainen et al. / Agricultural and

etween these deterministic trends at the canopy scale and theechanisms governing the well-behaved leaf-level responses of gs

o variations in its local microenvironment.The investigation of gs responses to environmental stimuli at

he leaf scale appeared to have followed one of two broad modellasses. The first describes gs as an empirical response to changesn photosynthetically active radiation, atmospheric vapor pres-ure deficit (D), air temperature, leaf water potential, and the ratef leaf carbon assimilation (Jarvis, 1976; Ball et al., 1987; Collatzt al., 1991; Leuning, 1995). These empirical and semi-empiricalesponses are often ‘assigned’ to a plant functional type therebyllowing their routine use in climate or hydrological models (Sellerst al., 1996; Lai et al., 2000). The second approach is based on the so-alled ‘economics of gas exchange’ and assumes that the regulatoryoles of the stomata are to autonomously maximize their carbonains while minimizing water losses (Givnish and Vermeij, 1976;owan, 1977, 1982; Cowan and Farquhar, 1977; Hari et al., 1986;erninger and Hari, 1993; Mäkelä et al., 1996). There are appealingttributes about this latter approach. First, the optimality hypothe-is provides closed form analytical expressions for gs, assimilationate and intercellular CO2 concentration. Second, these expressionsequire only one physiological parameter – the marginal water usefficiency, �. Third, the derived gs responses to several environ-ental stimuli can be viewed as an outcome of the optimization

ypothesis thereby offering more transferability across plant func-ional types than empirical approaches, at least if the plant isperating its stomata optimally (Hari et al., 2000; Katul et al., 2009,010). Although the behavior of gs to environmental stimuli areell described by both model classes, any general responses at the

anopy scale can be obscured by highly inhomogeneous foliage dis-ribution, complex wind and radiation fields, as well as the irregularariation in the mean atmospheric CO2 concentration within theanopy volume.

The links between the responses of gs to variations in its localnvironment as described by these two model classes and theanopy scale processes (including scalar fluxes and mean scalaroncentration distribution within the canopy) is explored. Threenter-related questions, labeled Q.1–Q.3, frame the scope of thisxploration:

(Q.1) How sensitive are the modeled vertical CO2 and H2Osink/source distributions, scalar fluxes, and mean ambient scalarconcentrations within the canopy to the two commonly usedmodel classes of gs?(Q.2) How well do the up-scaled scalar fluxes compare againstmeasured whole-canopy fluxes if only shoot-scale measurementsare available and, in particular, how do models based on the opti-mality principle contrast against semi-empirical descriptions?(Q.3) What are the effective properties of a ‘big-leaf’ representa-tion of this eco-system and how do they relate to the shoot-scalemeasurements?

Q.2 is a generalization of a recent effort by Schymanski et al.2007, 2008), who explored it in the context of H2O and CO2 fluxesithout considerations to predicted mean scalar concentrationrofiles – used here as additional independent measures to evaluatehe skill of the model.

These three questions are addressed by combining a leaf pho-osynthesis model (Farquhar et al., 1980) and four commonly usedescriptions of gs at the leaf scale, as recently done in Katul et al.2010). The up-scaling from leaf to canopy is carried out by incor-orating a layer-resolving light attenuation model and a closure

odel for turbulent fluxes within the canopy air space, both depen-

ent on the vertical leaf area density distribution. The parametersf the leaf-level photosynthesis and stomatal control models arendependently inferred from leaf- and shoot-scale gas exchange

Meteorology 151 (2011) 1672– 1689 1673

measurements. The turbulent closure scheme and light attenuationmodel are tested against detailed velocity and radiation profilescollected in a Scots pine forest located at the SMEAR II-stationin Southern Finland (Vesala et al., 2000; Hari and Kulmala, 2005;Launiainen et al., 2007). The novelty of the present approach isthat it deconstructs the predictive skills within and between-modelvariability using two scalars (CO2, H2O) and turbulent scalar fluxesas well as mean scalar concentration profiles. The advantages ofincluding the scalar concentration profiles in such comparisons aregenerally attributed to the high vertical resolution that far exceedsthose of eddy-covariance (EC) measurements within canopies.Hence, these measurements do serve as independent checks onhow well the CO2 and H2O source/sink distributions within thecanopy are reproduced. The full scale-independency is retainedby assessing the model predictions against independent EC andmean scalar concentration profile measurements noting that thephysiological parameters are also independently inferred at theshoot-scale.

2. Theory

The up-scaling from leaf to canopy using the mean scalar con-tinuity equation and physiological principles that include the fourgs models are considered next. The abbreviations used in the equa-tions are listed in Table 1.

2.1. The CO2 budget within a canopy layer

For a stationary and planar homogeneous high Peclét numberflow in the absence of subsidence, the one-dimensional mean CO2continuity equation reduces to

∂w′c′(z)∂z

= Sc(z), (1)

where w′c′ is the turbulent flux of CO2, overbar represents timeand planar averaging (Raupach and Shaw, 1982; Finnigan, 2000),primed quantities represent excursions from this space-time aver-age, Sc is the mean uptake or emission rate (sink or source strength)of CO2 from the foliage at layer z, and z is the vertical direction withz = 0 being at the forest floor.

Representing the turbulent flux by first order closure principlesresults in

w′c′(z) = −Kt(z)∂ca(z)∂z

, (2)

where Kt is the turbulent diffusivity for CO2, and ca is the meanatmospheric CO2 concentration. The necessary conditions for theapplication of such closure scheme inside canopies are discussedwithin the context of predictions from second and third - orderformulations in Juang et al. (2008) and Bash et al. (2010).

Assuming Fickian diffusion from the atmosphere into the leafintercellular pores results in an expression for Sc given by (Katulet al., 2000)

Sc(z) = a(z)gs(z)ca(z)(ci(z)ca(z)

− 1), (3)

where a(z) is the leaf area density, gs is the total leaf conductance,which is dominated by stomatal regulation (vis-à-vis aerodynamic

and mesophyll conductance), and ci is the mean intercellular CO2concentration.

Defining R(z) = ci(z)/ca(z) − 1 so that Sc(z) = a(z) [gs(z)R(z) ca(z)] and combining Eqs. (1) and (2) results in a homoge-

1674 S. Launiainen et al. / Agricultural and Forest Meteorology 151 (2011) 1672– 1689

Table 1List of abbreviations (excluding those used only in Appendixes).

Abbreviation Variable Value/units Source/reference

z Height above ground mh Canopy height 15 mLAI Leaf-area index m2 m−2

a(z) Leaf area density m2 m−3 Launiainen et al. (2007)ac Ratio of molecular diffusivity of H2O to CO2 in air 1.6 Campbell and Norman (1998)gs Leaf-scale stomatal conductance mol m−2(leaf) s−1

Gs ‘Big-leaf’ conductance mol m−2(ground) s−1 Eq. (17)Amax Light-saturated assimilation rate �mol m−2(leaf) s−1

Acan Net canopy assimilation �mol m−2(ground) s−1 Eq. (18)Sc(z) The net uptake (Sc < 0) or emission of CO2 from the foliage at layer z �mol m−2 (leaf) s−1

fc Net shoot CO2 flux �mol m−2(leaf) s−1

fe Shoot transpiration mol m−2(leaf) s−1

Fc Canopy net CO2 flux �mol m−2(ground) s−1

Fe Canopy latent heat flux W m−2 (ground)Es Soil evaporation W m−2 (ground)Rsoil Soil respiration �m olm−2(ground) s−1 Soil chambers (Pumpanen et al., 2001)Rstem Trunk and branch respiration �mol m−2(ground) s−1 Kolari et al. (2009), Hölttä and Kolari (2009)R10 Trunk respiration at reference temperature 10 ◦C �mol m−2(trunk) s−1 Kolari et al. (2009)Q10 Temperature sensitivity of trunk respiration – Kolari et al. (2009)WUE Water use efficiency (leaf or canopy scale) mol CO2/mol H2OQp Photosynthetically active radiation �mol m−2(ground) s−1 Measured above the canopy at 22 m heightU(z) Mean horizontal wind speed m s−1

u* Friction velocity at highest grid-point m s−1 Eddy-covariance, 23.3 m heightw′c′(z) Turbulent vertical flux at height zl Mixing length m Appendix AKm(z) Turbulent eddy diffusivity for momentum m2 s−1 Appendix AKt(z) Turbulent eddy diffusivity for scalars m2 s−1 Appendix ASN Turbulent Schmidt number, Km/Kt – Appendix Aca Ambient CO2 mixing ratio ppmci Internal CO2 mixing ratio ppmS Long-term mean ci/ca ratio in linearized optimality model (Eq. (15)) 0.74 Shoot chambers, Appendix Bea Ambient H2O mixing ratio mmol mol−1

ei Leaf internal H2O mixing ratio mmol mol−1

D Ambient vapor pressure deficit, es(Ta) − ea kPa or mol mol−1

DTs ‘True’ vapor pressure deficit, es(Ts) − ea kPa or mol mol−1

RH Ambient relative humidity –Ta Air temperature ◦CTs Leaf temperature ◦CTstem Trunk temperature ◦C� Volumetric humus moisture content m3 m−3

MLM Multi-layer modelBB Ball–Berry model Eq. (11a), Ball et al. (1987)

nC

((

K

mila

tcs

Leu Leuning model

Opti Optimal stomatal control model

OptiL Linearized optimal stomatal control model

eous second-order ordinary differential equation for the meanO2 concentration given by (Siqueira and Katul, 2010)

∂2ca∂z2

+ 1Kt

(∂Kt∂z

∂ca∂z

+ a(z) gs R ca

)= 0 (4)

Invoking first-order closure principles for momentum transfersee Appendix A) and using a mixing length hypothesis leads toHarman and Finnigan, 2007),

t =(

1SN

)l2∣∣∣∣∂U∂z

∣∣∣∣ , (5)

where SN is the turbulent Schmidt number defined as the ratio ofomentum to scalar turbulent diffusivities (and need not be unity

nside canopies), U is the mean velocity, and l is the effective mixingength. These first-order closure results link the mean momentumnd mean scalar concentration budgets via

∂2ca∂z2

+(∂2U/∂z2

∂U/∂z

)∂ca∂z

+(a (z) gsRKt

)ca = 0 (6)

To solve Eq. (6) for ca at a given layer inside the canopy, addi-ional equations describing the vertical variations of gs and R (ori/ca) must be formulated from leaf-gas exchange principles con-idered next.

Eq. (11b), Leuning (1995)Eq. (14), Katul et al. (2009)Eq. (15), Appendix B

2.2. Leaf- gas exchange principles

As noted earlier, mass transfer of CO2 and H2O through leaves(i.e. expressed in m2 of leaf area) occur via Fickian diffusion:

fc = gs(ca − ci)

fe = acgs(ei − ea) ≈ acgsD (7)

where fc and fe are the CO2 and H2O fluxes from the leaf sur-face (i.e. Sc(z) = a(z)fc), ac=1.6 is the relative molecular diffusivityof water vapor with respect to carbon dioxide in air, ei is the inter-cellular and ea the ambient water vapor mixing ratio and D is thevapor pressure deficit representing ei − ea when the leaf is well-coupled to the atmosphere. Furthermore, when leaf respiration issmall with respect to fc and the mesophyll conductance is muchlarger than the stomatal conductance, the biochemical demand forCO2 is described by the Farquhar photosynthesis model given by(Farquhar et al., 1980):

fc = a1(ci − cp), (8)

a2 + ci

where cp is the CO2 compensation point, a1 and a2 are coef-ficients whose values depend on whether the photosyntheticrate is restricted by electron transport or Ribulose bisphosphate

Forest

(tKCt2raltf

f

fpaate

2f

msmbef

g

g

spfiE(aap1b

2

faaa2

S. Launiainen et al. / Agricultural and

RuBP) carboxylase (or Rubisco). Under light-saturated condi-ions, a1 = Vc,max (maximum carboxylation capacity) and a2 =c

(1 + Coa/KO

), where Kc and Ko are the Michaelis constants for

O2 fixation and oxygen inhibition, and Coa is the oxygen concen-ration in air. When light is limiting, a1 = ˛pemQp = �Qp and a2 =cp, where ˛p is the leaf absorptivity of photosynthetically activeadiation (Qp), em is the maximum quantum efficiency of leavesnd � is the apparent quantum yield determined from empiricalight-response curves. Hence, expressed in terms of leaf stoma-al conductance, gs, Eqs. (7) and (8) can be combined to yield theollowing leaf-level expressions:

cica

= 12

+ −a1 − a2gs +√

(a1 + (a2 − ca)gs)2 + 4gs(a1cp + a2cags)

2gsca(9)

and

c = 12

(a1 + (a2 + ca)gs −

√(a1 + gs(a2 − ca))

2 + 4gs(a1cp + a2cags)

). (10)

These expressions do not assume any functional relationshipor gs with its local environment. They show how fc and ci can beredicted from gs if ca and the physiological attributes of the leaf (a1,2, and cp) are known or can be inferred from the local light regimend temperature. Eq. (10) also shows that fc is non-linearly relatedo gs, with a convexity that is necessary for optimality solutions toxist as discussed later.

.3. The closure problem at the leaf scale – semi-empiricalormulations for stomatal conductance

To mathematically ‘close’ Eqs. (9) and (10), an independent for-ulation for gs is needed so that fc and ci can be computed. The

o-called Ball–Berry (Ball et al., 1987) and the Leuning (1995) for-ulations can be used for such ‘closure’ schemes in multi-layer

iosphere-atmosphere models (Baldocchi and Meyers, 1998; Lait al., 2000; Siqueira and Katul, 2002; Juang et al., 2008). Theseormulations are, respectively:

s = m1

ca − cpfc RH + b1; (11a)

and

s = m2

ca − cpfc ILEU(D) + b2; ILEU(D) =

(1 + D

Do

)−1(11b)

where RH is the mean air relative humidity, ILEU reflects theensitivity of the stomata to vapor pressure deficit, Do is a vaporressure deficit constant, and m1, m2, b1, and b2 are empiricaltting parameters (slope and residual conductance, respectively).qs. (11a) and (11b) provide the necessary mathematical closurei.e., three equations and three unknowns: fc ci and gs) if ca, a1, a2,nd cp are known. The formulations in Eqs. (11a) and (11b) assume

priori that gs is linearly related to fc/(ca − cp), a result that is sup-orted from a large number of experiments (e.g. Palmroth et al.,999). However, Eq. (10) already suggests that a linear relationshipetween gs and fc can only be approximate.

.4. An optimality model

An alternative formulation to Eqs. (11a) and (11b) can be derivedrom an optimality principle, originally proposed by Cowan (1977)

nd Givnish and Vermeij (1976) and retained in the work by Cowannd Farquhar (1977), Hari et al. (1986), Berninger and Hari (1993),nd more recently by Konrad et al. (2008), and Katul et al. (2009,010). Assuming each leaf is autonomous, an objective function to

Meteorology 151 (2011) 1672– 1689 1675

be maximized at the leaf scale can be defined as

f (gs) = fc − �fe = 12

(a1 + (a2 + ca)gs

−√

(a1 + gs(a2 − ca))2 + 4gs(a1cp + a2cags)

)− �acgsD

(12)

where Eq. (10) was used to determine fc and Eq. (7) was used todetermine fe. The premise here is that stomata regulate their aper-ture so as to maximize the carbon gain while minimizing water loss(in units of carbon) for a leaf specific cost parameter � (note thatthe notation by Hari et al. (1986) is used here instead of the orig-inal notation proposed by Cowan and Farquhar, 1977). Provided∣∣d�/�∣∣<< ∣∣∂fe/fe∣∣, the gs that maximizes this objective functioncan be determined from ∂f (gs)/∂gs = 0 (i.e., f (gs)) is an extremumguaranteed to be a maximum given the expected convexity of f (gs)with respect to gs in Eq. (12). The optimization problem framed inEq. (12) does not a priori assume any functional response of gs to Dor RH nor does it assume a priori any linear dependence betweengs and fc/(ca − cp) as in the case of Eqs. (11a) and (11b). Instead, itassumes that excessive water losses can induce stomatal closure,which is consistent with the experimental findings of Mott andParkhurst (1991) who, using a ‘helox’ gas medium, demonstratedthat stomata appear to respond to transpiration rates rather thanmeasures of air humidity.

Differentiating Eq. (12) with respect to gs results in (Katul et al.,2010)

∂f (gs)∂gs

= 12

(a2 + ca + a1(−a2 + ca − 2cp) − gs(a2 + ca)

2√(a1 + (a2 − ca)gs)

2 + 4gs(a1cp + a2cags)

−2�acD

). (13)

When setting ∂f (gs)/∂gs = 0 and solving for gs, we obtain:

gs = −a1(a2 − ca + 2cp)

(a2 + ca)2

+

√acD�a2

1(ca−cp)(a2+cp)(a2+ca−2 acD�)2(a2+ca−acD�)

acD�(a2+ca)2(a2+ca−acD�).

(14)

This gs formulation is explicit in relating conductance to caand D. However, unlike the empirical formulations in Eqs. (11a)and (11b), both relating gs to the photosynthetic rate leaving two(Ball–Berry) or three (Leuning formulation) unknown parameters,this formulation relates conductance to the three parameters of thephotosynthesis model and leaving only one unknown physiologicalparameter (�) related to conductance. Moreover, the model formu-lation in Eq. (14) suggests that � cannot be entirely ‘free’ and mustbe bounded to ensure real and positive conductance (Katul et al.,2010).

Linearizing the biochemical demand function in Eq. (8) resultsin a much simpler (and insightful) model for the optimal gs, fc , andsubsequently ci given as (see Appendix B for discussion)

gs = a1

a2 + sca

(−1 +

√ca − cpac�D

)

f = a1(ca − cp)[

1 −√

ac�D]

(15)

c a2 + sca (ca − cp)

cica

= 1 −√ac�D

ca

(ca − cpca

).

1676 S. Launiainen et al. / Agricultural and Forest Meteorology 151 (2011) 1672– 1689

Table 2Photosynthesis model and stomatal control model parameters. All values are given per projected leaf area.

Abbreviation Variable Value Units Source

Vcmax,25 Maximum carboxylationvelocity at 25 ◦C

37 (pine),31 (understory)

�mol m−2(leaf) s−1 Wang et al. (1996), Kolari et al. (2006)

Vcmax Vc max,25C exp[(Ha/RT0)(1−T0/T)]

1+exp[(SvT−Hd )/RT], C =

1 + exp[

(SvT0 − Hd)/RT0

] �mol m−2(leaf) s−1 Wang et al. (1996)

˛p Absorptivity of leaves for PAR 0.8 – Campbell and Norman (1998)� Apparent quantum yield 0.042 (pine)

0.055 (understory)mol(CO2) mol (PAR)−1 Ciras-2 measurements (pine), Kolari et al. (2006)

rd,25 Dark respiration at 25 ◦Crd,25 = 0.015Vcmax,25

�mol m−2(leaf) s−1 Collatz et al. (1992)

rd rd =rd,25

exp(13.157−32648/(RT)exp(13.157−32648/(R×298.15)

�mol m−2(leaf) s−1 Wang et al. (1996)

Kc Michaelis–Menten constant forCO2 fixation

460 ppm Aalto (1998)

Ko Michaelis–Menten constant foroxygen inhibition

330,000 ppm Aalto (1998)

Coa Oxygen consentration in air 210,000 ppm Aalto (1998)cp CO2 compensation point

cp = Coa2×2.6×exp[−0.056×(T−T0)]

ppm Campbell and Norman (1998)

Do Vapor pressure deficit constantin Leuning model (Eq. (11b))

1.3 kPa Shoot chambers

T0 Scaling temperature 25 ◦C Campbell and Norman (1998)m1 Slope in Ball–Berry model (Eq.

(11a))– Shoot chambers

m2 Slope in Leuning model (Eq.(11b))

– Shoot chambers

b1, b2 Residual conductance for CO2

(Eqs. (11a) and (11b))0.001 mol m−2(leaf) s−1 Shoot chambers

�, �LI Marginal water use efficiency(cost parameter) for non-linearand linear optimality model,respectively

mol(CO2) mol (H2O)−1 Shoot chambers

a1 Coefficient of Farquhar-model �mol m−2(leaf) s−1 Eq. (8)

Hmm

ugf(ewet

g

gLt(ic

taapa

a2 Coefficient of Farquhar-model

The gs expression here is similar to the one in Hari et al. (1986).ereafter, we refer to the solution in Eq. (14) as the ‘nonlinearodel’ (denoted by Opti) and the result in Eq. (15) as the ‘linearodel’ (OptiL).Eq. (15) reproduces a number of known stomatal responses doc-

mented across many species. For example, when expressed ass/gs,ref with gs,ref defined as gs for D = 1.0 kPa, Oren et al. (1999)ound that gs/gs,ref = 1 − mo log(D) with mo ∈ [0.5, 0.6]. Katul et al.2009) showed that OptiL is mathematically equivalent to thismpirical formulation across a wide range of D values. Moreover,hen � increases linearly with ca (i.e. � = �oca/co, where ca is a ref-

rence CO2 concentration at which �o is known to have acclimatedo), Eq. (15) can be re-arranged so that

s = 1√ac�o

(fc

ca − cp

)1√D. (16)

This result shows that the assumed linear relationship betweens and fc/(ca − cp) in Eq. (11a) and (11b) for the Ball–Berry or theeuning models is an outcome of OptiL. Moreover, it shows thathere is a one-to-one mapping between OptiL and the Leuning1995) model of Eq. (11b) when ILEU(D) = D−1/2 and b2 ≈ 0 resultingn m2 = (ac�o)

−1/2. In other words we do expect m to be inverselyorrelated with �1/2 in the OptiL framework.

For the calculation of a1, the vertical attenuation of photosyn-hetically active radiation and the estimation of the fraction of sun

nd shaded leaves at various depths within the canopy are needednd discussed in Appendix C. The gas-exchange calculations wereerformed separately for sunlit and shaded leaves to yield weightedverages at each layer. Here after, we refer to this combined radia-

ppm Eq. (8)

tive – turbulence closure – physiological–conductance biosphere–atmosphere transfer scheme as multi-layer model (MLM).

2.5. Effective “big-leaf” representation using MLM

To derive the effective parameters of the ‘big-leaf’ representa-tion of ecosystem level fluxes from MLM, an integrated big-leafconductance (Gs) and ci/ca ratio are computed using the approachof Lai et al. (2000). We assume that D does not vary appreciablywithin the canopy volume when compared to the light environ-ment and thus Gs for CO2 transport is

Gs =h∫0

a(z)gs(z)dz, (17)

where gs is given by the leaf-level solution derived by couplingthe Farquhar photosynthesis model, Fickian diffusion, and one ofthe four stomatal conductance models described in previous sec-tion.

Similarly, the net canopy assimilation (Acan) is

Acan = Gsca(h)

[1 −(cica

)eff

]=

h∫0

Sc(z)dz

h∫

=

0

a(z)gs(z)ca(z)[

1 −(ci(z)ca(z)

)]dz, (18)

Forest Meteorology 151 (2011) 1672– 1689 1677

Wb

(

uQ

3

peLai2lm∼∼bb2

3

tgpfecsicjoovbtmetob

tsbuesdigcam

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

a(z)/amax

, Qp(z)/Q p

(h)

z/h

a(z)Q

p(z)/Q p

(h)

S. Launiainen et al. / Agricultural and

where ca(h) is the ambient CO2 mixing ratio at canopy top (h).ith Gs inferred from Eq. (17), the ci/ca ratio characteristic for the

ig-leaf canopy is

cica

)eff

= 1 −

h∫0

a(z)gs(z)ca(z)(1 − ci(z)/ca(z))dz

Gsca(h). (19)

How Gs and the big-leaf (ci/ca)eff vary with environmental stim-li will be discussed using above formulations (in the context of.3).

. Measurements and model parameterization

The leaf-photosynthetic and stomatal conductance modelarameters required in the MLM were determined using shoot gas-xchange measurements conducted in a Scots pine (Pinus sylvestris.) stand. This stand was sown in 1962 and is currently servings one of the long-term flux-monitoring sites (SMEAR II station)n southern Finland (61◦51′N, 24◦17′E, 181 m above sea level). In006, the main canopy was characterized by the following: total

eaf area index (LAI) ∼6.5 m2 m−2, stand density 1400 ha−1, theean tree height (h) ∼ 15 m and the mean diameter at breast height16 cm. The forest floor vegetation is relatively low (mean height0.2–0.3 m) but dense (total LAI ∼ 1.5 m2 m−2) and is dominatedy dwarf shrubs, mainly lingonberry (Vaccinium vitis-idaea), blue-erry (V. myrtillus), and mosses (Kolari et al., 2006; Kulmala et al.,008).

.1. Leaf-level physiological measurements

Photosynthetic parameters: The parameters a1, a2, and cp wereaken either from the literature or defined using shoot-levelas-exchange measurements made at the site (Table 2). Thehotosynthetic light response of the Scots pine was estimatedrom needle gas-exchange measurements. In July 2006, shoot gasxchange measurements were made at different levels (shadingonditions) in the canopy using a portable photosynthesis mea-uring device (CIRAS-2, PP Systems, Hitchin, UK). Apart from therradiance, the conditions within the measuring cuvette were keptlose to the ambient conditions. The quantum efficiency (� , per pro-ected leaf area) was determined by a non-linear regression fittingf a rectangular hyperbola (Michaelis–Menten-equation) to eachf the measurements and then averaged to obtain a representativealue to be used in computing a1 from �Qp(z). The maximum car-oxylation velocity (Vcmax,25) and its temperature dependency isaken from Wang et al. (1996) (Table 2). The light response of the

ain forest floor species were measured in a previous study (Kolarit al., 2006). We use their biomass-weighted averages as an effec-ive value in the MLM for ground vegetation. They reported valuesf the light-saturated assimilation rate (Amax) instead of Vcmax,25,ut we approximated the latter by 2 × Amax (Leuning, 1997).

Stomatal conductance parameters: The necessary parameters forhe four leaf-scale gs models were determined using continuoushoot gas exchange chamber measurements (Table 2). Two cham-ers, acrylic plastic boxes with a volume of 1 dm3, are located in thepper part of the canopy (∼0.9 h) and open most of the time therebyxposing the shoot in the chamber to ambient conditions. To mea-ure the gas exchange, the chambers were closed 70–100 times aay for 1 min. The details of chamber measurements are described

n Hari et al. (1999) and are not repeated here. The shoot-level

s parameters were estimated as follows: First, all the availablehamber data in dry daytime conditions (Qp > 200 �mol m−2 s−1

nd RH < 90%) were pooled together and the Ball–Berry and Leuningodel parameters (Eqs. (11a) and (11b), respectively) were deter-

Fig. 1. Profiles of normalized leaf area density, a(z), and radiation attenuation (Qp).The circles with horizontal bars represent measured profiles (mean ± Std) based onVesala et al. (2000).

mined using linear least square regression (i.e. regressing gs againstfc/(ca − cp)). Then, the intercepts (b1, b2) (and Do for Leuning-model) were fixed to their long-term mean values and the slopeestimation was done separately for each measurement solving m1and m2 from Eq. (11). We further averaged these to obtain dailyvalues. Similarly, the average cost parameter � was estimated frommeasured shoot water use efficiency (WUE) by regressing WUEagainst D1/2 and solving for � from the slope of the expression

WUE = fcfe

=ca(

1 − cica

)acD

= caacD

√ac�D

ca

(ca − cpca

)

=[√

�

ac

(ca − cp

)]D1/2. (20)

In this computation, ca and cp were taken equal to their meanvalues over the period. Several studies have indicated that � isnearly constant during a day whereas optimality can be achieved atlonger time scales only if � varies as a response to changing envi-ronmental conditions, drought stress in particular (Mäkelä et al.,1996; Schymanski et al., 2008; Manzoni et al., 2011). Therefore,we determined the temporal variability of � as in Katul et al.(2010, their Eqs. (17–20)), which included several approximations:First, in abundant light (Qp > 600 �mol m−2 s−1) the photosynthe-sis is Rubisco-limited and the parameter a1 ∼ Vcmax in the Farquharmodel (Eq. (8)). Second, we assumed that a1 and a2 follow their“generic” temperature dependencies, and cp is only a function oftemperature (Table 2). Using these simplifications, a1 can be solved

for each gas exchange measurement by inverting Eq. (8). Finally,inverting Eq. (14) (or Eq. (15) when determining �LI from the lin-earized optimality model) provides instantaneous �. Also � wasfurther averaged to daily mean values to be used in MLM.

1678 S. Launiainen et al. / Agricultural and Forest Meteorology 151 (2011) 1672– 1689

0 0.005 0.010

0.02

0.04

0.06

0.08

0.1

0.12

g s (m

ol m

−2 s−

1 )

An/(C

a−C p

)*RH

gs=0.001

m1=6.89

R2 = 0.88

a

0 0.005 0.010

0.02

0.04

0.06

0.08

0.1

0.12

g s (m

ol m

−2 s−

1 )

An/(C

a−C p

)*(1+D/Do

)−1

g0=0.001

m2=6.46

Do=1.3

R2=0.84

b

5 10 150

0.005

0.01

0.015

WU

E (

mol

CO

2 (m

ol H

2O)−

1 )

D1/2 (kPa)1/2

λ = 1.9x10−3c

160 180 200 2203

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8

8.5

m1

DOY

d

160 180 200 220

2

3

4

5

6

7

8

m2

DOY

e

160 180 200 2200

1

2

3

4

5

6

7x 10

−3

λ LI

DOY

f

F d e) ant aged

b d of t

3

aaaaamt1atahbloht

ig. 2. Inferred shoot-scale parameter values for Ball–Berry (a and d), Leuning (b anhe dots and circles represent data from the two shoot chambers. Bottom: daily averecause of gas-analyzer malfunction. The vertical dashed line in (d–f) represents en

.2. Boundary conditions for the MLM

The upper boundary conditions (values at the highest layerbove the canopy) are based on time series of half-hourly aver-ged friction velocity u* (m s−1, measured by EC at 23.3 m height),mbient CO2 mixing ratio (ca, ppm), atmospheric pressure (P, kPa),ir temperature (Ta, ◦C), air relative humidity (RH, %) and diffusend direct PAR (Qp, �mol m−2 s−1) determined from measurementsade above the canopy. Instruments and details of the data acquisi-

ion are described elsewhere (Hari and Kulmala, 2005; Vesala et al.,998). During the model runs, Ta and ambient RH (and D) weressumed to be vertically uniform within the canopy each 1/2 hhereby eliminating the need for a full leaf energy balance. Thisssumption is justified for the range of air temperatures consideredere. Moreover, in this geographic region, the leaf satisfies its car-on demand first thereby incurring a concomitant water loss, with

eaf temperature being a ‘by-product’ (via the leaf energy balance)f these carbon uptake priorities. In other words, we are assumingere that gs is not operating to control the temperature status ofhe leaf. Moreover, measured mean air temperature and H2O con-

d linear optimality models (c and f). Top: data from days 152–231 pooled together,slopes (m1, m2) and �LI. Shoot gas-exchange data during days 179–192 was missinghe period considered in the latter analysis.

centration profiles suggest that the vertical variability in D is about0.05 kPa and hence does not exceed 5% in typical conditions withinthe stand.

The woody biomass (Rstem) and soil (Rsoil) respiration, and soilevaporation (Es) provide the necessary boundary conditions on thescalar budgets. The Rsoil was taken from the automated soil chambermeasurements (Pumpanen et al., 2001). The modeled stem respi-ration was computed as (Kolari et al., 2009):

R = R10Q(T−10)/1010 (21)

where R10 = 0.3 �mol m−2 s−1 is the base respiration rate at 10 ◦Cper unit trunk area in July and Q10 the temperature sensitivity (∼2)during typical June–August. The stem temperature lags air temper-ature and was approximated as

dTstem = Ta − Tstem (22)

dt �

where � is a time constant estimated to be 4 h and discussedelsewhere (Kolari et al., 2009). The obtained Rstem per m2 of stemsurface area was scaled to per m2 ground area by multiplying it by

S. Launiainen et al. / Agricultural and Forest Meteorology 151 (2011) 1672– 1689 1679

2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

An (μmolm-2 s-1 )

z/h

a

-4-200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Sc (μmolm-2 s-1 )

z/h

b

0.02 0.04 0.060

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

gs (molm-2 s-1 )

z/h

c

0.65 0.7 0.750

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

c/ci a

z/h

d

-10 -5 0 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Fc (μmolm-2s-1 )

z/h

e

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Fe (Wm-2 )

z/h

f

4 5 6 7

x 10-3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

WUE (mol CO

2/mol H

2O)

z/h

g

366 368 3700

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Ca (ppm)

z/h

h

BB Leu OptiL Opti

8.7 8.8 8.9 90

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

H2O (ppth)

z/h

i

Fig. 3. Within-canopy daytime (08:00–18:00) profiles of (a) assimilation rate (An), (b) CO2 sink/source (Sc), (c) stomatal conductance of CO2 (gs), (d) ratio of leaf internal toa use efF (c), ang

tcHsem

Jt

mbient CO2 (ci/ca), (e) CO2 flux profile (Fc), (f) latent heat flux profile (Fe), (g) wateror reference, the leaf area density a(z) (dashed line) is superimposed in panels (a)–iven. Only periods when the canopy was dry are included.

he total stem and branch surface area (∼0.5 m2 m−2). The verti-al distribution of Rstem was based on measurements reported byölttä and Kolari (2009). The CO2 efflux from the stem roughly

cales with a(z) in the crown peaking where a(z) is at its high-st value. The Rstem below the foliage layer is roughly 1/2 of the

aximum value.The Es was modeled assuming equilibrium evaporation (e.g.

arvis and McNaughton, 1986) driven by radiation load computed athe forest floor. However, when the measured volumetric humus

ficiency (WUE), (h) ambient CO2 mixing ratio (ca) and (i) ambient H2O mixing ratio.d measured canopy scale fluxes (mean ± Std) and concentrations (mean ± S.E.) are

water content (�) was below saturation, Es was reduced linearlywith the decreasing water content.

3.3. Canopy-scale flux and concentration measurements modelevaluation data

The MLM fluxes and corresponding mean scalar concentrationprofiles are compared against fully independent canopy-scale mea-surements made at the site. Continuous EC measurements of CO2

1680 S. Launiainen et al. / Agricultural and Forest Meteorology 151 (2011) 1672– 1689

0 4 8 12 16 20 24

0

2

4

6

f c (μm

olm

-2s-1

)

time

0.97 h

b

0 4 8 12 16 20 24

0

2

4

6

f c (μm

olm

-2s-1

)

time

0.80 h

e

0 4 8 12 16 20 24

0

1

2

3

4

5

f c (μm

olm

-2s-1

)

time

0.60 h

h

0 4 8 12 16 20 24

0

1

2

3

4

5

f c (μm

olm

-2s-1

)

time

0.50 h

k

0 4 8 12 16 20 240

0.04

0.08

0.12g s (

mol

m-2

s-1)

time

0.97 h

a

0 4 8 12 16 20 240

0.04

0.08

0.12

g s mol

m-2

s-1)

time

0.80 h

d

0 4 8 12 16 20 240

0.02

0.04

0.06

g s (m

olm

-2s-1

)

time

0.60 h

g

0 4 8 12 16 20 240

0.02

0.04

0.06

g s (m

olm

-2s-1

)

0.50 h

time

j

BB Leu OptiL Opti

0 4 8 12 16 20 240.6

0.7

0.8

0.9

1

c/c i

a

time

0.97 h

c

0 4 8 12 16 20 240.6

0.7

0.8

0.9

1

c/c i

a

time

0.80 h

f

0 4 8 12 16 20 240.6

0.7

0.8

0.9

1

c/c i

a

time

0.60 h

i

0 4 8 12 16 20 240.6

0.7

0.8

0.9

1

c/c i

a

time

0.50 h

l

Fig. 4. Predicted stomatal conductance of CO2 (gs , left) and leaf net CO2 exchange (fc , middle) and ci/ca (right) at various levels within the canopy (h = 15 m). All values areg

(3atdffa2e1

iven per unit leaf area.

Fc) and latent heat (Fe) fluxes were carried out at 23.3, 11.7 and.0 m (1.55, 0.78 and 0.2 times h, respectively) using closed-pathnalyzers (LI-6262 and LI-7000, Li-Cor Inc., Lincoln, NE, USA) andhe1/2 h average fluxes were calculated according to the stan-ard FluxNet methodology (Aubinet et al., 2000) and correctedor storage below the measurement height to allow comparisonor nighttime conditions. The details of the EC-measurements both

bove and below the canopy can be found in Launiainen et al. (2005,007) and Launiainen (2010) and are not repeated here. The ambi-nt mean concentrations of CO2 and H2O were measured at 33.6.,6.8, 8.4 and 4.2 m heights and are described in Rannik et al. (2004).

3.4. Model calculations

The MLM was run for the period of 1st of June–20th of August,2006. The results were derived as follows: The canopy was firstdivided into 200 horizontal layers and the radiative environmentat each layer was solved. Second, initial guesses for the com-bined assimilation – stomatal conductance – transpiration were

computed separately for sunlit and shaded leaves at each layer,assuming a constant ca and D profile (values set to the measuredones above the canopy for each 1/2 h period). Then, the turbulentclosure scheme was applied and the vertical ca profile following

S. Launiainen et al. / Agricultural and Forest Meteorology 151 (2011) 1672– 1689 1681

−20 −10 0 10

−20

−15

−10

−5

0

5

10

Fc B

B (μ

mol

m−

2 s−1 )

Fc EC (μmolm−2 s−1 )

y= 0.94 x − 0.39 , R2= 0.83 a

−20 −10 0 10

−20

−15

−10

−5

0

5

10

Fc L

eu (μ

mol

m−

2 s−1 )

Fc EC (μmolm−2 s−1 )

y= 0.96 x − 0.47 , R2= 0.84 b

−20 −10 0 10

−20

−15

−10

−5

0

5

10

Fc O

ptiL

(μ

mol

m−

2 s−1 )

Fc EC (μmolm−2 s−1 )

y= 1.03 x − 0.87 , R2= 0.84 c

−20 −10 0 10

−20

−15

−10

−5

0

5

10

Fc O

pti (μ

mol

m−

2 s−1 )

Fc EC (μmolm−2 s−1 )

y= 1.00 x − 0.66 , R2= 0.84 d

F net Ce (13:0c

frcabewn(cBtta

4

aNtetasvtm

ig. 5. Comparison of measured (abscissa) and modeled (ordinate) 1/2 h averagexplained variance (R2) are given for comparison. The squares represent afternoonanopy was dry are included.

rom the sink/source distribution was calculated and then used toefine estimates on the fc , gs and fe iteratively. The iterations wereontinued until the results converged. Convergence was defineds the maximum difference between two successive iterations toe within 0.1% for all the state variables and across all the lay-rs. Within each iteration, the semi-empirical gs models (Eq. (11))ere also solved iteratively while the two optimality models doot require any further iterations given their analytical form (Eqs.14) and (15)), a decisive advantage in such model applications. Foronsistency between gs predictions of the optimality models andall–Berry–Leuning, the dark respiration (rd) was neglected whenhe fc to gs-relationship was iteratively “optimized”; thus, rd wasaken into account only in the sink/source profiles and neglecteds the leaf’s internal CO2 source.

. Results and discussion

Prior to addressing Q.1–Q.3, the environmental conditionsbove the canopy during the experiment period are first discussed.ext, the shoot-scale parameters and their daily variation reflecting

he physiological states as computed from the shoot-scale gas-xchange are presented. With these results, the predictions fromhe four leaf-level stomatal control models are integrated into MLMnd compared against independently measured canopy-level mean

calar concentration and turbulent flux data. Note that the fourersions of the gs models only differ in their canonical responseso humidity or vapor pressure deficit. Hence, differences among

odels in terms of canopy-scale predictions ‘fingerprint’ the role of

O2 exchange (Fc). The linear least-squares regression equation and proportion of0–17:00) values and the dashed line gives 1:1 relationship. Only periods when the

stomatal controls and any concomitant two-way coupling betweenthe leaf and its micro-environment. All model versions signifi-cantly overpredicted the canopy scale gas exchange in August whendrought stress was strongest, which hints that down-regulation ofthe photosynthetic capacity had occurred. Therefore, the analysisof MLM results are restricted to period during which the photosyn-thetic machinery was not significantly mediated by drought (1stJune–31st July).

4.1. Environmental conditions

The growing season of 2006 was very dry in Southern Finland;the cumulative precipitation from June to August was only abouthalf of the typical 250 mm long-term value. Consequently, volumet-ric soil moisture content decreased from 0.4 m3 m−3 in early June to0.1 m3 m−3 at end of July. The stand showed clear signs of droughtstress in late July and during first half of August, manifested inleaf- and canopy scale reductions of photosynthesis, transpirationand also decreased rates of the decomposition of organic matter(Duursma et al., 2008; Kolari et al., 2009; Launiainen, 2010). Theperiod of progressing drought provides a rather unique possibilityto explore the stomatal control theories for a wide range of hydro-climatic conditions occurring in a boreal Pine forest. During theperiod of this study, air temperature varied between 10 and 28 ◦C

and typical diurnal amplitude was ∼8–10 ◦C. The first week of Junewas cooler and Ta remained below 15 ◦C. The diurnal amplitude of Dwas on the order of 1.5 kPa providing sufficient stimuli for stomatalopening and closure.

1682 S. Launiainen et al. / Agricultural and Forest Meteorology 151 (2011) 1672– 1689

0 100 200 300

0

50

100

150

200

250

300

Fe B

B (

Wm

−2 )

Fe EC (Wm−2 )

y= 0.75 x + 12.31 , R2= 0.76 a

0 100 200 300

0

50

100

150

200

250

300

Fe L

eu (

Wm

−2 )

Fe EC (Wm−2 )

y= 0.89 x + 12.15 , R2= 0.78 b

0 100 200 300

0

50

100

150

200

250

300

Fe O

ptiL

(W

m−

2 )

Fe EC (Wm−2 )

y= 1.01 x + 17.08 , R2= 0.77 c

0 100 200 300

0

50

100

150

200

250

300

Fe O

pti (

Wm

−2 )

Fe EC (Wm−2 )

y= 0.90 x + 9.22 , R2= 0.72 d

Fig. 6. Comparisons between measured (abscissa) and modeled (ordinate) 1/2 h latent heat fluxes (Fe) above the canopy. The linear least-squares regression equation andp t aftew

4

vhaedJlaho12

mmhtss(t(fP

roportion of explained variance (R2) are given for comparison. The squares represenhen the canopy was dry are included.

.2. Radiation attenuation and leaf-level parameters

Because within-canopy radiation is needed to infer a1, a briefalidation of the radiation attenuation model used is presentedere. Fig. 1 shows the leaf area density profile and the light regimend demonstrates good agreement between measured and mod-led Qp throughout the canopy. The measurements were conducteduring one clear and one cloudy day (from 10:00 to 16:00) in late

une of 1998 using a multi-point sensor system (20–48 sensors eachevel) described in Vesala et al. (2000). The measured a(z) was notvailable for that 1998 period and in the model data comparisonere, the shape function of a(z) was assumed to be stationary andnly h was adjusted for the recorded changes from 13 (in 1998) to5 m (in 2006). The stand was partly thinned in 2002 (Vesala et al.,005) but in 2006 the leaf area index had recovered to its 1998 level.

The quantum yield estimated from shoot gas-exchangeeasurements was nearly constant (0.042 ± 0.012 mol mol−1,ean ± Std) irrespective of the shading conditions the shoot

ad acclimatized to. The light-saturated assimilation (Amax) athe needle level showed marked difference between the light-hoots (7.6 ± 3.4 �mol m−2 s−1, mean ± Std) and shoots growing inhaded conditions (4.7 ± 2.1 �mol m−2 s−1) in the lower canopythe degree of shading was determined from hemispherical pho-

ographs) but the difference was statistically non-significantp > 0.05) justifying the use of uniform photosynthesis parametersor this Scots pine canopy. The average � matches the values inalmroth and Hari (2001) measured in June. However, they found

rnoon (13:00–17:00) values and the dashed line gives 1:1 relationship. Only periods

higher � (0.054) in July, a similar value to the one (0.057) reportedin Leverenz (1987) for Scots pine.

The stomatal conductance parameters were determined fromlinear regression analysis for all models as described in Section 3.1and the results are shown in Fig. 2. There was distinct temporal(Fig. 2d and e) and shoot-to-shoot variability both in m1 and m2. Asa response to a progressive drying of the soil, � increased and m1and m2 decreased significantly during the last weeks of the exper-iment when � decreased below ∼0.12 m2 m−2 (not shown). Thesimilarity in the temporal courses of � and m2 is expected giventheir unambiguous dependency as earlier discussed (Eq. (16)). Incontrast, there was no significant correlation between m1 andm2 for the same period. The marginal water use efficiency deter-mined from OptiL, �LI, was ∼7% higher than the � obtained fromthe non-linear model (R2 = 0.98), in qualitative agreement withKatul et al. (2010) in a Loblolly pine stand in the SoutheasternU.S.

4.3. Addressing Q.1

To address Q.1, modeled scalar sources, fluxes and mean con-centration profiles are considered in detail followed by a directcomparison between the aggregated and eddy-covariance mea-

sured canopy-level fluxes. In addition, differences between themodels and the measurements are discussed. Fig. 3 shows theensemble-averaged daytime canopy profiles between 8.00 and18.00. The net leaf CO2 exchange (fc) profile (Fig. 3a) shows rapid

S. Launiainen et al. / Agricultural and Forest

0 4 8 12 16 20 24

−16

−12

−8

−4

0

4

Fc (μ

mol

m−

2 s−1 )

time

a

+/−σEC

ECBBLeuOptiLOpti

0 4 8 12 16 20 24

0

50

100

150

200

Fe (

Wm

−2 )

time

b

Fig. 7. Ensemble diurnal variation of canopy-scale fluxes: (a) CO -flux (F ) and (b)le

dWt“ci‘smTattsmcnfatd

2 c

atent heat flux (Fe). The grey dots and shaded area represent the above-canopyddy-covariance data (mean ± Std). Only dry-canopy conditions are included.

ecrease in photosynthesis rates (per leaf area) below z = 0.8 h.ithin the pine canopy, the fc decrease more than by a factor of

wo in all models, while Qp is reduced by a factor of 5 (Fig. 1). Thekink” closest to the ground results from the higher quantum effi-iency (�) of the shade-adapted forest floor vegetation. When fcs multiplied by a(z) and Rstem is taken into account, the verticalcanopy’ source/sink distribution of CO2 can be computed and ishown (Fig. 3b). The Ball–Berry model and the linear optimalityodel predict the lowest and highest fc at all levels, respectively.

his is caused by differences in gs and consequently the ci/ca (Fig. 3cnd d) between the models. The gs of OptiL is significantly higherhroughout the canopy than predicted by other models and thushe modeled ci/ca ratio becomes fairly invariant with height. Thehapes of gs, fc and ci/ca profiles predicted by OptiL and Leuningodels are close to each other but OptiL predicts higher gs, fc andi/ca at each height. The vertical gradient of gs is strongest in theon-linear optimality model, which predicts that gs decreases by a

actor of 3.0 between the uppermost and lowermost shoots whilell other models estimate a reduction factor of ∼1.9–2.1. In addition,he most striking difference is that non-linear optimum model pre-icts the opposite curvature of the ci/ca ratio inside the pine canopy.

Meteorology 151 (2011) 1672– 1689 1683

In contrast, the other formulations predict increased ci/ca whendescending deeper into the canopy volume. Recall that the non-linear model is the only model that does not assume (or produce)a linear dependence between gs and fc . The increase in ci/ca withdepth inside the canopy is however what can be expected basedon carbon isotope data presented in the literature. For instance,Ellsworth (1999, 2000) explored ci/ca ratios in Loblolly pine (Pinustaeda) and found that it varied between 0.66 and 0.75, the lattermeasured in the shaded shoots in the lower canopy.

Despite these differences, the flux profiles (i.e. scalar flux ateach height represents the net effect of the sources and sinks perm2 ground below the height of interest) behave consistently andcompare well with the EC data (Fig. 3e and f). The forest floorcontributes, in an ensemble sense, about 12% of the canopy scalephotosynthesis despite the fact that it represents some 20% of thetotal LAI. This MLM estimate is similar to what was reported byKolari et al. (2006) who concluded that forest floor GPP was 13% ofthe stand GPP during the period from 20th of April to November20th in 2003. All the models overpredicted the ensemble averagedcanopy net CO2 exchange (Fc) by 0.5–1.9 �mol m−2 s−1 (∼6–20% ofthe measured). The crown and trunk-space EC measurements werewell matched (Fig. 3e). During nocturnal conditions, the ensemblesof measured and modeled Fc were rather well balanced (averagedifference <0.2 �mol m−2 s−1) indicating that the respirative com-ponents were adequately described in the model and thus themismatch of daytime fluxes was mainly due to assimilation rates.Because of the assumption of a depth-constant D (set to its valueabove the canopy), the water vapor sources are simply ‘slaved’ tothe carbon fluxes and modeled as the leaf conductance profile mul-tiplied by a(z) and D. Hence, the models predicting highest gs alsoresulted in highest water vapor fluxes, shown here in terms of latentheat flux (fe, Fig. 3f). In relative terms, the combined understorytranspiration and soil evaporation was 17–19% of the stand-scalevalues, slightly less than the fraction (∼23%) measured by the ECsystems. The correspondence between the ensemble averaged EC-data and MLM results was encouraging considering that the modelswere only calibrated with the shoot chamber data independent ofany canopy level mean concentration or EC flux measurements.

The water use efficiency profiles show highest WUE at thetop, ranging from 4.8 × 10−3 to 5.8 × 10−3 mol CO2(mol H2O)−1 fol-lowed by estimates of the order of ∼5 × 10−4 drop within the crownindicating that assimilating a unit of CO2 becomes more uneconom-ical in terms of water loss deeper in the canopy (Fig. 3g). Again,the non-linear optimum model is an exception and predicts strongWUE increase within these canopy layers. At the shaded forest floor,the higher quantum efficiency enhances fc per leaf area and WUEbecomes comparable to its upper canopy value except for Opti.The various stomatal closure schemes only resulted in minor dif-ferences for ambient mean scalar concentrations. The predicted caand H2O profiles (Fig. 3h and g) show rather good agreement withthe measured concentrations considering the small scalar gradientscaused by strong mixing during daytime.

Although the MLM assumes the photosynthesis and stomatalconductance model parameters remain uniform with height withinthe canopy, the importance of stomatal control on regulating fcdiminishes deeper inside the canopy (Fig. 4). The fraction of sunlitfoliage degreases rapidly in the crown and the transition from pri-marily temperature to light limited assimilation occurs accordingly.Thus, in the deeper layers, the diurnal pattern of gs becomes more‘symmetrical’ since any alteration in Qp creates a linear change in fcthereby explaining the lesser WUE in the lower canopy predictedby the majority of the gs-schemes. Because of the compensating

effects of ci, the diurnal patterns of fc are more symmetrical dur-ing the day than those of gs. The fingerprint of stomatal regulationis hardly visible in gs below ∼0.6 h. A notable difference is that thenon-linear optimality model predicts smaller gs during the morning

1684 S. Launiainen et al. / Agricultural and Forest Meteorology 151 (2011) 1672– 1689

0 1 2 30

0.5

1

1.5

2

2.5

3

Gs/G

1 EC

D (kPa)

−mo= 0.73

−mo= 0.59

a De

s(T s

) −ea

0 1 30

0.5

1

1.5

2

2.5

3

g s/g1 le

af

D (kPa)

−mo= 0.48

b

0 1 2 30

0.5

1

1.5

2

2.5

3

Gs/G

1 BB

D (kPa)

−mo= 0.71

c

0 1 2 30

0.5

1

1.5

2

2.5

3

Gs/G

1 Leu

D (kPa)

−mo= 0.61

d

0 1 2 30

0.5

1

1.5

2

2.5

3

Gs/G

1 Opt

iL

D (kPa)

−mo= 0.60

e

0 1 2 30

0.5

1

1.5

2

2.5

3

Gs/G

1 Opt

i

D (kPa)

−mo= 0.54

f

F h lightf tance

b + 1 is

hptp

4

cacflssTp01mtso

ig. 8. Sensitivities of ‘big-leaf’ conductance (Gs) to vapor pressure deficit (D) in higorce equal to D and DTs = es(Ts) − ea , respectively, (a), measured shoot-scale conducy their value at D = 1.0 kPa (G1) and the slope mo of equation Gs/G1 = −m0 × ln(D)

ours, when D is typically low, than the other models. At the shadedortions of the crown, the ci/ca ratio predicted by Opti remains lowhroughout the day (Fig. 4), while in the upper layers, the modelredicts gs, fc and thus ci/ca similar to OptiL.

.4. Addressing Q.2

To address Q.2, ensemble averaged daytime flux profiles wereomputed from MLM and compared with the EC measurementsbove the canopy. This comparison indicated reasonably goodomparison between the measured and modeled whole-canopyuxes. The linear least squares regressions of modeled canopycale CO2 fluxes (Fc) on measured 1/2 h fluxes yield regressionlopes of 0.94–1.03 and intercepts from −0.87 to −0.39 (Fig. 5).he data-model agreement of latent heat flux (Fe) was marginallyoorer and the scatter was larger; regression slopes were between.75 (Ball–Berry) and 1.01 (OptiL) and intercepts between 12 and7 W m−2 (Fig. 6). The ensemble diurnal cycles indicate that the

easured Fc and Fe are more asymmetric than their modeled coun-

erparts (Fig. 7), also evident in Figs. 5 and 6. The measured canopycale fluxes peak around noon and decline thereafter, indicativef stomatal closure as a response to changes in microclimate. The

conditions (Qp > 600 �mol m−2 s−1): Measured Gs inferred by assuming the driving(gs) (b) and modeled Gs . (c)–(f) For direct comparison, the Gs and gs were normalized

shown for reference (Oren et al., 1999).

modeled Fc and Fe using Ball–Berry, Leuning and Opti follow closelythe measured values during morning and evening hours but do notreproduce as strong reduction during the afternoon as recorded bythe EC system. During this time of day, the difference between D andthe ‘true’ driving force (DTs = es(Ts) − ea, where Ts is needle surfacetemperature) are likely to be greatest. However, a separate anal-ysis (not shown) indicated that this finding alone cannot explainthe slower decrease of modeled canopy-scale fluxes, which partlyhas to be attributed to different D sensitivities at canopy and shootscale. Unlike the other models and measurements, the OptiL pre-dicts steeper flux increases during the morning and these predictedfluxes remain higher throughout the afternoon when compared tothe EC measured values. The difference is because the OptiL is basedon linearized fc(ci) response and thus the predictions are sensitiveto the chosen value of the long-term mean ci/ca ratio s (Appendix B).Moreover, s should equal the ci/ca ratio which, on average, balancesthe fc/ci –ratio based on the full (Eq. (8)) and linearized CO2 demandfunction (Appendix B). Here, s was determined on this premise and

was taken equal to 0.74 based on shoot gas-exchange measure-ments. The sensitivity of OptiL predictions of s are greater whenthe leaf operates in light-limited regime and also the differencebetween the linearized and full CO2 demand function increases in

S. Launiainen et al. / Agricultural and Forest Meteorology 151 (2011) 1672– 1689 1685

0 0.5 1 1.5 2 2.50.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

y= −0.18 x + 0.94 , R2= 0.49

y= −0.22 x + 0.93 , R2= 0.53

(ci/c

a) eff B

B

D1/2 (kPa1/2 )

a

0 0.5 1 1.5 2 2.50.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

y= −0.17 x + 0.91 , R2= 0.41

(ci/c

a) eff L

eu

D1/2 (kPa1/2 )

b

0 0.5 1 1.5 2 2.50.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

y= −0.14 x + 0.91 , R2= 0.38

(ci/c

a) eff O

ptiL

D1/2 (kPa1/2 )

c

0 0.5 1 1.5 20.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

y= −0.22 x + 0.93 , R2= 0.53

(ci/c

a) eff O

pti

D1/2 (kPa1/2 )

d

F s. In p

tctpoE

EAEMfiwsaavttRto

ig. 9. Big-leaf (ci/ca)eff against D1/2 in high light (Qp > 600 �mol m−2 s−1) condition

hese conditions. Hypothesizing that ci/ca exceeds s in light limitedonditions during morning and evening hours when D and therebyhe degree of stomatal closure are low, the linear optimality modelrovides higher fc and gs than the full optimality model. Unlike thether models, the Ball-Berry model predictions remain below theC recorded midday maximum.

The MLM assumes space-time averaged canopy properties (e.g.q. (1)) while the EC measurements represent only time averages.ccording to Katul et al. (2004a), the time average inferred from theC system may converge to a space-time average (computed fromLM) in the canopy sub-layer when ensemble-averaging over suf-

ciently long periods are employed to cover natural variations inind direction (e.g. ergodicity in the flow statistics). Hence, the

catter in Figs. 5 and 6 is not surprising given that MLM resultsre for space-time averaged quantities while the EC measurementsre for individual 1/2 h periods. Moreover, the scatter in modeledersus measured canopy Fe is, depending on the magnitude of Fe, ofhe same order or smaller than typical scatter (∼100–50 W m−2) in

he measured energy balance closure (i.e. whether Rn − G = H + Fe;n is net radiation, G ground heat flux and H sensible heat flux) athe site. Thus, it can be argued that some of the scatter in Fig. 6riginates from uncertainties in the short term energy balance clo-

anel (a) the closed gray circles and dashed line show measured shoot-scale data.

sure and measured Fe. However, the biases do not. The measuredensemble-averaged energy-balance was not closed (absolute bias<50 W m−2) in the morning until noon but good ‘closure’ (∼90%)was obtained in the early afternoon, where vapor pressure deficitis at its maximum (not shown).

4.5. Addressing Q.3

Thus far, the comparison between MLM-derived canopy levelfluxes revealed that differences in the leaf-level stomatal controlmodels alone can have as large as 10% influence on canopy scaleCO2 and 25% on water fluxes. Concurrently, the vertical mean con-centration profiles indicated that the contribution of a single layerto the upscaled fluxes differs somewhat between models – spawn-ing a question as to how these differences manifest themselvesif leaf-level theories are used in a big-leaf framework, which isthe basis of Q.3. The effective big-leaf conductance for CO2 (Gs)and (ci/ca)eff are considered next to address this question. Fig. 8

shows the sensitivities of canopy conductance to D in ample light(Qp > 600 �mol m−2 s−1). The ‘measured’ Gs is inverted from EC datain two ways: The first assumes that the canopy is ‘well-coupled’ tothe atmosphere (i.e. effective Ts equals air temperature) so that

1 Forest

Das(tcmmomtrmawcw1

iogoaolOls−tsbw

rfitg

5

fllwrewbmstasdmtptsa

686 S. Launiainen et al. / Agricultural and

approximates the driving force for transpiration. The secondssumes the canopy is not entirely well-coupled to the atmo-phere necessitating an estimate of ‘true’ vapor pressure gradientDTs). Here, an approximation of ‘effective’ Ts was inferred fromhe measured emitted upwelling long-wave radiation above theanopy (not shown). Roughly, the long-wave radiation measure-ents suggest that the difference between skin and air temperatureeasured near the canopy top (Ts − Ta(h)) are <1 ◦C during most

f the daytime conditions. For direct comparison, the Gs was nor-alized by its value at vapor pressure deficit of 1.0 kPa (G1) and

he slope mo of equation Gs/G1 = −m0 × ln(D) + 1 is shown foreference (Oren et al., 1999). Recall that this formulation witho ∈ [0.5 − 0.6] is consistent with predictions from OptiL (i.e. D−1/2)

s discussed in Katul et al. (2009). This logarithmic formulationas selected here because the sensitivity parameter mo can be

ompared across scales (and models) and was shown to be robusthether leaf, tree, or canopy scales were considered (Oren et al.,

999).The canopy conductance Gs scales well with D−1/2, and results

n mo ranging from 0.54 (Opti) to 0.71 (Ball–Berry). The sensitivityf measured Gs is stronger than the modeled, while the shoot-scales appears least sensitive to D(mo = 0.48) (Fig. 8). The sensitivityf Gs to D varies by ∼25% and the optimality models, where no

priori sensitivity to D (or RH) was assumed, match rather wellther formulations and measurements. The (ci/ca)eff scales non-inearly with D in high light (Qp > 600 �mol m−2 s−1), particularly ifpti and OptiL are considered (recall that the leaf-scale ci/ca scale

inearly in D or RH in the Leuning and Ball-Berry models). Fig. 9hows (ci/ca)eff as a function of D1/2. The sensitivity to D varies from0.14 (OptiL) to −0.22 (Ball–Berry, Opti) which closely resembles

he behavior of measured shoot-scale ci/ca. This behavior is con-istent with that observed by Katul et al. (2009) who showed thatased on the optimization hypothesis ci/ca should vary predictablyith D, namely

cica

= 1 −(ac�

ca

)1/2

D1/2. (23)

In further support of the conclusions in Katul et al. (2009), theestricted range in ci/ca often masks such non-linearity, and lineartting to the data is often no worse. In fact, it is this near-linearrend between ci/ca and D that was the basis of the Leuning (1995)s model (Eq. (11b)).

. Conclusions

We showed that the up-scaled canopy-level CO2 and H2Oux and concentration profiles compare well against the canopy

evel eddy-covariance measurements by predicting the variabilityithin 6% (Fc) and 25% (Fe) (Figs. 5 and 6). The correspondence was

emarkable considering that the models rely largely on the param-ter values obtained from the literature and only few parametersere inferred using measurements collected at the shoot-scale

eing hence entirely independent of the canopy scale measure-ents. Thus, the model-data inter-comparison implies that the

implified MLM proposed here was sufficient to capture the essen-ial interplay between the vertical variation of foliage distributionnd its local microclimate and any resulting controls on the meancalar concentration profiles and fluxes. The observed inter-modelifferences between Ball-Berry, Leuning and the two optimality for-ulations were on the order of 10% (Fc) and 25% (Fe). In addition,

he sensitivity of each stomatal control scheme to their respective

arameters was conservative and of similar magnitude in each ofhe models. The median and 25th/75th percentile values of dailytomatal sensitivities provide a legitimate estimate of the vari-bility of these skewed distributions and were therefore used to

Meteorology 151 (2011) 1672– 1689

examine the effect of parameter values on the up-scaled fluxes.Use of leaf-scale gs model parameter values corresponding to the25th/75th percentiles lead to under- or overestimation of Fc by3–5% and Fe by 14–16%. This implies that calibration errors or biasedliterature values of stomatal control parameters can significantlyalter the predicted transpiration rates from a specific ecosystem.However, the photosynthesis remains less affected because of thecompensating effect of ci/ca. The influence of varying stomatal con-trol parameters is comparable to the effect of varying the quantumefficiency and maximum carboxylation capacity. The MLM predic-tions of canopy CO2 uptake and transpiration are affected by ∼5%when � is adjusted by 20% of the original value. For similar change inVcmax,25, the MLM prediction change by ∼10%; hence the up-scaledresults are more sensitive to the exact value of Vcmax,25 rather thanquantum efficiency. Nevertheless, the results of this study call for athorough research of both within and across-species variability ofstomatal control parameters. Moreover, operating in the Farquhar-framework, the marginal water use efficiency (�) required by theoptimality models should be determined independently of pho-tosynthesis parameters to allow transferability across sites. Wealso showed that the canopy scale bulk conductance appears moresensitive to variations in vapor pressure deficit than that of theuppermost shoots, even when ‘true driving force’ (DTs) is used inlieu of D in the bulk canopy conductance computations. Moreover,the effective ‘big-leaf’ and the shoot-scale ci/ca do scale well withD1/2 as predicted by stomatal optimization theories. To summa-rize, the choice of a stomatal control scheme can have as largeas ∼25% impact on up-scaled canopy fluxes even when gs modelparameterization is done against the same shoot-scale dataset.

Acknowledgements

The financial support by the Academy of Finland Centre of Excel-lence program (project no. 1118615) is gratefully acknowledgedalong the EU projects IMECC and ICOS. Petri Keronen, Veijo Hiltunenand Toivo Pohja are acknowledged for designing and maintain-ing the field measurements utilized in this study. We are gratefulto Tiia Grönholm for providing the measured light distributiondata. Katul acknowledges partial support from the National ScienceFoundation (NSF-EAR-10-13339), the U.S. Department of Agricul-ture (2011-67003-30222), the United States Department of Energy(DOE) through the Office of Biological and Environmental Research(BER) Terrestrial Carbon Processes (TCP) program (NICCR grant: DE-FC02-06ER64156), and the Department of Physics at University ofHelsinki for its support during his 3-month summer leave fromDuke University.

Appendix A. Generation of the mean flow and turbulentdiffusivity

In a stationary and planar-homogeneous flow at high Reynoldsnumber and with no subsidence, the mean momentum budgetreduces to:

∂u′w′

∂z= −Cda(z)U2 (A.1)

where Cd is the foliage drag coefficient (here 0.15), usually between0.1 and 0.3 (Katul et al., 2004b). Using first order closure principlesfor simplicity,

u′w′ = −Km ∂U∂z

(A.2)

resulting in a second-order nonlinear ordinary differential equa-tion, given as

Km∂2U

∂z2+ ∂Km∂z

∂U

∂z− Cda(z)U

2 = 0, (A.3)

Forest

w

K

l

wnmiagiftnav

w

wttslS

A

ictta

f

woeiaabig

c

f

ay

i

S. Launiainen et al. / Agricultural and

here the eddy diffusivity for momentum (Km) is

m = l2∣∣∣∣∂U∂z

∣∣∣∣ (A.4)

The mixing length (l) is given as

={kvz, z < (˛′h/kv)˛′h, ˛′h/kv ≤ z < hkv(z − d), z ≥ h

. (A.5)

here the parameter ˛′ = kv(1 − d/h) to ensure continuity (butot smoothness) in the mixing length, d is the zero-plane displace-ent height (here 0.7 h), kv = 0.4 is the von Karman constant and h

s the canopy height. This equation can be solved when two bound-ry conditions on the mean velocity are imposed (zero-stress at theround given the density of the canopy, and measured mean veloc-ty above the canopy). The eddy diffusivity of scalars may differrom Km (the turbulent Schmidt number SN = Km/Kt does not haveo be unity). In near-neutral surface layer, neglecting the rough-ess sub-layer effects (or assuming them similar for both scalarsnd momentum), setting Km = kv(z − d)u∗, where u* is the frictionelocity, and applying the first-order closure gives

′s′ = −Kt ∂s∂z

= −ku∗SN

∂s

∂ ln z, (A.6)

hich allows estimation of SN from the direct flux and concentra-ion gradient measurements made above the forest. According tohe analysis for the site of this study, SN was around unity for thetudied scalars with a median and 25th to 75th percentiles as fol-ows: 0.92 (0.68/1.20) for CO2, 1.01 (0.77/1.33) for H2O. Therefore,N was in this analysis set to 1 for both scalars considered.

ppendix B. A linearized optimality model

Linearizing the biochemical demand function in Eq. (8) resultsn a much simpler (and insightful) model for the optimal gs, whichan be readily implemented as well. The linearization requireshe assumption that the variability of ci affects only marginallyhe denominator of Eq. (8), leading to an approximation a2 + ci =2 + (ci/ca) ca = a2 + sca. As a result,

c = a1(ci − cp)a2 + s ca

, (B.1)

here s is treated as a constant set equal to the long-term meanf ci/ca. It must be stressed here that only in the denominator ofquation (8) s is treated as a model constant, while in Eq. (9) ci/cas allowed to vary (and hence R does vary within the canopy). Notelso that the geometric interpretation of the group of parameters1/(a2 + sca) in Eq. (B.1) is simply the slope of the fc(ci) curve. Com-ining this linearized photosynthesis model with Eq. (7) results

n an expression for f (gs) = fc − �fe = a1(ca−cp)gsa1+gs(a2+sca) − �acgsD, and fc

iven by

i(gs) = a1cp + a2cag + ca2gs

a1 + a2g + gcas, fc(gs) = a1(ca − cp)gs

a1 + gs(a2 + sca). (B.2)

The objective function in Eq. (12) simplifies to

(gs) = fc − �fe = a1(ca − cp)gsa1 + gs(a2 + sca)

− �acgsD, (B.3)

nd upon differentiating this objective function with respect to gs

ields

∂f (g ) a2(ca − cp)

s

∂gs= −ac�D + 1

[a1 + gs(a2 + sca)]2

(B.4)

Note that the convexity of f (gs) versus gs ensures that a max-mum exist and can be determined by setting ∂f (gs)/∂gs = 0 (i.e.

Meteorology 151 (2011) 1672– 1689 1687

maximum carbon gain while minimizing water losses). Solving forgs results in

gs = a1

a2 + sca

(−1 +

√ca − cpac�D

)(B.5)

Apart from the compensation point (cp), this expression is iden-tical to the one derived in Hari et al. (1986). Replacing Eq. (B.5) intoEqs. (B.2) and (B.3) provides closed form of expressions for ci andfc given by

cica

= 1 −√ac�D

ca

(ca − cpca

), . (B.6)

and

fc = a1(ca − cp)a2 + sca

[1 −√

ac�D

(ca − cp)

](B.7)

Hereafter, we refer to the solution in Eq. (B.7) as the ‘linearmodel’ (denoted by OptiL). The derivation of linearized optimal-ity model here is identical to Katul et al. (2010) except that theyassumed ca � cp and thus neglected the latter.

Appendix C. Radiation attenuation within the foliage

A simple horizontal slab model was used to provide the within-canopy light regime needed in gas-exchange calculations. Theattenuation of the direct beam (�b) and diffuse (�d) radiation weremodeled, respectively, as (Campbell and Norman, 1998)

�b( ) = exp[−Kb( ) Lt(z) ˘] (C.1)

�d = exp[−Kd Lt(z) ˘] (C.2)

where Lt(z) is the cumulative plant area density above z and ̆ isa clumping factor which accounts for the shading effect by otherleaves (Stenberg, 1998). The extinction coefficient of direct PAR,defined here as Kb, for an ellipsoidal leaf distribution can be deter-mined as (Campbell and Norman, 1998):

Kb( ) = (x2 + tan2 )0.5

x + 1.774(x + 1.182)−0.733(C.3)

where is the solar zenith angle and x is the leaf angle distri-bution index. Here we assume x to be unity (spherical leaf angledistribution), which is reasonable for the coniferous foliage. Theattenuation of diffuse PAR is independent on direction and we setKd = 0.7 based on the measurements.

At each layer the fraction of sunlit leaves is given by Eq. (C.1)and PAR at sunlit (Qsl) and shaded (Qsh) leaves, respectively, by

Qsl = Qb,0 + Qsc + �dQd,0 (C.4)

Qsh = �dQd,0 + Qsc (C.5)

where Qb,0 and Qd,0 are the beam and diffuse radiation above thecanopy. Qsc is the down-scattered beam radiation

Qsc = exp[−√˛Kb( ) Lt(z) ˘]Qb,0 (C.6)

where ̨ is the PAR absorptivity of the leaf (0.8).

References

Aalto, T., 1998. Carbon dioxide exchange of Scots pine shoots as estimated by abiochemical model and cuvette field measurements. Silva Fennica 32, 321–337.

Aubinet, M., Grelle, A., Ibrom, A., Rannik, Ü., Monchrieff, J., Foken, T., Kowalski, A.S.,Martin, P.H., Bergibier, P., Bernhofer, C., Clement, R., Elbers, J., Granier, A., Grün-wald, T., Morgenstern, K., Pilegaard, K., Rebmann, C., Snicders, W., Valentini,R., Vesala, T., 2000. Estimates of the annual net carbon and water exchange offorests: the EUROFLUX methodology. Adv. Ecol. Res. 30, 113–178.

1 Forest

B

B

B

B

B

C

C

C

C

CC

C

D

E

E

F

F

FG

G

H

H

H

H

H

H

J

J

J

K

K

K

K

K

688 S. Launiainen et al. / Agricultural and

all, J.T., Woodrow, I.E., Berry, J.A., 1987. A model predicting stomatal conductanceand its contribution to the control of photosynthesis under different environ-mental conditions. In: Biggens, J. (Ed.), Progress in Photosyntehsis Research.Martinus Nijhoff, CZ Zoetermeer, Netherlands, pp. 221–224.

aldocchi, D.D., Meyers, T., 1998. On using eco-physiological, micrometeorologialand biogeochemical theory to evaluate carbon dioxide, water vapor and tracegas fluxes over vegetation: a perspective. Agric. For. Meteorol. 90, 1–25.

ash, J., Walker, T.J., Katul, G.G., Jones, M.R., Nimetz, E., Robarge, W., 2010. Estimationof in-canopy ammonia sources and sinks in a fertilized Zea mays field. Environ.Sci. Technol. 44, 1683–1689.

erninger, F., Hari, P., 1993. Optimal regulation of gas-exchange – evidence fromfield data. Ann. Bot. – Lond. 71, 135–140.

lackman, F.F., 1895. Experimental researches on vegetable assimilation and res-piration II. On the paths of gaseous exchange between aerial leaves andatmosphere. Philos. Trans. R. Soc. Lond. B 186, 503–562.

ampbell, G.S., Norman, J.M., 1998. An Introduction to Environmental Biophysics,2nd ed. Springer, p. 281.

ardon, Z.G., Mott, K.A., Berry, J.A., 1994. Dynamics of patchy stomatal movements,and their contribution to steady-state and oscillating stomatal conductance cal-culated with gas-exchange techniques. Plant Cell Environ. 17, 995–1008.

ollatz, G.J., Ball, J.T., Grivet, C., Berry, J.A., 1991. Physiological and environmen-tal regulation of stomatal conductance, photosynthesis and transpiration –a model that includes a laminar boundary-layer. Agric. For. Meteorol. 54,107–136.

ollatz, G.J., Ribas-Carbo, M., Berry, J.A., 1992. Coupled photosynthesis-stomatalconductance model for leavesof C4 plants. Aust. J. Plant Physiol. 19, 519–538.

owan, I.R., 1977. Stomatal behavior and environment. Adv. Bot. Res. 4, 117–228.owan, I.R., 1982. Regulation of water use in relation to carbon gain in higher plants.

In: Lange, O.L., Nobel, P.S., Osmond, C.B., Ziegler, H. (Eds.), Physiological PlantEcology II; Water Relations and Carbon Assimilation. Springer-Verlag, Berlin,pp. 589–613.

owan, I.R., Farquhar, G.D., 1977. Stomatal function in relation to leaf metabolismand environment. Symp. Soc. Exp. Biol. 31, 471–505.

uursma, R.A., Kolari, P., Perämäki, M., Nikinmaa, E., Hari, P., Delzon, S., Loustau,D., Ilvesniemi, H., Pumpanen, J., Mäkelä, A., 2008. Predicting the decline in dailymaximum transpiration rate of two pine stands during drought based on con-stant minimum leaf water potential and plant hydraulic conductivity. Tree Phys.28, 265–276.

llsworth, D.S., 2000. Seasonal CO2 assimilation and stomatal limitations in a Pinustaeda canopy with varying climate. Tree Phys. 20, 425–435.

llsworth, D.S., 1999. CO2 enrichment in a maturing pine forest: are CO2 exchangeand water status in the canopy affected? Plant Cell Environ. 20, 537–557.

an, L.M., Zhao, Z.X., Assmann, S.M., 2004. Guard cells: a dynamic signaling model.Curr. Opin. Plant Biol. 7, 537–554.

arquhar, G.D., Caemmerer, S.V., Berry, J.A., 1980. A biochemical model of photosyn-thetic CO2 assimilation in leaves of C3 species. Planta 149, 78–90.

innigan, J., 2000. Turbulence in plant canopies. Annu. Rev. Fluid Mech. 32, 519–571.ivnish, T.J., Vermeij, G.J., 1976. Sizes and shapes of liane leaves. Am. Nat. 110,

743–778.ower, S.T., Norman, J.M., 1991. Rapid estimation of leaf area index in conifer and

broad-leaf plantations. Ecology 72, 1896–1900.ari, P., Kulmala, M., 2005. Station for measuring ecosystem–atmosphere relations

(SMEAR II). Bor. Environ. Res. 10, 315–322.ari, P., Mäkelä, A., Pohja, T., 2000. Surprising implications of the optimality hypoth-

esis of stomatal regulation gain support in a field test. Aust. J. Plant Physiol. 27,77–80.

ari, P., Mäkelä, A., Berninger, F., Pohja, T., 1999. Field evidence for the optimalityhypothesis of gas exchange in plants. Aust. J. Plant Physiol. 26, 239–244.

ari, P., Mäkelä, A., Kospilahti, E., Holmberg, M., 1986. Optimal control of gasexchange. Tree Physiol. 2, 169–175.

arman, I.N., Finnigan, J.J., 2007. A simple unified theory for flow in the canopy androughness sub-layer. Boundary-Layer Meteorol. 123, 339–363.

ölttä, T., Kolari, P., 2009. Interpretation of stem CO2 efflux measurements. TreePhys. 29, 1447–1456.

arvis, P.G., 1976. The interpretation of the variations in leaf water potential andstomatal conductance found in canopies in the field. Philos. Trans. R. Soc. B 273,593–610.

arvis, P.G., McNaughton, K.G., 1986. Stomatal control of transpiration: scaling upfrom leaf to region. Adv. Ecol. Res. 15, 1–49.

uang, J.Y., Katul, G.G., Siqueira, M.B., Stoy, P.C., McCarthy, H.R., 2008. Investigating ahierarchy of Eulerian closure models for scalar transfer inside forested canopies.Boundary- Layer Meteorol. 128, 1–32.

atul, G.G., Manzoni, S., Palmroth, S., Oren, R., 2010. A stomatal optimization theoryto describe the effects of atmospheric CO2 on leaf photosynthesis and transpi-ration. Ann. Bot. 105, 431–442.

atul, G.G., Palmroth, S., Oren, R., 2009. Leaf stomatal responses to vapor pressuredeficit under current and CO2-enriched atmosphere explained by economics ofgas exchange. Plant Cell Environ. 32, 968–979.

atul, G.G., Cava, D., Poggi, D., Albertson, J.D., Mahrt, L., 2004a. Stationarity, homo-geneity, and ergodicity in canopy turbulence. In: Lee, X., et al. (Eds.), Handbookof micrometeorology, 29. Kluwer Academic Publishers, pp. 161–180.

atul, G.G., Mahrt, L., Poggi, D., Sanz, C., 2004b. One and two equation models forcanopy turbulence. Boundary-Layer Meteorol. 113, 81–109.

atul, G.G., Ellsworth, D., Lai, C.T., 2000. Modeling assimilation and intercellular CO2

from measured conductance: a synthesis of approaches. Plant Cell Environ. 23,1313–1328.

Meteorology 151 (2011) 1672– 1689

Kolari, P., Kulmala, L., Pumpanen, J., Launiainen, S., Ilvesniemi, H., Hari, P., Nikinmaa,E., 2009. CO2 exchange and component CO2 fluxes of a boreal Scots pine forest.Bor. Environ. Res. 14, 761–783.

Kolari, P., Pumpanen, J., Kulmala, L., Ilvesniemi, H., Nikinmaa, E., Grönholm, T., Hari,P., 2006. Forest floor vegetation plays an important role in photosynthetic pro-duction of boreal forests. For. Ecol. Manage. 221, 241–248.

Konrad, W., Roth-Nebelsick, A., Grein, M., 2008. Modeling stomatal density responseto atmospheric CO2. J. Theor. Biol. 253, 638–658.

Kulmala, L., Launiainen, S., Pumpanen, J., Lankreijer, H., Lindroth, A., Hari, P., Vesala,T., 2008. H2O and CO2 fluxes at the floor of a boreal pine forest. Tellus 60B,167–178.

Lai, C.T., Katul, G.G., Oren, R., Ellsworth, D., Schäfer, K., 2000. Modeling CO2 andwater vapor turbulent flux distributions within forest canopy. J. Geophys. Res.105, 26333–26351.

Launiainen, S., 2010. Seasonal and inter-annual variability of energy exchange of aboreal Scots pine forest. Biogeosciences 7, 3921–3940.

Launiainen, S., Vesala, T., Mölder, M., Mammarella, I., Smolander, S., Rannik, Ü., Kolari,P., Hari, P., Lindroth, A., Katul, G.G., 2007. Vertical variability and effect of stabil-ity on turbulence characteristics down to the floor of a pine forest. Tellus 59B,919–936.

Launiainen, S., Rinne, J., Pumpanen, J., Kulmala, L., Kolari, P., Keronen, P., Siivola,E., Pohja, T., Hari, P., Vesala, T., 2005. Eddy covariance measurements of CO2

and sensible and latent heat fluxes during a full year in a boreal pine foresttrunk-space. Bor. Environ. Res. 10, 569–588.

Leuning, R., 1997. Scaling to a common temperature improves the correlationbetween the photosynthesis parameters Jmax and Vcmax. J. Exp. Bot. 48, 345–347.

Leuning, R., 1995. A critical appraisal of a combined stomatal-photosynthesis modelfor C3 plants. Plant Cell Environ. 18, 339–355.

Leverenz, J.W., 1987. Chlorophyll content and the light response curve of shadeadapted conifer needles. Physiol. Plant. 71, 20–29.

Manzoni, S., Vigo, G., Katul, G., Fay, P.A., Polley, H.W., Palmroth, S., Porporato, A.,2011. Optimizing stomatal conductance for maximum carbon gain under waterstress: a meta-analysis across plant functional types and climates. Funct. Ecol.,doi:10.1111/j.1365-2435.2010.01822.x.

Mott, K.A., Parkhurst, D.F., 1991. Stomatal responses to humidity in air and helox.Plant Cell Environ. 14, 509–515.

Mott, K.A., Peak, D., 2007. Stomatal patchiness and task-performing networks. Ann.Bot. 99, 219–226.

Mäkelä, A., Berninger, F., Hari, P., 1996. Optimal control of gas exchange duringdrought: theoretical analysis. Ann. Bot. 77, 461–468.

Oren, R., Sperry, J.S., Katul, G.G:, Pataki, D.E., Ewers, B.E., Phillips, N., Shafer, K.V.R.,1999. Survey and synthesis of intra- and inter-specific variation in stomatalsensitivity to vapor pressure deficit. Plant Cell Environ. 22, 1515–1526.

Palmroth, S., Hari, P., 2001. Evaluation of the importance of acclimation of needlestructure, photosynthesis, and respiration to available photosynthetically activeradiation in a Scots pine canopy. Can. J. For. Res. 31, 1235–1243.

Palmroth, S., Berninger, F., Nikinmaa, E., Lloyd, J., Pulkkinen, P., Hari, P., 1999. Struc-tural adaptation rather than water conservation was observed in Scots pine overa range of wet to dry climates. Oecologia 121, 302–309.

Parkhurst, D.F., 1994. Tansley review 65: diffusion of CO2 and other gases insideleaves. New Phytol. 126, 449–479.

Pumpanen, J., Ilvesniemi, H., Keronen, P., Nissinen, A., Pohja, T., Vesala, T., Hari, P.,2001. An open chamber system for measuring soil surface CO2 efflux: anal-ysis of error sources related to the chamber system. J. Geophys. Res. 106,7985–7992.

Rannik, Ü., Keronen, P., Hari, P., Vesala, T., 2004. Estimation of forest–atmosphereCO2 exchange by eddy-covariance and profile techniques. Agric. For. Meteorol.126, 141–155.

Raupach, M.R., Shaw, R.H., 1982. Averaging procedures for flow within vegetationcanopies. Boundary-Layer Meteorol. 22, 79–90.

Schulze, E.D., Buschbom, U., Evenari, M., Lange, O.L., Kappen, L., 1972. Stomatalresponses to changes in humidity in plants growing in desert. Planta 108,259–270.

Schymanski, S.J., Rodrick, M., Sivapalan, M., Hutley, L., Beringer, J., 2008. Acanopy scale test of the optimal water-use hypothesis. Plant Cell Environ. 31,97–111.

Schymanski, S.J., Rodrick, M., Sivapalan, M., Hutley, L., Beringer, J., 2007. A test ofoptimality approach to modelling canopy properties and CO2 uptake by naturalvegetation. Plant Cell Environ. 30, 1586–1598.

Sellers, P.J., Randall, D.A., Collatz, G.J., Berry, J.A., Field, C.B., Dazlich, D.A., Zhang, C.,Collelo, G.D., Bounoua, L., 1996. A revised land-surface parameterization (SiB2)for atmospheric GCM:s. Part 1: model formulation. J. Climate 9, 676–705.

Siqueira, M.B., Katul, G.G., 2010. An analytical model for the distribution of CO2

sources and sinks, fluxes, and mean concentration within the roughness sub-layer. Boundary-Layer Meteorol. 135, 31–50.

Siqueira, M.B, Katul, G.G., 2002. imating heat sources and fluxes in thermally strati-fied canopy flows using higher-order closure models. Boundary-Layer Meteorol.103, 125–142.

Stenberg, P., 1998. Implications of shoot structure on the rate of photosynthesis atdifferent levels in a coniferous canopy using a model incorporating groupingand penumbra. Funct. Ecol. 12, 82–91.

Vesala, T., Suni, T., Rannik, Ü., Keronen, P., Markkanen, T., Sevanto, S., Grönholm,T., Smolander, S., Kulmala, M., Ilvesniemi, H., Ojansuu, R., Uotila, A., Levula, J.,Mäkelä, A., Pumpanen, J., Kolari, P., Kulmala, L., Altimir, N., Berninger, F., Nikin-maa, E., Hari, P., 2005. The effect of thinning on surface fluxes in a boreal forest.Global Biogeochem. Cycl. 19, GB2001, doi:10.1029/2004GB002316.

Forest

V

V

micrometeorology, aerosol physics and atmospheric chemistry. Trends Heat,

S. Launiainen et al. / Agricultural and

esala, T., Markkanen, T., Palva, L., Siivola, E., Palmroth, S., Hari, P., 2000. Effect ofvariations of PAR on CO2 exchange estimation for Scots pine. Agric. For. Meteorol.

100, 337–347.

esala, T., Haataja, J., Aalto, P., Altimir, N., Buzorius, G., Garam, E., Hämeri, K.,Ilvesniemi, H., Jokinen, V., Keronen, P., Lahti, T., Markkanen, T., Mäkelä, J.M.,Nikinmaa, E., Palmroth, S., Palva, L., Pohja, T., Pumpanen, J., Rannik, Ü., Siivola,E., Ylitalo, H., Hari, P., Kulmala, M., 1998. Long-term field measurements of

Meteorology 151 (2011) 1672– 1689 1689

atmosphere-surface interactions in boreal forest combining forest ecology,

Mass Momentum Transfer 4, 17–35.Wang, K.-Y., Kellomäki, S., Laitinen, K., 1996. Acclimation of photosynthetic param-

eters in Scots pine after three years exposure to elevated temperature and CO2.Agric. For. Meteorol. 82, 195–217.