influences of local feedbacks on land–air exchanges of energy and carbon

Global Change Biology (1998) 4, 477–494

Influences of local feedbacks on land–air exchanges ofenergy and carbon

M . R . R A U PA C HCSIRO Land and Water, GPO Box 1666, Canberra, ACT 2601, Australia

Abstract

Land–air exchanges of energy and matter are modulated by several feedback processesat both small and large space and time scales, with implications for the linked carbon,water and energy cycles. This paper studies the influences of four local feedbacks,occurring at single-patch spatial scales and subdiurnal temporal scales, on the surfaceenergy balance (SEB) and land–air carbon fluxes. The feedbacks are: (i) radiativefeedback, the modulation of available energy through the effect of surface temperature,Ts, on outgoing longwave radiation; (ii) physiological feedback, the interaction betweenvegetation physiology and the SEB through Ts; (iii) aerodynamic feedback, the modula-tion of turbulent heat and moisture transfer by atmospheric stability; and (iv) ConvectiveBoundary Layer (CBL) feedback, the coupling between the daytime evolution of theSEB and CBL through saturation deficit.

It is found that radiative feedback is significant only over very smooth surfaces.Physiological feedback is positive with respect to Ts at moderate to high temperatures,pushing stomata towards complete closure and the SEB towards very low evaporationrates. The SEB is quite sensitive to whether or not such closure occurs. Aerodynamicfeedback, on the other hand, is negative with respect to Ts at these temperatures,reducing Ts and attenuating the tendency for heat-induced stomatal closure. CBLfeedback alone does not dampen the sensitivity of the SEB to physiological feedbackand stomatal closure. However, when aerodynamic feedback is included, this sensitivityis greatly reduced.

Keywords: land–air exchanges, feedbacks, biosphere–atmosphere interactions

Introduction

Land–air exchanges of energy and matter form acrossroads in the climate system: they provide interfacesbetween its meteorological, hydrological and ecologicalcomponents, involving all three of the carbon, water andenergy cycles. In the carbon cycle, land–air carbon fluxesform the principal link between terrestrial carbon poolsand the global carbon cycle. The strength of the linkat any particular location depends on the three majorresources for plant growth: light, water and nutrients. Inthe water cycle, land surface evaporation (the commonterm between terrestrial energy and water balances)determines soil moisture dynamics following rainfall,thus influencing local and continental runoff. This, inturn, influences the carbon cycle through the role ofwater as a plant resource. In the energy cycle, direct energyexchanges between land surfaces and the atmosphereinfluence large-scale atmospheric dynamics. Energy

Correspondence: e-mail [email protected]

© 1998 Blackwell Science Ltd. 477

exchanges also exert strong local effects on the microcli-mates in which ecosystems live, thus influencing long-term ecosystem evolution.

The present aim is to explore just a small part ofthis vast array of interactions: the local physical andphysiological feedbacks which control the fluxes of energyand carbon between vegetation and the atmosphere, andthus the surface energy balance (SEB) of a vegetatedlandscape. The focus is on short time scales, of the orderof a day, and small space scales appropriate for a singlehomogeneous patch. The vegetation and soil moisturestatus of the patch are assumed to be specified andconstant. Naturally, both vegetation and soil moisture aredetermined by processes at larger time and space scalesin which land–air interactions play a significant role, butthose are outside the present scope. Even this restrictedscope calls for an examination of the interactions betweenseveral surface and atmospheric factors: the availableenergy from incident radiation and thermal storage, land–

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air turbulent exchanges, the physiological state of thevegetation, and air temperature and humidity.

We focus upon four main interactions or feedbacks.First, radiative feedback involves the modulation of avail-able energy through the effect of surface temperatureon outgoing longwave radiation. Second, physiologicalfeedback involves the interaction between vegetationphysiology and the SEB, through surface temperatureand humidity and the availability of soil moisture. Third,aerodynamic feedback involves the modulation of turbulenttransfer of heat and moisture by atmospheric stabilityand therefore by the SEB, feeding back upon the SEBitself. Fourth, CBL deficit feedback (henceforth abbreviatedto CBL feedback) occurs because the saturation deficit inthe air above the surface is not independent of the SEBduring the day, but evolves in response to the growth ofthe atmospheric Convective Boundary Layer (CBL),which in turn is determined by the SEB (together withthe temperature and humidity structure of the overlyingtroposphere). This constitutes a regulation mechanism onthe daytime behaviour of both the SEB and the CBL. Itis not the only short-term atmospheric feedback actingon the SEB; others occur through local convective cloudprocesses, particularly the effects of clouds on surfaceradiation. However, these are outside the present scope.

To make an integrated assessment of the relativestrengths of these feedbacks, a ‘‘roughness-wetnessplane’’ is used as a general 2D parameter space on whichto represent the state of a land surface. Hydrometeorolo-gical indicators such as the energy partition ratio (theratio of latent heat flux to available energy flux) can beplotted on this plane to show the behaviour of the land–air system over essentially a complete range of landsurface conditions, for given meteorological conditions.Such plots are used here to quantify the effect of each ofthe above four feedbacks, by progressively incorporatingthem into models of the terrestrial SEB. The approachdiffers from that of McNaughton & Jarvis (1991), whostudied feedbacks using control theory. The aim here isto obtain a quantitative picture of the effects of all fourfeedbacks across the roughness–wetness plane.

The modelling framework is based on well-knownmaterial summarized in Sections 2 and 3, with detailsin Appendices. These sections also discuss radiativefeedback. Physiological and aerodynamic feedbacks areanalysed in Section 4, and CBL feedback in Section5. The following notational conventions are used: fluxdensities at canopy and leaf or element scales are denotedF and f, respectively, with a subscript to indicate thetransferred entity. Following meteorological conventions,conductances and resistances are expressed kinematically(with the dimensions of velocity and inverse velocity,respectively). Conductances are denoted by G and g atcanopy and leaf scales, respectively, and resistances by

© 1998 Blackwell Science Ltd., Global Change Biology, 4, 477–494

R 5 1/G canopy scale, and r 5 1/g at leaf scale.Conductances and resistances are used interchangeably.Carbon fluxes are expressed in kg CO2 m–2 s–1, positiveupwards.

The Surface Energy Balance

The bulk SEB of a vegetated land surface is:

FN – FG 5 FA 5 FH 1 FE, (1)

where FH and FE are the outgoing sensible and latent(evaporative) heat fluxes, FN is the incoming net irradi-ance, FG is the heat flux into thermal storage, and FA isthe available energy flux. The task is to determine allterms in the SEB. The externally specified quantities arethe incoming global shortwave and longwave irradiances(FS↓ and FL↓); the air temperature (T), humidity (Q) andwind speed (U) at a reference level above the surface;and surface properties including vegetation height, leafarea index and soil moisture content.

Each of the four fluxes in (1) depends on the surfacetemperature, Ts. For the net irradiance (FN), this depend-ence is given by

FN 5 (1–as)FS↓ 1 esFL↓ – esσ Ts4, (2)

where as is the surface albedo, es is the surface emissivity,and σ is the Stefan–Boltzmann constant (5.67 3 10–8 Wm–2 K–4). The sensible and latent heat fluxes (FH and FE)depend on Ts through

FH 5 ρcpGaH (Ts – T) (3)

ρλ(Qsat(Ts)–Q)FE 5 ρ λGtE (Qsat(Ts) – Q) 5 , (4)

RaE 1 Rs

where ρ is air density, cp is the isobaric specific heat ofair, λ is the latent heat of vaporization of water, Qsat(T)is the saturation specific humidity at temperature T,GaH 5 1/RaH and GaE 5 1/RaE are the bulk aerodynamicconductances for heat and water vapour transfer; Gs 5

1/Rs is the surface or bulk stomatal conductance; andGtE 5 (RaE 1 Rs)–1 is the total conductance for watervapour transfer (the series sum of RaE and Rs). Finally,the heat flux into thermal storage (FG) depends on Ts in away which can only be determined precisely by solvinga differential equation, the heat equation within thethermal store. This is routinely done in approximate formin the land-surface schemes used in atmospheric models,but for the present discussion, FG is simply ignored.This is a fair approximation for leaves, and for denselyvegetated surfaces without large thermal inertia such asgrasslands and crops.

Equations (1) to (4) are to be solved for the fourunknowns FN, FH, FE and Ts. Two approaches are com-

L O C A L F E E D B A C K S O N L A N D – A I R E X C H A N G E S 479

monly used: either (1) is solved directly for Ts usingnumerical iteration (as in most land-surface schemes foratmospheric models), or all terms in Ts are linearized andTs eliminated algebraically. The latter route leads tothe ‘‘isothermal’’ or ‘‘radiatively coupled’’ form of thePenman-Monteith (PM) equation for FE, and its counter-part for FH:

εRtHFA* 1 ρλDFE 5 ,

εRtH 1 RtE

RtEFA* – ρλDFH 5 (RtH/RaH) . (5)( )ε RtH 1 RtE

Here D 5 Qsat(T)–Q is the saturation deficit of the ambientair; ε 5 (λ/cp)dQsat/dT is the dimensionless slope ofthe saturation specific humidity Qsat(T); and FA* is theradiatively isothermal available energy flux, equal to FA

with the surface temperature Ts replaced by the airtemperature T in the outward longwave term. Thus, FA*is defined by

FA* – FG 5 FN* 5 (1–as)FS↓ 1 esFL↓ – esσ T4, (6)

where superscript asterisks denote radiatively isothermalquantities. The total thermal resistance RtH is defined byRtH

–1 5 GtH 5 GaH 1 Gr, where Gr 5 4esσT3/(ρcp) isthe radiative conductance, a quantity determining thedifference in outward longwave fluxes between bodiesat temperatures Ts and T (Monteith 1973). Since RtH isthe parallel sum of RaH and Rr 5 1/Gr, it is less than RaH.

Equation (5) enables an assessment of radiative feed-back, the effect on Ts of the dependence of the outgoinglongwave irradiance on Ts itself. This can be quantifiedby the ratio p 5 RtH/RaH 5 GaH/(GaH 1 Gr) in (5), a‘‘radiative decoupling factor’’ between 0 and 1 whichdetermines the role of radiative feedback in the SEB.Equation (5) suggests that when p 5 1 (or GtH 5 GaH),radiative feedback has a negligible effect on the SEBfluxes, and that its effect increases as p decreases toward0. Since Gr is about 0.005 m s–1 in practice, radiativefeedback is expected to be significant when GaH is of thisorder or lower. A quantitative comparison can be madeby using the PM equation in both its radiatively-coupledand nonradiatively-coupled forms. The radiatively-coupled form is (5), while the (widely used) nonradiat-ively-coupled form replaces RtH with RaH, thus ignoringradiative feedback. The SEB predictions obtained fromthese two versions of the PM equation are significantlydifferent only in conditions of low GaH, less than around0.01 m s–1 (Raupach 1998). In practice, such low GaH

values are found only over very smooth natural surfaces(momentum roughness length , 0.01 m). For these low-GaH conditions, the nonradiatively-coupled PM formleads to overprediction of FE, FH and surface temperature


by day. However, for conditions where GaH takes moretypical values (around 0.02 m s–1 or greater), radiativefeedback has only a small effect.

All the ensuing SEB calculations are based on exactsolution of (1) to (4) for Ts rather than the PM equation,to eliminate errors caused by the PM linearization. Theseexact solutions include radiative feedback naturally.

Aerodynamic and surface conductances

Whether the SEB is solved using iteration for Ts or thePM equation, critical parameters are the aerodynamicconductance Ga and the surface conductance Gs. Hence-forth GaH and GaE are assumed equal, and denoted byGa 5 GaH 5 GaE. From the point of view of land–airexchanges, Ga and Gs are measures of the ‘‘roughness’’and ‘‘wetness’’ of the system, respectively. These conduct-ances are sometimes taken as specified surface properties,particularly when discussing the implications of the PMequation (e.g. Thom 1975; Raupach & Finnigan 1988).However, they are actually dependent variables deter-mined by many other properties of the land–air system,with consequences which are a central concern of thispaper.

Aerodynamic conductance

It is usual to express Ga using Monin–Obukhov similaritytheory (Garratt 1992):

FH κ2UGa 5 5 , (7)

ρcp (Ts – T) χH(z-d,z0H,L) χM(z-d,z0M,L)

where z0H and z0M are roughness lengths for momentumand heat, respectively, z is the reference height at whichU, T and Q are measured, d is the zero-plane displacementof the surface, κ 5 0.4 is the von Karman constant, L isthe Monin–Obukhov length, and χH and χM are dimen-sionless temperature and velocity profiles. The necessarytheory is summarized in Appendix A. Through L, Ga

depends on the SEB fluxes FH and FE. This makes (1) to(5) implicit and is the source of aerodynamic feedback(see Introduction).

An uncertainty in this formulation for Ga is the assign-ment of the thermal roughness length z0H. In themomentum case, there exist fairly well-defined relation-ships linking z0M to the height and leaf area index of avegetated surface, spanning a large range of roughnessdensities (Raupach 1994). The same is not true of z0H,which is a function of both surface geometry and heatsource distribution. It is often assumed that z0H is aroundz0M/5 (Garratt & Hicks 1973), but this is likely to be trueonly of relatively dense vegetation (leaf area index around1 or greater). For more sparse vegetation, the ratio of z0H

480 M . R . R A U PA C H

to z0M is likely to become very small because the sparseroughness elements absorb momentum very efficientlywhile also decreasing aerodynamic transfer to the ground,which is usually the dominant heat source. In the follow-ing, the simple assumption z0H 5 z0M/5 is used despitethese reservations.

Although the theory underlying (7) is familiar, it isworth recalling that Ga is a strongly variable functionof both roughness length and L. Figure 1 shows thedependence of Ga on both these parameters, as a reminderthat each of them can cause Ga to vary by more than anorder of magnitude in practice.

Surface conductance

The bulk surface conductance (Gs) describes evaporationfrom both vegetation elements (mainly leaves) and soil.Because of averaging complications, its relationship toleaf stomatal conductance (gs) is not straightforward ingeneral. However, when the soil contribution to totalevaporation is small and in the absence of free water onthe vegetation, Gs is satisfactorily approximated (Raupach1995) by a parallel sum over all leaf gs:

Gs 5 ∫Λ

0

gs(ζ) dζ, (8)

where Λ is the leaf area index and ζ is a cumulative leafarea index coordinate. The present paper is restricted toconditions where this is a reasonable approximation.

The role of physiological feedback in land–airexchanges is determined by the behaviour of gs andthence Gs. This will be quantified by using four modelsfor the response of Gs to environmental influences:(a) ‘‘Constant’’: a specified Gs is imposed, independentof atmospheric influences. Hence, there is no short-termphysiological feedback through light, temperature orhumidity. Variation of Gs with soil wetness is introducedby setting Gs 5 GsmaxsW(η), where sW is a soil waterstress function (between 0 and 1) of soil water content η.(b) ‘‘Jarvis’’: A model first proposed by Jarvis (1976) hasbeen used widely to describe both gs and Gs. The approachis to discount a maximum or optimal conductance by amultiplicative series of stress functions s, each between0 and 1. For gs, this gives:

gs 5 gsmaxsPAR(φ) sD(Ds) sT(Ts) sW(η), (9)

where gsmax is a maximum stomatal conductance (Kelliheret al. 1995) and the functions sPAR, sD, sT and sW accountfor physiological stress (departure from optimal function)resulting, respectively, from low absorbed PAR (photo-synthetically active radiation) flux φ; high leaf-surfacesaturation deficit Ds; high or low leaf temperature Ts;and low soil water content η. Appendix B summarizes


Fig. 1 Scalar aerodynamic conductance determined from (7)and Appendix A, as a function of stability (z–d)/L, for threemomentum roughness lengths z0M. Thermal roughness lengthz0H 5 z0M/5. The reference height z–d is 10 m, and the windspeed U at that height is 4 m s–1.

the details of the model as applied in this paper for bothgs and Gs.

(c,d) ‘‘Assimilation-F’’ and ‘‘Assimilation-H’’: Assimila-tion-based models for C3 vegetation which yield leafstomatal conductance (gs) have been developed byLeuning (1990, 1995), Collatz et al. (1991) and others.They provide a description of gs which is more firmlygrounded in physiological mechanisms than the stress-function model, starting from the recognition that gs isclosely coupled with photosynthetic carbon uptake. Threerelated variables are determined together: the net carbonflux or assimilation rate an (positive into a leaf), theintercellular CO2 concentration Ci, and gs itself. Thenecessary three independent equations are: a biochemicalmodel for an as a function of Ci, net leaf PAR flux (φ) andleaf temperature (Ts); a semiempirical model for theresponse of gs to an and leaf-surface saturation deficit Ds;and the definition of gs. Details of the assimilation-based model for gs, as implemented here, are given inAppendix C.

The integration of leaf gs to canopy Gs, using (8), is quitestraightforward for the Jarvis model with appropriatesimplifying assumptions (see Appendix B). However, thisscaling-up requires a little more care in the case of theassimilation model. Appendix D outlines a simple methodfor doing this.

The choice of parameters in assimilation models for gs

and Gs is a difficult matter, as many of the parametersdepend on plant nutritional state and species. The choicesused here are given in Appendix C, Table 1. To examinesome of the possible variation, we use two sets ofpublished coefficients for the temperature dependenciesin the assimilation model: ‘‘assimilation-F’’ denotes theuse of coefficients from Farquhar et al. (1980), and ‘‘assim-ilation-H’’ denotes coefficients from Harley et al. (1992).As shown in Appendix C, the main difference betweenthe models is that model H produces higher values of an


and gsC, peaking at a higher temperature, than doesmodel F. Model H therefore describes vegetation whichis more adapted to high temperatures than model F.

Physiological and aerodynamic feedbacks

Sections 2 and 3 have summarized a simple modellingframework for the terrestrial SEB, which is conventionalapart from Appendix D. In passing, radiative feedbackhas been quantified as small in most circumstances(Section 2). The framework is now applied to assess theeffects of physiological and aerodynamic feedbacks, theshort-term feedbacks taking place at surface-layer orcanopy scale. Let us begin with physiological feedback.

Figure 2 shows the variation of latent heat flux FE ona roughness–wetness plane, calculated using the fourdifferent models for Gs outlined above, without aero-dynamic feedback in Ga (that is, taking (z–d)/L 5 0 in(7)). The assumed leaf area index throughout is 4. Theabscissa is ‘‘roughness’’, for which a convenient measureis the momentum roughness length z0M. This is anexternally specified quantity (or is derivable from otherexternally specified quantities) and is independent of theSEB, unlike Ga. The ordinate is the evaporative energyfraction, defined here as the latent heat flux FE normalizedwith FA* from (6). This normalization is used because FA*is a convenient externally specified energy scale, unlikeFA which depends on the SEB through radiative feedback.‘‘Wetness’’ is varied parametrically, so Fig. 2 shows FE

along a sequence of constant-wetness slices through theroughness–wetness plane.

The ‘‘wetness’’ parameter is the soil moisture stressfunction sW rather than the soil water content η. Thewetness parameter sW is therefore a model-dependentindex describing the effect of soil water stress on Gs,rather than of soil water content itself. This devicesidesteps the difficult question of how soil water contentis related to stomatal conductance. The same four valuesof sW (1.0, 0.6, 0.3 and 0.15) are used in all four modelsfor Gs, though they do not necessarily induce the samestomatal closure in each model. The four sW values aresupplemented with two additional lines, one for a fullywet surface (Rs 5 1/Gs 5 0; long dashed line) and theother for a nearly completely dry surface (sW 5 0.01;short dashed line). These provide reference cases andindicate the directions of trends with wetness.

The calculations were carried out by solving (1) to (4)exactly, using Newton–Raphson iteration for 10 steps.The assumed conditions are typical of a temperate sum-mer day near solar noon: shortwave FS↓ 5 1000 W m–2,longwave FL↓ 5 350 W m–2, T 5 25 °C, relative humidity50% and wind speed 4 m s–1 at a reference level 50 mabove the zero-plane displacement of the surface.

Figure 2 reveals very large differences in the SEB


Fig. 2 Evaporative energy fraction (latent heat flux FE normalizedwith isothermal available energy FA*) plotted against momentumroughness length z0M, using four different models for bulksurface conductance Gs, without aerodynamic feedback (nostability dependence in aerodynamic conductance Ga). Differentcurves represent different values of soil water stress parametersW. Dashed line: wet surface (1/Gs 5 0); dotted line: dry surface(sW 5 0.01). Canopy leaf area index 4; other conditions givenin text.

behaviours predicted by the four schemes. With theconstant-Gs model, FE tends with decreasing roughnesstowards a well-defined limit, closely related to the ther-modynamic equilibrium evaporation rate [pε/(pε 1 1)]Fa* (Raupach 1998), where p is the radiative decouplingfactor (Section 2). However, with all three of the respons-ive Gs models (Jarvis, assimilation-F and assimilation-H),FE cuts off rapidly as roughness decreases, reverting tothe value for a dry surface. This complete stomatal closureoccurs as a result of a positive feedback involving surfacetemperature, identified by Collatz et al. (1991). In each ofthe responsive Gs models, Gs/Ts , 0 above a Ts valuearound 25–30 °C. Therefore, a decrease in Gs leads tolower FE, higher FH (to conserve energy), higher Ts

and yet lower Gs, until complete stomatal closure hasoccurred. The association between stomatal closure and

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Fig. 3 Evaporative energy fraction FE/FA*, surface–airtemperature difference Ts–T, bulk surface conductance Gs andCO2 flux FC (positive upward), plotted against z0M. Gs fromassimilation-H model; Ga calculated without aerodynamicfeedback (no stability dependence). Other conditions as in Fig. 2.

Ts is made clear in Fig. 3, where FE, Ts, Gs and the carbonflux FC are plotted on slices through the roughness-wetness plane for the assimilation-H model. Stomatalclosure occurs as roughness is decreased (with all externalatmospheric properties held constant) because Ts

increases as roughness and Ga decrease. The cutoff issharper for the Jarvis model than for the assimilation-For assimilation-H models (Fig. 2) because the assimilationmodels predict that leaves at different light levels in thecanopy close their stomata at different temperatures,leading to a smeared canopy cutoff after integrationthrough the canopy by the method of Appendix D.

This stomatal closure is a threshold phenomenondepending on the interaction of the SEB and stomatalresponses, so it is not surprising that it is stronglysensitive to both atmospheric and physiological proper-ties of the system. Sensitivities appear in several ways.First, the location of the cutoff points on the roughnessaxis depends strongly on soil water stress sW, as shownin Figs 2 and 3. Secondly, the cutoff points also depend


strongly on atmospheric temperature and humidity, suchthat drier conditions and warmer air temperatures inducestomatal closure at higher sW and z0M-values (detailsare not shown here). Thirdly, the cutoff points dependsignificantly on the model used for Gs: for instance, theassimilation-H model predicts closure over less of theroughness–wetness plane than the assimilation-F model(Fig. 2), which is consistent with the parameterization ofthe assimilation-H model for heat-adapted vegetation.Fourthly, the cutoff points depend on the parameterchoices in any particular model: for instance, in theassimilation-F and assimilation-H models, the choice ofg0 in (C11) (details not shown here).

Despite these sensitivities, a general conclusion can bedrawn: all three responsive stomatal models predictclosure in the assumed near-noon temperate summerconditions, for vegetated surfaces with significant rough-ness lengths. The assumed meteorology represents typicalrather than extreme conditions, and the roughness lengthsat which closure occurs (around 0.03–0.1 m) are common.In other words, Fig. 2 predicts stomatal closure in partsof the roughness–wetness plane which are well populatedby the earth’s vegetation, under commonly occurringrather than extreme weather conditions. Plausible vari-ations in the parameters for the three responsive Gs

models do not alter this general conclusion.Complete stomatal closure on this scale clearly does

not occur. What, then, is wrong with Fig. 2? It is possiblethat the parameterizations in the Gs models are incorrectin this respect, and should be revised to move theshutdown points further to the left in Fig. 2, out ofwell-populated parts of the roughness–wetness plane.However, an additional important factor which helps tosave the situation (for the model rather than for theearth’s vegetation, which functions very well regardless)is aerodynamic feedback.

Figures 4 and 5 repeat the calculations of Figs 2 and 3,this time with Ga properly dependent on stability using(7) and Appendix A. This calculation requires an extraiteration for stability, which was done using the methodoutlined in Raupach & Finnigan (1995, appendix B). Theinclusion of aerodynamic feedback has two main effects.First, it decreases Ts by 10 °C or more at low roughnessand by several °C at high roughness, under the assumedmeteorological conditions (see the Ts panels in Figs 3 and5). Second, and in consequence, it moves the points ofstomatal closure on the roughness axis towards lowerroughness lengths, by about 1.5 orders of magnitude. Inthe temperate summer conditions assumed, stomatalclosure is now predicted by all schemes to occur onlyover extremely smooth surfaces with roughness lengthssubstantially less than 0.01 m. Such vegetation isuncommon.

The reason for this behaviour is that aerodynamic


Fig. 4 As for Fig. 2, but with aerodynamic feedback (stabilitydependence included in Ga).

feedback on Ts is negative in thermally unstable conditions(Ts . T). A small increase in Ts causes an increase in FH,making conditions more unstable and Ga larger (Fig. 1),thereby acting to decrease Ts through more effectiveaerodynamic transfer (3). The role of aerodynamic feed-back alone can be seen by comparing the top (constantGs) panels in Figs 2 and 4, in which physiological feedbackis absent. This shows that aerodynamic feedback inisolation has moderate effects on the SEB. Its effect is toincrease Ga in unstable conditions (Ts . T) and reduce itin stable conditions (Ts , T). Analysis of the radiativelycoupled PM equation (Raupach 1998) shows that the effectof increasing Ga depends on the modified thermodynamicequilibrium evaporation [p2ε/(p2ε 1 1)] Fa*, where p isdefined in Section 2: increasing Ga causes FE to rise whenFE is larger than this modified equilibrium rate, and viceversa. The net result is that aerodynamic feedback shiftswetness isolines away from the thermodynamic equilib-rium isoline (on a (z0M, FE) plot) in unstable conditions,and pushes them towards the equilibrium line in stable(high FE and negative FH) conditions. This behaviour isevident in the constant-Gs panels of Figs 2 and 4.


Fig. 5 As for Fig. 3, but with aerodynamic feedback (stabilitydependence included in Ga).

CBL feedback

We turn now to feedback at a larger scale than the surfacelayer, that of the entire daytime Convective BoundaryLayer (CBL). McNaughton & Spriggs (1986), McNaughton(1989) and Raupach & Finnigan (1995) argued that a‘‘slab’’ model of the CBL provides a rational link betweenthe SEB and larger-scale tropospheric dynamics. The slabmodel represents the CBL as a well-mixed air layer whichprogressively deepens through the day and is boundedabove by a sharp capping inversion. This is usually areasonable idealization of the terrestrial daytime CBL inthe absence of atmospheric disturbances such as frontsor deep cumulus convection, to which the slab model isinapplicable. The simplifications of the CBL slab modelare appropriate for studies of land–air energy and matterexchanges, because the key properties of the CBL whichinfluence the SEB are the mixed-layer scalar (temperature,humidity and CO2) concentrations. These are governedby the surface fluxes and the rate of entrainment ofoverlying (generally dry) air into the CBL, and can bedescribed quite realistically by a slab model with anappropriate entrainment parameterization to describe theCBL growth rate.

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Fig. 6 Time evolution of latent heat fluxFE, CO2 flux FC and surface temperatureTs, calculated for a vegetated surfacewith leaf area index 4, roughness lengthz0M 5 0.1 m, and other conditionsspecified in Table E1. Different curvesrepresent different values of water stressparameter sW. Gs from assimilation-F andassimilation-H models; Ga calculatedwithout aerodynamic feedback (nostability dependence).

The CBL slab model used here is standard, followingMcNaughton & Spriggs (1986) and Raupach & Finnigan(1995), and is summarized in Appendix E. Table 1 givesthe assumed atmospheric conditions, described by pro-files of potential temperature, humidity and CO2 concen-tration in the free troposphere above the CBL. In the CBLslab model, these replace specifications of reference-levelscalar concentrations in the surface layer. They are chosen,as before, to be typical of temperate summer conditions.Many broad features of the behaviour of this model arefamiliar: the mixed layer deepens rapidly after 08.00 has convection breaks through the shallow nocturnal inver-sion and entrainment begins; the onset of entrainment isaccompanied by a rapid increase in latent heat flux; andthe CBL depth ceases growing in mid-afternoon. Thesefeatures are shown, for example, in fig. 2 of Raupach &Finnigan (1995).

Here the focus is on land–air feedbacks. Figures 6 and7 show the predicted daytime evolution of the latent heatand CO2 fluxes and the surface temperature Ts, for z0M 5

0.1 m and for six values of the moisture stress parameterranging from sW 5 1 to sW 5 0. Results are shown forboth the assimilation-F and assimilation-H models forGs (left and right panels, respectively). Aerodynamicfeedback is suppressed in Fig. 6 by setting (z-d)/L 5 0in (7), but is incorporated in Fig. 7 by including thedependence of Ga on (z–d)/L using Appendix A. Withoutaerodynamic feedback (Fig. 6), there is clear evidence ofthe effects of stomatal closure in reducing the magnitudesof FE and FC in the warmest part of the day, when surfacetemperatures exceed roughly 35 °C. This occurs overprogressively more of the day as sW decreases, and ismore pronounced with the assimilation-F model forGs than the assimilation-H model. When aerodynamic


feedback is included (Fig. 7), surface temperatures aresubstantially lower and the effects of stomatal closureare almost entirely removed.

Figures 8 and 9 provide a more integrated picture overthe entire roughness–wetness plane, focusing on day-averaged water and carbon fluxes. These are shown byplotting the evaporative energy fraction (the normalized,day-averaged evaporation (FE/FA*) and the day-integ-rated CO2 uptake IC against the soil water stress para-meter sW, for six roughness lengths z0M. The overbarsrepresent averages through the daytime integrationperiod of the CBL slab model (06.00–18.00 h solar time),and the integrated CO2 uptake IC (in g CO2 m–2) is theintegral of FC (t) (in g CO2 m–2 s–1) through the daytimeintegration period. Thus, these graphs represent transectsthrough the roughness–wetness plane along lines ofconstant roughness. Figure 8 is calculated without aero-dynamic feedback by setting (z–d)/L 5 0 in (7), whileFig. 9 includes aerodynamic feedback.

We first consider the evaporative energy fraction calcu-lated without aerodynamic feedback, the left panels inFig. 8. With constant Gs, this ratio passes through a‘‘focus’’ close to the thermodynamic equilibrium evapor-ative fraction pε/(pε 1 1), at which the evaporativefraction is nearly independent of Ga and therefore of bothwind speed and z0M (Raupach 1998). This occurs at aparticular equilibrium value of Gs, and therefore (underthe assumptions used in the constant-Gs model) at aparticular sW. As sW increases through this focus, evapora-tion increases at a rate which rises with z0M. In contrast,when Gs is evaluated with any of the responsive models(Jarvis, assimilation-F or assimilation-H), this focus disap-pears or at least occurs at a quite different and lowerevaporation, eliminating interpretation in terms of


Fig. 7 As for Fig. 6, but with aerodynamicfeedback (stability dependence includedin Ga).

thermodynamic equilibrium evaporation. For all three ofthese Gs models, the curves of evaporative fraction againstsW spread widely as z0M is varied. The curves also exhibitslope discontinuities induced by the onset of stomatalclosure during the warmest part of the day, as in Figs 6and 7. Broadly similar features are evident in the curvesof IC against sW, shown in the right panels of Fig. 8.

The picture changes substantially when aerodynamicfeedback is introduced (Fig. 9). Its effect is to cluster thewater and carbon fluxes much more tightly under variationof z0M, so that the land–air exchanges of water vapour andCO2 are much weaker functions of z0M when aerodynamicfeedback is included than when it is ignored. Figures 6 and7 provide evidence that the main reason for this is the roleof aerodynamic feedback in moderating surface temper-atures, thereby decreasing the incidence of heat-inducedstomatal closure in the warmest part of the day.

It is important to note that changes in atmosphericconditions which warm the system (for instance, a risein the temperature of the overlying troposphere) willinduce stomatal closure near noon even in the presenceof aerodynamic feedback, thus causing the curves ofwater and carbon fluxes against sW in Fig. 9 to spreadunder variation of z0M as in Fig. 8. Aerodynamic feedbacktherefore promotes the collapse shown in Fig. 9, but doesnot ensure it.

These results imply a substantial modification to thesuggestion of McNaughton & Spriggs (1989) andMcNaughton & Jarvis (1991) that CBL feedback mightcause curves of evaporative fraction against Gs to collapsefairly tightly under variation of weather conditions andaerodynamic conductance (or z0M). That suggestion wasbased on CBL slab model calculations using (in effect) theconstant-Gs model as defined here, without aerodynamic


feedback. The present results modify the suggestion inthree ways. First, under the constant-Gs model with noaerodynamic feedback, there is in fact a significant spreadin curves of evaporative fraction against sW, with variationof z0M (Fig. 8, top left panel). Second, such a collapsedoes indeed occur approximately (though not exactly)under variation of roughness when all four feedbacksare considered, provided that stomatal closure does notoccur. Third, a major factor destroying the collapse is theonset of near-noon stomatal closure in warm conditions,so that the collapse is likely to disappear in heat-stressedconditions.

Summary and conclusions

Four feedback processes acting on land–air energy andcarbon exchanges have been studied by progressivelyadding these processes into a model of the surface energybalance (SEB) and assimilative carbon flux at a vegetatedland surface. Radiative feedback has a modest quantitativeeffect on the SEB, except over very smooth surfaces(momentum roughness length z0M , 0.01 m) where it cansignificantly reduce the latent heat flux FE. Physiologicalfeedback is positive with respect to the surface temper-ature Ts at moderate to high temperatures, pushingstomata towards complete closure and the SEB towards alow evaporation rate. Because this heat-induced stomatalclosure is a threshold phenomenon, it is sensitive to waterstress, atmospheric temperature and humidity, the choiceof model for the surface conductance Gs, and modelparameters (Section 4). Nevertheless, accounting for radi-ative and physiological feedbacks alone would suggestthat this stomatal closure is widespread, occurring intypical summer conditions over much of the roughness–

486 M . R . R A U PA C H

Fig. 8 Day-average evaporative energyfraction, equal to day-average FEnormalized with day-average FA* (leftpanels), and day integrated CO2. flux IC(g m–2, positive upward) (right panels),plotted against soil water stressparameter sW, using four differentmodels for bulk surface conductance Gs,without aerodynamic feedback (nostability dependence in aerodynamicconductance Ga). Different curvesrepresent different momentumroughness lengths z0M.

wetness plane, encompassing vegetation with z0M com-parable with or smaller than that of a field crop, in mostsoil moisture states. Aerodynamic feedback, on the otherhand, acts negatively on Ts in the unstable conditionstypically encountered by day over land, reducing Ts tothe point where stomatal closure does not occur widely,being confined to very smooth surfaces (z0M , 0.01 m),water-stressed conditions, or high temperatures.

Convective boundary layer or CBL feedback providesan additional constraint by regulating the diurnal courseof saturation deficit in the CBL, which becomes a depend-ent rather than an independent variable when the SEBand CBL are considered together. However, the surfacefeedbacks (radiative, physiological and aerodynamic) stillexert major influences on the coupled SEB–CBL system.The cases studied here suggest that, without the moderat-


ing influence of aerodynamic feedback, the system isstrongly subject to perturbation by positive physiologicalfeedback at moderate to high temperatures. Plots ofdaytime-average evaporative fraction against soil mois-ture stress (sW) show a strong dependence on roughness(z0M) when aerodynamic feedback is not included,because of the occurrence of heat-induced stomatal clos-ure near noon over smoother surfaces. However, whenaerodynamic feedback is included, this acute dependenceis largely removed.

The general conclusion is that the physical environmentin which plants grow is controlled quite tightly at localscales by three major feedbacks: physiological, aero-dynamic and CBL. This has implications for modellingand for the natural world itself. In modelling, an implica-tion is that terrestrial microclimates and land–air


Fig. 9 As for Fig. 7, but with aerodynamicfeedback (stability dependence includedin Ga).

exchanges of energy and carbon can only be modelledproperly in general by accounting for all the feedbacks.In particular, because physiological and aerodynamicfeedbacks have opposite influences on Ts at moderate tohigh temperatures (respectively positive and negative),incorporation of one without the other in a model maybe counterproductive. For the natural environment itself,it may be surmised that plants have adapted their physi-ology for optimal performance in a physical environmentcreated substantially by themselves. Responses to temper-ature, for instance, are tuned for optimal performance inthe terrestrial microclimate created by a vegetated surface,which includes the negative feedbacks on Ts induced byaerodynamic and CBL processes.

These arguments have been guided by illustrationsbased on particular models with their own parameter


choices. Many details of modelled system behaviourcertainly depend on these choices: for instance, the onsetof (modelled) stomatal closure (see Section 4). However,these sensitivities do not affect the general conclusionsof this work about the roles and magnitudes of thefeedback processes, and the part played by aerodynamicfeedback in reducing the tendency for heat-inducedstomatal closure. I have endeavoured to guard againstmodel-dependent conclusions in two ways: by surveyingthe behaviour of the system across the complete rough-ness-wetness plane, and by the use of several differentmodels for the environmental response of the surfaceconductance Gs.

Finally, it is noted that the focus here has been onmoderate to high temperatures. In cold conditions, thesituation reverses: aerodynamic feedback acts positively

488 M . R . R A U PA C H

on Ts rather than negatively, and physiological feedbackacts negatively rather than positively, provided frostingdoes not occur. Thus, these feedbacks remain opposingbut are reversed.

Acknowledgements

I am indebted to several colleagues, particularly Ray Leuningfor his guidance on stomatal physiology and for comments ona draft of the paper, and to Helen Cleugh, Dean Graetz andChris Field for raising questions which prompted this work.Peter Briggs provided skilful and speedy assistance with thepreparation of the figures.

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Appendix A: Monin–Obukhov similarity theoryfor aerodynamic conductance

The dimensionless temperature and velocity profiles χH

and χM are

χX (zR – d, z0X, L) 5

zR–d zR–d z0Xln – ΨX 1 ΨX , (A1)( ) ( ) )(z0X L L

where X stands for either H or M, and ΨH,M are diabaticprofile influence functions defined by

ΨX (ζ) 5 ∫ζ

0

(1 – φX (ζ1))dζ1/ζ1. (A2)

The dimensionless wind and temperature gradients φM (ζ)and φH (ζ) are defined by

dU u*φM (ζ) dT θ*φH (ζ)5 , 5 – (A3)

dz κ (z – d) dz κ (z–d)

with u* the friction velocity, θ* 5 FH/(ρ cpu*) the frictiontemperature and ζ 5 (z-d)/L. The Monin–Obukhov lengthis defined by

z–d κ g (z–d) (FH 1 0.07FE)5 – , (A4)

L ρ cpTu*3

where g is gravitational acceleration and T0 a referenceabsolute temperature. The functions φM,H(ζ), and thenceψM,H(ζ), are specified empirically, with the aid ofconstraints from dimensional analysis (Kader & Yaglom1990, Garratt 1992, Hogstrom 1996). The forms used hereare φM,H (ζ) 5 1 1 5 ζ on the stable side (ζ . 0), andφM,H(ζ) 5 (1 1 A |ζ|n)/ (1 1 B |ζ|n) on the unstable side(ζ , 0), with (A, B, n) 5 (0.05, 0.6, 0.72) for momentum,and (0.07, 1.2, 0.9) for heat. The unstable formulation isanalytically integrable (Brutsaert 1992) and gives a goodfit to data summarized by Kader & Yaglom (1990).


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Thom AS (1975) Momentum, mass and heat exchange of plantcommunities. In: Vegetation and the Atmosphere, Vol. 1 (ed.Monteith JL), pp. 57–109. Academic Press, London.

Wullschleger SD (1993) Biochemical limitations to carbonassimilation in C3 plants – a retrospective analysis of theA/Ci curves from 109 species. Journal of Experimental Botany,44, 907–920.

Appendix B: Stress function model for stomataland surface conductance

Leaf stomatal conductance is assumed to be given by (8).The stress functions sPAR, sD, sT and sW are given by:

sPAR(φ) 5 φ/ (φ 1 φ*) (B1)

sD(Ds) 5 (1 1 Ds/Ds*)–1 (B2)

sT(Ts) 5 1– (Ts–Tm) 2/ (Th–Tm)2 (B3)

sW(η) 5 (η–ηwilt)/ (ηonset–ηwilt), (B4)

where φ*, Ds*, Tm, Th and ηonset are empirical parametersdefined in Table 1, ηwilt is the volumetric soil water contentat wilting, and the additional constraint 0 ø s ø 1 isimposed as necessary. The bulk surface conductance isobtained by parallel summation of gs values for leavesobeying (8), with two additional assumptions. First, theleaf- level absorbed PAR is exponentially distributed overall leaves, as in (B5) below. Second, the leaf-surfaceproperties Ts and Ds are set to uniform, average values, onthe grounds that they vary through the canopy much lessthan the leaf PAR, φ.

Let ζ be a cumulative leaf area index from high-PARleaves to low-PAR leaves; φ (ζ) the leaf-scale absorbed PARflux (the net flux density of PAR on a leaf, measured inmol quanta per unit leaf area per unit time); and Φ(ζ) thecanopy-scale absorbed PAR flux (the net flux density ofPAR reaching leaves beyond leaf area ζ, measured in molquanta per unit ground area per unit time). Then Φ0 5

Φ(ζ 5 0) is the net PAR flux (per unit ground area) at thetop of the canopy, and

dΦ Φ(ζ)φ(ζ) 5 – , 5 τ (ζ) , τ (ζ) 5 exp (– c ζ), (B5)

dζ Φ0

where τ is the canopy transmission and c is an extinctioncoefficient. The first of these follows from conservation,and an exponential distribution is assumed in the lastequation. Incident PAR is assumed to be related to incident

490 M . R . R A U PA C H

Table B1. Parameters used in Jarvis model for bulk sufrace conductance Gs.

Parameter Name Units Value

gsmax maximum leaf stomatal conductance m s–1 0.01c canopy extinction coefficient for PAR – 0.6φ* PAR at which sPAR (φ) 5 0.5 W m–2 100Ds* saturation deficit at which sD(Ds) 5 0.5 kg kg–1 0.01Tm median (optimal) leaf temperature deg C 25Th high-cutoff leaf temperature deg C 50Ci assumed constant intercellular [CO2] ppm 300ηonset soil water content at onset of water stress – unspecified

solar irradiance by: Φ↓ [µmol quanta m–2 s–1] 5 2Fs↓[W m–2] at canopy scale, and similarly at leaf scale. Withthe assumption that sD, sT and sW are independent of ζ, (8),(9), (B1) and (B5) imply (Saugier & Katerji 1991; Dolman &Wallace 1991):

gsmaxGs 5 3

c

1 1 X3 ln sD(Ds) sT(Ts) sW(η), (B6)( )exp (– c Λ) 1 X

where X 5 φ*/(cΦ0). The parameters used to implementthis model are given in Table 1, and are consistent withchoices commonly used in models of this type (Kowalczyket al. 1991; McNaughton & Jarvis 1991).

A canopy carbon flux is derived from this Gs model withthe crude assumption that Ci is constant and given. ThenFC (kg CO2 m–2 s–1, positive outward) is given by

(MC/V) (Ci–C)FC 5 , (B7)

1.6Rs 1 Ra

where MC is the molecular mass of CO2 and V is themolar volume.

Appendix C: Assimilation model for leaf stomatalconductance

Background, notation and units

The model outlined here is adapted with minormodifications from Leuning (1990, 1995). The principle isto solve three coupled equations for leaf net assimilation(an), intercellular CO2 concentration (Ci) and stomatalconductance for CO2 (gsC). This Appendix follows plantphysiological conventions which differ slightly from therest of the paper: carbon and PAR fluxes are expressed inmol m–2 s–1, and the stomatal conductance for CO2 (gsC) inmol m–2 s–1. The kinematic stomatal conductance for H2Oused elsewhere in the paper (gs, in m s–1) is related to gsC

by gs 5 1.6VgsC, where V is the molar volume (about


0.025 m3 mol–1) and 1.6 is the ratio of molecular diffusivitiesof H2O and CO2 in air. The assimilation (an, in mol m–2 s–1)is related to the outgoing CO2 flux (fC, in kg CO2 m-2 s–1)by fC 5 –MCan (MC is CO2 molecular mass).

Assimilation equation

Net C3 leaf assimilation is modelled by (Farquhar et al.1980):

an 5 min(ac, aφ) – rd, (C1)

where ac is the gross assimilation rate limited by thecatalytic activity of the enzyme Rubisco (Ru), aφ is thegross assimilation rate limited by electron transport(PAR), and rd is day respiration. Both ac and aφ are given by

Ci – Γ*ac,φ 5 βc,φ , (C2)( )Ci 1 γc,φ

where Γ* is the CO2 compensation point without dayrespiration, and

βc 5 Vcmax, γc 5 Kc (1 1 [O2]i/Ko) (C3)

βφ 5 J/4 , γφ 5 2 Γ*. (C4)

Here Vcmax is the maximum catalytic activity of Rubiscowhen RuP2 and CO2 are saturating; Kc and Ko areconstants (Table 1); [O2]i is intercellular O2 concentration;and J is the electron transport rate for a given absorbedleaf PAR flux, φ. The function J(φ) is the lesser root of thequadratic (Farquhar & Wong 1984):

θ J2 – (αφ 1 Jmax)J 1 αφJmax 5 0, (C5)

where Jmax is the maximum electron transport rate, α isthe quantum yield, and θ is a shape factor for thenonrectangular hyperbola J(φ).

Dependences of parameters on temperature and waterstress

The behaviour of the model depends critically on thetemperature dependences of Kc, Ko, Γ *, rd, Jmax and Vcmax.


Fig. C1 Predictions of the assimilation-Fand assimilation-H models for leaf-levelassimilation (an), intercellular CO2concentration (Ci) and stomatalconductance for CO2 (gsC), plottedagainst leaf temperature Ts, at severalPAR levels φ (20, 100, 200, 400, 800, 2000µmol m–2 s–1). Other conditions: Ds 5

500 Pa, Cs 5 360 ppm, sW 5 1.

Table C1. Parameters used in assimilation-based model for stomatal conductance gsC. Values except for parameters marked # are fromLeuning (1995, 1997), with temperature coefficients from Farquhar et al. (1980) (model F) and Harley et al. (1992) (model H, identicalto model F except where shown). Reference temperature T0 is 20 °C.

Parameter Name Unit Model F values Model H values

Jmax0 # potential electron transport rate at T0 µmol m–2 s–1 100Hv

J Jmax activation energy J mol–1 37 000 79 500Hd

J Jmax deactivation energy J mol–1 220 000 201 000Sv

J Jmax entropy term J mol–1 K–1 710 650Vcmax0 maximum catalytic activity of Rubisco at T0 µmol m–2 s–1 Jmax0/2.44Hv

c Vcmax activation energy J mol–1 58 520 116 300Hd

c Vcmax deactivation energy J mol–1 undefined 202 900Sv

c Vcmax entropy term J mol–1 K–1 undefined 650rd0 day respiration at T0 µmol m–2 s–1 0.32H1

R # temperature coefficientJ mol–1 53 000for rd

H2R # temperature coefficient for rd J mol–1 200 000

T2 # high-T rolloff for rd deg K 313γ0 coefficient determining Γ* µmol mol–1 34.6γ1 coefficient determining Γ* K–1 0.0451γ2 coefficient determining Γ* K–2 0.000 347α quantum yield mol mol–1 0.20θ shape factor for J – 0.95aPAR leaf albedo for PAR – 0.14Kc0 Michaelis constant for CO2 mol mol–1 300 3 10–6

Ko0 inhibition constant for oxygen mol mol–1 256 3 10–3

HKc activation energy for CO2 J mol–1 59 430HKo activation energy for O2 J mol–1 36 000[O2] intercellular O2 concentration mol mol–1 0.209b gsC slope parameter – 5.0g0 # gsC intercept parameter µmol m–2 s–1 0.005Γ # CO2 compensation point mol mol–1 1.1Γ*Ds0 Vapour pressure deficit parameter Pa 1500

The temperature dependence of Kc and Ko is representedby a normalized Arrenhius equation:

Kx 5 Kx0 exp[(HKx/RT0) (1–T0/T)], (C6)


where x 5 c or o; Ts is leaf temperature and T0 a referencetemperature (both in °K); and other quantities are definedin Table 1. For Γ*, the temperature dependence isrepresented empirically by (Brooks & Farquhar 1985):

492 M . R . R A U PA C H

Γ * 5 γ 0[1 1 γ 1 (TS – T0) 1 γ 2 (Ts–T0) 2] (C7)

For rd, the temperature dependence is expressed by:

exp[(HR0/RT0) (1-T0/T)]rd 5 rd0 , (C8)

1 1 exp[(HR2/RT2) (1-T2/T)]

where rd0 is the leaf respiration at T0, HR0,2 are empiricalcoefficients, and T2 is a high-temperature rolloff point.This is a slight modification of the proposal by Leuning(1995), in which a high-temperature attenuation term (theexponential in the denominator) is included to attenuaterd progressively above T2. The expressions for Jmax andVcmax are:

exp[(HvJ/RT0) (1-T0/Ts)]

Jmax 5 Jmax0sW(η) (C9)1 1 exp[(Sv

J Ts – HdJ)/RTs]

exp[(Hvc/RT0) (1-T0/Ts)]

Vcmax 5 Vcmax0sW(η) , (C10)1 1 exp[(Sv

c Ts-Hdc)/RTs]

where sW (η) is a soil water stress function (0 , sW ø 1)of soil moisture η, similar to (B4), and other quantitiesare defined in Table 1. It is assumed that Jmax0 5 2.44Vcmax0

at a reference temperature of 20 °C (Wullschleger 1993,Leuning 1997), so that only one of Jmax0 and Vcmax0 needsto be set as a model parameter. Both Jmax and Vcmax

depend on temperature through (C9) and (C10), but theyalso depend on species and leaf nitrogen status. Equations(C9) and (C10) also include a dependence of Jmax andVcmax on water stress, following Sellers et al. (1996),though there is no known mechanistic basis for such adependence. This is one of two ways of introducing thenecessary water stress dependence into the system (theother being through stomatal conductance directly; seebelow).

Stomatal conductance model

Following Leuning (1995), gsC is described by the semi-empirical equation

bsD(Ds)angsC 5 g0 1 , (C11)

Cs – Γ

where g0 and b are intercept and slope parameters,respectively; Γ is the CO2 compensation point; Cs is leaf-surface CO2 concentration; and sD is a dimensionlessfunction (between 0 and 1) of the leaf-surface saturationdeficit Ds. An equation of this type was first suggestedby Ball et al. (1987). Leuning (1995) showed that (C11) with

sD(Ds) 5 (1 1 Ds/Ds0)–1 (C12)

(where Ds0 is an empirical parameter) fits observationsbetter than several alternatives. This ‘‘Lohammer’’


equation has the property that gs falls linearly with risingevaporation, as Ds is raised (Monteith 1995). A similarform is used in the present implementation of the Jarvismodel, in Equation (B2).

Solution procedure, parameters and model performance

To solve for an, Ci and gsC, three independent equationsare required. Two are provided by the assimilation andstomatal conductance models, (C2) and (C11). The thirdis the definition equation

an 5 gsC (Cs–Ci). (C13)

Combining these three equations leads to a quadratic foran which is solved for both Ru-limited and PAR-limitedconditions, using (C3) and (C4), respectively. The lesservalue of an is chosen in accordance with (C1), and finallygsC and Ci are calculated using (C11) and (C13).

Table C1 gives parameter values, mainly adopted fromLeuning (1995, 1997). The choice of Jmax0 (dependent onspecies and nutrient status) is typical of values in Leuning(1997). Two sets of values for the Jmax and Vcmax

temperature coefficients are used: ‘‘assimilation modelF’’ denotes the use of coefficients from Farquhar et al.(1980), and ‘‘assimilation model H’’ denotes coefficientsfrom Harley et al. (1992). Figure 1 shows the performanceof both models, by plotting leaf an, Ci and gsC against Ts

for several values of PAR flux φ. The main differencebetween the models is that model H produces highervalues of an and gsC, with more high-temperaturetolerance, than does model F.

An alternative formulation of the water stressdependence was examined, in which the stress functionsW appears in (C11) for gsC (as a multiplier in the secondterm along with sD), rather than in (C9) and (C10) forJmax and Vcmax. Compared with the formulation in whichsW multiplies Jmax and Vcmax, the alternative formulationproduces lower Ci values in water-stressed conditions(, 200 ppm) and requires lower sW values to inducesignificant reductions in gs due to water stress. The useof this alternative would not change the conclusions ofthe present work. It is not clear at this stage whichformulation is preferable.

Appendix D: Integration of assimilation modelto canopy scale

The assimilation model for leaf gs (Appendix B) permitscalculation of leaf assimilation and stomatal conductanceas functions an(φ,Ts,Ds,Cs) and gs(φ,Ts,Ds,Cs) of net leafPAR flux φ and leaf-surface temperature Ts, saturationdeficit Ds and CO2 concentration Cs. Net canopy


assimilation and surface conductance Gs are given fromthese as

An 5 ∫Λ

0

an(φ,Ts,Ds,Cs)dζ, Gs 5 ∫Λ

0

gs(φ,Ts,Ds,Cs)dζ, (D1)

where ζ is cumulative leaf area index. The first of theseis a conservation requirement, while the second is theparallel-sum approximation valid under conditionsdefined in Section 3. As in Appendix B, it is assumedthat (i) φ is exponentially distributed, and (ii) Ts, Ds andCs can be set to uniform, average values, since they varymuch less than φ. The first assumption ignores (forsimplicity) the question of separate populations of sunlitand shaded leaves. The second assumption means thatthe only quantity varying with ζ is φ, so (omitting thedependences on Ts, Ds and Cs)

an(φ) dφAn 5 ∫

Λ

0

an(φ(ζ)) dζ 5 ∫φ(Λ)

φ(0)

(D2)dφ/dζ

and similarly for Gs. Using the exponential distributionfor φ(ζ) specified in (B5) and Table 1, it is found that

an(φ) dφAn 5 ∫

φ(0)

φ(Λ)

, (D3)cφ

where φ (0) 5 cΦ0 is the net PAR flux (per unit leafarea) on the best-lit leaves in the canopy, and φ(Λ) 5

cΦ0exp(–cΛ) is the PAR flux on the least-lit leaves (Φ0

being the net PAR flux per unit ground area at the topof the canopy). A similar equation describes Gs.

Equation (D3) and its counterpart for Gs express An

and Gs as weighted integrals over the individual-leaflight response curves an(φ) and gs(φ), the weighting factorbeing 1/(cφ). In practice, these integrals can be evaluatedsufficiently accurately by calculating an(φ)/φ and gs(φ)/φ

Table E1. Meteorological conditions and computational parameters for CBL slab model.

Downward shortwave irradiance FS↓ FSmax sin(2πt/Tday)with FSmax 5 1000 W m–2, Tday 5 86400 s

Downward longwave irradiance FL↓ Swinbank (1963) formula:FL↓ (W m–2) 5 5.31 3 10–14T6 (T in deg K)

Wind speed in CBL 4 m s–1

Conditions above CBL Θ1(z) 5 15 1 0.005z CQ1(z) 5 0 g kg–1

C1(z) 5 350 ppmInitial conditions Θm(0) 5 10 C

Qm(0) 5 QsatCm(0) 5 500 ppmZi(0) 5 50 m

Reference level (depth of surface layer and height at which zm 5 50 m(z-d)/L is calculated)Integration period 12 hr (0600 to 1800 local time)Integration time step 15 min


at only three points (φ(0), (φ(0) 1 φΛ))/2, φ(Λ)), andintegrating the parabola passing through the resultingvalues. A canopy model is thus obtained by determiningthe assimilation and conductance of three individualleaves: the brightest leaf, the dimmest leaf, and themedian leaf. The mass CO2 flux is FC 5 –MCAn.

Appendix E: The CBL slab model

The CBL slab model used here follows McNaughton &Spriggs (1986) and Raupach & Finnigan (1995). TheCBL is assumed to be uniformly mixed by convectiveturbulence, except within a relatively thin surface layer,and bounded above by a capping inversion at heightZi(t). The conservation equation for a scalar S, height-integrated from z 5 0 to z 5 Zi in an air column movingwith the mean wind field, is:

dSm FS S1(Zi) – Sm dZi5 1 – W1 , (E1)( ) ( )dt Zi Zi dt

where Sm(t) is the mixed-layer scalar concentration(uniform in the CBL), W is mean vertical velocity, the 1

subscript denotes conditions in the troposphere just abovez 5 Zi, and FS is the scalar flux at the surface. Thisequation is written for potential temperature (S 5 Θ 5

T 1 Γaz, Γa being the dry adiabatic lapse rate), humidity(S 5 Q) and CO2 (S 5 C). The fluxes FΘ 5 FH/(ρcp)and FQ 5 FE/(ρλ) are specified by (1) to (4), and the CO2

flux FC by Appendix B (Jarvis model) or C and D(assimilation model). Entrainment is parameterized by:

dZi CKW*3

5 , (E2)dt CT W*

2 1 gZi (Θv1 – Θvm)/T0

where g is gravitational acceleration, Θv 5 Θ(1 1 0.61Q)

494 M . R . R A U PA C H

is potential virtual temperature, W* 5 (hgFΘv/T0)1/3 isthe convective velocity scale, FΘv is the land–air flux ofΘv, and CK 5 0.18 and CT 5 0.8 are empirical coefficients(Rayner & Watson 1991). The model is integrated throughthe day from dawn (t 5 0), using the Runge–Kutta


method with a time step of 15 min. Initial conditionsΘm(0), Qm(0), Cm(0) and Zi(0) are given, together withthe tropospheric temperature and humidity profiles Θ1(z)and Q1(z) above the CBL. Assumptions used for theseconditions are specified in Table E1.

influences of local feedbacks on land–air exchanges of energy and carbon

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