A coupled model of stomatal conductance, photosynthesis and transpiration

Authors


Correspondence: Andrée Tuzet. Fax: 33 1 30 81 55 63; e-mail: tuzet@grignon.inra.fr

ABSTRACT

A model that couples stomatal conductance, photosynthesis, leaf energy balance and transport of water through the soil–plant–atmosphere continuum is presented. Stomatal conductance in the model depends on light, temperature and intercellular CO2 concentration via photosynthesis and on leaf water potential, which in turn is a function of soil water potential, the rate of water flow through the soil and plant, and on xylem hydraulic resistance. Water transport from soil to roots is simulated through solution of Richards’ equation. The model captures the observed hysteresis in diurnal variations in stomatal conductance, assimilation rate and transpiration for plant canopies. Hysteresis arises because atmospheric demand for water from the leaves typically peaks in mid-afternoon and because of uneven distribution of soil matric potentials with distance from the roots. Potentials at the root surfaces are lower than in the bulk soil, and once soil water supply starts to limit transpiration, root potentials are substantially less negative in the morning than in the afternoon. This leads to higher stomatal conductances, CO2 assimilation and transpiration in the morning compared to later in the day. Stomatal conductance is sensitive to soil and plant hydraulic properties and to root length density only after approximately 10 d of soil drying, when supply of water by the soil to the roots becomes limiting. High atmospheric demand causes transpiration rates, LE, to decline at a slightly higher soil water content, θs, than at low atmospheric demand, but all curves of LE versus θs fall on the same line when soil water supply limits transpiration. Stomatal conductance cannot be modelled in isolation, but must be fully coupled with models of photosynthesis/respiration and the transport of water from soil, through roots, stems and leaves to the atmosphere.

INTRODUCTION

Although the response of stomata to environmental and physiological factors is complex, we know that stomatal conductance varies with leaf irradiance, leaf temperature, atmospheric water vapour pressure deficit and CO2 concentration (Cowan 1977; Buckley & Mott 2002). We know additionally that stomatal conductance depends on guard cell and epidermal turgor (Wu, Sharpe & Spence 1985; Mencuccini, Mambelli & Comstock 2000; Franks et al. 2001), and that regulation of turgor in these cells requires metabolic energy (Farquhar & Wong 1984; Assman 1999; Blatt 2000; Netting 2000). Leaf turgor also depends on the balance between loss of water through transpiration and supply of water to the leaf from the soil (Cowan 1977; Mott & Parkhurst 1991; Maier-Maercker 1999; Mott & Franks 2001). A major challenge is to develop a model which accounts for all the factors which control stomatal conductance.

There are presently few, if any, models which combine our mechanistic understanding of stomatal function at the cellular level with descriptions of the coupled fluxes water and CO2 in the soil–plant–atmosphere continuum (SPAC). Instead, the dependencies of stomatal conductance on environmental and physiological factors are included in semi-empirical models, such as the response-function approach of Jarvis (1976), which relates stomatal conductance to leaf temperature, irradiance and leaf and soil water potentials. More recently, Ball, Woodrow & Berry (1987) and Leuning (1990, 1995) developed relationships between stomatal conductance, CO2 assimilation rate and air humidity which summarized successfully the results of many observations of the stomatal behaviour of well-watered plants. Such models have concentrated on leaf-level process, but to account for the various environmental and physiological controls of stomatal conductance, models of photosynthesis, respiration, leaf energy balances and the transport of water from soil to leaves should be fully integrated with stomatal models.

This continuum approach has been adopted by many workers, but there are marked differences in the way stomatal control of water and CO2 fluxes have been modelled. Some emphasize plant physiology (e.g. Leuning et al. 1995) whereas others provide a greater balance between soil, plant and atmospheric processes affecting carbon and water fluxes (e.g. Sellers et al. 1992; McMurtrie et al. 1992; Jensen et al. 1993; Tardieu & Davies 1993; Williams et al. 1996; Baldocchi & Meyers 1998; Whitehead et al. 2002; Dewar 2002). Many such models are satisfactory when soil moisture is adequate, but some do not predict correctly the patterns in conductance, transpiration and CO2 assimilation when soil moisture availability becomes limiting (e.g. Calvet et al. 1998; Grant et al. 1999). Measured fluxes show a marked daytime asymmetry at low soil moisture contents, with higher fluxes of water and CO2 in the morning than the afternoon (Schulze & Hall 1982; Olioso, Carlson & Brisson 1996; Prior, Eamus & Duff 1997; Grant et al. 1999; Eamus, Hutley & O’Grady 2001). Whereas some models are able to capture the observed asymmetry (Tardieu & Davies 1993), those of others are not (Leuning et al. 1995; Leuning, Dunin & Wang 1998; Grant et al. 1999; Ronda, de Bruin & Holtslag 2001), and we suggest this may result from inadequate coupling of stomatal conductance to the dynamics of water transport from soil to the roots and leaves. Many SPAC models which do incorporate the effects of soil moisture on stomatal conductance use an electrical resistance analog for the transport of water from the soil to the roots (Tardieu & Davies 1993; Amthor 1994; Olioso et al. 1996; Williams et al. 1996; Baldocchi & Meyers 1998). Hydraulic resistance to water flow in the soil varies strongly with soil moisture. Cowan (1965) showed that water content, and hence hydraulic resistance, varies significantly with distance from the root surface to the bulk soil once soil water supply starts to limit transpiration. Thus use of mean soil moisture content, rather than the value near to the roots, may lead to incorrect values for the resistance to water flow in the soil. The spatial distribution in moisture content also changes through the day and hence the dynamics of water flow to the roots will not be captured when mean soil moisture content is used to calculate soil hydraulic resistance (Perrier & Tuzet 1998). To overcome this problem, the model presented in this paper solves Richards’ equation for water flow from the soil to roots numerically (Philip 1957; Gardner 1960), rather than using hydraulic resistances.

Water potentials within the leaf are linked to potentials at the soil–root interface by the flux of water across hydraulic resistances in the plant. We assume that leaf water potential controls stomatal conductance (Buckley & Mott 2002; Leuning, Tuzet & Perrier 2003), an assumption that accounts for the effects of supply and demand for water at the evaporating sites on guard cell and epidermal turgor. This removes the explicit dependence of stomatal conductance on atmospheric humidity used by Ball et al. (1987) and by Leuning (1990, 1995), since their models account only for water losses from the leaves by transpiration.

Supply and demand for CO2 by photosynthesis and respiration also affects stomatal conductance through variations in intercellular CO2 concentration (Mott 1988; Assmann 1999). These considerations, along with the above arguments concerning leaf water potentials, have led in this paper to a revision of the stomatal model of Leuning (1995). While retaining the explicit dependence of conductance on CO2 assimilation (and hence on absorbed light and temperature), the original humidity deficit term is replaced by a function of leaf water potential, and CO2 concentration at the leaf surface is replaced by intercellular CO2 concentration. This approach ensures a complete coupling between stomatal conductance, the flux of water through the plant and soil, and CO2 exchange between leaves and the atmosphere.

In the following two sections we briefly describe the new stomatal model and the equations for uptake of water by roots and transport of water through the plants. Appendix A contains details of the rest of the model which describes exchanges of radiant energy, sensible heat, water vapour and CO2 between the soil, plant and the atmosphere. After presenting the initial and boundary conditions, the full model is used to predict diurnal patterns of leaf water potential, water absorption by roots, and canopy transpiration during a drying cycle for soils with varying hydraulic characteristics. Sensitivity of the model to variations in key controlling parameters is investigated, followed by an examination of the relationships between stomatal conductance, CO2 assimilation, transpiration, intercellular CO2 concentrations and atmospheric humidity deficit.

MODEL DESCRIPTION

Stomatal conductance

Several studies have used various forms of the Ball–Berry model for stomatal conductance (Ball et al. 1987; Collatz et al. 1991; Sellers et al. 1992; Leuning 1995; Leuning et al. 1995). Although this model is satisfactory for describing the gas exchange of well-watered plants, it is inadequate for plants subjected to some degree of water stress. Wang & Leuning (1998) modified the stomatal model of Leuning (1995) to account for slow changes in soil water content, but that function cannot describe the diurnal response of stomata to the competing demand of water vapour from the atmosphere and supply of water to leaves by the soil via the roots and xylem. To overcome this deficiency, we propose a variant of Leuning's (1995) model which incorporates stomatal response to leaf water potential, namely:

image(1)

where gCO2 is stomatal conductance for CO2, g0 is the residual conductance (limiting value of gCO2 at the light compensation point), A is assimilation rate, ci is CO2 concentration in the intercellular spaces, Γ is the CO2 compensation point, and a is an empirical coefficient. (see Appendix d for a full list of symbols). The use of ci in Eqn 1, rather than CO2 concentration at the leaf surface, cs (Ball et al. 1987; Leuning 1995), agrees with observations that stomata respond to substomatal CO2 concentration, ci (Mott 1988; Assmann 1999). Dewar (2002) also used ci in his model.

An empirical logistic function is used to describe the sensitivity of stomata to leaf water potential:

image(2)

where ψv is the bulk leaf water potential, ψf is a reference potential and sf is a sensitivity parameter. With this function we assume that stomata are insensitive to leaf water potential at values close to zero and that stomata rapidly close with decreasing ψv (see Fig. 6a). The value of the parameters sf and ψf depend on morphological adaptations in different species (Schulze & Hall 1982; Henson, Jensen & Turner 1989; Sperry 2000) and with environmental conditions experienced during growth (e.g. Saliendra, Sperry & Comstock 1995). Inclusion of leaf water potential in the stomatal model is consistent with the hypothesis of Tyree & Sperry (1988), Jones & Sutherland (1991), and Sperry (2000) who argue that stomata regulate transpiration rates to avoid cavitation in the xylem, and with Buckley & Mott (2002) who postulate that water potential at the evaporating sites within the leaf controls stomatal conductance. Maier-Maercker (1998), Bond & Kavanagh (1999) and Leuning et al. (2003) provide further support for this approach to modelling interactions between leaf water potential and stomatal conductance.

Figure 6.

(a) Three different functions describing the sensitivity of stomatal conductance to ψv[(I) sf = 4.9, ψf = −1.2; (ii) sf = 3.2, ψf = −1.9; (iii) sf = 2.3, ψf = −2.6]. (b) Predicted diurnal variations in leaf water potential ψv as the soil dries for the three stomatal functions. (c) The relationship between gCO2 and ψv for days 6, 15 and 20, for each of the fψv functions. The first 5 d of the simulation have been omitted in (b) because during that period the model predicts no differences between the different functions.

Water uptake by roots and transport through the canopy

Cylindrical geometry and the concept of effective soil volume per unit root length have been used to model uptake of water by roots (Philip 1957; Gardner 1960). The roots are taken to be long cylinders with uniform radius and water-absorbing properties, and water movement in the soil is assumed to occur only in the radial direction. The absorbing roots are assumed to be uniformly distributed throughout the entire soil volume. Under these conditions, Richards’ flow equation is then:

image((3a))

where θs is the soil water content; t is time; r is radial distance from the axis of the root; Ks is soil hydraulic conductivity and Ψs is soil water potential. Using the ‘Kirchhoff transform’, an integral transform that uses a ‘matric flux potential’, inline image as a driving force for flow, Equation 3a becomes:

image((3b))

Both Ψs and Ks depend on soil water content θs according to the empirical relationships (Campbell 1985):

image((4a))
image((4b))

where ψe is the air entry water potential of the soil, θsat is the saturation water content, Ksat is the saturated conductivity and b is an empirical parameter. Calculations in this paper are based upon data for three soils: sandy loam, silt loam and loam, and values of ψe, Ksat, θsat and b for these soils are listed in Table 1. Initial water content is assumed to be uniform throughout the soil volume with some value θ0, corresponding to a soil water potential ψ0. The radial flow of water is set to zero at the half average distance between neighbouring roots.

Table 1.  Parameter values to describe soil hydraulic properties using Eqn 4
Typebθsat
(m3 m−3)
ψe
(MPa)
Ksat
(s−1 MPa−1 m2)
Sandy loam3.310.4−0.91 × 10−3957.6 × 10−6
Silt loam4.380.4−1.58 × 10−3217.8 × 10−6
Loam6.580.4−1.88 × 10−3228.6 × 10−6

Water uptake by the roots is assumed equal to the flow of water through the plant canopy, F, given by:

image(5)

where ψr is the water potential at the boundary between the plant roots and the soil and χv is the leaf-specific resistance to water flow through plant from the roots to the stomata. Reid & Huck (1990) and Sperry (2000) found that χv is correlated to flow rate, but their findings could result from uncertainties in the determining ψr experimentally. In practice it is extremely difficult to separate plant and soil resistances to water flow because the latter depends on microgradients around the root zone. Thus χv is taken as constant in our calculations.

The second part of Eqn 5 equates the steady-state flow of water through the plant with transpiration; rsH2O=β/(1.6gCO2) and rbv are the resistances to water transport through the stomata and boundary layer, respectively. The factor 1.6 accounts for the ratio of molecular diffusivities for water and CO2 in air and β (= P/RT) is the coefficient used to convert the molar units (mol−1 m2 s) into physical resistance units (s m−1) and P is the atmospheric pressure, Mv is the molecular mass of water, R is the ideal gas constant, ρv is the liquid water density, hi= exp(Mvψv/ρvRTsv) is the fractional relative humidity in the intercellular spaces. This expression links the thermodynamic potential of water in the liquid and gas phases at the evaporating sites within the leaf. The functions P(Tsv) and P(Trv) are the saturation water vapour pressure at foliage temperature Tsv and dewpoint temperature of the air surrounding leaves Trv, while Tav is air temperature.

System modelled

The full model framework is shown in Fig. 1. Fluxes, water contents, and associated quantities are expressed on the basis of unit area of ground, the plant is divided into shoots and roots and there is only one soil layer. Although the model is relatively simple, its main purpose is to demonstrate the intimate connection between stomatal conductance, CO2 exchange by the leaves and the flow of water through the soil and plants to the atmosphere. All simulations were for a crop with a leaf area index of 3 and a height of 0.8 m. Reference simulations used a uniform root length density of 4 km m−3, a root depth of 1 m, root radius of 0.1 mm, a sandy loam soil and a constant plant hydraulic resistance of 1 × 107 MPa s m−1 (Sperry 2000). All model parameters are presented in Table 2.

Figure 1.

Schematic diagram of the latent heat, sensible heat and CO2 exchange system described by the model. The cylindrical geometry used to calculate the flow of water to the root is also shown.

Table 2.  Parameter values used to describe the vegetation and stomatal conductance in standard model runs
ParameterSymbolValueUnits
Vegetation
 Leaf area indexLAI3m m−1
 Canopy heighthc0.8m
 Root depthzroot1m
 Root radiusrroot0.1 × 10−3m
 Plant hydraulic resistanceχv1.107MPa s m−1
 Average leaf widthdl0.01m
 Canopy mixing lengthlc0.2m
Stomatal conductance
 Residual conductance   for CO2g00.3 × 10−3mol m−2 s−1
 Parameter in gCO2Eqn 1a2 

Figure 2 shows the 24 h cycle of climatic data of air temperature (Ta), dew point temperature (Tr), wind speed (U), global and atmospheric long wave radiation (Rg, Ra) which were used in the simulations. This is an idealized day representative of typical clear summer conditions in temperate agricultural regions. During the day, Rg advances and declines symmetrically around noon; Ra remains essentially constant throughout the day and night period; Ta and Tr follow the course of the sun with a time lag of about 2 h and U, which increases after sunrise, remains nearly constant throughout the day and decreases at sunset. All these climatic data refer to conditions 3 m above the ground.

Figure 2.

The 24 h cycle of climatic data of air temperature (Ta), dew point temperature (Tr), wind speed (U), global and atmospheric long wave radiation (Rg, Ra) used in the simulations.

Simulation results are presented for a 30 d drying cycle with an initial root zone uniformly wet (Ψs = −0.05 MPa). The meteorological forcing was repeated each day to simplify interpretation of predicted diurnal variations in stomatal conductance, assimilation, transpiration and leaf, root and soil water potentials.

RESULTS OF SIMULATIONS AND DISCUSSION

Time-course of the potentials ψs, ψr and ψv

The simulated diurnal variations in the water potentials of soil ψs, root ψr and leaf ψv, during a 30 d drying cycle are presented in Fig. 3. Initially, only the leaf water potential shows a marked variation during the day, accompanied by a steady decline in root and soil water potentials. Soil water supply starts to limit transpiration after approximately 10 d of drying and values of ψr start to decrease rapidly due to a decrease of soil water content and soil hydraulic conductivity close to the roots. There is a lesser rate of decline in the minimum ψv, so the difference between ψv and ψr values gradually decrease throughout the drying cycle, leading to a reduction in water flow through the plants. The system equilibrates at night so that soil, root and leaf water potentials are equal. This value of ψs at night, which corresponds to the predawn water potential, decreases progressively throughout the drying cycle from the initial value of −0.05 MPa, to about −0.54 MPa after 30 d.

Figure 3.

Time-course of leaf, root and soil water potentials during a 30 d drying cycle. The arrows indicate days of the drying cycle selected subsequently to show in more detail the diurnal changes of different variables.

Predicted variations in soil water potential, ψs, as a function of distance from the roots, r, are shown in Fig. 4a at 3 h intervals for day 10 of the drying cycle. There is little variation in ψs with r around dawn (0600 h), but the potential near the roots progressively decreases during the day, reaching a minimum between 1200 and 1500 h. Matric potentials near the roots start to increase in the afternoon as evaporative demand diminishes in late afternoon, and the ψs curves cross those of the morning as water in the root zone redistributes from the wetter regions in the bulk soil to the zone close to the roots. By late evening the potential is approximately uniform with distance but is at a lower value than the previous morning, commensurate with the extraction of soil water during the day. It is clear that the greatest changes in matric potential (and θ) occur close to the root surfaces. Hydraulic conductivity and resistance to water flow varies strongly with moisture content (Eqn 4b). Hence use of mean soil matric potentials will lead to incorrect values of resistance to water flow and will not capture the dynamics of the water transport as the soil dries.

Figure 4.

(a) The distribution of soil water potential, ψs, as a function of radial distance from the root, r, surface at 3 h intervals on day 10 of the drying cycle. (b) The distribution of ψs at noon as a function of r over a 10 d drying cycle (top to bottom)

Matric potentials decrease progressively as the soil dries, as shown in Fig. 4b for the predicted ψs profiles at noon during the first 10 d of the dry-down cycle. Initially there is little variation in ψs with distance, but as the soil dries ψs in the bulk soil decrease progressively whereas the potential near the roots decreases more strongly. These profiles are similar to those presented by Cowan (1965).

Diurnal changes of gCO2, LE, A and ci/cs

Predicted diurnal variations in gCO2, LE, A and the ratio ci/cs are shown in Fig. 5 for days 1, 10, 13 and 16 of the drying cycle. On day 1 when soil water availability is non-limiting, all quantities show a smooth and symmetrical variation around noon, with little or no asymmetry in these quantities caused by the peak in temperature and humidity deficit in mid-afternoon (Fig. 2) As soil moisture declines there is an increasingly asymmetrical pattern in gCO2, A and LE, with values always higher in the morning than in the afternoon. This asymmetry is due to the lower leaf water potentials in the afternoon than in the morning, resulting from a higher atmospheric demand and a reduced ability of the soil to supply water to the roots because of lower matric potentials and hydraulic conductivity (Fig. 4). There is a close correlation in the patterns for gCO2 and A, which both showing a midday depression that becomes more pronounced as soil drying continues. The model shows that the proportional variation in A is less than in gCO2. This is because the CO2 concentration difference (cs − ci) increases as the soil dries out, hence damping the variation in A = gCO2(cs − ci). The predicted midday depression is even less evident for LE than for A because the humidity deficit increases steadily with rising temperature from morning to mid-afternoon, counteracting the decrease in midday stomatal conductance. These predictions conform with the analysis of McNaughton & Jarvis (1983) which showed that LE is insensitive to canopy conductance for dense canopies. The general patterns in gCO2, A and LE in response to soil drying shown in Fig. 5 are similar to reports in the literature as summarized by Schulze & Hall (1982) and Maier-Maercker (1998).

Figure 5.

Diurnal variations of: (a) CO2 stomatal conductance, (b) latent heat flux, (c) assimilation rate, and (d) the ratio of the CO2 concentration in the intercellular spaces and the leaf surface for days 1, 10, 13 and 16 of the drying cycle. The asymmetry in the each quantity increases with the progression of the drying cycle and there is a marked midday depression predicted for gCO2 and A.

Soil drying also affects ci/cs. There is a small amplitude in ci/cs when soil water is adequate but a particularly strong draw-down occurs later in the drying cycle (Fig. 5d). Such variation of ci/cs with increasing drought has been observed in the field for some species (Henson et al. 1989; Brodribb 1996; Eamus et al. 1999; Giorio, Sorrentino & d’Andria 1999). Patchy stomatal conductance or variation in photosynthetic capacity in response to drought causes ci/cs to remain constant in some other species (Forseth & Ehleringer 1983; Brodribb 1996). Such responses are not considered in this paper.

Sensitivity analysis

Sensitivity of stomatal conductance to leaf water potential

Stomata of different species vary in their sensitivity to leaf water potential, and three typical response curves are given in Fig. 6a. Curve (i) is typical of plants which are highly sensitive to leaf water potential, whereas curves (ii) and (iii) represents plants progressively more tolerant to water stress (cf. lupins and wheat in Henson et al. 1989). Evidence for the general shape of the functions may be found in Schulze & Hall (1982) and in the review by Leuning et al. (2003) which supports the hypothesis of Tyree & Sperry (1988), Jones & Sutherland (1991), Sperry (2000) and others, that stomata regulate leaf water potential to avoid xylem cavitation. Figure 6b. shows that species sensitive to leaf water potential [curve (i)] will have a smaller diurnal variation in ψv as the soil dries than less sensitive plants [curve (ii) or (iii)]. This figure shows that minimum leaf potentials are almost constant for type (i) plants, whereas the minimum ψv continues to decrease throughout the drying cycle for type (iii) plants.

The effect of variations in fψv on the relationship between gCO2 and ψv is presented in Fig. 6c for days 6, 15 and 20 of the drying cycle. We see that conductance is not a monotonic function of leaf water potential as may be expected from Fig. 6a, but follows a hysteresis loop, with higher gCO2 in the morning than in the afternoon for a given ψv. Hysteresis arises because stomatal conductance responds to multiple environmental and physiological factors (light, temperature, humidity deficit, intercellular CO2 concentration) and because of the lower values of water potential at the root surfaces in the afternoon than in the morning as discussed above. The model predicts a lower maximum stomatal conductance, and a tighter hysteresis loop between gCO2 andψv on each day for plant type (i) than for (ii) and (iii). There is also a progressive decrease in the maximum stomatal conductance as soil drying proceeds, but differences in gCO2, max for the various fψv are small by day 20 (Fig. 6c).

Sensitivity of leaf water potential and stomatal conductance to plant hydraulic resistance

Figure 7 shows the sensitivity of ψv and gCO2 to three values of plant hydraulic resistance: χv = 0.25 × 107, 0.5 × 107 and 1.0 × 107 MPa s m−1. Early in the drying cycle when soil moisture is not limiting, the simulations indicate that the minimum daytime value ψv depends strongly on χv. For a given transpiration rate, lower hydraulic resistance results in less negative leaf water potentials and higher stomatal conductances. The relatively small response of gCO2 to variation in ψv at high soil moisture contents is because of the ‘shoulder’ in the function fψv at potentials near zero (Fig. 6a). The rate of soil water supply to the roots starts to limit transpiration around day 11 and there is a progressive decrease of both ψv and gCO2, with midday depressions in the latter becoming increasingly evident (Fig. 7). At this stage of the drying cycle, there is little effect of hydraulic resistance on leaf water potential and stomatal conductance, because soil moisture limits supply. Leaf and root potentials also become very similar at this time (Fig. 3).

Figure 7.

Sensitivity of (a) leaf water potential and (b) stomatal conductance for CO2 to three values of plant hydraulic resistance. The first 5 d of the simulation have been omitted in (b) because the model predicts no differences between plant hydraulic resistances during that period.

These simulations are examples of ‘anisohydric’ plants as defined by Tardieu & Simonneau (1998). In these plants, minimum daytime leaf water potential continues to decrease when the soil dries out. In contrast, ‘isohydric’ plants appear to maintain an approximately constant minimum daytime leaf water potential throughout the drying cycle. Anisohydric plants are expected to have low hydraulic resistance, χv, and/or low-to-moderate stomatal sensitivity to leaf water potential, sf (Fig. 6) whereas isohydric plants are expected to have high χv, and/or higher sf. Tardieu & Davies (1993) and Dewar (2002) explained these results through the combined roles of the hormone abscisic acid (ABA) and the supply and loss of water to/from the leaves on stomatal conductance. While not denying a role of ABA in regulating stomatal aperture at the cellular level, the present model is able to explain the contrasting behaviour of anisohydric and isohydric plants in terms of differing plant hydraulic resistances and stomatal sensitivities of stomata to leaf water potential, without explicitly invoking hormonal control.

Sensitivity of stomatal conductance and cumulative transpiration to soil characteristics

Predictions for gCO2 and cumulative soil water consumption during a 30 d drying cycle are shown in Fig. 8 for three soil types: loam, silt loam and sandy loam (see Table 1 for soil parameter values). Stomatal conductance is insensitive to soil type during the first 10 d of drying (results not shown). Conductances for plants in sandy loam then begin to respond to soil drying, showing a rapid decrease each day and marked midday depressions. However, reduction in cumulative transpiration rates are observed only 4 d later. A similar picture is seen for the silt loam and loam soils, except that the onset of soil limitation to conductance and transpiration occurs progressively later. Differences in the responses of stomatal conductance to soil type are a function of their moisture retention characteristics, with matric potential dropping more rapidly at low moisture contents in the order sandy loam, loam and then silt loam. This causes leaf water potential and stomatal conductance to decrease in the same order.

Figure 8.

Sensitivity of (a) stomatal conductance, and (b) cumulative transpiration to soil type with hydraulic characteristics given in Table 1. The first 10 d of the simulation have been omitted in (a) because the model predicts no differences between soil types during that period.

Sensitivity of stomatal conductance and cumulative transpiration to root densities

The sensitivity of gCO2 and soil water consumption to root length density is presented in Fig. 9. These results correspond to three values of root length density: 2, 4 and 6 km m−3 in silty loam during a 30 d drying cycle. The first 5 d of the simulation have been omitted from the top panel because the model predicts no differences between root densities during that period. Small differences in conductance appear after day 7 of drying, but differences in transpiration are evident only after 11 d. Higher root length densities maintain higher stomatal conductances only from days 8–17, after which lack of soil water causes similar limitations to stomatal conductance, irrespective of root length density.

Figure 9.

Sensitivity of (a) stomatal conductance, and (b) cumulative transpiration to root densities. The first 5 d of the simulation have been omitted in (a) because the model predicts no differences between root densities during that period.

Transpiration rates as a function of soil moisture and atmospheric demand

So far we have focused on relationships between stomatal conductance, transpiration and water potential gradients between the soil, root and leaves, during a soil drying cycle. However, it is also possible to consider the role of soil water content in controlling transpiration, since soil water potential and water content are related through the moisture retention curve (Eqn 4). In a classic paper, Denmead & Shaw (1962) hypothesized that leaf wilting, and hence stomatal closure, will occur at higher soil moisture contents when atmospheric demand for water vapour increases. This is expected to occur because the moisture content immediately around the roots will be rapidly depleted at high atmospheric demand and the decrease of soil conductivity will mean that the soil will not be able to supply the water sufficiently quickly (Fig. 4). Leaf water potential will also be lower at high transpiration rates. To support this hypothesis, Denmead & Shaw (1962) showed that transpiration rates declined at higher soil moisture contents when atmospheric demand was high, compared to when demand was low.

We attempted to simulate these results using the current model and the results are shown in Fig. 10a for transpiration rate as a function of soil moisture content and evaporative demand. Because the transpiration rate varies during the day, only the midday values are plotted, to emphasize the changes from day to day. Contrary to the findings of Denmead and Shaw, the model shows there is no cross-over of the curves, although there is a suggestion that LE does start to decline at a higher soil moisture content when demand is high compared to when it is low. However, the predicted transition from energy-limited transpiration (approximately horizontal part of curve) to limitation by soil water supply occurs over a rather narrow moisture content range. Cross-over of the curves is found when LE is plotted as a function of time (Fig. 10b). Longer times correspond drier soils, so it appears that the results of Denmead & Shaw (1962) can be explained if their graph is a composite derived from plants which started transpiration at different initial moisture contents.

Figure 10.

Maximum daily transpiration rates as a function of (a) soil moisture content, and (b) time, for three different levels of atmospheric evaporative demand. The diurnal variation of climatic demand used in these simulations have a mean daytime value equal to: 201, 237 and 274 W m−2.

Response of gCO2, LE, A and ci/cs to humidity deficit at the leaf surface

It was seen in Fig. 6c that a plot of the diurnal variation gCO2 versus ψv shows hysteresis, even though the relationship between gCO2 and ψv used in the stomatal model is monotonic (Eqn 2, Fig. 6a). Hysteresis occurs because conductance depends simultaneously on multiple factors such as light, temperature, intercellular CO2 concentration and water flow through the soil and plant. Similarly, no unique relationships are expected between gCO2, LE, A and ci/cs and humidity deficit at the leaf surface, Ds (Fig. 11) when multiple factors vary concurrently. Each curve shows hysteresis, the degree of which depends on the above factors. During a daily cycle, the model predicts that gCO2 and Ds are positively correlated at low values of Ds when absorbed light largely controls conductance through photosynthesis, whereas there is a negative correlation between gCO2 and Ds later in the day (Fig. 11a). Conductances are higher in the morning than in the afternoon at the same value of Ds because soil matric potential around the roots differs from morning and afternoon (Fig. 4). Such hysteresis has been observed by Pereira, Tenhunen & Lange (1987), Takagi, Tsuboya & Takahashi (1998), Eamus et al. (1999) and by Meinzer et al. (1997) for a variety of plant species when measurements were made in the field. According to the model presented here, the hyperbolic relationship between gCO2 and Ds used by Lohammer et al. (1980) and by Leuning (1995) is apparent only at high values of Ds once soil moisture starts to limit water supply to the roots. Note that the maximum value of Ds increases from day 1 to day 16, despite the use of the same maximum atmospheric humidity deficit at the reference height. This occurs because Ds depends on interactions between stomatal conductance, transpiration rate and leaf temperature through the leaf energy balance.

Figure 11.

Predicted relationships between humidity deficit at the leaf surface and: (a) stomatal conductance; (b) transpiration rate; (c) assimilation rate; and (d) the ratio of the CO2 concentration in the intercellular spaces and the leaf surface for days 1, 10, 13 and 16 of the drying cycle. The model predicts hysteresis in all the relationships shown as a result of multiple factors such as light, temperature, intercellular CO2 concentration and water flow through the soil and plant.

Transpiration rates (LE) are positively correlated with Ds throughout the day early in the drying cycle, but once soil water supply becomes limiting (days 13 and 16 in Fig. 11b), LE is negatively correlated to Ds from mid-morning to mid-afternoon and then insensitive to Ds later in the day. Similar trends are observed in the relationship between A and Ds, except that the negative correlation is more apparent at low water contents than it is for LE (Fig. 11c). Intercellular CO2 concentrations are relatively independent of Ds at high soil moisture contents, but are predicted to decrease markedly later in the drying cycle (Fig. 11d). Evidence of diurnal hysteresis in some or all of gCO2, LE, A and ci/cs under field conditions when plotted against, time of day, humidity deficit and leaf water potential may be found in Schulze & Hall (1982), Henson et al. (1989), Brodribb (1996) and Meinzer et al. (1997).

Dependence of A and LE on gCO2

The predicted dependence of A and LE on gCO2 is shown in Fig. 12 for several days during a drying cycle when all factors controlling stomatal conductance co-vary. The relationship between A and gCO2 is approximately linear when soil water is non-limiting (Fig. 12a), implying an approximately constant value of ci (Fig. 11d). Hysteresis appears in the relationship by day 13 and the spread in the hysteresis curve increases as the soil dries out, yielding results similar to those presented by Schulze & Hall (1982) and Pereira et al. (1987) for observations made under field conditions.

Figure 12.

Predicted relationships between stomatal conductance for CO2 and: (a) assimilation rate and (b) transpiration rate for days 1, 10, 13 and 16 of the drying cycle. Hysteresis occurs in the relationships because of interactions of multiple factors.

There is also hysteresis in the predicted relationship between LE and gCO2 (Fig. 12b), very similar to results presented by Meinzer et al. (1997). Transpiration and gCO2 are positively correlated throughout the day when soil moisture is adequate, but for moderately dry soils LE remains approximately constant whereas gCO2 declines between mid-morning and mid-afternoon. There is some recovery of stomatal conductance after the midday depression in these simulations, but LE continues to decline with reduced atmospheric demand late in the afternoon (cf. Figs 5 & 12b). Farquhar (1978) argued that simple feedback between gCO2 and Ds was unable to account for a decrease in LE with decreasing gCO2 and postulated that stomata must sense water stress by mechanism(s) that involve more than transpiration (so-called ‘feed-forward’ response). Our model shows that it is unnecessary to invoke a feed-forward response of stomata to Ds, when stomatal conductance is a function of leaf water potential and the dynamics of the resistance to water transport from soil to roots is included in the model. Hydraulic resistance can vary due to changes in soil hydraulic conductivity with soil moisture content around the roots and with water potential-dependent cavitation in the xylem (Sperry 2000; Buckley & Mott 2002).

CONCLUSIONS

A fully coupled soil-plant-atmosphere continuum model has been developed in which stomatal conductance is a function of photosynthesis, intercellular CO2 concentration and leaf water potential. The latter in turn depends on soil water potential, the flux of water through the soil and plant, xylem hydraulic resistance and hence stomatal conductance. Diurnal patterns of leaf water potential, absorption of water by roots, canopy transpiration and photosynthesis are predicted by the model using specified meteorological variables, soil water content and canopy and soil characteristics.

The model reproduces the observed diurnal variation in stomatal conductance, assimilation rate and transpiration for a typical crop. It successfully captures the daytime asymmetry in conductance, assimilation and transpiration under water stress conditions and it describes successfully the observed hysteresis in stomatal conductances versus leaf water potential. Hysteresis occurs because leaf water potential is a function of the dynamics in the distribution of soil matric potential around the roots. Once soils start to dry, matric potentials close to the roots are less negative in the morning than in the afternoon and thus, for a given level of atmospheric demand for water from the leaves, stomatal conductances are higher earlier in the day.

The model shows that ‘isohydric’ plants, which maintain constant minimum leaf water potentials, are expected have high xylem hydraulic resistance and/or high stomatal sensitivity to leaf water potential. ‘Anisohydric’ plants, which have variable minimum leaf water potentials, are expected to have lower hydraulic resistances and/or lower stomatal sensitivity to leaf water potential.

Model sensitivity analysis for a soil dry-down cycle of 30 d shows that stomatal conductance depends on soil and plant hydraulic properties and root length density. Variation in these model parameters cause differences in stomatal conductance and leaf water potential to appear only after approximately 10 d of drying, when supply of water by the soil to the roots becomes limiting. High atmospheric demand causes transpiration rate, LE, to decline at a slightly higher soil water content, θs, than at low atmospheric demand, but all curves of LE versus θs fall on the same line when soil water supply limits transpiration. The model emphasizes that stomatal conductance cannot be modelled in isolation, but must be fully coupled with photosynthesis/respiration and the transport of water from soil, through roots, stems and leaves to the atmosphere.

The above results may be criticised because the model is too simple or that no direct experimental evidence has been presented in support of the model. The model has deliberately been kept simple to emphasize the key processes involved in stomatal control of the coupled exchanges of CO2 and water vapour between plants and the atmosphere. For example, the model does not separate the canopy into sunlit and shaded leaves (cf. DePury & Farquhar 1997; Wang & Leuning 1998), and uses a single, homogeneous soil layer with a uniform root distribution. Phenomena such as patchy stomatal conductance or reductions in photosynthetic capacity with drought have also not been considered. Despite these limitations, the model successfully predicts patterns in the diurnal variation of gCO2, LE, A and ci/cs widely observed under field conditions for plants in well-watered and drying soils. However, it is not possible to verify the model with a single experiment. The details of the response for a given experiment will vary considerably, depending on a combination of meteorology, plant physiology (e.g. photosynthetic capacity, hydraulic resistance), canopy density, soil moisture content and soil hydraulic properties. The key conclusion of the paper is that stomatal conductance cannot be modelled using leaf-level processes alone, but must be incorporated into a comprehensive model of CO2 exchange by leaves and water flow from soil through the plants to the atmosphere.

APPENDIX A

Models for canopy energy balance and CO2 exchange.

Canopy energy balance

The conceptual structure of the canopy energy balance model is shown in Fig. 1. The model is a two-source (soil and vegetation) model of heat, water and carbon exchange between the canopy and the atmosphere (Choudhury & Monteith 1988; Norman, Kustas & Humes 1995; Anderson et al. 2000). Each flux from the canopy to the atmosphere (H, LE, FCO2) consists of a contribution from the soil (subscript ‘s’) and from the foliage layer (subscript ‘v’). Here rbv is the bulk boundary layer resistance, while rsH2O and rsCO2 are the bulk stomatal resistances for H2O and CO2, respectively. The resistances Rs and Rv describe turbulent transport from the ground to the canopy source-sink height, hs, and from hs to the top of canopy (see Appendix c). An aerodynamic resistance ra describe heat and mass (H2O and CO2) transport above the canopy (from the top of the canopy to a reference level, zr). The foliage layer is characterized by a leaf surface temperature, Tsv (K) a bulk leaf water potential, Ψv (MPa) and a CO2 concentration in the intercellular spaces, ci (mol CO2 mol−1 air). At hs the characteristics of the air surrounding the leaves (temperature, Tav, vapour pressure, P(Trv), and CO2 concentration, cv) result from turbulent transport through Rs and Rv. The soil surface is characterized by a surface temperature, Tss.

The sensible heat fluxes from the soil surface Hs, the foliage layer Hv and the canopy (total flux) H are calculated, respectively, as:

image((A1))

where ρ is the air density, cp is the specific heat of air at constant pressure, and Ta is the air temperature at the reference level, zr, above the canopy.

Soil between the surface and the damping depth is divided into an upper, completely dry layer and a lower wet layer (Seginer 1971; Sifaoui & Perrier 1978; Choudhury & Monteith 1988). The damping depth ds is related to the thermal properties of the soil and the frequency of the temperature variation as follows:

inline image

where λw is the thermal conductivity of wet layer, C is the volumetric heat capacity, ω equals 2π times the frequency of the temperature variation, thus for the diurnal variation ω = 2π/86 400 = 7.27 × 10−5 s−1. Soil evaporation takes place at the top of the wet layer, at depth dm, and the transfer of water vapour through the upper soil layer is regulated by a resistance rm given by:

image((A2))

where Dv is the water vapour diffusivity in air, f is the dry layer porosity and ξ is the tortuosity factor. As soil evaporation proceeds, the site of evaporation moves progressively deeper and the thickness of the dry layer changes according to[Δdm = EΔt/θs], where E is the total evaporation flux density, Δt is the model time step, θs is the soil water content.

The latent heat fluxes from the soil surface LEs, the foliage layer LEv and the canopy LE are given by:

image((A3))

where L is the latent heat of vaporization of water, Mv is the molecular weight of water vapour, R is the gas constant. Soil at the wet–dry interface is assumed to be saturated at a temperature, Tm and in the foliage elements, the vapour pressure of air in contact with the substomatal cavities is expressed as a function of the bulk leaf water potential, Ψv through hi = exp(MvJv/ρvRTsv), the fractional relative humidity in the intercellular spaces.

Energy exchange between the canopy and the atmosphere satisfies the conservation equations:

image((A4))

where Rnv and Rns are the net absorption of radiation in the foliage layer and at the soil surface, respectively; and G0 and Gs are the conduction downwards from the soil surface and into the wet soil layer, respectively. Partitioning of total net radiation between Rnv and Rns is written as:

image((A5))

where Rg is the global solar radiation above the canopy, av is the average canopy albedo, Ra is the long-wave radiation of the atmosphere, k is the extinction coefficient (k = 0.7) and LAI is the leaf area index of the foliage. Soil heat fluxes G0 and Gs are calculated by assuming linear temperature gradients:

image((A6))

where λd is the thermal conductivity of the dry layer extending from the surface to dm, Tm is the soil temperature at the dry–wet layer interface and Td is the temperature at the damping depth dd.

Canopy CO2 exchange model

The CO2 flux between the reference height and the canopy air space, FCO2, arises as the effect of net canopy photosynthesis, A (positive toward the canopy in units of mol m−2 s−1) and soil respiration, Fsoil (positive upwards):

image((A7))

The net canopy photosynthesis, A is calculated using the detailed biochemical model of photosynthesis of Farquhar (Farquhar, von Caemmerer & Berry 1980) which is widely employed in both leaf and canopy-scale gas studies (Harley et al. 1992; Leuning et al. 1995; Baldocchi & Meyers 1998; Wang & Leuning 1998). In this model CO2 uptake is either limited by Rubisco activity and the respective partial pressures of the competing gases CO2 and O2 at the sites of carboxylation Vc or by electron transport, Vj which limits the rate at which Ribulose bisPhosphate (RuBP) is regenerated. Net canopy photosynthesis A may then be expressed as:

image((A8))

where Rd is the rate of CO2 evolution from processes other than photorespiration. The term min(Vc, Vj) represents the minimum of Vc and Vj. The assimilation rate, Vc, is given by:

image((A9))

where Vlmax is the maximum carboxylation rate per unit leaf area when RuBP is saturated, Γ* is the CO2 compensation point in the absence of day respiration, oi is the intercellular oxygen concentration, and Kc and Ko are Michaelis coefficients for CO2 and O2, respectively. The rate of carboxylation limited solely by the rate of RuBP regeneration due to electron transport is calculated as:

image((A10))

where J is the electron transport rate for a given absorbed photon irradiance. Both Vc and Vj are functions of ci and temperature, and Vj also depends on absorbed radiation. Respiration is also dependent on leaf temperature. Parameter values and their dependence on leaf temperature are from Leuning (1995) and are summarized in Appendix B.

The maximum canopy photosynthetic capacity, Vcmax is calculated by an integral of the leaf photosynthetic capacity and the leaf nitrogen profile over the entire canopy (De Pury & Farquhar 1997):

image((A11))

where Vlmax is the maximum photosynthetic capacity per unit leaf area, LAI is the leaf canopy area index and kn is the coefficient of leaf nitrogen allocation.

The CO2 flux, from the soil surface, Fsoil, is computed using the Arrhenius equation:

image((A12))

where F0,soil is the CO2 flux from the soil at T0 = 298 K (F0,soil = 3 µmol m−2 s−1), Ea,soil is the activation energy (Ea,soil = 53000 J mol−1), R is the universal gas constant, Ts10 (K) is soil temperature at 10 cm. In this model Ts10 is replaced with the soil temperature at the dry–wet layer interface Tm.

Canopy photosynthesis, A and net CO2 flux from the canopy plus soil, FCO2, are also governed by the biophysical control of canopy resistances and they are parameterized, respectively, as:

image((A13))

where µ is the ratio of diffusivity of CO2 and H2O appropriate to leaf boundary layer diffusion and β (= P/RT) is the coefficient used to convert the molar units (mol−1 m2 s) into physical resistance units (s m−1) and P is the atmospheric pressure.

APPENDIX B

Parameters in the biochemical model of photosynthesis

The dependence of electron transport rate J on absorbed photon irradiance Q was modelled by Farquhar & Wong (1984) using the non-rectangular hyperbolic function:

image((B1))

where θ determines the shape of the non-rectangular hyperbola, Jlmax is the potential rate of whole-chain electron transport per unit leaf area and α is the quantum yield of whole-chain electron transport. Parameters values used in the photosynthesis model are given in Table 3.

Table 3.  Valuesof parameters used in the model of photosynthesis presented in AppendixB
SymbolParameterValueUnits
Trefreference temperature293.2K
inline imagemax photosynthetic capacity at Tref100 × 106mol CO2 m−2 s−1
F0,soilCO2 flux from the soil at 298 K3µmol CO2 m−2 s−1
Ea,soilactivation energy for soil CO253000J mol−1
oiintercellular oxygen concentration210 × 10−3mol mol−1
Ko,refMichaelis coefficient for O2256 × 10−3mol mol−1
Kc,refMichaelis coefficient for CO2302 × 10−6mol mol−1
Eoactivation energy for Ko36 000.J mol−1
Ecactivation energy for Kc59 430.J mol−1
EaVactivation energy for Vcmax58 520.J mol−1
EdVdeactivation energy for Vcmax220 000.J mol−1
EaJactivation energy for Jmax37 000.J mol−1
EdJdeactivation energy for Jmax220 000.J mol−1
Sentropy term for Jmax and Vcmax700.J mol−1 K−1
αquantum yield0.2 
θparameter of the hyperbola0.9 
γ0parameter in Γ*Eqn B428 × 10−6mol mol−1
γ1parameter in Γ*Eqn B40.0509 
γ2parameter in Γ*Eqn B40.0010 

The temperature sensitivity of the Michaelis–Menten constants Kc and Ko, the maximum photosynthetic capacity per unit leaf area Vlmax and the CO2 compensation point in the absence of day respiration Γ* were calculated following Leuning (1995). A normalized Arrhenius equation was used to represent the temperature dependence of Kc and Ko:

image((B2))

in which R is the universal gas constant, Ex is the activation energy for Kx, where the subscript x stands for either c or o, and Kx,ref is the value of Kx at a reference temperature Tref.

The temperature function used for Vlmax is:

image((B3))

where inline image is the value of Vlmax at Tref, Ea is the energy of activation, Ed is the energy of de-activation and S is an entropy term. The potential rate of electron transport per unit leaf area at the reference temperature inline image is assumed to be proportional to inline image and the proportionality coefficient is equal to 2.0 (Leuning 1997).

A second-order polynomial was used to describe the temperature dependence of Γ*, namely

image((B4))

where γ0, γ1, and γ2 are empirical coefficients, T is leaf temperature and Tref is a reference temperature.

APPENDIX C

Aerodynamic resistances

The aerodynamic resistances Rs, Rv and ra (Fig. 1) are calculated assuming an exponential profile of wind speed u(z) and eddy diffusivity K(z) inside the canopy:

image((C1))

where u(hc) and K(hc) are the wind speed and eddy diffusity at the canopy top, respectively, and η is the extinction factor. Assuming a vertically homogeneous canopy with a constant mixing length lc, η is expressed as:

image((C2))

where cd is the leaf drag coefficient (Inoue 1963). Thus, Rs and Rv and ra are obtained using:

image((C3))

where z0g is the roughness length of soil surface.

Finally, the bulk boundary layer resistance rbv can be written as:

image((C4))

where dl is the average leaf width and Ct is the transfer coefficient (Ct = 156.2).

APPENDIX D

List of variables, categorized by the models in which they are used

SymbolUnitsDescription
Canopy energy balance model
 CJ m−3 K−1soil volumetric heat capacity
 cdleaf drag coefficient
 cpJ kg−1 K−1specific heat of air at constant pressure
 ddmdamping depth of the thermal wave
 dlmaverage leaf width
 dmmthickness of the upper dry soil layer
 Dvm2 s−1water vapour diffusivity in air
 fdry layer porosity
 G0W m−2conduction downwards from the soil surface
 GsW m−2conduction into the wet soil layer
 HW m−2sensible heat flux from the canopy (total flux)
 hcmheight of the canopy
 hsmcanopy source-sink height
 HsW m−2sensible heat flux from the soil surface
 HvW m−2sensible heat flux from the foliage layer
 kextinction coefficient of radiation
 K(z)m2 s−1eddy diffusivity at level z
 LJ kg−1latent heat of vaporization of water
 LAIleaf area index of the foliage
 lcmcanopy mixing length
 LEW m−2latent heat fluxes from the canopy
 LEsW m−2latent heat fluxes from the soil surface
 LEvW m−2latent heat fluxes from the foliage layer
 Mvkg mol−1molecular weight of water vapour
 PMPaatmospheric pressure
 P(T)MPasaturation water vapour pressure at temperature T
 RJ K−1 mol−1ideal gas constant
 ras m−1aerodynamic resistance for heat and mass above the canopy
 RaW m−2long-wave radiation of the atmosphere
 rbvm s−1bulk boundary layer resistance
 RgW m−2global solar radiation above the canopy
 rms m−1resistance to water vapour transfer through the dry soil layer
 RnW m−2net absorption of radiation in the canopy
 RnsW m−2net absorption of radiation at the soil surface
 RnvW m−2net absorption of radiation in the foliage layer
 Rss m−1turbulent transport resistance from the ground to hs
 rsCO2s m−1bulk stomatal resistance for CO2
 rsH2Os m−1bulk stomatal resistance for H2O
 Rvs m−1turbulent transport resistance from hs to the top of canopy
 TaKair temperature at the reference level, zr
 TavKtemperature of the air surrounding leaves
 TdKtemperature at the damping depth
 TmKsoil temperature at the dry–wet layer interface
 TrKdew point temperature at the reference level, zr
 TrvKdew point temperature of the air surrounding leaves
 Tss.Ksoil surface temperature
 TsvKfoliage temperature
 u(z)m s−1wind speed at level z
 z0gmroughness length of soil surface
 zrmreference level above the canopy
 avaverage canopy albedo
 ξdry layer tortuosity factor
 ωs−12π times the frequency of the temperature variation
 λdW m−1 K−1thermal conductivity of the of the upper dry soil layer
 λwW m−1 K−1thermal conductivity of the wet layer
 ηextinction factor of wind
 ρkg m−3air density
 Δtsmodel time step
Stomatal conductance model
 aempirical coefficient
 fψvempirical function (sensitivity of stomata to leaf water potential)
 g0mol m−2 s−1limiting value of gCO2 at the light compensation point
 gCO2mol m−2 s−1stomatal conductance for CO2
 sfMPa−1sensitivity parameter
 Γmol mol−1CO2 compensation point
 ψfMPareference potential
 ψvMPaleaf water potential
Plant water transfer model
 bempirical parameter
 Fm3 m−2 s−1flow of water through the plant canopy
 hirelative humidity in the intercellular spaces
 Kss−1 MPa−1 m2soil hydraulic conductivity
 Ksats−1 m2matric flux potential
 Lrm m−3root length per soil volume
 rmsoil radial distance from the axis of the root
 tstime of day
 zrootmroot depth
 ψeMPaair entry water potential of the soil
 ψrMPawater potential of the roots
 θsm3 m−3soil water content
 ΨsMPasoil water potential
 θsatm3 m−3saturation water content of the soil
 ρvkg m−3liquid water density
 χvMPa s m−1plant hydraulic resistance
Canopy CO2 exchange model
 F0,soilmol m−2 s−1CO2 flux from the soil at T0 = 298 K
 inline imagemol m−2 s−1max carboxylation rate per unit leaf area at Tref
 Amol m−2 s−1net canopy photosynthesis
 cimol mol−1CO2 concentration in the intercellular spaces
 Ea,soilJ mol−1soil activation energy
 EaJJ mol−1activation energy for Jmax
 EaVJ mol−1activation energy for Vcmax
 EcJ mol−1activation energy for Kc
 EdJJ mol−1deactivation energy for Jmax
 EdVJ mol−1deactivation energy for Vcmax
 EoJ mol−1activation energy for Ko
 FCO2mol m−2 s−1CO2 flux between the reference height and the canopy air space
 Fsoilmol m−2 s−1CO2 flux, from the soil surface
 Jmol m−2 s−1electron transport rate for a given absorbed photon irradiance
 Vlmaxmol m−2 s−1maximum carboxylation rate per unit leaf area
 Kcmol mol−1Michaelis coefficient for CO2
 Kc,refmol mol−1Michaelis coefficient for CO2 at Tref
 kncoefficient of leaf nitrogen allocation
 Komol mol−1Michaelis coefficient for O2
 Ko,refmol mol−1Michaelis coefficient for O2 at Tref
 oimol mol−1intercellular oxygen concentration
 SJ mol−1 K−1entropy term for Jmax and Vcmax
 TrefKreference temperature
 Ts10Ksoil temperature at 10 cm
 Vcmol m−2 s−1Rubisco activity at the sites of carboxylation
 Vjmol m−2 s−1electron transport
 αquantum yield
 µratio of diffusivities appropriate to leaf boundary layer diffusion
 βmol m−3coefficient to convert molar units into physical resistance units
 θparameter of the hyperbola
 Γ*mol mol−1CO2 compensation point in the absence of day respiration
 γ0mol mol−1parameter in Γ* equation
 γ1parameter in Γ* equation
 γ2parameter in Γ* equation

Ancillary