Modelling advection and diffusion of water isotopologues in leaves

Authors

  • MATTHIAS CUNTZ,

    Corresponding author
    1. Environmental Biology Group, Research School of Biological Sciences, Australian National University, Canberra, Australia,
      M. Cuntz. Fax: +49 3641 57 7353; e-mail: mcuntz@bgc-jena.mpg.de
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    • *

      Present address: Max-Planck-Institut für Biochemie, Jena, Germany.

  • JÉRÔME OGÉE,

    1. Functional Ecology and Environmental Physics (EPHYSE), INRA, Villenave d'Ornon, France and
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  • GRAHAM D. FARQUHAR,

    1. Environmental Biology Group, Research School of Biological Sciences, Australian National University, Canberra, Australia,
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  • PHILIPPE PEYLIN,

    1. BiOEMCO, CNRS/INRA/UPMC, Grignon, France
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  • LUCAS A. CERNUSAK

    1. Environmental Biology Group, Research School of Biological Sciences, Australian National University, Canberra, Australia,
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    • Present address: Smithsonian Tropical Research Institute, Republic of Panama.


M. Cuntz. Fax: +49 3641 57 7353; e-mail: mcuntz@bgc-jena.mpg.de

ABSTRACT

We described advection and diffusion of water isotopologues in leaves in the non-steady state, applied specifically to amphistomatous leaves. This explains the isotopic enrichment of leaf water from the xylem to the mesophyll, and we showed how it relates to earlier models of leaf water enrichment in non-steady state. The effective length or tortuosity factor of isotopologue movement in leaves is unknown and, therefore, is a fitted parameter in the model. We compared the advection–diffusion model to previously published data sets for Lupinus angustifolius and Eucalyptus globulus. Night-time stomatal conductance was not measured in either data set and is therefore another fitted parameter. The model compared very well with the observations of bulk mesophyll water during the whole diel cycle. It compared well with the enrichment at the evaporative sites during the day but showed some deviations at night for E. globulus. It became clear from our analysis that night-time stomatal conductance should be measured in the future and that the temperature dependence of the tracer diffusivities should be accounted for. However, varying mesophyll water volume did not seem critical for obtaining a good prediction of leaf water enrichment, at least in our data sets. In addition, observations of single diurnal cycles do not seem to constrain the effective length that relates to the tortuosity of the water path in the mesophyll. Finally, we showed when simpler models of leaf water enrichment were suitable for applications of leaf water isotopes once weighted with the appropriate gas exchange flux. We showed that taking an unsuitable leaf water enrichment model could lead to large biases when cumulated over only 1 day.

INTRODUCTION

Leaf water isotope enrichment is a cornerstone of a variety of isotopic applications. Leaf water imprints the oxygen isotopic composition on different substances such as atmospheric CO2 and O2, and plant organic matter. This happens in different parts of the leaf and at different times. For example, leaf organic matter is formed in different intracellular compartments in the leaf so that the overall isotopic enrichment is thought to be more determined by bulk mesophyll water than by water at the evaporative sites (e.g. Barbour & Farquhar 2000; Cernusak et al. 2003). Leaf organic matter is mostly formed while there is energy input from photosynthesis so that the leaf organic matter reflects the assimilation-weighted average bulk mesophyll water enrichment. Atmospheric CO2 and O2, however, are probably determined by the isotopic composition of leaf water near the evaporating sites, at least in C3 plants. Oxygen formation, for example, occurs at photosystem II, that is, in the chloroplasts, and the isotopic composition of the released oxygen is therefore close to the leaf water enrichment at the evaporative sites (Hill 1965; Helman et al. 2005). Water is only split in photosystem II, and oxygen evolved, if there is photosynthesis. The distribution of atmospheric oxygen isotopologues (cf. Dole effect, e.g. Bender, Sowers & Labeyrie 1994) is therefore determined by the electron transport-weighted average leaf water enrichment at the evaporative site. Atmospheric CO2, however, can enter the leaf through the stomatal pores, exchange its isotopic composition with leaf water at the evaporative sites and exit the leaf through the stomatal pores again (Farquhar et al. 1993). This process occurs if stomata are open, regardless of whether there is photosynthesis or not. The isotopic composition of oxygen in atmospheric CO2 is therefore determined by the leaf water enrichment at the evaporative sites weighted by the product of stomatal conductance and the CO2 mixing ratio in the sub-stomatal cavity (one-way CO2 flux from the stomata to the atmosphere) (Cernusak et al. 2004). Thus, it is essential to understand the time course of leaf water enrichment at both the evaporating sites and in the leaf mesophyll in order to harness the different isotopic signals.

Leaf water isotope enrichment is conventionally described by the steady-state Craig and Gordon equation (Craig & Gordon 1965 and Appendix B, Craig and Gordon model)

image(1)

where Rs is the isotopic ratio of plant available source water; Rv is the water vapour isotope ratio; α+ (>1) is the equilibrium fractionation factor between liquid water and water vapour; αk (>1) is the total kinetic fractionation factor associated with diffusion, and h = wa/wi is the relative humidity corrected to leaf temperature, where wi and wa (mol mol−1) denote air moisture in the stomatal cavity and in ambient air, respectively.

A number of studies have reported that observed leaf water is less enriched than the prediction of the steady-state Craig and Gordon equation, even after removing the unenriched veins before the measurements (e.g. Leaney et al. 1985; White 1989; Yakir, Deniro & Rundel 1989; Bariac, Jusserand & Mariotti 1990; Flanagan, Bain & Ehleringer 1991; Flanagan, Comstock & Ehleringer 1991; Flanagan 1993; Wang, Yakir & Avishai 1998; Roden & Ehleringer 1999). The cause of this observation is thought to relate to the following two explanations: (1) the leaf represents a continuum or a combination of unenriched (source) and enriched (evaporative sites) water, and (2) the steady-state assumption of the Craig and Gordon equation is only valid during certain periods of the day (e.g. midday) or over longer time scales, that is, days to weeks.

The former explanation has been termed the Péclet effect (Farquhar & Lloyd 1993): water isotopologues in the mesophyll are subject to advection from the xylem to the evaporative sites and back-diffusion of the enriched water at the evaporative sites to the xylem. The relative importance of advection and diffusion is characterized by a Péclet number ℘: if ℘ > 1 advection is dominant and if ℘ < 1 diffusion is the dominant transport process. Farquhar & Lloyd (1993) assigned the following Péclet number for advection–diffusion of water isotopes in leaves:

image(2)

where E is transpiration rate (mol m−2 s−1); D (m2 s−1) is the tracer diffusivity in liquid water; C = 106/18 = 55.6·103 mol m−3 is the molar water concentration, and Leff (m) is the effective length of water movement in the leaf mesophyll. The argument of Farquhar & Lloyd (1993) follows the idea outlined in Fig. 1a. The advection–diffusion in this picture happens along the path of the water molecules. Because of tortuosity of the water path through the mesophyll, the advection speed v along the path is k times greater than the slab velocity E/C, that is, v = kE/C. The effective length is therefore seen as k times the actual distance between the leaf xylem and the evaporative sites. In this picture and in the steady state, the mesophyll water isotope ratio increases exponentially from the xylem (Rs) to the Craig and Gordon value RC at the evaporative sites. Bulk steady-state mesophyll water enrichment inline image(expressed relative to source water) can then be written as

Figure 1.

Idealized one-dimensional leaf geometry of (a) Farquhar & Lloyd (1993) and subsequent publications, and (b) of this study [and Ogée et al. (2007)]. The lower end of the inner box represents the interface between xylem and mesophyll (rx), and the upper end represents the evaporating sites in the stomata (re). In (a), water isotopologues move along the grey tortuous path with advection velocity kE/C and diffusivity D. An isotopic gradient builds individually in each tortuous path, that is, individual paths are independent of each other. In (b), water moves along different tortuous paths (black lines) that cross and intermingle. The isotope gradient builds effectively in the r-direction (grey shading, darker means more enriched), with velocities reduced by tortuosity factor κm and volumetric mesophyll water content Θm. White circles represent air spaces that lead to Θm < 1. The figure thus represents a conceptualization of the difference between the present modelling approach and that proposed previously by Farquhar & Lloyd (1993). Refer to Appendix A for definition of symbols.

image(3)

with ΔC the Craig and Gordon steady-state enrichment expressed relative to source water (i.e. Eqn 1 expressed relative to source water Rs).

Regarding the second explanation mentioned earlier, a non-steady-state leaf water enrichment model was first presented by Dongmann et al. (1974). By taking fixed environmental conditions between subsequent measurements, they found an iterative solution to compute leaf water enrichment over the diurnal course. However, Dongmann et al. (1974) and subsequent publications (Bariac et al. 1994;Cernusak, Pate & Farquhar 2002) did not distinguish between bulk leaf (mesophyll) water enrichment and enrichment at the evaporative sites. [In the following, we refer to this model as the Dongmann et al. (1974) non-steady-state model, but actually use the more rigorous formulation of Bariac et al. (1994).]

Farquhar & Cernusak (2005) combined both approaches in a non-steady-state model of bulk mesophyll water enrichment. They distinguished between bulk mesophyll water isotopic enrichment and enrichment at the evaporative sites, and they did not assume isotopic steady state. However, the model still contains a variety of assumptions. The model assumes, for example, that there is an exponential profile in the isotope ratio from the xylem to the evaporating sites and that this exponential profile has the same form as in steady state (Eqn 3 or cf. Eqns A14–A16). Another assumption is that a single effective mixing length Leff describes the movement of water isotopes through the leaf mesophyll. Yet the leaf veins are generally in the middle of the leaf and, at least in amphistomatous leaves, there are two evaporative sites, one on each side of the leaf. So, there should be two isotope gradients in leaves, one at the adaxial (upper) side and one at the abaxial (lower) side. The measured mesophyll water enrichment Δm is then the water volume (Vm,up, Vm,down) weighted enrichment of both sides (Δm,up, Δm,down), that is, Δm = (Vm,upΔm,up + Vm,downΔm,down)/Vm. In steady state, this means that Eqn 3 should rather be

image(4)

which cannot be expanded to give a sensible combination of one single ℘m or Leff. This equation becomes that of Farquhar & Lloyd (1993) (Eqn 3) for leaves with a symmetrical cross-section (e.g. wheat). A third and maybe less critical assumption is that the tracer diffusivity D is constant. Yet it is shown in Appendix B (Diffusivity D) that the temperature dependence of D is significant.

We present here a description of advection and diffusion of water isotopologues in leaves that allows us to test some of the underlying assumptions of the Farquhar & Cernusak (2005) non-steady-state model. It is conceptually similar to the Péclet description, in that it assumes that water isotopologues are subject to advection and diffusion. We further compare the advection–diffusion model to observations and to other existing models in order to test the circumstances under which assumptions and simplifications are appropriate. A description of the advection–diffusion model is given subsequently. Following the model description, we explain how the current model relates to previously articulated models of leaf water enrichment. We begin the discussion of the model results with a detailed account of how the model was parameterized. We then discuss the predictive capability of the model with regard to both bulk leaf water isotopic enrichment and enrichment at the sites of evaporation in leaves. Next, we present a sensitivity analysis, focusing on model parameters that were assumed, but that could potentially be measured in future investigations. Finally, we discuss the implications of the current modelling approach for the various applications of leaf water enrichment that we detailed earlier.

MODEL DESCRIPTION

We describe water transport in the leaf mesophyll in a similar way to that for saturated soils (Zimmermann, Ehhalt & Münnich 1967). This is outlined in Fig. 1b. Water enters the mesophyll at r = rx, moves along tortuous paths and finally transpires at r = re. These tortuous paths cross each other and mix their isotopic composition so that an effective gradient develops between the xylem and the evaporating sites. This is described by the advection–diffusion equation for porous media with an effective advection velocity vr and an effective diffusivity Dr (cf. Ogée et al. 2007).

Advection–diffusion equation

Water moves from the xylem to the evaporating sites. Any tracer in the water is thus transported by advection with the effective water velocity vr. If the tracer concentration χ is higher at one point compared with another, diffusion tries to adjust this imbalance. Diffusion is therefore an opposing effect that is proportional to the tracer concentration gradient (henceforth referred to as back-diffusion). The flux of a tracer at a certain point can thus be written as

image(5)

The local continuity equation is then

image(6)

We first take for χ the molar water concentration of H216O molecules, which is the most abundant isotopologue and whose concentration C is constant and uniform. We also take the volumetric water content Θm as uniform over the mesophyll. This implies that it changes simultaneously in the whole mesophyll, which is likely a simplification of reality. For example, a sharp increase in transpiration leads to an advection front that moves from the sites of evaporation to the xylem. However, the associated time constant is around 1 min, and it is therefore appropriate in this context to neglect this effect (Proseus, Ortega & Boyer 1999;Passioura & Munns 2000). This leads to

image(7)

If the volumetric water content Θm does not change with time (hydraulic steady state), the water velocity vr in the mesophyll is uniform, that is, vr(r,t= vr(t).

Identifying now χ with the molar water concentration of H218O (or HDO) molecules, which, at natural abundance, can be expressed as χ = RC, leads to

image(8)

Combining both Eqns 7 and 8 gives the advection–diffusion equation in porous media:

image(9)

The effective diffusivity Dr is thereby related not only to the tracer diffusivity in liquid water D but also to the tortuosity factor κm (<1) of the water path through the mesophyll and the volumetric water content in the leaf mesophyll Θm (<1): Dr = ΘmκmD (Shurbaji & Phillips 1995; Melayah, Bruckler & Bariac 1996). The effective velocity vr will be treated in the next section.

Boundary conditions

The boundary condition for the continuity equation of H216O molecules (Eqn 7) is that of the velocity at the evaporating sites vr(re,t) = ve(t). The effective advection velocity vr is usually not uniform along the r-axis. It is, however, uniform when hydraulic steady state is reached, that is, leaf water volume is not changing anymore. In hypostomatous leaves vr is then equal to the slab velocity E/C, and it is about half of the slab velocity in amphistomatous leaves: E/2C. This is because E is usually expressed relative to one-sided leaf area, regardless of the distribution of stomata over the leaf surface. The velocity profile in the mesophyll is thus (solution of Eqn 7)

image(10)

The volumetric water content in the leaf mesophyll Θm is simply related to the water volume Vm (per unit leaf area) and the mesophyll thickness rm = re − rx through Θm = Vm/Crm. The mesophyll water volume Vm changes with time, and so does volumetric water content Θm and/or the distance to the evaporative sites re. Changing dimensions (re) are difficult to deal with in numerical simulations. Because the observed quantity is mesophyll water volume Vm rather than leaf thickness, one can simulate the change in water volume solely by a change in volumetric water content Θm, fixing the spatial dimensions re, rx and hence rm, that is, Vm∝Θm. This workaround should have, however, negligible influence because the Péclet number ℘m is the same if one includes variable dimensions or solely changes volumetric water content Θm. Hence, the velocity profile in the mesophyll can be written as

image(11)

with the velocity at the evaporating sites vr(re,t) = ve(t) equal to E/2C for amphistomatous leaves. This shows the influence of changing leaf water status and that the water velocity in the mesophyll is uniform (E/2C) if mesophyll water volume Vm is constant.

The boundary conditions for the continuity equation of H218O molecules (Eqn 9) are given by the continuity of the flux. The water isotope flux that enters the mesophyll from the xylem must equal vr(0,t)Rs, and the water isotope flux that leaves the stomata must equal ve(t)RE:

image(12)
image(13)

The last equation can be rewritten as [with ve(t) = E/2C and Eqn A3]:

image(14)

Note that as h → 1, E/(1 − h) must be replaced by gtwi.

Remarks

For amphistomatous leaves with an asymmetric cross-section, like most dicotyledonous leaves with palisade cells on the adaxial side and spongy mesophyll on the abaxial side such as tobacco or spinach leaves, Θm and κm should be different on each side of the leaf, even with an equally distributed stomatal density. The mesophyll thickness rm is also likely to be different. The advection–diffusion model can take this effect into account by solving two separate advection–diffusion equations (one for each side of the leaf) and then averaging the leaf water enrichment, weighted by water volume.

There are still potentially a variety of complications in leaves that do not allow such a simplified description of leaf water isotope enrichment. Effects such as hydrodynamic and mechanical dispersion should be negligible in this context with the relevant advection speeds (Passioura 1971; Gvirtzman & Gorelick 1991).

Other effects such as isotopic transport in the vapour phase instead of the liquid phase may not be negligible, though, because tracer diffusivity in the vapour phase is about 105 higher than that in the liquid phase. However, the vapour molar concentration is also about 105 smaller in the vapour compared with liquid water so that the diffusive flux should be similar in the vapour and in liquid phases.

Finally, it is the current understanding that there are three parallel pathways for liquid water movement in leaves: symplastic (through the plasmodesmata), transcellular (across cell membranes via aquaporins) and apoplastic (along cell walls), with κsymplastic < κapoplastic < κtranscellular (Barbour & Farquhar 2004). It is very possible that the relative importance of these three pathways for total water movement varies with transpiration rate E. For example, a high transpiration rate will most likely thin the water film along cell walls and diminish the influence of the apoplastic pathway. However, we take κm to be constant because the study of water movement in plants is not sufficiently advanced to allow such a partitioning. We do, however, emphasize that it is likely that κm changes with time and environmental conditions.

Relationships with previous models and model limitations

We show in Appendix B[Relating advection–diffusion to Farquhar & Lloyd's (1993) steady-state Péclet effect] that the advection–diffusion equation (Eqn 9) leads directly to the steady-state model of Farquhar & Lloyd (1993) (Eqn 3). This is evident because the advection–diffusion equation is simply the generalization of the Farquhar and Lloyd steady-state Péclet formulation to non-steady state. However,Farquhar & Lloyd (1993) use a constant effective length Leff because they are in steady state. We do not consider Leff to be constant here but define it as Leff = rm/2Θmκm (i.e. k = 2/Θmκm). Θm obviously changes with leaf water volume so that Leff is changing because of the leaf water volume change but becomes constant in steady state.

We also show in appendix B[Relating advection–diffusion to the Farquhar & Cernusak (2005) non-steady-state model] that the advection–diffusion equation (Eqn 9) conforms to the mass-budget derivation of Farquhar & Cernusak (2005). However, it does not assume the form of the isotope gradient from the xylem to the evaporative sites but rather calculates this gradient at any given time.

The present model does not take into account varying source water Rs along the leaf length. Farquhar & Gan (2003) showed that accounting for this effect, which is prominent in long monocotyledon leaves, leads to an enriched bulk xylem water in the steady state (inline image), but that the enrichment for bulk mesophyll water (inline image) was still described by the Farquhar & Lloyd (1993) Péclet description (Eqn 3). Using a full two-dimensional treatment of advection and diffusion in leaf water, Ogée et al. (2007) have further shown that a bulk description of leaf water enrichment such as the one presented in Farquhar & Cernusak (2005) was still valid in non-steady state even in long monocotyledonous leaves. The present one-dimensional advection–diffusion description is of intermediate complexity between the bulk (‘zero-dimensional’) model ofFarquhar & Cernusak (2005) and the two-dimensional treatment of Ogée et al. (2007). It can be derived from the Ogée et al. (2007) model for constant leaf water volume Vm by assuming a well-mixed uniform xylem isotope ratio Rs. However, the present model also treats changing leaf water volume Vm and can deal with asymmetric leaf cross-sections, which were not implemented in Ogée et al. (2007).

We believe that our model is well adapted to amphistomatous leaves. Our model is less well adapted to needles because water movement is then better described by a radial symmetry along the needle axis rather than by the planar symmetry adopted here (Eqn 5et seqq.). To treat this case, the model of Ogée et al. (2007) could be used because it already includes a radial description of xylem water movement that can easily be expanded to the mesophyll. The present model is probably not well adapted to hypostomatous leaves either. Indeed hypostomatous leaves have two distinctively different leaf sides: one side gets enriched directly by evaporation, and the other side gets enriched only by diffusion. There may be advection in the stomata free side as well, especially when leaf water status changes, but it is most probably possible to treat the adaxial site as a water buffer reservoir with no advection but diffusion. This would lead to two coupled advection–diffusion equations (one having no advection) and is beyond the scope of this study.

Numerical solution

The numerical model discretizes the advection–diffusion equation for porous media, Eqn 9, including the boundary conditions, Eqns 12 and 14. For that, the velocity profile vr(r,t) is first calculated, according to Eqn 11, and using measured values of mesophyll water content Θm(t) = Vm(t)/Crm.

Two advection–diffusion equations, one for the adaxial and one for the abaxial leaf side, are solved numerically with an algorithm implicit-backward in time and centred in space, which is unconditionally stable, that is, independent of step sizes in time and space. We took a 10 min time step and a 1 µm resolution in the r-direction.

We arbitrarily take the first measured value as initial condition everywhere in the leaf.

RESULTS AND DISCUSSIONS

Model parameterization

We applied the advection–diffusion model to two independent data sets of Lupinus angustifolius (Cernusak et al. 2002) and Eucalyptus globulus (Cernusak, Farquhar & Pate 2005). Both species are amphistomatous dicotyledons. The adaxial and abaxial parts of the leaf have different leaf geometries and thus, for example, different volumetric water contents Θm. The adaxial (upper) and abaxial (lower) leaf parameters for the two advection–diffusion equations were estimated for lupin from cross-sections of typical dicotyledonous leaves (e.g. Atwell, Kriedemann & Turnbull 1999) and were taken from James & Bell (2001) for Tasmanian blue gum (see Appendix B, Leaf parameters for the adaxial and abaxial leaf sides). Because of lack of knowledge about water flow in the mesophyll, we assumed the same tortuosities in the adaxial and abaxial leaf parts, that is, κm,up = κm,down = κm. But we emphasize that the relative importance of the above-mentioned three water pathways (symplastic, transcellular, apoplastic) could be, and probably is, different in the adaxial and abaxial part of the leaf, so that κm,up and κm,down are most probably different. With these estimates, all the relevant parameters can be calculated from the measured intermediate effective length Leff and mesophyll water volume Vm (the values of the parameters and details of the calculations are given in Appendix B, Leaf parameters for the adaxial and abaxial leaf sides). If one cannot estimate the leaf parameters mentioned earlier, we suggest using the observed Leff and Vm to calculate one single advection–diffusion equation, which is equivalent to a symmetric cross-section. However, one still has to measure/estimate leaf thickness to get Θm from the water volume Vm.

The effective lengths Leff in the original publications were estimated with a fixed value of D = 2.66·10−9 m−2 s−1 for H218O, which would correspond to a leaf temperature of 32.5 °C (see Appendix B, Diffusivity D). This leaf temperature would be 5–10° higher than the maximum daytime temperature in the data sets. However, the ‘real’ tortuosity κm is not known for any species, and κm or Leff are pure fitting parameters. We show the sensitivity to Leff of the advection–diffusion model in Fig. 3c and discuss it in the next section. For L. angustifolius, we took as standard Leff the values of Farquhar & Cernusak (2005) corrected to D of the mean midday temperature of ca. 25 °C (D = 2.2·10−9 m−2 s−1): Leff = 12 mm (using the fractionation factors of Cappa et al. 2005) and 7 mm (using the fractionation factors of Merlivat 1978), respectively. We took the standard Leff for E. globulus so that the advection–diffusion model with constant mesophyll water volume Vm best fitted the observations: Leff = 60 mm (with Cappa et al. 2005) and 50 mm (with Merlivat 1978). We show in the following results obtained with Merlivat's fractionation factors. Results for lupin Δ18O and blue gum Δ18O are basically identical to the results obtained with Cappa et al. (2005) because of the adapted Leff. However, using Cappa et al. (2005) in the case of lupin Δ2H leads to Craig and Gordon steady-state estimates that are lower than the measured mesophyll bulk water enrichment. This comes most probably from the fact that vapour isotopes were not measured but taken to be constant at the mean value of Perth, Western Australia (Rich 2004). This underlines the importance of measuring water vapour isotopes.

Figure 3.

Sensitivity of the bulk mesophyll enrichment Δ18Om of lupin in the advection–diffusion model. (a) The two individual advection–diffusion models for the adaxial (dotted, upper) and the abaxial (dashed, lower) sides and the water volume weighted mean (solid, mean). (b) Same as (a), but with extreme estimates of leaf geometry for the adaxial and abaxial sides. (c) Sensitivity of the symmetric advection–diffusion model to different effective lengths Leff (10 lines equally spaced between Leff = 1 mm and Leff = 31 mm). (d) Sensitivity of the symmetric advection–diffusion model to different night-time stomatal conductances gs,night (10 lines equally spaced between gs,night = 10 mmol m−2 s−1 and gs,night = 110 mmol m−2 s−1). The four panels illustrate therefore the sensitivity of the advection–diffusion model to key parameters. Refer to Appendix A for definitions of symbols.

Bulk mesophyll enrichment Δm

Figure 2 shows the comparison between measured and modelled Δm. It compares the advection–diffusion model calculated independently for the adaxial and abaxial sides (named ‘2m’ for two mesophylls) with the observations, and also with the Farquhar & Cernusak (2005) non-steady-state model and an advection–diffusion model for a symmetrical leaf, that is, the adaxial and abaxial sides are the same (named ‘1m’ for one mesophyll). It is noticeable that the advection–diffusion model is less sensitive to noise in the input data and gives smoother results than the Farquhar & Cernusak (2005) non-steady-state model. A perturbation in the input data has to propagate from the evaporative sites throughout the mesophyll in the advection–diffusion model, whereas it acts instantaneously on the whole mesophyll in the Farquhar & Cernusak (2005) non-steady-state model. It is also apparent that taking just one single set of leaf parameters (1m) performs equally satisfactorily as making the distinction between adaxial and abaxial sides. There is even no apparent difference between 2m and 1m in the case of Tasmanian blue gum. However, this depends strongly on the morphological and physiological differences between the adaxial and abaxial sides. Figure 3a shows the two independent advection–diffusion models for Δ18Om of both leaf sides of lupin, and the weighted mean. Figure 3b shows the same model results calculated with extreme and unrealistic differences between adaxial and abaxial sides, that is, close to 90% of the water volume is in the adaxial side, whereas it is only ca. 65% in the standard case. One can see that the weighted mean is heavily biased towards the adaxial side, and it compares quite favourably still with the observations in the extreme case. But the water volume in the adaxial side is too big in the extreme case in order for the mesophyll water to descend to the steady-state value during night. This strongly depends on the night-time stomatal conductance gs,night. This was not measured for either of the data sets, and standard values were taken in the original publications (Cernusak et al. 2002, 2005). We took the same values as in the original publications but show the sensitivity of the lupin Δ18Om values to gs,night in Fig. 3d. This shows that there has to be a non-zero stomatal conductance at night; otherwise, the mesophyll water would only relax to its mean isotopic composition just before the stomata closed. This exposes the night-time stomatal conductance to be a mere additional fitting parameter if it is not measured, as is the case in our data sets. Figure 3c shows the sensitivity of the lupin Δ18Om values to Leff. Leff is important mostly during the day, while gs,night is determinant during the night. One has, in summary, two fitting parameters, Leff to get the amplitude of Δm right and gs,night to get the right time course of Δm during the night.

Figure 2.

Validation of the present model with observations of the diurnal course of bulk mesophyll enrichment Δm of lupin 18O (a), lupin 2H (b) and blue gum 18O (c). Filled circles are observations, and lines represent the Craig and Gordon steady-state prediction (solid), the Farquhar & Cernusak (2005) non-steady-state model [dotted, F&C (2005)], the advection–diffusion model for a symmetric leaf [dashed, adv.-diff. (1m)], and the advection–diffusion description with different adaxial and abaxial leaf sides [dash dot, adv.-diff. (2m)]. The insets are the predictions of the full advection–diffusion model [adv.-diff. (2m)] and the observed leaf water enrichments. The line is the 1:1 line and the coefficient of determination ℜ2 between model and data is given. The figure illustrates the predictions of the advection–diffusion model in relation to observations and to predictions of previous leaf water models. Refer to Appendix A for definition of symbols.

Both oxygen (Fig. 2a) and deuterium (Fig. 2b) isotopes were measured in L. angustifolius, and they follow the same diurnal cycle and are both well fitted with the same chosen Leff and gs,night. This indicates that there is at least some physical reality in the approach of enriched non-steady-state leaf water with influx of unenriched vein water. It does, however, neither support nor disprove the Péclet effect because one can get comparably good results by taking different well-mixed metabolic pools of water (Yakir 1997, results not shown).

Enrichment at the evaporative sites Δe

The isotopic composition of atmospheric O2 and CO2 are probably influenced by leaf water close to the evaporating sites Δe rather than by bulk mesophyll water Δm, at least in C3 plants. One can calculate the enrichment at the evaporative sites Δe from the change in time of observed bulk leaf mesophyll water Δm (cf. Eqn A9 and Harwood et al. 1998). Figure 4 shows the isotopic compositions at the evaporative sites Δe predicted from the measured bulk mesophyll water isotopic compositions Δm. The derived values at night depend strongly on the measurements of mesophyll water volume Vm, at least for Tasmanian blue gum, which are very noisy because they were measured at leaves adjacent to the leaves of the gas exchange and isotope measurements. We therefore plotted also the derived values of Δe with constant mean leaf water volume Vm. We compared the derived values of Δe to the symmetric advection–diffusion model (1m) and to the Farquhar & Cernusak (2005) non-steady-state model, but taking a mean leaf water volume Vm. We also included the Dongmann et al. (1974) non-steady-state model, which does not distinguish between bulk leaf water and leaf water at the evaporative sites and takes a mean leaf water volume as well. All models are quite similar and perform very well during the day because both data sets reach steady state during early afternoon. This can be tested more critically on leaves with higher leaf water volumes such as needles (Seibt et al. 2006). All models perform well for the whole diurnal cycle of lupin. There are notable differences at night that can be caused by missing atmospheric water vapour isotope measurements. Water vapour isotopic composition for lupin was assumed constant at the mean value for Perth, Western Australia (Rich 2004). However, water vapour isotopes were measured in the blue gum experiment and did not improve the comparison between the models and the derived values. The model values seemed not to decrease enough during the night to explain the derived values. This would indicate that the night-time stomatal conductance could be larger (independent of the Péclet effect), but this would lead to great discrepancies in the Δm values.

Figure 4.

Model performance at the sites of evaporation Δe compared to derived diurnal courses of Δe of lupin 18O (a), lupin 2H (b) and blue gum 18O (c). Open circles are derived from observations of bulk mesophyll water enrichment Δm and mesophyll water volume Vm, while open squares are derived from the same observations of Δm but with a mean observed water volume Vm. Lines represent the Craig and Gordon steady-state (solid), the Farquhar & Cernusak (2005) non-steady-state [dotted, F&C (2005)], the symmetric advection–diffusion [dashed, adv.-diff. (1m)] and the Dongmann et al. (1974) non-steady-state [D (1974)] models. The figure illustrates predictions of Δe by the advection–diffusion model and previous models in relation to calculated values for Δe, which are based on observations of mesophyll leaf water enrichment and leaf water content. Refer to Appendix A for definition of symbols.

It is noteworthy that the Dongmann et al. (1974) non-steady-state model performs as well as the other models for Δe, which indicates that leaf water volume Vm measurements are not very crucial observation for Δe. The Dongmann et al. model is normally used in global models of δ18O in atmospheric CO2 (Cuntz et al. 2003). But this has to be confirmed, for example, at species with higher leaf water volumes that do not reach steady-state during midday, such as needles (Lai et al. 2006; Seibt et al. 2006).

Sensitivity analysis

We performed a variety of sensitivity analyses to assumptions and methods that are used in other leaf water enrichment models: (1) symmetric cross-section of leaf (1m), (2) constant diffusivity at certain leaf temperatures D(°C), (3) constant mesophyll water volume Vm,const, (4) same isotopic gradient in non-steady state as in steady state (Farquhar & Cernusak 2005), (5) solving algorithms (Dongmann-style, Excel Solver). The sensitivity runs are summarized in the form of Taylor diagrams in Fig. 5. The Taylor-diagram (Taylor 2001) shows in polar coordinates the correlation coefficient ℜ and the normalized standard deviation (SD) inline imageof sensitivity runs versus standard model run (advection–diffusion, 2m). The point at unit distance from the origin along the abscissa is the reference point where a perfect model–model match would be situated. In summary, the correlation coefficient ℜ gives information about the right phasing (the better the phasing, the closer is the symbol to the abscissa), and the normalized SD inline imagegives information about the amplitude of the sensitivity run compared to the standard model (the better the amplitudes agree, the closer the symbol is to the dashed line of unity in Fig. 5). We show here rather a model versus model than a model versus data comparison because of the two fitting parameters Leff and gs,night, which were chosen so that one particular model performs best compared with the observations. For example, in the case of Tasmanian blue gum, Leff and gs,night were chosen with the symmetrical advection–diffusion model (1m) with a fixed mean water volume Vm,const (filled standing hourglass), which consequently performed best among the model runs compared with the observations.

Figure 5.

Sensitivity of modelled bulk mesophyll enrichment Δm (Adv.-Diff. 2m) of lupin 18O (a), lupin 2H (b) and blue gum 18O (c) to assumptions and methods that are used in other leaf water model descriptions. The polar diagrams show the normalized SD inline imageof the model runs (normalized to the full advection–diffusion model SD) in radial direction and the correlation coefficient ℜ between the full advection–diffusion model and the sensitivity run on the azimuth scale. Note that the circular ℜ-axis is in inverse cosine (arccosine) scale (cf. Taylor 2001). Open symbols denote the Farquhar & Cernusak (2005) non-steady-state model [F&C (2005)], and closed symbols denote the advection–diffusion model, where ‘1m’ stands for the symmetric advection–diffusion model with same leaf parameters for the adaxial and abaxial leaf sides. Half circles mark runs with fixed diffusivities D at different temperatures and hourglasses mark runs with fixed mesophyll leaf water volume Vm, where Vm,const stands for the mean, Vm,const-min for the minimum and Vm,const-max for the maximum observed values in the data sets. Open hats mark different solving algorithms for the Farquhar & Cernusak (2005) non-steady-state model and are explained in the text. In summary, the aggregation or disaggregation of points on the charts shows the similarity or dissimilarity of model runs with respect to predictions of the full advection–diffusion model. Refer to Appendix A for definition of symbols.

In all three cases (lupin Δ18Om, lupin Δ2Hm and blue gum Δ18Om), the advection–diffusion model shows slightly less sensitivity than the Farquhar & Cernusak (2005) non-steady-state model. As already stated earlier, the advection–diffusion model performs similarly in the case of two distinct leaf halves (standard model, 2m) and with symmetric leaf halves (filled square, 1m), that is, the symmetric advection–diffusion model or filled square is very close to the point of unity at the abscissa. The Farquhar & Cernusak (2005) non-steady-state model (open square) is also very similar in the case of lupin, but deviates substantially in the case of blue gum. This comes from the slightly different sensitivities of the advection–diffusion and the Farquhar & Cernusak (2005) non-steady-state models to leaf water volume Vm changes. The hourglasses show the models' sensitivity to fixed water volume Vm. Apparently, both the advection–diffusion and Farquhar & Cernusak (2005) non-steady-state models are very similar with fixed mean leaf water volume Vm (standing hourglasses). The models show an influence of mesophyll water volume Vm on amplitude and phase. If the water volume Vm taken is too high, the leaf water enrichment Δm decreases too slowly during the night, whereas a low water volume Vm leads to a decrease of leaf water enrichment Δm at night, which is too rapid. However, there is interplay between night-time stomatal conductance gs.night and leaf water volume Vm because the time constant for leaf water enrichment changes is proportional to Vm/gtwi[cf. Appendix B, Solution to the Farquhar & Cernusak (2005) and earlier non-steady-state models]. But if night-time stomatal conductance gs,night is measured, it becomes essential to get a realistic estimate of the mean leaf water volume Vm. It is impossible to determine with our data sets how important it is to measure leaf water volume changes during the day. Our data sets indicate that a mean leaf water volume Vm,const is sufficient, but measuring water volume can be more important if night-time stomatal conductance gs,night is measured.

The half circles show the models' sensitivities to fixed diffusivities D at different temperatures. The diffusivity D is most important during daytime so that the models show most sensitivity in the amplitude rather than the phase. An erroneous diffusivity D can bias the model results by up to 10% (e.g. Fig. 5c).

The Farquhar & Cernusak (2005) non-steady-state model was solved with a fourth-order Runge–Kutta algorithm (e.g. Press et al. 1992). Dongmann et al. (1974) showed an iterative solution to their ordinary differential equation (ODE). To our knowledge, one can solve all published non-steady-state models of bulk leaf water enrichment with the same iterative solution [see Appendix B, Solution to theFarquhar & Cernusak (2005) and earlier non-steady-state models], which we call ‘Dongmann-style solution’. This is strictly incorrect for the Farquhar & Cernusak (2005) non-steady-state model but the smaller the time step dt, the smaller the error in the iterative solution (cf. Appendix B, Solution to the Farquhar & Cernusak (2005) and earlier non-steady-state models). The Dongmann-style solution is appealing because of its simplicity and because it can be calculated in a simple spreadsheet. We included in Fig. 5 two Dongmann-style solutions of the Farquhar & Cernusak (2005) non-steady-state model, once calculated only at times of leaf water enrichment measurements (ca. 2–4 h apart, dt>, hat down) and once calculated in 10 min time steps (dt<, hat up). We also included the solution obtained with the Microsoft Excel Solver (hat right), as suggested by Farquhar & Cernusak (2005). Both Dongmann-style solutions perform as well as the more rigorous Runge–Kutta solution. There is no preference here whether Dongmann-style should be solved with small or large time steps. The Microsoft Excel Solver is in the range of the other sensitivity runs with a tendency to deviate from the Runge–Kutta solution. But our data sets indicate that the errors as a result of the solving algorithm are less important than other unknowns.

Applications of leaf water enrichment

Different applications of leaf water enrichment use enrichment in different parts of the leaf and weighted by different fluxes: (1) leaf organic matter isotopic composition is determined by bulk mesophyll water Δm and, because it is mostly formed while there is energy input from photosynthesis, the leaf organic matter reflects the assimilation-weighted average bulk mesophyll water enrichment Δm; (2) molecular oxygen evolution from the leaf to the atmosphere is the electron transport-weighted leaf water enrichment at the evaporative sites Δe; (3) the isotopic composition of atmospheric CO2 is determined by the gtCi weighted average leaf water enrichment at the evaporative sites Δe, where Ci is the CO2 mixing ratio in the stomatal air space; (4) the atmospheric water vapour receives the signal EΔE, which is determined by the one-way isotope flux of H2O from the leaf stomata into the atmosphere gtwiΔe (cf. Eqn A9). These different applications of leaf water enrichment are used from the leaf level up to global scales. It is therefore crucial to ask how much error one introduces by using a simpler model of leaf water enrichment such as the Craig and Gordon steady-state prediction or the Dongmann et al. (1974) non-steady-state model. We thus calculated 1 day averages of leaf water enrichment, weighted with different CO2 and H2O fluxes, shown in Table 1. Assimilation rate is closely linked to electron transport (Farquhar, von Caemmerer & Berry 1980) so that we take assimilation as a surrogate for electron transport.

Table 1.  Average weighted bulk mesophyll water enrichment Δm and enrichment at the evaporative sites Δe, relevant for different leaf water isotope applications, of the advection–diffusion description (adv.-diff.) compared to simpler leaf water enrichment models: the Farquhar & Cernusak (2005) non-steady-state [F&C (2005)], the Dongmann et al. (1974) non-steady-state [D (1974)] and the Craig and Gordon steady-state model
 Weighting variable
NonAgtwigtCi
  1. The weighting is by different gas exchange parameters for 1 day, where ‘Non’ stands for no weighting, A for assimilation, gtwi for the one-way H2O flux and gtCi for the one-way CO2 flux from the stomata into the atmosphere. The three values at each weighting are from left to right: lupin Δ18O, lupin Δ2H and blue gum Δ18O.

Δm adv.-diff.14.6|44.7|12.917.3|50.5|14.716.1|47.9|14.212.7|40.2|13.9
Δm F&C (2005)14.3|46.2|11.616.9|51.0|12.715.7|48.7|12.312.5|42.3|12.1
Craig and Gordon11.1|40.8|09.918.9|57.1|16.617.2|53.4|15.212.4|43.5|14.0
Δe adv.-diff.15.1|46.3|13.718.5|53.8|17.017.3|51.1|16.213.5|42.5|15.6
Δe F&C (2005)15.2|48.9|12.918.6|56.2|16.117.3|53.7|15.313.6|46.0|14.8
Δe D (1974)15.7|50.6|13.418.6|55.9|15.817.3|53.2|15.113.6|45.9|14.7

All weighted averages are very different from the non-weighted averages, and they are also quite different from each other. The non-weighted averages lie in-between the assimilation weighted and the CO2 one-way flux weighted averages. This would make a difference between leaf water isotopic composition relevant to atmospheric O2 and that relevant to atmospheric CO2 of ca. 5‰ in lupin and ca. 1.3‰ in blue gum. The discrepancy between global leaf water δ18O estimates based on budget calculations of δ18O in atmospheric CO2 and δ18O in atmospheric O2 (Bender et al. 1994; Keeling 1995) can well be just a matter of different weightings (cf. Cernusak et al. 2004). The weighted averages of bulk mesophyll water Δm and water at the evaporating sites Δe are all very different, and the Craig and Gordon steady-state prediction is naturally closer to the evaporating sites Δe. Applications that involve bulk mesophyll water enrichment Δm (e.g. tree rings) therefore introduce a bias towards higher values while not taking the Péclet effect into account.

The Farquhar & Cernusak (2005) and the Dongmann et al. (1974) non-steady-state models are very similar in the averages of the enrichment at the evaporating sites Δe. So it would probably not be very important to take the Péclet effect into account in our data sets if we are not dealing with organic matter.

It is noteworthy that the model calculations with mean volumetric leaf water Vm deviate less than 0.5‰ in Δ18O and less than 1‰ in Δ2H from the calculations with varying leaf water (not shown).

CONCLUSIONS

We have developed an advection–diffusion description of water isotopologues in leaves, similar to Ogée et al. (2007). We have shown it underlies the Farquhar & Cernusak (2005) non-steady-state model. But the present model can deal with asymmetric leaf cross-sections and does not assume the shape of the isotopic gradient in non-steady state, contrary to the Farquhar & Cernusak (2005) non-steady-state model. The latter model shows more sensitivity to input parameters than the advection–diffusion description because the input parameters act on the whole mesophyll at once in the Farquhar & Cernusak (2005) non-steady-state model, whereas any change has to propagate first through the mesophyll in the advection–diffusion description. However, the Farquhar & Cernusak (2005) non-steady-state model performs equally well compared to observations given the uncertainties in the model parameters.

Night-time stomatal conductance gs,night is essentially a tuning parameter in our analysis (and also in the original publications) but could potentially be measured. Water transport in leaves is subject of considerable debate so that the effective length Leff[in the Farquhar & Cernusak (2005) non-steady-state model] or the tortuosity factor κm (in the advection–diffusion model) are unknown and are therefore fitted parameters for the models. Measuring both δ18O and δ2H on the same leaves should however allow the uncertainty to be narrowed for estimates of Leff or κm.

Our data sets together with the advection–diffusion model suggest that simpler leaf water enrichment models such as the Farquhar & Cernusak (2005) or the Dongmann et al. (1974) non-steady-state model or even the Craig and Gordon steady-state prediction might be sufficient in particular applications of leaf water enrichment if weighted with the appropriate fluxes.

ACKNOWLEDGMENTS

M. Cuntz is sponsored by a European Union Marie-Curie Fellowship (contract number MOIF-CT-2004-008087). G.D. Farquhar acknowledges support from the Australian Research Council.

Appendix

Appendix A

List of symbols
a1Constant in temperature dependent description of diffusivity
a2Constant in temperature dependent description of diffusivity (K)
a3Constant in temperature dependent description of diffusivity (K2)
ANet assimilation rate (mol m−2 s−1)
CMolar water concentration (mol m−3)
c1Constant in Dongmann-style solutions
CiCO2 mixing ratio in stomata (ppm)
d(Apparent) Leaf thickness (2 mesophyll thickness) (m)
DDiffusivity (m2 s−1)
DrEffective diffusivity in r-direction (m2 s−1)
inline imageH2O self-diffusivity (m2 s−1)
inline imageTracer diffusivity of H218O in ‘normal’ water (m2 s−1)
inline imageTracer diffusivity of HDO in ‘normal’ water (m2 s−1)
inline imageTracer diffusivity of HTO in ‘normal’ water (m2 s−1)
dttime step (s)
ETranspiration rate (mol m−2 s−1)
femFactor relating bulk mesophyll water with leaf water enrichment at the evaporating site in the steady-state Péclet model
fm,downfraction of abaxial to apparent total mesophyll thickness
fm,upfraction of adaxial to apparent total mesophyll thickness
gbBoundary-layer conductance (mol m−2 s−1)
gsStomatal conductance (mol m−2 s−1)
gs,nightNight-time stomatal conductance (mol m−2 s−1)
gtTotal conductance (mol m−2 s−1)
hRelative humidity corrected to leaf temperature
JrFlux of tracer in mesophyll (mol m−2 s−1)
JxFlux from xylem to mesophyll (mol m−2 s−1)
kFactor for water velocity in Farquhar & Lloyd's (1993) Péclet description
LeffEffective length of water isotopologue flow in mesophyll (m)
m0Mass of one mole H2O (kg)
miMass of one mole H2O, HDO or HTO (kg)
Péclet number
lLongitudinal Péclet number in xylem
mPéclet number in mesophyll
m,downPéclet number in abaxial mesophyll
m,upPéclet number in adaxial mesophyll
rRadial coordinate (m)
reDistance from xylem to evaporative site (m)
Correlation coefficient
2Coefficient of determination
RIsotope ratio
RCCraig and Gordon steady-state isotope ratio at evaporative site
ReIsotope ratio at evaporative site
REIsotope ratio of transpiration
inline imageSteady-state isotope ratio at evaporative site
rmMesophyll thickness (m)
rm,downAbaxial mesophyll thickness (m)
rm,upAdaxial mesophyll thickness (m)
RmIsotope ratio of bulk mesophyll
RsIsotope ratio of plant available source water
RvIsotope ratio of ambient water vapour
rxRadial coordinate at xylem-mesophyll interface (m)
RxIsotope ratio of xylem to mesophyll flux
ttime dimension (s)
vAdvection velocity (m s−1)
veAdvection velocity at the evaporative sites (m s−1)
VmMesophyll water volume (mol m−2)
Vm,constMean observed mesophyll water volume (mol m−2)
Vm,const-maxMaximum observed mesophyll water volume (mol m−2)
Vm,const-minMinimum observed mesophyll water volume (mol m−2)
Vm,downWater volume of abaxial mesophyll (mol m−2)
Vm,upWater volume of adaxial mesophyll (mol m−2)
vrEffective advection velocity in r-direction (m s−1)
waAmbient humidity in mol(H2O) mol(air)−1
wiHumidity in the stomatal cavity mol(H2O) mol(air)−1
α+Equilibrium water-vapour fractionation factor
αkKinetic fractionation factor
ΔIsotope ratio relative to source water
ΔCCraig and Gordon steady-state isotope ratio at evaporative site relative to source water
ΔeIsotope ratio at evaporative site relative to source water
ΔEIsotope ratio of transpiration relative to source water
ΔmIsotope ratio of bulk mesophyll relative to source water
Δm,downIsotope ratio of abaxial bulk mesophyll relative to source water
Δm,upIsotope ratio of adaxial bulk mesophyll relative to source water
Δ2HmHydrogen isotope ratio of bulk mesophyll relative to source water
Δ18OmOxygen isotope ratio of bulk mesophyll relative to source water
inline imageSteady-state isotope ratio of bulk mesophyll relative to source water
inline imageSteady-state isotope ratio of bulk xylem relative to source water
ΘmVolumetric liquid water content
Θm,downVolumetric liquid water content of abaxial mesophyll
Θm,upVolumetric liquid water content of adaxial mesophyll
κapoplasticTortuosity factor of apoplastic water pathway
κmTortuosity factor of in mesophyll
κm,downTortuosity factor of in abaxial mesophyll
κm,upTortuosity factor of in adaxial mesophyll
κsymplasticTortuosity factor of symplastic water pathway
κtranscellularTortuosity factor of transcellular water pathway
inline imageNormalized SD
µireduced mass of water isotopologue in ‘normal’ water (kg)
χTracer concentration (mol m−3)

Appendix B

Craig and Gordon model

Considering the stomatal opening as a resistance to diffusion of water vapour, the transpiration rate E (mol m−2 s−1) can be written as (for symbol definition, see the list of symbols in Appendix A):

image((A1))

The same flux scheme can be written for the heavier isotopologue E′ = REE (with RE << 1, that is, inline image):

image((A2))

with α+ > 1 and αk > 1. This leads to the Craig and Gordon model without the steady-state assumption:

image((A3))

In the steady state, it is apparent that transpiration must have the same isotopic composition as the supplying water, that is, RE = Rs, so that this becomes the so-called Craig and Gordon steady-state model:

image((A4))
Farquhar & Cernusak (2005) non-steady-state model

Writing the mass balance equations for the different water isotopologues in the mesophyll gives for H216O:

image((A5))

This means that mesophyll water volume Vm (mol m−2) will be diminished by transpiration E, but it will be refilled by a flux from the leaf veins (or xylem) Jx. Writing the analogous mass balance for H218O (or HDO) gives

image((A6))

There is a continuity of the flux at the xylem–mesophyll interface (Farquhar & Gan 2003), that is, the incoming flux is RsJx. With this boundary condition and the H2O mass balance (Eqn A5), this becomes

image((A7))

or in Δ notation,

image((A8))

The isotopic composition of transpiration ΔE is directly linked to the isotopic composition at the evaporating sites by the Craig and Gordon equation without the steady-state assumption (Eqn A3), and also substituting in the Craig and Gordon steady-state equation (Eqn A4), this can be written as

image((A9))

where gtwi can be seen as the one-way flux from the stoma into the atmosphere. This formulation links the change in isotopic bulk mesophyll water Δm to the isotopic composition at the evaporating site Δe. To solve this (Eqn A9), one needs to know how both quantities are linked together in time. The idea of Farquhar & Cernusak (2005) was to assume that in non-steady state, there is still an exponential isotope profile in the mesophyll from the xylem to the evaporating sites, and that this profile has the same form as in steady state:

image((A10))
image((A11))

This assumption links the isotopic composition at the evaporating sites Δe with the bulk isotopic composition Δm, so that the Farquhar & Cernusak (2005) non-steady-state model can be written as an ordinary differential equation (ODE):

image((A12))
Solution to the Farquhar & Cernusak (2005) and earlier non-steady-state models

Farquhar & Cernusak (2005) give a closed analytical solution for their ODE (Eqn A12) if one knows the time course of Vm, E, h and ΔC. But even linear evolutions of these variables lead to integrals that are almost impossible to solve. Another possibility for solving this equation is to assume that at time t, the environmental conditions change to a new value and stay constant for a certain time dt, that is, making a step change in the environmental conditions (Dongmann et al. 1974). This is strictly incorrect because, for example, either mesophyll water volume Vm is constant and then dVm/dt must be zero or dVm/dt is not zero and then Vm cannot be constant. However, the smaller the time step dt, the smaller the error in this form of solution. The general iterative solution of Eqn A12 is

image((A13))

We call this type of iterative solution the ‘Dongmann-style solution’. It turns out that to our knowledge, all earlier formulations of one-dimensional non-steady-state leaf water enrichment can be solved in this iterative form with different constant c1:

  • 1Dongmann et al. (1974) and later Bariac et al. (1994) assume constant water volume Vm and make no distinction between bulk mesophyll water Δm and leaf water at the evaporating site Δe, that is, c1 = 1.
  • 2Cernusak et al. (2002) assume Δm = Δe but varying water volume Vm, that is,inline image
  • 3In the case of Farquhar & Cernusak (2005), Vm is varying and Δm = femΔe, that is, inline image.
  • 4For completeness, one can construct the case when Vm is constant and Δm = femΔe, that is, c1 = fem.
Relating advection–diffusion to Farquhar & Lloyd's (1993) steady-state Péclet effect

The Péclet effect as written by Farquhar & Lloyd (1993) is only valid in the steady state, that is, the isotopic composition of water leaving the leaf by transpiration is equal to that entering the mesophyll from the xylem, and the bulk mesophyll water isotopic composition is constant. The term ∂R/∂t in the advection–diffusion equation (Eqn 9) is then zero. The original equation of Farquhar & Lloyd (1993) assumed also no change in leaf water content. If leaf water content does not change, there is no velocity profile and vr is constant. The solution of Eqn 9 with ∂R/∂t = 0, and the boundary conditions Eqns 12 and 13 with vr = E/2C is

image((A14))

with ℘m = Erm/2CDr. This means that the isotope ratio in steady state diminishes from RC at the evaporating site (r = re) to inline image, and not Rs, at the xylem–mesophyll interface (r = rx). Written in Δ notation this is

image((A15))

The mean Δ between rx and re is then just the Péclet formulation of Farquhar & Lloyd (1993):

image((A16))
Relating advection–diffusion to the Farquhar & Cernusak (2005) non-steady-state model

To relate the present advection–diffusion description to the formulation of Farquhar & Cernusak (2005), we integrate Eqn 9 from rx to re, divide by rm (averaging) and use the relation between ∂vr/∂r and ∂Θm/∂t (Eqn 7):

image((A17))

where we integrated by parts the first term of the right-hand side of Eqn 9. Using the boundary conditions (Eqns 12 and 13) gives

image((A18))

Using the linear velocity profile (Eqn 10) and switching to Δ notation leads almost to Eqns A8 (and A9):

image((A19))

The xylem is situated rather in the middle of the leaf. The advection–diffusion equation calculates from the xylem (r = rx) to the evaporating side (r = re), that is, it calculates only one side of the mesophyll. ΘmCrm is therefore the water volume of half the mesophyll. If both halves of the leaf (adaxial and abaxial) are the same, each is Vm/2, and this yields exactly Eqns A8 (and A9).

The advection–diffusion equation (Eqn 9) does not lead to the ODE of Farquhar & Cernusak (2005) (Eqn A12) because this includes the assumption about the shape of the isotope profile. The advection–diffusion equation makes no assumption about the isotope profile from the xylem to the mesophyll but rather calculates exactly this profile.

Leaf parameters for the adaxial and abaxial leaf sides

Mesophyll water volume Vm is linked to volumetric mesophyll water content Θm (for symbol definition, see the list of symbols in Appendix A):

image((A20))

with rm,up = fm,upd, rm,down = fm,downd and fm,up + fm,down = 1. The tortuosity factor κm is taken to be the same in the adaxial and abaxial mesophyll so that it relates to half the leaf thickness d:

image((A21))

In order to calculate one advection–diffusion equation for each mesophyll side separately, one needs to know or estimate the relative thickness of both mesophylls, fm,up = rm,up/d and fm,down = rm,down/d, and the volumetric water contents Θm,up and Θm,down. The water variables and (apparent) leaf thickness follow then from Eqn A20 with the measured mesophyll water volume Vm. One needs to estimate an effective length Leff or tortuosity factor κm. Because Leff is a tuning parameter for the amplitude of the model, this is carried out with Θm at midday. The advection–diffusion model uses a fixed tortuosity factor κm and a fixed length rm instead of an effective length Leff. The apparent Leff in the Péclet number ℘ depends then on the varying Θm and is therefore not constant.

We estimated the leaf parameters for Lupinus angustifolius from cross-sections of typical dicotyledonous leaves (e.g. Atwell et al. 1999) to fm,up = 0.6, fm,down = 0.4, Θm,up = 0.9 and Θm,down = 0.5. We took the values of James & Bell (2001) for Eucalyptus globulus: fm,up = 0.6, fm,down = 0.4, Θm,up = 0.9 and Θm,down = 0.7. The estimated effective lengths Leff are explained in the main text.

Our extreme estimates for the lupin simulations of Fig. 3b are fm,up = 0.7, fm,down = 0.3, Θm,up = 1.0 and Θm,down = 0.3.

Diffusivity D

In the literature of leaf water enrichment, the diffusivity D is normally taken as constant: 2.66·10−9 m2 s−1 for H218O and 2.34·10−9 m2 s−1 for HDO, respectively. The values are supposed to come from a paper of Wang (1954), but this was a mistaken citation, which was carried forward through subsequent publications. The values come from Wang, Robinson & Edelman (1953) and were measured at 25 °C. However, diffusivity D is not constant but is temperature dependent. Additionally, the turbulent stirring method of Wang et al. (1953)‘tends to give values that are too high’ (Weingärtner 1982).

Measurements of H2O self-diffusivity (inline image) and tracer diffusivity of tritiated water in ‘normal’ water [inline image] were performed by Mills (1973) at different temperatures. He also reports measurements of tracer diffusivity of deuterated water in ‘normal’ water (inline image) of Longsworth (1954, 1960). Easteal, Edge & Woolf (1984) reported values for the diffusivity of H218O in ‘normal’ water [inline image] between 0 and 50 °C with an accuracy of about 0.3%. The exact form of the temperature dependence of tracer and self-diffusivity is unknown, but two functional forms that work well are firstly a derivative of an Arrhenius plot (Mills 1973; Woolf 1975):

image((A22))

and secondly, a Vogel–Tamman–Fulcher relationship (VTF) (Price, Ide & Arata 1999; Price et al. 2000):

image((A23))

where a1, a2 and a3 are constant parameters. We show in Fig. A1b,c the measurements and fitted curves of the tracer diffusivities of water isotopologues in ‘normal’ water. Both functional forms perform equally well and are indistinguishable on the plot (so only the VTF is shown as the solid line). Table A1 records the parameters a1a3 for both functional forms including their uncertainties. The values of the Arrhenius type relationship can be compared to the values obtained by Easteal et al. (1984).

Table A1.  Fitted parameters to the Arrhenius derived and the Vogel–Tamman–Fulcher (VTF) relationship of tracer and self-diffusivity of water
 ArrheniusVTF
a1a2a3a1a2a3
  1. Number in parentheses give the uncertainties of the parameters. H2O stands for H2O self-diffusivity, and H218O, HDO and HTO stand for tracer diffusivities in ‘normal’ water.

H2O−1.2 (0.2)2055 (137)−634783 (20167)100 (6)577 (18)145 (2)
H218O−0.4 (1.6)1528 (959)−554368 (143446)119 (55)637 (151)137 (19)
HDO−0.7 (1.4)1729 (807)−586977 (118620)116 (45)626 (121)139 (15)
HTO−1.3 (0.2)2082 (142)−638900 (21006)097 (6)574 (19)146 (2)

Tracer diffusivities of HDO and HTO, and H2O self-diffusivity seem to be proportional to each other with the square root of the reduced masses, at least under atmospheric pressure (Mills & Harris 1976; Harris & Woolf 1980; Harris 1996):

image((A24))

the index i denoting 0 ≡ H2O, 1/2 ≡ HDO/HTO. This means that the constants a2 and a3 should be the same in Fig. A1c,d, which they are only approximately. The data of Harris & Woolf (1980) suggest that there is ‘a slightly greater coupling of translational and rotational motion of HTO relative to H2O in pure water at low pressures’. However, the pressure dependency effect of Harris & Woolf (1980) is rather small and most likely negligible at atmospheric pressures.

For H218O, tracer diffusivity slightly deviates from the pure mass relationship with other diffusivities (Easteal et al. 1984). But this effect is small at atmospheric pressure. However, the mean ratio of inline imageand inline imagedoes not follow Eqn A24: inline imageinstead of 1.026 (Eqn A24).

However, all measurements of water isotopologue tracer diffusivity were carried out in pure water. It is likely that the tracer diffusivities have lower values in solutions (such as in plant cells), whereas the temperature dependence should be very similar (Harris, personal communication).

We therefore fitted one single function to all tracer and self-diffusivities, where the tracer diffusivities were multiplied by the ratio of the reduced masses to put them on the same scale as the H2O self-diffusivity (Fig. A1a & Table A1). This means that the measurements of inline imagewere multiplied by [19/18·(18+18)/(18+19)]0.5 ≈ 1.013, and the measurements ofinline image and inline imagewere multiplied by [20/18·(18+18)/(18+20)]0.5 ≈ 1.026. Note that we included only the measurements of H2O self-diffusivity of Mills (1973) and not of Price et al. (1999) because of the uncertainty of temperature measurements in magnetic resonance systems. This gives an excellent fit where the small uncertainties in the parameters a1a3 stem mostly from the small uncertainties in inline image. We included in Fig. A1c,d the overall VTF fits with a2 = 577, a3 = 145 and a1 = 100/1.026, 100/1.013 and 100/1.026 for H218O, HDO and HTO, respectively (dashed lines). The values of the Arrhenius type relationship can be compared to Braud et al. (2005).

image

Figure A1. Temperature dependence of tracer and self-diffusivity of water: (a) combined H2O self-diffusivity inline image, (b) H218O tracer diffusivity inline image, (c) HDO tracer diffusivity inline image and (d) HTO tracer diffusivity in ‘normal’ water inline image. Diamonds are measurements of Easteal et al. (1984); triangles are measurements of Longsworth (1954, 1960) as cited by Mills (1973), and squares (HTO) and circles (H2O) are measurements of Mills (1973). Measurement precision is, in all cases, smaller than the symbol dimensions. Solid lines are Vogel–Tamman–Fulcher (VTF) relationships to the individual tracer diffusivities, and dashed lines are the VTF relationship for all data where the tracer diffusivities are multiplied by the ratio of the reduced masses. The figure illustrates the non-linear nature of tracer and self-diffusivity of water across a biologically relevant range of temperatures.

We think that this level of detail is sufficient in the context of leaf water isotope enrichment, because firstly, only the effective diffusivity Dr is important, and this includes the unknown tortuosity factor κm; secondly, because the measured diffusivities were carried out in pure water and are likely to be lower in leaf water; and thirdly, because diffusivities are much more sensitive to temperature than to an eventual bias. So in this study, we take the VTF relationship fitted to all tracer and self-diffusivities with the values of a1a3 given in the previous paragraph.

Ancillary

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