## 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 CO_{2} and O_{2}, 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 CO_{2} and O_{2}, however, are probably determined by the isotopic composition of leaf water near the evaporating sites, at least in C_{3} 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 CO_{2}, 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 CO_{2} is therefore determined by the leaf water enrichment at the evaporative sites weighted by the product of stomatal conductance and the CO_{2} mixing ratio in the sub-stomatal cavity (one-way CO_{2} 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)

where *R*_{s} is the isotopic ratio of plant available source water; *R*_{v} is the water vapour isotope ratio; *α*^{+} (>1) is the equilibrium fractionation factor between liquid water and water vapour; *α*_{k} (>1) is the total kinetic fractionation factor associated with diffusion, and *h* = *w*_{a}/*w*_{i} is the relative humidity corrected to leaf temperature, where *w*_{i} and *w*_{a} (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:

where *E* is transpiration rate (mol m^{−2} s^{−1}); *D* (m^{2} s^{−1}) is the tracer diffusivity in liquid water; *C* = 10^{6}/18 = 55.6·10^{3} mol m^{−3} is the molar water concentration, and *L*_{eff} (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 (*R*_{s}) to the Craig and Gordon value *R*_{C} at the evaporative sites. Bulk steady-state mesophyll water enrichment (expressed relative to source water) can then be written as

with Δ_{C} the Craig and Gordon steady-state enrichment expressed relative to source water (i.e. Eqn 1 expressed relative to source water *R*_{s}).

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 *L*_{eff} 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 Δ_{m} is then the water volume (*V*_{m,up}, *V*_{m,down}) weighted enrichment of both sides (Δ_{m,up}, Δ_{m,down}), that is, Δ_{m} = (*V*_{m,up}Δ_{m,up} + *V*_{m,down}Δ_{m,down})/*V*_{m}. In steady state, this means that Eqn 3 should rather be

which cannot be expanded to give a sensible combination of one single ℘_{m} or *L*_{eff}. 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.