## INTRODUCTION

The isotopic enrichment of leaf water during transpiration is most commonly associated with the Craig–Gordon model (Craig & Gordon 1965), which can be modified, to a reasonable approximation, to describe the enrichment of leaf water at the sites of evaporation (Dongmann *et al*. 1974; Farquhar *et al*. 1989):

where Δ_{C} denotes the Craig–Gordon estimate of the isotopic enrichment of leaf water above source water as described by this equation; Δ_{v} the enrichment of atmospheric water vapour above source water; *e*^{+} and *e*_{k} the equilibrium and kinetic fractionation factors, respectively; *e*_{a} and *e*_{i} the water vapour pressures in the atmosphere and intercellular spaces, respectively. A full list of symbols and their definitions is given in the Appendix. As more measurements of leaf water were made, experimenters noted that the Craig–Gordon model tends to overestimate the isotopic enrichment of leaf water, Δ_{lw} (Allison, Gat & Leaney 1985; Walker *et al*. 1989; Yakir, DeNiro & Gat 1990; Flanagan, Bain & Ehleringer 1991; Flanagan, Comstock & Ehleringer 1991; Flanagan *et al*. 1994; Wang, Yakir & Avishai 1998; Roden & Ehleringer 1999) and is not able to account for the spatial variation of Δ_{lw} (Luo & Sternberg 1992; Bariac *et al*. 1994; Wang & Yakir 1995; Helliker & Ehleringer 2000, 2002; Gan *et al*. 2002). To explain these observations, more leaf water models have been proposed in conjunction with the Craig–Gordon model. The two-pool model (Leaney *et al*. 1985) describes leaf water as a composite of enriched leaf tissue water and non-enriched vascular water. The Péclet model (Farquhar & Lloyd 1993) accounts for the ‘micro’ variation of leaf water enrichment between the vein and the sites of evaporation by considering the relative effects of advection and diffusion. The ‘string-of-lakes’ model (Gat & Bowser 1991; Yakir 1992; Helliker & Ehleringer 2000, 2002) predicts a ‘macro’ spatial variation across the entire leaf surface based on the progressive enrichment of leaf water along the path of evaporation. Comprehensive reviews of the various leaf water models can be found (Yakir 1992, 1998; Gan *et al*. 2002).

From a study of the ^{18}O spatial patterns of vein xylem water, leaf water and dry matter in cotton leaves grown at different humidities, Gan *et al*. (2002) demonstrated that leaf water in dicotyledoneous plants has different isotopic enrichment patterns at different humidities, resembling the behaviour of a string of interconnected evaporating lakes. The analogy of the leaf water pathway to the movement of water through a sequence of lakes in series was previously noted in monocotyledoneous plants (Yakir 1992; Helliker & Ehleringer 2000) but deemed to be less appropriate in dicotyledoneous leaves (Helliker & Ehleringer 2000). To account for the observations that vein xylem water is partly enriched in comparison with source water, and bulk leaf water less enriched than the Craig–Gordon prediction, Δ_{C}, Gan *et al*. (2002) highlighted the need to include Péclet effects (advection from the vein opposing diffusion from the sites of evaporation) in the ‘string-of-lakes’ model for its application to leaves. Gan *et al*. (2002) envisaged the competing effects of the transpiration flux and the back-diffusion of enriched water to occur in two dimensions, longitudinally along the leaf length as well as radially in the plane of the leaf lamina, from the vein network to the leaf evaporative sites. Gan *et al*. (2002) also identified the presence of capacitance in the ground tissues of vein ribs to be one reason for the lower enrichment of bulk leaf water.

On the basis of these findings (Gan *et al*. 2002), Farquhar & Gan (2003) developed a leaf water model featuring aspects of progressive enrichment, two-dimensional Péclet effects and ground tissue capacitance. As a first step in the leaf water modelling, Farquhar & Gan (2003) showed how the ‘string-of-lakes’ model could be modified to describe a continuous treatment of evaporative sites in leaves, as opposed to the discrete number of evaporating elements in the Gat–Bowser formulation where

where *δ*_{v} and *δ*_{n} are the isotopic composition of atmospheric water vapour and of water entering the *n*th evaporating element, respectively, *F*_{+} and *E* represent the influx and evaporative efflux of the *n*th element, *h* is the relative humidity and *e* = *e** + (1 − *h*)*e*_{k}[note that *e** = *e*^{+}/(1 + *e*^{+})]. Continual loss of water by evaporation in leaves resembles a desert river system, first introduced by Fontes & Gonfiantini (1967). According to Farquhar & Gan (2003), the enrichment of leaf water, Δ_{lw}, at a relative distance *l*/*l*_{m} from the leaf base of a leaf *l*_{m} long, can be written with respect to the maximum possible enrichment, Δ_{M}, as follows:

where

and

For the same order of approximation as Eqn 1, *h*′ ≈ *h*, where *h* is the relative humidity at the leaf temperature (equal to *e*_{a}/*e*_{i}). For a more realistic modelling of the enrichment of leaf water, Farquhar & Gan (2003) also took into account the presence of Péclet effects along the longitudinal and radial dimensions of a leaf blade and the significant pool of ground tissue water in the vein ribs. Considering first the modelled radial Péclet effect, an isotopic gradient, on the plane of the leaf lamina, develops from the major vein xylem, through the leaf mesophyll and onwards to the evaporative sites. The enrichment of the vein xylem water, Δ_{x}, is lower than the radial average enrichment of water in the mesophyll of the lamina, Δ_{l}, which in turn is lower than the enrichment at the sites of evaporation, Δ_{e}. The enrichment of these water compartments (Δ_{x}, Δ_{l} and Δ_{e}) varies with the longitudinal length, *l*, along the main axis of the leaf, due to longitudinal Péclet effects sustaining the progressive enrichment along a line of evaporating cells. Note that unlike the conventional treatment of leaves, which takes Δ_{e} to be equal to Δ_{C}, the present model allows Δ_{e} to vary spatially, and so to differ from Δ_{C}. For ease of comparison with experimental data, the modelled Δ_{x}, Δ_{l} and Δ_{e} are normalized against the maximum possible enrichment, Δ_{M}, and their variations are expressed with respect to *l*/*l*_{m} as follows (Farquhar & Gan 2003):

where ; *P*_{r}, *P* and *P*_{l} are the total radial Péclet

number, lamina radial Péclet number and longitudinal Péclet number, respectively. The expression _{1}F_{1}[*a*, *b*; *z*] is the Kummer function describing a confluent hypergeometric series and can be computed using mathematical software packages, for example MATHEMATICA, MAPLE and DERIVE 5 (Farquhar & Gan 2003). In the work described in this paper, MATHEMATICA (Version 4.1, Wolfram Research, Champaign, IL, USA) was used for the computation of the Kummer function.

As illustrated by Gan *et al*. (2002), the enrichment of leaf water in cotton leaves does not occur in an uni-dimensional direction from the leaf base to the tip. Distinct regions of higher or lower enrichment have been identified. Such complex enrichment patterns are attributed to the reticulate venation of cotton leaves. This venation complexity makes the measurement of vein water path-length (*l*/*l*_{m}) near impossible, and deters us from fitting the observed values of leaf water (Δ_{lw}) and vein xylem water (Δ_{x}) enrichment to Eqn 6 for quantitative assessment of the Farquhar–Gan model. The maize leaf, on the other hand, has long and minimally branched parallel veins typical of a monocotyledoneous plant. This enables the total water pathway along the longitudinal direction to be represented by the full length of the leaf blade. The maize leaves could thus form a good system for studying water distribution and isotopic fluxes. Nevertheless, the Farquhar–Gan model is a simplification of what happens in maize leaves where there are many longitudinal xylem elements of different length, *l*_{m}, in a single leaf.

In this paper, we present experimental results on the spatial variation of leaf water enrichment along a maize leaf, at different humidities, for comparison with the predicted values of the leaf water models described above. Concurrently, the enrichment of vein xylem water was also studied. Next, Péclet numbers along the longitudinal and radial dimensions were estimated from anatomical dimensions, vein xylem water and bulk leaf water isotope measurements. Farquhar & Gan 2003) have indicated that the radial Péclet number (*P*_{r}) is likely to be of order one whereas the longitudinal Péclet number (*P*_{l}) estimated from the anatomical dimensions of vein xylem could be of order 10^{7}. They also highlighted that with the consideration of advection and diffusion in the longitudinal dimension in the lamina (a feature yet to be incorporated in their model), the effective *P*_{l} value should be smaller and somewhere between this xylem-based value and *P*_{r}. This article aims to assess the extent of agreement between the anatomical estimate of *P*_{l} in the xylem, and the effective *P*_{l} as observed from the experimental data of leaf water enrichment of maize leaf segments.