## INTRODUCTION

As postulated by the cohesion-tension theory, the flow of water from soil to leaf represents a ‘tug-of-war’ on a hydraulic rope. If the hydraulic continuum breaks, the plant cannot access atmospheric CO_{2} without desiccating to death. There are two weak spots in the continuum: at the rhizosphere where steep water potential gradients may create dry non-conductive zones (Newman 1969), and in the xylem where cavitation can eliminate water transport (Zimmermann 1983). While earlier studies have considered the limitation of water uptake by one or the other of these processes (Newman 1969; Bristow, Campbell & Calissendorff 1984; Tyree & Sperry 1988), it is an open question how rhizosphere and xylem properties interact to limit water uptake. In this paper, we answer this question with a model.

The theory of hydraulic limits on water uptake begins with Darcy's law, which can be applied to steady-state flow through the soil–plant hydraulic continuum:

where *E* is the transpiration rate (per leaf area), *dΨ*/*dx* is the water potential gradient driving flow, and *K* is the hydraulic conductivity expressed per leaf area (Table 1 lists symbols, definitions, and units). Figure 1 shows the steady-state relationship between *E* and leaf *Ψ* for a constant bulk soil *Ψ* (*Ψ*_{s} = the *Ψ* intercept). If *K* is a constant, *E* is directly proportional to the decrease in leaf *Ψ* and there is no hydraulic limit to *E* or leaf *Ψ* (dashed line 4 in Fig. 1).

**.**List of major symbols and their definitions. Units are those used in equations. Values cited in text or figures may have different units

Hydraulic limits arise because *K* is not constant, but instead decreases in xylem and soil as a function of decreasing *Ψ*. In the xylem, the decline in *K* is caused by cavitation, and the *K* (*Ψ*) function is described by a ‘vulnerability curve’ (e.g. Fig. 3). In the soil, the decrease in *K* occurs by the same mechanism causing cavitation in xylem: the displacement of water-filled pore space by air as capillary forces fail (Hillel 1980; Pockman, Sperry & O’Leary 1995). The *K*(*Ψ*) function for soil depends largely on soil texture, with more sensitive functions for coarser soils (Hillel 1980).

When *Ψ*-dependent *K* is incorporated into Darcy's law, there is no longer a directly proportional relationship between *E* and *Ψ* (Fig. 1, curves 1–3). Instead, increases in *E* are associated with progressively disproportionate decreases in *Ψ* because of declining *K*. The *E* reaches a maximum (*E*_{crit}) at the corresponding minimum leaf *Ψ* (*Ψ*_{crit}). At these critical values, *K* (*Ψ*) has approached zero somewhere in the hydraulic continuum (Appendix). As *Ψ*_{s} decreases, *E*_{crit} declines (Fig. 1, compare curves 1–3). When *Ψ*_{s} = *Ψ*_{crit}, the plant cannot transport water.

If stomata allow *E* to exceed *E*_{crit} long enough for steady-state conditions to develop, the positive feedback between decreasing *K* and *Ψ* becomes unstable: a phenomenon dubbed ‘runaway cavitation’ when it occurs in xylem (Tyree & Sperry 1988). Runaway cavitation breaks the hydraulic rope and eliminates water transport by driving *K* to zero. A model by Tyree & Sperry (1988) predicted an *E*_{crit} that was only slightly greater than actual maximum *E* in four diverse tree species, suggesting stomatal regulation of *E* was adaptive in avoiding hydraulic failure of the xylem. The gas exchange capacity of plants may have hydraulic constraints.

The Tyree and Sperry model, however, did not incorporate the *K* (*Ψ*) relationship for the soil. Transpiration-driven decreases in *Ψ* soil are accentuated in the rhizosphere because of the cylindrical geometry of water uptake (Cowan 1965; Newman 1969; Bristow, Campbell & Calissendorff 1984), and ‘runaway cavitation’ can potentially occur at the soil–root interface. Rhizosphere limitations should be especially important for coarse soils and plants with less absorbing root area relative to their transpiring leaf area (Newman 1969). Although variable rhizosphere conductance has been incorporated in water uptake models (Cowan 1965; Bristow *et al.* 1984), none have incorporated variable xylem conductance. It is not clear whether below-ground hydraulic constraints are more or less important than those of the xylem.

The model presented in this paper shows how three causal factors – (1) cavitation resistance, (2) root:leaf area ratio (*A*_{R}:*A*_{L}), and (3) soil type (specifically, soil texture) – interact to set hydraulic limits on water transport. The analysis of cavitation resistance includes the influence of root xylem, which is more vulnerable than canopy xylem in many species (Sperry & Saliendra 1994; Alder, Sperry & Pockman 1996; Hacke & Sauter 1996; Mencuccini & Comstock 1997). The purpose of the model is to obtain a better understanding of biophysical limits on water uptake and their relevance to physiological responses of plants to water availability.