## Introduction

How does tree water use scale with tree size, and how does it differ across species? Given the essential role of water, this question is fundamental to understanding the metabolic scaling of individual trees, species, forest communities and ecosystems. Predicting the answer from vascular anatomy is the subject of this study. Modelling water use from vascular properties has a long history dating at least to da Vinci's rule of area-preserving branching (Richter 1970), continuing with the Ohm's law analogy of van den Honert (1948; Richter 1973) and culminating in the concept of ‘hydraulic architecture’ (Zimmermann 1978) represented in contemporary models (e.g. Tyree 1988; Sperry *et al*. 2002; Macinnes-Ng *et al*. 2011). At the heart of these complex models is a simple relationship for whole-tree sap flow at steady state (*Q*):

where *K* is tree hydraulic conductance, ∆*P* is soil to canopy pressure drop, and ρg*H* is the pressure required to offset the force of gravity on the water column (*ρ* = density of water; *g*, acceleration of gravity; *H*, tree height). Canopy xylem pressure regulation (via stomatal control of *Q*) constrains the (∆*P* – *ρgH*) term, and most of the uncertainty in hydraulic modelling lies in representing *K* which depends mostly on the complex anatomy of the flow path from soil to leaf.

Until the revolutionary approach of West, Brown and Enquist (‘WBE’; West, Brown & Enquist 1997; Enquist, West & Brown 2000; West, Brown & Enquist 1999), most hydraulic modelling was based on specifying what *K is* from empirical inputs. In contrast, the WBE model derives what the allometric scaling of *K should be* by assuming a universal set of optimization criteria and an intentionally minimalist representation of plant vasculature. The WBE goal is to predict universal expectations for how *K*, and hence *Q*, and all dependent metabolic processes, should scale with plant size. The focus is on predicting the power function scaling exponent (*b*):

where *Y* is the variable of interest (*K*,* Q*, rates of metabolism or growth) and *M* is plant mass.

The result is a metabolic scaling theory that emphasizes the unifying consequences of selection for optimal vascular transport under overarching constraints. Savage *et al*. (2010) have recently extended the theory with important improvements in how it represents vascular architecture.

In this study, we present a model that strikes a middle ground between the structure-to-function optimization approach of Savage *et al*. (and its WBE predecessors) and the descriptive – empirical approach of more complex numerical models. We add a minimal set of hydraulic inputs to the Savage *et al*. analytical model with the goal of predicting the actual value of *K* and *Q* rather than proportional proxies that are sufficient for predicting scaling exponents. Our species-level model turns the proportionality in (eqn 2) (*Y* ∝ *M* ^{b}) into an equality (*Y* = *k*_{0} *M* ^{b}) by specifying scaling multipliers (*k*_{0}). The additional complexity requires a numerical approach, but is justified because selection for optimal vascular function should concern traits underlying the multiplier as well as the exponent. Furthermore, variation in scaling multipliers across species could influence interspecific exponents (*b*) independently of the intraspecific value of *b*. We relax any *a priori* optimization criteria and allow key hydraulic inputs to be empirical, so that we can predict the ‘scaling space’ defined by variation in *k*_{0} and *b* across species.

Figure 1 provides a roadmap of the Savage *et al*. (2010) model. The branching architecture component (Fig. 1, left) specifies that the tree has symmetric, self-similar branching architecture that preserves the cross-sectional area of branches across each branching junction (da Vinci's rule; Horn 2000). Hence, the tree can be represented by a column (Fig. 1, center). The mass allometry module predicts the best-fit power-law scaling between trunk diameter (*D*_{B0}; 0 denotes trunk branch rank; symbols in Table 1) and tree mass (*M*):

Symbols | Definitions |
---|---|

A _{ Si } | Sapwood area, branch level i |

A _{ S } /A _{ T } | Sapwood area/basal area for reference tree size with D_{B0} = 72 cm |

C _{ F } | Fraction of wood occupied by conduit lumens (conduit lumen fraction) |

C | Xylem hydraulic conductance/Hagen–Poiseuille conductance (end-wall correction) |

D _{ B0 } | Trunk diameter (branch rank 0) |

D _{ Bi } | Stem diameter for branch rank i |

D _{ C } | Xylem conduit diameter |

D_{C} max | Maximum allowable conduit diameter |

D_{C} twig | Conduit diameter in the distal-most branch rank (twigs) |

D _{ P } | Pith diameter |

F | Number of conduits per wood area |

g | acceleration of gravity |

G | biomass growth rate of shoot |

H/H _{ B } | tree height/Euler buckling height |

K | tree hydraulic conductance |

K_{L} /K_{T} | Leaf hydraulic conductance/supporting twig conductance |

K/K _{ S } | Tree conductance/shoot conductance |

k _{ 0 } , b | Generalized scaling multiplier and exponent (e.g. Y = k_{0 }M^{b}) |

k _{ 1 } , c | Mass scaling multiplier and exponent (D_{B0} = k_{1 }M^{c}) |

k _{ 2 } , q | Water use scaling multiplier and exponent (Q = k_{2} D_{B0}^{q}) |

k _{ 3 } , p | Taper function multiplier and exponent (D_{C} = k_{3} D_{Bi}^{p}) |

k _{ 4 } , d | Packing function multiplier and exponent(F = k_{4} D_{C}^{d}) |

k _{ 5 } , a | Bark thickness function multiplier and exponent (T_{Bi} = k_{5} D_{Bi}^{a}) |

k _{ 6 } , s | Sapwood area function multiplier and exponent (A_{Si} = k_{6} D_{Bi}^{s}) |

L _{ i } | Branch segment length, level i |

M | Shoot (above-ground) mass |

N _{ C } | Conduit number |

n | Daughter/mother branch number ratio |

Q | Steady-state tree water transport rate at mid-day |

Q _{ ref } | Q for ‘reference’ tree size of trunk diameter D_{B0} = 72 cm |

T _{ Bi } | Bark thickness, branch level i |

V | Shoot (above-ground) volume |

β | Daughter/mother branch diameter ratio |

∆P | Total soil to canopy water potential difference |

γ | Daughter/mother branch length ratio |

η | Viscosity of water |

ρ | Density of water |

where *k*_{1} is the scaling multiplier and *c* the scaling exponent. The value of the exponent *c* is derived from well-tested theory that *H* must scale with *D*_{B0}^{2/3} for trees to maintain a constant safety margin from buckling under their own weight (‘elastic similarity’, McMahon 1973). An elastically similar column has a mass exponent of *c* = 3/8 in (eqn 3) (West, Brown & Enquist 1997, 1999; Enquist, West & Brown 2000; Savage *et al*. 2010).

The water use allometry module predicts how the steady-state rate of midday xylem transport (*Q*) scales with trunk diameter:

with multiplier *k*_{2} and water use exponent, *q*. To obtain *Q*, the Hagen–Poiseuille equation (Zimmermann 1983) is used to calculate tree hydraulic conductance (*K*) from the number and dimensions of the xylem conduits in the tree sapwood, given by the xylem architecture module (Fig. 1, right). The prediction of *K* yields *Q* by (eqn 1), and the scaling of *Q* with tree size yields the water use allometry of (eqn 4). Previous derivations of the water use exponent *q* in (eqn 4) have assumed that selection for transport efficiency has driven it to its theoretical maximum of *q* = 2 (for the assumed xylem architecture; West, Brown & Enquist 1999; Enquist, West & Brown 2000; Savage *et al*. 2010). At this point, the rate of whole-tree water transport depends solely on its trunk basal area and is not negatively influenced by tree height or transport distance (*Q* ∝ *D*_{B0}^{2}/*H*^{0}).

The fifth metabolic isometry component of the Savage *et al*. model is a fundamental assumption of metabolic scaling theory: because photosynthetic CO_{2} flux and transpirational water flux are both limited by stomatal diffusion, gross photosynthesis and potential isometric surrogates such as total respiration and growth rate (*G*) should scale proportionally with *Q* (Enquist *et al*. 2007a). Combining metabolic isometry with the mass and water use allometries predicts metabolic scaling: *G* ∝ *Q* ∝ *M* ^{cq}, where the metabolic scaling exponent is the product of the exponents for mass (*c*; (eqn 3)) and water use (*q*; (eqn 4)) scaling.

If *c* = 3/8 (from elastic similarity) and *q* = 2 (the theoretical Savage *et al*. maximum), the metabolic exponent *c·q* = 3/4 (West, Brown & Enquist 1999; Enquist, West & Brown 2000). This prediction has provoked debate, partly over the validity of metabolic isometry and in partly regarding *q* (*e.g*. Meinzer *et al*. 2005; Reich *et al*. 2006; Enquist *et al*. 2007a; Sperry, Meinzer & McCulloh 2008). Metabolic isometry is addressed in the second paper of this series (von Allmen *et al*. 2012). Here, we focus on the derivation of *q*.

For *q* ≥ 2 the negative effects of tree height and distance on *Q* must be eliminated. Height is negated if the drop in xylem pressure from soil to canopy (∆*P*) compensates for gravity (ρ*gH*), making the driving force (∆*P* – ρ*gH*; (eqn 1)) height-invariant. However, the (∆*P* – ρ*gH*) term often declines with height (Mencuccini 2003; Ryan Phillips & Bond 2006).

Transport distance can be negated by the ‘bottleneck effect’ where high flow resistance at the end of the xylem pipeline restricts the flow rate regardless of pipeline length. A bottleneck effect is consistent with the tapering of xylem conduits from trunk to terminal twig (West, Brown & Enquist 1999; Enquist, West & Brown 2000; Sperry, Meinzer & McCulloh 2008). This narrowing is captured in the Savage *et al*. model by a ‘taper function’: the conduit diameter inside the terminal twigs is assumed size-invariant and conduits widen proximally as the stems themselves widen across branch ranks (Fig. 1, downward ‘axial taper’ arrow).

The bottleneck effect is also influenced by how the number of conduits running in parallel changes across branch ranks. The Savage *et al*. model uses a ‘packing function’ (Sperry, Meinzer & McCulloh 2008) to govern the number of conduits that fit in a specified portion of wood space. Consequently, as conduits become narrower towards the twigs, their number per wood area increases (Fig. 1, upward ‘conduit packing’ arrow). To optimize space-filling, Savage *et al*. assume a universal packing function that allocates a constant fraction of wood space to transport vs. across all branch ranks. Savage *et al*. then solve for optimal conduit taper on the basis of an efficiency vs. safety trade-off (see also Enquist, West & Brown 2000). Taper is increased just enough to yield *q* = 2 (to maximize transport efficiency), but no more. Excessive taper would continue to widen conduits proximally, but to no effect other than to compromise safety from cavitation (larger conduits tend to be more vulnerable; Hacke *et al*. 2006).

Is the bottleneck effect enough to yield *q* = 2? Savage *et al*. recognize that not all species have identical taper and packing functions (McCulloh *et al*. 2010), suggesting that the space-filling and efficiency vs. safety trade-offs they invoke may have diverse context-dependent optima (Price, Enquist & Savage 2007). The intentional simplicity of the Savage *et al*. model also excluded additional variables that potentially influence the bottleneck effect such as the terminal resistance of leaves and the presence of nonconducting heartwood and bark.

The Savage *et al*. model also considers a basic issue in the derivation of *q*: the water use allometry only becomes a pure power function (e.g. (eqn 4)) at the limit of infinite tree size (Mencuccini *et al*. 2007). Thus, best-fit power functions across different size ranges yield different *q* (and *c*) exponents. For example, Savage *et al*. solve for the rate of conduit taper that is just sufficient to make *q* = 2 at the limit of infinite tree size, while the same taper yields only *q* ≈ 1·86 for finite-sized trees. This leads to their prediction of a metabolic scaling exponent of *c·q* ≈ 0·70 (3/8·1·86) in trees of actual size, with *c·q* = 0·75 as an upper bound (Savage *et al*. 2010).

Our species model attempts to clarify some of the uncertainty in metabolic scaling theory by revisiting the derivation of the water use allometry component ((eqn 1), (eqn 4)). New inputs of xylem architecture and function (asterisks in Fig. 1, see 'Model description') are added to the Savage *et al*. framework to improve *q* estimation and to enable the prediction of the *k*_{2} multiplier so that actual flow rates, *Q*, can be estimated. We focus on how specific hydraulic traits can effect the scaling of water use. For simplicity, we do not alter the branching architecture of the Savage *et al*. model (2010). We apply the new model to four objectives. (i) Using the simpler Savage *et al*. parameterization, we quantify the effects of finite tree size and gravity (i.e. the [∆*P* – ρ*gH*] term) on intraspecific scaling. (ii) We determine the influence of new inputs and variable taper and packing on the water use exponent (*q*) and multiplier (*k*_{2}). (iii) We translate how interspecific variation in wood traits translates into a map of ‘scaling space’ – defined by all possible combinations of multipliers (*k*_{2}) and the exponents (*q*) across species. The scaling space was simulated for four major functional tree types: conifers, ring-porous- and tropical and temperate diffuse-porous-angiosperms. (iv) Ecological drivers of scaling diversity are discussed, as are the implications for ¾ power metabolic scaling within vs. across species. The second paper tests the model against empirical measurements (von Allmen *et al*. 2012).