## 1. Introduction

[2] Over the past two decades, the role of photosynthetically active radiation (PAR) and vapor pressure deficit (VPD) in regulating stomatal conductance [e.g., *Leuning*, 1995; *Oren et al.*, 1999a] and photosynthesis [*Farquhar et al.*, 1980; *Collatz et al.*, 1991] has been made clearer, and progress has been made in scaling these processes to the canopy level. In particular, the nonlinear interaction between light and leaf photosynthesis necessitates a multilevel description of the canopy to scale leaf-level processes up to the canopy scale [*Baldocchi*, 1992; *Baldocchi and Meyers*, 1998; *Lai et al.*, 2000b, 2002; *Schäfer et al.*, 2003]. There is a growing recognition that plant hydraulics also play a central role in linking root water uptake to transpiration and carbon uptake [e.g., *Sperry et al.*, 1998; *Katul et al.*, 2003; *Brodribb and Holbrook*, 2004]. A mechanistic understanding of the effects of soil-plant hydraulics on transpiration must account for water movement within the tree system and the potential onset of xylem cavitation [*Sperry*, 2000; *Sperry et al.*, 2002].

[3] However, unlike canopy radiation models, in which plant area density can be treated as a random medium (with clumping), plant hydrodynamics must resolve the plant architecture, allometry, and branching. Several factors frustrate modeling water movement within the tree system, as summarized below:

[4] The first factor is computational. Although it is possible to derive point equations for flow within one organ element, solving a transient nonlinear diffusion equation in high-resolution 3-D on an unstructured mesh needs a large computational resource.

[5] The second factor is data availability. Although few studies quantify the hydraulic properties of trees [*Maherali and DeLucia*, 2001; *Domec and Gartner*, 2003; *Brodribb and Holbrook*, 2004; *Chuang et al.*, 2005], measurements conducted with the intent to model water flow through all the plant organs are scarce.

[6] The third factor is relevance to ecohydrologic models. Little information exists as to whether differences in tree level structure lead to consistent biases that accumulate at scales relevant to the canopy [e.g., *Goldstein et al.*, 1998; *Ewers et al.*, 2001; *Köstner*, 2001; *Maherali and DeLucia*, 2001; *Domec and Gartner*, 2003; *Meinzer et al.*, 2003]. An essential question is whether these effects can be formalized into scaling laws, and whether the architecture of the crown is beneficial for diagnostic and prognostic hydrologic models. This is the subject of this study.

[7] Although this study does not offer new empirical data, it makes use of published empirical parameters to derive a novel numerical model, the finite element tree crown hydrodynamics (FETCH) model. This model resolves plant hydrodynamics across a range of realistic tree structures. This model is coupled to tree microclimate through radiative transfer and simplified turbulent transport theories. Using this new framework, we explore the sensitivity of plant hydrodynamics to canopy structure via parametric variations of simple scaling laws. The goal of this study is to highlight the advantages of this new modeling approach relative to the commonly used models of plant contribution to land-surface energy balance. These advantages include better temporal description of water fluxes from a tree crown and detailed vertical distribution. We suggest that by adopting this hydrodynamic description of the plant system, it is possible to better predict nonlinear, hard-to-parameterize phenomena such as midday stomata closure and differential stomatal response along a branch and incorporate the effects of tree structure and physiology in the prediction of changes in transpiration fluxes following environmental changes such as fertilization.

[8] The common models of tree-water systems incorporated within meteorological land-surface models are based on equivalence to an electric circuit (Figure 1). The most simple of these water flow models are the single (1R) or multiple (2R) resistor models [*Jones*, 1992]. These lump all vegetation below the grid resolution to a single effective “big leaf,” or to a low number of subgrid patches. Some examples for “big leaf” resistor models include SiB2 [*Sellers et al.*, 1996a, 1996b] and LEAF2 [*Walko et al.*, 2000]. The resistor models take no account of water storage in the plant and therefore assume constant equilibrium between transpiration and the energy gradients between the soil and the atmosphere (see Figure 1). The resistance-capacitance (RC) models are another type of electric circuit equivalence model developed to account for water storage (or “charge capacitance”), which is presumably important in tall forested ecosystems [*Jones*, 1992; *Schulte and Costa*, 1996; *Phillips et al.*, 1997]. The resistance and capacitance are empirical properties often calibrated to match observations. It is possible to represent a crown structure using a RC model, but that would require branch level or crown/canopy-layer level empirical calibration [*Jones*, 1992].

[9] A more physically based approach is to combine the continuity equation with a physical transport law applied to an elemental organ, which leads to a nonlinear partial differential equation (PDE) resembling Richards' equation for soil water movement including sources and sinks. In essence, this approach assumes that water movement through a collection of interconnected tracheids in the hydroactive xylem (typical to conifers) resembles porous media flow [*Siau*, 1983; *Früh and Kurth*, 1999; *Kumagai*, 2001; *Chuang et al.*, 2005]. This view is perhaps indirectly supported by the wealth of hydraulic cavitation curves collected for various plant organs that are analogous to hydraulic losses in porous media (e.g., *Sperry* [2000] and review by *Feild et al.* [2002]). As discussed by *Chuang et al.* [2005], the mathematical properties (and boundary conditions) of the resulting porous-media PDEs may significantly diverge from RC models whose mathematical properties are first-order ordinary differential equations for the water flux.

[10] Treating sap flow in wood as porous media flow to apply Darcy's law was first suggested by *Siau* [1983]. *Arbogast et al.* [1993] developed a Richards-type equation as a way to numerically solve water movement in lateral roots as part of a soil water system. A branch system was suggested by *Früh and Kurth* [1999], who generated a splitting-branch tree-grid system and solved it using finite differences. A similar but somewhat simplified approach was also developed by *Kumagai* [2001] and later adopted by *Chuang et al.* [2005], who revised it to include nonlinear terms for the conductance and capacitance. All of the above require specified transpiration from each branch as a model input. The stem systems in the latter two studies were limited to a single stem without branching (see Figure 1).

[11] An even more complete model of the tree transport system was suggested by *Aumann and Ford* [2002a, 2002b]. They observed that a tracheid-level model could lead to a better representation of the flow in nonsaturated stages, for example, when recovering from cavitations. The need, though, for many cell-level parameters and explicit cell-structure element construction makes it less attractive as a general simulation tool. Furthermore, none of the above models considers interaction between the tree and its microclimate.

[12] Here we adopt the formulation of *Chuang et al.* [2005] and adjust it to handle a nonvertical branching system, similar to that proposed by *Früh and Kurth* [1999]. We also added a multiple-layer canopy microclimate model using leaf-level physiological, radiative transfer, and canopy drag properties as discussed by *Lai et al.* [2000a] with a first-order turbulent closure scheme [*Poggi et al.*, 2004], and add a stomatal response that links pressure in the branches to stomatal resistance. The resulting FETCH model simulates the hydrodynamics of a branching pipe system, constructed in a manner to be consistent with the plant area density. The model also utilizes the “aorta” or “fully coupled” tree transport system formulation [*McCulloh et al.*, 2003; *Schulte and Brooks*, 2003]. This approach assumes free flow along the radial axis of the branches.

[13] By using such a physically based plant-atmosphere-hydraulics model with explicit three-dimensional (3-D) representation of the tree structure, we can confront the question of how the variability in crown architecture (i.e., interspecific, age, or ecosystem dependent structure) would lead to different transpiration responses. We compare the results from such a detailed model with the equivalent resistance and capacitance parameters that would have been obtained by 2R and RC models (see Figure 1). This comparison is used to test the sensitivity of the tree system to structure and its consequent effects on 2R or RC parameterizations that do not and in most cases cannot include structural aspects of the tree.