The porous media model for the hydraulic system of a conifer tree : linking sap flux data to 1 transpiration rate

1 Linking sap flow in tree boles to plant transpiration continues to be a fundamental and 2 practical research problem in physiological ecology and forest hydrology. Many models have 3 been proposed to describe water movement within trees with varying degrees of success. The 4 prevailing resistance-capacitance (RC) circuit models have the advantage of being easy to 5 implement. However, RC models are ordinary differential equation (ODE) models that reduce the 6 spatial-temporal dynamics of a tree hydraulic system to the temporal variation of simplified 7 quantities; thus, the RC parameterization is more empirical and open to various interpretations. 8 For coniferous trees, a reasonable alternative to RC-circuit models is a porous media (PM) model, 9 which is a partial differential equation (PDE) model that describes the spatial-temporal dynamics. 10 The model more closely represents the physical elements of the conifer hydraulic system but also 11 requires a direct estimation of its properties. Our proposed PM model is original in that it 12 formulates a theoretical link between measured quantities (i.e., sap flux density and tree structure) 13 and model parameters, obtained during nighttime, which permits direct numerical conversion of 14 sap flow to transpiration rate during daytime. In addition to fully simulating the PDE, we propose 15 an alternative method to transform the PDE into a set of ODE's, to significantly reduce 16 computational demands. Although the ODE results are noisy, the transpiration pattern produced 17 by the ODE, once filtered, is similar to that of the PDE. We demonstrate that measurements of the 18 sap flux in multiple positions below and within the crown can be used to compute the height-19 dependent transpiration rate; but if rates of bulk crown transpiration are of primary interest, 20 readily obtainable measurements at two heights, at the base of the tree and below the crown, are 21 sufficient for the computation.