Sap flow and heat transport in trees: An asymptotic and numerical study

Transport of fluid and heat inside a tree, and the interchange of water and energy between the tree and the environment, are topics that have been and continue to be areas of active research in plant physiology, agriculture and environmental studies. Many models have been proposed to describe the flow of sap inside the tree, and to connect it to the driving transpiration rate, with various levels of complexity, and with different levels of abstraction. Most existing models are 1D models and many only attempt to get numerical results, without much analysis. For our work, we adopt a porous medium model that has been verified experimentally [Chuang et al., Ecological Modelling, 191(3):447-468, 2006]. We generalize this 1D model to a 3D axisymmetric geometry, where flow is transpiration driven and has anisotropic and spatially dependent hydraulic conductivity. Through asymptotic analysis, we derive approximate solutions that produce the axial and radial trunk sap fluxes for a given transpiration function. We validate the analytical solutions using a second order finite difference scheme. Next we use our solution formulas to tackle the inverse problem of determining spatial and temporal components of transpiration given a discrete set measurements of the trunk sap flux. Finally, we compare our results to some experimental data on radial variations of sap flux. As for the heat transport problem, previous work related to trees discuss special cases of the problem, while giving detailed accounts and specific formulas of the boundary conditions, like wind and solar radiation effects. Most of this work does not include the possible effects of advection owing to sap flux, and does not discuss the effects of spatial variation in saturation on the thermal diffusivity. Assuming local thermal equilibrium for porous media, we propose a simple advection-diffusion model, with general boundary conditions, and derive Fourier-Bessel series solutions for the various possible cases suggested by dimensionless parameters.