Pacific Institute for the Mathematical Sciences - PIMS

Mathematical modelling of sap flow in maple trees

    The flow of sap in trees is such a common everyday phenomenon that it is hard to believe that there is a lack of understanding in several fundamental aspects of sap flow. This talk will demonstrate the role that mathematics can play in dealing with the complex coupled physics that govern sap flow in trees. More specifically, I will explain how an improved understanding of fundamental aspects of sap flow in sugar maple trees (Acer saccharum) can be applied to answer pressing questions in the maple syrup industry. This talk will focus on two mathematical modelling efforts. The first aims to develop a macroscopic model for sap flow and heat transport in a tree during the growing season when sap flow is driven by the process of "transpiration". The tree is treated as an anisotropic porous medium through which sap flow is driven by a given transpiration flux, and heat transport is driven by daily variations in ambient temperature and solar radiation. The second project aims to explain the phenomenon of "sap exudation", in which sugar maple (and a few related species) generate a positive stem pressure during the spring thaw in the absence of leaves. Many (bio-)physical mechanisms have been proposed over the past century to explain this phenomenon, yet there remains a great deal of controversy over the precise mechanism driving sap exudation. We consider the prevailing hypothesis due to Milburn and O'Malley that treats sap as a two-phase (gas/liquid) mixture whose dynamics are governed by the combined effects of porous media flow, freezing/thawing, gas dissolution, and osmotic pressure. We develop a nonlinear system of differential equations that captures these effects at the cellular scale, and we demonstrate through a combination of analytical and numerical methods that the model is capable of reproducing qualitatively many of the behaviours observed in maple trees.

Location: ESB 2012