A Mathematical Model of Sap Exudation in Maple Trees Governed by Ice Melting, Gas Dissolution, and Osmosis

Abstract

We develop a mathematical model for sap exudation in a maple tree that is based on a purely physical mechanism for internal pressure generation in trees in the leafless state. There has been a long-standing controversy in the tree physiology literature over precisely what mechanism drives sap exudation, and we aim to cast light on this issue. Our model is based on the work of Milburn and O'Malley [Can. J. Bot., 62(10): [2101][2102][2103][2104][2105][2106] 1984] who hypothesized that elevated sap pressures derive from compressed gas that is trapped within certain wood cells and subsequently released when frozen sap thaws in the spring. We also incorporate the extension of Tyree [in Tree Sap, pp. 37-45, eds. M. Terazawa et al., Hokkaido Univ. Press, 1995] who argued that gas bubbles are prevented from dissolving because of osmotic pressure that derives from differences in sap sugar concentrations and the selective permeability of cell walls. We derive a system of differential-algebraic equations based on conservation principles that is used to test the validity of the Milburn-O'Malley hypothesis and also to determine the extent to which osmosis is required. This work represents the first attempt to derive a detailed mathematical model of sap exudation at the micro-scale.

A two-scale Stefan problem arising in a model for tree sap exudation

“…The modified Milburn-O'Malley description just presented (with the exception of root pressure) was employed by Ceseri & Stockie (2013) and Graf et al (2015) to derive a mathematical model for celllevel processes governing sap exudation during a thawing cycle. In this study, we study the same problem, including the effect of the gas phase in both cell chambers (fiber and vessel), but we will assume for the sake of simplicity that the effects of gas dissolution and nucleation are negligible.…”

“…Note that during a thawing cycle, we are only concerned with a vessel containing liquid sap (no ice) because of the effect of freezing point depression, which ensures that any given vessel thaws before the adjacent fiber(s). This reference cell geometry should be contrasted with that depicted in Ceseri & Stockie (2013, Fig. 3.1).…”

“…For the moment, we will consider the four solution variables s iw , s gi , r, U as depending on time t only, with an additional dependence of temperature on the microscale spatial variable; however, beginning in Section 3 when we derive macroscale equations for the homogenized problem, these variables will also depend on the global spatial variable x that denotes the location of the reference cell within the tree stem. Within a reference cell, the dynamics for s iw (t), s gi (t), r(t) and U(t) are governed by four differential equations whose derivation can be found in Ceseri & Stockie (2013). The first is the Stefan condition for the ice/water interface in the fiber…”

“…This paper is motivated by the study of sap flow in sugar maple trees that are subject to repeated cycles of thawing and freezing during the sap harvest season in late winter (Ceseri & Stockie, 2013). We seek insight into the phenomenon of sap exudation, which refers to the generation of elevated sap pressure within the maple stem when the tree is in a leafless state and no transpiration occurs to drive the sap flow.…”

“…We seek insight into the phenomenon of sap exudation, which refers to the generation of elevated sap pressure within the maple stem when the tree is in a leafless state and no transpiration occurs to drive the sap flow. Our work is based on the model derived in Ceseri & Stockie (2013) that captures the physical processes at the microscale (i.e., at the level of individual wood cells) and includes multiphase flow of ice/water/gas, heat transport, porous flow through cell walls, and osmosis. There is an inherent repeating structure in sapwood at the cellular scale that lends itself naturally to the use of homogenization ideas that we exploited in Graf et al (2015) to obtain a multiscale model for the macroscale temperature that is coupled to a corresponding system of equations governing the microscale cellular processes.…”