Multiscale model of a freeze-thaw process for tree sap exudation

TL;DR

This paper develops a multiscale mathematical model to explain sap exudation in trees during winter, demonstrating that physical processes like freeze-thaw and osmosis can generate positive stem pressure.

Contribution

It introduces the first comprehensive mathematical model of tree sap exudation, integrating phase change, transport, and temperature dynamics across multiple scales.

Findings

Model simulations match experimental stem pressures.

Physical mechanisms alone can explain exudation.

Provides new insights into winter tree physiology.

Abstract

Sap transport in trees has long fascinated scientists, and a vast literature exists on experimental and modelling studies of trees during the growing season when large negative stem pressures are generated by transpiration from leaves. Much less attention has been paid to winter months when trees are largely dormant but nonetheless continue to exhibit interesting flow behaviour. A prime example is sap exudation, which refers to the peculiar ability of sugar maple (Acer saccharum) and related species to generate positive stem pressure while in a leafless state. Experiments demonstrate that ambient temperatures must oscillate about the freezing point before significantly heightened stem pressures are observed, but the precise causes of exudation remain unresolved. The prevailing hypothesis attributes exudation to a physical process combining freeze-thaw and osmosis, which has some support from experimental studies but remains a subject of active debate. We address this knowledge gap by developing the first mathematical model for exudation, while also introducing several essential modifications to this hypothesis. We derive a multiscale model consisting of a nonlinear system of differential equations governing phase change and transport within wood cells, coupled to a suitably homogenized equation for temperature on the macroscale. Numerical simulations yield stem pressures that are consistent with experiments and provide convincing evidence that a purely physical mechanism is capable of capturing exudation.