  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.
