Nav: Home

Mathematical analysis explains transpiration-driven sap flow in coniferous trees

July 26, 2018

The exact science of tree sap transport has puzzled plant physiologists for many years. Sap's migration throughout tree trunks and branches is linked heavily to transpiration, the movement and subsequent evaporation of moisture from plants. As carbon dioxide diffuses inward from the air to plant leaves, a vapor pressure deficit between the leaf interior and surrounding atmosphere causes evaporation. This generates tension within leaf cell walls that is then transmitted via sap to tracheids -- conductive hollow wood cells with vertical grooves that comprise the trunk, stem, and branches of trees and are collectively called sapwood. The resulting negative sap pressure draws water from roots to leaves, sometimes to heights of over 300 feet.

Tracheids are the primary conductive elements in coniferous trees, and resemble tubes with small holes (or pits) that connect them both vertically and radially. Substances travelling in the radial direction must pass through many of these pits; thus, radial travel is more difficult than vertical travel. As a result, hydraulic conductivity is highly anisotropic (direction-dependent) and liquid movement is easier in the vertical direction.

In an article publishing this week in the SIAM Journal of Applied Mathematics, Bebart M. Janbek and John M. Stockie present a multidimensional porous medium model that measures sap flow within a tree stem. "I became interested in tree sap flow about seven years ago when I started studying the freeze-thaw mechanism that governs exudation--a fancy name for oozing--of maple sap from sugar maple trees during harvest season in late winter," Stockie said. "I grew up in Ontario and visited sugar bushes as a child, so I was thrilled by the opportunity to apply mathematical techniques to the study of the iconic sugar maple." His work with Janbek expands upon an existing one-dimensional model, and notably includes a nonlinear parabolic partial differential equation (PDE) with a transpiration source term. Researchers frequently use mathematical models to study the flow of sap within conductive sapwood. Electric circuit analogy and porous medium models--which model sap flow quite well due to the simple, repeating microstructure of sapwood--are both popular approaches. Unfortunately, most PDE-based porous models are one-dimensional, thus ignoring the radial variations within plant stems that make sapwood anisotropic.

The authors' extended multidimensional model of a tree trunk records radial velocity and allows for the study of radial flow patterns within the stem. It also includes a more realistic tapered axisymmetric stem geometry. In this geometry, an outer layer of conducting sapwood--containing both liquid sap and air-- surrounds a core region of non-conducting heartwood (the dense, inner part of a tree trunk) that is resistant to flow. An imposed transpiration flux along the outer surface drives the flow of water from the roots through the stem and branches to the leaves or needles. "The main advantage of this model is that it captures radial variations through the stem," Stockie said. "This is important when studying effects of geometry, which lead to significant differences between very young trees, which are cylindrical columns of conducting sapwood, and more mature trees, where a 'dead' sapwood core means that flow is constrained to a thin annular-shaped layer. One-dimensional models can only capture transport between roots and branches in an averaged sense, and cannot distinguish radial flows or geometric effects."

Janbek and Stockie employ realistic coefficient functions fit to experimental data on Norway spruce, a conifer native to Northern, Eastern, and Central Europe. However, they note that their model is not limited to any particular tree species. "We choose the Norway spruce for three main reasons," Janbek said. "First, there is a great deal of experimental data available that can be compared to results from our original one-dimensional porous medium model. Secondly, the stem anatomy in conifers like spruce is much simpler, and so we were much more confident in applying our model. Finally, Norway spruce grows in temperate regions where there is sufficient rainfall to ensure that our key assumption of a well-hydrated tree is valid; this spares us the extra complications arising from formation of embolisms (air bubbles) under very dry conditions." As with most spruce trees, the Norway spruce's stem resembles a circular cylinder that tapers from base to crown. Because its branches occur densely and consistently throughout the entire trunk and stem, the authors can postulate the transpiration flux as a complimentary distribution in the axial direction and include a sap outflow with a subsequent flux boundary condition. They then conduct asymptotic analysis.

"The asymptotic analysis helped us reduce the number of model parameters to a manageable set of dimensionless parameters that allows us to interpret results on tree hydraulics in a meaningful way," Janbek said. "We can capture many essential observations, like the finite speed at which disturbances travel through the stem or the effect of high anisotropy on radial variations in sap flow." Janbek and Stockie validate their findings via a numerical method with a cell-centered finite volume approximation, which confirms the accuracy of their analysis for a large range of saturations. "Our asymptotic results provide new insights into various flow regimes that occur in tree hydraulics and how this behavior depends on easily-measured physical parameters," Stockie said. "One interesting and somewhat surprising result is that the stem aspect ratio has a much greater influence on sap transport than the degree of anisotropy in the hydraulic permeability, which is often emphasized in other studies. We also derived approximate formulas describing how certain flow variables depend on parameters, which could provide tree physiologists with new opportunities for experimental studies."

The authors' findings enable future study of additional model parameters and inverse problems related to transpiration functions. Future work includes a plan to extend the model to a more general nonsymmetric, three-dimensional geometry to yield a solution with angular variations, and to account for a more complicated branching distribution along the stem. These types of expansions would allow Janbek and Stockie to examine the interplay between transpiration and embolism formation under more extreme conditions. "There are many interesting questions that can be studied using such a model, such as 'what happens when a tap-hole is drilled into the stem of a maple tree, thus breaking radial symmetry?'" Stockie said. "Or, 'how can we explain the known correspondence between temperature fluctuations and small expansion/contraction in stem diameter, and how does this affect sap transport?' The long-term goal of our research is to develop a comprehensive model for tree sap flow that incorporates a whole array of physical and biological mechanisms taking place over multiple spatial scales."
Source article: Janbek, B.M., & Stockie, J.M. (2018). Asymptotic and Numerical Analysis of a Porous Medium Model for Transpiration-driven Sap Flow in Trees. SIAM J. Appl. Math. (to be published).

Society for Industrial and Applied Mathematics

Related Geometry Articles:

Bubble dynamics reveal how to empty bottles faster
Researchers from the Indian Institute of Technology Roorkee have discovered how to make bottles empty faster, which has wide-ranging implications for many areas beyond the beverage industry.
How to break new records in the 200 metres?
Usain Bolt's 200m record has not been beaten for ten years and Florence Griffith Joyner's for more than thirty years.
GIS-based analysis of fault zone geometry and hazard in an urban environment
Typical geologic investigations of active earthquake fault zones require that the fault can be observed at or near the Earth's surface.
Strange warping geometry helps to push scientific boundaries
Princeton researchers have built an electronic array on a microchip that simulates particle interactions in a hyperbolic plane, a geometric surface in which space curves away from itself at every point.
The geometry of an electron determined for the first time
Physicists at the University of Basel are able to show for the first time how a single electron looks in an artificial atom.
Hopping bacteria
Scientists have long known that key models of bacterial movement in real-world conditions are flawed.
To replicate physical objects for virtual reality, just turn on your smartphone
A global team of computer scientists have developed a novel method that replicates physical objects for the virtual and augmented reality space just using a point-and-shoot camera with a flash, without the need for additional, and oftentimes expensive, supporting hardware.
Novel method for investigating pore geometry in rocks
In a fusion of mathematics and earth science, researchers in Japan proposed a novel method for characterizing pore geometry in rock, based on persistence diagram analysis and a newly proposed parameter, the distance parameter H.
Geometry is key to T-cell triggering
A new study reveals the geometric underpinnings of T-cell triggering through the precise engineering of T-cell receptor geometry in all three dimensions.
New quantum materials offer novel route to 3-D electronic devices
Researchers have shown how the principles of general relativity open the door to novel electronic applications such as a three-dimensional electron lens and electronic invisibility devices
More Geometry News and Geometry Current Events

Trending Science News

Current Coronavirus (COVID-19) News

Top Science Podcasts

We have hand picked the top science podcasts of 2020.
Now Playing: TED Radio Hour

Climate Mindset
In the past few months, human beings have come together to fight a global threat. This hour, TED speakers explore how our response can be the catalyst to fight another global crisis: climate change. Guests include political strategist Tom Rivett-Carnac, diplomat Christiana Figueres, climate justice activist Xiye Bastida, and writer, illustrator, and artist Oliver Jeffers.
Now Playing: Science for the People

#562 Superbug to Bedside
By now we're all good and scared about antibiotic resistance, one of the many things coming to get us all. But there's good news, sort of. News antibiotics are coming out! How do they get tested? What does that kind of a trial look like and how does it happen? Host Bethany Brookeshire talks with Matt McCarthy, author of "Superbugs: The Race to Stop an Epidemic", about the ins and outs of testing a new antibiotic in the hospital.
Now Playing: Radiolab

Speedy Beet
There are few musical moments more well-worn than the first four notes of Beethoven's Fifth Symphony. But in this short, we find out that Beethoven might have made a last-ditch effort to keep his music from ever feeling familiar, to keep pushing his listeners to a kind of psychological limit. Big thanks to our Brooklyn Philharmonic musicians: Deborah Buck and Suzy Perelman on violin, Arash Amini on cello, and Ah Ling Neu on viola. And check out The First Four Notes, Matthew Guerrieri's book on Beethoven's Fifth. Support Radiolab today at