Xylem Sap Ascent Lab (3D)

Water hangs, it is not pushed

Evaporation from the leaf cell walls pulls on a continuous column of water that runs through the xylem to the root. Water is cohesive (hydrogen bonds) and sticks to the wall of the conduit (adhesion), so the column can sit at a pressure far below zero. Plant physiologists use the water potential psi in megapascals; a static column has psi(z) = psi_soil - rho g z with rho g = 0.00981 MPa/m, so a 100 m tree already needs about -1 MPa before a drop of sap moves.

Darcy flow with gravity

For steady transpiration E (mmol per square metre of leaf per second) the potential falls along the trunk as

dpsi/dz = - rho g - E / (ks f(z))

where ks is the conductivity of the intact xylem per unit leaf area and f the fraction of the conductivity still working in that layer. The model integrates this upward through 24 layers starting from psi(0) = psi_soil - E R_root; the leaf potential is the trunk-top value minus E R_leaf.

Vessels are tubes: conductance goes as r⁴

Each layer holds 16 vessel classes with lognormal radii. The flow in a tube is Hagen-Poiseuille, Q = pi r4 dP / (8 mu L), so one vessel twice as wide carries 16 times as much sap. Check 1 reproduces the factor numerically from the Stokes equation. The consequence: losing a few wide vessels costs far more conductivity than losing many narrow ones.

Air-seeding and the vulnerability curve

When the tension in a conduit exceeds the capillary strength of a pit-membrane pore, dP = 2 sigma / r_pore (surface tension sigma = 0.072 N/m; a 0.05 micrometre pore holds about 2.9 MPa), an air bubble is pulled in and the vessel cavitates. The population of thresholds follows a Weibull curve: the percentage loss of conductivity is PLC = 100 (1 - exp(-(-psi/b)c)) with b = -P50 / (ln 2)1/c. P50 is the potential at 50 % loss, c the steepness. In the model each vessel gets its own threshold from this distribution, optionally correlated with its size (the "rare pit" idea that wide vessels are weaker). A cavitated vessel stays dead until it refills: here at night, in wet soil (psi_soil above -0.7 MPa), 12 % of the dead vessels per hour.

Stomata and the supply-demand balance

Demand is E = Emax L(t) [0.03 + 0.97 g(psi_leaf)] with L(t) a daylight curve, Emax set by the dryness of the air and g a linear stomatal response between -0.6 MPa (open) and the closing potential you set (3 % cuticular leak remains). The model finds the unique E where supply and demand meet; if a vessel then exceeds its threshold, the conductivity drops, the potential falls further, and the loop repeats: the runaway that the Sperry supply curve shows as a collapse of the curve (graph 3). Closing the stomata early (isohydric) protects the xylem and costs carbon; keeping them open (anisohydric) is a gamble.

Numerical checks on this page

(1) A finite-difference solution of Poiseuille flow in a tube against the analytic pi R4 G / 8 mu. (2) The layered potential integration for an intact trunk against the closed form psi_soil - E R_root - rho g H - E H / ks. (3) With gravity off and a reversible curve, the Kirchhoff transform: E H / ks = integral of f(psi) dpsi over the potential drop, evaluated by midpoint quadrature.

Honest simplifications

One-dimensional steady flow for every time step (no capacitance, no refilling by root pressure, no embolism-driven stomatal feedback beyond the potential); the vessel thresholds are a statistical population, not individual pit membranes; pressure in all vessels of a layer is taken equal; leaf area to sapwood area ratio and the vessel lumen fraction (25 %) used for the sap speed are typical values, not measurements of a species. Trees above about 100 m are limited by exactly this arithmetic in real life, but real redwoods also use hydraulic segmentation and foliar water uptake that are not modelled. Numbers are plausible orders of magnitude for teaching, not data for a particular tree. The tree in the picture is drawn at a fixed size and scaled gently with H.

What to try

• Load Oak in a wet summer: the potential line (graph 1) is a gentle slope; at night it falls back to the static column.
• Load Slow summer drought and watch graph 4: the stomata close first, then the embolism creeps up. Hit Rain to re-wet the soil; the vessels refill at night.
• Load Anisohydric: stomata stay open and compare: the supply curve (graph 3) collapses and the crown dies in a cascade.
• Set the correlation slider to 0 vs 1 and look at which colours of vessel die first in the trunk window.
• Load Giant redwood, 110 m: gravity alone puts the top layer near -1.1 MPa before any flow.