Theory: inference of boundary layer conductance from leaf
temperature dynamics
Leaf boundary layer conductance to heat (g bh) can
be estimated from the time constant for leaf cooling (τ ) using a
model of leaf energy balance, in which the rate of change of leaf heat
content, and thus temperature, is proportional to the difference between
energy gains and losses (radiative, convective and latent) and inversely
proportional to the leaf’s heat capacity. A full derivation is presented
in Supporting Information Methods S1; only key results are given here.
The derivative of leaf temperature (T , K) with respect to time
(t ) is
\begin{equation}
\frac{\text{dT}}{\text{dt}}=\frac{Q-2\epsilon\sigma T^{4}-c_{\text{pa}}g_{\text{bh}}\left(T-T_{a}\right)-\lambda g_{\text{tw}}\Delta w}{k}\ ,\ \ \ \ \ \ \ \ Eqn\ 1\nonumber \\
\end{equation}where k is the leaf heat capacity (J m-2K-1), Q is absorbed radiation, including both
shortwave and longwave (J m-2 s-1),ε is leaf thermal emissivity, σ is the Stefan-Boltzmann
constant (5.67⋅10-8 J m-2s-1), c pa is air heat capacity
(29.2 J mol-1 K-1),g bh is (2-sided or whole-leaf) boundary layer
conductance (mol m-2 s-1),T a is air temperature in kelvins, λ is the
latent heat of vaporization (44000 J mol-1),g tw is total leaf conductance to water vapor (mol
m-2 s-1), and Δw is the
leaf-to-air water vapor mole fraction difference (mol
mol-1). Δw equalsw s(T ) – w a, wherew s and w a are saturated
and ambient water vapor mole fractions, respectively, the former
calculated at the leaf temperature. The terms that are nonlinear in leaf
temperature (T 4 andw s(T )) can be expressed in terms of the
leaf-to-air temperature difference, δ ≡ T –T a, by approximations given in Supporting
Information Methods S1 (Eqns S2-S6). Assuming T ais constant, the result is a differential equation for δ :
\begin{equation}
k\frac{\text{dδ}}{\text{dt}}=a-\text{bδ}\ ,\ \ \ \ \ \ \ Eqn\ 2\nonumber \\
\end{equation}where a ≡ Q –
2εσT a4 –λg twD a, b ≡
8εσT a3 +c pag bh +λg tws , and s is the derivative ofw s with respect to T . Integrating Eqn 2
leads to the solution
\begin{equation}
\delta\left(t\right)=\delta_{f}-\left(\delta_{f}-\delta_{i}\right)e^{-\frac{t}{\tau}}\ \ \ ,\ \ \ \ \ \ Eqn\ 3\nonumber \\
\end{equation}where δ i and δ f are values
of δ at t = 0 and in the limit of large t , and the
cooling time constant, τ , equals k /b . τ is
thus a function of g bh,T a, stomatal conductance (viag tw) and leaf heat capacity (k ). We fitted
Eqn 3 to each cooling curve using the nls() function in R. To provide
the algorithm with initial estimates for the parameters
(δ i, δ f and τ ), forδ i we used the first three measurements ofδ (which spanned the first 0.6 s of cooling), forδ f we used the minimum value of δ (at or
near the end of the cooling curve), and for τ we used the
observed halftime for cooling (the time at which δ first dropped
below δ f + 0.5(δ i –δ f)) divided by the natural logarithm of 2. We
used the mean value of air temperature during each cooling curve asT a, and computed s at the midpoint between
leaf and air temperatures when δ was halfway betweenδ i and δ f. The resulting
fitted model produced an estimate for τ for each patch in each
cooling curve. We then estimated g bh fromτ by solving τ = k /b =k /(8εσT a3 +c pag bh +λg tws ) for g bh to
give
\begin{equation}
g_{\text{bh}}=\frac{\frac{k}{\tau}-8\epsilon\sigma T_{a}^{3}-\lambda g_{\text{tw}}s}{c_{\text{pa}}}\text{\ \ .\ \ \ \ \ \ \ }Eqn\ 4\nonumber \\
\end{equation}To estimate g tw for application to Eqn 4, we
measured surface conductance to water vapor, g sw,
in each patch immediately before each experiment, for the abaxial leaf
surface, using a leaf porometer (AP4, Delta-T devices, Cambridge, UK).g tw is not independent ofg bh, because g tw includes
components of g bh relevant to the transpiring
surface(s); accounting for this interaction leads to a quadratic
expression for g bh (Eqn S15). We assumed thatg sw at the adaxial surface was either equal to
the value measured by porometry at the abaxial surface (in
amphistomatous species) or zero (in hypostomatous species), and that the
resistance added by hairs was either equal on both leaf surfaces (in
SOLY, which had hairs on both surfaces) or was zero on the adaxial
surface (for the other species).
Estimation of g bh as described above also
requires estimates of leaf thermal emissivity (ε ) and leaf heat
capacity (k ). We assumed ε = 0.98 (e.g., Chen, 2015), and
addressed the effect of uncertainty in ε as described below. We
estimated k as (4.184⋅WM + 1.5⋅DM)/LA, where WM, DM and LA are
leaf water mass, dry mass and leaf area, respectively; this assumes heat
capacities of 4.184 J g-1 K-1 for
water and 1.5 J g-1 K-1 for leaf dry
matter (Samarasekara and Coorey, 2011). We measured WM, DM and LA as
follows: immediately after each experiment, we photographed each leaf to
measure leaf area in ImageJ, then weighed the leaf using a 5-point
digital balance (XS225DU, Mettler-Toledo, Columbus, OH), dried it in a
drying oven at 65 oC for at least 24 hours, and then
reweighed it after mass stopped changing. DM was taken as the final leaf
mass, and WM as the difference between fresh and dry masses.