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, δTT 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 aQ – 2εσT a4λg 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 iδ 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.