The functional dependence of canopy conductance on water vapor pressure deficit revisited

Abstract

Current research seeking to relate between ambient water vapor deficit (D) and foliage conductance (g_F) derives a canopy conductance (g_W) from measured transpiration by inverting the coupled transpiration model to yield g_W = m − n ln(D) where m and n are fitting parameters. In contrast, this paper demonstrates that the relation between coupled g_W and D is g_W = AP/D + B, where P is the barometric pressure, A is the radiative term, and B is the convective term coefficient of the Penman-Monteith equation. A and B are functions of g_F and of meteorological parameters but are mathematically independent of D. Keeping A and B constant implies constancy of g_F. With these premises, the derived g_W is a hyperbolic function of D resembling the logarithmic expression, in contradiction with the pre-set constancy of g_F. Calculations with random inputs that ensure independence between g_F and D reproduce published experimental scatter plots that display a dependence between g_W and D in contradiction with the premises. For this reason, the dependence of g_W on D is a computational artifact unrelated to any real effect of ambient humidity on stomatal aperture and closure. Data collected in a maize field confirm the inadequacy of the logarithmic function to quantify the relation between canopy conductance and vapor pressure deficit.

Appendix

The calculation of transpiration uses the approach and numerical values given in (Campbell and Norman 1998). Sunlit and shaded leaves were treated separately because of differences in the way they intercept radiation and differences in leaf conductance values (Jagtap and Jones 1989). In the following development index, j designates the class to which the considered leaves belong:

$$ {\displaystyle \begin{array}{l}j=0;\kern1em \mathrm{shaded}\kern0.5em \mathrm{leaves}\\ {}j=1;\kern1em \mathrm{sunlit}\kern0.5em \mathrm{leaves}\end{array}} $$

(A.1)

Radiation balance

The global radiation intercepted by the foliage in class j, R_{g j} is:

$$ {R}_{g\;j}=j\kern0.5em f\kern0.5em {R}_r\;{L}_j+x\frac{L_j}{L}\left({R}_d+{S}_d\right) $$

(A.2)

The symbols of Eq. (A.2) are defined here below:

The ray interception factor f by leaves with random inclination and orientation is:

$$ f=0.5\;\sec \left(\eta \right) $$

(A.3)

where η is the solar zenith angle.

L is the leaf area index, L_j is:

$$ {\displaystyle \begin{array}{l}{L}_j\kern0.5em \mathrm{for}\kern0.5em j=1\to {L}_1=\frac{\left[1-\exp \left(-f\;L\right)\right]}{f}\\ {}{L}_j\kern0.5em \mathrm{for}\kern0.5em j=0\to {L}_0=L-{L}_1\end{array}} $$

(A.4)

The diffuse interception x of uniformly distributed hemispherical rays is:

$$ x=\frac{1}{\pi }{\int}_0^{2\pi }{\int}_0^{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left[1-\exp \left(-f\;L\right)\right]\sin \left(\eta \right)\cos \left(\eta \right)\partial \eta\;\partial \varphi $$

(A.5)

(η = zenith angle, φ = azimuth angle, of an incident ray).

The beam R_r and diffuse R_d components of global radiation through a cloudless sky are calculated from the extraterrestrial solar constant R_o = 1356 W m⁻² (List 1966):

$$ {\displaystyle \begin{array}{l}{R}_r={R}_o\cos \left(\eta \right)\times {0.65}^{\raisebox{1ex}{$P\sec \left(\eta \right)$}\!\left/ \!\raisebox{-1ex}{$101.3$}\right.}\\ {}{R}_d={R}_o\cos \left(\eta \right)\times 0.34\times \left(1-{0.65}^{\raisebox{1ex}{$P\sec \left(\eta \right)$}\!\left/ \!\raisebox{-1ex}{$101.3$}\right.}\right)\\ {}P=\mathrm{baromertic}\ \mathrm{pressure},\left(\mathrm{kPa}\right)\end{array}} $$

(A.6)

The foliage also intercepts S_d, the secondary scattering of global radiation approximated as:

$$ {S}_d\approx 0.22\times \left\{{R}_r\left[1-\exp \left(-f\;L\right)\right]+x\;{R}_d\right\} $$

(A.7)

Assigning a value of 0.55 to leaf solar radiation absorptivity, ${\varepsilon}_s=0.767\times {e}_a^{1/7}$ (e_a = air vapor pressure in kPa) to sky emissivity of terrestrial radiation (Brutsaert 1982) and assuming the canopy emissivity of terrestrial radiation is close to unity (Fuchs and Tanner 1966), R_{n j} the net radiation of leaves in class j is:

$$ {\displaystyle \begin{array}{l}{R}_{n\;j}=0.55{R}_{g\;j}+x\frac{L_j}{L}\left({\varepsilon}_s\sigma {T}_{a\kern0.24em K}^4-\sigma {T}_{F\;K\;j}^4\right)\\ {}\sigma {T}_{F\;K\;j}^4\approx \sigma {T}_{a\kern0.24em K}^4+4\sigma {T}_{a\;K}^3\left({T}_{F\;j}-{T}_a\right)\end{array}} $$

(A.8)

Here, σ = 5.67 × 10⁻⁸ W m^‐2 K^‐4; T_{a K} and T_{F K j} are the Kelvin air and foliage temperatures; T_a and T_{F j} are the Celsius air and foliage temperatures.

We define an intermediate radiation term R_{m j}:

$$ {R}_{m\;j}=0.55{R}_{g\;j}+x\frac{L_j}{L}\left({\varepsilon}_s-1\right)\sigma {T}_{a\;K}^4 $$

(A.9)

Equation (A.8) becomes:

$$ {R}_{n\;j}={R}_{m\;j}-4x\frac{L_j}{L}\sigma {T}_{a\;K}^3\left({T}_{F\;j}-{T}_a\right) $$

(A.10)

Energy balance

We define a radiative conductance g_{r j} as:

$$ {g}_{r\;j}=4x\frac{L_j}{L}\frac{\sigma {T}_{a\;K}^3}{c_p} $$

(A.11)

Here, c_p is the specific heat of air (= 29.3 J mol⁻¹ C⁻¹). Setting radiative flux density H_{r j} as:

$$ {H}_{r\;j}={c}_p{g}_{r\;j}\left({T}_{F\;j}-{T}_a\right) $$

(A.12)

leads to the energy balance of the leaves in class j:

$$ {R}_{m\;j}=\lambda {E}_j+{H}_{c\;j}+{H}_{r\;j} $$

(A.13)

where λ is the latent of vaporization (= 43,384 J mol⁻¹ at T = 300 K) and λE_j is the latent heat flux density:

$$ \lambda {E}_j=\frac{\lambda }{P}{g}_{V\;j}\left[e\left({T}_{F\;j}\right)-{e}_a\right] $$

(A.14)

where e(T_Fj) is the saturated vapor pressure at T_Fj and e_a is the vapor pressure in the air.

H_{c j} is the convective sensible heat flux density:

$$ {H}_{c\;j}={c}_p{g}_{H\;j}\left({T}_{F\;j}-{T}_a\right) $$

(A.15)

Transport coefficients

In Eq. (A.14), g_{V j} is the conductance of the vapor path from the evaporating locations within the plant tissues to the free air above the canopy:

$$ {g}_{V\;j}=\frac{L_j}{r_{V\;c}+{r}_{L\;j}} $$

(A.16)

r_{V c} is the convective resistance per unit leaf area (r_V c = 1/g_V c) of air for water vapor transport from the outer surface of a leaf to freely moving air above the canopy.

r_{L j} is the parallelly connected diffusive leaf resistance per unit leaf area through stomata and cuticle of the adaxial and abaxial faces of the leaves (r_L j = 1/g_L j, g_Lj is the leaf conductance).

The foliage conductance g_Fj is then defined as:

$$ {\displaystyle \begin{array}{l}{g}_{Fj}={L}_j{g}_{Lj}=\frac{L_j}{r_{Lj}}\\ {}{g}_F={g}_{F0}+{g}_{F1}\end{array}} $$

(A.17)

The convective resistance r_{V c} is made of:

$$ {r}_{V\;c}={r}_a+0.5\kern0.5em {r}_{V\;b} $$

(A.18)

Here, r_a is the convective resistance of the turbulent air layer above the canopy assigned to a unit leaf area:

$$ {r}_a=\frac{\ln \left(\frac{z-d}{z_0}\right)\ln \left(\frac{z-d}{0.2\;{z}_0}\right)L}{\rho\;{k}^2u(z)} $$

(A.19)

where z is the level above the canopy taken from the ground at which wind speed u(z) is measured, d = 0.66 × Z is the wind profile displacement level, z₀ = 0.13 × Z is the roughness length, Z is the canopy height, k is the von Kármán constant (= 0.40), and ρ is the air molar density [=44.6 × (T_a K/273.2) × (P/101.3) mol m⁻³.]

r_{V b} in Eq. (A.18) is the one-sided convective resistance per unit leaf area for vapor transport in the laminar air boundary layer of a leaf surface. The factor 0.5 halves the resistance because leaves are two-sided.

$$ {r}_{V\;b}=6.8\left(\raisebox{1ex}{${l}^{0.5}$}\!\left/ \!\raisebox{-1ex}{$\overline{u_{\mathrm{\ell}}^{0.5}}$}\right.\right) $$

(A.20)

l is the characteristic length of the leaf and $\overline{u_{\mathrm{\ell}}^{0.5}}$ is defined in Eq. (A.25).

Heat convective conductance g_{H j} in Eq. (A.15) is:

$$ {g}_{H\;j}=\frac{L_j}{r_a+0.5\;{r}_{H\;b}} $$

(A.21)

where r_{H b} is the one-side convective resistance per unit leaf area for heat transport in the laminar air boundary layer surrounding a leaf:

$$ {r}_{H\;b}=7.4\left(\raisebox{1ex}{${l}^{0.5}$}\!\left/ \!\raisebox{-1ex}{$\overline{u_{\mathrm{\ell}}^{0.5}}$}\right.\right) $$

(A.22)

The combined conductance of heat g_{T j} is:

$$ {g}_{T\;j}={g}_{H\;j}+{g}_{r\;j} $$

(A.23)

The boundary-layer convective resistance is proportional to $1/\overline{u_{\mathrm{\ell}}^{0.5}}$. The mean value $\overline{u_{\mathrm{\ell}}^{0.5}}$ in the canopy is derived from the shear that foliage skin friction and form drag exert on wind:

$$ \frac{\partial {u}_{\mathrm{\ell}}}{\mathrm{\partial \ell }}=a\;{u}_{\mathrm{\ell}} $$

(A.24)

Here, ℓis the leaf area index (LAI) depth at which the wind has penetrated (at the top of the canopy ℓ = L = LAI; ℓ = 0 at the bottom of the canopy), u_ℓis the wind speed at ℓ, and a is the combined drag and friction coefficient set at 0.4. The mean $\overline{u_{\mathrm{\ell}}^{0.5}}$ adjusted to the wind profile in the canopy is:

$$ \overline{u_{\mathrm{\ell}}^{0.5}}=\frac{2{u}_Z^{0.5}}{a\;L}\left[1-\exp \left(-a\;L/2\right)\right] $$

(A.25)

Here, u_Z is the wind speed at canopy height Z:

$$ {u}_Z=u(z)\frac{\ln \left(\frac{Z-d}{z_0}\right)}{\ln \left(\frac{z-d}{z_0}\right)} $$

(A.26)

Additional definitions

The additional elements needed to complete the calculation of E_j are listed here.

The saturated water vapor pressure function of temperature T is (Tetens 1930):

$$ e(T)=0.6018\times \exp \left(\frac{17.27\times T}{T+237.3}\right);\kern1em \left(\mathrm{kPa}\right) $$

(A.27)

The slope of the molar fraction vapor saturation function of temperature is:

$$ \Delta =\frac{1}{P}\frac{\partial e\left({T}_a\right)}{\partial {T}_a}=\frac{e\left({T}_a\right)}{P}\times \frac{17.27\times 237.3}{{\left({T}_a+237.3\right)}^2}\approx \frac{e\left({T}_{F\;j}\right)-e\left({T}_a\right)}{P\left({T}_{F\;j}-{T}_a\right)};\kern1em \left({\mathrm{C}}^{-1}\right) $$

(A.28)

Finalizing

Combining Eqs (A.12), (A.13), (A.14), (A.15), and (A.28) leads to the P-M equation:

$$ {E}_j=\frac{\raisebox{1ex}{$\Delta\;{R}_{m\kern0.24em j}$}\!\left/ \!\raisebox{-1ex}{$\lambda\;{\Gamma}_j$}\right.+{g}_{V\;j}\raisebox{1ex}{$D$}\!\left/ \!\raisebox{-1ex}{$P$}\right.}{1+\raisebox{1ex}{$\Delta $}\!\left/ \!\raisebox{-1ex}{${\Gamma}_j$}\right.} $$

(A.29)

Here, Γj is the psychrometric constant (C⁻¹) adjusted to the conductance of heat g_{T j} and vapor g_Vj for the leaves in class j.

$$ {\Gamma}_j=\raisebox{1ex}{${c}_p\;{g}_{T\;j}$}\!\left/ \!\raisebox{-1ex}{$\lambda\;{g}_{V\;j}$}\right. $$

(A.30)

The resulting value of foliage transpiration is:

$$ E={E}_0+{E}_1 $$

(A.31)

The coupled model of transpiration is obtained when u(z) → ∞. This condition draws in Eqs. (A.20), (A.19), (A.18), (A.17), (A.16), r_Vb, r_a and r_Vc → 0, and g_Vj → g_Fj.

Coupling also implies that in Eq. (A.22), r_Hb → 0 and in Eq. (A.21) gHj → ∞. Consequently in Eq.(A.23), g_Tj → ∞ leading to Γ_j → ∞ in Eq.(A.30). The implications on convection deriving from coupling transforms Eq. (A.29) into:

$$ {E}_j\to {E}_{\infty j}={g}_{Fj}\frac{D}{P} $$

(A.32)

leading through Eqs. (A.17) and (A.31) to Eq. (6).