Canopy stomatal conductance is commonly estimated from
eddy covariance measurements of the latent heat flux (

Leaf stomata are a key coupling between the terrestrial carbon and water
cycles. They are a gateway for carbon dioxide and transpired water and often
limit both at the ecosystem scale (Jarvis and McNaughton, 1986). Although
the many stomata in a plant canopy experience a wide range of
micro-environmental conditions and therefore exhibit a wide range of
behaviors at any given moment in time, it has proven useful in many contexts
to approximate the canopy as a single “big leaf” with a single stoma
(Baldocchi et al., 1991; Wohlfahrt et al., 2009; Wehr et al., 2017). That
stoma is characterized by the canopy stomatal conductance to water vapor
(

When the aerodynamic conductance to water vapor outside the leaf
(

In general, neither

The iPM equation is further impacted by how the underestimation of

To deal with the energy budget closure problem, Wohlfahrt et al. (2009)
considered various schemes for correcting the fluxes in the iPM equation,
following earlier recommendations that EC fluxes be corrected to close the
energy budget in a more general context (Twine et al., 2000). All but one of
the schemes in Wohlfahrt et al. (2009) involve attributing the half-hourly
budget gap entirely to

Here we use data simulations to show that regardless of whether the energy
budget gap is due to

By definition, conductance is the proportionality coefficient between a flux
and its driving gradient. In the case of

Finally, the stomatal conductance to water vapor (mol m

The above FG theory is also the basis of the Penman–Monteith equation for a
leaf (Monteith, 1965) and its inverted form (Grace et al., 1995), which can
be expressed as

The inverted PM equation is usually expressed in a slightly simpler form by
neglecting the distinctions (a) between transpiration and evaporation and
(b) between the leaf boundary layer resistances to heat and water vapor. We
retain those distinctions here in order to highlight two important points.

Absent a means to accurately partition the measured eddy flux of water vapor into transpiration and non-stomatal evaporation (e.g., from soil or wet leaves), the FG and iPM equations are applicable only when evaporation is negligible, which is a difficult situation to verify but does occur at particular times in particular ecosystems (see, e.g., Wehr et al., 2017).

Setting

Note that the iPM equation can be derived from the FG equations by invoking
energy balance to replace

Our analysis consisted of two parts: simulations and real data analysis. The
simulations were designed to unambiguously demonstrate the impact of flux
measurement biases and the resultant energy budget gap on FG and iPM
calculations of

We assessed the proportional bias in

The simulations began by setting the “true” target values of all the
variables involved; in other words, their values without any simulated
measurement error. To keep these values realistic, we started with
approximate observed fluxes and conditions obtained from the papers cited
above or from our own work at the Harvard Forest (Table 1), with the precise
values of

Values of environmental and biological variables used in the error simulations (representing midday).

For all sites,

Next, we simulated a wide range of measurement bias scenarios, each with a
20 % gap in the energy budget (the FLUXNET average). The simulations were
explored along three main axes of variation.

For each measurement bias scenario, we used the FG and iPM formulations to
calculate

In order to show how the FG and iPM methods differ in a real forest over the
diurnal cycle, we calculated time series of

To minimize the influence of non-stomatal evaporation, we focused on 2
sunny midsummer days more than 24 h after the last rain (25–26 July
2014). Because our aim was to show the relative bias between the FG and iPM
methods rather than to obtain the most accurate possible estimate of

Our simulations indicate that the flux–gradient formulation is substantially more accurate than the inverted Penman–Monteith equation regardless of the cause and magnitude of the energy budget gap and regardless of the ecosystem type.

Figure 1 shows bias in

Proportional bias in canopy stomatal conductance obtained from the flux–gradient (FG, black) and inverted Penman–Monteith (iPM, red) formulations versus the fraction of the hourly energy budget gap caused by bias in the eddy fluxes rather than by bias in the available energy. Solid lines show results without eddy flux correction and dashed lines show results with perfectly corrected eddy fluxes. The average estimated contribution of eddy flux bias to the budget gap across FLUXNET is indicated by the grey vertical line (Leuning et al., 2012). Circles highlight where the various lines cross the FLUXNET average.

Comparison of Fig. 1a, b, and c reveals that the qualitative
relationships in Fig. 1 do not depend on the values of the environmental and
biological variables in Table 1, but the severity of the bias in

As noted in the introduction, pervasive eddy flux biases likely preserve the
true Bowen ratio in some but not all circumstances. Thus Fig. 3 shows bias
in

Inverted Penman–Monteith results from Fig. 1a, annotated to indicate the various sources of bias.

Proportional bias in canopy stomatal conductance obtained from the flux–gradient (FG, black) and inverted Penman–Monteith (iPM, red) formulations versus proportional bias in the measured Bowen ratio. Solid lines show results without eddy flux correction, and dotted lines show results with the eddy fluxes adjusted to close the long-term energy budget while preserving the (erroneously measured) Bowen ratio. The unshaded region denotes the plausible range of pervasive bias, which is bounded by the buoyancy-flux-based and Bowen-ratio-preserving limits (see text).

Aside from the energy budget gap, another potentially important source of
bias in the FG and iPM equations is the aerodynamic resistance (

Proportional bias in canopy stomatal conductance obtained from the flux–gradient (FG, black) and inverted Penman–Monteith (iPM, red) formulations versus proportional bias in the estimated boundary layer resistance. Solid lines show results without eddy flux correction, and dashed lines show results with perfectly corrected eddy fluxes.

If the aerodynamic resistance outweighs the stomatal resistance, then
transpiration is insensitive to the stomata and it is inadvisable to try to
retrieve

Same as Fig. 4a, but with true boundary layer resistance increased to make the aerodynamic and stomatal conductances to water vapor equal, simulating very calm atmospheric conditions and increasing the sensitivity of the FG and iPM equations to the value used for the boundary layer resistance.

Figure 6 compares the diurnal patterns of

We have shown that for the purpose of determining canopy stomatal
conductance at eddy covariance sites, the inverted Penman–Monteith equation
is an inaccurate and unnecessary approximation to the flux–gradient
equations for sensible heat and water vapor. Incomplete measurement of the
energy budget at EC sites causes substantial bias and misleading spatial and
temporal patterns in canopy stomatal conductance derived via the iPM
equation, even after attempted eddy flux corrections. The biases in iPM
stomatal conductance vary between 0 % and

In theory, the FG equations are mathematically equivalent to the iPM
equation aside from the relatively minor psychrometric approximation in the
latter. In practice, however, errors in

Unfortunately, there does not appear to be a universally appropriate method
for correcting the eddy fluxes at present. When the Bowen ratio is low or
moderate in tall vegetation like forests, the published evidence supports
increasing

Our results suggest that future studies should use the FG equations in place of the iPM equation and that published results based on the iPM equation may need to be revisited. It also motivates further work to determine a general and reliable framework for correcting the measured fluxes of sensible and latent heat at eddy covariance sites.

The canopy flux-weighted leaf boundary layer resistance to heat transfer
from all sides of a leaf or needle (s m

The R code used for the simulations and the Igor Pro code used for the
Howland Forest data analysis are freely available in the Dryad data archive
under the digital object identifier

RW conceived and designed the study, wrote the software code, performed the simulations, and prepared the manuscript with contributions from SRS.

The authors declare that they have no conflict of interest.

Funding for AmeriFlux data resources was provided by the U.S. Department of Energy's Office of Science. The Howland Forest data were produced under the supervision of David Hollinger.

This research has been supported by the National Science Foundation, Division of Environmental Biology (grant no. 1754803).

This paper was edited by Christopher Still and reviewed by Bharat Rastogi and one anonymous referee.