The Penman–Monteith (PM) equation is commonly considered the most advanced
physically based approach to computing transpiration rates from plants
considering stomatal conductance and atmospheric drivers. It has been widely
evaluated at the canopy scale, where aerodynamic and canopy resistance to
water vapour are difficult to estimate directly, leading to various empirical
corrections when scaling from leaf to canopy. Here, we evaluated the PM
equation directly at the leaf scale, using a detailed leaf energy balance
model and direct measurements in a controlled, insulated wind tunnel using
artificial leaves with fixed and predefined stomatal conductance.
Experimental results were consistent with a detailed leaf energy balance
model; however, the results revealed systematic deviations from PM-predicted
fluxes, which pointed to fundamental problems with the PM equation. Detailed
analysis of the derivation by

A vast number of current global land surface models, hydrological models and
inverse approaches to deduce evaporation from remote sensing data employ the
analytical solution for the latent heat flux from plant leaves derived by

A number of authors have focused on biases introduced by the simplifications
inherent in the PM equation, such as the linearisation of the saturation
vapour pressure curve and the neglect of dependency of net irradiance on
surface temperature, and proposed various approaches to reduce such biases

develop an experimental set-up allowing direct and independent measurement of all components of the energy balance of a single leaf and the relevant boundary conditions,

compare different analytical and numerical leaf energy balance and gas exchange models with experimental results and

derive an improved analytical representation of latent and sensible heat fluxes at the leaf scale.

The detailed derivations are described in the Appendix, while the
experimental methods are discussed in detail in a technical note submitted to
HESS

The detailed leaf energy balance model used here is based on derivations
published previously

The leaf energy balance is determined by the dominant energy fluxes between
the leaf and its surroundings, including radiative, sensible and latent
energy exchange (linked to mass exchange). These are illustrated in Fig.

Components of the leaf energy balance and their thermodynamic
drivers. Bent arrows indicate fluxes that are directly affected by wind
speed. Table at bottom illustrates the drivers for each flux (temperature
differences for sensible and radiative heat exchange, water vapour
concentration differences for mass exchange and hence latent heat flux).
Additional equations below the table illustrate that the driver for latent
heat flux is also related to temperature differences and that the transfer
coefficients for both latent and sensible heat flux depend on wind.

Latent heat flux (

Both the one-sided leaf convective heat transfer coefficient
(

Figure

Flow chart of computation procedure for different leaf energy balance components. Dashed pink boxes with rounded corners indicate external input, while solid blue rounded boxes indicate computed variables. Note the central role of leaf temperature, which needs to be computed by iteration against the leaf energy balance.

The PM equation derived by

Equations (

This gives four equations (Eqs.

In the original formulations by Penman and Monteith, the term

To solve Eqs. (

From the general form (Eqs.

Monteith pointed out that the ratio between the conductance to sensible heat
and the conductance to water vapour transfer, expressed in the psychrometric
constant (

To test whether Eq. (

However, this was done by specifying

Different representations of energy partitioning into sensible and
latent heat flux.

The above analytical solutions eliminated the non-linearity problem of the
saturation vapour pressure curve, but they do not consider the dependency of
the long-wave component of the leaf energy balance (

We can now use a similar procedure as in Sect.

Artificial leaf and wind tunnel. (a): cross section of artificial
leaf; (b): leaf image before full assembly; (c): topography of
laser-perforated foil with 60

Variations in leaf temperature and leaf energy balance components were
simulated using a detailed numerical model (Sect.

To separate the physical aspects of leaf energy and gas exchange from complex
biological control, we used artificial leaves with laser-perforated surfaces
representing fixed stomatal apertures and continuous water supply monitored
by micro-flow sensors (Fig.

The artificial leaves were constructed of a core made of porous filter paper
(Whatman no. 41), glued onto aluminium tape and connected to a water supply
by a thin tube, flattened at one end and tightly glued between the aluminium
foil and the filter paper, using Araldite epoxy resin
(Fig.

Different laser perforations were performed by Ralph Beglinger (Lasergraph
AG, Würenlingen, Switzerland), Robert Voss (ETH Zurich, Switzerland) and
Rolf Brönnimann (EMPA, Zurich, Switzerland), and the geometry of laser
perforations was measured using a confocal laser scanning microscope (CLSM
VK-X200, Keyence, Osaka, Japan). See Fig.

The stomatal conductance resulting from a particular perforation size and
density was computed following the derivations presented by

Leaf energy and gas exchange were measured in a thermally insulated wind
tunnel with full control over energy and mass exchange
(Fig.

The sensible heat flux (

The leaf wind tunnel was used to measure steady-state conditions under given
forcing (air temperature, humidity, wind speed and irradiance). Sensible heat
exchange between the leaf and the surrounding air was computed from total
chamber heat exchange, using monitored flow rate and temperature of incoming
and outgoing air (Fig.

Simplified energy balance of insulated wind tunnel. Latent heat flux
(

Experiments were performed for various artificial leaves with different
stomatal conductances under varying air humidity or varying wind speed, in
the absence of short-wave radiation. Stomatal conductance was deduced form
confocal laser scanning microscope (CLSM) images of the perforated foils, as
described above. The ranges of stomatal geometries and deduced conductances
for the two different leaves presented here are given in
Table

Perforation characteristics and resulting stomatal conductances,
computed using Eqs. (

The numerical model reproduced observed sensible and latent heat fluxes very
accurately (Fig.

Numerical simulations vs. observed fluxes of sensible, latent and radiative
heat in response to varying wind speed and vapour pressure. Numerical model results (lines) are based on observed boundary conditions representative of observations (dots).
The boundary conditions are summarised as follows:

The analytical models generally underestimated latent heat flux, but the
model based on linearised

Since we were not able to systematically assess the effects of irradiance and
air temperature in our lab experiments, we conducted a numerical experiment
where we compared simulations by the numerical model with simulations by the
best analytical model and the PM equation. The results shown in
Fig.

Analytical simulations vs. observed fluxes of sensible and latent
heat in response to varying wind and vapour pressure. Numerical model results
(lines) are based on observed boundary conditions representative of observations
(dots). Conditions are the same as in Fig.

Numerical vs. analytical simulations of sensible and latent heat in
response to varying irradiance and air temperature. Crosses represent
numerical solution of leaf energy balance model (S-mod.), solid lines our
new analytical solution based on linearised long-wave balance (Rlin;
Eq.

This age values usefulness more highly than correctness, and the making of
money more highly than both. In fact, there is definitely something suspect
about an examiner who would bother at all with whether an idea is correct or
not.

The widespread use of the PM equation is mainly due to its simplicity and usefulness, the latter of which is contingent on its ability to accurately represent the sensitivity of evapotranspiration to atmospheric variables and surface properties (boundary layer and bulk stomatal conductances).

In our rederivation and subsequent analyses, we have identified two errors
in the PM equation and in the “corrected” MU formulation by

Although the upscaling of a physically based leaf-scale model to a canopy or
land surface is fraught with various challenges, including characterisation
of the stomatal or canopy conductance, canopy-scale boundary layer
conductance, consideration of canopy storage and distinction between
radiative and aerodynamic surface temperatures

In the present study, we have developed an experimental set-up allowing to
control all relevant boundary conditions at the leaf scale, including
stomatal conductance, and measuring (to our knowledge, for the first time) all
components of the leaf energy balance. In contrast to previous tests of the
PM equation, which were conducted at the canopy scale, where boundary layer
and canopy conductances could not be measured directly, we have been able to
greatly constrain model parametrisation by independent measurements of
stomatal conductance. This has led to the discovery that the PM equation, in
its original formulation and common use, does not accurately represent
leaf-scale processes. Our newly derived analytical solutions (Eqs.

Given the widespread and successful use of the PM equation, the question
arises whether common practice, which relies on parameterisation by fitting
resistance terms that provide a match with observations, somehow compensates
for the errors we identified in the present study. The answer is “yes and
no”. As shown in Fig.

In this study, we revisit the governing equations for the exchange of water
vapour and energy between a planar leaf and a surrounding air stream under
forced convection. We derived general analytical solutions for steady-state
sensible and latent heat fluxes from a leaf and the corresponding leaf
temperature (Eqs.

All code and data used to generate the results presented in this paper are
available online at

Table of symbols and standard values used in this paper. All area-related variables are expressed per unit leaf area.

Continued.

Dependence of the leaf–air water vapour concentration difference
(

Example confocal laser scanning microscope (CLSM) images of
perforated foils summarised in Table

Net long-wave radiation away from leaf as a function of leaf
temperature. Solid line represents Eq. (

The total leaf conductance to water vapour is determined by the boundary
layer and stomatal conductances and equal to 1 over the sum of their
respective resistances (

The concentration difference in Eq. (

Note that the dependence of the leaf–air water concentration difference
(

Note that

Given climatic forcing as

The vapour concentration in the free air can be computed from vapour pressure
analogously to Eq. (

The heat transfer coefficient (

For sufficiently high wind speeds, inertial forces drive the convective heat
transport (forced convection) and the relevant dimensionless number is the
Reynolds number (

In the absence of wind, buoyancy forces, driven by the density gradient
between the air at the surface of the leaf and the free air dominate
convective heat exchange (free or natural convection). The relevant
dimensionless number here is the Grashof number (

For

The average Nusselt number under forced convection was calculated as a
function of the average Reynolds number and a critical Reynolds number
(

In order to simulate steady-state leaf temperatures and the leaf energy
balance terms using the above equations, it is necessary to calculate

In order to obtain analytical expressions for the different leaf energy
balance components, one would need to solve the leaf energy balance equation
for leaf temperature first. However, due to the non-linearities of the
blackbody radiation and the saturation vapour pressure equations, an
analytical solution has not been found yet.

In order to eliminate

Substitution of Eq. (

Equation (

To account for stomatal resistance to vapour diffusion,

In accordance with Eqs. (

Equation (

Comparison of Eq. (

To find a solution for

As opposed to the formulations in Sect.

Division of
Eq. (

Comparison of Eq. (

From the general form (Eq.

The Penman equation for a wet surface (Eq.

Similarly, the Penman–Monteith equation (Eq.

In the main text, Eq. (

At least three confocal laser scanning images of each perforated foil were
analysed and average pore area (

Stanislaus J. Schymanski performed the mathematical derivations, designed and carried out the experiments and wrote the paper. Dani Or was involved in the design of the experimental set-up, interpretation of the results and writing the paper.

The authors declare that they have no conflict of interest.

The authors are very grateful to Dani Breitenstein for his assistance in designing and constructing the wind tunnel, to Stefan Meier and Joni Dehaspe for assistance in constructing artificial leaves and to Hans Wunderli for assistance in the lab. We also wish to acknowledge technical advice by Roland Kuenzli (DMP AG, Fehraltorf, Switzerland), laser perforation services by Ralph Beglinger (Lasergraph AG, Würenlingen, Switzerland), Robert Voss (ETH Zurich, Switzerland) and Rolf Brönnimann (EMPA, Zurich, Switzerland), helpful feedback from Tim Reichenau and Claire Zimmermann (Univ. Köln, Germany) and constructive reviewer comments by Stefan Dekker and an anonymous reviewer. The project benefited from funding by the Swiss National Science Foundation (project 200021 135077). Edited by: P. Gentine Reviewed by: S. C. Dekker and one anonymous referee