Elevated atmospheric CO

Elevated atmospheric CO

There is empirical evidence for the compensatory effects of stomatal closure
and leaf area increase on canopy-level transpiration under elevated
CO

In line with these empirical results, a detailed process-based model
predicted that the direct effect of elevated atmospheric CO

Compared to complex process-based models, parsimonious analytical models can
provide more direct understanding and theoretical insight into this
question. Analytical models of plant gas exchange have been formulated based
on different assumptions, ranging from heuristic relationships to
eco-evolutionary theory. An example of the first type is the heuristic
Partitioning of Equilibrium Transpiration and Assimilation (PETA) model,
which describes how leaf area index (LAI), canopy and leaf transpiration,
and CO

Optimal stomatal conductance models are sensitive to changes in atmospheric
CO

In this contribution, the PETA model and the three optimization model
variants are compared, providing a set of predictions in the form of compact
analytical equations. These equations, in turn, quantify the sensitivity of
gas exchange rates (especially transpiration) to changing climatic
conditions and thus address the following questions:

How do physiological (stomatal conductance) and morphological (leaf area)
adjustments coordinate to determine leaf and canopy gas exchange rates under
atmospheric CO

How do these physiological and morphological adjustments vary under combined
changes in CO

Both the PETA and optimization models describe leaf and canopy exchanges of
water vapour and CO

Conceptual representation of the PETA and stomatal
optimization models used to assess gas exchange responses (transpiration

Definitions of symbols (including units) and subscripts and superscripts.

Leaf-level transpiration rate

Therefore, increasing atmospheric CO

Compared to the equations above, nonlinear models of assimilation accounting
for RuBisCO or RuBP regeneration limitation
(Farquhar
et al., 1980; Vico et al., 2013; Katul et al., 2010) would yield a more
complex relation between

Further assuming the big-leaf approximation and that the canopy is
well-coupled with the atmosphere, the canopy-level transpiration (

Hence, by promoting plant growth and larger LAI, elevated atmospheric
CO

Knowing transpiration and CO

The PETA model is formulated as a set of relations between the relative
changes of variables related to leaf gas exchange and the relative change in
atmospheric CO

In Eq. (5) and in the following, the symbol

When changes in

In an open canopy far from the maximum

In a closed canopy (i.e.

Equations (6) and (7) link
the changes in gas exchange rates to the changes in atmospheric CO

A simplified version of the PETA model is described in Appendix A and used to develop analytical arguments in the “Discussion” section.

The optimal stomatal conductance model is formulated as an optimal control
problem with the objective to maximize net CO

In versions OPT2 and OPT3, a model of soil moisture dynamics needs to be
added to the gas exchange equations. Neglecting evaporation from the soil or
canopy surface, the soil water balance during a dry-down with negligible
precipitation can be written (in units of metres per day) as

If stomatal conductance is allowed to vary through time but independently of
soil moisture, the Lagrange multiplier of the optimization is
time-invariant. Substituting Eqs. (1) and
(3) in Eq. (B2) in Appendix B1 and solving for

A more realistic approach that overcomes the limitation of a freely
adjustable

Equation (11) could be also found by simply imposing
that the stomatal conductance adjusts to use all the water in the allotted
time (details are shown in Sect. 3.1). Therefore,
assuming optimal stomatal control and a finite amount of plant-available
water results in a stomatal conductance equation that is independent of the
atmospheric CO

Temporal trajectories of

The equations of OPT2 can be used in two ways. Environmental conditions and
soil parameters can be set to the long-term mean values and

Different from OPT1 and OPT2, we now consider soil moisture limitations on
gas exchange (dashed lines in Fig. 2). Stomatal conductance is reduced as
soil moisture decreases during a dry period because of the combined effect
of lowered water pressures along the soil–plant system and reduced
conductance to water transport in the soil and the plant xylem
(Cruiziat
et al., 2002; Klein, 2014). As a result, transpiration rate proceeds at a
high and stable rate in well-watered conditions but decreases approximately
linearly as soil moisture declines due to stomatal closure and limited water
supply from the soil
(Sadras
and Milroy, 1996). Based on this assumption, stomatal conductance decreases
linearly with

In contrast, in well-watered conditions, stomatal conductance can be
optimized. The optimal stomatal conductance is calculated with Eq. (10) after finding the Lagrange multiplier specific
to model OPT3, which differs from that in OPT2 because the boundary
conditions of the optimization have changed. Therefore, when the soil is
relatively moist, optimal stomatal conductance is found with an equation
similar to OPT2 but modified to account for the fact that stomatal
conductance will become water-limited when

The specific value of

Predictions of the OPT3 model are functions of time and must be interpreted
as time series, different from the time-invariant gas exchange rates of the
other models (OPT1, OPT2, and PETA). Thus, to compare results of OPT3 to those from
the other models, the time-averaged gas exchange rates are calculated as

To compare the results of the optimization models with those of the PETA
model, the relative changes in leaf transpiration and assimilation rates are
calculated as

The relative changes for transpiration can be re-written in a compact form
at both the leaf and canopy levels for OPT2 and OPT3 (after some algebraic
manipulation of Eqs. (1, 4, and 11),

In particular, Eq. (16) shows that changes in
canopy transpiration are predicted to be independent of changes in LAI or
atmospheric CO

While in the PETA model the water use efficiency

Similarly, the variations in intrinsic water use efficiency are found using
the definition

In scenarios in which VPD does not change in the future (i.e.

Baseline parameter values (relative variations in

The models are parameterized for a generic vegetation type and a baseline
climate (Table 2), from which variations in gas exchanges for a wide range
of future climate conditions are evaluated. In both the PETA and
optimization models, LAI varies with atmospheric CO

Relative change in leaf area (

In the PETA model,

The

Effect of atmospheric CO

The VPD can be changed by letting relative humidity vary at constant
temperature or by letting temperature vary at constant relative humidity.
The first scenario allows isolation of the effect of VPD on stomatal
conductance and transpiration alone. In the second scenario, VPD affects
both water and CO

Relative changes in leaf-level

Dry-period lengths during the growing season have been shifting towards
either longer or shorter lengths depending on location, with historical
variations up to

We start by comparing the effects of atmospheric CO

Different variants of the optimization model predict contrasting responses
to atmospheric CO

The mean stomatal conductance (

Using the definition of temporal average, Eq. (19)
can be written as

Canopy-level net CO

Therefore, based on the results in Fig. 4, the inclusion of the dynamic
feedback (OPT2 and OPT3) in the stomatal optimization model produces
plausible responses to elevated

The relative variations in gas exchange rates and water use efficiency
predicted under elevated CO

The predicted sensitivity of the gas exchange responses varies between the PETA
and optimization models, depending on the canopy status (i.e.

Contour plots of relative changes in leaf-level

The gas exchange patterns driven by

While the responses of transpiration rates are the same regardless of how
the variation in VPD is produced, patterns in net CO

Contour plots of relative changes in leaf-
(

Changing the length of the mean dry period leads to contrasting responses of
the PETA and optimization models (Fig. 8), mostly because PETA does not
include any effect of soil moisture on the CO

Vegetation acclimates and adapts to increasing atmospheric CO

Contour plots of relative changes in leaf-level

Both the PETA and dynamic feedback optimization models predict that in fully
acclimated plants and for a given soil water availability and VPD,
increasing atmospheric CO

If indeed plants adjust leaf area and stomatal conductance to use the
available water, in semiarid or seasonally dry ecosystems, soil moisture
values should be stable in long-term CO

Both the PETA and optimization models predict increasing leaf- and canopy-level
net CO

The effect of elevated atmospheric CO

Reductions in

Increasing VPD (driven by either temperature or relative humidity) in
conjunction with

The dry-down duration affects the gas exchange response to elevated

While typical rain exclusion experiments alter rewetting intensities more
than dry-period durations, rainfall manipulations where the same amount of
water is concentrated into fewer, more intense events could provide a
suitable testing ground for these predictions. The advantage of these
experiments compared to observations along a natural climatic gradient is
that all conditions except rainfall event timing and amount are the same, as in
our numerical experiments, where we let one or two factors vary at a time.
Consistent with model results, both net CO

The choice of the specific limiting factor for photosynthesis leads to a
range of optimal stomatal conductance solutions as a function of the
Lagrange multiplier

As long as the Hamiltonian of the optimization problem is independent of
soil moisture (i.e.

Other models based on instantaneous maximization of C gains for given costs
offer alternative frameworks to predict responses to atmospheric CO

In more complex models, it was assumed that not only stomatal conductance,
but also LAI or rooting depth were optimized to reach a certain objective
(typically maximize long-term productivity)
(Schymanski et al., 2015). Here
instead, LAI was prescribed – not optimized – as a function of

Besides root allocation, we also neglected evaporation from the soil or canopy surface. Changes in LAI do not affect strongly the partitioning of evapotranspiration into transpiration and evaporation, thanks to two compensating mechanisms: with increasing LAI, interception and subsequent evaporation from leaf surfaces increase, while heating of the soil surface is reduced, thus also reducing evaporation (Fatichi and Pappas, 2017; Paschalis et al., 2018). Therefore, even without explicitly modelling evaporation from the soil, the relative changes in gas exchange (as presented here) should be correctly predicted.

For simplicity, we restricted our analysis to deterministic conditions – a
single “representative” dry-down with prescribed initial and final soil
moisture states and duration. All these features of dry periods
should be treated as stochastic because rainfall timing and amounts are
inherently stochastic (Rodriguez-Iturbe and Porporato, 2004).
Stomatal optimization can be studied also in a stochastic rainfall scenario
consisting of consecutive dry-downs of random initial states and durations,
where rainfall is characterized by a constant mean event frequency and daily
intensity. Under long-term steady-state conditions, the optimization of
CO

Despite increasing atmospheric CO

To support the arguments in Sect. 4.2, a
simplified version of the PETA model is derived here considering that, in
free-air CO

This simplified model can be used to separate the effects of diffusion
limitations to gas exchange from either diffusion and biochemical
limitations (using the full PETA model with

To set up the optimal stomatal conductance model, we start from the
assumption that plants regulate stomatal conductance (

Because soil moisture (

A more realistic approach that overcomes the limitation of a freely
adjustable

The linear scaling of

Using the optimal stomatal conductance in Eq. (11),
the soil water balance of Eq. (9) can be solved to obtain the time trajectory of
soil moisture during the dry-down (solid line in Fig. 2b),

The decrease in transpiration during drying is often included in
soil–plant–atmosphere models through a piecewise linear function,
representing water-stress-induced reductions in

Here, the subscript “

Since

Next, we can determine

The system of Eqs. (B9)–(B11)
can be solved to obtain the unknowns

To summarize the solution of the OPT3 model (dashed lines in Fig. 2),
optimal stomatal conductance is initially constant and equal to

In this Appendix, we explore the consequences of coordination between
rooting depth (

Aboveground biomass (including leaves) and

Allometric theory predicts that plant leaf area scales as plant height to
the third power and that root extent (lateral and vertical) scales linearly
with height (Kempes et al., 2011). It follows that

Equation (11) shows that the optimal stomatal
conductance scales as the ratio of

This equation indicates that optimal stomatal conductance is inversely
related to

Maximum rooting depth as a function of leaf area during
plant growth, as measured

Relative changes in leaf-level

Data shown in Fig. 3 are reported in the Supplement.

The supplement related to this article is available online at:

SM, GGK, and GV designed the study, with feedback from all co-authors. SM developed the model, produced the results, and drafted the manuscript. All co-authors commented on the draft and contributed to the manuscript.

The contact author has declared that none of the authors has any competing interests.

This article is part of the special issue “Global change effects on terrestrial biogeochemistry at the plant–soil interface”. It is not associated with a conference.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We thank Stanislaus J. Schymanski for his in-depth comments on an earlier version of the manuscript and Benjamin Stocker and an anonymous reviewer for insightful comments during the discussion phase. The “Discussion” section also benefitted from comments by Yair Mau and Yuval Bayer.

This research has been supported by the European Research Council, under the European Union’s Horizon 2020 research and innovation programme (grant no. 101001608 – SMILE to Stefano Manzoni); the Svenska Forskningsrådet Formas (grant nos. 2018-01820 and 2018-02787); the United States National Science Foundation (CAREER award DEB-2045610 to Xue Feng and AGS-2028633 to Gabriel Katul); and the United States Department of Energy (DE-SC0022072 to Gabriel Katul).The article processing charges for this open-access publication were covered by Stockholm University.

This paper was edited by Emily Solly and reviewed by Benjamin Stocker and one anonymous referee.