Okay, now that Hoppe is here, we can start.
Well, thank you all for coming.
Today we will continue with a sequence of lectures that are dedicated
to the inner workings of the plant.
And the goal of my lecture is to try to bridge some of the ideas that
have been presented by Wilfred Conrad and Nadine and as well as Kari
Jensen a little bit down the road.
So perhaps you could view this lecture as more of a stitching lecture
between the various phenomena that have been presented in the other
two lectures.
Okay, so to start with, I'd like to thank my collaborators from Duke
University, Jean-Christophe Domecq, Ram Oren and Sari Palmaroth, from
ETH Zurich, Gabriele Manoli, who was a previous postdoc at Duke, from
Nagoya University, Taro Nakai and Tomo Kumagai, although Tomo just
moved to Tokyo University, from University of Helsinki, Samori
Loinanen, who is now at Duke, from University of New Mexico, Shengwei
Huang, who just finished his PhD with me and has started his postdoc
there, from Padova, Valeria Volpe, who visited Duke University for
about a year in 2011, and Marco Marani, who is a colleague of mine who
has a joint appointment between the two institutes.
Tanos Pashalis, who was a former postdoc of mine and now he's at the
Imperial College in London, has also done some work that I'm
presenting.
Stefano Manzoni, who was a student postdoc at Duke University, now at
University of Stockholm.
Wilfred Conrad, Danielle Wei, who is now at University of Western
Ontario, and she used to be a postdoc at Duke University.
And Giulia Vico, who is now at the Swedish University of Agricultural
Sciences, and she used to be a postdoc with me and Amilcare Porporati.
Okay, so to start this discussion, it's hard to come up with a good
starting point for biosphere-atmosphere exchange.
The study of evaporation probably dates back thousands of years, and
it is almost impossible to do justice and cover all the literature
that is relevant to at least that topic on its own, just evaporation.
But perhaps a major turning point has occurred when the kinetic theory
of gases was proposed, and what we will do is try to start the
discussion here from this point onwards, realizing that by no means
this is doing justice to the whole literature that has been developed
up to the point of the kinetic theory.
The reason this point is chosen is that it's a point that is basically
associated with the phase transition, and it's the first time we
started viewing the phenomena of biosphere-atmospheric change perhaps
from a microscopic view.
So again, a brief summary of accomplishments that have occurred.
In 1738, Daniel Bernoulli published his book Hydrodynamica, and the
basis for the kinetic theory of gases was actually proposed in Chapter
7, and I should say that the current textbooks that we see today are
still teaching this topic exactly like Daniel Bernoulli did in 1738.
In 1801, Dalton, actually building on some of the ideas of Daniel
Bernoulli, proposed the law of partial pressures.
Also, the first evaporation equation was proposed by Dalton then.
In 1857, Clausius took the work of Daniel Bernoulli one step further.
He noted that particle motion actually has both translational,
rotational, and vibrational components.
So that allowed actually the kinetic theory of gases to be extended to
basically all matter, including liquids and solids.
And in this paper in 1857, Clausius actually was the first to
introduce the concept of mean free path.
Maxwell derived the statistical distribution of the kinetic energies
themselves, and it was Boltzmann who put all the pieces together where
the connection between entropy and probability was first stated.
And in it, we finally had a unified picture of the classical laws of
thermodynamics and statistical mechanics.
So when you discuss evaporation to colleagues in physics, they say,
what's new?
We almost know everything.
And at some point, if you have a closed system, we're subject to a
constant temperature, so this is an equilibrium system.
That has liquid and gas drawn over there.
Or I should use this one drawn here.
We know that the kinetic energy distribution of these molecules
follows the Boltzmann, Maxwell-Boltzmann distribution.
And so, you know, there is always a finite probability that one of
those molecules is going to jump out.
And if you increase the temperature, you basically increase the
kinetic energy of those molecules, so more of them can jump out.
And at some point, you're going to reach a phase where the number of
molecules that are jumping out is going to be balanced by the number
of molecules that are going to reach.
They're going to recolide back with the water surface.
And hence, you reach a phase where at a given temperature, the net
evaporation rate is zero.
And in fact, it is the saturation vapor pressure of these molecules
that follows the Clausius-Clapeyron equation.
That's why it's dependent on temperature.
Okay, so in a closed system, we seem to understand what's happening,
but that's not what occurs in nature.
Why is that?
Well, two things.
First, the system that we're dealing with is actually open to mass and
open to energy exchanges.
Okay, so let's start with the opening of this box to flow.
The idea then is that these molecules are no longer just jumping based
on mean free paths, but now there are coherent eddies that are moving
them.
This is by no means a new concept.
Boussinesq, in 1877, attempted to study the assemblage of swirling
eddies, as we call them, swirling an agitated liquid.
Now we call them eddies.
And it was a major hallmark in the study of turbulent flows, even
though the word turbulence was not yet invented.
It was not even invented at the time of Reynolds.
In fact, it was proposed after Reynolds experiments.
So we didn't have even a word for turbulence.
But in this paper of Boussinesq, he proposed the concept of eddy
diffusivity or eddy viscosity.
And that actually happened way before Reynolds famous experiments,
because in fact Boussinesq's paper proposed a local averaging concept.
And he also proposed the concept of mixing lengths to resemble
basically Clausius mean free path lengths.
And that idea actually inspired later von Kármán and Prandtl to
propose their log rule.
And in fact, if you look at Boussinesq's paper, what is interesting
about it is that he attempted to formulate it in tensor notation.
So in other words, it was the first attempt to actually look at these
equations using tensor point of view.
And he showed that the Reynolds stress, what we call now Reynolds
stress, and the mean strain rate tensors are actually aligned.
And that was the first result.
And in fact, if one reads Boussinesq's paper, there are plenty of
mistakes and left and right assumptions that are contradictory.
But these ideas have been certainly put forth in a major way.
And they were maybe, if you like, cleaned up a little bit by Oswald
Reynolds years later.
Okay, so now we have a basic idea that allows us to think of movement
in this box when it is opened up, rather than having a mean free path
that is based on molecular kinetics, they are now based on turbulent
transport processes.
Now the box has to be opened up also for energy exchanges.
And perhaps the work of Ira Bowen in 1926 is not the first, but
certainly the most influential.
The concept of what we call now Bowen ratio has been around for a long
time.
But Bowen was the one who pushed it on force with a study on leg
evaporation.
And in fact, just to bring up the issue in terms of graduate students
here, you see Ira Bowen was not even interested in this project.
He was told by his advisor, Robert Millikan at Caltech, that he needed
to help a student finish his PhD.
And so the student was working on evaporation from legs.
And so basically Ira Bowen had to drag the student and figure out how
to estimate evaporation from legs.
But Robert Millikan was quite astute about energy and ions and so
forth.
And so Bowen had that foresight and of course he immediately
approached the problem from the perspective of an energy balance.
And he basically wrote the net radiation minus the total heat flux, or
at the time water flux, the heat flux that enters the water surface,
must be balanced by the sensible heat flux and the latent heat flux.
And then he used ideas similar to Boussinesq where he introduced the
eddy diffusivity.
So he assumed that the sensible heat flux somehow scales with the
temperature gradient multiplied by an eddy diffusivity for heat.
Likewise for the latent heat flux, scales with the water vapor
concentration gradient multiplied by the eddy diffusivity for water
vapor.
And he immediately realized that if he divides the two, he gets the
Bowen ratio as a function of the change of water vapor concentration
with the change in temperature.
And he was able then to reason, incorrectly in fact, because he made
the argument that if the eddy diffusivity of heat and water vapor
resembles those of molecular diffusion.
So in fact he assumes that the ratios of eddy diffusivities of heat
and water vapor is the same as the molecular diffusion coefficients of
heat and water vapor.
And he was lucky because they are not that far off in this case, but
they could have been far off in other cases.
So he made this analogy, canceled the ratio of the eddy diffusivities
and ended up with what we now call the Bowen ratio.
And that allowed the estimation of the latent heat flux to be
dependent on the measured available energy and the Bowen ratio that
now can be inferred from atmospheric measurements.
Okay, so that allowed the box to be open for energy.
But then for operational purposes, in fact Penman took it one step
further.
He looked at Bowen's analysis, basically replaced anything that has
derivatives with differences, did a little bit of linearization and
came up with what we now call the combination equation.
And interestingly enough, a few years later, the literature in the
West translated the work of Monin and Aboukhov that was probably
written in the late 30s.
And in it, they actually proposed, as John Finnegan mentioned
yesterday, a revision to the eddy diffusivity to include now stability
correction functions.
So up to this point, we seem to have an analog to molecular motion,
except that now the eddies are much bigger and the effect of thermal
stratification could be taken into account.
But up to this point, there is no plants in the picture or the story
that we are weaving.
The plants actually were introduced by Monteith in 1965, where he in
fact revised Penman's equation to include the stomatal conductance on
top of the aerodynamic conductance.
But how do you go about specifying the stomatal conductance turned out
to be a major, major hassle.
Jarvis took this task upon himself to figure out how to do that.
And as is the case in fields that are in their embryonic form, to
borrow a word from Wilfrid, what you typically do is you have a
maximum rate and a bunch of reduction functions, and you basically try
to figure out what these reduction functions are from some maximum
rate that you think you could estimate.
And that turned out to be the case of stomatal conductance independent
on photosynthetically active radiation or light, vapour pressure
deficit, leaf pressure, atmospheric CO2, etc.
And if you track actually a lot of the history of science, a lot of
the assemblage of information typically is done like this.
This equation, if you ever studied erosion, resembles the universal
soil loss equation, if you ever heard of that.
And in fact the universal soil loss equation for erosion looks exactly
like this.
The erosion rate is normalized by some maximum theoretical value, and
there are a bunch of reduction functions.
That depends on the intensity of rainfall, how loose the sediments
are, etc.
So you could see that the approach is typically to collect a lot of
field data or a lot of data on this phenomena, and organize it in a
certain way, catalog it in a certain way.
This is a very useful approach, and in fact Jarvis did exactly that.
Now I should also mention the book by Wilfrid Brunser, Evaporation in
the Atmosphere, because this was one of the first books on this topic
that dealt with it exhaustively, from the days pre-Greek up to the
present, at the time it was 1982.
Okay, but then what happened is that by the early 60s and onwards,
numerical experiments were being conducted on the planet.
And in fact the realization from these climate models is that there is
quite a bit of sensitivity of what's going to happen to the climate in
terms of how surface evaporation and transpiration are specified.
No surprise, because if you're interested in precipitation, the
dynamics of the boundary layer are going to be important, and of
course those are going to be highly dependent on how you specify the
evaporation rate and the sensible heat flux that drives the growth of
the boundary layer.
And in fact this sensitivity to the land surface led to perhaps the
first large-scale field experiment in our field, which is called the
Hopex -Mobilier experiment in 1986, conveniently located in a nice
place in France, south of France.
But this was the experiment that was conducted on what was perceived
to be at the time a GCM grid cell, a 100 km by 100 km grid cell.
So they actually took 100 km by 100 km and instrumented the living
daylights out of it, including flux measuring stations, aircraft
sounding, remote sensing products that were pretty primitive at that
time.
But the era of large-scale field experiments started basically from
this experiment onwards.
One was in the U.S.
and then there was a whole bunch in many other places.
And in fact this push to gather information on such large scales
actually forced the remote sensing community to start developing
products that can be interfaced with those sorts of experiments, with
the plant that it will feed into climate models.
And in parallel for the water vapor, of course the Hopex-Mobilier had
no CO2, it was all transpiration and evaporation and so forth, because
they were interested in boundary layer dynamics.
Now for the CO2, there was a parallel story that was developing.
In fact, this parallel story perhaps might be traced back to Jean
-Baptiste Van Helmond, who coined the word gas in the 17th century and
noted that the gas silvestri, which is carbon dioxide, is given off by
burning charcoal.
He also in fact investigated water uptake by a willow tree, so in fact
he might have been thinking about this connection way back.
And some people credit him for pioneering some of the earliest
experiments on gas transfer.
Now you could give the Belgians a thumbs up, this was major.
But centuries later, in fact, Van Helmond's activities almost
converged into a modern day story.
Atmospheric CO2 is largely rising because of fossil fuel combustion,
and one of the leading mitigations that we have at our hand is the
ability of plants to uptake CO2.
So ironically, we are back to Van Helmond.
But the important point here is that the role of stomata became more
evident and its control on the climate system became even more
pressing as the carbon cycle is being coupled to the water cycle.
So this coupling between the two alerted basically the climate
community that carbon dioxide and the global carbon cycle and the
global water cycle are so tightly coupled because of what the plants
are doing.
And you see very high profile papers, this is just a summary of a few
of them, on the carbon cycle.
It was recognized that it reduced the model conductance because of
elevated atmospheric CO2 would lead to saturation of uptake by plants,
but of course you could still have, because temperature goes up,
respiration processes are going up, CO2 concentration goes up, so
warming will have a positive effect.
You increase temperature, you increase CO2 emission, temperature goes
up.
And that's because primarily the biosphere has saturated in its
ability to uptake carbon dioxide.
And that was kind of the idea behind a very influential paper by Peter
Cox from the Hadley Center in 2000.
The water cycle, there is a whole bunch of them as well, this is just
a few of them I picked here, that they deal with the continental scale
runoff.
And primarily the argument is that if plants open their stomata less,
precipitation statistics are probably not changing, transpiration will
go down, so continental scale runoff will go up.
And there are many other papers that have popped up since then.
I'm showing here a figure from a paper actually led by the Dutch
group, primarily by Susan Steele Dunn, and they actually attempted to
investigate the recycling ratio of water.
And what you see here is how much of the water that you see in
precipitation actually came locally from the land surface, and
primarily more than half of the water that we have, or actually we
use, is recycled water, except if you go, of course, to the edges,
where you have now water vapor coming from some oceans.
But the majority of the land surface actually sees precipitation as an
outcome of recycled evaporation.
Okay, so this is all fine and tandy, and we are beginning to
understand better the role of stomates.
But recently, in fact, we are now confronted with more pressing
problems on top of that, which is now the effect of climate on plants.
So we've been dealing primarily with the effect of plants on climate,
but there is another side to this coin, which is the effect of climate
on plants.
And that tended to be accentuated primarily in the news after a
sequence of droughts have hit many areas around the world, and the
question of forest mortality and, you know, food security and so forth
started popping up.
And this is just a picture taken from the Food and Agricultural
Organization review, where they were documenting the intensification
of droughts in many long-term field experiments.
And clearly, any attempt to study now the effect of climate on plants,
not the other way around, would require a much deeper probing into how
carbon and water moves into the plants.
And that's what we're going to do now as a complement or a supplement
to Wilfred's talk and Nadine's talk.
Okay, so this, we have seen a little bit this aspect of these
processes before.
So what's happening inside the plants?
Well, at the leaf scale, we know that when stomata opens to uptake CO2
and photosynthesize, water vapor is escaped.
Now, this water vapor is basically supplied from the soil through the
xylem all the way up to the leaf under tension, as we have seen
yesterday.
But ultimately, what happens is that the products of photosynthesis
are primarily carbohydrates and sugars, are then pushed back down to
where they are needed, basically to cover the sinks, to grow biomass,
etc.
And in each one of those compartments, the leaf, the xylem, and the
phloem, we have a set of governing equations that describe water
movement or sugar movement or the coupling between water vapor and
CO2.
And these ideas are by no means new.
So, for example, in the case of the leaf, it had been recognized by
Francis Darwin way back in 1898 that stomata actually controls
transpiration rates.
That was a major result, but it was known even before Francis Darwin
that this was the case.
And in fact, the amount of water vapor that can leave the stomate or
the amount of CO2 molecules that can enter the stomate are primarily
described by Adolph Hicks's approach.
And this was known since 1855.
In terms of water movement and the xylem, Wilfred mentioned yesterday
cohesion tension theory.
This was somewhat known here and there before, but it was formalized
by the work of Dixon and Jolly in 1894 to the horror, by the way, of
Francis Darwin.
He was entirely not convinced by this idea.
And for good reasons at his time, he actually called cohesion tension
theory as ropes of sand, basically, because the tensile strength of
water was not known at the time.
So the idea that water can withstand tension was completely off the
wall.
It was one of those miracles to him, you know.
How could that be?
So he was quite critical of the Irish duo, Dixon and Jolly, but they
persisted and continued on it.
Now, in the case of phloem, we have a different mechanism.
In fact, rather than water moving up the xylem under tension, in fact,
now water is moving down with the carbohydrates or the sugars in the
phloem under positive pressure.
And that phenomena was recognized by Munk in 1930, and now it is
called the Munk hypothesis, where basically you have osmoregulation.
You build up sugar, water is pulled in, it gets pressurized, and it
pushes basically water down into the phloem.
And so you have positive pressure down, negative pressure up, and the
exchange processes are happening at the least.
So clearly, if we want to now think of a general theory for water
movement in plants, we have to almost couple these processes.
And that is an important question.
Are they really coupled or are they really operating autonomously?
And that's an interesting problem on its own.
It's been recognized now in physics for some time that when you have
networks on networks, that goes for power grids, that goes for many
other types of networks, when you have networks on networks operating,
in fact, the resilience diminishes.
Because if something goes wrong with one network, all the other
connected networks might actually collapse.
However, it has been also recognized that natural networks that co
-evolve appear to actually be much more robust.
And Eugene Stanley, who did quite a bit of work in the past on
diffusion -limited aggregation, very famous in physics, is now almost
picking this topic of networks as a career and trying to open up this
new field in physics of how do you deal with coupled networks
together.
So there is much more to this topic than just water moving in plants.
Okay, so what we have now done is basically stated the basic
mechanisms that are responsible for moving water in the xylem, in the
phloem, and in the leaf.
And in each one of those compartments, there are, of course,
parameters that need to be inferred so that you could use these
concepts operationally.
And in the case of the leaf, you know, one of the concepts that was
put forth by Ian Cowan and Troughton in a major paper in 1971, that
stomata may have evolved so that it can maximize its carbon gain while
preventing excessive loss of transpiration, means that now the leaf is
somehow operating as an engineered system.
It knows when to open the stomates and when to close them.
And it wants to open the stomates in a way that maximizes the carbon
gain.
Now, of course, this is a macroscopic description.
It's an outcome of many, many microscopic processes.
But it looks like the plant has figured out a way to do that.
In the case of the xylem, well, Wilfred mentioned a number of
phenomena that are operating in terms of cavitation and safety, and
same thing with Nadine's talk.
And basically, the concept there is that plants have to deal again
with another optimization problem.
If they build very big vessels, they are more likely to cavitate.
If they build very small vessels, they cannot transport water very
effectively, but they are more robust to cavitation.
So there goes another optimization problem.
And in fact, in this case, the problem is framed more as a safety
efficiency tradeoff or runaway embolism versus hydraulics efficiency.
But it's another optimization problem, that plants have to engineer a
solution for it.
And likewise, in the case of the phloem, as Jensen will discuss when
he comes, but I will touch a little bit upon his topic, so we will
make some contact with the phloem too, that the phloem vasculature is
optimized for efficient sugar transport.
And there again, there is another optimization problem.
If you ram too much sugar, because of photosynthesis, the viscosity
actually will go up.
Maple syrup.
You now are dealing with maple syrup.
Very busy.
So it becomes much harder to push the water in the phloem.
In fact, that's how you extract maple syrup.
So that's the problem.
So if you photosynthesize a lot, you produce a lot of sugar, that
means that the viscosity goes up.
But if you want to generate a large concentration gradient to push the
water down in the phloem, you want that sugar concentration to be
high.
So again, you are faced with another optimization problem.
So in each one of those compartments, what we're going to do next is
discuss briefly these optimization problems, and we're going to end
this lecture with a stitching mechanism between all these
compartments.
Hoping here, you know, we're from the US, faith-based initiatives are
becoming the norm in science.
We're hoping that by stitching them together, we can add more
constraints on the system.
So in other words, if we treat each system in isolation, we have a
large number of parameters.
By actually coupling them all together, they will enforce certain
constraints on each other, and that might make actually the problem
more practical.
So are we... did I set the background okay?
Okay.
And Wilfrid would say, good!
Let's start with the lead.
So what's the problem there?
I mean, at the mathematical level, let's say, you know, at the
mathematical level, we do have a decent description of photosynthesis
as a function of the internal CO2 concentration in the lead.
And that equation is often written as a biochemical demand.
Photosynthesis is some saturation function of the internal CO2
concentration.
And that saturation function depends on two parameters, alpha 1 and
alpha 2, and those actually are described by how photosynthesis is
limited.
It could be limited by enzymatic activity, or it could be limited by
electron transport activity.
But mathematically, alpha 1 and alpha 2 will vary in a very generic
way with temperature and light, if you like, in more common terms.
Okay.
Sounds good.
That's helpful.
So this is the demand by the plant.
Where is the supply?
The supply comes from the atmosphere.
So you have CO2 molecules bombarding the lead surface.
Its stomates open.
Some of them are going to enter the stomata.
And how much will enter in terms of rate is going to be dictated by,
well, some information about the porosity of this medium, multiplied
by the difference in concentration between atmospheric CO2 and
internal CO2 concentration.
So even if we know the environment around the plant, say temperature
and light so that we can estimate alpha 1 and alpha 2, and even if we
know the atmospheric CO2 concentration around the plant so that we can
estimate atmospheric CO2, the problem is very much like what people in
turbulence call the closure problem.
We have two equations and three unknowns.
The unknowns are photosynthetic rate, internal CO2 concentration, and
unfortunately stomata contact.
So we need the closure model.
We need the closure model.
And there is one approach to do that.
You could say, hey, there is a wealth of empirical data that supports
the argument that perhaps conductance, the stomatal conductance, must
be proportional to photosynthesis.
After all, stomates are opening up to uptake CO2.
And in a very famous paper by Ball and Berry and later revised by
Leunig, it was suggested that stomatal conductance must be
proportional to atmospheric CO2, must be inversely proportional to the
photosynthetic rate, inversely proportional to the atmospheric CO2
concentration, and somehow a measure of aridity, as Nadine Rouh
mentioned yesterday, somehow a measure of aridity had to be taken into
account.
The original work of Ball and Berry, later popularized by Collatz,
suggested relative humidity.
Rayleigh-Leunig, realizing the weakness of the relative humidity as a
metric of atmospheric aridity, suggested vapor pressure deficit, and
his paper actually has a good following, so to speak.
And both formulations have received quite a bit of support in the
literature, especially the connection between stomatal conductance and
photosynthesis normalized by atmospheric CO2.
So this approximate linear relation between stomatal conductance,
photosynthesis, and atmospheric CO2 is actually supported by a large
wealth of lead-gas-exchange measurements.
And this version, in fact, of closure, if you like, when combined with
the Fickian mass transport and the Farquhar photosynthesis model was
the basis of the first greening up of general circulation models in
the mid-1990s.
And it was described at length by a paper by Pierre Sellers.
And, in fact, Takahashi did a nice survey where he took all the Earth
system models and the land surface models and tried to characterize
how they are actually describing their stomatal conductance.
And the ones that were, in fact, colored with purple are basically
based on relative humidity, whereas the ones that are in somewhat
green are based on the reuning approximation.
So you can see that the majority of these models are in use.
They are quite popular.
And so those closure assumptions are quite useful.
Okay, so what's the problem?
The problem is can we do better?
And part of the reason we would like to do better is that whether you
describe this aridity index of the atmosphere based on vapor pressure
deficit or relative humidity is a major factor.
Why is that?
Because we know that climate models predict roughly a constant
relative humidity, but they predict warming patterns.
So if you are forcing the atmospheric aridity to be described by
relative humidity, it means that the term in this equation is not
changing much with climatic conditions.
However, if relative humidity is constant and temperature goes up,
vapor pressure deficit is going up exponentially.
And so in another case, it would suggest that the models for vapor
pressure deficit are in fact shutting down the stomata too much maybe.
So almost these two approaches might bind the end members of whether
aridity of the atmosphere has a major impact on stomatal behavior or
not.
And that point was recognized way back by a number of collaborators of
mine, including Tom Okumagai from Nagoya University.
Okay, now a different line of attack was taken on this problem
initiated by Ian Cowan, who John Finnegan just told me passed away
recently, unfortunately.
Ian actually came up with a very nice idea of formulating this
problem, this disclosure idea, in a way that seems to be less ad hoc
and perhaps is rooted maybe in a more general ecological hypothesis.
He actually argued that plants must be somehow operating in a way to
maximize their carbon gain while trying to minimize their water loss.
And so there are many ways you could write this sort of optimization
problem.
But the idea is that as CO2 enters the plant, water vapor escapes
through the same route.
And if this is the case, if the stomata are somehow smart enough to
maximize the carbon gain over a certain time period T, then there is
an objective function for the opening and closure of the stomata.
The objective function is to maximize the carbon gain, but of course
there are some limitations on this objective function.
And those come in the form of availability of water.
Now at that time, the availability of water was surrogated basically
to transpiration rates because there was no clear way of describing
water availability in the soil in a clean way that then interfaces
with a single equation for describing the fluxes of water in and out
of the leaves, basically out of the leaves.
So rather than force the objective function to be subjected to a soil
moisture availability, what Ian Cowan did cleverly is he bypassed this
problem and said hey, what I really need to make sure is that the
transpiration rate becomes finite.
So that is basically suggesting that there must be enough water so
that the transpiration rate is maintained finite.
So that was basically the condition that he put.
And you can always convert a constrained optimization problem into an
unconstrained optimization problem using basically the approach of
Hamilton and Lagrange.
So that's a well-established trick.
Constrained optimizations are no different than unconstrained
optimizations if you introduce the Lagrange multiplier.
And so basically one can write the Hamiltonian as photosynthesis that
varies with the control parameter, which is stomatal conductance.
Water vapor losses now can also be expressed as a function of this
control parameter.
And now we have basically an objective function that is strictly
formulated with respect to the control parameter, in this case
stomatal conductance.
So if we can maximize this Hamiltonian with respect to this control
variable GF, then we are basically in business.
And that might give us, in fact, a formulation for stomatal
conductance with this goal in mind, okay, maximization of this
Hamiltonian, which is carbon gain subject to finite water losses.
Clear?
Okay.
So let's recap.
We have one equation for the biochemical demand.
We have another equation for the atmospheric supply.
When we combine, when we balance biochemical demand with atmospheric
supply, because that's already a major assumption, meaning that every
single CO2 molecule that goes into its fate is basically to be
consumed by photosynthesis.
We have two equations and three unknowns.
If we add this optimality hypothesis of maximizing the Hamiltonian
with respect to the control variable, then the problem can be
mathematically closed.
So that's an interesting result.
But that makes the assumption that the Lagrange multiplier, which is
essentially needed for this optimization problem, needs to be at least
constant if it is going to vary.
If it is going to vary, it has to vary on timescales much slower than
the timescales at which Somata conductance changes.
So if Somata conductance, if Somata opens and closes, as Nadine has
shown, you know, in timescales of minutes, then whatever boundary
conditions that are going to influence this Lagrange multiplier, they
have to change on timescales way longer, way longer than minutes.
Okay?
So this problem is mathematically closed.
There are analytical solutions for it, and I'm not going to deal with
it too much.
But what I would like to do is give you a flavor of what these
analytical solutions do.
So to do that, we will look at a linearized biochemical demand
function, just because the equations become cleaner and everything
becomes evident.
And, in fact, if you like, the linearity of the biochemical demand
function would be actually way appropriate, way appropriate, if
Rubiscu limits photosynthesis.
So now if photosynthesis is linked to internal T2 concentration and
the linearization happens here, basically, rather than have the
denominator of this equation dependent on Ci, we are replacing it by
some sort of a long-term Ci over Ca quantity multiplied by Ca.
So that's how the linearization is done.
Now, with this estimate of Fc versus Ci, in fact, of Ci, and combine
the outcome with the Fickian diffusion model to write photosynthesis
as a function of the control variable that we see, which is the
stomatal conductance.
Okay, so now we have an expression for photosynthesis as a function of
stomatal conductance.
And there are a couple of features that we might point out.
Despite this horrible assumption that we made about the linearization,
there are a couple of features that should be pointed out.
The first one is that beyond Fe being zero, which is going to be
precluded by our condition that transpiration has to be finite, if you
have a concave function like that, then you are going to intersect the
transpiration rate.
So your supply and demand will have an intersection point, which would
be the optimal intersection point.
So that is guaranteed, basically, by the curvature of this function.
This is what some politicians in the United States would argue, that
if you keep increasing atmospheric CO2, the plants will be happy.
No, there is apparently a finite limit that you could set on the
biosphere, and that finite limit comes about also from this
calculation that if CA goes to infinity, Fc of GS does not
monotonically increase with atmospheric CO2.
It actually levels off at alpha 1 over S,
which is the maximum capacity, and S is the long-term CI over CA
concentration.
So you could set a finite limit, in fact, on the ability of the
biosphere to uptake CO2, even in the limit of infinite CO2
concentration.
Okay, so these are just two points to point out.
But now that we have the supply function dependent on the control
variable, stomatal conductance, based on spatial deficit, now we have
the Hamiltonian completely described by stomatal conductance.
Now we can do some math on it, basically.
And the math is pretty straightforward.
All we need to do is just differentiate it with respect to stomatal
conductance, set the outcome to zero, and solve for the stomatal
conductance that makes the Hamiltonian maximum.
And if you do that, you find an explicit expression for the stomatal
conductance.
And this is clearly now an outcome, an outcome of the optimization
theory.
There is a minus one plus this quantity CA over A lambda D.
CA is the atmospheric CO2 concentration, D is the vapor pressure
deficit, and lambda is the Lagrange multiplier.
I will show later that this minus one that arises from the optimality
solution is entirely connected to the apparent feedforward mechanism
for the stomatal conductance.
We can predict what the long-term CI over TA is, and that itself
varies with the vapor pressure deficit to the one-half, and we will
explore a lot, actually, what that means.
And also we finally have an expression for photosynthesis as a
function of CA, the marginal water use efficiency or the Lagrange
multiplied lambda, the vapor pressure deficit, and atmospheric CO2.
It is clear to us that there is a clean relation also between
photosynthesis and stomatal conductance.
Notice the slope here and the slope here are the same.
So if we take the slope here and set it equal to FC divided by this
quantity here, take this out and plug it back and do a little bit of
algebra, lo and behold, you find that photosynthesis divided by CA
emerges on the right-hand side, stomatal conductance is
here, that is now multiplying FC over CA, and there is an aridity
index, d to the minus one-half that falls out.
This looks exactly like the equations of Bol and Berry and Leunig,
except that now the linearity of FC over CA emerges or comes out of
the optimization.
It's not a priori assumed.
The aridity index that we have been looking at, relative humidity or
vapor pressure deficit that was set in an ad hoc way by
one -half from stomatal optimization theory.
So these are all predictions that were made and can be compared, in
fact, to these equations.
Now Belinda Medlin in 2011 had a variant of this derivation, slightly
different result here, but by and large the result is very similar,
and she actually came up with this result assuming RUBT regeneration
rather than a linear ACI curve.
So her result is a little bit more general than this one, but
the tendency of stomatal conductors to scale linearly with FC over CA
is, as we say in mathematics, a strong result.
Okay.
Now, is there support for this?
Sari Palbrot did a very nice experiment on Scott Spine.
She actually did lead gas exchange measurements across a climatic
gradient going from Finland all the way down to southern Europe, and
she assembled a wealth of datasets on transpiration rate, vapor
pressure deficit, and photosynthesis.
And when she plotted transpiration rate on the y-axis against square
root of vapor pressure deficit multiplied by 1.6 multiplied by this
constant, lambda, and against photosynthesis, the datasets seem to
have collapsed all on the same line despite the fact that they have
varied quite a bit in terms of climatic gradient, and that was
somewhat an effective support for the fact that if stomatal
conductance is multiplied by d, stomatal conductance varies as d to
the minus one-half multiplied by d.
So, in fact, based on these optimality results, the transpiration rate
should vary with the square root of vapor pressure deficit.
Okay.
So, with this, we might ask, is this consistent with other results?
Now, I will finish this and then we take a break.
My colleague at Duke University, his office is next door to mine,
thought that this problem was important, and so he, in 1999, had a
paper that came up with an interesting result.
So, what Ram Orel noticed is that if he plotted stomatal conductance
against some reference conductance, and he called this reference
conductance as conductance derived at one kilopascal vapor pressure
deficit, but with well-watered conditions, high light, he found that
in fact the vapor pressure deficit response is almost general for all
species.
He analyzed 30 or 40 of them for all species.
It seems that GS over this reference conductance seemed to follow an
expression of the form 1 minus m log d.
And so he inferred m from 30 or 40 species, well, maybe 31 here, and
found that m is between 0.5 and 0.6, highly constant.
But these were all well-watered conditions.
Nadine actually showed the graph similar to this one in her talk
yesterday.
And so it suggests that irrespective of what the species is, it seems
that for well-watered conditions there may be at least one of the
Jarvis functions may actually be more universal, may actually be more
universal than previously thought.
Okay, so that's an important result
because if a generalization cannot predict this summary of many, many
species lumped together, then it's worthless.
Can we do that?
So this is basically Ram's argument is that m is between 0.5 and 0.6,
and when you plot stomatal conductance divided by this reference, this
slope is very stable.
So, okay, this is the solution that we have.
So what's the connection between this and this?
We
know that this is still the same.
So if you do that, you automatically notice that any Taylor series
expansion of d to the minus one-half will have its first term to be
one minus one-half log d plus a bunch of higher-order corrections.
And actually this series converges quite rapidly in three terms.
You are almost there.
So if you take this linearization of d to the minus one-half and you
do a little bit of algebra, you could show quickly that the d to the
minus one-half results from stomatal optimization theory is exactly
predicting a slope of one-half multiplied by this quantity.
And this quantity is going to be slightly bigger than one because the
numerator and the denominator are the same except that the denominator
has a minus one in it.
And what this quantity is, it turns out to be atmospheric CO2 divided
by the marginal
denominator, which is going to be slightly bigger than one, and that
might explain why the result from Oren's work is between 0.5 and 0.3.
So this is great.
That doesn't prove that the stomatal optimization theory is correct,
but at least it is getting the right vapor pressure deficit response.
Now, typically if you try to tease apart from data sets what happens,
is there any check whether a log d or from a fitting perspective, the
data is not going to tell you which is better, but the two are very
close.
For the range of vapor pressure deficits that we deal with, the two
are pretty close.
Okay, so I will take a break here and give you a few minutes to recoup
and then we continue on the apparent feedforward mechanism and then we
finish the story on the leads and then we can continue with the
discussion about what to do for the panel meetings.
Yes.
That's the next part, Harvey.
That is the question that we're going to try to offer some ideas, not
answers, just ideas.
Yes, Nadim?
Yes.
No, this was all assuming stomata.
You could revise this, but then you would need to specify it.
Of course, if you want to do that, of course.
I totally agree.
It's just that if it's not controlled by stomates, that would be the
issue.
Of course.
Sure.
No, no, this is, I mean, also the nighttime transpiration is not
considered.
All of that.
But what... it's been done.
Actually, it's been done.
So you could add particular conductors and you could even add
mesophilic conductors, but from a stomatal optimization perspective,
you could ask what happens if there is an extra water loss that is not
controlled by the stomates.
That can be done.
But theoretically, you could re-derive the same result by breaking
actually the water flux into two compartments, stomatically controlled
and non-stomatically controlled.
As long as you can make
that is also an assumption that may be wrong, as you have shown
yourself, because you have shown that there is excessive drying.
In fact, even the curticular conductors don't.
So there may be some artificial correlation between stomatal
conductors and curticular conductors.
But if you are able to separate them and say that the rate of change
of stomatal conductors to curticular conductors is zero, then there
are, you could re-derive exactly the same argument with the curticular
conductors embedded in it, but it has to be specified.
Right.
So the separation is... but in the end it becomes almost uncontrolled
by the plants, right?
I mean, the stomatal conductor almost becomes its own beast and the
plant does not give it orders anymore.
It's leaking.
Yeah, yeah, yeah.
Oh, absolutely.
But in fact there is another way to do this, Nadim, just not...
Yeah, yeah.
In fact, under those conditions it will be interesting to see what
happens to the Hamiltonian.
The assumption here is that the Hamiltonian is positive.
If the Hamiltonian drops to zero, you are dead.
Okay, should we resettle again and continue?
So we left this discussion about the vapor pressure deficit response
to...
of plants, and this by no means is a new topic.
It's been almost on vogue every 20 years or so.
Perhaps the idea of somehow plants anticipating atmospheric aridity
has been around for a long time.
It was perhaps articulated in a number of papers, including one by
Dedlef Schulze, where in 1972 he stated that stomata responses to
changes in humidity in plants growing in the desert somehow provide
some sort of predictions.
The plants, almost stomata, can predict that aridity in the atmosphere
is going to continue and they are going to shut down before they
desiccate.
And the idea of the apparent feed forward mechanism is pretty
straightforward.
If you plot transpiration on the y-axis against vapor pressure deficit
on the x-axis, and what you find is that up to a certain point, the
transpiration against vapor pressure deficit seems to be linear.
As vapor pressure deficit increases and increases, it becomes more and
more curvilinear.
And so at some point, transpiration starts dropping with increasing
vapor pressure deficit, despite the fact that the driving force for
transpiration, which is vapor pressure deficit, continues to go down.
So that would suggest that the stomatal conductance, as a function of
vapor pressure deficit, is dropping faster than the driving force.
And that generates this sort of curvature.
And this sort of pattern has been observed before in many data sets.
Again, these are some papers that are around that have reported those
sorts of graphs, so you could see the exact source.
And the question is, can stomatal optimization theories predict this
sort of pattern?
Or at least, what do they give for this sort of pattern?
Now, we have just established that if leaf conductance is proportional
to some function of vapor pressure deficit, in this particular case it
is minus one half, the driving force for leaf transpiration is
proportional to vapor pressure deficit.
These are all simplifications.
Then the leaf transpiration, of course, has to be proportional to
their product.
So the apparent feedforward mechanism has been viewed as a way that
stomata is shutting down faster to prevent leaf dehydration.
Now, some people have claimed that, hey, if this is truly an apparent
feedforward mechanism, it should also be reversible.
In other words, if you reduce vapor pressure deficit, it should follow
the same trajectory as increasing it.
OK, so this topic, as mentioned, started in the 70s.
Then in the mid-90s, so that's, you know, 20 years, this popped up.
And then it popped up again.
Monteith came about to reinterpret the stomata responses to humidity.
And then a whole sequence of papers followed later by Meinzer and many
others as to whether this apparent feedforward mechanism exists and is
real and can be quantified, etc.
OK.
So, from the stomatal optimization perspective, again, let's, for
mathematical brevity and compactness, let's just look at the result
for the linear ACI curve, where Rubiscu limits for the synthesis.
We have shown that the transpiration rate actually is written like
this.
This is basically taking the stomatal conductance result, multiplying
it by vapor pressure deficit.
And I've mentioned this minus one before, that this is primarily
connected to the apparent feedforward mechanism.
Now I'm going to show you how.
If you multiply the conductance by the vapor pressure deficit, you get
exactly the apparent feedforward mechanism.
You have a minus D here, you have D to the one half here, and this
quantity here.
So if you differentiate the transpiration rate with respect to vapor
pressure deficit, set it to zero, you could basically solve where this
maximum is, what would be the vapor pressure deficit where this
maximum is.
And it turns out that this maximum turns out to be at one-fourth the
atmospheric CO2 normalized by the marginal water use efficiency.
And so if the vapor pressure deficit exceeds this critical vapor
pressure deficit, then we will see some apparent feedforward
mechanism.
If the vapor pressure deficit is less than this D critical, your data
set is not going to resolve it.
And so the interesting thing is that whether you see apparent
feedforward mechanism or not has to be a reference to what you predict
a priori the apparent feedforward mechanism critical vapor pressure
deficit is.
Some data sets push the vapor pressure deficit quite a bit and they
don't see it but they have never checked whether this condition is
actually satisfied.
And that's one case that popped up.
For example, James Buns in 1997 made the claim that the apparent
feedforward mechanism is all a load of crap, basically.
But he had some good points.
If you look at the data sets, yes, the conductance does drop with
vapor pressure deficit.
However, if you look at the transpiration rate as a function of vapor
pressure deficit, you don't see much apparent feedforward mechanism.
Now, if you use this data set to predict where this critical vapor
pressure deficit is, it's hardly satisfied.
So, while he may be correct, it's difficult to judge from the data set
alone whether the vapor pressure deficit was strong enough to see it
or not.
So, this is just a cautionary point that sometimes you have to almost
use models to plan experiments.
If you are going to test a hypothesis, you better be able to make sure
your experiment goes to the correct limits to be able to test it.
Now, the more interesting one actually comes about from whether the
apparent feedforward mechanism is somehow connected to temperature
responses.
This is basically work that Julia did in 2013 primarily to address
whether the apparent feedforward mechanism is consistent with the full
optimality solution.
And what she did is she plotted from the optimality solution with no
assumptions the transpiration rate as a function of vapor pressure
deficit for Rubiscu limitations where she saw actually a little bit of
an apparent feedforward mechanism and for RUBP regeneration which she
could show theoretically that the apparent feedforward mechanism
cannot exist.
Remember that the linear ACI curve is associated with Rubiscu
limitations not light limitations because in fact for light
limitations A becomes almost independent of CI.
And in fact she did something also quite clever she changed the vapor
pressure deficit in two ways.
This is all modern calculations.
In one case she actually held relative humidity constant and increased
temperature so she allowed all the photosynthetic parameters to adjust
for these temperature changes.
In another case she actually held temperature constant and varied
relative humidity.
Now, if this result is general and only dependent on vapor pressure
deficit you should have seen the same apparent feedforward mechanism
curve.
She does not.
So that proves that a lot of the accentuation in the data sets of the
apparent feedforward mechanism may be temperature driven, and that's
why I'm saying that maybe James Bunce is correct, but not because of
the experiment that he showed.
However, even if you... and so this is actually the result here.
So you can see that in the case of
Rubisco limitations will still exhibit apparent feedforward mechanism,
but you could prove mathematically that the light limitations or the
RUBP mechanism will prevent it from happening.
So this is an interesting result.
Strictly based on how you change temperature or relative humidity, you
could get an apparent feedforward mechanism in your data or not.
And that would suggest that if you take into account the temperature
response of your photosynthetic parameters, you will blur basically
the apparent feedforward mechanism.
Another thing that we have followed up on the stomatal optimization
work was done by Valeria Volpe.
She was visiting us at the time and the Italians loved their olive oil
and spinach and so there was some concern about what salinity might do
to stomatal conductances.
By no means this is a new topic, but attacking the effect of salinity
using stomatal optimization is an interesting one.
And salinity is quite different from droughts in the sense that
salinity in fact changes so much the osmotic pressure that the
mesophyll system here where photosynthesis happens can contract or
expand depending on the salt concentrations and that is a structural
change.
So in other words you could material wise you could create an
irreversible change in the mesophyll structure.
So basically expansion and contraction become irreversible.
But the idea is as follows.
Stomatal conductance of course is primarily driven by what is
happening in the sub- stomatal cavity and above it.
However photosynthesis does not happen in the sub- stomatal cavity.
It happens in the mesophyll.
And so you typically can add another conductance that goes from the
sub - stomatal cavity to the mesophyll using the mesophyll
conductance.
Now the problem is that mathematically you are adding another
resistance but you are not adding new equations to close it.
So somehow to close the problem mathematically you need another
expression for the mesophyll conductance.
But that doesn't mean that you cannot say what the mesophyll
conductance might be doing indirectly from stomatal optimization
theory if you could predict a correlation between the mesophyll
conductance and the other parameters.
And that's exactly what Valeria started.
She actually wrote the stomatal optimization theory as a combination
of mesophyll and stomatal and made the assumption that the variations
in stomatal conductance are not connected to the mesophyll
conductance.
This is a very hard assumption.
And I could easily punch holes in it by simply saying that if the leaf
pressure is somehow connecting the mesophyll and the stomatal, you are
not going to satisfy this assumption.
But that doesn't mean that we should not explore the consequence of
this assumption on the relation between mesophyll conductance and
stomatal conductance, even though we know that this may be approximate
and not entirely correct.
So if you assume that the stomatal conductance does not vary with the
mesophyll conductance, they may be somewhat connected, but apparently
Lian Wangu agreed that this may be a reasonable assumption.
Then you could solve for the stomatal conductance now as a function of
the mesophyll conductance.
OK, so you could find an expression between the two.
And based on some measurements of cell CO2 concentrations that have
their own problems, but still allow you to independently estimate the
mesophyll conductance, you could plot the mesophyll conductance on one
axis and the stomatal conductance on another axis and actually see for
the two species that Valeria has studied, olives and spinach, and
those were selected for good reasons.
One is salt tolerant, the spinach, and the olives are not very salt
tolerant.
And so what you find is that in the case of olives being irrigated
with fresh water, of course stomatal conductance increases, and as
mesophyll conductance increases, so there is a little bit of
connection between the two.
These are all predictions.
But for salt conditions, you could...
the mesophyll conductance can increase a lot, but primarily you are
limited by stomatal conductance.
In the case of...
this is for olives, in the case of spinach, you find that almost the
relation...
the effect of salt is not as big as in the case of olives, but still
the same patterns are correct, and the solid lines here, or the dashed
lines, are model calculations.
So, you don't predict the mesophyll conductance, but you could still
predict the relation between stomatal conductance and mesophyll
conductance that seem to be reasonably followed by the data.
And she also did quite a bit of calculation of how salt might impact
the marginal water use efficiency, the Lagrange multiplier, and again,
what she found is that for olives, there is a big separation, for
spinach, there is not.
And this is basically the expression that she had.
What she also found rather interesting is that if you plot now
stomatal conductance somewhat affecting the outcome of the flux.
So the interesting thing is that stomatal conductance seems to be
robust despite the fact that mesophyll conductance still enters into
the equations.
Now, the problem with all of these optimality theories is that how do
you go about specifying the marginal water use efficiency or the
Lagrange multiplier?
Lambda.
That's a big issue.
And what we are going to do now is suggest few lines of attack on
this.
And there are many others including throwing away the maximization of
carbon and simply assuming that plants operate to maximize
transpiration or plants maximize sugar transport rather than all this
operation in the leaf.
And of course this is important.
If you look at time series of volumetric soil moisture content, this
is a graph taken from the Duke forest where soil moisture was measured
on half hourly time scale.
You
find that the resource availability that we have bypassed by
specifying the whole optimization of transpiration fluctuates a lot on
many time scales.
So the question is, are these fluctuations so important that they
would completely nullify the whole approach of Cowen and later
revisions to it.
And the answer is a defense.
If you look at the spectra of soil moisture which describe how much
variability you have at different time scales, you find that the
spectra of decay pretty rapidly at sub daily time scales.
And that would suggest that perhaps the variance there is not so
crucial at very small time scales, but of course the variance is quite
high at larger and larger time scales.
And that has been tested for many, many forests.
And with Taro and others, there are models that can predict the
spectrum of precipitation and some information about water uptake by
plants and drainage.
And the excess that you see beyond minus 2 tends to come from
precipitation spectra.
And that tends to actually steepen the spectrum of soil moisture.
And that will be a
time scale separation that would allow us to assume that perhaps
lambda is constant on sub daily time scale but would vary on longer
time scales.
So with this, Stefano Manzoni decided to revisit the whole problem but
now looking at it from the available water itself rather than
transpiration.
So he wrote a hydrologic balance that said the rate of change of
stored water in the rooting zone is rainfall minus transpiration minus
other losses.
And these other losses could be anything.
could be drainage, it could be competition with another rooting
system, whatever you like.
The objective function does not change, you are still interested in
maximization of photosynthesis over a certain time period t, but now
we have to specify a boundary condition on the objective function.
So what do you do at the end of this time interval t?
And that is often referred to as the terminal gain.
And in fact for our purposes this terminal gain is basically telling
us what is the value of this water that you will leave in the rooting
zone in carbon units, if you want an ecological interpretation.
So if a plant is aggressively using water, it doesn't care about
leaving any water in the rooting zone, but if a plant is adopting a
very conservative strategy, because of uncertainty in rainfall, you
probably want to leave some water in the rooting zone.
So in fact, this terminal gain allows us to at least pit different
strategies of plant water use in this optimality theory.
The
Hamiltonian still follows the same thing, but now the Lagrange
multiplier has to vary on timescales, but again, these timescales are
going to vary a little by differentiating this Hamiltonian with
respect to the control variables with the condition that the time rate
of change of the Lagrange multiplier is now dependent on the
Hamiltonian with respect to the state variable, in this case, soil
moisture.
So this, under some conditions of losses, yeah, and some conditions of
terminal gain, Stefano Manzoni is very good with this stuff.
He was able to solve this problem for dry downs, single dry downs, and
I will not show all the formulations just to highlight again a few
features about this outcome.
He actually showed that if you drop the terminal gain, so you don't
care about what you leave as water in the rooting zone, and basically
you assume that lambda is separated completely from the transpiration
rate.
If the soil moisture content is roughly constant, you recover the
classical approach.
But if you allow soil moisture content to vary gradually, you discover
that the marginal water use efficiency must go up in time.
So the cost of water becomes more and more valuable as the dry down
progresses.
But the canonical features do not change.
But in this case, if lambda is still a boundary condition on the
optimization that you need, specify different models that would vary
with the state variable.
So you could make g of t big or small as a function of soil moisture
content that you leave at the end.
And he showed that in all cases, you start with well water conditions
for lambda, which is now a boundary condition, and that of course
increases with time if leaf pressure becomes more and more negative or
stomatic conductance drops with time.
And in this bigger and bigger at an exponential rate.
These are all analytical results.
He was able to compare these calculations with Avogadro's number of
data points for herbaceous plants under arid and wet environments as
well as evergreen and conifers as well.
So these are a collage of probably 550 data points from different
experiments normalized together.
So now the problem is how do we determine the marginal water use
efficiency for well water plants and wet Well, Wilfred Conrad solved
this problem for me, so we don't have to worry too much about it.
Basically, you could derive it based on the geometry of the pores.
By equating the two formulations, one, how much you get in water
solution of Jean-Yves Pallange and Paul Wagner who first derived this
result.
Okay.
We've talked a lot about the
hydraulic transport system.
Now, the hydraulic transport system, the fact that there is some
capitation occurring, as Wilfred has mentioned and as Nadine has
mentioned before, is not a new result.
In fact, this is the quotation taken from Haberland's work in 1914.
He actually stated that vessels and plants are substances varying
according to the season and the time of day.
So the fact that capitation existed in plants was known way before any
new ideas about how to deal with it came about.
So there was clear recognition that yes, plants operate in a normal
way and they do cavitate.
So this was basically coming from Haberland's work.
So, okay, so now we understand that capitation and the formation of
gas emboli are common occurrences in the xylem and many plant species.
Okay, so now if we put the most detailed approach that we could put in
terms of modelling water movement in the trees based on whatever we
know about the capitation curves or the vulnerability curves, put them
all how they might vary with scale, put a realistic radiative forcing
on the tree, have a fractal tree so there is some branching and so
forth, and assume that the pressure in the roots are close to zero and
so therefore the pressure in the plant is going to drive everything.
And these are simulations done by Gil Bohrer and the model is
published in Water Resources Research in 2005.
Gil did this as part of his watershed hydrology project for my course
and basically the goal was to try to assess what would be the maximum
hydraulic carrying capacity that can support photosynthesis under well
water conditions.
And so I will just run this movie.
What you are going to see now are basically pressure distributions in
the plants.
And the red is something that crosses 1.5 megapascals negative
pressure and the blue is well water conditions.
Also I should also note that capacitance or storage of water is taken
into account in these simulations.
So now the day starts.
Now we are close to noon and then now we are going into the afternoon.
Evening is the recharge.
And so you could see that basically the stresses in the plant are
happening at the most distant part of the plants and then later the
refilling mechanism fixes everything and then the plants the soil.
So this can be all modeled but the point from these simulations is
that can we at least simplify them and be able to interpret them in
ways that would allow us to put many many data sets together.
And that is basically what we are going to now do.
The premise here is that it seems that because the most distant parts
of the plants have the most critical pressures we are going to assume
that the controlling pressure of water movement in the xylem is driven
by the leaf pressure.
And that is an assumption.
So basically what we are going to do now is say hey let's look at this
problem from a single resistor.
We have pressure in the soil.
We have pressure in the leaf.
And we have the xylem hydraulic conductivity varying with the leaf
pressure.
Because primarily if the soil pressure is constant you vary this one.
That's the only thing that is going to change.
So probably that's not a horrible assumption.
Now because the soil pressure is small relative to the leaf pressure
under well water conditions we're going to throw the soil pressure
out.
So the driving gradient to the whole problem is leaf pressure.
And if we assume that the xylem hydraulic conductivity is primarily
driven by this leaf pressure, this is not a new idea, this has been
around for a while, by Zimmermann it was called the hydraulic
segmentation hypothesis in that cavitation occurs first in the most
distant parts of the plants and the simulation by McGill do provide
evidence for that.
And if we assume that we're not dealing with trees that are 120
meters, as Nadine mentioned, but we're dealing with trees that are
more like 20 meters, maybe, we can drop the gravitational potential
relative to the pressure potential relative to So under those
conditions we can say that the maximum transpiration rate, because the
soil is wet, the maximum transpiration rate that can be supplied to
the leaf is dictated by the vulnerability curve, GP,
that you haven't figured out the theme of this lecture.
It's all about simplification to the max.
Now, the vulnerability curve often is expressed as such.
The vulnerability curve normalized by some maximum value drops as a
function of leaf pressure, meaning you lose hydraulic conductivity,
and there are many, many forms.
Now, I have a student working on theoretical models to predict the
forms of these vulnerability curves from some basic thermodynamics and
basic hydraulics, but basically a good fit to these models are power
laws or Weibull functions or logistic curves, and basically they all
say the same thing.
That if you normalize the leaf pressure with respect to psi 50, which
is the point at which hydraulic conductivity is impaired to 50% of its
value, this is exactly the same shapes that Nadine has presented
yesterday.
They are whole-plant vulnerability curves, so they're not
incorporating one bubble in one element like Wilfred has mentioned,
but they are a collage of many, many elements.
The parameter a that you see here basically dictates how sharp the
drop in conductivity is as a function of pressure, so if you want a
very sharp drop, so you are either conductive or not conductive, then
a is big.
If a is small, you have a more gradual transition, so this is a very
flexible function to play with.
And if you do that, you find that the vulnerability curve requires
three parameters, this curvature a, psi 50, and the maximum
conductivity that you could allow.
So if we put this formulation for vulnerability curve here and try to
ask what would be the maximum possible transpiration rate that you
could supply to the leaf.
As you increase the leaf pressure, of course transpiration rate goes
up, but the xylem hydraulic conductivity drops, so again we have
almost this interplay between increasing driving force and reduced
conductivity.
Very much like the vapor pressure deficit problem.
So you could actually proceed and find the maximum by differentiating
e with respect to psi L, set it to zero, solve for the leaf pressure
at maximum that turns out to be dependent on psi 50 and the curvature
parameter.
And what would be the maximum e associated with this leaf pressure now
can be computed and you could also calculate gp at this pressure and
it turns out to be basically given by gp max and this curvature
parameter.
And so the percent loss of conductivity is 1 over a.
So now we have three parameters that would allow us to diagnose how
maximum transpiration rate occurs in plants and that is giving us at
least an anchor on how much water can be delivered to the leaves by
the xylem system.
So the solution is e max is gp max times psi 50 times a minus 1 to the
1 minus a over a based on this model.
Now if you do a sensitivity analysis, and Stefano has done this, so
you take all the vulnerability curves that have been published in the
literature and you look at the value of a, it turns out to vary from 2
to 10, which is a lot, but this f of a that I show here varies from 0
.5 to 0.7.
So in the business of plants, between friends, we're not going to
worry about this variation, so we're going to treat it roughly as a
constant.
If you do that, the concept between safety and efficiency immediately
emerges.
It's actually the product of the maximum transpiration rate, it's
actually the product of safety by efficiency.
Efficiency is encoded in the maximum conductance.
Safety is encoded in psi 50, and it is their product, ironically, that
dictates the maximum transpiration rate, as simple as this.
GPmax, basically, if you build an efficient xylem system, you will
have to have big vessels, your psi 50 is going to be probably lower.
So we don't have yet a super plant where we can increase psi 50 and
GPmax.
We have not yet designed it, but basically it does suggest that for a
maximum transpiration rate, there are a whole slew of strategies that
you could achieve this maximum.
You could be safe, so you have a high psi 50, but then you are not
very efficient.
Or you could be extremely efficient, but you are going to pay the
price if there is uncertainty in pressure.
So how well does this model compare to data sets?
So again, Stefano is able to mine the literature big time and he was
able to derive something like 550 vulnerability curves and then he
went and assessed for different species in climatic zones, boreal,
temperate, Mediterranean, tropical, tropical moist and tropical dry.
And then what he did is he basically found what would be the maximum
transpiration rates for those species.
So this was completely independent measurements, not even the same
experiment, not even the same location.
And he plotted the model Emax from this equation against the measured
Emax, where GPmax and Psi50 were inferred from cavitation curves for
species collected in those zones.
And this is the comparison.
So here Emax is plotted in cubic meters per day.
So I told Stefano this is nonsense, sorry to tell you man, nice work,
but this just tells me that big trees transpire more than small trees.
So he came back to me and said no, I actually looked at maximum
transpiration rate also normalized by sapwood area and the agreement
seems to be still okay.
So that does suggest that at least GPmax and Psi50 may reflect this
tradeoff between safety and efficiency, but you have to condition it
on maximum transpiration.
So alone on their own they don't say much.
But for a given maximum transpiration rate you could compare now
safety versus efficiency based on what GPmax is and Psi50.
And these are by the way measured from vulnerability curves.
So we have a lot of that information.
Okay, now I would like to touch a little bit on the floam transport
problem.
This again was tackled in a very similar way by Jensen, so that's why
I thought it would be nice to show it here.
What Jensen did is he says, hey, look, if the sugar mass flux out of
the loading floam is given by the water flow rate multiplied by the
density of water that may vary with sugar concentration multiplied by
sugar concentration, so that's the water flux down through the floam
to the roots, it's an advective flux, the water flow rate is Q, and of
course the density still varies a little bit with the sugar
concentration, but the driving force for that is the sugar
concentration itself.
Now, at low Reynolds number, which is the case for flow in the floam,
the flow rate is dependent on the pressure difference divided by the
water viscosity.
This is basically the Hagen-Poiseuille equation with just the
geometric factors absorbed in one multiplier.
Now the water viscosity varies a lot with sugar concentration.
Think of maple syrup, very sugary but very viscous.
So, and the pressure difference could have a hydrostatic component but
an osmotic component.
And if we assume a linear osmotic pressure, which is what Jensen did
using the Van Hoff equation, then it is basically driven by the
density multiplied by the concentration itself.
So you take this pressure difference, you plug it here, you take this
flow rate, you plug it here, now you have the flux strictly as a
function of concentration of sugars.
And you ask the question, what would be the sugar concentration in the
flow that would maximize this flux?
Again, there is a trade-off, yeah, because you increase sugar
concentration, you increase the flux, but you decrease the viscosity
in a very non-linear way.
This is very non-linear.
So I did this calculation again, and what you find is that if you plot
the sugar concentration on the x-axis, the flux normalized by its
maximum value, so we're not worried about units, on the y-axis, and
you look at different approximations for the pressure drop, only
hydrostatic, only osmotic, or their combination, you find that, yes,
sure enough, there is a local maximum, and then, of course, if the
sugar concentration drops, the ability to actually transport sugars in
the flow drops very rapidly.
So this is a very touchy balance, and the optimum sugar concentration
that you have is somewhere around here, based on these model
calculations, very trivial, very trivial calculations.
In fact, if you look at the data set that have come about in terms of
what the sugar concentrations are for many, many species, they're here
broken up into averages for passive loaders and active loaders, you
find that this is the range that they exist.
This is the maximum that we are predicting, and the two are pretty
close.
This perhaps suggests that yes, plants have optimized their ability to
transport sugar as well.
But the approach is very similar to what we have done for the water,
and it is very similar to what we have done for the leaf.
Putting this my former student did this, and he was able to show that
if you combine water movement in the xylem, the leaf optimization and
transport in the phloem, and you maximize the sugar transporting
system alone, so you let everything be slave to it, because he found
that the sugar concentration is the most restricted, this is the most
touchy, you could actually re-derive the marginal water use efficiency
as a function of all the loading phloem properties and the maximum
hydraulic conductivity in the xylem.
So this is actually strictly based on the water balance, based on
demand, supply, and transport.
So I'm not going to go through this whole derivation, but basically
now you could show that you could link it to.
And other things pop up is that this marginal water use efficiency
might actually scale with the square of the optimum sugar
concentration and you could predict analytically that the marginal
water use efficiency has to drop with the conductance.
And both of those, there is some support in the literature for them.
So now I will conclude.
We have shown that the control of transpiration allows maximum fluxes
in well-watered conditions despite moderate cavitation.
So this is basically suggested that there is some coordination between
the liquid and vapor conductances.
We have also shown that optimal stomatal behavior is still possible
despite down regulation of transpiration during droughts and that is
coming at the expense of a marginal water deficiency increase.
So the price of water becomes higher.
But it doesn't affect the general solution of the optimality
arguments.
Now I should also comment on a recent paper that was published by
Gleason in 2015 where they had a huge data set where they tried to
find whether GPmax and Psi50 are somehow correlated and they found
nothing.
They conditioned the data on everything temperature soil biome species
you name it none of these correlations popped up.
They just did not do one thing.
They did not condition their data on maximum transpiration.
That was the only thing they did not do.
Whereas our suggestions would be that if you are able to rank these
data sets with maximum transpiration rate for a given transpiration
rate you could see some relation between GPmax and Psi50.
So that would be an interesting inquiry to look into.
But I did not do it yet.
And then the last point is this connection between the N-member
correction and the stomatal optimization theories and how leaves might
operate.
As I mentioned before, the objective function now has a terminal gain
basically.
And one could simply link this terminal gain to the soil moisture
state at the end of the integration period.
And one could argue that if you want to assign a conservative water
use strategy you make this gain big and if you want an aggressive
water use efficiency you could set this gain to zero.
And then you could ask what would be the optimal path that the stomata
has to take for conservative water use strategies and aggressive water
use strategies.
Are they very different or not?
What would be the impact on concentrations in the flume?
More important, what is their impact on this so to speak spectrum of
isohydric to anisohydric behavior in terms of looking at relations
between leaf pressure and stomata conductors.
So now analytically or well semi-analytically one can explore these
questions by looking at how to connect water use strategies with the
operation of stomata.
And with this I will stop.
Thank you.