Ankur Desai
Written Comprehensive Exam
August 6, 2003
Start time: 8:45 a.m.
End time: 4:45 pm
1. Oxygen as a tracer in studying the global carbon cycle.
The atmospheric oxygen/nitrogen ratio (O2/N2) has been measured over the past 15 years or so
with sufficient accuracy to be useful for providing constraints on the global carbon cycle.
Please be as specific and as quantitative as possible in answering the following questions about
atmospheric O2/N2.
Oxygen and carbon are intimately linked through photosynthesis, respiration, and
combustion. Photosynthesis takes up carbon and produce oxygen. Aerobic respiration does the
opposite. Combustion also consumes oxygen. This does not mean that oxygen and carbon are
interchangeable when making measurements of the global carbon cycle. Both carbon and
oxygen are used in other processes (e.g. ocean solubility and biology differs for the two species –
leading to different equilibrium times, the production of O2 from ozone and hydroxyl reactions,
the methane cycle, anaerobic respiration, etc…). The ratio of moles of CO2 consumed or
released versus O2 released or consumed varies for the different processes. Also, oxygen is not
measured directly, but rather the O2/N2 ratio (which as we know is roughly 20/80 by volume). I
am no expert on the use of oxygen measurements, and I draw most of my conclusions about this
method from Battle et al (2000) and discussions from the global carbon cycle class (Davis et al,
2002). I also point out that we owe respect to Ralph Keeling who is the father of the O2/N2
measurements (Keeling and Shertz, 1992).
For terrestrial plants, about 1 mol of CO2 is produced for 1.1 mol of O2 (and vice versa
for plant/animal respiration). The ratio is 1:1.4 for the ocean due to different carbohydrate
synthesis products. The number is around 1:1.43 for fossil fuels depending on what you are
burning. The neat thing about the ratio of O2 to N2 is that that change in this ratio is not affected
by ocean uptake (for the most part, there is some uncertainty about this – in the spring,
atmospheric O2/N2 increases due to ocean biology as net production supersaturates the ocean
surface layer, and falls in the winter to resaturate - but this process, despite have a large seasonal
signal, is assumed to be in balance over the course of several years). Basically, the oxygen
exchange with the ocean equilibrates faster than the carbon exchange with the ocean. Thus you
can use O2/N2 ratio measurements (along with other carbon measurements like CO2 and
13
C) to
separate terrestrial and ocean sinks. Fossil fuel combustion will decrease O2/N2 in the
atmosphere, while terrestrial growth will increase.
(a) Do N2 variations substantially affect the O2/N2 ratio? If so, through what process or
processes?
This is not something I’ve really thought about until this moment. Generally, we assume
that the total atmospheric nitrogen content does not change that much – as the total nitrogen
fluxes of the nitrogen cycle are small compared to the reservoir size. Nitrogen fluxes arise from
breakdown of plant litter and soil organic matter by microbes and algae (nitrogen mineralization
of lignin N into inorganic N) and the uptake of atmospheric nitrogen by nitrogen-fixing soil
microbes (i.e. microrhyzzae). Plants uptake nitrogen (an essential nutrient for amino acid
(protein) synthesis). Soils degass and leach immobilized nitrogen to the atmosphere and water.
Lightening also breaks down atmospheric nitrogen. These processes, when not altered by
humans, are in relative balance. The oceans are also a small flux to the atmosphere of nitrous
oxide (N2O) (Najjar, 1992). Human production and application of fertilizer, NOx byproducts
from combustion (and the release of previously – ancient – fixed nitrogen), and cultivation of
nitrogen fixing plants (soybeans, legumes) have led to a steady increase in the nitrogen cycle rate
and total nitrogen amount in the soil, water and atmosphere (especially N2O). The increasing
nitrogen deposition rate from atmosphere to soil is implicated by many researchers for
explaining part of the enhanced observed biological sink (Schimel, 1994).
I found one website (http://www.dpw.wau.nl/pp/research/example2.htm) that showed
nitrogen fluxes around 1 Pg for decomposition and uptake, and around .1 Pg for N-fixation and
denitrification. Compared to the size of the total N pool – around 4 x 106 Pg (assuming 78% of
atmosphere in nitrogen), these are small fluxes. Any imbalance, then, is not going to change the
total N2 concentration that much. Meanwhile, because of the dominance of photosynthesis and
respiration across our planet, it is certainly the case that the fluxes of oxygen (in relation to the
size of the total oxygen pool – 1 x 106 Pg) are larger (though the net flux might be small – but
even if the net N and O flux were the same, the relative change in O would be more). The
change in O2/N2 seasonally is around 100 per meg.
So, overall, I don’t think variations in nitrogen would affect the ratio all that much. It is
possible that at individual flasks sites that may be near large sources or sinks of nitrogen, there
could be some biasing in the data. However, I wouldn’t be overly concerned with it. Keeling’s
dissertation goes into a bit, I think.
(b) Describe the annual cycle (in both hemispheres) of atmospheric O2/N2. How and why does it
differ from the annual cycle in atmospheric CO2? What useful information does the annual cycle
in atmospheric O2/N2 provide about the global carbon cycle?
The oxygen cycle has a strong oceanic biology imprint on it not seen in CO2 since the
equilibrium time for CO2 in longer (1 year) than for O2 (weeks, I think). This tends to smear out
any seasonal biological cycle in CO2 (also winter ocean respiration producing carbon is
cancelled out by increased solubility due to colder water). Conversely, atmospheric O2 content is
less influenced by air-sea exchange than CO2 (which is driven by the solubility pump which
dissolve CO2 into carbonate ions). This means that changes in the atmospheric O2 content
indicate carbon uptake and release by photosynthesis, respiration and fossil fuel combustion
better than changes in the atmospheric CO2 content itself. Thus, the O2/N2 provides useful
information about separating the influence of ocean and land on the global carbon cycle. The
ocean biology signal on oxygen occurs because there is supersaturation of oxygen in the surface
ocean during spring due to net production and warming temperatures (lower solubility), and
subsaturation in the winter.
This large signal (due to small buffering) is thus imprinted on top of the seasonal O2
cycle that would be produced by terrestrial photosynthesis, respiration and combustion. These
signals would be opposite the changes occurring in CO2 (decrease in CO2 due to photosynthesis
in the growing season, increase as photosynthesis declines) – and at similar magnitudes (based
on stochiometry). However, the ocean signal is 15 times larger than the biosphere signal,
according to Battle (2000). So the annual ocean signal would dominate the time series of O2/N2
(increase in spring, decline in winter). This is opposite the CO2 signal which tends to increase in
the fall and decline in summer, i.e. they have small lags but are generally anticorrelated. Both
signals would have smaller seasonal amplitudes in the southern hemisphere (less land, more
ocean, less combustion, smaller sink), but the presence of ocean biology would still leave an
imprint on the oxygen cycle, whereas the carbon cycle seasonal variation would be much
smaller.
There is also an interhemispheric gradient (O2/N2 is 15 meg lower and CO2 is larger in
the northern hemisphere), which implies that fossil fuel combustion occurs primarily in the north
(along with the land sink). For a graph of the cycles of carbon and oxygen, refer below to the
figure below that I shamelessly took (for educational use only, thus a proper use of copyright)
from Battle’s paper.
Figure 1. Battle et al. (2000) figure of O2/N2 and CO2 flask measurements in the northern and
southern hemispheres. Alert and Barrow are in the northern hemisphere. Cape Grimm, Baring
Head and Syowa are in the southern hemisphere. The oxygen pattern still has large seasonal
variation in the southern hemisphere due to ocean biology, whereas carbon dioxide does not. In
the northern hemisphere, carbon and oxygen are essentially anticorrelated.
(c) A method has been developed to use the long-term trends in atmospheric O2/N2 and CO2 to
separate terrestrial and oceanic sinks of carbon. Describe in detail how this method works and
what assumptions underlie it. Describe at least one way in which these assumptions can be
reasonably violated, and provide methods for correcting the errors introduced by such a
violation.
Basically, knowing the long term trends in CO2 and O2/N2 with time along with some
oxygen and carbon stochiometry (derived from photosynthesis, respiration and combustion
equations) gives us these two equations. First, an equation for the time rate of change of global
CO2:
CO 2
 0.471( f fuel  f cement  fland  f ocean )
t
The change in CO2 with time (ppm/yr) is a function of the net fluxes of fossil fuel
combustion, cement production, land sink and ocean sink in Gt C/yr. Ffuel and Fcement are known
from combustion inventories. The latter two are unknown. The time rate of change of carbon
dioxide is measured with the global flask network. The -0.471 converts ppm to Gt C and
provides the right sign.
Since, we have two unknowns, we can have another equation to describe the change in
O2/N2:
O2
 0.471 4.8  (1.43 f fuel  1.1 fland )
N 2
The 0.471 is the same conversion factor as before. 4.8 converts per meg to ppm. 1.43 and 1.1
are the stochiometric conversion factors (amount of carbon produced for oxygen consumed).
The ocean is assumed to have no impact on the O2/N2 ratio when averaged over several years
(net balance in oxygen cycle of ocean biology).
Two equations, two unknowns – solve for Ffuel and Fland. This seems pretty reasonable,
and the results show a moderate ocean sink and a large northern hemisphere land sink in the
1980s. There are several assumptions though. First, we have to assume that the flask networks
properly samples the global variation in CO2 and O2/N2. Network and model tracer-transport
tests can show whether this is the case. The lack of measurements in terrestrial regions
(complicated by the presence of biology) could bias the results. We could also add some
information about hemispheric mixing times since we are assuming they are the same for oxygen
and carbon with these equations (or at least less than 2 years). Also, instrumental error can be
accounted for to provide reasonable uncertainty estimates in the results. Second, to remove the
ocean biology effect from the O2/N2, we average over 2 years, assuming that over long time
periods, the ocean biology effect would cancel out. However, there is evidence that the ocean
has a net outgassing of O2 due to global warming and changes in ocean stability (mixing). We
could fix that by either trying to measure this value and adding it to the O2/N2 equation, or by
further constraining the two equations with another measure that may be less affected by ocean
biology, for example the isotope 13C. Another assumption is that flask measurements in the
marine boundary layer are unaffected by the covariance between seasonal changes in vertical
mixing (due to changes in mean boundary layer depth) and CO2 or O2 concentrations. Actually,
if both are affected equally by this “rectifier” effect, then there is no problem. But we don’t
know that and it is something that could be modeled and the measurements corrected for.
2. Temporal variability in ecosystem-atmosphere CO2 fluxes: Hours to decades.
(a) Describe and explain the temporal variability of ecosystem-atmosphere carbon dioxide fluxes
observed at the WLEF tower in Wisconsin, focusing on diurnal, synoptic, and seasonal
variability. Briefly compare and contrast these temporal patterns to the temporal patterns of two
other Ameriflux sites—the Sky Oaks/Chaparral site in southern California and the
Barrow/Tundra site on the Alaska North Slope. (If you do not know the variability patterns for
either of these sites, try to reason them out; see part b.)
Surface ecosystem-atmosphere carbon dioxide fluxes in the terrestrial biosphere arise
from photosynthesis and respiration. Photosynthesis occurs in the chloroplasts of leaf cells
which remove carbon from the atmosphere to be used in the creation of leaf, stem and root
tissue. Photosynthesis requires water and light. Respiration occurs in all living cells that are
found throughout the ecosystem from microbes in the soil to the stems of trees. The bulk of
respiration occurs in the soil (around 75%) as microbes decompose leaf litter and organic matter
and roots respire. Respiration requires oxygen and its rate is generally a first-order kinetic
reaction dependent on temperature. Thus, the daily, synoptic and seasonal patterns of carbon
dioxide flux within any ecosystem are going to be modulated by the responses of plant, soil and
atmosphere to environmental factors. These environmental factors include light (radiation),
temperature, precipitation, light quality (i.e. diffuse vs direct radiation), nutrient cycling
(primarily nitrogen and phosphorous, but also Mg, Cd, Mn, K), soil moisture availability, etc…
There are also spatial factors. Within the “footprint” of the measured flux, there is an
assemblage of plants and soil types. This assemblage could change over time due to disturbance
(natural or human) or long-term change (i.e. succession). The timescales that these
environmental factors and disturbances change (hours, days, weeks, years, or decades) will
determine the timescale that they effect carbon dioxide flux.
The WLEF tower is a northern mixed (deciduous and coniferous) forest. I refer to Davis
et. al. (in press), who describes the fluxes at WLEF. There is also a lot of wetland in the
footprint (maybe 1/3 of the landscape). The diurnal pattern of flux is governed by changes in
solar radiation (sun rise/set). During the growing season (late spring to early fall), carbon
dioxide flux reaches a maximum negative in the middle of the day – around solar noon (possibly
slightly earlier if water stress occurs later in the day) – anywhere from -5- -15 mol CO2/m2/s.
By nighttime, photosynthesis stops, and soil respiration dominates the signal, leading to a net
positive flux to the atmosphere for 5-15 umol/m2/s. In between (evening and dawn) there is a
rapid transition between these two states, as the day progresses from a stable, low turbulence
night, to an unstable, convective day. The daily amplitude reaches a peak early in the growing
season as leaves are growing quickly and there is rapid decomposition of leaf litter that levels off
later in the summer. Once the leaves fall off, there is no significant daily amplitude, and the net
carbon dioxide flux is a small positive value reflecting net respiration of the soil (the effect of
coniferous uptake is very small).
Carbon dioxide flux also responds on the synoptic scale (3-4 days). At the synoptic
scale, weather fronts change the bulk environmental conditions (cloudiness, precipitation,
temperature). These changes effect the net carbon flux on the weekly timescale. Hurwitz et. al.
(in press) reports on some the synoptic variability seen at WLEF on mixing ratios. This
variability also can effect fluxes. A strong cold front may have an instantaneous effect at frontal
passage of decreasing net uptake (cloudiness, precipitation), but over the weekly timescale, may
actually increase net uptake (cooler conditions, more water availability, more optimal for
uptake).
Seasonal variability at WLEF is driven primarily the presence or absence of leaves,
which is primarily determined by mean air and soil temperatures (can be modeled with growing
degree days). I have already described how carbon dioxide amplitude changes with time due to
seasonal changes. If you look at the cumulative sum of carbon dioxide exchange over time, you
would see a slow increase in total net flux to the atmosphere due to winter time respiration, then
a sudden drop (net uptake) over the course of a month or two as the leaves come out and
temperature becomes warm enough for photosynthesis. Eventually as photosynthesis declines,
but respiration stays more constant, carbon dioxide flux is closer in balance. In autumn, there is
rapid return of carbon to the atmosphere as leaf decomposition continues until the soil freezes, at
which point we return to winter.
I don’t know much about Sky Oaks chaparral site. There is both a young and old stand
there according to FLUXNET. They are evergreen sites with a Mediterranean climate at
moderately high elevation (around 1600 m). I would suspect that the uptake would could occur
year round, though the higher elevation may preclude much winter uptake. I would suspect
smaller rates of flux due to water limitations, but a longer growing season. A table in Falge et al.
(2002), shows both maximum carbon uptake and release are almost half as small as WLEF,
consistent with smaller vegetation and greater water stress. Over longer timescales, I imagine
fire frequency is higher (hence the existence of chaparral), and so disturbance is regular, causing
both carbon loss at fire and a large source and then rapid uptake after reestablishment. The
Tundra site in Barrow would have different carbon flux dynamics than either sites. Here,
vegetation is limited in photosynthesis by temperature year round. Baldocchi et al (2001) notes
that the tundra site have a long term carbon flux to the atmosphere (net source) possibly due to
disturbance of the long term carbon storage (global warming leading to permafrost melt and soil
decomposition). This net source is not very big, and not much bigger than the small
source/balance seen at WLEF. I imagine the growing season in Barrow is very short and the
amplitude of the fluxes very small due to small, low leaf area, shrub (wetland, sedge) vegetation.
Precipitation is low causing another stress. Diurnal variation would be nonexistent since the sun
shines 24 hours a day (albeit at very low angle) in the summer, so net photosynthesis could
possibly occur all day in the summer.
(b) Outline a model of temporal (diurnal, synoptic, seasonal) variability of ecosystematmosphere carbon dioxide fluxes that could be applied to any CO2 flux site. That is, explain
what variables would be needed to predict the temporal variability in CO2 flux at any site and
suggest a quantitative relationship, or reference literature that would provide a quantitative
relationship.
In part a, I already outlined how environmental factors affect carbon dioxide fluxes at
multiple time and spatial scales. There have been numerous attempts by modelers to encapsulate
these effects and attempt to explain carbon dioxide flux at any one point due to environmental
factors. This is not trivial, it turns out. Foresters and biologists have long known that light,
temperature and precipitation certainly affect plant growth and soil organic matter decay.
Photosynthesis (carbon uptake) has a initially linear increase with photosynthetic active radiation
(PAR) and then saturates. Photosynthesis can be moisture limited (from the soil).
Photosynthesis is also sensitive to vapor pressure deficit (VPD), since a dry atmosphere would
lead to rapid evaporation and loss of turgor pressure, so stomata close and carbon flux uptake
declines. Finally, photosynthesis is a function of carbon dioxide concentration, both outside and
inside the leaf. Changes in carbon concentration leads to short term direct changes (i.e.
fertilization), but over the longer term, plants can adapt to changing environments. Respiration
(carbon source) has an exponential relationship with temperature. It is also dependent on the size
and quality of the carbon pool, thus reflecting seasonal variability in the rate relationship.
By modeling the net uptake of carbon and the net release of carbon, you can attempt to
model the net flux of carbon at a flux site. On short time scales (hours or days), photosynthesis
and respiration can be empirically modeled with PAR and temperature (Falge et al., 2001; Law
et al., 2002). On longer timescales, these simple relationships breakdown (Yi et al., submitted).
In arid environments, soil moisture deficit (water stress) can significantly reduce carbon dioxide
flux (Law et al., 2000). There is strong correlations between water and carbon cycling in all
environments. These relationships can be incorporated into standard biophysical relationships
based on carbon biochemistry in plants and soil (i.e. the Farquhar model (Farquhar and Sharkey,
1982), Ball-Woodrow-Berry model (Ball et al., 1986)) that use information about CO2
concentration, stomatal diffusion rates, Rubisco kinetics to derive carbon uptake rates in plants.
These models along with soil bucket models (i.e. CENTURY) and environmental weather and
site information (nutrient, vegetation type, leaf area, etc…) have then been implemented in
biophysical big leaf models (also known as gap models) to predict fluxes of CO2 (i.e., SiB
(Sellers et al., 1986) or BIOME-BGC (Thornton et al., 2002)). The effect of fire, disturbance
and stochastic spatial events (tree falling) are needed to scale beyond from the gap to flux or
GCM fooprint (i.e., the Ecosystem Demography model (Moorcroft, 2001)). Most of these
models rely on parameterizing the “resistance” or “conductance” to carbon and water exchange
between various carbon pools (slow soil, labile soil, leaf litter, root, stem, leaf, atmosphere).
(c) The discussion above considers temporal variability within the annual cycle. Could this
model be extended to describe interannual to decadal-scale variability in net ecosystematmosphere flux of CO2? Suggest a simple model for this scale of temporal variability. Explain
the biophysical basis for this model, if it differs from the shorter-term model.
Some of the models described above already deal with interannual variability.
Interannual variability is driven by longer-term changes in 1.) climate, 2.) vegetation function,
type, and structure, 3.) disturbance. All three occur at multiple spatial and temporal scales. Over
decadal timescales, site history essentially is more important in determining fluxes than daily
weather patterns.
Climate change can be fed directly into most biophysical models, however, you would
need to additionally parameterize vegetation and soil adaptation to long term change. For
example, you could take GCM outputs of changes in temperature, precipitation and carbon
dioxide concentration and find out how the flux would change (Sellers et al., 1992). Carbon flux
could be influenced, for example, by climate changes driven by ENSO, especially in the tropics
(Rayner et al., 1999).
Vegetation, however, also changes with age. As most vegetation matures, photosynthetic
capacity declines, while total respiration increases (due to increased coarse woody debris CWD) (Kira and Shidei, 1967). Both of these effects can be modeled by including the age of the
stand in your model and having a growing CWD pool on the ecosystem surface. The ecosystem
demography model and BIOME-BGC have stochastic mortality functions and birth functions (as
a function of seed dispersal) to account for birth and death of vegetation. Few models to date
deal with longer term vegetation succession. This is more difficult since you need prior data on
how vegetation changes as climate changes due to changes in mortality and biotic competition.
Biotic competition could influence abiotic factors in the environment such as light availability,
thus this has to feedback into your radiation module. Thus, there are many feedbacks which get
more complicated (and unknown) at increasing time and spatial scales. To accurately model this
kind of change, there needs to be feedback between the climate/meteorology module and the
biophysical model.
Finally, disturbance is another complicated beast to try to incorporate into a model. Tree
mortality can appear random – thus stochastic or Monte Carlo methods can sometimes produce
the best results in predicting vegetation change due to disturbance. Fire has a wide range of
spatial effects (splotchy fires that burn 10% of canopy to large fires that burn the whole crown) –
though the space, time and intensity scales are correlated, which can help in determining the
effect of fire on your ecosystem (usually parameterized as a simple vegetation loss). Human
impacts (deforestation, fire) are difficult, too.
Ultimately, if all you want your model to do is produce carbon flux (i.e. not too worried
about the details of the vegetation structure or other nutrients), then the goal is to parameterize
these long term effects and feedbacks loops in terms of their effect on your carbon pools and
fluxes. One major difference is incorporating stochastic effects – some of which can be
prescribed (like a crown fire) and some of which can be randomly distributed (mortality and
small fires). The ecosystem demography (ED) model, for example, attempts to models the
differential equations describing the change of a biophysical state (certain vegetation, climate,
flux) to another state, with rate equations for fire, mortality, etc... Since over longer timescales,
vegetation site history determines fluxes, it is important to properly characterize the vegetation
dynamics, which may not be important for short time scale flux models.
(d) Globally there are probably no more than 5-10 old-growth flux towers, each with a flux
footprint of about 1 km2. One rationale for establishing these sites is to provide insight into the
carbon sequestration of North American and European forests if they are allowed to continue to
grow undisturbed, thus become “secondary old growth” forests. Comment on the strengths and
weaknesses of this scientific rationale.
Let me first describe old-growth forests. As forests age, there is changes in the net uptake
and release of carbon. Frelich (1995) describes the changes that lead a forest to become “oldgrowth” or multi-aged. After an initial disturbance, a stand is initiated. Over time, the canopy
closes up. Then the self-thinning process occurs, as stems die. Gaps form from dead trees are
filled in by neighbor trees. New stems in the understory are excluded from the canopy.
Eventually, the tree gaps get larger, and the understory is reinitiated. Neighbor trees cannot fill
up the gaps, so new stems grow up. Over time, these leads to a quasi-stable multi-aged stand of
trees, with saplings filling in the gaps, until the next disturbance occurs. This occurs in both
primary and seral old-growth forests.
Most flux sites are located in young or middle-aged forest stands that have not reached
the multi-aged, multi-species stage. This is due to the need for selected homogenous sites
(disturbance can be patchy spatially and temporally) and for selecting sites that are representative
of the current vegetation. However, with time and decreased logging, increased agricultural
abandonment and woody encroachment, forests will get old. Caspersen et al (2001) shows that
early successional species are more productive (more uptake) and late successional species are
slower growing, although the total carbon is higher, and higher diversity (multi-aged) forests
store more carbon. With increasing stand age, Murty et al. (1996) and Gower et al. (1996)
conjecture that sapwood maintenance respiration increases, stomatal conductance declines
(larger hydraulic resistance), and soil nutrient availability declines (due to nitrogen
immobilization). Meanwhile, belowground net primary productivity often increases with stand
age. Thus, understanding both past and future carbon flux changes will require understanding
how this quantity changes with stand age. We do not know quantitatively these effects.
Models have shown conflicting results that depends especially on how we parameterize
disturbance and fire (i.e., Hurtt et al., 2002). Although stand-level eddy covariance
measurements do not represent very large regions, they are larger than those measured by
biometry, which can also underestimate the important of belowground processes. The Kyoto
protocol (UN Climate change Secretariate, 1992) has provisions to account for managed forests’
increases in carbon storage with time which can be used against net carbon emissions. Schulze
et al. (2002) argues that the protocol this will lead to a rapid turnover of forests by forest
managers since early successional forests grow faster and store carbon faster, despite that
realization that older forests altogether have larger total carbon pools and the harvest of old
forest leads to rapid large carbon source (in addition to a large carbon export not accounted for in
the protocol) before becoming a large transient sink (Thornton et al., 2002).
The primary weakness of this effort of quantifying stand-level old-growth flux is the
small footprint of the flux. Process-based models can deal with larger scales, along with remote
sensing. However, they need to be parameterized with what we observe on the ground, by
biometry, chamber flux, eddy flux and forest inventory analysis. Old-growth forests are
currently not spread out equally among all vegetation types and are not found in all climate
spaces. Thus, there is no way to know is what we observe from the fluxes is statistically the
same as what all old-growth vegetation does. This is why we need models and more sites.
Remote sensing can also help bridge the gap between the flux footprint scale and the regional
flux of carbon dioxide.
3. Water use efficiency
Attached is a copy of a celebrated paper written in 1977 by Cowan and Farquhar (call if CF) in
which the concept of water use efficiency was introduced and dissected. CF concede that the
water use efficiency parameter, , varies in time, with environmental conditions, with the type of
plant and the season. They attempt to attach an importance to the parameter as a minimization
factor in water loss from plants. Yet in their discussion they hardly mention it.
Read through the paper and provide a coherent and illuminating synopsis of what CF are really
trying to say in the last few pages of the article. In particular, you should address yourself to
such issues as (1) the value of such a  parameter; (2) does nature try to optimize water loss and
carbon gain; (3) does their analysis accommodate the idea of midday partial stomatal closure.
The idea here is not to become lost in detail in reading the article but be able to skim through it,
concentrate on the important passages that are relevant to the problem and emerge with a
reasonably clear impression of what the article is about and what it signifies.
Water use efficiency is a handy little ratio that tells us the efficiency of carbon
assimilation with respect to water use. Briefly, in photosynthesis:
6CO2 + 6H2O + light -> C6H12O6 + 6O2
This equation encapsulates the photosynthesis reaction for terrestrial C3 plants. Water is
taken up from the soil and into the leaves (through a water potential gradient driven by
evaporation and osmosis) where some of it is hydrolyzed by Photosystem II, producing a proton
and oxygen. The oxygen is released to the atmosphere by diffusing out of the plants through
openings in the leaves called stomata. About 90% of the water taken up by the plant is directly
transpired as water vapor to the atmosphere through the stomata. Carbon dioxide is diffused into
plants through these same openings and into the chloroplasts where it is broken down with the
aide of that proton (stored in NADPre AND ATP) and an enzyme known as RuBP (Rubisco) in
the Calvin cycle to produce a carbohydrate, namely glucose (along with electron-poor ADP and
NADPox).
Stomatal guard cells open and close in response to an osmotic potential driven by
potassium ions and controlled by plant hormones. If the plant is well-watered, stomata will also
swell with water (to smooth out the osmotic gradient in potassium) and tend to open to allow
carbon to enter and water to escape. When leaf or soil water potential drops (in various ways
and with various levels of debate), stomata deflate and assimilation and evaporation both decline.
Stomata may also be sensitive to light and close at night.
I give this above description only to illuminate the reason that water and carbon are
linked and this linkage is intimate with stomatal control (written out in equations in terms of
resistance or conductance). Cowan and Farquhar (1978) explore the meaning of their parameter
 for plants. This parameter is the optimal water use efficiency for a plant (the minimum slope
of the change in evapotranspiration rate for a given change in the rate of carbon assimilation).
Cowan and Farquhar argue that plants, namely via stomata, minimize their water loss while
maximizing carbon assimilation. They use a model of leaf energy balance to model leaf
evapotranspiration and a light-saturating carbon assimilation model to calculate how this
parameter might be given certain climatic conditions (temperature, radiation, humidity). Then,
they argue that the optimal trajectory is one where a uniform slope of E/A (equal to , the
minimum slope) is maintained, unless it is impossible to do so, in which case assimilation and
transpiration drop to zero (in theory only, since gas exchange can occur with closed stomates).
They argue (at least initially) that plants have evolved to follow a certain , although this
 varies by species and season. It is assumed that uptake at any one point in time is completely
determined by the environmental factors at that instant. The authors note that they do not
consider how uptake in the morning, for example, can alter the uptake later in the day. And
while that may not be a big deal, it certainly is over longer courses of time, since, for example,
the carbon taken up in spring determines the plant size and thus its ability to uptake carbon in the
summer. Finally, they argue that a stochastic climate would preclude a plant from ever fixing on
a certain  value. In other words, their optimization problem has instabilities.
Nevertheless, if one were to know this parameter, and one were to assume plants
followed it, you could have a very simple and straightforward model of carbon assimilation
based not on plant biochemistry, but primarily on environmental conditions and gas exchange. I
wonder if this paper (along with technological improvements in the measuring of gas exchange)
helped lead the revolution in plant ecophysiology over the last several decades toward
observations of gas exchange in the field and away from lab based plant physiology (meanwhile,
models seem to have gone the other way).
I am not sure if nature tries to optimize water loss and carbon gain. It is certainly in
plants best interest (if the plants best interest is to grow and propagate to produce more of itself).
At the end of the paper, Cowan and Farquhar argue that on the evolutionary timescale, stomatal
regulation of gas exchange probably had little to do with the form and function of plants. In
other words, they assume that stomata are always efficient in the environment they are in - they
are optimized for our climate (they are more adaptable than the plants they are on – i.e. because
of feedbacks for example between leaf temperature and stomatal closure). But, climate changes.
There is evidence, for example, that the number of stomata on a plant may change over time with
changing carbon dioxide concentration (Schlesinger, 1997). Many studies show WUE changes
due to stomatal conductance changes with climate and carbon dioxide concentration (Cao and
Woodward, 1998). Some plants have stomata on both side and some on one side. I think
measuring water use efficiency (WUE) is valuable and I can see how one might assume that the
measured WUE is optimal. Law et al. (2002) argue that when vapor pressure deficit is high,
stomata adjust to maintain sustainable water flow and minimize xylem cavitation. Certainly
biological competition (especially in a water stressed environment) would lead to optimizing
WUE over evolutionary timescales. Water is tricky, though. Stomata respond to various deficits
in water: in the soil (soil water potential), leaf (leaf water potential) and the atmosphere (vapor
pressure deficit). This is not a simple one-way optimization problem. Although the three are
correlated, it seems there are other concerns to optimize, too, such as light use efficiency, seed
dispersal and minimizing maintenance respiration. Net primary production (NPP) ultimately
determines total plant growth – you can have all the assimilation you want, but if you want to
grow, you can’t lose it all in the poker table known as aerobic respiration.
Midday partial stomatal closure occurs when transient stress happens in a plant due to
leaf water potential reaching a critical value (as temperature rises, transpiration increases)
(Carlson, 2003). Instead of a collapse of transpiration, however, it instead plateaus due to a
negative feedback response. Stomata partially close, transpiration declines, but leaf temperature
goes up (since less heat can escape) – increasing the vapor pressure deficit between leaf and
atmosphere, and so transpiration increases. The two effects can nearly cancel, so transpiration
plateaus in the middle of the day when stressed. However, this effect is not seen in carbon
assimilation, since there is no such feedback for carbon dioxide (there is a small effect – namely
internal CO2 concentration declines, but it is not enough). Thus plant water use efficiency
changes in the middle of the day (E/A increases). I am not sure if their model accounts for this
change. Some of the figures certainly show a decline in assimilation in the middle of the day.
They argue that their model does account for this midday decline that occurs as the humidity
gradient from leaf to atmosphere increases. They show that the increase in leaf resistance with
increases VPD is small (at large E/A), but transpiration also increases, and the feedback
mechanism described above occurs. However, this doesn’t work in their model when E/A is
small. The reason their model captures this effect is in the energy balance equation (equation 4).
They do question the physiological effectiveness of this stomatal closure – since there are other
feedbacks (changes in internal carbon dioxide) – and sometimes multiply constrained
optimization problems do not produce a consistent solution.
Overall, this paper, appears to be an important article in the ecophysiology and
bioclimatology literature. I think that understanding carbon assimilation using a synthesis of the
gas exchange point of view and a biochemical point of view allows for the greatest
understanding of the interaction of stomata and carbon and water. Cowan and Farquhar make a
strong theoretical case for attempting to understand the use of water use efficiency and its ability
to predict carbon assimilation and evapotranspiration, and paving the way for models of the like
of Farquhar and Sharkey (1982) and Ball et al. (1986).
4. Drainage flow.
(a) Drainage flows, as you know, are thought to cause problems at eddy covariance flux towers
used to monitor ecosystem-atmosphere exchange of CO2. First, explain why this is believed to
be a problem (observational and/or theoretical evidence), and explain the implications of this
problem for a tower, such as the Sylvania site, which is attempting to observe gross and net
ecosystem-atmosphere CO2 fluxes with a fairly high degree of precision and accuracy.
Drainage flows occur at night under weak winds and typically clear skies. In these
conditions, the stable night atmosphere produced a strong stratification (temperature increases
with height). Cold (i.e. colder than surroundings) air would thus experience downward
buoyancy forces. Over even small slopes, this can lead to a downslope flow of cold air.
Wyngaard (2002) shows how for mean momentum balance in the stable layer, anything larger
than a very small slope (1m per kilometer – typical of flat parts of the southern great plains) can
alter this momentum balance. Mahrt et al. (2001) provides observations of a wide range of
scales of nighttime and evening drainage flow in the southern great plains. Flow down a slightly
sloping gully developed even with a strong wind at upper levels (i.e. a nocturnal jet).
Turbulence, meanwhile, can be very intermittent at night, since much of the stable boundary
layer is near the critical Richardson number where turbulence cannot continue against
destruction of buoyant eddies. Drainage flows are typically not a problem in the convective
daytime boundary layer since buoyancy and shear production are large.
A drainage flow develops due to the effect of gravity and friction on the sinking air
(Stull, 1988). Near the ground, drag lowers the wind speed, while a maximum is reached just
above the ground where the air is still cold. Above this layer, shear at the top of the gravity
induced flow can generate turbulence. The depth of a drainage flow can increase as the air flows
downhill and speed can also increase. Typical speeds are 0.5 to 3.5 m/s. Mean winds can
enhance or suppress this flow. Turbulent modeling schemes can be used to model these drainage
flows (Heilman and Dobosy, 1985).
Sinking cold air draining out the sides is a problem for measuring fluxes, due to the
problem of not measuring advection (and even if you did, the depth of this drainage may be too
small). The Reynolds-decomposed and ensemble-averaged equation for the conservation of a
scalar, neglecting viscous dissipation is:
C
C
C uc  wc
U
W


0
t
X
Z x
z
(Wyngaard, 2002), where capital letters are ensemble averages and lower case letters are
turbulent deviations from the average.
Using the ergodic hypothesis, we use time averages to stand-in for ensemble averages.
We also use Taylor’s frozen field hypothesis to be able to have point measurements represent an
area. To make flux measurements, we then make two major assumptions: quasi-steadiness (over
our averaging period) and local homogeneity (in the horizontal within the flux footprint). These
assumptions allow us to measure the variations of vertical velocity and scalar above the surface
at one point (at a height high enough for a large footprint and fully developed turbulence, but low
enough to assume a linear flux divergence profile) along with a scalar vertical profile to compute
carbon atmosphere-ecosystem exchange (neglecting advection terms):
C
dz
t
sfc
z
NEE  ( wc) sfc  ( wc) z 

(Yi et al., 2000).
This is essentially a one-dimensional assumption for fluxes. Generally, this is an okay
assumption since 1.) average turbulence should be homogenous (random) over an area and 2.)
even with a strong mean wind, in homogenous terrain, the gradient of mean CO2 with horizontal
space should be negligible. We can think of the NEE being measured at a point and a column
below it. However, at nighttime, under cold air drainage conditions, carbon dioxide respired
from the surface would advect in the horizontal downslope, never being measured by the two
terms. If there was no slope, there would be no problem, since carbon lost by horizontal wind
would be replaced by carbon respired upwind, which eventually, during bouts of turbulence
(when shear is high), will be mixed up into the sensor footprint. With a slope, however, all the
respired carbon will slowly settle down into the lowest points (and most flux towers are on high
points to minimize tower height and avoid wetland disturbance). When turbulence occurs, this
carbon could be advected upwards but not near the sensor. Even if it was measured by the
sensor, the results would be a very large flux (essentially a spike), which may explain odd fluxes
seen at one of our hardwoods sites (Cook et. al., in prep.)
When the momentum flux is large (and thus friction velocity, u* is high), downslope
advection of carbon at night should not be a problem. The turbulence produced by shearproduction (divergence of shear in the vertical) would mix respired carbon vertically before
being advected downslope, assuming the depth of this advection is small. Plots of friction
velocity versus nighttime flux at Sylvania show a loss of respired flux (lower than the mean flux
over a time period) at low u*, suggesting drainage flow is a problem at our site when u* is less
than 0.3 (Desai et al., in prep.). Thus, we throw out fluxes measured during low friction velocity,
under the assumption of drainage flow. To get accurate long-term net ecosystem exchange
values, we have to fill these values using biophysical models and correlated variables of carbon
exchange at night.
(b) Find or derive a means of estimating the speed of a drainage flow as a function of the slope.
Feel free to refer to an expression derived in the literature.
I would expect that the drainage flow would be a function of the strength of the buoyancy
(either by temperature/density gradient or heat flux), gravity, the slope. It would probably also
be a function of any mean wind speed, but I think it would be easier at first to neglect the mean
wind (i.e. assume it is zero). I think we don’t need to consider the strength of the shear stress (or
possibly the Richardson number), since basically we are assuming a drainage flow is already
occurring and we mainly want to know its speed (and depth).
Flipping through one of my favorite books (Stull, 1988), I find that our friend Larry
Mahrt did all the hard work for us 20 years ago (Mahrt, 1982). He writes an equation for the
downslope component of the wind (where x and z are parallel and perpendicular, respectively, to
the slope) as:
U
U
U
U

g  hd
u ' w '
U
V
W
g
sin( )  cos( )
 f cV 
t
x
Y
Z

 x
z
where the terms are acceleration, downslope advection, cross slope advection, vertical advection,
buoyancy, thermal wind, coriolis, and stress divergence.  is slope angle (with respect to a
horizontal plane), hd is the depth of the flow,  is potential temperature difference between the
cold air and air above it, and  with two bars is the average potential temperature depression of
the flow. This is pretty much the equation of motion with a “hidden” pressure term (the thermal
wind term) and with a buoyancy term (Boussinesq-like).
For gravity driven advective flow, only the downslope advection and the buoyancy terms
are important, as I previously conjectured. Thus, you can easily solve for the mean velocity over
the depth of the flow as:
 g

U 
 x  sin( ) 
 

1
2
(Mahrt, 1982). Stull also presents a solution that relies instead on heat flux, but for this exam, I
think the above equation suffices. We see that drainage flow increases with the mean difference
in temperature (larger buoyancy) between the surface flow and the flow above it, with the slope,
and the total length of the slope. It decreases with increasing temperature.
(c) Based on your observational knowledge of the nocturnal surface layer and the expression
from (b), estimate a typical drainage flow speed that you might find around the Sylvania tower.
Looking at the topographic contour map I have for Sylvania and assuming the drainage
flows occur most often in the direction of the steepest gradient (from tower to the lake), I assume
that the mean slope, very roughly, is about 20 meters down over 200 meters. This is in fact, a
pretty steep slope, which is typical of one direction (it’s a workout on your knees to get down to
the lake), but not typical of all directions. But since we want to estimate the worst case scenario
here, I will use this slope, which comes to about 5.7 degrees. Based on a quick look at the
nighttime temperature gradient in July 2003 at Sylvania, I will use a temperature depression of
around 3 degrees K and a mean temp of around 294 K (so I can get nice round numbers!). I get a
mean velocity of 1.4 m/s. These seems pretty reasonable.
(d) Given this velocity (assume something you believe is reasonable if either you did not get very
far with (b) and (c), or you don’t believe the results), estimate the impact of such a flow on a
nocturnal measurement of ecosystem respiration from an eddy covariance tower such as
Sylvania. Put your results into context by comparing to typical hourly, daily and annual net
fluxes observed at Sylvania.
The nighttime flux at Sylvania during turbulent conditions during the growing season is
somewhere between 5-10 mol CO2/m2/s (let’s pick 5). Typical nighttime carbon dioxide
mixing ratios near the surface (summer) when drainage is negligible is around 450 ppm. The
typical depth of the drainage flow (according to an equation from Stull) would be around 1.5 m
deep with a 1.4 m/s velocity. I use a 30 minute averaging time for fluxes. Actually, after trying
to do the math, I’m a bit stumped at how to get a carbon dioxide mixing ratio gradient (to
calculate advection due to drainage) with these numbers. (My approach was to try to predict the
CO2 concentration measured near the surface when there is drainage using a box model).
Instead, I’ll look at a specific day: July 24, 2003. On that day, mixing ratio hit around 450 ppm
in the middle of the night when turbulence was strong (high u*) and the NEE was 10 mol/m2/s.
Earlier in the evening, the flux was small (almost 0), u* was very small (almost 0), and the
mixing ratio at the surface was instead 410 ppm.
So, let’s say that due to drainage flow, the carbon dioxide mixing ratio at the low point is
450 ppm (representing the mixing ratio developed when the carbon flux stayed where it was
supposed to stay). Then the advection term U * dC/dx is around .3 ppm CO2 m/s, which
translates to about 12 mol CO2/m2/s, which is quite large, but with large uncertainty due to my
methodology. Still, this would suggest that during drainage, NEE would be around 0, which is
what is observed. This is significant on an hourly basis. If we assume nighttime drainage occurs
about half the time (more than actually occurs according to the data) during a typical night
(mostly in the evening, less in the night as turbulence or a nocturnal jet may form, maybe more
again toward morning), then the average nighttime flux would be reduced by 50%. If I assume
that nighttime fluxes are typically 10 mol CO2/m2/s and daytime fluxes are typically -20
mol/m2/s, then the mean flux (12 hours of each) is around -5 mol CO2/m2/s. With drainage
causing 50% of the nighttime flux to go to 0, net daily flux is -7.5 mol CO2/m2/s, 50% larger.
If we assume again that this drastic drainage occurs when the wind is only from the opposite
direction (the southwest), which occurs about 40% of the time, and the growing season in 100
days, this could make annual NEE values off by 100 mol CO2/m2/s (which is about 101
gC/m2/yr), not insignificant (about 20% larger than predicted based on this simple thought
experiment). This is a bit more offset than I actually see (though not far off). Drainage causes
somewhere around 50 gC/m2/yr lost nighttime carbon based on analysis of Sylvania data. Since
this forest is almost in carbon balance, this uncertainty is relatively large, and larger than random
errors due to eddy covariance method.
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