ESDEarth System DynamicsESDEarth Syst. Dynam.2190-4987Copernicus PublicationsGöttingen, Germany10.5194/esd-9-1127-2018Diurnal land surface energy balance partitioning estimated from the thermodynamic limit of a cold heat engineThermodynamics and energy balance partitioningKleidonAxelakleidon@bgc-jena.mpg.dehttps://orcid.org/0000-0002-3798-0730RennerMaikhttps://orcid.org/0000-0002-2992-8414Biospheric Theory and Modelling Group, Max-Planck-Institut für Biogeochemie, Jena, GermanyAxel Kleidon (akleidon@bgc-jena.mpg.de)21September2018931127114017April201823April20189August201829August2018This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://esd.copernicus.org/articles/9/1127/2018/esd-9-1127-2018.htmlThe full text article is available as a PDF file from https://esd.copernicus.org/articles/9/1127/2018/esd-9-1127-2018.pdf
Turbulent fluxes strongly shape the conditions at the land surface, yet they
are typically formulated in terms of semiempirical parameterizations that
make it difficult to derive theoretical estimates of how global change
impacts land surface functioning. Here, we describe these turbulent fluxes as
the result of a thermodynamic process that generates work to sustain
convective motion and thus maintains the turbulent exchange between the land
surface and the atmosphere. We first derive a limit from the second law of
thermodynamics that is equivalent to the Carnot limit but which explicitly
accounts for diurnal heat storage changes in the lower atmosphere. We
call this the limit of a “cold” heat engine and use it together
with the surface energy balance to infer the maximum power that can be
derived from the turbulent fluxes for a given solar radiative forcing. The
surface energy balance partitioning estimated from this thermodynamic limit
requires no empirical parameters and compares very well with the observed
partitioning of absorbed solar radiation into radiative and turbulent heat
fluxes across a range of climates, with correlation coefficients r2≥95 %
and slopes near 1. These results suggest that turbulent heat fluxes on
land operate near their thermodynamic limit on how much convection can be
generated from the local radiative forcing. It implies that this type of
approach can be used to derive general estimates of global change that are
solely based on physical principles.
Introduction
The turbulent fluxes of sensible and latent heat play a critical role in the
land surface energy balance during the day as these fluxes represent the
principal means by which the surface cools and exchanges moisture, carbon
dioxide and other compounds with the atmosphere. Due to their inherently complex nature, these fluxes are typically described by semiempirical
expressions e.g.,.
Yet representations of this exchange in land surface and climate models are
still associated with a high degree of uncertainty. This uncertainty results,
for instance, in biases in evapotranspiration and surface temperatures across
different models , in empirical relationships
of land surface exchange outperforming land surface models
, and in biases in boundary layer heights
. The semiempirical and highly coupled nature of land–atmosphere exchange seems to make it almost impossible to derive
simple, physically based estimates of the magnitude of turbulent exchange and
how it changes with land cover change or global warming.
Schematic diagram of the land–atmosphere system where turbulent
heat fluxes from the surface, Jin, act as the driver of an
atmospheric heat engine that generates convective motion and which sustains
the heat fluxes. The heat source of the engine is the absorption of solar
radiation at the surface, Rs, reduced by the net exchange of
terrestrial radiation, Rl,net, which depends on surface
temperature. The two critical effects that set the limit on how much work the
engine can perform are illustrated in panel (b): diurnal changes in heat
storage in the lower atmosphere due to the diurnal variation of solar
radiation and the reduction in surface temperature, Ts, due to
greater turbulent heat fluxes both lower the work output of the
engine.
An alternative approach to describing surface–atmosphere exchange can be based
on thermodynamics , an aspect that is
rarely considered in the description of surface–atmosphere exchange. In this
approach, turbulent exchange is formulated as a thermodynamic process by
which turbulent heat fluxes drive a convective heat engine within the
atmosphere that does the work to maintain convection and thus the turbulent
exchange near the surface. This approach specifically invokes the second law
of thermodynamics as an additional constraint on atmospheric dynamics
similar to previous approaches, such as the maximization of
material entropy production (MEP); e.g.,. The
second law sets a limit on how much work can be derived from the local
radiative forcing of the system. The dynamics associated with convection are
then essentially captured by the implicit assumption that convection works as
hard as it can, so that the use of the thermodynamic limit approximates the
emergent convective dynamics. Previous applications of this thermodynamic
approach have shown that it can successfully describe the broad
climatological variation of surface energy balance partitioning on land and
ocean , the strength and sensitivity of
the hydrologic cycle and surface temperatures to global change
,
and the dynamics of the Earth system in general .
Here we extend this approach to the diurnal variation of the surface energy
balance on land and compare its estimated partitioning to observations across
different climates. As in the previous applications of thermodynamics to
land–atmosphere exchange, the starting point is to view turbulent fluxes as
the result of a heat engine that is driven by these heat fluxes (Fig. ).
The limit on how much work this heat engine can
maximally perform is set by the first and second law of thermodynamics, from
which the well-known Carnot limit of a heat engine can be derived e.g.,.
When applied to the setting of the diurnal cycle of the land–atmosphere
system, two key aspects need to be considered as these shape the
thermodynamic limit (as illustrated by the two
boxes in Fig. b). First, the strong diurnal variation of solar radiation
causes strong changes in heat storage within the system that result in a much
less varying emission of terrestrial radiation to space. In the absence of
such heat storage changes, nighttime temperatures would be much lower than
those found on Earth. In the ideal case that is being considered here, the
strong variation of solar radiation is completely leveled out to yield a
uniform emission of radiation to space, as indicated by the blue line in the
graph at the top of Fig. labeled Rl,out. While these
heat storage changes predominantly take place below the surface for open
water surfaces such as the ocean and lake systems reflected in nearly
uniform turbulent fluxes during night and day; see, e.g., measurements
by, the land–atmosphere system accommodates these changes
mostly in the lower atmosphere because heat
diffusion into the soil is slow (e.g., ). The relevance of this
different way of accommodating heat storage changes over land is that it takes
place within the heat engine that we consider. The heat storage change is
associated with a heating of the engine during the day, which represents an
additional term in the entropy balance of the engine. What we show
here is that the resulting thermodynamic limit is somewhat different to the
common Carnot limit. We refer to this limit as the Carnot limit of a
cold heat engine. Our motivation to refer to this limit as the limit of a
cold heat engine is the behavior of a cold car engine in winter. When the
car engine is still cold just after it has been started, one needs to hit the
gas pedal harder to get the same power. As we will show below, the expression
we derive here shows the same effect, that is, that a heat gain inside the
engine reduces the work output of the engine. We will show that this enhanced
heat flux is consistent with observations, so that this effect of heat
accumulation during the day is an important factor that shapes the magnitude
of turbulent fluxes on land.
The magnitude of the diurnal variation in heat storage is well constrained
when assuming that the radiative heating by solar radiation and the emission
to space are roughly balanced over the course of day and night. The temporal
change in heat storage during the day can then be inferred from the imbalance
of radiative fluxes at the top of the atmosphere indicated in the
upper panel of Fig. b, and as described by.
The second aspect that shapes the thermodynamic limit is the reduction in
surface temperature in the presence of greater turbulent fluxes at the
surface (lower panel at the right of Fig. ). This
reduction in surface temperature reduces the temperature difference that is
utilized by the heat engine to derive power, thus setting a limit of maximum
power of the heat engine as in, e.g.,. (This
maximum power limit is very closely related to the proposed principle of
maximum entropy production (MEP), as maximum power equals maximum dissipation
in steady state, and entropy production is proportional to dissipation. An
example of the application of MEP to convection is given by
.) We then combine the thermodynamic limit of a cold
heat engine with the energy balances of the surface and of the whole
surface–atmosphere system and maximize the power output of the heat engine
to get a fully constrained description of the system that can, in first
approximation, be solved analytically. It yields a description of the
turbulent exchange between the land surface and the atmosphere that is fully
constrained by thermodynamics and free of empirical turbulence parameterizations.
In the following, we first derive the thermodynamic limit of a cold heat
engine, combine it with the energy balances of the system and maximize the
power output to estimate surface energy balance partitioning based on the
solar forcing of the system. The estimated partitioning is then tested with
observations across field sites of contrasting climatological conditions. We
then discuss how our thermodynamic approach compares to the common approaches
in boundary layer meteorology and consider the utility of our approach for future work as well as potential implications.
Thermodynamic formulation of the land surface energy balance
We consider the land–atmosphere system as a thermodynamic system in a
steady state when averaged over the diurnal cycle. Surface heating by
absorption of solar radiation, Rs, causes the surface to warm, while the
atmosphere is cooled by the emission of radiation to space, Rl,out
(Fig. ). The surface and atmosphere are linked by the net
exchange of terrestrial radiation, Rl,net, and turbulent heat fluxes,
Jin, that result from convective motion. We consider this system to be a
locally forced system with no advection. Convective motion within the
boundary layer is seen as the consequence of a heat engine that generates
motion out of the turbulent heat fluxes, where, for simplicity, we do not
distinguish between the effects of the sensible and latent heat flux and the
associated forms of dry and moist convection. The steady-state condition is
used for the radiative forcing of the whole system by requiring that the mean
radiative fluxes taken over the whole day are balanced such that
Rs,avg=Rl,out (with Rs,avg being the average of Rs).
Furthermore, we assume that the generation of turbulent kinetic energy, or
power G (or work per time), and its frictional dissipation, D, are in
balance, so that G=D. In the following, we derive the limit on how much
power can be derived from the forcing of the system directly from the first
and second law of thermodynamics in a general way, so that we do not need to
make the assumption that the atmosphere operates in a Carnot-like cycle. All
variables used in the following are summarized and described in Table .
Carnot limit with heat storage changes
We first derive a thermodynamic limit akin to the Carnot limit from the
energy and entropy balances of the heat engine, which specifically includes
the change in heat storage within the engine. The first law of thermodynamics
applied to this setup is given by
dUedt=Jin+D-Jout-G,
where dUe/dt is the change in heat storage within the heat
engine, Jin represents the addition of heat by the turbulent heat
fluxes from the surface and Jout is the rate by which the heat engine is
being cooled, which is accomplished by radiative cooling. Note that this
formulation differs from the derivation of the Carnot limit by accounting for
changes in internal energy on the left-hand side and for dissipative heating,
D, on the right-hand side as frictional dissipation takes place within the
system. As we consider a steady state with G=D, note that the
contributions of these terms in Eq. () cancel out so that
the equation reduces to dUe/dt=Jin-Jout. Also
note that at this point, we neglect the effects of radiative energy transport
from the surface to the atmosphere that would contribute to dUe/dt in the
application to the surface–atmosphere system. As it turns out, this
contribution by radiation does not alter the limit, as shown in Appendix .
Variables and parameters used in this study.
SymbolVariableUnitsUse or assumptionDFrictional dissipationW m-2Assumed to be in steady state, with D=GGConvective powerW m-2Eqs. (), () and ()JinTurbulent fluxes of sensible and latent heatW m-2Eqs. (), () and ()JoptTurbulent fluxes Jin optimized to yield max. powerW m-2Eq. ()JoutCooling rate of the heat engineW m-2Eqs. () and ()kRadiative parameterization constantW m-2 K-1Used in linearization of Rl,netRl,outFlux of terrestrial radiation to spaceW m-2Assumed to be in steady state, with Rl,out=Rs,avgRsSurface absorption of solar radiationW m-2ForcingRs,avgSurface absorption of solar radiation (average)W m-2Eq. ()TaAtmospheric temperatureKAssumed to be the radiative temperatureTeTemperature of the heat engineKAssumed to be similar to the surface temperatureTsSurface temperatureK–dUa/dtChange in atmospheric heat storageW m-2Eq. ()dUe/dtChange in heat storage within heat engineW m-2Eqs. ()–()(assumed to be the same as dUa/dt in Sect. )dUs/dtChange in ground heat storageW m-2Prescribed from observations, Eq. ()(or ground heat flux)dUtot/dtChange in total heat storageW m-2Eq. ()
The associated entropy budget of the heat engine is given by a change in
entropy associated with the change in heat storage, dUe/dt, at an
effective engine temperature Te, the entropy input by Jin at a
temperature Ts, the entropy export by Jout at a temperature Ta,
frictional dissipation that is assumed to occur at temperature Te, and
possibly some irreversible entropy production σirr within the engine,
i.e. irreversible losses that are not accounted for by the frictional dissipation term,
D/Te:
1TedUedt=JinTs+DTe-JoutTa+σirr.
Note that this entropy budget is the entropy budget for thermal entropy, not
for radiative entropy. This is an important distinction. A contribution by a
radiative flux, e.g., a flux Rl,out/Ta, represents a flux of radiative
entropy (and would require an additional factor of 4/3 as it deals with
radiation); i.e., it is entropy reflected in the composition of radiation but not associated with the thermal motion of molecules that describes heat
or thermal energy. As we deal with a convective heat engine, we must not
include radiative terms as such but only when radiation is absorbed and
heats air and water (adds thermal energy) or when the net emission of
radiation cools (removes thermal energy). Radiative terms and radiative
entropy production are typically much larger in the Earth system than
non-radiative contributions (easily by a factor of 100, e.g.,
). Yet any form of motion is associated with the much
smaller but relevant thermal entropy terms.
For the atmospheric temperature, Ta, we use the radiative temperature
associated with Rl,out (i.e., we use the Stefan–Boltzmann law,
Rl,out=σTa4, to infer Ta, with σ=5.67×10-8 W m-2 K-4
being the Stefan–Boltzmann constant). This is the most
optimistic temperature for the entropy export from the heat engine as it is
the coldest temperature possible to emit radiation at a rate Rl,out to
space, and it thus represents the highest entropy export from the heat engine
(note that blackbody radiation represents the radiative flux with maximum
entropy). Note also that this temperature is not bound to a particular height
within the atmosphere but is instead inferred from the energy balance
constraint. The effective engine temperature, Te, essentially represents
the potential temperature of the lower atmosphere as the temperature
variation within the lower atmosphere is shaped by convection and is thus
approximately adiabatic.
The thermodynamic limit on how much power, G, can maximally be derived by
the engine is obtained from the entropy budget using the ideal case in which
σirr=0 (the second law of thermodynamics requires σirr≥0).
This ideal case implies that the only source of entropy production is the
frictional dissipation term, D/Te (cf. Eq. 2). Using Eq. () to replace Jout in Eq. (), we obtain
G=Jin⋅TeTs⋅Ts-TaTa-dUedt⋅Te-TaTa.
In this expression, the temperature of the heat engine, Te, plays an
important role. In the limiting case of Te≈Ta, this expression
reduces to the common Carnot limit as the effect of the change in heat
content is indistinguishable from the waste heat flux, Jout, of the heat
engine. As the engine temperature essentially represents the potential
temperature of the lower atmosphere, it is much closer to the surface
temperature, so that the approximation Te≈Ts is better justified.
With this approximation, the thermodynamic limit of power then reduces to
G≈Jin-dUedt⋅Ts-TaTa.
In the absence of heat storage changes, the term dUe/dt vanishes
and yields, again, the common Carnot limit, except that Ta appears in the
denominator of the Carnot efficiency rather than Ts, an aspect that has
previously been derived in the context of a “dissipative” heat engine
. Note that in the presence of
positive heat storage changes, as is the case during the day, the maximum
power that can be derived from the heat flux Jin is reduced. That is,
the increase in heat storage within the engine (dUe/dt>0) results
in a lower efficiency in converting heat into power (with the efficiency
given by the ratio G/Jin), consistent with our explanation in the
Introduction of why we refer to this effect as that of a cold heat engine.
Energy balance constraints
We next use the energy balance constraints of the surface and the whole
system to express dUe/dt and Ts-Ta in terms of the absorption
of solar radiation at the surface, Rs, and the turbulent heat flux Jin.
This will allow us to replace these two terms in
Eq. (), so that the power G only depends on Rs
and Jin. Note that we refer to the atmospheric heat storage
change, dUa/dt in the following rather than the engine heat storage
change, dUe/dt. The difference is that when we apply the thermodynamic
limit on the atmosphere, the heat storage is also affected by the net
exchange of longwave radiation, which adds another term to the energy and
entropy budget but which does not go through the engine as a heat flux.
However, the resulting limit remains unaffected, as shown in
Appendix .
The surface energy balance constrains the relationship between the heat
flux Jin and the temperature difference that drives the heat engine,
Ts-Ta. We express this balance by
Rs-kTs-Ta-Jin-dUsdt=0,
where we linearize the net longwave radiative exchange, Rl,net=k(Ts-Ta),
between the surface and the atmosphere and where dUs/dt describes
heat storage changes below the surface, which is represented by the ground
heat flux. This formulation of the surface energy balance can be used to
express the temperature difference, Ts-Ta, as a function of Rs,
Jin, and heat storage changes below the surface, dUs/dt.
The energy balance of the whole system, neglecting heat advection terms,
yields a constraint of the form
dUtotdt=dUadt+dUsdt=Rs-Rl,out=Rs-Rs,avg,
where dUtot/dt is the total change in heat storage within the
surface–atmosphere system. We assume this balance to be in a steady state
when averaged over day and night, so that on average, Rl,out=Rs,avg,
where Rs,avg is the temporal mean of Rs taken over the
whole day. The energy balance of the whole system provides an expression for
dUa/dt as a function of the instantaneous value of absorbed solar
radiation, Rs, the mean absorption of solar radiation, Rs,avg, and
the ground heat flux, dUs/dt.
Maximization of convective power
The surface energy balance (Eq. ) can now be used to
express the temperature difference that drives the heat engine, Ts-Ta,
in the thermodynamic limit given by Eq. (), while the
energy balance of the whole system (Eq. ) can be used to
constrain the terms describing the changes in heat storage, dUa/dt. As the
power G is an increasing function of Jin, but the temperature
difference declines with greater values of Jin, the power has a maximum,
which is referred to as the maximum power limit. This limit can be derived
analytically by ∂G/∂Jin=0 and is associated with an
optimum heat flux of the form
Jopt≈12Rs-dUsdt+dUadt.
This expression is consistent with previous work where the optimum
heat flux is given by Jopt=Rs/2 in the absence of heat storage
changes . It is, however,
modulated by heat storage changes, and it matters whether these changes take
place below the surface or in the lower atmosphere as the two forms of heat
storage change enter Eq. () with a different sign.
Diurnal changes in heat storage are reflected in variations of soil
temperature near the surface and in variations of air temperature and
humidity in the lower atmosphere. Panel (a) shows a schematic diagram of
these heat storage changes. It shows a typical, colder nighttime profile
with an inversion near the surface and a warmer daytime profile. The
difference between the extremes of these temperature (and humidity) profiles
(area shaded in light red) corresponds to the change in diurnal heat storage
change in the lower atmosphere, dUa/dt. Typical changes in
belowground temperature profiles are also shown, with the heat storage change
dUs/dt being marked in dark red. Panel (b) shows
observations from Lindenberg, Germany, for the mean diurnal variation of
absorbed solar radiation (shifted by its mean),
Rs-Rs,avg, averaged for the month of June over the
years 2006–2009 (red line, n=480), the diurnal variation in heat storage
in the lower atmosphere derived from 6-hourly radio soundings,
dUa/dt (blue boxes represent the interquartile range and the
horizontal thick blue line the median) and the ground heat flux,
dUs/dt (orange line).
We next consider the two limiting cases. The first limit is when the heat
storage changes take place primarily below the surface, like an open water
surface of a lake. In this case, dUs/dt≈dUtot/dt (and
dUa/dt≈0), and the optimum heat flux reduces to
Jopt≈Rs,avg2.
The other limiting case is when the heat storage changes take place above the
surface. Then, dUa/dt≈dUtot/dt (with
dUs/dt≈0), and the optimum heat flux is
Jopt≈Rs-Rs,avg2.
This expression implies that the optimum value of the turbulent heat flux
varies directly with the absorbed solar radiation, Rs, but has a constant
offset given by half of the mean absorption, Rs,avg/2. This offset
should be a comparatively small value of about 80–100 W m-2, given a
global mean value of surface absorption of solar radiation of 165 W m-2. Note that the power, however, does not
differ between the two cases and yields the same value of Gmax=(Rs,avg/2)⋅(Ts-Ta)/Ta.
Hence, the information on absorbed solar radiation (and the ground heat flux to
account for dUs/dt) is sufficient to estimate surface energy balance
partitioning from the thermodynamic limit of maximum power.
Evaluation of the approach
Evaluating our estimate requires observations of absorbed solar
radiation during the day, Rs, and the ground heat flux, dUs/dt. From
the diurnal course of Rs, the mean value of Rs,avg can be calculated,
which in turn yields an estimate for dUtot/dt. Taken together with the
ground heat flux, this yields the value of dUa/dt, so that all terms in
Eq. () can be specified. The resulting estimate
of Jopt can then be compared to observations of the turbulent heat fluxes or to the available energy, i.e., net radiation reduced by the ground heat flux.
Data sources
We use two types of data sources to test our approach. To test how reasonable
the estimates are for the diurnal heat storage changes in the lower
atmosphere, we first use 6-hourly radiosonde data from the DWD
meteorological observatory Lindenberg in Brandenburg, Germany (data available
at http://weather.uwyo.edu/upperair/sounding.html, last access: 16 April 2018). These observations
allow us to derive an estimate of the diurnal variations in temperature (and
moisture) in the lower atmosphere and thus of dUa/dt
(Fig. a). We use data from this site because this
observatory provides a long and consistent record of four vertical profiles a
day as well as surface energy balance components, while typically only two
vertical profiles a day are taken during routine radiosonde measurements. We use observations from June for the
years 2006 to 2009 and calculate the moist static energy at each 6 h interval and then take the difference over the time interval to obtain
estimates for changes in atmospheric heat storage. These differences are then
compared to the change in atmospheric heat storage expected from solar
radiation, as described by Eq. ().
Site description of the six sites used for evaluating the estimations
of the maximum power limit (with the letters referring to the graphs shown in
Fig. ). Also shown are the correlation statistics of the comparison
to observations. The adjusted squared explained variance of the linear regression
of Jopt to observed net radiation (Rn=Rs-Rl,net)
minus ground heat flux Rn-dUs/dt is reported
as r2. Standard error of slope and intercept of the regression are derived
by a pre-whitening procedure to reduce the effect of serial correlation of the
residuals .
SiteDescriptionr2SlopeInterceptReferenceATundra (open shrubland), USA0.9721.138-54.80Anaktuvuk River (unburned site)±0.019±5.1710.17190/AMF/124614468∘56′ N, 150∘16′ WData used for June 2009, n=1392BCropland, USA0.9931.106-73.30, ARM Southern Great Plains site±0.015±5.0410.17190/AMF/124602736∘36′ N, 97∘29′ WData used for June 2009, n=1060CTemperate grassland, Germany0.9821.099-36.38DWD Falkenberg boundary layer site±0.012±3.5652∘10′ N, 14∘7′ EData used for June 2009, n=1440DPine forest, Germany0.9821.023-37.87DWD Falkenberg boundary layer site±0.011±4.3452∘11′ N, 13∘57′ EData used for June 2009, n=1438EPine forest, Israel0.9981.086-53.87Yatir Forest±0.006±2.2231∘20′ N, 35∘3′ EData used for June 2006, n=1440FTropical rain forest, Brazil0.9990.995-59.82Santarem km83 logged forest±0.003±1.2310.17190/AMF/12459953∘1′ S, 54∘58′ WData used for June 2002, n=1053
We then use observations of absorbed solar radiation (Rs) and the ground
heat flux (dUs/dt) at six field sites in highly contrasting climatological
settings (listed in Table ) to calculate the turbulent
heat fluxes from maximum power (Eq. ). The six sites
include a grassland and a forested site at Lindenberg, Brandenburg, Germany
; three AmeriFlux sites (a tundra site at Anaktuvuk
River, Alaska ; a grassland site at Southern Great
Plains, Oklahoma ; and a tropical
rain forest site at Tapajos National Park, Brazil, );
and a site in a planted pine forest at Yatir Forest in Israel
. For each site, we use 1 month of observations for a summer period in which solar radiative heating of
the surface is highest and the effects of heat advection are minor; we estimate
turbulent fluxes associated with maximum power (using Eq. ) and compare these to the observed fluxes.
Results
We first evaluate the extent to which diurnal variations in solar radiation
are buffered by heat storage changes in the lower atmosphere. To do so, we
use the diagnosed variations of moist static energy from the radiosoundings
in Lindenberg, Germany and compare these to the mean variation in absorbed
solar radiation at the surface as well as variations in the ground heat flux
at the site in Fig. b. The comparison shows that
the heat storage variations in the lower atmosphere are substantially greater
than the ground heat flux so that the diurnal variations in solar radiation
are mostly buffered by the lower atmosphere. Although there is considerable
variation (as indicated by the blue boxes), mostly due to pressure
changes and advective effects, these variations follow the temporal course of
what is expected from the variation in absorbed solar radiation (as described
by Eq. ). This confirms our conjecture that the diurnal
variations in solar radiation on land are buffered primarily by heat storage
changes in the lower atmosphere. This buffering of the diurnal variations
over the land surface is rather different to how an open water surface
buffers these variations (as also shown by observations;
; this is an aspect used previously to explain the difference in climate
sensitivity of land and ocean surfaces; ).
Mean diurnal cycle of the absorption of solar radiation at the surface
(Rs, red line, observed), ground heat flux (dUs/dt,
orange line, observed), and turbulent heat fluxes estimated by maximum power
(Jopt, black line, estimated) and observations (Jobs,
black circles, observed) for a selected month for six field observations in
(a) a tundra ecosystem in Alaska, (b) a cropland in the
midwestern US, (c, d) a grassland and pine forest in a temperate
environment in Germany, (e) a planted pine forest in an arid
environment in Israel, and (f) a tropical rain forest in the humid
Amazon Basin in Brazil. The comparison of the turbulent heat fluxes estimated
from maximum power to energy balance measurements is shown for 30 min
observations in the right panel for each site for two cases of thermodynamic
limits that differ by their consideration of heat storage changes (dark blue:
with storage, as in Eq. (); light blue: without
storage, i.e., dUs/dt=0 and dUa/dt=0 so that
Jopt=Rs/2). More information on the sites and the correlation
statistics are provided in Table .
The comparison of the estimated surface energy balance partitioning from
maximum power to observations at the six sites is shown in Fig. .
The correlations are summarized in Table in terms of
the correlation coefficient as well as the
slope and intercept. During nighttime, there is a mismatch between our
approach and observations, which is represented by the intercept shown in
Table . This mismatch may be explained by the prevalent
stable nighttime conditions in which the atmosphere does not act as a heat
engine, an aspect that we did not consider in our approach. During daytime,
we find very high correlations of above 95 % between the estimated turbulent
fluxes from the maximum power limit with observed net radiation (reduced by
the ground heat flux), with a very good match of the estimated slopes in the
correlation within 15 % of the observed. This high level of agreement is
found across the range of climatological settings shown in Fig. .
Also note that the maximum power limit without an explicit consideration of
heat storage changes (i.e., with dUs/dt=0 and dUa/dt=0 in
Eq. (), as in , and as indicated by
light blue points in Fig. ) estimates turbulent fluxes that also result in a high
correlation but with a magnitude that is too low compared to observations.
This high level of agreement of the maximum power limit with diurnal heat
storage changes suggests that it is an adequate description of surface energy
balance partitioning and land–atmosphere exchange at the diurnal timescale, so that turbulent fluxes appear to operate near their thermodynamic
limit. It further shows that it is critical to account for diurnal variations
in heat storage in the thermodynamic limit to adequately represent the
magnitude of the observed turbulent fluxes.
Discussion
Our approach, of course, only represents a general description of the full
dynamics of surface–atmosphere exchange. Notable effects not considered in
our approach that could alter the results and potentially modulate the
outcome of the maximum power limit include a more detailed representation of
radiative transfer, a distinction between the sensible and latent heat fluxes
which result in different forms of storage changes in the atmosphere,
entrainment effects at the top of the boundary layer, advection and coupling
to large-scale atmospheric processes, and a better representation of
nighttime processes, particularly regarding the formation of stable
conditions at night that prevent convection to occur. These aspects can be
explored further in future extensions. Yet even at this highly simplified
level, the agreement of the estimated flux partitioning with observations is
rather remarkable, indicating that the dominant forcing and the dominant
constraints are captured by our approach.
Our results emphasize the importance of considering the constraint imposed by
the second law of thermodynamics on land–atmosphere exchange. While the
complex, turbulent nature of this exchange makes it seem almost impossible to
describe its outcome in simple terms, the generation of turbulent kinetic
energy that drives the diurnal development of the convective boundary layer
is nevertheless constrained by thermodynamics. The very good agreement of our
results with observations suggests that this constraint imposed by
thermodynamics is relevant to this generation, and land–atmosphere exchange
appears to operate near this thermodynamic limit. This is consistent with
previous research that applied thermodynamics and/or heat engine frameworks
to atmospheric motion, for instance approaches using the proposed principle
of maximum entropy production or
applications to hurricanes and atmospheric convection . Note that our
maximization of power is almost identical to the maximization of material
entropy production, as we assume a steady state in which power equals
dissipation (G=D), and entropy production by turbulence is then given by D/T,
where T is the temperature at which dissipation occurred (with T≈Ts).
Yet our approach differs in that it specifically considered
the effect of heat storage changes in altering the thermodynamic limit and
feedbacks with the surface energy balance that altered the driving
temperature difference of the heat engine. The heat storage changes in the
lower atmosphere result in an additional term in the Carnot limit, and this
can explain why the land–atmosphere system functions quite differently with
its pronounced diurnal variations in turbulent fluxes than the temporally
much more uniform turbulent fluxes over open water surfaces
e.g.,. Thermodynamics
combined with these two additional factors then provide sufficient
constraints on the magnitude of turbulent heat fluxes. It would seem that
this could provide valuable information to better parameterize turbulent
fluxes within the Monin–Obukhov similarity theory for unstable conditions,
specifically regarding the stability functions that are used in this approach
e.g., as in.
This insight that surface energy balance partitioning is predominantly
determined by the local partitioning of the absorbed solar radiation is
rather different than the way this exchange is commonly represented in
climate models. In these models, surface exchange is parameterized using the
aerodynamic bulk approach, in which the aerodynamic drag of the surface and
horizontal wind speeds play a dominant role that is modulated by stability
functions. Our approach differs in that solar radiation plays the dominant
role in surface exchange by the local generation of buoyancy and power to drive
convection, rather than wind speed and aerodynamic roughness as what the bulk
method would suggest. A recent intercomparison between a number of commonly
used land surface models shows, however, that land
surface models using the bulk method generally underestimate the strong
correlation of turbulent fluxes with downward solar radiation found in
observations. Our approach can resolve this bias and suggests that the bulk
method may underestimate the effect of the local forcing by solar radiation
on surface–atmosphere exchange.
We think that our approach provides ample opportunities for future
applications and research. First, the simple expression of how
turbulent heat fluxes on land vary during the day, as given by
Eq. (), provides an easy way to get a first-order
estimate. It could serve as a baseline estimate that is solely based on
physical principles, specifically, the first and second law of
thermodynamics, and does not require tuning. This expression should
nevertheless be further evaluated in a broader range of climatological
conditions and over extended time periods to identify possible shortcomings,
for instance with respect to the simple parameterization of longwave
radiation or regarding the omission of advective effects. For a broader range
of applicability, the approach would need to be extended further to derive an
expression for near-surface air temperature, which would be related to the
changes in atmospheric heat storage (dUa/dt), for the aerodynamic
conductance, and for boundary layer development, and the turbulent heat
fluxes should be separated into the fluxes of sensible and latent heat.
It would also be instructive to compare the power associated with the
limit with estimates of the turbulent kinetic energy generation rate from
observations to develop another possibility for testing the maximization approach.
Our approach can then be used to evaluate aspects of global change analytically, such as
land cover change or global warming, providing an alternative
approach to these topics that complements complex, numerical modeling
approaches. More generally, the success of our approach in reproducing
observations very well constitutes another example of processes in complex
systems appearing to evolve to and operate at their thermodynamic limit
.
This, in turn, encourages the application of thermodynamics to a broader
range of questions and topics to understand the evolution and emergent
dynamics of complex Earth systems.
Conclusions
We formulated a Carnot limit which accounts for heat storage changes within
the atmospheric heat engine and used this limit to estimate the partitioning
of the solar radiative forcing into radiative and turbulent cooling at the
diurnal timescale. In contrast to common approaches to describe near-surface
turbulent heat transfer into the atmosphere, we explicitly consider the
thermodynamic constraint imposed by the second law of thermodynamics by
treating turbulent heat fluxes and convection as the result of a heat engine.
The maximization of the work output of this convective heat engine then
yields estimates of turbulent fluxes that compare very well to observations
across a range of climates and do not require empirical parameterizations.
This demonstrates that our approach represents an adequate, general
description of the land surface energy balance that only uses physical
concepts and that does not rely on semiempirical turbulence parameterizations.
We conclude that turbulent fluxes over land appear to operate near its
thermodynamic limit by which the power of the convective heat engine is
maximized. This limit is shaped by the second law of thermodynamics, as in
the case of the Carnot limit of a heat engine in classical thermodynamics,
but also requires the consideration of two additional factors that relate the
heat engine to its environmental setting. The first factor relates to the
strong diurnal variation of solar radiation, which results in diurnal heat
storage changes. Over land these changes are buffered primarily in the lower
atmosphere and these modulate the Carnot limit, resulting in a reduced
efficiency and in what we referred to as a cold heat engine. Second, the
limit of maximum power of the atmospheric heat engine is shaped by the
trade-off in the driving temperature difference between surface and
atmosphere, which decreases with greater turbulent heat fluxes. This trade-off
results in the maximum power limit and represents a strong coupling between
surface conditions and the lower atmosphere.
Overall, our study shows that thermodynamics adds a highly relevant
constraint to land–atmosphere coupling. This thermodynamic approach
to the surface energy balance and land–atmosphere interactions should help us
to better understand the role of the land surface and terrestrial vegetation
in the climate system and how they interact with global change.
All data used in this study did not originate from the authors
but were obtained from other studies. The sources for the data are provided in
Table 2. For reproducing the results of this study, the data is available from
the corresponding author upon request.
Effects of radiative exchange on the limit of a cold heat engine
The derivation of the Carnot limit with heat storage changes in
Sect. assumed in the first law that the heat
storage change within the heat engine is entirely caused by the heat flux Jin.
When applying this approach to turbulent fluxes between the land
surface and the atmosphere, one also needs to consider the net transport of
energy by radiative exchange between the surface and the atmosphere. In the
derivation above, this net exchange is represented by the flux Rl,net.
This flux contributes to the heat storage change in the lower atmosphere, but
it is not driven by the heat engine. This results in a small inconsistency
when applying the limit of Sect. to the lower
atmosphere. In the following, we show that the limit derived in
Sect. is still valid. However, whether the lower
atmosphere is opaque to longwave radiative transfer and absorbs Rl,net or whether it is instead transparent makes a difference in the justification,
which is why we included this derivation here rather than in the main text.
In the following, we assume that the radiative–convective layer of the lower
atmosphere is sufficiently opaque and absorbs the net longwave radiation of
the surface, Rl,net. Then, the first law described by
Eq. () becomes the energy balance of the lower atmosphere
and changes to
dUadt=Jin+Rl,net-Rl,out-G+D,
where G=D and Rl,out=Rs,avg in steady state.
The second law (Eq. ) obtains another term related to the
entropy being added by the warming due to the absorption of the net flux of
longwave radiation, Rl,net. As this warming takes place at the
prevailing physical temperature of the atmosphere (rather than the potential
temperature), its temperature is likely closer to Ta rather than Te
or Ts. Hence, the entropy budget changes to
1TedUadt=JinTs+DTe-Rl,outTa+Rl,netTa+σirr.
As in Sect. , we can combine
Eqs. () and (), solve them for D (=G),
and obtain a limit on the power (G) by assuming that the entropy production σirr=0:
G=Jin⋅TeTs⋅Ts-TaTa-dUedt⋅Te-TaTa.
This is the same expression as Eq. (), so that
the effect of net longwave radiative transfer actually cancels out.
In the case in which the lower atmosphere is comparatively transparent for
longwave radiation, the flux Rl,net passes through the lower
atmosphere without being absorbed. In this case, Eqs. ()
to () remain unaffected.
AK and MR jointly developed the idea for this paper.
AK performed the theoretical derivation and MR the data analysis. AK led the
writing of the manuscript with input from MR.
The authors declare that they have no conflict of interest.
This article is part of the special issue “Thermodynamics and
optimality in the Earth system and its subsystems (ESD/HESS inter-journal SI)”.
It is not associated with a conference.
Acknowledgements
We thank two anonymous reviewers for their helpful reviews and Henk de Bruin,
Andreas Chlond, Pierre Gentine and Aljosa Slamersak for constructive
discussions on land–atmosphere exchange. This research contributes to the
“Catchments As Organized Systems (CAOS)” research group (FOR 1598) funded
by the German Science Foundation (DFG). We acknowledge the University of
Wyoming for making the radio sounding data available at
http://weather.uwyo.edu/upperair/sounding.html (last access: 16 April 2018).
Data from Site A were funded by NSF grant 1556772 to the University of Notre Dame. Data from Site B
were supported by the Office of Biological and Environmental Research of the
US Department of Energy under contract no. DE-AC02-05CH11231 as part of the
Atmospheric Radiation Measurement Program, ARM). Data for Site C and D were
provided by the Deutscher Wetterdienst (DWD) –Meteorologisches Observatorium
Lindenberg/Richard-Assmann Observatorium. They were obtained in the context of
the Coordinated Energy and Water Cycle Observation Project (CEOP), which was
initiated as an international effort in 1998 by the World Climate Research
Programme (WCRP) Global Energy and Water Cycle Experiment (GEWEX)
Hydrometeorology Panel (GHP) in support of global climate research interests.
Data for Site E were kindly provided by Eyal Rotenberg and Daniel Yakir. Data
for Site F were provided by the AmeriFlux data server:
http://ameriflux.ornl.gov (last access: 16 April 2018).
The article processing charges for this open-access publication
were covered by the Max Planck Society.
Edited by: Michel Crucifix
Reviewed by: two anonymous referees
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