The hydrological sensitivity to global warming and
solar geoengineering derived from thermodynamic
constraints
1
2
1
Axel Kleidon, Ben Kravitz, and Maik Renner
Corresponding author: Axel Kleidon, Biospheric Theory and Modelling Group, Max-PlanckInstitute for Biogeochemistry, Jena, Germany. ([email protected])
1
Biospheric Theory and Modelling Group,
Max-Planck-Institute for Biogeochemistry,
Jena, Germany.
2
Atmospheric Sciences and Global
Change Division, Pacific Northwest
National Laboratory, Richland, WA, USA.
This article has been accepted for publication and undergone full peer review but has not been through
the copyediting, typesetting, pagination and proofreading process, which may lead to differences between this version and the Version of Record. Please cite this article as doi: 10.1002/2014GL062589
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We derive analytic expressions of the transient response of the hydrological cycle to surface warming from the surface energy balance in which turbulent heat fluxes are constrained by the thermodynamic limit of maximum
power. For a given steady-state temperature change, this approach predicts
the transient and steady-state response of surface energy partitioning and
the hydrologic cycle. We show that the predicted hydrological sensitivities
to greenhouse warming and solar geoengineering are comparable to the results from climate model simulations of the Geoengineering Model Intercomparison Project (GeoMIP). Although not all effects can be explained, our
approach nevertheless predicts the general trend as well as the magnitude
of the changes in the global-scale hydrological cycle surprisingly well. This
implies that much of the global-scale changes in the hydrologic cycle can be
robustly predicted by the response of the thermodynamically-constrained surface energy balance to altered radiative forcing.
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1. Introduction
The hydrological sensitivity, defined by the increase in global mean precipitation (or
evaporation) for a given change in global mean temperature, is a key component in assessing the hydrological cycle response to global warming. The response of climatic variables
to change have previously been separated into a fast response that describes the rapid
adjustment of the climate system to the altered radiative forcing, and a slow response that
is associated with changes in surface temperature that lags the response due to storage
effects [Andrews et al., 2009; Bala et al., 2010]. This separation of the temporal adjustments to an altered radiative forcing has been shown to be highly useful in providing
information on climate sensitivity [Gregory et al., 2004]. This separation also shows that
rapid hydrologic changes may have the opposite sign of the slow, steady state hydrologic
sensitivity [Bala et al., 2010]. A conceptual understanding of fast and slow adjustment
processes is critical in attributing observed changes to transient or steady state effects.
These responses are typically evaluated using state-of-the-art, complex climate system
models. What we ask here is how much of these changes can be attributed to basic physical constraints that are imposed by the surface energy balance and by thermodynamic
limits to turbulent heat fluxes. The application of thermodynamic limits has been rather
successful, for instance, for estimating the intensity of hurricanes [Emanuel , 1999] or hydrologic cycling [Pauluis and Held , 2002a, b]. When thermodynamic limits are combined
with the effects of the associated heat fluxes on temperature differences, the resulting
maximum power limit yields a thermodynamic constraint on the turbulent heat fluxes
(see also Supplemental Online Material). This maximum power limit, or, closely related,
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a maximum in the associated entropy production, has successfully predicted the magnitude of hemispheric heat transport [Paltridge, 1975; Lorenz et al., 2001], and the value of
empirical parameters in climate models [Kleidon et al., 2003; Pascale et al., 2012]. Recently, we showed that this thermodynamic limit can explain the steady-state sensitivity
of the global hydrological cycle [Kleidon and Renner , 2013b] and the geographic mean
variations of the surface energy balance on land [Kleidon et al., 2014] rather well.
Here we evaluate how much these physical constraints alone shape the fast and slow
responses to global climate change. Our approach yields analytic expressions, which we
then compare to climate model results of the Geoengineering Model Intercomparison
Project (GeoMIP) [Kravitz et al., 2013a].
2. Methods
2.1. Steady state surface energy balance
We consider the global, annual mean surface energy balance, assuming rapid energy
exchange between the surface and the atmosphere to achieve a steady state, while the
slow adjustment is primarily due to oceanic heat uptake. The surface energy balance is
expressed as
Rs − Rl − H − λE − Ho = 0
(1)
where Rs is the net absorption of solar radiation by the surface (including albedo effects),
Rl is the net thermal (longwave) radiative exchange between the surface and the atmosphere (i.e., surface emission and greenhouse effect combined; positive Rl indicates net
upward longwave flux), H and λE are the sensible and latent heat fluxes, respectively,
from the surface to the atmosphere (λ ≈ 2.5 × 106 J kg−1 is the latent heat of vaporization
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and E is evaporation), and Ho is the heat flux associated with ocean heat uptake [e.g.,
Boer , 1993]. In a climatic steady state, Ho = 0.
Following the formulations of Kleidon and Renner [2013a, b], we use a linear approximation for Rl that parameterizes the strength of the greenhouse effect via a radiative
“conductance”: Rl = kr (Ts − Ta ) where kr is the radiative conductance, Ts is the surface
temperature, and Ta is the radiative temperature of the atmosphere, assuming that all
emitted radiation from the surface is absorbed in the atmosphere. The temperature Ta is
determined by the planetary energy balance, Rs,toa − Ho = σTa4 , where Rs,toa is the total
absorption of solar radiation at the surface and in the atmosphere and σ = 5.67 · 10−8 W
m−2 K−4 is the Stefan-Boltzmann constant.
The thermodynamic constraint is derived from the assumptions that the turbulent exchange at the surface is mostly driven by surface heating and that the power G to generate motion by dry and moist convection (G = (H + λE) · (Ts − Ta )/Ta ) is maximized
(∂G/∂(H + λE) = 0). Because power balances dissipation in steady state, this maximization of power corresponds to the strongest possible turbulent dissipation of kinetic
energy that can be generated from the vertical difference in radiative heating and cooling.
Note that this approach does not account for power generated by horizontal differences
in heating, although this power is also thermodynamically constrained. As dissipation
happens mostly near the surface, this maximum power limit provides a constraint for
the magnitude of the surface turbulent heat fluxes. The optimum partitioning (indicated
by the subscript opt) yields an equal partitioning of net solar heating (i.e., absorbed solar radiation, reduced by the transient oceanic heat uptake, Rs − Ho ) into radiative and
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turbulent cooling:
Rl,opt =
Rs − Ho
2
Hopt =
γ Rs − Ho
γ+s
2
λEopt =
s Rs − Ho
γ+s
2
(2)
where γ = 65 Pa K−1 is the psychrometric constant (evaluated at sea level pressure,
globally averaged) and s = 111 Pa K−1 is the slope of the saturation vapor pressure curve
for the present-day global mean temperature of Ts = 288 K. For more details on the
derivations of these relationships and limitations, see Kleidon and Renner [2013a, b]. In
the following, we assume that this optimum partitioning is achieved by the climate system
at the fast, convective time scale of the atmosphere. We then use these expressions for
deriving the transient response by considering the changes in this partitioning due to Ho .
2.2. Transient changes
We consider the transient change from the initial, unperturbed climatic state with
temperatures Ts,0 and Ta,0 to a new steady state of the perturbed radiative forcing that
is characterized by Ts,1 and Ta,1 . We use ∆Ts,1 = Ts,1 − Ts,0 to describe the steadystate temperature difference associated with the difference in radiative forcing between
the initial and perturbed states (see SOM Fig. 2). The transient response in the energy
balance components is determined by deriving the corresponding value of Ho for a specific
value of ∆Ts = Ts − Ts,0 . We then derive the transient change of the energy balance
components as functions of ∆Ts .
Using the energy balance partitioning constrained at maximum power, the ocean heat
uptake, Ho , can be written as a function of the altered radiative forcing and the temperature differences ∆Ts,1 and ∆Ts :
Ho = 2(F − α∆Ts )
(3)
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The terms 2F and 2α describe the intercept and slope of the linear relationship, with the
instantaneous response given by ∆Ts = 0 so that Ho = 2F . This formulation is similar to
the approach described by Gregory et al. [2004]. The parameters α and F are given by
α=
2Rs,toa,0 Rs,0
Ta,0 Rs,0 + 4Rs,toa,0 (Ts,0 − Ta,0 )
F = α · ∆Ts,1
(4)
With this formulation for Ho , we can derive linearized expressions of the changes in surface
energy balance partitioning from Equation 2 as functions of ∆Ts . For the deviation in net
longwave radiation ∆Rl , we obtain
∆Rl (∆Ts ) = ∆Rl (0) +
∂Rl ∂Ho
∆Rs
∆Ts =
− F + α∆Ts
∂Ho ∂Ts
2
(5)
For an instantaneous, greenhouse-induced change (i.e., ∆Rs = 0 and ∆Ts = 0), ∆Rl = −F
so that F corresponds to the reduction in net radiative cooling of the surface due to a
stronger greenhouse effect. However, in a geoengineering scenario in which the steadystate temperature sensitivity is ∆Ts,1 = 0, there is no net radiative change at the surface
and F = 0 (cf. Equation 4).
Equivalently, we obtain for the deviations of the sensible and latent heat fluxes
γ
∆Rs
γ
∆H(∆Ts ) =
−F −
γ+s
2
γ+s
!
∂s/∂Ts Rs,0
− α ∆Ts
(γ + s) 2
(6)
and
s
∆Rs
s
∆λE(∆Ts ) =
−F +
γ+s
2
γ+s
!
γ ∂s/∂Ts Rs,0
+ α ∆Ts
s (γ + s) 2
(7)
where we consider only constant and first-order terms. The first term on the right hand
side corresponds to the intercept, while the term associated with ∆Ts describes the slope.
Note that ∆Rs − ∆Rl − ∆H − ∆(λE) − Ho = 0, so these expressions satisfy the surface
energy balance under transient conditions.
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2.3. Fast and steady-state responses
Using Equations (5) - (7), the fast response is given by the intercept at ∆Ts = 0, and
the steady-state response by ∆Ts = ∆Ts,1 = F/α, with the inherent approximation that
the slopes are constant. For the fast response, the oceanic heat uptake has a maximum
value of Ho = 2F , while in the steady-state response, Ho = 0. We then obtain the fast
and steady-state responses for Rl , H, and λE in relative terms:
∆Rl
Rl
=
f ast
∆Rs
2F
−
Rs,0
Rs,0
∆H
∆Rs
2F
=
−
H f ast
Rs,0
Rs,0
∆E
∆Rs
2F
=
−
E f ast
Rs,0
Rs,0
∆Rl
Rl
=
steady
∆Rs
Rs,0
∆H
∆Rs
s 1 ds
=
−
∆Ts,1
H steady
Rs,0
γ + s s dTs
∆E
∆Rs
γ 1 ds
=
+
∆Ts,1
E steady
Rs,0
γ + s s dTs
(8)
(9)
(10)
Note that the fast responses for all three fluxes are identical in relative terms. The fast
response to radiative forcing changes is assumed to be partitioned by maximum power,
so that the change in radiative forcing is partitioned in equal proportions. Hence, the
maximum power constraint predicts that the relative changes in these fluxes should be
identical. The temperature-dependent terms in the steady state responses describe the
changes in the partitioning between the sensible and latent heat fluxes related to the
change in water vapor saturation. The expressions for the steady-state responses are
equivalent to those of Kleidon and Renner [2013b].
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3. Evaluation
3.1. Setup
We use present-day global mean (land+ocean) climatic conditions of Rs,toa,0 = 240 W
m−2 , Rs,0 = 165 W m−2 [Stephens et al., 2012], Ts,0 = 288K and Ta,0 = 255K, which yields
a value of kr,0 = 2.50 W m−2 K−1 and α = 1.07 W m−2 K−1 .
Using these values, our expressions are evaluated for three scenarios: greenhouse warming (4xCO2), solar cooling (S), and solar geoengineering (G). For 4xCO2, we assume that
∆Ts,1 = 5 K and ∆Rs = 0, yielding a value of F = 5.37 W m−2 , which is approximately
the predicted radiative forcing for a quadrupling of CO2 [e.g., Myhre et al., 1998]. For
S, we use ∆Ts,1 = −5 K and ∆Rs = −10.75 W m−2 (using supplementary Equation 8
with ∆kr = 0, approximately 6.5% of Rs,0 or 4.4% of Rs,toa,0 ), which yields a value of F
equivalent in magnitude to the 4xCO2 experiment, but of opposite sign (F = −5.37 W
m−2 ). For G, greenhouse warming is offset by solar cooling such that ∆Ts,1 = 0 K; then
∆Rs = −10.75 W m−2 and F = 0 W m−2 .
In all calculations, we use s = desat /dT = 111 Pa K−1 and ds/dTs = d2 esat /dTs2 = 6.59
Pa K−2 , both derived with Ts,0 = 288 K.
3.2. Results and Discussion
The transient changes of the surface energy balance for the three scenarios that are
expected from the thermodynamic constraint are shown in Figure 1. The changes vary
linearly between the fast response due to the applied forcing at ∆Ts = 0 K and the steady
state response at ∆Ts = ∆Ts,1 .
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For the 4xCO2 scenario, the fast responses are ∆Rl = −F , ∆H = −γF/(γ + s), and
∆λE = −sF/(γ+s); the reduction in net longwave flux from the surface to the atmosphere
is compensated by partitioned reductions in sensible and latent heat fluxes. The relative
changes in longwave radiation, sensible and latent heat flux are −2F/Rs,0 = −6.5%. This
reduction in λE is the fast response to greenhouse warming, which reflects the buffering
effect of the oceanic heat uptake Ho . However, with time, surface temperature increases
and oceanic heat storage change reduces to zero. The steady-state longwave response is
∆Rl = 0. Sensible and latent heat fluxes change due to the increase in temperature, so in
steady state, the sensible heat flux is reduced by 19%, and latent heat flux increases by
11%. For a 5 K change in temperature, this corresponds to a 2.2% K−1 increase in latent
heat flux, which is consistent with the hydrologic sensitivity derived from climate models
[Allen and Ingram, 2002; Held and Soden, 2006; Kleidon and Renner , 2013b].
In the S scenario, ∆Rs = 2F , so the fast responses are ∆Rl = ∆H = ∆λE = 0. In
steady state, ∆Rl = ∆Rs /2 and ∆H + ∆λE = ∆Rs /2. The cooling that results from a
decrease in Rs causes a shift from latent to sensible heat, so λE is decreased by 17.5%
while H is increased by 12.2%.
The geoengineering scenario G is a combination of scenarios 4xCO2 and S. Because
our expressions are linear by design, the response in G is simply the sum of 4xCO2
and S. As such, the fast and steady-state responses of G are ∆Rl = ∆Rs /2, ∆H =
(γ∆Rs )/(2(γ + s)), and ∆λE = (s∆Rs )/(2(γ + s)), i.e., the sensitivities are entirely
determined by the changes in shortwave radiation. The relative changes for both the fast
and steady-state responses for Rl , H, and λE are ∆Rs /Rs,0 = −6.5%. These results are
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consistent with the findings of Bala et al. [2010] that the slow response of the hydrological
sensitivity in climate model simulations is independent of the forcing mechanism. In our
Equations 9 and 10, the rapid response depends on the radiative terms ∆Rs and F , while
the slow response is represented by the second term and depends on the temperature
difference ∆Ts,1 only, independent of the type of radiative change.
We next compare these differences to the GeoMIP simulations. The mean temperature
increase in the 4xCO2 simulations is ∆Ts = 6.8K (ranging from 3.9 - 10.7K), with an
implied value of F = 8.0 W m−2 . These values are similar, although slightly larger
than our inferred values from the setup. The change in solar absorption in the GeoMIP
simulation experiment G1 [Huneeus et al., 2014] ranges between 3.5 and 5.0% in Rs,toa
[Kravitz et al., 2013b], which is consistent with our value of ∆Rs /Rs,toa ≈ 4.4%. Our
inferred values for the radiative parameters in F and ∆Rs are thus consistent with the
climate model simulations.
To compare the transient behavior, it is sufficient to compare the fast and steady state
responses as these define the linear, transient relationships shown in Fig. 1. The fast
and steady state responses are derived from the GeoMIP simulations by regressing the
annual mean radiative flux quantities against annual mean temperature changes. The fast
response is taken to be the y-intercept, and the steady-state response is taken to be the
x-intercept, as in Gregory et al. [2004].
Figure 2 shows the comparison of the fast and steady-state responses of the hydrologic
changes to the GeoMIP simulations. Although we would not expect our expressions to
exactly match the results from the GeoMIP ensemble, they reproduce the hydrological
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fast response of the GeoMIP ensemble reasonably well. The fast response is reasonably
captured for the 4xCO2 and G scenario in terms of sign and magnitude, although the
magnitude for scenario G is larger than in the simulations. Note that scenario S is inferred
from the climate model difference G1 - 4xCO2, so that the bias shown in scenario S
reflects the bias for scenario G. For the steady state response, our estimated hydrological
sensitivity of 2.2% K−1 reproduces the relative sensitivity of the GeoMIP simulations
remarkably well. This value of sensitivity is derived from the second part of Equation
10 when applied to the 4xCO2 scenario. The changes in scenario G are represented by
∆Rs /Rs,0 (cf. Equation 10), which matches the climate model simulations very well.
The fast and slow responses of net longwave radiation and the sensible heat flux are
also reasonably well captured (see also SOM Fig. 3). One particular aspect noted in the
GeoMIP ensemble is that the fast responses of experiment G are similar to the steady
state responses [Kravitz et al., 2013c], which is also predicted by our approach. A notable
deviation of our approach is that it poorly describes the relative changes in net longwave radiation in steady state for scenarios 4xCO2 (SOM Fig. 3b.). We attribute this
shortcoming to the highly simplified treatment of longwave radiation in our approach.
For simplicity, we also did not consider changes in Rs that are due to changes in cloud
cover, water vapor, or albedo in our approach although such changes are included in the
responses simulated by the climate models.
4. Summary and Conclusions
Our approach formulates the fast and steady state response of the surface energy balance and evapotranspiration solely in terms of physical constraints, which, importantly,
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include the thermodynamic limit on turbulent heat fluxes. In this approach, we do not
need empirical parameters, but only four basic climate characteristics: the absorbed solar
radiation at the surface, Rs,0 , and in total, Rs,toa,0 , the surface temperature, Ts,0 , and
the magnitude of steady-state surface temperature change, ∆Ts,1 . By using only these
four properties, we inferred the transient and steady state energy balance changes and
changes in the hydrologic cycle to greenhouse-induced warming and its compensation by
a reduction of solar radiation. We showed that this approach is able to reproduce the first
order changes of the transient response and the different sensitivity of evapotranspiration
to solar vs. greenhouse radiative changes of the GeoMIP simulations very well. This suggests that a large part of the climate model sensitivities reflects solely a change in these
physical constraints, even when these are formulated in an extremely simple way.
We see the main utility of our approach as setting a reference case that formulates
how much of the anticipated climate change can solely be attributed to a change in the
physical constraints. These changes have a much more robust foundation than some of the
more empirically-based formulations in climate models, for instance regarding turbulent
exchange fluxes at the surface or parameterizations of atmospheric convection. We thus
think that the separation of climate model responses into those that can be explained
by constraints and others that cannot should help to better identify which aspects of
climate change are physically robust. Obviously, our formulation of physical constraints
may still be improved, particularly regarding the representation of net longwave radiation,
as it currently does not capture the changes seen in the climate models. This may point
towards the need of a better, yet still simple formulation of net radiative transfer in our
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approach. Nevertheless, our approach represents an alternative way based on physical
first principles that should help to robustly predict the dominant features of global climate
change.
Acknowledgments. We thank two anonymous reviewers for their helpful comments.
Ben Kravitz is supported by the Fund for Innovative Climate and Energy Research (FICER). The Pacific Northwest National Laboratory is operated for the U.S. Department of
Energy by Battelle Memorial Institute under contract DE-AC05-76RL01830. We thank
all participants of the Geoengineering Model Intercomparison Project and their model
development teams, CLIVAR/WCRP Working Group on Coupled Modeling for endorsing GeoMIP, and the scientists managing the Earth System Grid data nodes who have
assisted with making GeoMIP output available.
References
Allen, M. R., and W. J. Ingram (2002), Constraints on future changes in climate and the
hydrologic cycle, Nature, 419, 224–232.
Andrews, T., P. M. Forster, and J. M. Gregory (2009), A surface energy perspective on
climate change, J. Climate, 22, 2557-2570, doi:10.1175/2008JCLI2759.1.
Bala, G., P. B. Duffy, and K. E. Taylor (2008), Impact of geoengineering schemes on the
global hydrologic cycle, Proc. Natl. Acad. Sci. USA, 105, 7664–7669.
Bala, G., K. Caldeira, and R. Nemani (2010), Fast versus slow response in climate
change: implications for the global hydrological cycle, Clim. Dyn., 35, 423–434, doi:
10.1007/s00382-009-0583-y.
c
2014
American Geophysical Union. All Rights Reserved.
Boer, G. J. (1993), Climate change and the regulation of the surface moisture and energy
budgets, Clim. Dynam., 8, 225–239.
Emanuel, K. A. (1999), Thermodynamic control of hurricane intensity, Nature, 401, 665–
669.
Gregory, J. M., et al. (2004), A new method for diagnosing radiative forcing and climate
sensitivity, Geophys. Res. Lett., 31, L03,205, doi:10.1029/2003GL018747.
Held, I. M., and B. J. Soden (2006), Robust responses of the hydrological cycle to global
warming, J. Clim., 19, 5686–5699.
Huneeus, N., et al. (2014), Forcings and feedbacks in the GeoMIP ensemble for a reduction in solar irradiance and increase in CO2 , J. Geophys. Res., 119, 5226–5239,
doi:10.1002/2013JD021110.
Kleidon, A., and M. Renner (2013a), Thermodynamic limits of hydrologic cycling within
the Earth system: Concepts, estimates and implications, Hydrol. Earth Syst. Sci., 17,
2873–2892.
Kleidon, A., and M. Renner (2013b), A simple explanation for the sensitivity of the
hydrologic cycle to climate change, Earth Syst. Dynam., 4, 455–465, doi:10.5194/esd-4455-2013.
Kleidon, A., K. Fraedrich, T. Kunz, and F. Lunkeit (2003), The atmospheric circulation and states of maximum entropy production, Geophys. Res. Lett., 30, 2223, doi:
10.1029/2003GL018363.
Kleidon, A., M. Renner, and P. Porada (2014), Estimates of the climatological land surface
energy and water balance derived from maximum convective power, Hydrol. Earth Syst.
c
2014
American Geophysical Union. All Rights Reserved.
Sci., 18, 2201–2218.
Kravitz, B., A. Robock, P. M. Forster, J. M. Haywood, M. G. Lawrence, and H. Schmidt
(2013a), An overview of the Geoengineering Model Intercomparison Project (GeoMIP),
J. Geophys. Res., 118, 13,103–13,107, doi:10.1002/2013JD020569.
Kravitz, B., et al. (2013b), Climate model response from the Geoengineering
Model Intercomparison Project (GeoMIP), J. Geophys. Res., 118, 8320–8332,
doi:10.1002/jgrd.50646.
Kravitz, B. et al. (2013c), An energetic perspective on hydrological cycle changes in
the Geoengineering Model Intercomparison Project (GeoMIP), J. Geophys. Res., 118,
13087–13102, doi:10.1002/2013JD020502.
Lorenz, R. D., J. I. Lunine, P. G. Withers, and C. P. McKay (2001), Titan, Mars and
Earth: Entropy production by latitudinal heat transport, Geophys. Res. Lett., 28, 415–
418.
Myhre, G., E. J. Highwood, K. P. Shine, and F. Stordal (1998), New estimates of radiative
forcing due to well mixed greenhouse gases, Geophys. Res. Lett., 25, 2715-2718.
Paltridge, G. W. (1975), Global dynamics and climate – A system of minimum entropy
exchange, Q. J. Roy. Meteorol. Soc., 101, 475–484.
Pascale, S., J. M. Gregory, M. H. P. Ambaum, and R. Tailleux (2012), A parametric sensitivity study of entropy production and kinetic energy dissipation using the FAMOUS
AOGCM, Clim. Dyn., 38, 1211–1227, doi:10.1007/s00382-011-0996-2.
Pauluis, O., and I. M. Held (2002a), Entropy budget of an atmosphere in radiative convective equilibrium. Part I: Maximum work and frictional dissipation, J. Atmos. Sci.,
c
2014
American Geophysical Union. All Rights Reserved.
59, 126–139.
Pauluis, O., and I. M. Held (2002b), Entropy budget of an atmosphere in radiative convective equilibrium. Part II: Latent heat transport and moist processes, J. Atmos. Sci.,
59, 140–149.
Stephens, G. L., et al. (2012), An update on Earth’s energy balance in light of the latest
global observations, Nature Geosci., (5), 691–696.
Tilmes, S., et al. (2013), The hydrological impact of geoengineering in the Geoengineering
Model Intercomparison Project (GeoMIP), J. Geophys. Res., 118 (19), 11,036–11,058,
doi:10.1002/jgrd.50868.
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a.
∆J (W m-2)
12
10
5
10
0
c.
−5
0
−10
−5
0
1
−10
0
1
2
3
∆J (W m-2)
5
6
6
0
0
-2
-1
α = 0.853 W m K
-2
F=0Wm
-2
ΔR
=
-8
W
m
s
-2 -1
α
=
0.853
W
m
K
−12
-2
ΔRs
0 F = 01W m 2-2
3 ΔRl 4
5
ΔRs = -8 W m
Ho
12
G-C
ΔH
ΔRs
ΔλE
ΔRl
6
Ho
G - Control
ΔH
2
3
4
5
ΔλE
−6
0
4
G - Control
S-C
−6
−12
0
d.
−1
−2
−3
−4
−5
12
6
0
5
−6
−6
−12
−12
0
1
2
3
∆Ts (K)
Figure 1.
b. 12
4xCO2 - C
4
5
4xCO2 - G
0
1
2
3
4
5
∆Ts (K)
Differences in energy fluxes (∆J) of the surface energy balance due to transient
changes caused by (a.) a greenhouse-induced warming of 5K (4xCO2 - C), (b.) a solar-induced
cooling of 5K (S - C) (c.) a geoengineered climate in which the 5K warming is compensated by
reduced absorption of solar radiation at the surface (G - C), and (d.) the difference between the
5K warming scenario and the geoengineering scenario (4xCO2 - G). C indicates an unperturbed
control simulation. Note that (c.) experiences no surface temperature change, and the x-axis is
included here for comparison.
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a.
12
∆E/E (%)
6
S
0
C
G
−6
4xCO2
−12
−12
−6
0
6
12
(∆Rs - 2F)/Rs (%)
b.
30
∆E/E (%)
20
10
4xCO2
C
0
G
−10
S
−20
−30
−10
−5
0
5
10
∆Ts (K)
Figure 2. The (a.) fast response and (b.) steady state relative changes in evaporation (∆E/E,
which is equivalent to the relative change in precipitation in our model) for a greenhouse-induced
warming ∆Ts = 5K (4xCO2, red solid triangle), a solar-induced cooling of 5K (S, green solid
circle), and for a geoengineered scenario (G, blue solid square, overlaps with 4xCO2 in (a.))
predicted by our approach, with the white circle representing the unperturbed climatic state (C).
For comparison, the simulated responses from equivalent scenarios of GeoMIP climate models
[Kravitz et al., 2013c; Tilmes et al., 2013] are shown in equivalent, open symbols, with the green
open circles inferred from the difference “G1-4xCO2”. Note the difference in scale between (a.)
and (b.).
c
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