Toward Quantifying the Climate Heat Engine: Solar Absorption and Terrestrial Emission Temperatures and Material Entropy Production

JUNE 2017 B A N N O N A N D L E E 1721

PETER R. BANNON AND SUKYOUNG LEE Department of Meteorology and Atmospheric Science, The Pennsylvania State University, University Park, Pennsylvania

(Manuscript received 15 August 2016, in ﬁnal form 22 February 2017)

ABSTRACT

A heat-engine analysis of a climate system requires the determination of the solar absorption temperature and the terrestrial emission temperature. These temperatures are entropically defined as the ratio of the energy exchanged to the entropy produced. The emission temperature, shown here to be greater than or equal to the effective emission temperature, is relatively well known. In contrast, the absorption temperature requires radiative transfer calculations for its determination and is poorly known. The maximum material (i.e., nonradiative) entropy production of a planet’s steady-state climate system is a function of the absorption and emission temperatures. Because a climate system does no work, the material entropy production measures the system’s activity. The sensitivity of this production to changes in the emission and absorption temperatures is quantified. If Earth’s albedo does not change, material entropy production would increase by about 5% per 1-K increase in absorption temperature. If the absorption temperature does not change, entropy production would decrease by about 4% for a 1% decrease in albedo. It is shown that, as a planet’s emission temperature becomes more uniform, its entropy production tends to increase. Conversely, as a planet’s absorption temperature or albedo becomes more uniform, its entropy production tends to decrease. These findings underscore the need to monitor the absorption temperature and albedo both in nature and in climate models. The heat-engine analyses for four planets show that the planetary entropy productions are similar for Earth, Mars, and Titan. The production for Venus is close to the maximum production possible for fixed absorption temperature.

1. Introduction conduction, diffusion, and phase changes, and the loss of material entropy by the emission of terrestrial radiation The climate system of a planet is a heat engine that back out to space. In a steady state, the difference between absorbs solar radiation at a relatively high temperature the loss and the gain equals the material entropy pro- and emits terrestrial radiation to space at a lower tem- duction. Thus, the material entropy production budget is perature. Energy is conserved, and no work is done on the intimately connected to the solar and terrestrial radiative planet’s surroundings. The net result of this interaction is heating processes. the production of entropy that, in a steady state, must be The material entropy production is a fundamental exported to space. The entropy production has both a ra- measure of the activity of the climate system. Entropy is diation and a material component. The scattering of the produced by the transport of heat upward from the surface relatively focused, low-entropy, incoming solar radiation to the atmosphere and meridionally from the equatorial to beam into a more diffuse, reflected field and the creation of the polar regions. Entropy is also produced by the viscous an outgoing field of high-entropy terrestrial radiation dissipation of the winds and currents. Entropy is also comprises the radiation entropics of the planet. The ma- produced by the transport of water vapor from the oceans terial entropics involves the sum of the gain in material to the atmosphere and by nonequilibrium phase changes entropy by the absorption of the solar radiation, the irre- and chemical reactions. Thus, material entropy production versible production of entropy by the processes of thermal measuresthevibrancyoftheclimatesystem. Numerous authors have attempted direct quantification Corresponding author e-mail: Peter R. Bannon, bannon@ems. of the various material entropy production processes. An- psu.edu alyses of nonhydrostatic, radiative–convective equilibrium

DOI: 10.1175/JAS-D-16-0240.1 Ó 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses). Unauthenticated | Downloaded 10/01/21 05:37 PM UTC 1722 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 74 models (Pauluis and Held 2002a,b; Pauluis and Dias 2012; (1989, 2000) provides a clear procedure to incorporate Singh and O’Gorman 2016) indicate the importance of the spatially and temporally diverse heat inputs and viscous dissipation of the wind and of hydrometeor drag outputs into a global heat-engine analysis. Lucarini along with production due to moist processes, such as (2009) and Lucarini et al. (2014) review these contri- evaporation, diffusion of water vapor, and the cycle of butions. Differences in the analyses often result from melting and freezing. Pauluis and Held (2002a,b)and differences in the specification of the system of interest. Volk and Pauluis (2010) highlight the reduction of viscous Bannon (2015) articulates the need for clear definitions dissipation between dry and moist convection. These an- of the climate system. His example of a simple climate alyses emphasize the entropy production involved in the model yields six different results for the efficiency de- vertical transport of the absorbed solar energy from the pending on the definition of the system. The model surface to the upper atmosphere, where it is emitted back definition dictates the choice of the temperatures as- to space as terrestrial radiation. There is also entropy signed to the entropy produced by the heating. For ex- production associated with hydrostatic processes involved ample, the temperature associated with the solar heating in the meridional transport of energy poleward. Analyses canvaryfromthetemperatureofthesun(;6000 K) to a of hydrostatic, general circulation models include Goody representative tropospheric value (;270 K). Bannon (2000), Lucarini et al. (2011, 2014), and Laliberte et al. (2015) shows that the later temperature, the solar ab- (2015). The production associated with meridional pro- sorption temperature, is the one required to provide cesses appears secondary in importance compared to that direct information on the material entropy production associated with vertical processes. The total atmospheric in the system. entropy production lies in the range of 30–80 entropy The purpose of this manuscript is to provide the tools 2 2 production units (EPU; 1 EPU 5 1mWm 2 K 1). In necessary to quantify the entropy production of a cli- comparison, analyses of oceanic entropy production are mate system through the examination of the gain and relatively small, lying in the range of 1–2 EPU (Gregg loss in entropy due to the absorption of solar radiation 1984; Shimokawa and Ozawa 2001; Huang 2010; Pascale and the emission of terrestrial radiation. Section 2 re- et al. 2011; Bannon and Najjar 2016, manuscript submitted views the formulation of a climate system as a heat to J. Mar. Res.). engine. We present formal definitions of the solar ab- In addition to the direct quantifications, it is also of sorption temperature and the terrestrial emission tem- interest to examine how material entropy production perature. These temperatures define the Carnot efficiency changes as a climate system evolves. Climate studies in- of the system. Section 3 applies the formulation to an ideal dicate that Earth has been globally warmer and cooler climate system (Zircon) that emits uniformly to space at a than present but that the change in temperature is not single emission temperature. Appendix A shows that the necessarily uniform over the planet. In fact, various cli- lower bound for the emission temperature is the effective mate records indicate that, when Earth was warmer, the temperature associated with uniform blackbody emission Arctic was much warmer than the tropics (e.g., Budyko to space. Then use of this temperature provides an upper and Izrael 1991; Hoffert and Covey 1992). The current bound to the material entropy production of the system. warming also shows the same behavior: the Arctic region The maximum production is expressed as a function of the is warming at least twice as quickly as the global average. effective temperature and the absorption temperature. The warming of the Arctic also implies the melting of the The sensitivity of the production to changes in albedo and sea ice and hence a reduction in the polar albedo. In absorption temperature is presented. contrast, the tropical albedo may or may not decrease Section 4 presents an assessment of the role of regional because it is unclear how the tropical cloud cover will variability of the absorption and emission temperatures change. Climate change is not limited to Earth’s climate. on the material entropy production. The motivation for Theories of planetary evolution suggest that Venus once this assessment comes from the aforementioned non- had a more temperate climate before the onset of a uniformity in warming or cooling (e.g., the Arctic tem- runaway greenhouse led to extreme surface temperatures peratures respond much more strongly to global warming of over 700 K. This manuscript seeks to provide a struc- or cooling) and also in albedo changes. For this exercise, ture to quantify the entropy production of a climate sys- we introduce an ideal climate system (Janus) that has tem and its changes. The goal here is to gain insight, but two equal areas with different shortwave and longwave along the way some definitive answers will be provided properties. Janus is a straightforward extension of Zircon while some open questions will be raised. and is the simplest model that allows for the assessment of Heat-engine and entropy analyses of the climate sys- the regional variability. We show that if Janus has two tem have been presented by a number of authors. Peixoto different emission temperatures but the same total emis- and Oort (1992) review the early literature. Johnson sion as Zircon, then there will be less entropy production

Unauthenticated | Downloaded 10/01/21 05:37 PM UTC JUNE 2017 B A N N O N A N D L E E 1723 on Janus than Zircon, all else being equal. Then if Janus where V is the volume of the climate system and As is evolves with its emission temperatures becoming more the surface area of the planet. Then the total fluxes in uniform, its entropy production will increase. We call this watts are obtained by multiplying the fluxes by the process equalization. Section 4 quantifies this behavior area of the planet. The radiative gain in solar radiation and that for the equalization of the absorption tempera- is that due to the absorption of the incoming solar tures and the regional albedos. radiation. The radiative loss in longwave radiation is Section 5 applies the Zircon theory to other planets that due to the net emission of the outgoing terrestrial and estimates the radiation temperatures for Earth, Ve- radiation. nus, Mars, and Titan and then compares their entropy The budget for the total entropy S per unit area of the productions. Unlike the other planets, the production for planet is Venus is at a maximum for fixed absorption temperature. dS _ _ Section 6 summarizes the results and presents a call for 5 S 2 S 1 S_ , (2.3) climate modelers to include calculations of their model’s dt a e absorption and emission temperatures. These tempera- where S_ denotes the rate of entropy gain or loss by a tures are essential in analyzing the model’s heat engine heating process. The irreversible entropy production and the current and future entropy production. internal to the planet is S_ . 0. All quantities are area- Using trends in CERES data (analyzed in appendix B), 2 2 averaged quantities (W m 2 K 1). The entropy fluxes we offer some speculations on future trends in Earth’s are related to the energy fluxes by the relations entropy production. The purpose of these speculations is not to present hard forecasts but rather to demonstrate the ð _ _ 1 q_ (x, t) Q utility of the approach afforded by quantifying the climate S 5 a dV [ a and (2.4a) a A T(x, t) T system’s heat engine and its entropy production. s V a ð _ 1 q_ (x, t) Q S_ 5 e dV [ e , (2.4b) 2. Heat-engine analysis and material entropy e As V T(x, t) Te production which define the entropic absorption T and emission T A heat-engine analysis requires the merging of the a e temperatures. The formulation (2.4) of these radiation energy and entropy budgets of the thermodynamic sys- temperatures follows Johnson (1989). Using these tem- tem of interest. The thermodynamic system is taken to peratures, the entropy budget (2.3) may be written as be the material climate system of the planet and the terrestrial radiation contained therein. This situation _ _ dS Qa Qe _ corresponds to material climate system case 2 (MS2) in 5 2 1 S. (2.5) dt T T Bannon (2015) and provides direct calculation of the a e material entropy production. The total energy of this The result (2.5) indicates that the rate of storage of en- system (the sum of the thermal radiation, kinetic, in- tropy in the climate system is due to the imbalance be- ternal, and potential energies) increases because of the tween the diabatic entropy generation processes and the absorption of the flux of solar energy and decreases irreversible entropy production processes. because of the emission of terrestrial energy out of the For a climatological steady state, the time-derivative system. The energy budget for the total energy E per terms drop from the governing equations: there is no unit area is storageofenergyorentropyintheclimatesystem.In dE particular, the entropics is irreversible: the irrevers- 5 Q_ 2 Q_ , (2.1) dt a e ible entropy produced in the climate system is exported radiatively out to space. Then the mathematical where the terms on the right-hand side represent the net analysis is straightforward. The time-averaged energy _ absorbed solar flux Qa and the net emitted terrestrial and entropy equations [(2.1) and (2.5), respectively] _ 22 flux Qe. All fluxes (W m ) are area-averaged quantities. reduce to The energy fluxes are defined in terms of the local _ _ 2 _ 5 heating rates per unit volume for solar qa and terrestrial Qa Qe 0 and (2.6) _ qe radiation as _ _ Qe Qa _ ð ð 2 5 S, (2.7) T T _ 1 _ 1 e a Q 5 q_ (x, t) dV and Q 5 q_ (x, t) dV, a A a e A e s V s V which combine to yield an expression for the entropy (2.2) production as the difference between the net entropy

Unauthenticated | Downloaded 10/01/21 05:37 PM UTC 1724 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 74 loss by longwave emission and the net gain by solar absorption: Q_ Q_ S_ 5 e 2 a 5 1 2 1 _ Qa , (2.8) Te Ta Te Ta or

h _ 5 S_ Qa Te , (2.9) where the Carnot efﬁciency h is

T 2 T h 5 a e . (2.10) Ta The climate system does no external work on its surroundings. Any internal work manifests itself as material FIG. 1. Contour plot of the maximum material entropy pro- _ entropy production. A fraction h of the energy absorbed duction S* (EPU) as a function of the planetary solar absorption temperature Ta and terrestrial effective emission temperature is involved in the production of entropy. 1/4 22 Te*5 [(1 2 a)S0/4s] for Earth’s solar constant S0 5 1361 W m . The effect of energy storage may also be incorporated The emission temperature is a decreasing function of the plane- into the analysis. Hansen et al. (2005) suggest the ocean tary albedo a. The plus sign indicates the approximate situation 22 5 ; has been storing energy at the rate of about 1 W m . for Earth’s climate with Te* 255 K and Ta 275 K for a pro- _ Then we modify the analysis to include this storage, duction S ; 68 EPU. _ assuming a steady rate of energy Estor and entropy storage S_ . Then (2.1) and (2.5) are replaced by stor _ 5 _ 5 s *4 # s *3 Se Qe/Te Te /Te Te . (3.2) _ 2 _ 5 _ Qa Qe Estor and (2.11) _ _ Thus, the emission temperature is greater than or equal to Q Q $ e 2 a 5 S_ 2 _ the effective emission temperature Te Te*.Asaconse- Sstor , (2.12) Te Ta quence, the maximum material entropy production (2.8) is, for a given absorption temperature Ta, respectively. A positive (negative) energy storage may be subsumed into the absorption (emission) term to (1 2 a)S S_ * 5 sT*3 2 p . (3.3) deﬁne a net heating gain (loss). An entropy storage may e 4T be subsumed into the entropy production term to a deﬁne a net production. The effects of storage are not Using the energy balance statement, we can also write addressed further. this maximum as T* 3. Zircon: Maximum material entropy production S_ * 5 1 2 e sT*3 5 hsT*3 . (3.4) T e e We apply the formalism of section 2 to an idealized a planet, Zircon, whose climate engine is in a steady state. Figure 1 plots the dependence of the maximum en- Introducing the solar constant Sp and the planetary al- tropy production as a function of the solar absorption bedo a, the energy balance (2.6) is written as temperature and the effective temperature (or, equivalently, the albedo) for Earth’s solar constant 2 Sp 2 _ 5 2 a [ s 4 5 _ S0 5 1361 W m (Hartmann 2016). Physically re- Qa (1 ) Te* Qe , (3.1) 4 alistic values of positive entropy production (S_ . 0) are found in the lower-right portion of the diagram for where s is the Stefan–Boltzmann constant. Here, Te* is the effective emission temperature based on the energy absorption temperatures greater than the effective . balance (3.1). It is the blackbody emission temperature temperature Ta Te*. The production increases with for which the planet would be in radiative equilibrium. It increasing absorption temperature: is shown in appendix A that the maximum loss of en- ›S_ * sT*4 tropy from a planet is that due to uniform emission at 5 e . 0. (3.5) ›T T2 this effective temperature: a a

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TABLE 1. Total, tropical, and extratropical solar heating rates for the globe (GL) and for the Northern and Southern Hemispheres (NH and SH, respectively) based on 2000–16 CERES data, with the ENSO signal removed using a Mann–Kendall signiﬁcance analysis. The global tropics extend from 308Sto308N. The Northern (Southern) Hemispheric tropics extend from the equator to 308N (308S). The extratropics lie poleward of the tropics.

Value Variable Quantity GL NH SH

_ 22 Total solar heating Qa Mean (W m ) 240.56 240.30 240.83 2 2 Trend (W m 2 century 1) 1.47 1.87 0.20 Significance (%) 74 81 4 1 _ 22 Tropical solar heating Qa Mean (W m ) 304.90 301.63 308.64 2 2 Trend (W m 2 century 1) 20.70 22.96 1.66 Significance (%) 18 74 38 2 _ 22 Extratropical solar heating Qa Mean (W m ) 176.22 178.96 173.48 2 2 Trend (W m 2 century 1) 4.00 8.19 22.86 Significance (%) 99 99 90

But the production decreases with increasing effective ›S_ * sT*4 5 e 5 3:28 EPU K21 and (3.8) temperature ›T T 2 ! a !a _ 3 _ 3 ›S* 3T 4sT* ›S* 3T 4sT* 2 52 1 2 a e , (3.6) 52 1 2 a e 522:86 EPU K 1 , (3.9) › ›T* 4T* T Te* 4Te* Ta e e a provided Te* . 3Ta/4.Theenergybalanceequation › ›a 52 : 21 [see (3.1)] indicates that the effective temperature is only or, with Te*/ 0 91 K (%) , the sensitivity to a a function of albedo for fixed solar constant. Differenti- change in albedo is ! ating this relation yields ›Te*/›a 52Te*/[4(1 2 a)]. Then _ ›S* 3T S 2 the change in production due to a change in albedo is 5 12 a p 5 2:60 EPU(%) 1. (3.10) ›a 4T* 4T ! ! e a _ _ › ›S* ›S* T* 3T Sp 5 e 5 1 2 a , (3.7) The finite change in entropy production due to a change ›a ›T* ›a 4T* 4T e e a in the planet’s climate can be estimated by and the production increases with increasing planetary ›S_ * ›S_ * . DS_ * ’ DT 1 Da. (3.11) albedo, provided Te* 3Ta/4. ›T a ›a As a specific example of this analysis, we examine a conditions for present-day Earth (Hartmann 2016)with The CERES data (appendix B and Tables 1–3) indicate an albedo of 29.3%. Then the effective temperature is a slight increase in the absorbed solar radiation of 3 2 2 255 K and sTe* 5 940 EPU. Determination of the ab- 2 1 _ 1.47 W m century that corresponds to a trend in the sorption temperature requires calculation of both Q and 2 21 _ a albedo of 0.4% century . We caution the reader that Sa.ThestudyofGoody (2000) provides an example. The the solar radiation trend from CERES is likely affected _ 5 22 _ 5 data in his Table 5 (Qa 230.7 W m and Sa 842.5 by natural variability. Nevertheless, we use the trend EPU) implies an absorption temperature of 274 K. His derived from CERES data to demonstrate the utility of analysis of data from Peixoto et al. (1991) given in his using material entropy production to measure a climate’s Table 3 suggests an absorption temperature of 278 K. We vibrancy. We assume that, because the planet has been assume an absorption temperature of 275 K. Then the warming, there has been a general increase in the ab- planetary entropy production is 68 EPU, in broad agree- sorption temperature. If the surface and tropospheric ment with analyses of climate models (e.g., Lucarini et al. temperatures continue to increase while the albedo de- 2011). The cross in Fig. 1 indicates this estimate for Earth. creases as a result of a decrease in polar ice, we estimate a h 5 2 5 : We find the climate efficiency is 1 Te*/Ta 7 3%. change over the next century of Based on this analysis, we may also speculate on the DT ;12K, DT* ;10:4K, trends in entropy production. The production increases a e (3.12) with increasing absorption temperature and decreasing Da ;20:4%, DS_ * ; 5:5 EPU, effective temperature. The values for the sensitivities of the production to changes in absorption and effective where the change in absorption temperature has been temperatures are taken to be the same as the change in surface

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TABLE 2. Total, tropical, and extratropical terrestrial heating rates for the globe and for the Northern and Southern Hemispheres based on 2000–16 CERES data, with the ENSO signal removed using a Mann–Kendall signiﬁcance analysis.

Value Variable Quantity GL NH SH

_ 22 Total terrestrial heating Qe Mean (W m ) 239.69 240.37 239.02 2 2 Trend (W m 2 century 1) 0.09 1.01 21.43 Significance (%) 4 70 11 1 _ 22 Tropical terrestrial heating Qe Mean (W m ) 259.68 257.40 261.97 2 2 Trend (W m 2 century 1) 21.18 23.98 0.47 Significance (%) 66 93 4 2 _ 22 Extratropical terrestrial heating Qe Mean (W m ) 219.71 223.34 216.07 2 2 Trend (W m 2 century 1) 1.98 5.14 0.09 Significance (%) 90 99 4

temperature. These estimates suggest an increase in estimate the effect on the material entropy produc- the entropy production of about 10%. tion, as these radiative properties tend to equalize. We The Zircon model assumes that the emission from the demonstrate that equalization of the emission tempera- planet is uniform at the effective emission temperature. ture is a sign of increasing entropy production, but The next section examines the impact of horizontal equalization of the absorption temperature or albedo variations in the emission and absorption temperatures is a sign of decreasing entropy production. and in the associated energy fluxes. The global emission and absorption temperatures are defined in terms of the regional temperatures and heating by 4. Janus: Nonuniformity in emission temperature, ! absorption temperature, and albedo _ 1 2 Q 1 Q_ Q_ e 5 e 1 e and (4.1a) As was motivated in section 1, in order to analyze 1 2 Te 2 Te Te the effects of horizontal variations in the emission ! _ 1 2 Q Q_ Q_ and absorption processes on the global material en- a 5 1 a 1 a 1 2 , (4.1b) tropy production, we introduce a conceptual model: Ta 2 Ta Ta Janus. Janus is a hypothetical planet divided into two regions of equal area with distinctly different radia- where the superscript plus and minus signs denote the tive properties. In the longwave part of the spectrum, tropical and extratropical regions, respectively. Re- one half the planet emits blackbody radiation at a gional heating and entropy sources are defined as in higher temperature than the other half. Similarly, in (2.2), where the volume of integration is now over each the shortwave, one half the planet has a different region rather than the whole system. Absorption and absorption temperature and albedo than the other emission temperatures for the regions follow in analogy half. Janus is in a steady state with no storage. We with the global temperatures defined using (2.4). The

TABLE 3. Total, tropical, and extratropical effective emission temperatures for the globe and for the Northern and Southern Hemispheres based on 2000–16 CERES data, with the ENSO signal removed using a Mann–Kendall signiﬁcance analysis.

Value Variable Quantity GL NH SH

Total emission temperature Te* Mean (K) 254.99 255.17 254.81 2 Trend (K century 1) 0.02 0.27 20.38 Significance (%) 11 70 14 1 Tropical emission temperature Te Mean (K) 260.14 259.57 260.71 2 Trend (K century 1) 20.30 21.00 0.09 Significance (%) 66 92 4 2 Extratropical emission temperature Te Mean (K) 249.50 250.52 248.46 2 Trend (K century 1) 0.56 1.44 0 Significance (%) 90 99 0

Unauthenticated | Downloaded 10/01/21 05:37 PM UTC JUNE 2017 B A N N O N A N D L E E 1727 ! ! Q_ Q_ S_ 5 e 2 a [ S_ 1 S_ d d d d e d a . (4.3) Te Ta

Changes in the regional energetics sum to yield the global change. For example,

1 1 2 dQ_ 5 (dQ_ 1 dQ_ ) and (4.4) e 2 e e 1 1 2 dQ_ 5 (dQ_ 1 dQ_ ). (4.5) a 2 a a

a. Entropy production changes inferred from changes in emission The change in entropy production associated with the change in the planet’s longwave emission is FIG. 2. Difference in the regional emission temperatures of Janus 1 " !1 !2# as a function of the difference in tropical emission temperature Te _ _ 1 Q Q from the effective temperature Te*, subject to the constraint that the dS_ [ d e 1 d e . (4.6) total OLR is constant. The decrease in the extratropical temperature e 2 T T 2 e e Te (blue curve) is greater than the increase in the tropical temperature (red curve). The area average of the two emission tempera- Assuming blackbody radiation ﬂuxes, the change in tures (purple curve) decreases, but the global emission temperature emission (4.4) is Te (black curve) increases with increasing temperature difference. The large colored plus signs indicate the values from an analysis of 1 the CERES data (Table 1 and appendix B). dQ_ 5 (4sT13 dT1 1 4sT23 dT2), (4.7) e 2 e e e e

and the entropy change (4.6) is factor of 2 in (4.1) accounts for the equal areas of the two regions being half that of the whole system. 1 dS_ 5 (3sT12 dT1 1 3sT22 dT2). (4.8) Figure 2 illustrates some of the consequences of these e 2 e e e e deﬁnitions in terms of the entropic temperatures as in- If the total terrestrial emission does not change, then verse, heating-weighted averages. For example, the nu- 1 2 _ 5 _ 1 _ 5 merical average of the regional emission temperatures is dQe dQe dQe 0, and we have a statement of less than the global effective temperature but the emis- equalization: sion temperature is greater than the effective tempera- s 13 1 52 s 23 2 ture (Fig. 2). This latter fact is consistent with the proof in 4 Te dTe 4 Te dTe . (4.9) appendix A that the maximum entropy loss occurs for A decrease in tropical emission temperature compen- uniform blackbody emission at the emission temperature. sates for an increase in extratropical emission temper- The analysis of Fig. 2 shows the CERES values given in ature. Then the emission contribution to the change in Table 3 with the emission temperatures T*, T1,andT2 e e e entropy production (4.8) equal to 255.0, 260.1, and 249.5 K, respectively, while (4.1a) implies an emission temperature T 5 255.15 K. s 22 2 e S_ 5 3 Te 2 Te 2 We conclude that using the effective emission tempera- d e 1 1 dTe . (4.10) 2 Te ture Te* for the emission temperature in the Janus model is a good approximation for Earth. Assuming a lower emission temperature in the extra- 2 2 1 . The energy balance (2.6) implies that a change in the tropics [(1 Te /Te ) 0] and the term is positive for 2 . solar absorption is balanced by a change in the terres- polar warming (dTe 0), the entropy production would trial emission increase. This result proves the assertion that equalization in the emission temperature may be a sign of in- _ 2 _ 5 dQa dQe 0, (4.2) creasing entropy production. Analysis of the CERES data (appendix B and Table 3) where the change/differential in a quantity is denoted yields the values for the terms in (4.8). These results imply with the symbol d. The differential of the entropy an increase in entropy production (4.8) at the rate of about 2 budget (2.8) is 1.24 EPU century 1. Thus, the analysis suggests that

Unauthenticated | Downloaded 10/01/21 05:37 PM UTC 1728 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 74 equalization is having a small impact on entropy production. This result is consistent with Fig. 2, which shows the CERES global emission is close to being uniform. b. Entropy production changes inferred from changes in absorption The change in entropy production caused by changes in the solar absorption is

" !1 !2# Q_ Q_ S_ [21 a 1 a d a d d . (4.11) 2 Ta Ta

Using the energy balance (4.5), we find ! ! _ 1 _ 2 _ 1 _ 2 S_ 5 1 Qa 1 1 Qa 2 2 1 dQa 1 dQa d a 12 dTa 22 dTa 1 2 . 2 Ta Ta 2 Ta Ta FIG. 3. Difference in the regional absorption temperatures of Janus as a function of the difference in tropical absorption temper- (4.12) 1 5 ature Ta from the initial global temperature Ta 275 K subject to the constraint that the areal-average absorption temperature (pur- If the total solar absorption does not change, dQ_ 5 1 2 a ple curve) is constant. The decrease in the extratropical temperature _ 1 _ 5 2 dQa dQa 0, then we have Ta (blue curve) equals the increase in the tropical temperature (red ! curve). The area average of the two absorption temperatures (purple 1 2 _ _ curve) is constant, but the global temperature Ta (black curve) in- 1 Q 1 Q 2 1 1 1 2 S_ 5 a 1 a 2 2 _ creases with increasing temperature difference. d a 12 dTa 22 dTa 2 1 dQa . 2 Ta Ta 2 Ta Ta (4.13) Figure 3 illustrates some consequences of the defini- The first group of terms in (4.13) indicates an increase tion (4.1) on the absorption temperatures. Here, we use in entropy production with increasing regional absorp- the global tropical and extratropical values of Table 1 tion temperatures. We note that the absorption entropy 1 5 : 22 2 5 : 22 and define Qa 304 90Wm and Qa 176 22Wm , tendency (4.12) can also be expressed in terms of the respectively. The regional absorption temperatures are global absorption temperature as then varied linearly from the assumed global value of ! 275 K of Zircon. Even though the area-average tem- Q_ Q_ S_ [2 a 521 _ 1 a perature is that of Zircon, the global absorption tem- d a d dQa 2 dTa . (4.14) Ta Ta Ta perature is greater for Janus. Equalization of the regional temperatures reduces the global temperature Then using the differential of the expression (4.1b) yields and by (4.14) reduces the entropy production. an expression for the change in global temperature relative The second group of terms in (4.13) proves the asser- to the regional changes in temperature and heating: tion that albedo equalization may be a sign of decreasing ! entropy production. Assuming a lower absorption tem- _ 1 2 Q 1 Q_ Q_ perature in the extratropics, the second term is negative a dT 5 a dT1 1 a dT2 2 a 12 a 22 a for increasing polar absorption, and the entropy pro- Ta 2 Ta Ta duction would decrease. Because an increase in albedo 1 2 1 1 1 2 1 _ 2 1 1 2 1 _ results in a decrease in absorption, this result proves 1 dQa 2 dQa . 2 Ta Ta 2 Ta Ta the assertion. (4.15) We estimate the possible effect of changes in the distribution of solar heating and in the absorption tem- This result indicates that the change in global absorption peratures on the entropy production. We assume the temperature Ta depends on the change in the regional CERES solar heating rates of Table 1 and assume the 5 : 1 5 : temperatures and on the redistribution of the solar absorption temperatures Ta 275 0K, Ta 285 0K, 2 5 : heating. Increasing the regional temperatures will in- and Ta 259 3 K. For this case, the material entropy 1 . . 2 crease the global temperature. Because Ta Ta Ta , production is 68 EPU. The effects of changes in the ab- increasing the tropical (extratropical) absorption will sorption temperatures and heating on the entropy pro- increase (decrease) the global temperature. duction are summarized in Table 4. Increases in the

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TABLE 4. Effect of individual changes in regional absorption temperatures and heating rates on the global absorption temperature and entropy production. The base-state heating rates are given in Table 1, with the absorption temperatures taken to be Ta 5 275:0K, 1 5 : 2 5 : Ta 285 0 K, and Ta 259 3 K. For these settings, the entropy production is 68 EPU. 1 2 D 1 D 2 D _ 22 D _ 22 D DS_ Case Ta (K) Ta (K) Qa (W m ) Qa (W m ) Ta (K) (EPU) 1 1 0 0 0 0.59 1.88 2 0 1 0 0 0.41 1.31 3 0 0 1 0 0.02 21.75 40 0 0 1 20.03 21.93 50 0 21120.05 20.17

absorption temperatures increase the production, with material entropy production. By drawing an analogy a tropical increase having the larger effect. Increases between the spatial division of the tropics and extra- in the solar heating decrease the production, with an tropics and the temporal division of day and night, we extratropical increase having the larger effect. If an deduce that a decrease in the diurnal cycle in tempera- extratropical increase were compensated by a tropical ture would be a sign of increasing entropy production. decrease, then a small decrease in the production Similarly, by relaxing the gray-radiation approximation, would occur. we may draw an analogy between the spatial division Table 5 presents the effect of the entropy production and the variation in the absorption spectrum. We then due to one climate change scenario described by a 28C deduce that a change toward a more uniform absorption increase in the regional solar absorption temperatures spectrum (e.g., caused by an increase in greenhouse gas along with an increase in the extratropical heating rate loading in the atmosphere) would also be a sign of in- 2 of 4.00 W m 2 compensated by a tropical decrease of creasing entropy production. 2 0.70 W m 2. These heating trends are those currently evident in the global CERES data (Table 1). The emission 5. Earth, Venus, Mars, and Titan temperature is assumed to be 255.20 K. The impact of these changes on the global absorption temperature and We apply our formulation of Zircon (section 3)tofour entropy production is summarized in Table 5 for various terrestrial planets in the solar system whose basic radiative choices of the absorption temperatures. In each case, attributes have been recently summarized (Schubert and the global absorption temperature increases by slightly Mitchell 2013). Table 7 and Fig. 4 presents the comparison less than 28C, and the entropy production decreases by using a planetary temperature Tp, defined as less than 0.4 EPU. These results indicate that, for the scenario under consideration, the change in the entropy S 1/4 T 5 p , (5.1) production is relatively insensitive to the choice of ab- p 4s sorption temperature. Table 6 compares the change in entropy productions of which is the effective temperature of each planet with the Northern and Southern Hemispheres with that of the zero albedo. Then the maximum entropy production S_ [ s 3 global results given as case B in Table 5. The opposite (3.3) may be nondimensionalized using p Tp as tendencies of the solar fluxes between the two hemispheres imply large but opposite trends in the entropy production. It is unclear if the change in each production occurs or if TABLE 5. Effect on the global absorption temperature and entropy production for various temperatures. The global warming 2 there is an entropy exchange between the hemispheres. If scenario assumes an increase of 28Ccentury 1 in regional temper- the former case holds, then the trends suggest a large in- atures and extratropical and tropical heating rate trends of 4.00 2 2 crease in entropy production in the Southern Hemisphere and 20.70 W m 2 century 1, respectively. The extratropical absorp- with a large decrease in the Northern Hemisphere. tion temperature is adjusted so that the global emission temperature is fixed at 255.2 K. Case B is the benchmark case analyzed in Table 4. c. Consideration of temporal variation in the heating Case T (K) T1 (K) T2 (K) h (%) S_ (EPU) dT (K) dS_ (EPU) and of nongray absorption spectra a a a a A 270 280 254.3 5.48 52 1.85 0.00 It is also worth noting the implication of other cli- B 275 285 259.3 7.20 68 1.85 20.11 matically relevant situations, such as the diurnal cycle in C 280 290 264.2 8.85 83 1.85 20.21 temperature and the variations in the absorption spec- D 275 280 266.8 7.20 68 1.92 0.12 trum. The Janus analysis indicates that equalization of E 275 285 259.3 7.20 68 1.85 20.11 2 the emission temperatures could be a sign of increasing F 275 290 252.4 7.20 68 1.78 0.33

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TABLE 6. Trends in the total absorption temperature and entropy ‘‘ridge’’ of maximum entropy described mathematically production for the global (GL), Northern Hemisphere (NH), and by (3.6). This relation indicates that the entropy pro- Southern Hemisphere (SH) cases. The global warming scenario 2 duction for fixed absorption temperature is a maximum assumes an increase of 28Ccentury 1 in regional temperatures. The extratropical and tropical heating rate trends, based on the CERES for an emission temperature three-fourths the absorp- data (see Table 1), are indicated in the second and third columns. tion temperature: Te 5 (3/4)Ta. We note that for this The total and tropical absorption temperatures are 275 and 285 K in temperature relation the efficiency is exactly 25%. each case. The extratropical absorption temperature is adjusted so Physically the presence of the ridge results from that the global emission temperature is fixed. The efficiency and competition between the change in entropy production entropy production are 7.2% and 68 EPU for each case. due to a change in solar heating and the associated 1 2 D _ 22 D _ 22 D DS_ Case Qa (W m ) Qa (W m ) Ta (K) (EPU) change in the terrestrial emission temperature. From GL 4.00 20.70 1.85 20.11 (2.8), a change in production is described by NH 8.19 22.96 1.67 24.21 _ _ SH 22.86 1.66 2.14 8.99 Q Q S_ 5 1 2 1 _ 2 a 1 a d dQa 2 dTe 2 dTa . (5.3) Te Ta Te Ta _ S T T3 irr 5 1 2 e e . (5.2) Production increases because of an increase in solar S_ T T3 p a p heating, a decrease in emission temperature, and an increase in absorption temperature. In the steady state, The data for the solar constants, albedos, heating rates, an increase in solar heating implies an increase in the and atmospheric/surface temperatures are taken di- terrestrial emission and a concomitant increase in the _ 5 _ 5 s 3 rectly from Figs. 3–6 of Schubert and Mitchell (2013) for emission temperature. Then dQa dQe 4 Te* dTe*, Venus, Mars, and Titan. The absorption temperatures and taking Te* 5 Te, the change in production is are calculated following (2.4). For example, their Fig. 4 _ _ for Venus indicates solar absorption rates of 100, 30, and _ Te 3 Qa Qa 22 dS 524 2 dT 1 dT . (5.4) 20 W m at the temperatures of 250, 400, and 755 K, T 4 T2 e T2 a 2 a e a respectively. Then the total heating is 150 W m 2, and the entropy gain of the Venusian atmosphere is Changes in production due to solar heating dominate 502.2 EPU, implying an absorption temperature of those due to the increase in emission temperature, pro- 299 K. Schubert and Mitchell (2013) incorrectly average vided Te/Ta . 3/4. The coefficient multiplying the first the temperatures of absorption (rather than their in- term in (5.4) is negative for planets above the dashed line verse) to obtain an absorption temperature of 345 K. in Fig. 4 and positive below. As a consequence, their entropics and Carnot efficien- Consider a situation in which a planetary atmosphere cies differ from those in Table 6. We take the emission evolves while maintaining its level of entropy production. temperature to be the effective temperature for each As its albedo increases, its effective temperature would planet. This choice will slightly overestimate the entropy decrease and, by (5.4), its entropy production would tend production and efficiency. to increase. To maintain its entropy production, (5.4) Inspection of Table 7 and Fig. 4 indicates that the indicates that the absorption temperature would need to planets Earth, Mars, and Titan share general attributes decrease. Thus, a planet could maintain its entropy proof similar albedo and Carnot efficiency. Venus arises as a duction as its albedo increases by reducing its absorption distinctly different climate system with high albedo and temperature. Graphically the planet would evolve from large efficiency. Venus appears in Fig. 4 to lie close to a its initial position in Fig. 4 with a downward, leftward

TABLE 7. Comparison of the global energetics and entropics of four terrestrial planets.

Quantity Venus Earth Mars Titan

22 Solar constant Sp (W m ) 2500 1361 608 14.8 Planetary albedo a (%) 76 29.3 23 30 _ 22 Solar absorption QSW (W m ) 150 241 117 2.6 1/4 Planetary temperature Tp 5 (Sp/4s) (K) 324 278 228 89.9 Effective emission temperature Te*(K) 227 255 213 82.3 Solar absorption temperature Ta (K) 299 275 230 85.1 h Carnot efﬁciency max (%) 24.1 7.3 7.2 3.3 S_ Planetary entropy production p (EPU) 1929 1223 668 41 Entropy production S_ * (EPU) 159 68 39 1

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provide essential information and may be used to describe the evolution of a climate system. The comparative planetary climatology of section 5 presents an example. Another application lies in the area of climate change. Using a heat-engine analysis, the question of climate change can be displayed on a Zircon diagram like Fig. 1. In a steady state, the climate’s entropy production will evolve because of the changes in both the absorption and the emission temperatures. The effective emission temperature is directly related to the planetary albedo. The trend of the planetary albedo in climate model simulations (Bender 2011) indicates decreases in albedo of 2 1.5% century 1 that imply, using (3.10),a4-EPU(1EPU5 2 2 1mWm 2 K 1) or 6% decrease in the entropy production, provided there is no change in the terrestrial entropics. Estimates for global warming suggest surface temperatures FIG. 4. Entropy production in percentage of the planetary en- S_ increases in the range of 28–58C for the next century (IPCC tropy production constant p as a function of the effective emission temperature and absorption temperature. The temperatures are 2014). A concomitant increase in the absorption tempera- 5 s 1/4 normalized by the planetary temperature Tp (Sp/4 ) . The red ture would imply, using (3.8), a 6–16-EPU or 9%–24% in- 5 dashed line corresponds to Te* (3/4)Ta. For points above this crease in the entropy production. This line of speculation dashed line, the entropy production decreases with increasing emission temperature (decreasing albedo a). The solar constant is shows why it is important to compute both the absorption S_ 5 s 3 and emission temperatures in order to assess the evolution Sp and p Tp . of Earth’s climate system. The effective emission temperature provides a quick trajectory. At the ridgeline, a further increase in albedo estimate of a lower bound to the emission temperature. could be accomplished at constant production only if the Standard radiative calculations of the local solar heating _ absorption temperature did not change. Below the ridge- rates qa should be used along with temperature profiles to _ line, an increase in albedo would reduce the emission determine the local entropy gain sa due to the absorption. temperature but require an increase in absorption tem- Then the absorption temperature may be determined perature. Such an increase, however, would be unphysical. using (2.4a). Goody (2000) provides the only direct esti- If the atmospheric circulation maintains its intensity and its mate of this kind. We encourage scientists running cli- level of entropy production, the lapse rate of the atmo- mate models and reanalyses to make available and/or sphere is likely to remain constant. Therefore, at least from publish their results of similar calculations of the radia- the viewpoint of a one-dimensional radiative–convective tion heating and entropy exchanges. At present, only the model, if the emission temperature decreases, the ab- NASA MERRA climate simulations (Rienecker et al. sorption temperature must decrease (Earthlike condition). 2007) provide some of this essential information. This analysis suggests that the equilibrium states in the Complementary to the radiation studies, there should lower-right corner of the entropy production diagram are be direct analyses of the material entropy production unstable and that Venus’s atmosphere lies at the extremity budget of the climate simulation. The entropy production of the stable equilibrium regime. of the climate system is analogous to the gross domestic product (GDP) of a country’s economy and should be analyzed sector by sector. For example, the entropy 6. Conclusions production by viscous dissipation provides input on the A climate system is a heat engine governed by the laws strength of the kinetic energy generation rate of the of thermodynamics. Quantifying this engine in terms of its climate system. Similarly, the entropy production by efficiency and entropy production requires determining heat conduction provides input on the vertical and both the energetics and entropics of the system’s exchange horizontal transport of thermal energy from the regions of radiation with its surroundings. This exchange requires of solar absorption to the regions of terrestrial emission knowledge of the solar and terrestrial heating rates and to space. Last, the entropy production by the diffusion the associated entropy exchanges. Together, this infor- of water vapor, by nonequilibrium phase changes, and mation yields the entropically defined absorption and by hydrometeor drag provides input on the strength of emission temperatures (2.4) that quantify the engine’s the hydrologic cycle of the system. Budgets of these en- efficiency and entropy production. These temperatures tropy productions calculated directly from the climate

Unauthenticated | Downloaded 10/01/21 05:37 PM UTC 1732 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 74 simulation should be compared with the radiation gains zenith angle. We assume isotropic emission in local and losses along with the entropy storage. Such analyses thermodynamic equilibrium. The integration is over the will more fully quantify the simulation of a climate system’s parameters of time t, planetary surface area A, solid angle heat engine and its evolution. In order for the model’s V of outward directions, and emission optical pathlength entropy budget to be accurate, the physical parameteri- t. For the generic function g of these parameters p,we zations of the model physics should be made ‘‘second law’’ define its time-mean area-mean definite integral as compliant. For example (Gassmann and Herzog 2015), the ð ð ð ð ð t ‘ 1 2 1 turbulent fluxes of mass, momentum, and enthalpy should dp g(p) [ dt dA dV dt g(p). (A.2) t 2 t A p lead to positive definite terms in the model’s entropy 2 1 t1 2 0 equation (along with boundary flux terms). The emission and absorption temperatures of a cli- The limits of the integral with respect to time corre- mate system quantify its Carnot efficiency and material spond to the period of interest. entropy production. This information provides global Differentiation of I with respect to the Lagrange insight into a planet’s vibrancy and its evolution. multiplier, dI/dl 5 0, yields the constraint that the net mean emission is that of a blackbody at the effective Acknowledgments. We thank Eugene E. Clothiaux temperature Te*: and Jerry Y. Harrington for discussions on entropy loss ð by longwave emission and on the calculus of variations. 2t/cosu 5 s *4 dp B[T(x, t)]e Te . (A.3) We thank the three reviewers, Editor Olivier Pauluis, and Technical Editor Richard Brandt for their con- The variation of the integral I implies an extremum for structive comments. dI/dT 5 0, which yields the condition APPENDIX A d B d(B) 2 l 5 0. (A.4) dT T dT Maximization of the Material Entropy Loss by This condition implies that the temperature should be Longwave Emission constant: T 5 (3/4)l21. Then the energy constraint (A.3) We complement the analysis of O’Brien (1997) and implies that the uniform temperature is the effective present a yardstick for the maximum entropy loss of a emission temperature T 5 Te*. This analysis is limited climate system due to longwave emission subject to the to a nonscattering atmosphere with a blackbody lower constraint of energy conservation. Formally, we maxi- boundary. These constraints insure that only photons mize the time-mean loss subject to the constraint that emitted upward have a finite probability of escaping the the time-mean longwave radiation energy loss is the atmosphere. outgoing longwave radiation (OLR) defined by the ef- This analysis indicates that the maximum entropy loss fective emission temperature Te*. Mathematically, we for a given OLR is that from an isothermal climate sys- use the calculus of variations and define the integral I: tem. Then the emission temperature is a minimum. In ð reality, the system is nonisothermal and there are internal B[T(x, t)]e2t/cosu I 5 dp components of the longwave emission and absorption T(x, t) processes that comprise an exchange of radiation energy ð between the warmer and cooler parts of the system. This 1 l sT*4 2 d B[T( , t)]e2t/cosu . (A.1) e p x exchange involves the net heating of the cooler parts and the de-heating of the warmer parts. Thus, the exchange The first integral is the entropy loss due to longwave increases the entropy of the climate system. This entropy emission to space, while the term multiplied by the increase is misleadingly assigned as material entropy Lagrange multiplier l istheenergyconstraint.The production when the effective emission temperature is last integral in (A.1) definestheenergyloss.TheLa- used as the emission temperature. grange multiplier l is a constant with the dimensions of inverse temperature. The Planck irradiance is a APPENDIX B function of temperature T: pB(T) 5 sT4,wheres is the Stefan–Boltzmann constant. The gray, absorption path- Analysis Technique of CERES Data length t varies from the top of the atmosphere (t 5 0) to beyond the lower boundary of the climate system (t 5‘). The CERES data (EBAF-TOA Ed2.8; Loeb et al. It is a function of position x and time t. The angle u is the 2009) consist of the monthly mean outgoing longwave

Unauthenticated | Downloaded 10/01/21 05:37 PM UTC JUNE 2017 B A N N O N A N D L E E 1733 _ radiation QLW, the reflected solar radiation, and the incoming radiation as a function of latitude and longi- tude with a resolution of 18. The difference in the incoming and reflected radiation yields the net absorbed _ 3 solar radiation QSW. There are 360 180 data points centered on the 0.58 points. The data analyzed in this study span March 2000–February 2016, consisting of 192 months of data. To eliminate the annual variation, we obtain yearly time series by averaging consecutive 12-month values: March 2000–February 2001, March 2001–February 2002, etc. This averaging produces 16 data points in time. Hartmann and Ceppi (2014) use the same method in analyzing the CERES data. To apply the Janus model, we group the data spatially into relatively warm and cool pairings. Three pairings suggest themselves: a warm tropical and cool extratropical pairing for the globe and two similar pairings for each hemisphere. The data are averaged areally over the four regions corresponding to the latitude belts 8 8 8 8 8 8 8 8 FIG. B1. Monthly mean CERES (a) outgoing longwave radiation 90 –30 S; 30 S–0 ;0–30 N; and 30 –90 N. These regions _ _ Q and (b) absorbed shortwave radiation Q as a function of are the tropical and extratropical regions of each LW SW time (March 2000–February 2016) for the tropics (red) and the hemisphere. The heating for the global tropical and extratropics (blue). Mean values and trends over the 16-yr period extratropical pairings is displayed in Fig. B1 with are indicated. The trends are computed using the Mann–Kendall dashed lines. In the tropics (red dashed lines), there method. The superimposed straight lines are the slopes corre- are notable interannual variabilities, including the sponding to the trends. The leftmost values of the straight lines are the corresponding mean values minus the trends multiplied by one- anomalously large values during 2015–16. This time half of the period (8 yr). period corresponds to one of the strongest El Niño events in recent years. Therefore, for the purpose of evaluating decadal time-scale variations, it is prefer- equalizations occur during the analysis period. Evaluation able to eliminate the El Niño–Southern Oscillation using the Mann–Kendall test (Mann 1945; Kendall 1962), (ENSO) signal. which has been adopted for trend analysis of meteoro- The elimination of the ENSO signal is achieved by logical data (Shea 2014), indicates that the positive ex- linearly removing the signal using the same method as tratropical global trends are significant above the 90% L’Heureux et al. (2013). Specifically, the ENSO signal confidence level, while the negative trends in the tropics in a heating is calculated by linearly regressing the have lower confidence levels (Tables 1, 2). The trends and heating against the monthly Niño-3.4 index. [We use the the 16-yr-mean values are indicated in the figure. The index provided by the Climate Prediction Center (http:// superimposed straight line in Fig. B1 shows the slope ob- www.cpc.ncep.noaa.gov/data/indices/), and the Niño-3.4 tained from the Mann–Kendall analysis. data are derived from Extended Reconstructed Sea The emission temperatures indicated in Table 3 and Surface Temperature, version 4 (ERSST.v4) (Huang Fig. 2 are computed using the ENSO-removed Q_ as- et al. 2015).] The ENSO-removed heating is then ob- LW suming blackbody radiation. tained by subtracting the product of the linear regression coefficient and the Niño-3.4 index from the original time series. The resulting ENSO-removed heating is in- REFERENCES dicated in Fig. B1 with solid lines. In the tropics, there Bannon, P. R., 2015: Entropy production and climate efficiency. are often large differences between the original radia- J. Atmos. Sci., 72, 3268–3280, doi:10.1175/JAS-D-14-0361.1. tive energy fluxes and the corresponding ENSO- Bender, F., 2011: Planetary albedo in strongly forced climate, as removed fluxes. In the extratropics, the differences are simulated by the CMIP3 models. Theor. Appl. Climatol., 105, very small, as the blue dashed and solid lines are almost 529–535, doi:10.1007/s00704-011-0411-2. indistinguishable. Budyko, M. I., and Y. A. Izrael, 1991: Anthropogenic Climate Change. University of Arizona Press, 485 pp. The trends in Figure B1 show that, for both heatings, Gassmann, A., and H.-J. Herzog, 2015: How is local material en- the extratropical values (blue curves) increase, and the tropy production represented in a numerical model? Quart. tropical values (red curves) decrease, indicating that J. Roy. Meteor. Soc., 141, 854–869, doi:10.1002/qj.2404.

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