Entropy production rates of the climate
GGenerated using the official AMS L A TEX template—two-column layout. FOR AUTHOR USE ONLY, NOT FOR SUBMISSION! J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S
Entropy production rates of the climate G OODWIN G IBBINS ∗ Imperial College London, UK J OANNA
D. H
AIGH
Grantham Institute - Climate Change and the Environment, Imperial College London, UK
ABSTRACTThere is ongoing interest in the global entropy production rate as a climate diagnostic and predictor, butprogress has been limited by ambiguities in its definition; different conceptual boundaries of the climatesystem give rise to different internal production rates. Three viable options are described, estimated andinvestigated here, two of which – the material and the total radiative (here ‘planetary’) entropy productionrates – are well-established and a third which has only recently been considered but appears very promising.This new option is labelled the ‘transfer’ entropy production rate and includes all irreversible processes thattransfer heat within the climate, radiative and material, but not those involved in the exchange of radiationwith space. Estimates in three model climates put the material rate in the range 27-48 mW/m K, the transferrate 67-76 mW/m K, and the planetary rate 1279-1312 mW/m K.The climate-relevance of each rate is probed by calculating their responses to climate changes in a simpleradiative-convective model. An increased greenhouse effect causes a significant increase in the material andtransfer entropy production rates but has no direct impact on the planetary rate. When the same surfacetemperature increase is forced by changing the albedo instead, the material and transfer entropy productionrates increase less dramatically and the planetary rate also registers an increase. This is pertinent to solarradiation management as it demonstrates the difficulty of reversing greenhouse gas-mediated climate changesby albedo alterations. It is argued that the transfer perspective has particular significance in the climate systemand warrants increased prominence.
1. Introduction a. Motivation
The climate is, fundamentally, an entropy-producingsystem. The movement of energy from warmer regions,where it is supplied to the climate, to cooler regions, whereit leaves, is an inevitable consequence of the second law ofthermodynamics and drives the motion and activity of theclimate. The energy transfers are mediated by a myriad ofirreversible processes, for example wind, rain and radia-tion. Each process produces entropy, which must be ex-ported from the system in order to maintain a steady state.The export is by radiation; the supply of low-entropy solarradiation and loss of high-entropy outgoing thermal radi-ation has the net effect of carrying entropy away from thesystem and maintaining temperature gradients. Our cli-mate system exists in this balance.Although entropy in the climate system has been ex-plored in the literature for more than four decades, its util- ∗ Corresponding author address:
Department of Physics, ImperialCollege London, Prince Consort Rd, Kensington, London SW7 2BW,UKE-mail: [email protected] ity in studying the climate has not been established. A ma-jor limitation has been the difficulty in pinning down eventhe concept of a global entropy production rate, which hasbeen challenging both because an intuitive understand-ing of the entropics of the climate is difficult to developand because there are multiple candidate climate-relevantglobal entropy production rates that can be defined for dif-ferent notions of the system’s extent and boundaries, asrecently underlined by Bannon (2015).Our first purpose here is to clarify further the differentdefinitions of global entropy production rates. The twomain perspectives that have gained traction in the litera-ture are one that considers the entropy production due toall radiative and non-radiative irreversible processes (herelabelled planetary ) and one that restricts itself to the non-radiative irreversible processes only (labelled material ).We advance a third, labelled the transfer entropy produc-tion rate, which accounts for the entropy produced by allprocesses that transfer heat within the climate system –radiative and non-radiative – but not that associated withthe thermalization and scattering of incoming solar radia-tion or with the emission of outgoing thermal radiation tospace.
Generated using v4.3.2 of the AMS L A TEX template a r X i v : . [ phy s i c s . a o - ph ] A ug J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S
A summary of the historical evolution of the materialand planetary perspectives is given in Section 2, leadingto an argument for the addition of the transfer perspective.Definitions of the material, transfer and planetary entropyproduction rates are then offered in Section 3 using con-sistent notation, underlining their differences and interpre-tation. The measurement of all three entropy productionrates is demonstrated in Section 4 in three climate models:an energy balance model, an analytic radiative-convectivemodel and an observationally-based standard atmosphericcolumn.Our second purpose is to explore the behavior and pos-sible uses of these global entropy production rates in quan-tifying and understanding the climate. We approach thisin Sections 5a and 5b by demonstrating how each entropyproduction rate responds to greenhouse gas concentrationand albedo changes in the analytic radiative-convectivemodel. This leads to further insight into the physical sig-nificance of the three entropy production rate perspectivesin Section 5c. b. Using entropy production rates
What is an entropy production rate? Returning to thesecond law of thermodynamics, recall that the entropy ofthe universe increases due to each irreversible process. Ifthese processes occur inside an isolated, closed system,that increase must be reflected in the entropy ( S ) of thesystem and they are said to have ‘produced entropy’ insidethe system at a rate Σ : dS closed dt = Σ ≥ U , there is abalance between the fluxes, F , into and out of a steadysystem: dU open , steady dt = = F in − F out (2)This feature is already well-mobilized for simplifying,constraining and explaining aspects of the climate system,as in energy balance models and the concept of radiativeforcing.However, as entropy can be created but not destroyedwithin the system, a different kind of balance emerges:the cross-boundary flow must carry a net outwards flux Note that J is the positive rate of entropy change experienced by thesystem due to the cross-boundary flux. If the import or export processesare themselves irreversible, the synchronous entropy change perceivedby the surroundings due to the same cross-boundary flow will be differ-ent to that perceived by the system. of entropy ( J out − J in ), which equals the total internal pro-duction rate, Σ : dS open , steady dt = = J in − J out + Σ . (3)This entropy production rate can be identified (Peixotoet al. 1991) ‘directly’ by summing the entropy increases( σ i ) due to all the irreversible processes that occur withinthe system: Σ = ∑ i σ i . (4)or ‘indirectly’ as the difference of the cross-boundary en-tropy fluxes: Σ = J out − J in (5)This is a potentially useful constraint: although entropyis produced across a wide variety of processes within thesystem, in steady state it must be exactly exported by thecross-boundary fluxes. Note that there is a distinctionmade here between a flux, which changes the entropy ofthe system by moving energy across the boundary, and aproduction, in which movement of energy within a systemincreases its entropy.Heat delivered at rate F to a region at temperature T increases the entropy there at a rate F / T , which is iden-tified as an entropy flux into the system. This can begeneralized to give a temperature for each paired entropyand energy flux, T : = F / J , which represents an aver-age quality of the energy at the point of entry to or exitfrom the system. For a system with a steady flow of en-ergy F through it, the inflow and outflow temperatures, T in = F / J in and T out = F / J out , are sometimes combinedinto an ‘efficiency’ η = ( T in − T out ) / T in , as in the Carnotefficiency for the maximum work per unit heat input thatcan be extracted by a reversible engine operating betweentwo fixed temperatures. In our case, however, the valuedoes not represent work extracted (as there is no mecha-nism for any extraction), but instead can be interpreted asa summary metric of the irreversibility of the system orthe “lost work” (e.g. in this context, Bannon (2015)). Ifheat flows at a rate F from T hot to T cold , the entropy pro-duction will be exactly σ = F ( / T cold − / T hot ) , providedthe mechanism that mediates the transfer is returned to itsoriginal configuration. The entropy production rate of thesystem can then be related to the temperatures, efficiencyand energy flow by Σ = F ( / T out − / T in ) = F ( η / T out ) .
2. Context a. History
The idea that an entropy production rate might hold pre-dictive significance for the climate was initially “stumbledupon” by Garth Paltridge in the 1970s (his words, Pal-tridge (2005) describing Paltridge (1975)). Motivated bythe idea that the complexity of the climate might lead to
O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S ( F SW − F LW ) / T , summed across thezonal grid cells of his model.This was the observation that sparked the field ofclimate-entropy research. In Paltridge’s zonally-averagedmodel, the difference in radiative heating was interpretedas the meridional heat transfer rate, as further investigatedby Paltridge (1978); Nicolis and Nicolis (1980); Grassl(1981); Wyant et al. (1988), among others.Essex (1984), however, argued Paltridge’s characteri-zation of the Earth’s entropy production rate was funda-mentally “incorrect” as it had failed to account for entropyproduction in the radiation field. In doing so, Essex in-troduced the quantity labelled the planetary entropy pro-duction rate : “the entire entropy production of a steadystate climate is contained in the difference of the entropyof the outgoing terrestrial radiation from the entropy ofthe incoming solar radiation”. This planetary rate hasbeen further explored by, e.g. Lesins (1990); Stephens andO’Brien (1993); Pelkowski (1994); Li et al. (1994); Li andChylek (1994); Wu and Liu (2010).This insight into the role of radiation in producingentropy instigated the division of the climate into non-radiative (material) and radiative sub-systems and the sep-arate tabulations of the entropy produced in each (Essex1987; Goody and Abdou 1996; Goody 2000). Paltridge’smeridional view was mapped in higher-dimensional mod-els to the material entropy production rate , which includescontributions from horizontal and vertical sensible and la-tent heating (as established by Pujol and Llebot (1999)) aswell as all other non-radiative processes.By virtue of its connection with moving, tangible mat-ter, the material perspective has been preferred for appli-cation of a maximization principle, both in experimentaland theoretical studies (e.g. Ozawa and Ohmura (1997);Ozawa (2003); Dewar (2003); Fraedrich and Lunkeit(2008); Labarre et al. (2019)). The contributions dueto component processes have also been considered sepa-rately (e.g. Pauluis and Held (2002a,b); Volk and Pauluis(2010); Lembo et al. (2019)) and the response to changingclimate conditions studied (Singh and O’Gorman 2016;Bannon and Lee 2017). b. Why add the transfer perspective? The key insight that supports the transfer entropy pro-duction rate is that radiative processes can be further cate-gorized according to the roles they play within the climate.This is not a new perspective - Green (1967) argues for it explicitly - but it has not yet been discussed in detail in theentropy production literature.Energy is supplied to and exported from the planet byradiation; this external radiation on average heats warmplaces and cools cold places, continually driving tempera-ture differences in the system. By contrast, internal radia-tion , which is emitted and absorbed by the material of theEarth system, gives a net transfer of heat down-gradient,pulling the system towards thermal equilibrium. It is al-most coincidental that radiation occurs in both of theseroles. Internal radiation is an inherent feature of a translu-cent, warm atmosphere and would occur even if the sys-tem were driven by a non-radiative heat source and sink. Itis not fundamentally different nor necessarily distinguish-able from material heat transfer processes such as conduc-tion. While external radiation determines where and howmuch heating and cooling drives the system, internal ra-diation acts in parallel with material processes to transferenergy between where the external radiation delivers it toand takes it from.This similarity in function between internal radiationand material processes suggests that they might be bestconsidered together, as in the transfer entropy productionrate. This concept of global entropy production has ap-peared only occasionally in the literature, and has not yetbeen thoroughly explored. In Bannon (2015), one of thematerial entropy production rates discussed (his MS3) in-cludes the internal radiative heating processes and so is thetransfer rate discussed here. In Bannon and Lee (2017),the transfer rate is estimated as an upper bound for thematerial production rate. Most recently, Kato and Rose(2020) presents a study in which the transfer entropy pro-duction rate is estimated in Bannon’s simple model andfrom observational data. Here we explore all three en-tropy production rates and their definitions, estimates andresponses to climate change, in order to elucidate the sig-nificance of the transfer entropy production rate concept.
3. Definitions of entropy production rates
The multiplicity of entropy production rates for theEarth’s climate arises because there is not a self-evidentboundary of the system with respect to radiation. Differ-ent perspectives can give different extents of the system,not in terms of physical space (all extend from the litho-sphere to the upper atmosphere) but in terms of how orwhen radiation or heat crosses into and out of the system.The interaction of radiation with matter naturally sug-gests the three perspectives on the climate system thatwere introduced above and can be succinctly described bythe nature of the energy fluxes that cross their boundaries:
Planetary: photons carry entropy and energy into andout of the system as they cross a control volume sur-face beyond the top of the atmosphere. J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S
Transfer: energy enters the system when it is first ab-sorbed by matter on its way from the sun and leaveswhen it is last emitted on its way to space.
Material: every absorption or emission of radiation is acrossing of energy into or out of the material system.Energy is only in the system when it is in matter, notas photons.These systems are nested (listed here from large tosmall), have different cross-boundary entropy fluxes andinclude different sets of irreversible processes. Thus eachmeasures a different global entropy production rate, al-though they all, in some sense, self-consistently describe‘the climate’. This section explains their definitions, dis-tinctions, and interpretations; the question remaining forscientists trying to use entropy production rates as a pre-dictive or diagnostic variable will be their physical rele-vance.An energy balance model (EBM) adapted from Ban-non (2015) and shown graphically in Figure 1 is used todemonstrate which processes are included in each entropyproduction rate. It can be solved analytically (see Ap-pendix A) to give the surface and atmosphere temperaturesas a function of albedo, emissivities and latent and sensi-ble heat flux.The model broadly echoes the global energy budgetdiagrams of Wild et al. (2014), but with a non-standardrearrangement of the radiative energy fluxes to separatethe internal radiation (net flux between surface and atmo-sphere) from the external radiation (fluxes that leave to orenter from the surroundings), reminiscent of the Net Ex-change Formulation discussed by Herbert et al. (2011).The two material terms - latent and sensible heat trans-fer - should be taken as placeholders for the full range ofnon-radiative energy transfer mechanisms, including viacreation and dissipation of kinetic energy in the generalcirculation. Each down-gradient energy transfer causesan entropy production, which are labelled in rectangularboxes. Their values are calculated in Appendix B.For simplicity, heat from the planetary interior and theirreversibility of life are not explicitly treated here, but thedefinitions could be adapted to feature them. Note alsothat the entropy production rates and fluxes defined are forthe whole system, globally and not locally, although theyare quoted in per-area units ( mW/m K). The symbol σ refers to an entropy production rate due to a particular typeof irreversible process, while Σ is the aggregated globalvalue for the system in question. a. The planetary entropy production rate The planetary entropy production rate can be inter-preted as the entropy change of the universe due to the ex-istence of a planet interrupting and thermalizing the solar photons: it is the difference in entropy of the photons inci-dent on the Earth compared to those scattered and radiatedaway from it. The system is defined via a control volumethat surrounds the planet and includes the entropy produc-tion due to all irreversible processes that happen within it,radiative or otherwise (see CV1 in Bannon (2015)).Defined directly for the energy balance model of Figure1, the planetary rate is the sum of all the entropy budgetitems, radiative and non-radiative: Σ planet = σ scatSW + σ atmSW + σ sur fSW + σ LH + σ sens + σ intrad + σ sur fLW + σ atmLW (6)To define the production indirectly, the entropy content( L ν ) associated with the flow of photons of frequency ν (units of 1 / s) as they cross over the boundary of the con-trol volume can be calculated as a non-linear function oftheir spectral intensity I ν : L ν = ( + y ) ln ( + y ) − y ln ( y ) (7)where y = c n h ν I ν . (8)Here c is the speed of light and the number of polarizationstates n =
2. Equation 7 can be derived by analysis ofphotons as a boson gas (Planck (1914), further developedby Rosen (1954)) or from the relationship dL ν = dI ν / T where I ν = B ν ( T ) is the spectral Planck function intensity,which is integrated from zero to the relevant intensity (Ore1955).The entropy flux is then calculated by integrating thisover all wavelengths, in the inbound or outbound hemi-spheric directions and averaged over the surface area ofthe planet: J in / outplanet = A (cid:90) dA (cid:90) d ν (cid:90) hemi cos θ d Ω L ν . (9)This entropy flux in radiation simplifies in the case of ablack body to a simple dependence on source temperature T source , as explained by Planck (1914) and Essex (1984)and demonstrated in the appendix to Wu and Liu (2010): J BB = FT source = σ T source . (10)The factor of 4 / J mat = F / T source ) accounts for the irreversibility of radi-ating into a vacuum (Feistel 2011). From the planetaryperspective, it is this larger entropy flux in the radiationthat crosses the system’s boundary.Scattered solar photons carry much less entropy thanthose thermalized and re-emitted by the planet and so thealbedo of the Earth is a key determinate of Σ planet . Theplanetary perspective has been further subdivided in some O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S F IG . 1. A schematic of a simple two-layer zero-dimensional energy balance model with energy fluxes labelled as colored arrows F and theresulting entropy productions boxed, σ . The energy supplied to the system from the surroundings is shown entering or exiting from the top of thediagram. Three internal energy exchanges are shown: two material processes – latent and sensible heat fluxes – and internal radiation. Scatteringfrom the surface has not been shown but is implied. This diagram mirrors the energy balance study by Wild et al. (2014) and is constructed tomatch Bannon (2015). entropy studies to give related variables, for instance byexcluding the production from the fraction of the solar ra-diation that is scattered (CV2 in Bannon (2015)) or by fo-cusing on only the production which occurs within the at-mosphere (Peixoto et al. 1991). However, we would arguethat the definition as given here is more fundamental. b. The material entropy production rate The material entropy production rate avoids depen-dence on spectral properties of radiation and the solar tem-perature by excluding all radiative processes from the en-tropy tally. In this view, the system is exclusively the mat-ter. The photon gas permeating the atmosphere is con-sidered part of the surroundings and radiative heating andcooling supply the cross-boundary fluxes of energy (Ban-non 2015). The local temperature distribution of the mat-ter is unchanging at steady state and so the net heating ofeach parcel by material processes must be balanced by ra-diative cooling to space or within the climate system, andvice versa (Essex 1987; Goody 2000).The view that motivates this approach is that these ma-terial (or ‘molecular’) processes – such as phase changes,sensible heating, friction and diffusion – are the ones ofinterest in the dynamics of the weather and other tangi-ble aspects of the climate. The material entropy produc-tion has been linked to the kinetic energy conversion in theLorenz Energy Cycle (for example Lucarini et al. (2011))but includes processes that are not related to motion aswell, such as diffusion. In a moist atmosphere, the mate-rial entropy production is dominated by contributions fromthe hydrological cycle (Pauluis and Held 2002a). In our EBM (Figure 1), the material entropy productionrate can be directly specified as: Σ mat = σ LH + σ sens . (11)The net entropy flux into and out of the material systemdue to all radiative heating defines it indirectly: Σ mat = (cid:90) dV − ˙ Q rad ( x ) T mat ( x ) = J outmat − J inmat (12)where ˙ Q rad is the local radiative heating rate (in W/m )and T mat is the temperature of the material where the heat-ing or cooling occurs. There are multiple options for parti-tioning this net radiative heating into a J inmat and J outmat . Thetwo approaches described by Bannon (2015) are to con-sider all absorption of radiation separately from emission(his MS1), or alternatively to separate the solar SW heat-ing from the LW radiative heating and cooling (his MS2).Here we follow Lucarini (2009) in distinguishing regionsof net positive Q rad from those with net negative, whichtakes advantage of the fact (discussed in Goody (2000))that, in steady state, the net local radiative heating must bebalanced exactly by non-radiative cooling (and vice versa)and so the temperatures and energy fluxes calculated inthis way will closely reflect those experienced by the ma-terial processes. These approaches lead to different anal-yses of the energy and entropy fluxes, entropic tempera-tures and efficiency of the material system, but the sameproduction rate.Paltridge’s original meridional heat transport entropyproduction rate can be interpreted as the horizontal com-ponent of the material entropy production rate, which has J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S been estimated to be approximately 15% of the total (Pas-cale et al. 2012). c. The transfer entropy production rate
The transfer system includes matter and the internal ra-diation that travels between matter within the climate sys-tem but not the external radiation before it has interactedwith the matter or after it has been emitted for the last time,which is considered part of the surroundings.The observation that motivates the transfer perspective– that internal radiation plays a role in the climate thatis parallel to the material processes – is particularly evi-dent in the entropy productions. Since internal radiation isemitted and re-absorbed again within the system, only theheat transfer it causes, and not its entropy while a photongas, is relevant to the global entropy production budget.Therefore, the entropy production due to internal radia-tion, σ intrad , takes the same form as the entropy produc-tion due to the material processes in our two-layer model: F ( / T atm − / T sur f ) , where F is the rate of energy trans-port. By contrast, the five entropy productions relating toexternal radiation that are excluded directly depend on thedetails of the radiation spectra.Defined directly for our EBM (in Figure 1): Σ tran = σ LH + σ sens + σ intrad (13)which makes it intermediate in magnitude: Σ tran = Σ mat + σ intrad (14) = Σ planet − ( σ scatSW + σ atmSW + σ sur fSW + σ sur fLW + σ atmLW ) (15)as it includes contributions from the internal radiation,which are not in the material entropy production rate, andexcludes those from external radiation, which are includedin the planetary rate.In the transfer system, the cross-boundary fluxes are theheating or cooling of material upon absorption or emissionof external radiation and so the production can be written: Σ tran = (cid:90) dV − ˙ Q ext rad ( x ) T mat ( x ) = J outtran − J intran (16)The partitioning of the external radiative heating into J intran and J outtran components is, like in the material case, afurther definitional choice. One option would be to sep-arate areas of net positive external radiative heating fromareas of net negative, however we take the more straight-forward approach of separating the absorption of solar ra-diation from the emission of long-wave radiation to space.Then the incoming entropy flux is due to the absorptionof solar radiation: J intran = (cid:90) dV ˙ Q sw ( x ) T mat ( x ) (17) where ˙ Q sw is the heating rate due to SW radiation and T mat is the temperature of the absorbing material.The entropy flux out of the system is due to the LWemission of radiation that is not reabsorbed within the sys-tem. This is the cooling to space, ˙ Q cts , which can be calcu-lated by radiative transfer models if the optical depth andtemperature ( T ) are known (Rodgers and Walshaw 1966;Wallace and Hobbs 2006):˙ Q cts ( z ) = − π (cid:90) d ν B ν ( T ) d T ν ( z , ∞ ) dz (18)where T ν ( z , ∞ ) is the transmittance between z and the topof the atmosphere. Then the outwards entropy flux for thetransfer system becomes: J outtran = − (cid:90) dV ˙ Q cts ( x ) T mat ( x ) (19)Note that the cooling to space is necessarily negative andits sum is exactly the outgoing longwave energy flux leav-ing the planet. It is the temperature from which coolingto space occurs, along with the spectral properties of theradiatively active gases, that determines the shape of theoutgoing emission spectra. The average cooling-to-spacetemperature will generally be close to the emission tem-perature of the planet.In the horizontal, there is negligible net heat transfer byinternal radiation and so the horizontal components of thetransfer and material entropy production rates converge.Therefore, the meridional heat transfer entropy productionrate of Paltridge can equally be identified as the horizontalcomponent of Σ tran .
4. Estimates of the entropy production rates
We now use these definitions to estimate each entropyproduction rate and related variables in three model cli-mates of increasing complexity.For the EBM of Bannon (2015), the entropy produc-tion due to each process can be calculated separately(Appendix B) and the entropy production rates directlysummed. The indirect approach, which focuses on ra-diation fluxes, can equally be applied, as explored inAppendix C. The estimated entropy production rates arelisted alongside the fluxes, implied influx and outflux tem-peratures and resulting efficiencies in the first section ofTable 1.To increase the fidelity with which the climate is rep-resented while maintaining the possibility of analytic so-lutions, we use the analytic radiative-convective model ofTolento and Robinson (2019), which was originally de-signed for flexibility in capturing a wide range of plane-tary climates. It approximates the atmosphere as a radia-tively gray gas with a convective tropospheric region anda stratosphere in radiative balance with two channels of
O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S EBM RCM Std Atmos
Planet Tran Mat Planet Tran Mat Planet Tran Mat F in = F out (W/m ) 341 239 85 343 240 112 346 259 96 J in (mW/m K) 79 869 306 79 857 389 79 929 334 J out (mW/m K) 1358 936 336 1365 933 416 1391 999 382 T in (K) 4334 275 278 4334 280 288 4364 280 288 T out (K) 251 255 253 251 257 269 248 260 252 η (%) 94.2 7.2 8.6 94.2 8.1 6.5 94.7 7.0 12.5 Σ (mW/m K) 1279 67 30 1286 76 27 1312 70 48T
ABLE
1. The energy fluxes ( F ), associated entropy fluxes ( J ), temperatures ( T = F / J ), efficiencies ( η = ( T in − T out ) / T in ) and productionrates ( Σ ) for the energy balance model, radiative-convective model and for a clear-sky standard atmospheric column from each of the three systemperspectives: planetary, transfer and material. shortwave absorption, and is described in more detail inAppendix D. It is powerful because analytic expressionsfor the temperatures and radiative fluxes in the model al-low for exact calculation of the entropy production ratesvia the indirect definitions. The entropy production rateresults are quoted in the second section of Table 1.As the complexity of the climate model increases, thevalue of the indirect method becomes more apparent: itis much simpler to characterize the cross-boundary radia-tion or radiative heating than to quantify the contributionfrom every irreversible process within the system. Someof the necessary radiative information (such as shortwaveand longwave heating rates) is supplied as standard insome reanalysis products, but for calculation of the de-tailed spectra for the planetary rate or the cooling to spacefor the transfer rate offline radiative transfer calculationsare needed. We explore this approach in the clear-sky stan-dard atmospheric profile of Anderson (1986), using the ra-diative transfer software Libradtran (Emde et al. 2016) torecover spectrally and vertically resolved optical depths,irradiance and heating rates. The surface is treated as ablack body with temperature 288 .
15 K and the flux fromthe overhead sun is scaled such that the incoming and out-going energy fluxes balance. A standard aerosol profile isused (Shettle 1989) and the surface albedo is set to 0 . Σ mat ≈ K, Σ tran ≈ K and Σ planet ≈ K, which are broadly in agreement withthe values calculated in the literature. The material ratehas been estimated identically by Bannon (2015) at30 mW/m K in the EBM, and in more realistic models byPascale et al. (2011) at ≈
50 mW/m K, by Kato and Rose(2020) at 49 mW/m K and by Lembo et al. (2019) in therange 38 . − . K (outlier neglected) by theirdirect method. The planetary rate is also confirmed inthe EBM by Bannon (2015) (his CV1), and corroboratedby estimates of 1272 − K by Wu and Liu(2010). The transfer rate calculated by Kato and Rose(2020) is 76 mW/m K, and is estimated in Bannon andLee (2017) at 68 mW/m K. The transfer rate discussedhere differs slightly from the closest analogue in Bannon(2015) in the outflow temperature; for Bannon’s MS3 theatmospheric temperature (253 K) is used rather than thecooling-to-space weighted average (255 K).The thermalization of solar radiation is the largestsource of entropy production in the planetary view, ac-counting for more than 60% of the total (see AppendixB). The transfer entropy production rate sums the con-tributions from the material processes and internal radia-tive heat transfer, which are of similar orders of magni-tude: in the EBM, σ LH + σ sens = . K while σ intrad = . K. The flux F planet is the total in-cident solar flux, F tran is the fraction absorbed, and F mat focuses on the fraction of energy transferred by materialprocesses. The material T inmat is the surface temperature,as that is where there is net radiative heat input, while the T outmat is an atmospheric average. The transfer T intran reflectsthe average temperature of solar absorption and the T outtran is the average temperature of thermal emission to space,which is approximately the effective emission tempera-ture, ( F tran / σ ) / . The planetary T inplanet comes from thesolar temperature (scaled because of the 4 / J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S ciated with the emission of radiation) and the T outplanet re-flects an average of the scaled emission temperature andthe high temperature of the scattered solar photons. Theefficiencies capture the magnitude of these temperaturedifferences.
5. Sensitivity of EPRs to climate changes
With three different global entropy production ratesin hand, a natural next question is how they respond tochanges in the climate state. This offers insight about theirinterpretation as well as how they might be leveraged asdiagnostic variables. a. Experimental set-up
The analytic radiative-convective model (described inAppendix D) is well-suited for studying the effect of forc-ing in the longwave and shortwave on the entropy of the at-mosphere, as convection, internal radiation, stratosphericabsorption of shortwave radiation and thermal emissionof radiation are all handled explicitly and in a simplifiedmanner: convection by fixing a prescribed lapse rate andradiation by assuming a radiatively gray gas in one long-wave and two shortwave channels. This allows for climatechanges to be treated independently – with cloud and lapserate feedbacks suppressed – to isolate the first-order re-sponses in the vertical.The greenhouse effect is simulated by increasing thethermal optical depth by 25% while keeping the absorp-tion profile of solar radiation fixed, which results in an in-crease in the surface temperature by 5 . . . .
30 to 0 .
25 is compared.Although the surface temperature change is the samein these cases, other aspects of the climates differ. Thishas implications for the ability of uniform solar radia-tion management-type geoengineering to restore a climatewith a heightened greenhouse effect to its pre-industrialstate. To investigate this, a fourth case is modeled, whereto counteract the elevated greenhouse effect, the albedo isalso increased to 0 .
35, restoring the surface temperature. b. Results and discussion
The atmospheric profiles under these four climate con-ditions are represented in Figure 2, where the first columnshows the temperature profiles, the second the vertical en-ergy fluxes and the third the resulting heating rates. Identi-cal surface temperatures give rise to identical tropospherictemperature profiles because of the prescribed lapse rate,as in the greenhouse gas and increased solar absorptioncases (upper panels) and pre-industrial and solar radiationmanagement cases (lower panels). However, the heightof the radiative-convective boundary and the stratospheric temperature profiles differ depending on the nature of theclimate change, as do the vertical energy fluxes and theheating rates. This is what causes the difference in theentropy production rates.Table 2 lists energy fluxes, temperatures efficiency andentropy production rates for each climate change and sys-tem perspective. When the greenhouse gas concentrationis increased (second column of Table 2) relative to a pre-industrial control scenario (first column), the amount ofsolar energy absorbed by the system is unchanged, whichfixes the emission temperature and with it (approximately) T outmat , T outtran and the tropopause temperature T t p . Although F tran , the total energy transferred, is fixed, the increasedoptical thickness inhibits heat transfer by radiation, in-creasing the fraction of energy transferred by material pro-cesses ( F mat / F tran ) from 0 .
47 to 0 . Σ tran and η tran by 18%. The increased energy flux through the materialsystem results in an even more dramatic 29% increase inthe material entropy production rate, with only an 18% in-crease in η mat . The planetary entropy production rate is,by contrast, unaltered by this climate change as neither thesolar incoming nor the scattered spectra are altered and theshape of the outgoing spectrum can only change minutelygiven the requirement for a fixed total energy flux in thisgray atmosphere.The story is very different when the same surface tem-perature increase is achieved by reducing the albedo of theplanet (third column of Table 2). As the amount of solarradiation absorbed and thermalized rather than scatteredincreases by 8%, the effective emission temperature mustnecessarily increase by 1 . T outmat , T outtran and T t p ,while the surface and inflow temperatures increase onlymarginally more such that the material and transfer effi-ciencies η are conserved. The higher absolute tempera-tures alone would give a decrease in the entropy produc-tion rate by 2%, but the increase in energy flux dominatesfor an overall 6% increase in Σ mat and Σ tran , which is sig-nificantly less than in the greenhouse gas case. The de-creased scattered fraction causes an increase in Σ planet by4%, as thermalized and re-emitted radiation carries signif-icantly more entropy than scattered solar radiation.The difference in the entropy responses to surfacewarming by longwave and shortwave mechanisms reflectsthe different heating profiles by radiative and material pro-cesses in these two cases, even though the surface tem-perature change is the same. When the albedo change isreversed and combined with the greenhouse gas increase,the surface temperature is restored but not the heating ratesnor the entropy production rates, as shown in the fourthcolumn of Table 2 and in the lower panels of Figure 2.The total absorbed radiation ( F tran ) is reduced by 7% com-pared to the pre-industrial case, but the fraction of energy O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S F IG . 2. The temperature (first column), energy fluxes (second column) and heating rate (third column) profiles derived from the analyticradiative-convective model in a range of climate states. The solid lines show the control case (representing a pre-industrial scenario) with parametersfrom Tolento and Robinson (2019) chosen to match observation, with a surface temperature of 287 . . .
30 to 0 .
25 (dotted lines). In the lower panels, the surface temperature is restored byincreasing the albedo to 0 .
35 to balance the 25% increase in greenhouse effect, as in a globally-uniform solar radiation management intervention.In the second column, the longwave energy flux is separated into upwelling and downwelling components, while in the third column the longwaveheating rate is separated into that due to cooling to space (external radiative heating, blue) and that due to internal radiative heat transfer within theatmosphere and with the surface (magenta). transferred by material processes, F mat / F tran , is still ele-vated at 0 .
51 due to the greenhouse gases present, whichexplains the small overall increase in F mat . The reductionin absorbed solar radiation also lowers the effective emis-sion temperature of the planet relative to the pre-industrialscenario, and so there is a larger temperature differencebetween material and transfer influx and outflux temper-atures, explaining the higher efficiency. These result in amaterial entropy production rate that is 22% higher and atransfer entropy production rate that is 12% higher thanthe control case, although the surface temperature is unal-tered. The increased scattering fraction decreases Σ planet .These results are striking for three reasons. Firstly,we find that all three entropy production rates increase with solar absorptivity, which differs from the conclu-sions drawn by Kato and Rose (2020) upon regressionof observational transfer and material entropy productionrates against inter-annual variability of solar absorptiv-ity. Storage of entropy and energy in the oceans in high-absorptivity years could account for the decreases theynote, as a top of atmosphere imbalance in net energy flux will result in an imbalanced entropy flux that looks sim-ilar to a production. However, the increased greenhousegas results we find are consistent with previous studies.In a cloud-resolving model, Singh and O’Gorman (2016)find an increase in material entropy production rate withan increase in greenhouse gas concentration, as do Lu-carini et al. (2010) in a slab-ocean model. Bannon andLee (2017) also study the response to climate changes inan energy balance model, but without constraints connect-ing the surface and atmosphere temperatures, so that theirresults are not directly comparable to ours.Secondly, the material and transfer efficiencies mea-sured here vary with greenhouse gas concentration butnot albedo, as do the ratios F mat / F tran and Σ mat / Σ tran .That the fraction of the transfer entropy production thatis due to material processes remains fixed as the solar en-ergy absorbed and temperatures vary suggests that thereis physical significance in how internal energy transferis partitioned between radiative and non-radiative mech-anisms. The increased dominance of material processeswith a greenhouse gas increase is reminiscent of the ob-0 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S
Control GHG Increase Albedo Decrease SRM T sur f (K) 287.9 293.4 293.4 287.9 T tp (K) 224.1 223.9 228.3 219.7 T ef f (K) 255.1 255.1 259.9 250.3 F mat (W/m ) 111.9 123.5 120.6 114.6 T inmat (W/m ) 287.9 293.4 293.4 287.9 T outmat (W/m ) 269.1 270.8 274.2 265.7 η mat (%) 6.54 7.71 6.54 7.71 Σ mat (mW/m K) 27.2 35.1 28.7 33.2 F tran (W/m ) 240.0 240.0 258.6 222.7 T intran (W/m ) 280.0 284.7 285.2 279.4 T outtran (W/m ) 257.2 257.3 262.0 252.5 η tran (%) 8.13 9.64 8.13 9.64 Σ tran (mW/m K) 75.9 89.9 80.3 85.0 F planet (W/m ) 342.9 342.9 342.9 342.9 T inplanet (W/m ) 4334 4334 4334 4334 T outplanet (W/m ) 251.1 251.1 241.7 260.9 η planet (%) 94.21 94.21 94.42 93.98 Σ planet (mW/m K) 1286 1286 1340 1235T
ABLE
2. The values of the entropy-related variables estimated using the radiative-convective model in a unperturbed control climate (firstcolumn), under an increased greenhouse effect (second column), a decrease in the global albedo (third column, equivalently an increased solarabsorption) and a solar radiation management scenario (fourth column), corresponding to the scenarios plotted in Figure 2. The tropopausetemperature, T tp is defined as the temperature minimum, while the effective radiating temperature is a function of the absorbed solar radiation, T ef f = ( F tran / σ ) / . served increase in convective mean available potential en-ergy (Gertler and O’Gorman 2019) and so is not entirelyunexpected. Further investigation of this result in modelswith more precise treatment of humidity appears to be auseful area for further work.Thirdly, the fact that manipulating the planet’s albedoto balance a greenhouse gas change can restore surfacetemperature while not restoring these entropy metrics un-derlines that there is useful additional information in theseglobal scalar variables for climate change discussions anddecision-making. Entropy production rates have a moredirect relationship to the motion and flows in the climatethan does global mean surface temperature and, althoughthey are less familiar and so not as easy to interpret, theywarrant further exploration as a supplementary diagnosticto advance our understanding of the climate state. c. Advantages of the transfer perspective in capturing theclimate state As exemplified by the elevated greenhouse gas case, theplanetary entropy production rate is relatively insensitiveto climate changes in which the albedo remains fixed, asthe outgoing entropy flux is approximately determined bythe (unchanged) effective emission temperature and the in-coming entropy flux is set by the solar temperature. In fact,an atmosphere-less isothermal rock, with identical solarflux and albedo, will have a similar planetary entropy pro-duction rate to a planet with any greenhouse effect. Fur- thermore, the sun plays an inordinately significant role inthe planetary entropy production rate. The value is domi-nated by the entropy production due to the thermalizationof this solar radiation, σ atmSW + σ sur fSW (see Appendix B) andthe incoming entropy flux ( ≈ F / T sun ) depends explicitlyon the temperature of the sun, although this ought not toinfluence the climate separately from its role in deliveringenergy. Taken together, these arguments suggest that theplanetary perspective is not a good candidate for studyingthe climate.The material entropy production rate is, of the three, themost focused on processes relevant to human experience:it is material processes such as the hydrological cycle orconvective motion, and not the internal radiation, whichdirectly feature in the weather we experience. However,not all material processes are equally relevant, and it couldbe more meaningful to consider them separately; for ex-ample, the contribution from frictional dissipation aroundfalling precipitation is twice that from atmospheric mo-tions (Singh and O’Gorman 2016), but has a very differ-ent significance. The material sub-processes, and even thetotal material tally, are interdependent portions of a largersystem and the energy carried by them can vary becauseof changes in other parallel processes, such as internal ra-diation. This makes interpreting changes in the materialentropy production rate alone challenging, as they couldbe due to changes in the proportion of energy transferredrather than in the efficiency or temperature differences. O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S Σ mat would be powerless to resolve, ma-terial processes being identically null in both cases, eventhough it would seem to be a climatologically-relevantchange. Thus the total material entropy production rate,though meaningful, does not stand out as the most naturalclimate-summarizing global entropy variable, rather as avariable that describes important sub-processes.We argue that the transfer entropy production rate is ofmore compelling value from the perspective of the climatesystem. F tran , the flux of energy that is absorbed by theplanet, is a natural climate variable, as is the temperaturedifference between where shortwave radiation is absorbed, T intran , and where longwave radiation leaves the planet, T outtran . The transfer rate is sensitive to climate changes andreflects all processes that move heat in the planet, withoutdistinguishing between mechanisms. That the fraction ofthe transfer entropy production by material processes staysfixed as the albedo is changed but varies as the greenhousegas concentration changes suggests that, in the context ofthe transfer rate, the material entropy production rate andthe entropy production due to internal radiation gain addi-tional significance.
6. Conclusions
The total entropy production rate of the Earth is a tan-talizing physical concept without a simple interpretation,thanks to the ambiguity in defining the boundary of theclimate system with respect to the radiation that feeds it.In this paper we have laid out three options, and with themthree entropy production rates that account for the irre-versibility of the processes within each of these systems.The planetary perspective includes the entropy productionfrom all radiative and non-radiative processes, whereas thematerial perspective includes only non-radiative contribu-tions. The transfer perspective separates radiation accord-ing to its role within the climate, including only the pro-duction due to that which is emitted from and re-absorbedwithin the system. The exploration of this third option wasthe particular aim of this paper.We provide estimates of each entropy production ratein three model climates of varying complexity. The rangesuggests Σ mat ≈ K, Σ tran ≈ Kand Σ planet ≈ K, which are consistentwith the literature. The response of each entropy produc-tion rate to climate changes is also explored in a simplifiedradiative-convective model: the planetary entropy produc-tion rate is unchanged by changes in the greenhouse effectand increases with increased shortwave absorption, while the transfer and material entropy production rates increasewith surface temperature, but more significantly if that in-crease is mediated by the greenhouse effect rather thanalbedo. The fraction of entropy produced by material pro-cesses relative to internal radiation is unchanged by albedochanges but increases with greenhouse gas concentration.None of the entropy production rates is restored to pre-industrial levels by solar radiation management followinga greenhouse gas increase, if such an intervention restoresaverage surface temperature.The transfer view of the system has some immediatelyapparent physical elegance, but work is required to exploreits significance further. Although the entropy productionrate initially proposed by Paltridge (1975) as a climate-predicting variable is the horizontal component of boththe transfer and material perspectives, in the vertical onlythe extremization of the material entropy production ratehas, to our knowledge, been explored (e.g. Ozawa andOhmura (1997)). Heat transfer by internal radiation in thevertical is of a similar order of magnitude to that by ma-terial processes and acts alongside the material processesto transfer heat down-gradient. It is plausible that by con-sidering the sum of these radiative and non-radiative in-ternal heat transfer processes a more coherent view of theclimate as a self-optimizing system may emerge. In fact,the maximum flow theory known as the Constructal Law(discussed in Reis (2014)) appears to suggest a tendencyof systems with fixed energy flow, like the climate, to or-ganize to minimize the transfer entropy production rate inparticular. More generally, any theory of entropy produc-tion extremization must carefully address to which globalentropy production variable(s) it applies and why.If an entropy-extremization principle were understoodin the climate, it might also apply elsewhere. Non-equilibrium quasi-steady systems are common in otherareas of complexity, life being one example. In thisbroader context, the climate can be taken as a convenient,thoroughly-studied and modeled example.Fundamental research into the way we interpret andunderstand the climate has potential societal importanceas we wrestle with communication and decision-makingbased on the digestible knowledge gleaned from complexmodels. Entropy production rates offer another diagnosticfor comparing models to reality and to each other, and forsummarizing and tracking climate changes. A predictivetheory of entropy generation might also potentially help toconstrain climate predictions. This paper’s developmentof the concept of global entropy production is aimed atstimulating further research in this area.
Acknowledgments.
With thanks to Helen Brindley, Di-ane Barnett, Siarhei Barodka, Thomas Bendall, MichaelByrne, Jonathan Gibbins, Matthew Kasoar and BarnabasWalker for helpful discussions, to Bernhard Mayer for ad-vice on the utilization of Libradtran, to Tyler D. Robinson2 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S for example code for the radiative-convective model andto the reviewers for their helpful comments. This workwas supported by the EPSRC Mathematics of Planet EarthCentre for Doctoral Training, EP/L016613/1.APPENDIX A
Solution of the EBM
To make a numerical analysis of the energy balancemodel, we have used the same values as Bannon (2015).The script used in this study is available upon request.The EBM set-up is shown in Figure 1. Unlike in a stan-dard EBM, the radiation is separated between that whichis internally transferred within the climate and that whichis external, i.e. is emitted to space or absorbed from thesun.The solar flux is F sun = . with a fixed albedoof α = .
30 and a solar temperature of T sun = F scatSW = α F sun = . . The atmosphericabsorptivity in the shortwave is β = .
10, which sets F atmSW = β F sun = . and F sur fSW = ( − α − β ) F sun = . , where any surface scattering and subsequentabsorption by the atmosphere has been neglected, fol-lowing Bannon (2015). The surface-atmosphere mate-rial heat fluxes are represented by a sensible and a la-tent heat term that transfer heat as a fraction of the in-coming solar flux: F sens = γ sens F sun = . and F LH = γ LH F sun = . where γ sens = .
05 and γ LH = .
2, approximately following the energy budget of Wildet al. (2014) such that the total γ = .
25 matches Ban-non (2015). The emissivity of the atmosphere is ε = .
95 and so F atmLW = εσ T atm , while the surface is a blackbody such that F sur f = σ T sur f . Requiring energy bal-ance for the surface and atmosphere gives the temperatures T sur f = . T atm = . F intrad = εσ T sur f − εσ T atm = . , F sur fLW = ( − ε ) σ T sur f = . and F atmLW = ε T atm = . , as demon-strated in Bannon (2015). (It is coincidental that in this ex-ample the internal radiative heat transfer is the same valueas the scattered energy).APPENDIX B Calculation of EPRs by the direct method
Approximating to black body behavior, the absorptionof radiation results in an entropy production of the form ofa difference between the entropy of the heat in the materialand the entropy in the radiation: σ ( absorb ) = FT mat − FT source (B1) which is reversed for emission. For black body emission,the temperature of the relevant material T mat will also bethe source temperature T source .The entropy production due to an internal heat transferis: σ ( internal heat transfer ) = F (cid:18) T cold − T hot (cid:19) (B2)which applies both to the material and to the internal radi-ation terms. The spectral character of the internal radiationneed not be accounted for in the entropy production termbecause both the creation and destruction of those photonshappen within the planetary boundaries; only the resultantheating is relevant.These principles can now be applied to the EBM in or-der to calculate directly the three total entropy productionrates, beginning by calculating the contributions from eachprocess.The entropy produced upon scattering is generally afunction of the change in directional intensity of the radi-ation. Following Bannon (2015) and Wu and Liu (2010), J scat = . K so that the production rate is: σ scatSW = J scat − F scat T sun = . K . The thermalization of solar radiation in the atmosphereresults in an entropy production of: σ atmSW = F atmSW (cid:18) T atm − T sun (cid:19) = . Kand similarly, the thermalization of solar radiation at thesurface results in: σ sur fSW = F sur fSW (cid:18) T sur f − T sun (cid:19) = . K . The material transport of heat from the surface to the at-mosphere causes much smaller entropy productions, pro-portional to the reciprocal temperature difference: σ sens = F sens (cid:18) T atm − T sur f (cid:19) = . K σ LH = F LH (cid:18) T atm − T sur f (cid:19) = . Kas does the net transport of heat from the surface to theatmosphere via internal radiation σ intrad = F intrad (cid:18) T atm − T sur f (cid:19) = . K . Emission of longwave radiation from the surface re-sults in an entropy production because the radiation car-ries more entropy than the cooled matter loses. For the
O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S σ sur fLW = F sur fLW (cid:18)
43 1 T sur f − T sur f (cid:19) = . Kand for the emission from the atmosphere: σ atmLW = F atmLW (cid:18)
43 1 T atm − T atm (cid:19) = . K . From these constituent budget terms, the three entropyproductions rates can be calculated directly. The planetaryentropy production is the total from all processes: Σ planet = σ scatSW + σ atmSW + σ sur fSW + σ LH + σ sens + σ intrad + σ sur fLW + σ atmLW = . K (B3)which is in agreement with Bannon (2015) for CV1.The material entropy production rate excludes all radia-tive processes: Σ mat = σ LH + σ sens = . K . (B4)The transfer entropy production rate includes only theprocesses that involve energy exchange between parts ofthe material system, by both internal radiation and mate-rial processes: Σ tran = σ LH + σ sens + σ intrad = . K . (B5)APPENDIX C Comparison with the indirect method
These values can also be calculated via the indirectmethod, using the relationship Σ = J out − J in under the as-sumption of steady state.The sun as a black body at temperature T sun carries en-tropy towards the earth of J inplanet = F sun T s = . K (C1)and the outgoing radiation carries entropy according to itsemission temperature J outplanet = J scat + F sur fLW T sur f + F atmLW T atm = . K(C2)such that the difference is the planetary entropy produc-tion rate of Equation B3. The entropy fluxes can alsobe used to calculate representative temperatures via therelationship T = F / J where F planet =
341 W/m . Then T inplanet = . T outplanet = . J intran = F atmSW T atm + F sur fSW T sur f = . K . (C3)The flux of entropy out is the change of entropy of thematerial that is due to cooling to space: J outtran = F sur fLW T sur f + F atmLW T atm = . K . (C4)The difference, 67 . K, is identical to the resultof the direct calculation using Equation B5 above. Theflux of energy in this case is F tran = F atmSW + F sur fSW =
239 W/m , which results in representative temperatures T intran = . T outtran = . T ∗ e f f = (( − α ) F sun / σ ) / = . J inmat = F sur fSW − F sur fLW − F intrad T sur f = . K (C5) J outmat = F atmSW + F intrad − F atmLW T atm = . K . (C6)Again the difference in entropy flux agrees with the pro-duction calculated above (Equation B4). The flux throughthis version of the system is F mat = .
18 W/m and thetemperatures are accordingly T inmat = . T outmat = . Analytic radiative-convective model definition
The analytic radiative-convective model used here is theone described in Tolento and Robinson (2019), developedfrom earlier work in Robinson and Catling (2012, 2014). Itis designed to be simple enough to solve analytically andversatile enough to fit a range of planetary atmospheres.Here we describe its application to Earth in particular,using the parameter values from Tolento and Robinson(2019). Profiles of the temperature, energy fluxes andheating rates calculated from this model are shown in Fig-ure 2, solid lines. Our script to run this model is availableby request.4 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S
The vertical coordinate of the model, τ , is the gray ther-mal optical depth and the atmosphere is split into two por-tions: a stratospheric part, which is in radiative balance,and a tropospheric part where convection occurs. In thelower portion, the thermal structure is given by a modifiedadiabat up to the radiative-convective boundary at τ rc : T = T (cid:18) ττ (cid:19) β / n (D1)where T is the reference temperature at the τ , the sur-face of Earth, and n = τ / τ = ( p / p ) n , with p = β = a ( γ − ) / γ extends the dryadiabatic lapse rate (in terms of the ratio of specific heats γ = .
4) to account for latent heat release via the rescalingby a = .
6, which establishes the slope of the temperatureprofile shown in the first panel of Figure 2.The net solar radiative flux is given by two shortwavechannels, which attenuate as a function of the thermal op-tical depth: F (cid:12) = α (cid:16) F (cid:12) e − k τ + F (cid:12) e − k τ (cid:17) (D2)with F (cid:12) =
10 W/m , k = F (cid:12) =
333 W/m , k = . α the top of atmosphere albedo. The impact of thesetwo channels can be seen in the third panel of Figure 2: thestratospheric peak in the solar heating rate is accomplishedby F (cid:12) , while the tropospheric and surface solar absorp-tion is due to F (cid:12) . The entropy of the scattered flux isapproximated as in Stephens and O’Brien (1993); Bannon(2015); Wu and Liu (2010) by assuming isotropic (Lam-bertian) scattering, so that the J scat = σ T sun χ ( u L ) where χ ( u ) ≈ u ( − . u + . ) and u L = α Ω sun / π ,where Ω sun = . × − st and T sun = dF + d τ = D (cid:0) F + − σ T (cid:1) (D3) dF − d τ = − D (cid:0) F − − σ T (cid:1) (D4)where D = .
66 is the diffusivity factor and F + and F − arethe upwelling and downwelling longwave radiative fluxes.Integrating these two equations and plugging in the so-lar flux and the convective temperature profile provides,upon further manipulation, expressions for the upwellingthermal flux and the temperature in both the convectiveand non-convective region, in terms of incomplete gammafunctions that can be handled numerically (for derivation,see Robinson and Catling (2012); Tolento and Robinson(2019)). The two constraints – that the upwelling radiativeflux and temperature be continuous across the radiative-convective boundary – then allows the model to be solved for two free parameters (for example, τ rc and T ) by stan-dard root-finding methods. We take the total column op-tical thickness to be 1 .
96 and the global albedo to be0 . . F LWnet = F + − F − ) and downwelling net solar flux, F conv ( τ ) = F (cid:12) net ( τ ) − F LWnet ( τ ) . References
Anderson, G. P., 1986: AFGL atmospheric constituent profiles (0-120km).Bannon, P. R., 2015: Entropy production and climate efficiency.
J. At-mos. Sci. ,
72 (8) , 3268–3280.Bannon, P. R., and S. Lee, 2017: Toward quantifying the climate heatengine: solar absorption and terrestrial emission temperatures andmaterial entropy production.
J. Atmos. Sci. ,
74 (6) , 1721–1734.Dewar, R. C., 2003: Information theory explanation of the fluctuationtheorem, maximum entropy production and self-organized criticalityin non-equilibrium stationary states.
J. Phys. A: Math. Gen. ,
36 (3) ,631–641.Emde, C., and Coauthors, 2016: The libRadtran software package forradiative transfer calculations (version 2.0.1).
Geosci. Model Dev. , , 1647–1672.Essex, C., 1984: Radiation and the irreversible thermodynamics of cli-mate. J. Atmos. Sci. ,
41 (12) , 1985–1991.Essex, C., 1987: Global thermodynamics, the clausius inequality, andentropy radiation.
Geophys. Astrophys. Fluid Dyn. ,
38 (1) , 1–13.Feistel, R., 2011: Entropy flux and entropy production of stationaryblack-body radiation.
J. Non-Equilib. Thermodyn. ,
36 (2) , 131–139.Fraedrich, K., and F. Lunkeit, 2008: Diagnosing the entropy budget ofa climate model.
Tellus A: Dyn. Meteor. Oceanogr. ,
60 (5) , 921–931.Gertler, C. G., and P. A. O’Gorman, 2019: Changing available en-ergy for extratropical cyclones and associated convection in NorthernHemisphere summer.
Proc. Natl. Acad. Sci. U.S.A. ,
116 (10) , 4105–4110.Goody, R., 2000: Sources and sinks of climate entropy.
Quart. J. Roy.Meteor. Soc. ,
126 (566) , 1953–1970.Goody, R., and W. Abdou, 1996: Reversible and irreversible sources ofradiation entropy.
Quart. J. Roy. Meteor. Soc. ,
122 (530) , 483–494.Grassl, H., 1981: The climate at maximum entropy production byMeridional atmospheric and oceanic heat fluxes.
Quart. J. Roy. Me-teor. Soc. ,
107 (451) , 153–166.Green, J. S. A., 1967: Division of radiative streams into internal transferand cooling to space.
Quart. J. Roy. Meteor. Soc. ,
93 (397) , 371–372.Herbert, C., D. Paillard, M. Kageyama, and B. Dubrulle, 2011: Presentand Last Glacial Maximum climates as states of maximum entropyproduction.
Quart. J. Roy. Meteor. Soc. ,
137 (657) , 1059–1069.Kato, S., and F. G. Rose, 2020: Global and regional entropy productionby radiation estimated from satellite observations.
J. Climate ,
33 (8) ,2985–3000.
O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S Labarre, V., D. Paillard, and B. Dubrulle, 2019: A radiative-convectivemodel based on constrained maximum entropy production.
EarthSyst. Dynam. ,
10 (3) , 365–378.Lembo, V., F. Lunkeit, and V. Lucarini, 2019: A new diagnostic tool forwater, energy and entropy budgets in climate models.
Quart. J. Roy.Meteor. Soc. ,
126 (566) , 1953–1970.Lesins, G. B., 1990: On the relationship between radiative entropy andtemperature distributions.
J. Atmos. Sci. ,
47 (6) , 795–803.Li, J., and P. Chylek, 1994: Entropy in climate models. Part II: hori-zontal structure of atmospheric entropy production.
J. Atmos. Sci. ,
51 (12) , 1702–1708.Li, J., P. Chylek, and G. B. Lesins, 1994: Entropy in climate models.Part I: vertical structure of atmospheric entropy production.
J. Atmos.Sci. ,
51 (12) , 1691–1701.Lucarini, V., 2009: Thermodynamic efficiency and entropy productionin the climate system.
Phys. Rev. E ,
80 (2 Pt 1) , 021 118.Lucarini, V., K. Fraedrich, and F. Lunkeit, 2010: Thermodynamicsof climate change: generalized sensitivities.
Atmos. Chem. Phys. ,
10 (20) , 9729–9737.Lucarini, V., K. Fraedrich, and F. Ragone, 2011: New results on thethermodynamic properties of the climate system.
J. Atmos. Sci. ,
68 (10) , 2438–2458.Nicolis, G., and C. Nicolis, 1980: On the entropy balance of the earth-atmosphere system.
Quart. J. Roy. Meteor. Soc. ,
106 (450) , 691–706.Ore, A., 1955: Entropy of radiation.
Phys. Rev. ,
98 (4) , 887–888.Ozawa, H., 2003: The second law of thermodynamics and the globalclimate system: A review of the maximum entropy production prin-ciple.
Rev. Geophys. ,
41 (4) , 1075.Ozawa, H., and A. Ohmura, 1997: Thermodynamics of a global-meanstate of the atmosphere – a state of maximum entropy Increase.
J.Climate ,
10 (3) , 441–445.Paltridge, G. W., 1975: Global dynamics and climate - a system of mini-mum entropy exchange.
Quart. J. Roy. Meteor. Soc. ,
101 (429) , 475–484.Paltridge, G. W., 1978: The steady-state format of global climate.
Quart. J. Roy. Meteor. Soc. ,
104 (442) , 927–945.Paltridge, G. W., 2005: Stumbling into the MEP Racket: An HistoricalPerspective.
Non-equilibrium Thermodynamics and the Productionof Entropy , A. Kleidon, and R. D. Lorenz, Eds., Springer-Verlag,Berlin/Heidelberg, 33–40.Pascale, S., J. M. Gregory, M. H. P. Ambaum, and R. Tailleux, 2011:Climate entropy budget of the HadCM3 atmosphere–ocean generalcirculation model and of FAMOUS, its low-resolution version.
ClimDyn ,
36 (5-6) , 1189–1206.Pascale, S., J. M. Gregory, M. H. P. Ambaum, R. Tailleux, and V. Lu-carini, 2012: Vertical and horizontal processes in the global atmo-sphere and the maximum entropy production conjecture.
Earth Syst.Dynam. , , 19–32.Pauluis, O. M., and I. M. Held, 2002a: Entropy budget of an atmo-sphere in radiative–convective equilibrium. Part I: maximum workand frictional dissipation. J. Atmos. Sci. ,
59 (2) , 125–139. Pauluis, O. M., and I. M. Held, 2002b: Entropy budget of an atmospherein radiative–convective equilibrium. Part II: latent heat transport andmoist processes.
J. Atmos. Sci. ,
59 (2) , 140–149.Peixoto, J. P., A. H. Oort, M. De Almeida, and A. Tom´e, 1991: Entropybudget of the atmosphere.
J. Geophys. Res. Atmos. ,
96 (D6) , 10 981–10 988.Pelkowski, J., 1994: Towards an accurate estimate of the entropy pro-duction due to radiative processes: Results with a gray atmospheremodel.
Meteor. Atmos. Phys. ,
53 (1) , 1–17.Planck, M., 1914:
The Theory of Heat Radiation . P. Blackiston’s Sons& Co, Philadelphia.Pujol, T., and J. E. Llebot, 1999: Extremal principle of entropy produc-tion in the climate system.
Quart. J. Roy. Meteor. Soc. ,
125 (553) ,79–90.Reis, A. H., 2014: Use and validity of principles of extremum of entropyproduction in the study of complex systems.
Annals of Physics , ,22–27.Robinson, T. D., and D. C. Catling, 2012: An analytic radiative-convective model for planetary atmospheres . ApJ ,
757 (1) , 104.Robinson, T. D., and D. C. Catling, 2014: Common 0.1 bar tropopausein thick atmospheres set by pressure-dependent infrared trans-parency.
Nature Geosci , , 12–15.Rodgers, C. D., and C. D. Walshaw, 1966: The computation of infra-redcooling rate in planetary atmospheres. Quart. J. Roy. Meteor. Soc. ,
92 (391) , 67–92.Rosen, P., 1954: Entropy of radiation.
Phys. Rev. , , 555.Shettle, E. P., 1989: Models of aerosols, clouds, and precipitation foratmospheric propagation studies. In AGARD Conf Proc No. 454 .Singh, M. S., and P. A. O’Gorman, 2016: Scaling of the entropy budgetwith surface temperature in radiative-convective equilibrium.
J. Adv.Model. Earth Syst , , 1132–1150.Stephens, G. L., and D. M. O’Brien, 1993: Entropy and climate. I:ERBE observations of the entropy production of the earth. Quart.J. Roy. Meteor. Soc. ,
119 (509) , 121–152.Tolento, J. P., and T. D. Robinson, 2019: A simple model for radiativeand convective fluxes in planetary atmospheres.
ICARUS , , 34–45.Volk, T., and O. M. Pauluis, 2010: It is not the entropy you produce,rather, how you produce it. Philos. Trans. R. Soc., B ,
365 (1545) ,1317–1322.Wallace, J. M., and P. V. Hobbs, 2006:
Atmospheric Science: An Intro-ductory Survey . An Introductory Survey, Elsevier.Wild, M., and Coauthors, 2014: The energy balance over land andoceans: an assessment based on direct observations and CMIP5 cli-mate models.
Clim Dyn ,
44 (11-12) , 3393–3429.Wu, W., and Y. Liu, 2010: Radiation entropy flux and entropy produc-tion of the Earth system.
Rev. Geophys. ,
48 (2) , 1075.Wyant, P. H., A. Mongroo, and S. Hameed, 1988: Determination ofthe heat-transport coefficient in energy-balance climate models byextremization of entropy production.
J. Atmos. Sci. ,
45 (2)45 (2)